ā.…*-->± &*）'）;-ī£， º š, * · * *****Mae'_' ），--، ، ،• . . . . »-*...* -ae* *· – ） --- - - ~- - --° { * 7. “ - *· · · - - - -.• • • • •. -- - - - - - - - - - - “, “ ’ ‚+‘ · · · （, , ſg º ∞ : -> - .. sº - &sº sº ºw. º sº º º . ' a*， a ae ، ، ، ، ، ،° ſ√≠ √ |- §§§ºſſú： Ģ ×& !， T : .， …，» : （ ） ¿¿.* xae： : sºsºiriſ. 2S, S5. R N W } W. § TRANSPO RTATION LIBRARY ------ - * º iſſilſillſ|||ITITITITITIII: -ms----- A 758.064 THE FIELD ENGINEER: #antig 3300ft of practice J- IN THE SURVEY, LOCATION, AND TRACK-work OF RAILROADS; - CONTAINING A LARGE COLLECTION OF RULES AND TABLES, origiNAL AND SELECTED, APPLICABLE To BOTH THE STANDARD AND THE NARROW GAUGE, - AND PREPARED witH special, REFERENCE. To .* THE WANTS OF TEIE YOUNG ENGINEER. BY WILLIAM FINDLAY SHUNK, C.E. - > -ºº: •xt tº : SECOND EDITION, REVISED AND CORRECTED. . * q - & * - * * * . i NEW YORK: D. WAN NOSTRAND, PUBLISHER, 23 MURRAY AND 27 WARREN STREETS. 1882. CopyRIGHT, 1879, By D. VAN NOSTRAND. * "...,’” * & º *...ſ. l:| Afº §s. - & 2.3% #2 º iiansportation ...? & !/ 67 . . . . . 4 -& & #% Library '' . | F 9 O & \- ( & & 2- 2- THE AUTHOR, %ffectionatelg HBebicateg tijíg 350ck TO J * ALBERT J. SCHERZER, C.E., ©lº (Tamratic and Bear frient, IN TokEN of ESTEEM For HIS PROFESSIONAL ATTAINMENTS AND RESPECT FOR HIS MANLY CHARACTER. PIR E FA C E. THE author's principal aim in preparing this volume has been, as its title indicates, to serve that large class of young engineers who, like himself, have not had the advantage of a technical education before going out for their livelihood. The initial chapters are, therefore, given to a compendious exposition of those mathematical truths and methods which they must needs become familiar with from the beginning. S Plane Trigonometry, Logarithms, and propositions relating to the circle, are tools of the craft in constant use; ready han- dling of them is an indispensable condition of excellence. Be not discouraged by obscurities and difficulties at the outset; light will gradually break on the scrutinizing eye, and a way always open to manful effort. - These chapters are followed by instructions as to the adjust- ment and use of instruments, and hints concerning field rou- tine, which it is thought will be found acceptable to the inexperienced learner. The same may be said of the articles on staking out work, and those on track problems, with which the text of the book closes. They have been written with the author's own early ignorance in mind, and with a wish to set the subjects forth as plainly as possible, disembarrassed of hard words in the description, and of unpractical niceties in the operation. - - The chapter on field location is believed to include all the problems likely to occur. The author, in compiling it, has taken those only which have arisen in his own practice, and which, therefore, may arise in the practice of others. His - - -- V own practice having been unusually large and diversified, probably the examples given will prove adequate, directly or indirectly, to all contingencies. No attempt has been made to swell the bulk of the volume with imaginary cases; the object being, not to provide barren mathematical exercises, but to teach useful knowledge. - Problems, also, affecting location in its economical aspects, —- the balancing of physical and financial conditions, equating of alternative lines, and the like, – do not come within the scope of the work, and are therefore not treated. Considerable pains have been spent on the tables. However far the young engineer may eventually outgo his teacher as re- gards the text of the book, these are implements of his art which never become antiquated, and can never fall into dis- use. Those herein containod which are Uriginal W111, it is hoped, be esteemed worthy of place with their well-approved associates. * The author invites friendly criticism: he would be pleased to receive suggestions, both for the improvement of the book, and for the correction of possible errors in it, should another edition be called for. In dismissing the work from his hands, the precarious snatches of time occupied in its preparation, by day and by night, during the past two years, which might have been more agreeably spent in reading, talking, or musing, recur to the writer’s mind; and the thought arises, To what end or from what motive do people undertake these technical labors? Why should Forney and Bourne toil to simplify steam for our ap- prehension; Nystrom to compile mechanical, Molesworth and Trautwine to epitomize civil engineering; Henck to prepare his elegant manual of field mathematics; Box to illustrate hydraulics; and Shreve, with lucid pen, to make clear for us the strains in truss or arch? The ordinary motives to en- deavor here have no place. There is neither fame nor profit in these drudging enterprises. At best the author gives name PREFA CE,. Wii to his book; he remains impersonal, - known but indirectly, and but to a class. How, then, shall we account for his labors? I take it, the Father of mankind has not only made our minds to hunger for knowledge as our bodies for food, but has also imposed upon us a kindly law of communion, by virtue whereof we cannot do otherwise, without violence to generous nature, than share with our fellows whatsoever we have learned that seems new and useful. Under this law.these beneficial works would appear to have had their being, and thus pure are they from the stain of selfishness. - Though the present writer would not arrogate equal fellow- ship in the eminent brotherhood named, yet he may justly claim like pureness from unworthy motive, and certainly feels like comfort at heart to that which they must know, for having discharged, in what measure it has been laid upon him, the divine obligation. - WM. F. SHUNK. RAHWAY, N.J. ABBTREVIATIONS. Increased by. Diminished by. Multiplied by. Divided by. Equal to. Since, or seeing that. ..". Hence, or therefore. : Indicates the quotient of one divided by the other of the quantities it connects, called sometimes the ratio of the quan- tities. - ce - :: Indicates an equality of ratios, and connects equal ratios in a proportion. Thus, a . b : . c : d indicates that a + b = c —— d, or it may be read, a is to b as c is to d. ( ) Brackets indicate that the operations embraced by them shall first be performed, and the result treated as a single factor in the remaining processes required by a formula. Thus, (a X b) -- (a + b) requires that the product of a and b shall be divided by their sum. - A”. A small secondary figure annexed thus to an expression. is called its eacponent. It requires the principal to which it is attached to be used as many times in continued multiplication as there are units in the exponent. Thus, A* = A X A.; A* = A X A X A, which is called the cube, or third power, of A. ^/ This is called the square root sign: it signifies that the square root of the quantity covered by it is to be taken. */ If preceded by a small secondary figure, called the indew, as in the marginal figure, it indicates that the cube root of the quantity covered by it shall be taken; and so on. wº */ If the index be fractional, as in the marginal figure, it requires that the square root of the third power of the quantity covered shall be taken. - B. M. Bench-mark : any fixed reference point for the level, - IX : X ABBREVIATIONS. as outcropping ledge, water-table of building, or other perma- Inent object. Usually a blunt conical seat for the rod, hewn On a buttressed tree-base, having a small nail sometimes driven flush in the top of it, and a blaze opposite, on which the eleva- tion is marked with kiel. - - T. P. Turning-point : usually marked G) in the field-book. P. T. Point of intersection: as of tangents, which are to be connected by a curve. - A. D. Apex distance : i.e., the distance from the P.I. to the point where a curve merges in the tangent. P. C. Point of curve: the stake-mark at the beginning of 3, Clli VG. . . P. T. Point of tangent: the stake-mark at the end of a Cll]"We. P. C. C. Point "of compound curvature: the stake-mark where a curve merges in another of different curvature, turn- ing in the same direction. T. T. C. Polnt of reverse curvature: the stake-mark where a curve merges in another turning in the opposite direction. B. S. Backsight, in transit work; or the reading of the rod to ascertain the instrument height in levelling. - F. S. Foresight, in transit work; or the reading of the rod to ascertain elevations in levelling. II. I. Height of instrument: elevation of the level above the datum or zero plane. II. W. High water. L. W. Low water. TAIBLE OF CONTENTS. LOGARITHMS. I. Definitions and principles . º wº e tº II. Manner of using the tables . º g e e To find the logarithm of any number © e To find the number corresponding to a given logarithm . tº © * e e e tº Multiplication by means of logarithms e ſº Division by means of logarithms & ſº e To raise a number to any power by means of logarithms . d * e gº e To extract roots by means of logarithms te G PLANE TRIGONOMETRY. III. Definitions g º © tº g ge º © IV. Natural sines . g & g e • * * de V. Logarithmic sines, &c. . . º dº e VI. General propositions . g wº e & © VII. Solution of plane triangles . te & © o VIII. Right-angled plane triangles ADJUSTMENT AND USE OF INSTRUMENTs. IX. General remarks on adjustment . c & * X. The level. tº e te * e g º e To bring the intersection of the cross-hairs into the optical axis of the telescope &n te tº To bring the level bubble parallel with the tele- scope axis . e © e . º o e To adjust the wyes; or, in other words, to bring the telescope into a position at right angles to the vertical axis of the instrument . XI. Develling . e ſº * g e e tº º Correction for the earth’s curvature and refrac- tion . º e tº o te ſº e º To find differences in elevation by means of the barometer . e tº o * . tº g e Heights by the thermometer. . . . . --~~ XI 11 12 13 15 16 18 23 24 24 25 25 26 28 29 29 xii TABLE OF COW TENTS. XII. XIII. XIV. XV. Setting slope Stakes º º º tº s gº Vertical curves tº e g o gº tº The transit . & g © ge g G g To adjust the level tubes . & e tº & To adjust the vertical hair so that it shall re- volve in a plane, and mark backsight and fore- Sight points in the same straight line . tº To adjust the needle . . . G gº g & Miscellaneous. tº ſº e © dº g º The vernier . g * e e s § te To read an angle . g e tº g g g To re-magnetize a needle . o & e e To replace cross-hairs . s * • e To fix a true meridian * tº & PROPOSITIONS AND PROBLEMS RELATING TO THE CIRCLE. XVI. Propositions relating to the circle tº º º YVII. Circular curves on railroads e e g • . YVIII. To find the radius, the apex distance, the longth, the degree, &c., of a curve . © & Given the intersection angle I and radius R, to find the tangent T . º º e gº e Given the intersection angle I and tangent T, to find the radius R. § tº * Given the intersection angle I and chord AB=C, Connecting the tangent points, to find the radius R. © * gº e ſº iº tº Given the intersection angle I and the degree of curvature or deflection angle D, with 100-feet chords, to determine the length of the long chord C, the versed sine V, the externalsecant S, or the tangent T . & tº e & tº Given C, V, S, or T, of any curve, and D, the degree of curvature, to find the intersection angle I . e * e * * * * Given the intersection angle I and deflection angle D, to find the length of the curve. Given any radius R and chord C, to find the de- flection angle D . * • * g © e Given any radius R and chord C, to find the de- flection distance d te *... • g g © Given any radius R and chord C, to find the tangential angle T . . . . . . . Given any radius R and chord C, to find the PAGE 30 36 40 40 52 53 54 54 55 55 57 57 57 tangential distance t . tº & e tº g 58 TABLE OF CONTENTS. xiii - IPAGE YIX. Ordinates º & e e © © e . 58 Given any radius R and chord C, to find the middle ordinate M . • * g º Given the radius R, chord C, and middle ordi- W nate M, to find any other ordinate . º . 59 Ordinates of a 1° curve, chord 100-feet º . 60 TRACING CURVES AND TURNING OBSTACLES IN THE FIELD. XX. To trace a curve on the ground with the chain 58 only tº º e o º e . 63 XXI. To trace a curve on the ground with transit and 100-feet chain º e . 66 XXII. Turning obstacles to vision in tangent © . 71 XXIII. Turning obstacles to measurement in tangent . 73 SUGGESTIONS AS TO FIELD-WoRK AND Location-PROJECTs. XXIV. Suggestions concerning field-work • XXV. The curve-protractor and the projecting of loca- tions . & º * * > © © & . 84 Table showing the distance, D, in feet, at which a straight line must pass from the nearest point of any curve struck with radius r, in Order that a terminal branch having a radius R=2 r, and consuming a given angle, 2, may merge in said straight line º º º . 88 Table showing the distance, d, in feet, at which curves of contrary flexure must be placed asunder, in order that the connecting tangent, T, may be 300 feet long sº e º º . 89 PROBLEMS IN FIELD LOCATION. XXVI. How to proceed when the P. C. is inaccessible . 93 XXVII. How to proceed when the P. C. C. is inaccessible, 95 YXVIII. To shift a P. C. So that the curve shall termi- nate in a given tangent .. e e . . 96 XXIX. To substitute for a curve already located one of different radius, beginning at the same point, containing the same angle, and ending in a fixed terminal tangent e º e º XXX. Having located a curve A B C, to find the point B at which to compound into another curve of given radius, which shall end in tangent E F, parallel to the terminal tangent of the original curve, and a given distance from it . 98 XXXI. To shift a P. C. C. So that the terminal branch of a curve shall end in a given tangent .. . 99 79 97 xiv. TABLE or cowTENTS. PAGE XXXII. Having located a tangent, A B, intersecting a curve, C D, from the concave side, to find the point E on said curve, at which to begin a curve of given radius which shall merge in the located tangent . º tº g * ſº XXXIII. Having located a tangent, A B, intersecting a curve, C D, from the convex side, to find the point E on said curve at which to begin a curve of given radius which shall merge in the located tangent . tº e g & { } gº XXXIV. To locate a Y . e tº o e e g & XXXV. To locate a tangent to a curve from an outside - fixed point . tº e e e te g XXXVI. To substitute a curve of given radius for a tan- - gent connecting two curves * up e e XXXVII. To run a tangent to two curves already located. TRACK PROBLEMs. - - XXXVIII. Reversed curves . & g º © e gº XXXIX. To connect two parallel langents by a reversed curve having equal radii . e © e & XL. To connect two parallel tangents by a reversed curve having unequal radii • • tº © XLI. A reversed curve having unequal angles . g XLII. A reversed curve between fixed points XLIII. To connect two divergent tangents by a reversed Curve . & e & 2 e © e e tº XLIV. To shift a P. R. C. so that the terminal tangent shall merge in a given tangent . o e * XIV. To pass a curve through a fixed point, the angle of intersection being given o * e e XLVI. Frogs and switches . . . . . . . To find the radius of a turnout curve, the frog angles, and the distances from the toe of switch to the frog points . . & e ‘ e & To find the angle of switch-rail with main track. To find the distance from toe of switch to point of main frog. & iº º ſº e * tº To find the radius of outer rail of turnout curve, To find the main frog angle, the radius of Outer rail being known . { } t e © * © To find the angle of the middle frog in the case of a double turnout e º e e e To find the distance from toe of switch to point of middle frog . & ſº º & º 102 103 103 107 108 109 117 119 ... 120 123 125 127 129 129 130 130 131 131 131 131 Turnout tables º & 0. © º , 135 3. 136 TABLE OF CONTENT'S. XV XLVII. To locate a turnout. { } o e e º te XLVIII. Crossings on Straight lines . ge º te tº XLIX. Crossings on curves e †e L. Elevation of the Outer rail on Curves . Table for Same tº * * e * tº * LI. Trackmen’s table of curves and Spring of rails. Explanation of same . & tº ſº & LIST OF TABLES. I. II. III. IV. . Decimals of an acre in one chain length of 100 feet, VI. VII. VIII. IX. * . Radii and their logarithms, middle Ordinates, and YI. YII. YIII. |XIV. XVI. XV. XVII. Time of meridian passage of North Star above the pole for the year 1870, and on e * º Time of extreme elongation of North Star for the year 1870, lat. 40°, and on . e tº tº * * * Azimuths of the North Star, and their natural tan- gents e ſº & e © • e wº Boods and perches in decimal parts of an acre . and of various widths. & * ſº Acres, roods, and perches in square feet Circular arcs to radius of 1 . . tº e te e Eeet in decimals of a mile . º º g tº Inches reduced to decimal parts of a foot . * deflection distances . e tº © ſº te tº Squares, cubes, &c., of numbers from 1 to 1042 . e Logarithms of numbers, 1 to 1000. * e e e Logarithmic sines, cosines, tangents, and cotan- gentS . e e & e ſº Natural sines and tangents . g e Chords, versed sines, external Secants, and tangents of a one-degree curve . * g tº e Slopes for topography - Rise per mile of various grades . . . PAGE 137 139 139 141 142 143 144 149 150 150 151 151 152 152 153 153 155 161 179 197 243 269 315 317 LOGARITHMS. I.–II. LO G. A. RITEIM S. I. DEFINITIONS AND PRINCIPLES. 1. THE logarithm of a number is the exponent of the power to which it is necessary to raise a fixed number to produce the given number; that is to say, it represents the number of times a fixed number must be multiplied by itself in order to produce any given number. i The fixed number is called the base of the system. In the common system, this base is 10. It follows from the above, that the logarithm of any power of '10 is equal to the exponent of that spower. If, therefore, a number is an exact power of 10, its logarithm is a whole number. If a number is not an exact power of 10, its logarithm will not be a whole number, but will be made up of an entire part plus a fractional part, which is generally expressed decimally. The entire part of the logarithm is called the characteristic; the decimal part is called the mantissa. 2. The characteristic of the logarithm of a whole number is positive, and numerically 1 less than the number of places of figures in the given number. Thus, if a number lies between 1 and 10, its logarithm lies between:0 and 1; that is, it is equal to 0 plus a decimal. If a number lies between 10 and 100, its logarithm is equal to 1 plus a decimal; and so on. 3. The characteristic of the logarithm of a decimal fraction is negative, and numerically 1 greater than the number of 0's that immediately follow the decimal point. - - The characteristic alone, in this case, is negative, the man- 3 - 4 MANNER OF USANG THE TABLE'S. tissa being always positive. This is indicated by writing the negative sign over the characteristic: thus, 2.3802.11 is equiva- lent to — 2 + .3802.11. 4. The characteristic of the logarithm of a mixed number is the same as that of its entire part. Thus the mixed number 74.103 lies between 10 and 100; hence its logarithm lies be- tween 1 and 2, as does the logarithm of 74. 5. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. The logarithm of a quotient is equal to the logarithm of the dividend diminished by that of the divisor. The logarithm of any power of a number is equal to the loga- rithm of the number multiplied by the exponent of the power. The logarithm of any root of a number is equal to the loga- rithm of the number divided by the index of the root. 6. The preceding principles enable us to abridge labor in arithmetical calculations, by using simple addition and sub- traction instoad of multiplication and division. II. MANNER OF USING THE TABLES. TO FIND THE LOGARITHM OF ANY NUMBER. 1. First find the characteristic’ by rule 2, 3, or 4, given above. - 2. Then, if the number be less than 100, look in column N of the table for 10 times or 100 times the amount of it; oppo- site this multiple, in column O, will be found the mantissa. Thus the logarithm of 6 is 0.778151; that of 84 is 1.924279. 3. If the number lie between 100 and 10000, find the first three figures of it in column N: then pass along a horizontal line until you come to the column headed with the fourth figure of the number. At this place will be found the mantissa. Thus the logarithm of 7200 is 3.857332; that of 8536 is 3.931254. MANNER OF USING THE TABLES. 5 4. If the number be greater than 10000, place a decimal point after the fourth figure, thus converting the number into a mixed number. Find the mantissa of the entire part by the method last given. Then take from column D the correspond- ing tabular difference, multiply this by the decimal part, and add the product to the mantissa just found. The principle employed is that the differences of numbers are proportional to the differences of their logarithms, when these differences are small. - Thus the logarithm of 672887 is 5.827943; that of 43467 is 4.638160. - . . - 5. If the number be a decimal, drop the decimal point, thus reducing it to a whole number. Find the mantissa of the log- arithm of this number, and it will be the mantissa required. Thus the logarithm of .0327 is 2.514548; that of 378,024 is 2.577520. - x TO FIND THE NUMBER CORRESPONDING TO A GIVEN LOGARITHM. 6. The rule is the reverse of those just given. Look in the table for the mantissa of the given logarithm. If it cannot be found, take out the next less mantissa, and also the corre- sponding number, which set aside. Find the difference be- tween the mantissa taken out and that of the given logarithm; annex as many 0's as may be necessary, and divide this result by the corresponding number in column D. Annex the quo- tient to the number set aside, and then point off from the left hand a number of places of figures equal to the characteristic plus 1; the result will be the number required. ‘If the char- acteristic is negative, the result will be a pure decimal, and the number of 0’s which immediately follow the decimal point will be one less than the number of units in the charac- teristic. - * ... ? - t Thus the number corresponding to the logarithm 5.233568 is 171225.296; that corresponding to the logarithm 2.233568 is .0171225. - * * x - - MULTIPLICATION BY MEANS OF LOGARITHMS. 7. Find the logarithms of the factors, and take their sum; then find the number corresponding to the resulting logarithm, , and it will be the product required. * - 6 MANNER OF USING THE TABLES. Bacample. Find the continued product of 3.902, 597.16, and 0.0314728. operation. Log. 3.902 . . . 0.591287 Log. 597.16 . . . 2.76091 Log. 0.0314728 . 2.497936 1.865314 = log. 73.3354, the product. Here the 2 cancels the +2, and the 1 carried from the deci- mal part is set down. * IXIVISION BY MEANS OF LOGARITHMS. 8. Find the logarithms of the dividend and the divisor, and subtract the latter from the former; then find the number corresponding to the resulting logarithm, and it will be the quotient required. IEacample 1. Divide 24163 by 4567. Operation. Log. 24163 . . . 4.383151 Log. 4567 . . . 3.659631 0.723520 = log. 5.29078, the quotient. IEacample 2. Divide 0.7438 by 12.9476. , Operation. Log. 0.7438. . . 1.871456 Log. 12.9476. . . 1.112189 2.759267 =log, 0.057447, the quotient. Here 1 taken from - gives 2 for a result. The subtraction, as in this case, is always to be performed in the algebraic way. 9. The operation of division, particularly when combined with that of multiplication, can often be simplified by using the principle of the arithmetical complement. The arithmetical complement of a logarithm (written a... c.) MANNER OF USING THE TABLE'S. 7 is the result obtained by subtracting it from 10: it may be written out by commencing at the left hand, and subtracting each figure from 9 until the last significant figure is reached, which must be taken from 10. Thus 8.130456 is the arithmet- ical complement of 1.869544. To divide one number by another by means of the arith- metical complement, find the logarithm of the dividend and the arithmetical complement of the logarithm of the divisor; add them together, and diminish the sum by 10; the number corresponding to the resulting logarithm will be the quotient required. IEacample. Multiply 358884 by 5672, and divide the product by 89721. - Operation. Log. 358884 . . . 5.554954 Log. 5672 . . . 3.753736 (a.c.) Log. 89721 . . . 5.047106 4.355796 = log. 22688, the result. The operation of subtracting 10 is performed mentally. TO RAISE A. NUMBER TO ANY POWER BY MEANS OF LOGA- RITHMS. 10. Find the logarithm of the number, and multiply it by the exponent of the power; then find the number correspond- ing to the resulting logarithm, and it will be the power required. - Bacample. Find the 5th power of 9. Operation. Jog. 9 . . . . . 0.954.243 5 4.77.1215 = log. 59049, the power. TO EXTRACT ROOTS BY MEANS OF LOGARITHMS. 11. Find the logarithm of the number, and divide it by the index of the root; then find the number corresponding to the resulting logarithm, and it will be the root required. 8 MANNER OF USING THE TABLES. Fa'ample. Find the cube root of 4,096. Operation. - & Log. 4,096, 3.612360; one-third of this is 1.204120, to which the corresponding number is 16, which is the root sought. - - 12. When the characteristic is negative, and not divisible by the index, add to it the smallest negative number that will make it divisible, and then prefix the same number, with a plus sign, to the mantissa. { 6. . . . . IEacample. t - Find the 4th root of .00000081. The logarithm of this num- ber is 7.908485, which is equal to 8 + 1.908485, and one-fourth of this is 2.477121; the number corresponding to this logarithm is .03: hence .03 is the root required. PLANE TRIGONOMETRY. III. —VIII. PLANE TRIGONOMETRY. . III. DEFINITIONS. 1. Plane Trigonometry treats of the solution of plane tri- angles. - In every plane triangle there are six parts, – three sides and three angles. When three of these parts are given, one being a side, the remaining parts may be found by computation. The operation of finding the unknown parts is called the solu- tion of the triangle. 2. A plane angle is measured by the are of a circle included between its sides; the centre of the circle being at the vertex, and its radius being 1. The circle, for convenience, is divided into 360 equal parts called degrees; 90 of these parts are included in a quadrant, which includes one-quarter of the circle, and is the measure of a right angle. Each degree is further divided into 60 equal parts called minutes, and each minute into 60 equal parts called seconds. Degrees, minutes, and seconds are de- - noted by the symbols tº B T^ o, /, //; thus the ex- N º pression 7° 22' 33// is N M}. read, 7 degrees, 22 # `sº minutes, and 33 sec- onds. 3. The complement of an angle is the dif- ference between that a n gle and a right - angle. * 4. The supplement of an angle is the difference between that angle and two right angles. 11 e D~ A 12 NATURAL SINES, ETC. 5. Instead of employing the arcs themselves, certain func- tions of the arcs are usually employed, as explained below. A function of a quantity is something which depends upon that quantity for its value. - The sine of an angle is the distance from one extremity of the are enclosing it, to the diameter, through the other extrem- ity. Thus PM is the sine of the angle MOA. The cosine of an angle is the sine of the complement of the angle. Thus N M = O P is the cosine of the angle M O A. The tangent of an angle is a right line which touches the enclosing arc at one extremity, and is limited by a right line drawn from the centre of the circle through the other extrem- ity: the sloping line which thus limits the tangent is called the secant of the angle. A T is the tangent and OT the secant of the angle M O A. - } The versed sine of an angle is that part of the diameter AP which is intercepted between the foot of the sine and the ex- tremity of the enclosing arc. . The cotangent of an angle is the tangent of the complement of that angle; the co-versed sine and cosecant are similarly defined. Thus B TV, BN, and O TV are respectively the co- tangent, co-versed sine, and cosecant of the angle M O A. These terms are in practice indicated by obvious contractions; as, sin. A for the sine of A, cos. A for the cosine of A, &c. 6. The above definitions have been made with reference to a radius of 1. Any function of an arc whose radius is R is equal to the corresponding function of an arc whose radius is 1, multiplied by the radius R. So also any function of an arc whose radius is 1 is equal to the corresponding function of an are whose radius is R, divided by that radius. ' IV. NATURAL SINES, ETC. 1. Natural sines, cosines, tangents, or cotangents are those which are referred to a radius of 1. They may be used for all the purposes of trigonometrical computation; but it is found more convenient, in many cases, to employ a table of logarith- Imic sines. LOGARITHMIC, SIWES, ETC. 13 V. LOGARITHMIC SINES, ETC. . 1. Logarithmic sines, cosines, tangents, or cotangents are re- ferred to a radius of 10,000,000,000, of which the logarithm is 10. - TO FIND THE LOGARITHMIC FTINCTIONS OF AN ARC WHICH IS EXPRESSED IN DEGREES AND MINUTES. 2. If the arc is less than 45°, look for the degrees at the top of the page, and for the minutes in the left-hand column; then follow the corresponding horizontal line till you come to the column designated at the top by sine, cosine, tamg., or cotang., as the case may be; the number there found is the logarithm sought. Thus, log. sin. 190 55'. . . . 9.532312 log. tang. 19° 55'. . . . 9.559097 3. If the angle is greater than 45°, look for the degrees at the bottom of the page, and for the minutes in the right-hand column; then follow the corresponding line towards the left, till you come to the column designated at the bottom by sine, Cosime, tang, or cotang, as the case may be ; the number there found is the logarithm sought. - Thus, log. cos. 52° 18' . . . . 9.786416 log. tan. 52° 18' . . . . 10.11.1884 4. If the arc is expressed in degrees, minutes, and seconds, proceed as before with the degrees and minutes; then multiply the corresponding number taken from column D by the num- ber of seconds, and add the product to the preceding result, for the sine or tangent, and subtract it therefrom for the cosine Or Cotangent. Facample. Eind the logarithmic sine of 40° 26/28/. 14 . LoGAR1THMIC SINES, ETC. Operation. Log. sine 40° 26' . . . . . . 9.811952 Tabular diff. 2.47 No. of seconds, 28 Product . . 60.16 to be added . 69 Log, sine 40° 26' 28/ . . . . . 9.812021 5. If the arc is greater than 90°, find the required function of its supplement. Thus the logarithmic tangent of 118° 18' 25", is equivalent to that of its supplement, or 61° 41' 35", and is 10.268732. Also the logarithmic cosine of 95° 18' 24// is 8.966080, and the log, cot. of 126°23'50' is 9.851619. TO FIND THE ARC CORRESPONDING TO ANY LOGARITHMIC - FUNCTION. 6. This is done by a reverse process. Look in the proper column of the table for the given logarithm; if it is found there, the degrees are to be taken from the top or bottom, and the minutes from the left or right hand column, as the case may be. If the given logarithm is not found in the table, find the next less logarithm, take from the table the corresponding degrees and minutes, and set them aside. Subtract the loga- rithm found in the table from the given logarithm, and divide the remainder by the corresponding tabular difference. The quotient will be seconds, which must be added to the degrees and minutes set aside, in the case of a sine or tangent, and subtracted in the case of a cosine or cotangent. IEacample. IFind the arc corresponding to log. sin. 9.422248. Operation. Given logarithm . . . 9.422248 Next less in table . . . 9.421857 . . . .15° 19/ Tabular diff. . . 7.68) 391(51/ to be added. Hence theºrequired arc is 15° 19' 51/. 7. By analogous process, the are corresponding to log, cos. ‘9.427485 Will be found to be 74° 28/43/. . GENERAL PROPOSITIONS. 15 # WI. GENERAL PROPOSITIONS. 1. In any right-angled triangle the hypothen use is to one of the legs as the radius to the sine of the angle opposite to that leg. And one of the legs is to the other as the radius to the tan- gent of the angle opposite to the latter. 2. In any plane triangle, as one of the sides is to another, So is the sine of the angle opposite to the former to the sine of the angle opposite to the latter. -- 3. In any plane triangle, as the sum of the sides about the . vertical angle is to their difference, so is the tangent of half the sum of the angles at the base to the tangent of half their . difference. - 4. In any plane triangle, as the cosine of half the difference of the angles at the base is to the cosine of half their sum, so is the sum of the sides about the vertical angle to the third side, or base. Also, as the sine of half the difference of the angles at the base is to the sine of half their sum, so is the difference of the sides about the vertical angle to the third side, or base. 5. In any plane triangle, as the base is to the sum of the other two sides, so is the difference of those sides to the difference of the segments of the base made by a perpendicular let fall from the vertical angle. - 6. In any plane triangle, as twice the rectangle under any two sides is to the difference of the sum of the squares of those two sides and the square of the base, so is the radius to the cosine of the angle contained by the two sides. 16 - SOLUTION OF PLANE TRIANGLES. * VII. solution OF PLANE TRIANGLEs. 1. It is usually, though not always, best to work the propor- tions in trigonometry by means of logarithms, taking the logarithm of the first term from the sum of the logarithms of the second and third terms, to obtain the logarithm of the fourth term; or adding the arithmetical complement of the logarithm of the first term to the logarithms of the other two, to obtain that of the fourth. 2. There are three distinct cases in which separate rules are ... required. CASE I. 3. When a side and an angle are two of the given parts, the solution may be effected by proposition 2 of the preceding Section. - If a side be required, say, - As the sine of the given angle is to its opposite side, So is the sine of either of the other angles to its opposite side. 4. If an angle be required, say, - As one of the given sides is to the sine of its opposite angle, So is the other given side to the sine of its opposite angle. The third angle becomes known by taking the sum of the two former from 180°. IEacample 1. Given angle A=24° 26'; angle B=36° 43'; side b = 137.6: to find s side a. C -- - As sin. B . . log. 9.776598 Is to sin. A . log. 9.6.16616 So is b . . . log. 2.1386.18 11.755234, sum of 2d and 3d terms, . To a, 95.2 . . log. 1.978636 less 1st term. SOLUTION OF PLANE TRIANGLES. . . 17 Eacample 2. Given, sides a and b, as above, and angle A ; to find angle B. As side a . . . . . . (a. c.) log. 8.02.1364 Is to sin. A . . . . . . . log. 9.6.16616 So is side b . . . . . . . . log. 2.1386.18 To sin. B=36° 43' . . . . . log. 9.776598 sum. CASE II. 5. When two sides and the included angle are given, the solution may be effected by means of propositions 3 and 4. Thus, take the given angle from 1809; the remainder will be the sum of the other two angles. Then, by proposition 3, − As the sum of the given sides is to their difference, So is the tangent of half the sum of the remaining angles to the tangent of half their difference. Half the sum of the remaining angles added to half their difference will give the larger of them, and half their sum diminished by half their difference will give the lesser of them. The solution may be completed either by proposition 4, or by proposition 2, as in Case I. - Eaccºmple. Given side a =95.2, side b = 137.6, and the included angle c=418° 51/; to find the remaining angles. Here 180.00— 118° 51/=61° 09', the sum of the remaining angles. As sum of given sides, 232.8 . . . . . . log. 2.366983 Is to their difference, 42.4 . . . . . . log. 1.627366 So is tang. # sum of rem. angles, 30° 34' . log. 9.771447 To tang. , their difference = 6° 084' . . . log. 9.031830 Adding half the difference to half the sum, 30° 34'--6° 084/=369 43', = the larger angle, B. Deducting half the difference from half the sum =24° 26' = the smaller angle, A. This case is susceptible of Solution also by means of propo- sition 6. 18 RIGHT-ANGLED PLANE TRIA WGL ES. CASE III. 6. When the three sides of a plane triangle are given, to find the angles. - ** First Method. Assume the longest of the three sides as base; then say, conformably with proposition 5, – As the base is to the sum of the two other sides, So is the difference of those sides to the difference of the segments of the base. Half the base added to half the said difference gives the greater segment, and diminished by it gives the less; thus, by means of the perpendicular from the vertical angle, the original triangle is divided into two, each of which falls under the first case. Or they may be solved by the simpler methods applica- ble to right-angled triangles. & Second IMethod. 7. Find any one of the angles by means of proposition 6, and the remaining angles either by a repetition of the same rule, or by the relation of Sides to the sines of their opposite angles. VIII. BIGHT-ANGLED PLANE TRIANGLES. 1. Right angles may be solved by the rules applicable to all plane triangles; and it will be found, since a right angle is always one of the data, that the rule usually becomes simplified in its application. 2. When two of the sides are given, the third may be found by means of the rule that the square of the hypothen use is equal to the sum of the Squares of the remailling sides. 3. Another method for solving right-angled triangles is as follows: — To find a side. Call any one of the sides radius, and write upon it the word “radius.” Observe whether the other sides n RIGHT-ANGLED PLA WE TRIANGL ES. 19 become sines, tangents, cosines, or the like, and write upon them the propér designations accordingly. Then say, As the name of the given side is to the given side, So is the name of the required side to the required side. 4. To find an angle. Assume one side to be radius, and mark the remaining sides as before. Then say, As the side made radius is to radius, So is the other given side to the name of that side; Which determines the opposite angle. 5. Applying this method to the right- angled triangle A B C, and calling the hypothen use a radius, we shall have, b C A c = a sin. C –– R; hence sin. C = Rc -- a. b = a cos. C -- R; hence cos. C = Rb -- a. Then, assuming the side b to be radius, we shall have, c = b tang. C –– R; hence tang. C = Rc -i- b. If radius be called 1, the natural sines and cosines will be used in the application of these formulas; they are often more convenient than logarithms in railroad practice, especially when the numbers which measure the sides of the triangle are either less than 12, or are resolvable into factors less than 12. ADJUSTMENT AND USE . OF INSTRUMENTs. IX. —XV. AIDJUSTMENT AND USE OF * INSTRUMENTS. IX. GENERAL REMARKS ON ADJUSTMENTS. 1. Care should be taken in all instrumental adjustments, where screws work in pairs, to loosen one before tightening its Opposite. - 2. Remember that the eye-piece inverts the image of the cross-hairs, and that consequently any movement of it, by means of the small capstan head screws on the outside of the telescope-barrel, should be in the direction which would seem to increase the error requiring correction. 3. Before beginning the adjustments, screw the object-glass close home, and make a pin-scratch across its rim and the end of the tube, by which to mark its proper place; draw out the eye-piece until the cross-hairs are exactly in focus; that is to say, until no movement of the eye shall appear to displace them, and bring the object to be observed clearly into view. '4. Never permit the glasses to be rubbed with a gritty fabric. To remove the dust from them, use a Soft, clean handkerchief, and change often the part applied. 23 24 - TIVE L E WEL. Y tº THE LEVEL. To BRING THE INTERSECTION OF THE CROSS-HAIRS INTO THE OPTICAL AXIS OF THE TELESCOPE. *- 1. Set the instrument firmly, cast loose the wyes, and, by levelling and tangent screws, bring either of the cross-hairs to coincide with a well-defined object, distant from 400 to 600 feet, or as much farther as distinct vision can be had free from heat ripple. Gently rotate the telescope half-way around in the wyes. If the cross-hair selected for treatment then fails to coincide with the object, reduce the error one-half by means of the small capstan head screws at right angles to it on the telescope-barrel. Bring the spider-line again to coincide with the object by means of the levelling and tangent screws, and, if necessary, repeat the operation. Proceed in the same man- ner with the other cross-hair. If the error is large, bring both nearly right before undertaking their final adjustment. 2. Having thus adjusted the line of collimation upon a dis- tant point, requiring the object-tube to, be drawn well in, select a point close by, which shall require it to be thrust out almost to its limit. If any error appears, correct half of it with the small screws provided for the purpose, a little forward of the diaphragm, and usually protected by a movable sleeve on the outside; correct the other half with the levelling-screws. After completing this adjustment, test the former one on a distant object, and, if necessary, repeat the operations. 3. In the transit, the small guide-ring screws used for this adjustment are covered by the bulb of the cross-bar in which the telescope is fixed, and are therefore inaccessible. The adjustment, however, is one not liable to become deranged in either instrument, and, in the transit, is of comparatively Small importance. 4. The young practitioner should bear in mind that the intersection of the cross-liairs inay coincide with the optical axis of the telescope, and yet be out of centre as regards the field of view. Such eccentricity does not affect the working accuracy of the instrument, which depends upon the position THE LEVEL. 25 of the object-piece solely. It may be removed by manipulation of the small Screws securing the inner end of the eye-piece. TO TR RING THE LEVEL BUIBIBLE PAIRALLEL WITH THE TEI., E- SCOPE AXIS. 5. Clamp the instrument over either pair of levelling screws, and bring the bubble to the middle of its tube. Turn the tele- scope slightly on its bearings, so that the bubble-case shall project a little on one side or the other. If the bubble slips, correct half its movement by means of the small lateral capstan head Screws at one end of the case. Return the telescope to its first position, level up again, and repeat the operation until the erroneous movement ceases. This adjustment brings the telescope and level into the same vertical plane. 6. Next, the bubble being at the middle of its tube, carefully lift the telescope out of the wyes, turn it end for end, and replace it. If the bubble settles away from the middle, bring it half-way back by means of the capstan-heads, working up and down at one end of the case. Again middle it with the levelling Screws, and repeat the operation until the error is corrected. TO ADJUST THE WYES ; OR, IN OTHER WORDS, TO BRING THE TELESCOPE INTO A POSITION AT RIGLIT ANGLES TO THE VERTICAL AXIS OF THE INSTRUMENT. 7. Close the wyes. Unclamp. Set the telescope directly over two of the levelling screws, and with them bring the bubble to the middle of the tube. Then rotate the telescope horizontally, until it stands over the same pair of Screws, changed end for end. If the bubble errs, correct one-half of . the deviation with the capstan head nuts at the end of the bar, and one-half with the levelling screws. Place the tele- scope over the other pair of levelling screws. Repeat the operation there; and continue the corrections, over one and the other pair of levelling screws alternately, until the bubble stands without varying during an entire revolution of the instrument upon its vertical axis. 8. The capstan head nuts on the cross-bar should be moved by gradual stress, not by pounding. They are a rude device. With so short a leverage as the length of the common adjust- ing-pin supplies, it is almost impossible to give them a smooth, 26 LEVELLING. manageable motion. They reproach the instrument-maker’s art as unchecked hydrophobia and cancer do that of medicine, or mercenary villany that of law, and should be supplanted by better practice. - 9. Having thus completed the principal adjustments in their proper order, bring the telescope and its bubble-case as nearly vertical in the wye bearings as hand and eye can make them, and by reference to a plumb-line, or the corner of a well-built house, see if the vertical hair is out of true. If so, slightly loosen two opposite screws of the diaphragm, and correct the error by turning it. Try again the adjustment of the line of collimation before pinning up the Wyes. XI. LEVELLING. 1. Suppose O the starting-point; 1, 2, 3, &c., the stakes of survey; and A the initial bench-mark. Wherever convenient the elevation of A above mean tide should be ascertained. It is to be regretted that this was not done from the outset, º \º 2.64. 3.7 3- — — — — —---- 6 b 1.9 .8! //////777 % //7 //7 //7 / | * under statute provisions requiring maps and profiles also to be filed at the several State capitals. In that case, not only would much after labor and expense by way of duplicate sur- veys have been spared, but the older Commonwealths would now have in hand materials for the preparation of physio- graphical maps, the value of which to science, to the engineer, and to the economical geologist, it were hard to measure. LE VEL LING. - 27 2. For the purposes of a railroad-survey, however, such determination is not needful. Any elevation may be assumed for A, taking care only that it be large enough to avoid the possibility of having minus levels, which would be inconven- ient. Zero of the datum should be below the lowest probable ground on the contemplated line. 3. Let the elevation of the initial bench-mark, A, in the figure, be taken at +200. Set the level at B, and suppose the rod on the B M to read 2.22. The “instrument height'’ then is 202.22. If the rod at sta. O reads 8.4, the elevation at that point is 202.22 – 8.4 = 193.8. The rod being 1.9 at sta. 1, . the elevation there is 202.2 — 1.9 = 200.3. If desirable to turn at sta. 2, drive a pin nearly to the ground at that stake; sup- pose the rod on it to read 0.81. The elevation then is 202.22– 0.81 = 201.41. Now move the instrument to C, and, sighting back to sta. 2, let the rod standing on the pin read 2.64. This makes the new “instrument height'’ at C = 201.41, the height of sta. 2, + 2.64 = 204.05, and the elevations at 3, 4, 5, or other points observed from C are found by deducting the “rods” at those points from the ascertained instrument height at the new point of observation. 4. It thus appears how simple is the rule of levelling, namely: Find the “instrument height” by adding the “back- sight” to the elevation of the point upon which the rod stands. for that purpose: from the “instrument height” thus found deduct the “foresights,” severally, in order to find the eleva- tions of the points at which such foresights are taken. 5. The foregoing example would appear in the field-book as follows: — - STA. B. S. INST. F. S. ELEVA. REMARKS. |B M. tº wº e e gº º 200.00 || B M on W. Oak. tº & 2.22 || 202.22 tº gº is tº 40 ft. N. of Sta. O. O tº gº © & 8.4 193.8 1 1.9 200.3 2 tº gº © º 0.81 201.41 & de 2.64 204.05 e tº tº º 3 tº e tº º 3.7 200.3 4 3.2 200.8 5 10.36 193.69 6. In levelling where great exactness is necessary, the rod at turning-points should be read to thousandths, and the reading checked by the leveller. Before taking it down, after clamp- 28. J.E FTELLING. --, ing the target fast, it should be swayed slowly to and fro in the direction of the instrument to make sure of getting the full height. In foul weather the rodman should take care that the foot of the rod does not ball up with mud or snow. The leveller should have his cross-hairs free from parallax, the tar- get in focus, and see his bubble true at the moment of obser- vation. He should also set the instrument about half-way between turning-points when practicable, balancing largely 'unequal sights by Subsequent ones similarly unequal in the opposite direction; and his turning-points, even on favorable. ground. Ought not to be more than 600 or 800 feet asunder. 7. On ordinary railroad field work such nicety as is implied in most of these rules is not required. To read to the nearest tenth is sufficient, especially where the progress of the party depends in a good degree on the level; as, for example, in run- ning grade lines on preliminary survey. The location levels are usually carried along more carefully; but even then the writer’s practice has been to turn to hundredfhs only. 8. The Philadelphia Rod is the best for our service. The sliding halves are unconnected except by brass sleeves or clips, which guide them, and are therefore not liable to bind in wet weather. They are made by William J. Young’s Sons, who some years ago, at the Writer's suggestion, Supplied What seemed to be their only defect by adopting rivets for fastening the clips instead of wood screws: the screws had a tendency to work loose in the field, and cause the parts to chafe or jam.' These rods are clearly figured, so as to be legible at a distance of Several hundred feet; the leveller is thus enabled to take intermediate elevations rapidly, and, when necessary, to do his work with the aid of an unlettered rodman. 9. CORRECTION FOR THE EARTH's CURVATURE AND REFRAC- - TION. The correction for a 100-feet “station ” is .000215; for One mile, 0.6. It is to be added to the calculated elevation of the point observed, or to be deducted from the “rod” before calculating the elevation, in the case of a long unbalanced sight. It varies as the square of the distance. Calling the required correction A, for any given distance D, then A = .000215 × D 2 if D is in “stations,” and A = 0.6 X D 2 if-D is in miles. Thus the correction for 10 stations would be I, EVELLING. 29 .0215; for 50 stations, 0.5375; for 10 miles, 60 feet, and a spire or treetop apparently level with the instrument at that dis- tance would really be 60 feet above it. Transposing the equa- tion we have D = WA --0.6. In this form it is applicable to the determination of distances at sea. The Peak of Teneriffe, for example, 16,000 feet high, should be just visible from the sea-level at a distance= V 10000--0.6= say 163 miles. 10, TO FIND DIFFERENCES IN ELEVATION BY MEANS OF THE BAROMETER. Call the required difference D; the barometrical reading at the lower stand, IL; that at the upper stand, U. Then, D=[(L–U)+(L--U) | x 55000. rº. Bacample. L = 26.64; U = 20.82. Then, L – U = 5.82 . . . . log. 0.764923 L–H U = 47.46 . . . . log. 1.676328 0.1226 . . . . . . Diff. —1.088595 And 0.1226 × 55000 = 6743, the required difference of elevation in feet. -: - 11. A closer approximation is thought to be attainable by using a thermometer in connection with the mercurial barome- ter. In that case, having found the difference as above, add 14t of the result for each degree by which the mean tempera- ture of the air at the two stands exceeds 55°; subtract the like proportion if the mean temperature be below 550. When the upper thermometer reads highest, for “subtract ’’ say “add,” and vice versa in the foregoing rule. e 12. The naked formula, however, will usually be sufficient for the engineer. He can prescribe gradients by it for surveys, which shall develop the ground to be occupied, and can decide between summits well differenced in height. If not so differ- eliced, questions of detour, of approaches, and the like, will contribute to determine the expediency of making an instru- mental examination. - 13. HEIGHTS BY THE THERMOMETER. T = the difference, in degrees Fahrenheit, between 212°, the temperature of boiling water at the sea level, and that at the place of observation. 30 SETTING SLOPE STAKES. H = the height of place of observation above or below the Sea in feet. - H = 513 T + T2. IEacample. 20 — 20SO = 40. 21 (513 × 4) + 4* = 2068 feet. T = H = XII. SETTING SLOPE STAKES, 1. Like swallowing, this is more easily done than described. To no detail of field service does the proverb more fitly apply, that “work makes the workman.” - 2. The problem is, to find where a formation slope of given inclination, beginning at the side of the road-bed, must needs intersect the ground surface. Formation slopes are usually stated in parts horizontal to one part vertical. Thus a slope of 45° is “1 to 1.” A slope of “2 to 1” has a horizontal reach of two feet to each foot vertical. The carriages of a stairway with twelve-inch treads and eight-inch risers would have a slope of “1} to 1.” 3. To fix the point where any proposed formation slope must meet the surface on level ground, is quite simple; the distance from the centre line being obviously made up of half the width of road-bed added to the horizontal distance due from the slope, to the depth of cut or height of fill. Thus, with 20 feet road-bed, 9 feet cut, and slope of 1% to 1, the distance out would be 10 + 9 + 4} = 234 feet, as shown in the annexed diagram. - <. 23.5 × IO X Iº. 5 2. • * Cº. Co 2-9 Ríº, z X | O X | 0 * 4. On slant or broken ground, the Solution is more difficult: recourse must then be had to the level, with a rodman, a tape- man, and, for good Speed, an axeman to assist. SETTING SLOPE STAKES. 31 Pacample No. 1. 5. Let the accompanying figure represent the cross-section at any given point of a proposed excavation; road-bed 20 feet wide, cutting at centre Stake 12 feet, and formation slopes 1 to 1. Qy K------ tº e --ra * * * Wºw 2 O O – - *f * - † I f ! - - - CO 16:5 ! 12 rº i or) K---14.5 -------- s: il-o j < * ! Af x-r-ţ #Tº fºe ſ 2 N d; | ti } • ‘N #| ? i h &\ of ºp } 15 $2 X # N. LT ſ * * 39 º rº- 19, 2 I to tº S. | l I - f I § ; | - 5,8 ~!> J. Z. 18.9 <, łO x iO ...” -- *s a tº ºr - tº a wº 28. O ſº 6. The first step is to set the level, as at A, commanding, let us suppose, the lower slope, and to ascertain its height above grade at the proposed section. This is usually done by refer- ence to the nearest bench, and pegging from Stake to stake as the work progresses. Unless the ground is very steep, and the slope-stakes largely different in elevation, labor will be saved and likelihood of error reduced by levelling over the centre line beforehand, as a separate job, and marking on centre stakes the cuts, fills, and grade points, that is to say, the points where excavation passes into embankment. The rods should be taken carefully at the stakes, and the latter marked on their backs to the nearest tenth, as “grade,” “C 12,” signify- ing cut 12 feet, or “F 6.2,” signifying fill 6.2 feet, for ex- ample. This being done, each centre stake serves as a bench- ‘mark for slope staking at that section, and each cross section can be staked out independently. 7. Instrument height, in the example treated, being by either 'method fixed at 15.5 above grade, the next step is a guess how far out from the centre stake the formation slope would proba- ‘bly meet the ground surface. The closeness of the guess will correspond to the experience and natural skill of the leveller: the young engineer slıould not be discouraged if he misses the ‘mark rather widely in lais early trials. * 32 SETTING SLOPE STAKES. º 8. It is true, that, on a uniform declivity, he might aid con- jecture by taking a rod distant half the width of road-bed, or 10 feet, from the centre stake, ascertain thus the slope of the ground surface as well as the cutting at that point; and with these data, knowing also the formation slope, approximate the point sought by solving the terminal triangle of the pro- posed section, indicated by dotted lines in the figure. But, in practice, he will find it the quicker and better way to approxi- mate the point by vividly imagining the underground forma- tion lines; or by vividly imagining a level section, the upper surface of which shall coincide with his instrument height, 15.5 feet above grade . This gives him a point in the air, 10 — 15.5 = 25.5 feet out from the centre stake, level with the instrument, as the limit of the imaginary section; and from that point he can pretty well judge where a line corresponding to the formation slope must meet the ground. 9. Suppose him, by either method, or even by random guess, LU think that 1ſ) feet for half the road-bed, and 10 more for the slope, looks about right. The formation slope being 1 to 1, this implies a cutting of 10 feet at the side stake, and a rod, therefore, of 15.5 — 10.0 = 5.5 feet. Taking a rod accordingly, 20 feet out, measured horizontally from the centre stake, he finds it to be 11.0 instead of 5.5, indicating that he has gone too far down hill. Let him now reason that the rod of 11.0 corresponds to a cutting of 15.5 — 11.0 = 4.5 feet, and that a cutting of 4.5 feet corresponds to a distance out of 10 + 4.5 = 14.5 feet. Try, then, a rod 14.5 feet out. It proves to be 9.0, corresponding to a cutting of 15.5 – 9.0 = 6.5, instead of 4.5 feet, and a distance out of 16.5 instead of 14.5 feet. Try, next, 16.5 feet out; the rod there, of 10.0 instead of 9.0, shows him again to be in error on the down-hill side of his object; but the lessening error shows also that he is approaching it, and that a few more like trials will reach it. 10. Recurring to his first error with the 11.0 feet rod, he cannot fail to observe after a little practice, since the ground ascends thence toward the centre line, that the side stake must fall farther out than the point where his second trial was made; that its true position, in fact, divides the distance be- tween those points of observation into two parts which are to one another directly as the inclinations of the formation slope and the ground surface. By degrees he will grow skilful in divining this true position, and, becoming meanwhile quick in SETTING SLOPE STARTES. 33 observation, will place a slope stake on the second or third trial, without conscious effort of mind. 11. Next, suppose the level at B, 25.5 feet above grade, com- manding the upper slope. Note the change of ground 11 feet out, and take a rod there, recording the observation. The cutting at that point is 25.5 – 9.5 = 16 feet, corresponding to a distance out for the side stake of 10 + 16 = 26 feet, if the ground were level. A trial rod 26 feet out reads 7.8, corresponding to a cutting of 25.5 — 7.8 = 17.7 feet, and a distance out for the side stake of 10 + 17.7 = 27.7 feet, showing that the point sought is still beyond. A repetition of such trials will finally fix it; but, as in the case of the lower slope, practicé will speedily lessen the number and abridge the labor of them. 12. The foregoing section would be noted in the field book as follows: — - STA. - DIS. LEFT. CENTRE - RIGHT. AREA.!C. Y.D.s – pºw + 5.8 - + 16.0 | + 18.0 258 || 50 i5.8 + 12.0 11.0 | 28.0 Eacample No. 2. 13. In the annexed figure, representing an embankment 14 feet wide on top, with side slopes of 1% to 1, the first thing to attract attention is that the instrument is 1 foot below grade, a tº fº *** * * WW 2 ſ N cºco 'ojo 22.6 * and that, therefore, 1.0 is to be added to all rods, in order to find the height of embankment above the points at whicle rods are taken. 14. Consider the down-hill side. The engineer, with the ground in view, and with the height of embankment at the 34 SETTING SI, () I’E STARTES. centre stake to aid him in forming an airy image of the pro- posed section, judges that the natural surface and the forma- tion slopes will meet 30 feet out. Of this distance, 7 feet are due to half the road-bed, and 23 feet to horizontal reach of the enbankment slope. The slope being 1% to 1, or #, the hori- zontal reach of 23 feet corresponds to a vertical height of # of 23 = 15.3 feet; and, since the instrument is 1 foot be- low grade, to a rod at the supposed embankment base of 153 — 1.0 = 14.3 feet. But the rod at that point is only 11 feet, to which, if 1 foot, the distance of instrument below grade, be added, the height of embankment would be 12 feet. He may then, as in the case of the upper slope in Example No, 1, try a rod at the distance out corresponding to the 11 feet rod, or 12 feet embankment. This distance would be 7 -- 12 + 6 = 25 feet, where, on trial, the rod proves to be 10 feet, instead of 11 feet, corresponding to an embankment height of 10 —- 1 = 11 feet, and to a distance out of 7 -- 11 —- 5.5 = 23.5 feet. Approximating thus, by shorter and shorter steps, he finally reaches the point sought. 15. The process in fixing the upper slope stake is similar to that used in fixing the lower one in Example No. 1. The several steps are designated by small letters in the figure, and a detail of them is not thought necessary. 16. This section would be noted in the field book as fol- lows: — STA: DIS. LEFT. CENTRE RIGHT. AREA. C.Y.Ds. — 9.4 - — 3.2 2 - - - U - 140 6 22.6 6.3 12.7 IEacample INo. 3. 17. Here is a case, partly in rock excavation, slope } to 1; partly in embankment, slope 13 to 1; road-bed 17 feet wide, 9 feet of which are on the right of the centre line and 8 feet on the left. 18. For the lower slope suppose the instrument height at A to be 6.5 feet above grade; centre cutting 2.5 feet. Find first, with a 6.5 feet rod, the grade point, to left of centre line, which proves to be 2.5 feet out. Note it, and set a stake there marked “grade.” Note also the change of ground 5.5 SETTING SLOPE STARTES. 35 feet out and 10.0 – 6.5 = 3.5 feet below grade. Then set the lower slope stake as in Example No. 2, observing that in this * A ſº * -re v-z. k— — —--! 3. 1 — — — —-- B TWT fe. feQ * †- * sº -O § § : 8 O 4. tº ſº | 8- N. ==º .; ºf - º &;---10.3--- º & & * * * * * sº st > t; O) * 5.5 × t --ſ : ; 12 8 22:… _T case the instrument is above grade, and that its height above grade is to be deducted from the rod at any point in order to obtain the height of grade above such point. - 19. Move the instrument to B, say 22.5 feet above grade. This elevation, if the cutting on that side be deemed to equal it, corresponds to a distance out of 9 feet for road-bed added to . (22.5 –– 4) for slope; total, 14.6 feet. The trial rod, however, at that distance, instead of reading 0, reads 6 feet, indicating a cut 22.5 – 6.0 = 16.5 feet deep, and a distance out correspond- ing thereto of 9.0 + (16.5 -- 4) = 13.1 feet. Trying again at this distance out, the rod reads 7.6 instead of 6 feet, requiring a further movement towards the centre line of (7.6 – 6) –– 4 = 0.4 feet. Thus by approximations much more rapid than in the case of a flatter formation slope, the point is soon fixed. 20. The field record of the above is as follows: — I g ! $ 5. 5 STA. | DIS. LEFT. cºrne Right. AREA.C.Y.Ds. 0.0 ) – 6.9 2.5 © . + 15.0 sºs 40 T18.3 | – 3.5 } +2.5 TT2.3 - J 36 VERTICAL CUP VES. XIII. VERTICAL CURVES. I)IAGRAM. GIVING THE ORDINATES OF A PARABOLA AT IN- TERVALs of ºr To THE SPAN, THE MIDDLE or DINATE BEING UNITY. F. — | | | | | | T. OO | ..g4 .97 ° 7's .83 •89 f so .# .55 , Bö "* . a/º | | | | 12 !! IO 9. 8 7 6 5 4. 3 2 l D 1. Suppose gradients descending right and left at an equal rate from the summit B, and that it is required to truncate the summit with a vertical curve extending 150 feet each way. A circular arc consuming so small an angle may be treated as a parabola, in which the external secant B F is equal to the versed sine F.D. Referring to the above diagram, Ordinates 4 and 8 will be seen to correspond to the ordinates between *S* B 2^ A 2^ chord AC and the curve in this instance, which ordinates therefore will be equal to the middle ordinate FD multiplied by 0.89 and 0.55 respectively. Adding these multiples to the grade elevation at A, the elevations of the intermediate points 4. 4. D | 2. selected will be ascertained. -- PERTICAL CUP VES. 37 IEacample 1. Elevation at A = + 94.0; AB = + 1 in 100; B C = — 1 in 100; AID, DC, each = -150 feet or 1.5 stations of 100 feet each. Hence D D = 1.5; and F D = 0.75 feet. . Ordinate S = 0.75 × 0.55 = 0.41. Ordinate 4 = 0.75 X 0.89 = 0.67. Elevation of grade at 8–8 = 94.0 + 0.41 = 94.41. Elevation of grade at 4 — 4 = 94.0 –H 0.67 = 94.67. Elevation of grade at D = 94.0 + 0.75 = 94.75. Iºacample 2. T-is *— – — i t t h t | t 3. | 2^ f D Elevation at A = + 94.0. AB = + 1 in 100; B C = –0.4 in 100; A H, level; A D, D H, each = 200 feet, or 2 stations, divided into 50 feet spaces, the points of division correspond- ing therefore to ordinates 3, 6, and 9 of the preceding diagram. C ! I H C H = 1 × 2 – 0.4 × 2 = 2.0 — 0.8 = 1.2 feet. Ascent from A to C along chord AC = CH +8 = 1.2 -- 8 = 0.15 per 50 feet, *. BIE = B D = 3 C H = 2 – 0.6 = 1.4. .*. FE = 1.4 —- 2 = 0.7. - Ordinates at 9 — 9 = 0.7 × 0.44 = 0.31. Ordinates at 6 – 6 = 0.7 × 0.75 = 0.52. Ordinates at 3 — 3 = 0.7 × 0.94 = 0.66. Mid-ordinate = 0,70, / The elevations then along the chord A.C, ascending at the rate of 0.15 per 50 feet, will be:— A 9 6 3 0 3 6 9 C 94.0 94.15 94.30 94.45 94.60 94.75 94.90 95.05 95.20 3S VERTICAL CUT VES. to which add the ordinates just found:— 0.0 0.31 0.52 0.66 0.70 0.66 0.52 0.31 0.0 and the grade elevations on the curve will be: — 94.0 94.46 94.82 95.11 95.30 95.41 95.42 95.36 95.2 Eacample 3. Elevation at A = + 94.0; AB = + 1 in 100; BC, AH, level. A.D., B C, each 200 feet divided into 50-feet spaces, the points C 5 H of division corresponding therefore to ordinates 3, 6, and 9 of the ordinate diagram C H = 1 × 2 = 2 feet. Ascent from A to C along chord A C = C H -- 8 = 0.25 per 50 feet. B E = B D — ; C H = 1 foot. .*. FE = 1 + 2 = 0.5. Ordinates 9 – 9 = 0.5 × 0.44 = 0.22. Ordinates 6 — 6 = 0.5 × 0.75 = 0.37. Ordinates 3 — 3 = 0.5 × 0.94 = 0.47. . Mid. ordinate = = 0.50. The elevations then along the chord A C, ascending at the rate of 0.25 per 50 feet, will be: — A 9 6 3 0 3 6 9 C 94.0 94.25 94.5 94.75 95.0 95.25 95.5 95.75 96.0 to which add the ordinates just found:— 0.0 0.22 0.37 0.47 0.5 0.47 0.37 0.22 0.0 And the grade elevations on the curve will be:– 94.0 94.47 94.87 95.22 95.5 95.72 95.87 95.97 96.0 VER TICAL CUAE VES. 39 Bacample 4. Elevation at A = + 94.0; AB = — 1 in 100; BC, AH, level; A D, B C, each 150 feet, divided into 50-feet spaces, the points of division corresponding therefore to ordinates 8 and 4 of the initial diagram C H = 1 × 1.5 = 1.5. - ~A 8 4. - | | | D | | | IE F | | | B Descent from A to C along chord A C = C H -- 6 = 0.25. E B = D B — D E = 1.5 — 0.75 = 0.75 .*. FE = 0.75 –– 2 = 0.375 Ordinates S – 8 = 0.375 × 55 = 0.21 Ordinates 4 — 4 = 0.375 × 89 = 0.33 Mid ordinate _* = 0.37 The elevations then along the chord A C, descending at the rate of 0.25 per 50 feet, will be: — A 8 4 0 4. 8 C * 94.0 93.75 93.5 93.25 93.0 92.75 92.5 From which deduct the ordinates just found, 0.0 0.21 ().33 0.37 0.33 0.21 0.0. ºw And the grade elevations on the curve will be:– 94.0 93.54 93.17 92.8S 92.67 92.54 92.5 The figures are drawn much distorted, in order to make the illustration clear. 2. With profile paper at hand, a convenient and quite suf- ficient determination of the grade elevations on a vertical curve may be made by drawing the gradients to a scale of 2 feet to an inch vertical, and 50 feet to an inch horizontal. By means of the curve protractor (Art. XXV. 1) a suitable arc may then be fitted and struck in, and the elevations read off, 40 THE TRANSIT. XIV. THE TRANSIT. 1. Should the vernier and circle plates be out of parallel, - should one or the other be sprung, a defect shown by a slight rocking motion when the rims are pinched alternately on Op- posite sides, – the instrument must be sent to the shop for repair. This is a common disease of transits in their old age: instrument-makers need to study its cause and cure. 2. TO ADJUST THE LIEVEI, TUISES. Bring the bubbles to the middle of them by means of the levelling screws. Turn the top of the instrument horizontally half way round. If the bubbles then err, reduce the error one- half with the small adjusting screws attached to the tubes, and one-half with the levelling screws. Repeat until the ad- justment is perfect. • 3. To ADJUST THE VERTICAL HAIR SO THAT IT SHALL RE- VOLVE IN A PLANE, ANI). MARK IBACKSIGHT AND FORE- SIGHT POINTS IN THE SA M H STRAIGHT I,INE. Try, first, by reference to the corner of a well-built house, or to a plumb-line, whether the hair be truly vertical. If it is not, loosen the four small capsian head screws on the outside of the barrel slightly, and gently tap the topmost one right or left, until the adjustment is effected. 4. Then, after bringing the four screws to a snug bearing again, direct the cross-hair to the edge of some well-defined object, as a chain pin, or stake, placed 400 or 600 feet distant. Upset the telescope, and place a like mark at about the same distance, and level in the opposite direction. Unclamp. Re- volve the instrument horizontally on its spindle half way round, and direct the cross-hair to the point first observed. Again upset the telescope. If the cross-hair now strikes aside from the second mark, correct one-quarter of the error by means of the lateral capstan head screws on the barrel, and THE TRANSIT. '41 one-quarter with the tangent screw. Repeat until the adjust- ment is effected. An experienced transitman will generally prefer to make this adjustment without aid, points in range being readily found. 5. Having thus brought the cross-hair to revolve in a plane, it is next to be seen whether the plane in which it revolves is truly vertical. To do so, set the instrument near the base Of Some lofty point, as ā church spire or chimney, on which point direct the cross-hair, and thon, tilting the object end of the telescope downwards, set a pin, or make a pencil dot in line. Unclamp the spindle; turn the instrument horizontally half- way round; clamp fast; fix the cross-hair again on the lower point, and try it on the upper one. If it misses, correct half the error by means of the adjusting screws now usually pro- vided, at one of the bearings of the cross-bar; or, if these be lacking, by filing off the feet of the standard which supports the higher end of the cross-bar. - 6. TO ADJUST THE NEEDLE. Having removed the cap, and placed the instrument con- veniently in a still room, push one end of the needle a little aside from the point where it tends to settle, and exactly to some figured division line on the graduated circle. There gently stay it in position by means of a small wooden block, an ivory die, or the like. Observe where the opposite end Strikes. If between graduation lines, mark the precise spot with a sharp pencil. Turn the needle end for end, and stay the reverse point at the division line first observed. Again spot with the pencil where the opposite end stakes. Midway of these two pencil spots make another. Take the needle off the pivot, and bend it this way or that, until, by repeated trials, when replaced with one end stayed at the division line first observed, the other shall cut the midway pencil spot. 7. The needle being thus straightened, proceed to rectify the position of the centre pin, if necessary, by bending it with nip- pers so that the needle shall cut opposite degrees at the quarter points of the circle. 42. MiscºLLANEous. 3. ty E XV. MISCEDLANEOUS. THIE VERNIER. 1. The vernier in the transit is a short graduated arc, movable around the graduated circle of the instrument, by means of which Subdivisions of the circle graduation can be read. There are many varieties of the ver- nier; but a knowledge of the principle upon which one is made introduces the student to an easy acquaintance with all. 2. Suppose the tenth part of a foot to be marked off on a straight odge into ten cqual parts, and that on another straight edge a Space equal in length to nine of these parts is divided also into ten equal parts. The sub- divisions of the latter scale will then each be nine-tenths as large as the subdivisions of the former; and if the graduated edges are placed together, with the zero marks in both exact- ly lined, the first mark of the latter, of ver- Inier, scale will fall short of the first mark of the former, or limb, so to speak, by one-tenth part of the first space on the limb; that is to Say, by one-tenth part of one-hundredth of a foot, or one-thousandth of a foot. The sec- ond mark of the vernier will fall short of the second mark of the limb by two-thousandths of a foot, and so on. If, therefore, the ver- nier scale be moved slowly forward, the suc- cessive oppositions of the scale marks will indicate successive advances of the vernier, each equal to the one-thousandth part of a foot. The marginal example reads 6.217 =six feet, two-tenths, one-hundredth, and seven- thousandths. 3. The annexed figure represents the transit MISCEL LAWEC) U.S. 43 vernier, together with a part of the graduated circle. The vernier is a double one, for con- - venience in reading angles right or left. It will be observed that a Space, equal to twenty-nine half degrees on the limb, is laid off from zero each way on the ver- nier, and there subdivided, on both sides of zero, into thirty equal parts. If now the zeros are brought into line, the first marks of the vernier right and left will fall one-thirtieth part of a half degree short of the first, or half-degree marks on the limb; that is to say, one minute short. The vernier, therefore, is scaled to read minutes; and, if its zero mark be moved slowly half a degree on the limb, its several subdivision marks, one after an- other in arithmetical succession, will be seen to line with marks of the limb until the thirtieth is reached, when zero will be found to have traversed the half degree Space. 4. TO READ AN ANGLE. First note whether the vernier has been moved right or left ; then observe on the limb the number of full degrees, and the half-degree, if any, which zero of the vernier has passed; next, look along the vernier from its zero towards the right, if the movement has been towards the right, and from zero towards the left, if the movement has been - towards the left, until a “minute’’ mark is found exactly in line Willl. Some mark on the limb. Add the number of that 44. MISOEL LAWE'O US. minute mark on the vernier to the angle already ascertained within half a degree from the limb: the sum will be the angle sought. The vernier in the figure reads 1° 20' L. 5. In some respects a vernier graduated decimally would be more convenient on railroad locations, where the 100-feet chain is used; the calculation of engineers’ tables to sixtieths of a degree has prevented its adoption. 6. TO RE-MAGNETIZE A NEEDLE. Lay the north half flat on a smooth, hard surface, and With gentle pressure draw the south pole of a common magnet over it, from the centre outwards, withdrawing the magnet from it six or eight inches after each pass. Repeat ten or a dozen times. Treat the south half of the needle in the same manner with the north pole of the magnet. Replace the bal- ancing wire. If the needle yet proves to be sluggish, take out the centre pin, and newly point and polish it. 7. If the needle, by reason of electricity, clings to the cover- ing glass in the field, a touch of the moist finger to the top of the cover will release it. 8. Do not suffer idlers to play it about with knives, keys, and the like. 9. When the instrument is out of use, leave the needle free. 10. TO REPLACE CROSS-HAIRS. Take out the eye-glass tube. Remove the small lateral capstan head screws which hold the cross-hair ring athwart the barrel. Loosen the vertical Screws, and, taking care throughout to observe the position of the ring, in order that it may be got back again right side up and right face forward, turn it lengthwise of the barrel. Insert the end of a pine sliver into one of the side holes, take out the vertical screws, and withdraw the ring. Stretch across new hairs, in the scores traced for them, of the finest clean spider-line; secure them. with a touch of gum or wax, and put the ring in by a reverse process. 11. TO FIX A TIRUE MERIDIAN. By equal shadows of the sun. On level ground or ice, set up a pole. Two or three hours ſ' MISCELLA WEO U.S. 45 before noon, mark the extremity of its shadow. With radius reaching to that mark, from a centre on the surface vertically below the top of pole, strike an arc eastward. Two or three lmours after noon, watch for the moment when the extremity of the shadow touches the arc. There make another mark. The true meridian will pass from the centre midway between the two marks, if the observations be made about the period of the solstice, in June or December. The method gives a fair approximation at any time of year. 12. By observation of the North Star in meridian. Find the time of passage in Table I. Choosing still weather, hang a plumb-bob from some high place into a bucket of water, and establish a point of observation so far southward that the suspending line shall cover the breadth of sky be- tween the Dipper and the North Star. The point of observa- tion may be an upright bodkin, or compass-sight, fixed on a block movable horizontally east and west. Watch for the moment when, from the point of observation, the plumb-line covers the North Star and the first star in the handle of the Dipper; that is to say, the star nearest to the four which make the Dipper bowl. Exactly twenty-six minutes after- wards, bring the plumb-line in range with the North Star, by shifting the observation point laterally. That range will be the true meridian. Stakes may be set in it forthwith by . Imeans of candles. With a transit in good adjustment, the plumb-line is not necessary. Illuminate the cross hairs by reflecting light on the object glass from white paper. 13. IBy observation of the North Star at its eactreme elonga- diom. Find the time in Table II., and make the preparations above directed. Keep the plumb-line in range with the star until the star apparently ceases to move. Mark that range. Multi- ply the natural tangent of the azimuth, given in Table III., by the distance in feet from the point of observation to the mark in the northern range just set. The product will be the dis- tance from said northern range mark, square, right or left, to a point in the true meridian passing through the point of ob- servation. If the western elongation was observed, set off the calculated distance eastward from the northern range mark; if the eastern elongation was observed, set the distance off westward. If both the eastern and western elongations be 46 - MISCELLANEOUS. * observed, the true meridian will pass through the point of observation, bisecting the angle between the northern range marks. - With a verniers instrument, the azimuth can be laid off directly, in degrees and minutes. PROPOSITIONS AND PROBLEMS RELATING TO THE CIRCLE. IXVI. --YQIX. PROPOSITIONS AND PROBLEMs FELATING TO THE CIRCLE. XVI. - PROPOSITIONS RELATING TO THE CIRCLE. The following propositions, demonstrable by simple processes of geometrical reasoning, may be regarded as axiomatic.. B | - D H 1. In any circle a tangent is perpendicular to radius at the tangent point. Thus, B I is perpendicular to BC. 49 50 PropositroMS RELATING TO THE circle. 2. Tangents drawn to a circle from the same point are equal. Thus, I Ib = L E. - - 3. The angle I) I E, at the intersection of tangents, is equal to the central angle BC E, included between radii to the tan- gent points. -* 4. If a chord B E connect the tangent points, the angles I B.E, IE B, are equal: each of them is equal to half of the central angle B CE, or of the intersection angle DI E. 5. Any angle, B C E, at the centre, subtended by the chord BE, is double the angle B FE, at the circumference, on the same side of the chord B.E. º 6. Acute angles at the circumference, subtended by equal chords, are equal. - 7. An acute angle, KF H, between a tangent and a chord, is called a tangential angle, and is equal to the peripheral angle L FH subtended by an equal chord; each is equal to half the central angles FC H, or HC L, subdivided by the same chords. 8. The exterior angle L H N at the circumference, between two equal chords, is called a deflection angle : it is equal to the central angle, or to twice the tangential angle, subtended by either chord. Yºr - 9. If F K be made equal to FH, and HN be made equal to H L, H K is called the tangential distance, and L N the deflec- tion distance. - - 10. The exterior angle E H N at the circumference, between two unequal chords, is equal to the sum of their tangential angles, or to half the sum of their central angles. XVII. CIRCUIAR CURVES ON RAII, ROADS. 1. The circle is divided, for convenience, into 360 equal parts, called degrees. A circle 36,000 feet in circumference would be cut by such subdivision into 360 parts, each 100 feet long, and subtending an angle of one degree at the centre; its radius would be 5,729.6 feet, usually reckoned 5,730 feet. The CIRCULAR currºs ow. RAILROADS. 51 chain 100 feet long being the unit generally adopted by Ameri- cân engineers for field measurements, any circular arc having that radius, of 5,730 feet, is called a one-degree curve, for the I'eason that one clain is equivalent to an arc of one degree at the circumference. . . 2. The circumferences of circles vary directly as their radii: hence, in any circular arc struck with half that radius, or 2,865 feet, one hundred feet at the circumference would sub- tend an angle of two degrees at the centre. Such an arc is called a two-degree curve. If one-third of the primary radius of 5,730 feet, or 1,010 feet, be used, the arc is called a three- degree curve; and so on. 3. It should be borne in mind, however, that these measure- ments are supposed to be made around the are itself, and not on lines of chords. Since field measurements with the chain are always made on the lines of the chords, which are shorter between given points at the circumference than the lines of the arcs, as a bowstring is shorter than the bow, it is plain that, in advancing towards the centre of the one-degree curve by a series of concentric circles having radii equal to one-half, one-third, &c., of the primary radius, the chord 100 feet long differs more and more in length from the arc subtended by it, the bow being more and more arched in relation to the string. Thus, in the circle having a radius equal to one-twentieth of the primary radius, the chord 100 feet long subtends an angle of 20° 06', at the centre, instead of 20°, and the arc is 100.5 feet in length, instead of 100 feet. In order, therefore, that the chord of 100 feet may subtend arcs of 19, 29, 39, &c., in regular succession, the radii of these successive arcs must be somewhat greater than the above method by subdivision of the primary radius would make them; though, as might be inferred from the extreme case given by way of illustration, the dif- ference is not appreciable in ordinary field practice, and radii, together with all the functions dependent on them, may usually be held to vary as the degree of curvature, or central angle per 100 feet chord, varies. ~ 52 TO FIAWD TL/E /? A D / US OF A CUR WE. XVIII. TO FIND THE RADIUS, THE APEX. DISTANCE, THE LENGTH, THE DEGREE, ETC., OF A CURVE. 1. Let E IX, A, O be two straight lines intersecting at E. Lay off equal distances, EA, E B; erect perpendiculars at A and - * B, meeting at G, and con- nect A B, E G. From the centre G, with radius GA, draw the curve A H.B. - __ § * The point E will be the /* \ |W N P.I.; A and B, tangent \ / - points; EA, E B, the tan- gents, or apCx distances, - which denote by T.; E H, the external secant, or S; H N, the versed sine, or W. Let the long chord . A B, connecting the tan- gent points, be called C, and G. A or G B, the radius, R. Call the deflection angle to a chord of 100 feet D, as before. 2. By XVI. 3 and 4, angle E A B = E B A = A G E = E G B = # I. * - G 3. GIVEN THE INTERSECTION ANGLE I AND RADIUS R, To FIND THE TANGENT T. T = R X tam. 3 I. Iºacample. R = 1,910.1, I = 35° 24′. Then T = R tan. ; I = 1,910.1 × 0.3191 = 609.5. 4. Measure from the P.I. equal distances, E.M., E F, along the tangents. Measure, also, MF and E K, the distance from E to the middle point of MIT. Then, by reason of similarity in the triangles MEK, EA G, M K : E K :: A G : A E :: It : T ..'. T = R × E. K-- MI K. TO FIND THE RADIUS OF A CUR FE. , 53 * IEacample. Let M K = 190.5, E K = 60.8, R = 1910.1. Then R = 1910.1 . . tº g tº 3.28.1056 E K = 60.8 • , - ſº tº 1,783904 M K = 190.5 (a.c.) . e º 7.72010.5 T = 609.6 g e g & 2.785065 5. If 100-feet chords be used, find the tangent in Table XVI. corresponding to the given angle I. Divide that tabular tan- gent by the degree of curvature corresponding to the given radius: the quotient will be the required tangent. Thus, . Tab. Tan. corresponding to 35° 24′ = 1,828.7, which, divided by 3, the degree of curvature, gives 609.6, the tangent sought. * X 6. GIVEN THE INTERSECTION ANGLE I AND TANGENT T, To FIND RADIUS R. Transposing the equation in (3), R = T -- tan. # I = T X Col. 3 I. -- JEacample. T= 609.6, I=35°24' R = T cot. # I = 609.6 × 3.1334 = 1910.1. By a like transposition of the equation in (4), R = T X M K -- E K. v-vo 7. If 100-feet chords be used, find in Table xvi. the tangent corresponding to the given angle I. Divide that tabular tan- gent by the given tangent; the quotient will be the degree of curvature in degrees and decimals. The radius corresponding to this degree of curvature may be found by (12), by Table X., º \\ or, with sufficient accuracy for ordinary practice, by dividing 5,730, the radius of a 1° curve, by it. - - Thus, in the foregoing example, the tabular tangent cor- responding to 35° 24′ is 1,828.7. Dividing by 609.6, we have 3 for the degree of curvature; and 5,730 divided by 3 gives R = 1,910 feet. - - 54 TO FIND THE RADIUS OF A CUE. V.E. 8. Given THE INTERSECTION ANGLE I AND CHORD AB = 2 CoNNECTING THE TANGENT POINTs, TO FIND RADIUS R. - A N = . A B = | c. A G N = I. A G = A N -- sin. A G N ; i.e., R = # C –- sin. # I. IEa:(linple. I = 350 24', C = 1161.4. Then R = 4 C -- sin. I, + 580.7 -- 0.304 = 1910.2. 9. If 100-feet chords be used, find in Table XVI. the chord corresponding to the given angle I. Divide that chord by the given chord, for the degree of curvature in degrees and deci- mals. Determine the corresponding radius by (17), by Table X., or, for ordinary practice, by dividing 5,730 by it. Thus, in the foregoing example, the tabular chord corre- sponding to angle 35° 24′ would be 3,484.2, which, divided by the given chord, 1,161.4, gives 3 for the degree of curvature, and 5,730 divided by 3 makes the radius R = 1,910 feet. K 10. GIVEN THE INTERSECTION ANGLE T AND THE DEGREE of CURVATURE or DEFLECTION ANGLE D, witH 100-FEET CHORDs, To DETERMINE THE LENGTH OF THE LONG CHORD C, THE VERSED SINE W, THE EXTERNAL SECANT S, OR THE TANGENT T. Take from the proper column in Table XVI., the number corresponding to the intersection angle, and divide it by the degree of curvature: the quotient will be the length required. - Eacample. A 49 curve, I = 50° 16'; to find the several functions above named. - - - Table XVI. gives the designated functions of a 19 curve as follows: C 4,867.3, V 542.4, S 599.3, T 2,688.2. Dividing by 4 the degree of curvature, we have for the corresponding func- tions of a 49 curve as follows: C 1,216.8, W 135.6, S 149.8, T 672.0. - I&AD II, DEFLECTION ANGLES, ETC. 55 11. GIVEN C, V, S, OR T, OF ANY CURVE, AND D, THE DE- GREE OF CURVATURE, TO FIND THE INTERSECTION ANGLE, I. Multiply the given function C, V, S, or T, by the degree of curvature, D: the product will be found in the proper col- umn of Table XVI:, corresponding to the required angle. Bacample 1. Given T = 515, D = 5°; to find I. Then T X D = 2,575, which corresponds in Table XVI. to 480 24/ = I. * IEacample 2. Given C = 1,656, D = 39; to find I. Then Cx D = 4,968, which corresponds in Table XVI. to 510 23/ = I. 12. GIVEN C, V, S, OR T, OF ANY CURVE, AND THE INTER- SECTION ANGLE I, TO FIND THE DEGREE OF CURVATURE D. Take from the proper column of Table XVI. the number corresponding to the given angle I, and divide that tabular number by the length of the given part; the quotient will be D, in degrees and decimals. - Example 1. Given T = 587, I = 220 26/; to find D. The Tan. corresponding to I in Table XVI. is 1,136.3. Then 1,136.3 + 587 = 1.935 = 1° 56' = D. Eacample 2. Given S = 64, I = 30° 25', to find D. - - The Ea. Sec. corresponding to I in Table XVI. is 208. Then 208 —– 64 = 3.25 = 3° 15' = D. - 13. GIVEN THE INTERSECTION ANGLE I, AND DEFLECTION ANGLE D, TO FIND THE LENGTH OF THE CUIRVE. Divide I by D: the quotient will be the number of chord lengths in the curve. * If the degree of curvature is a whole number, the more con- vcnient method of effecting the division is, first, to reduce the 56 RADII, DEFLECTION ANGLES, ETC. minutes, if any, in I to decimals of a degree; then divide by the degree of curvature. Example 1. = 20° 40', I) = 39. 20° 40' is equivalent to 20.67 degrees. Dividing by 3, we have 6.89 chord lengths for the length of the curve. If the chords, as is usual, are each 100 feet long, the length of the curve in this case will be 689 feet. If the chord lengths were 50 feet each, the length of the curve would be half this number of feet. - 14. If the degree of curvature is fractional, the more con- venient method of effecting the division is, first, to reduce both I and D to minutes; then divide the former by the latter. - Example 2. I = 30° 22', D = 29 45'. ' These are equivalent, respectively, to 1,822 and 165 minutes. Dividing the former by the latter, we have 1,104 feet for the length of the curve. - 15. The ingenious assistant willo will attentively consider the preceding figures cannot fail to detect other obvious analo- gies which it has not been thought necessary to include in this compendium. - - 16. In railroad field practice it is usually sufficient to deter- mine angles to the nearest minute, and distances to the nearest foot. The nicety of seconds and tenths appears generally to be quite Superfluous; the time consumed on them were better employed in pushing ahead. H 17. GIVEN ANY DEFLECTION AN- :- - - - - - – B GLE D, AND CHORD C, TO FIND RADIUS R. F B -- sin. J. A. L. B = B L ; i.e., # C-- sin. ; D = R. - º IEa:ample. g Let C = 100 feet, D = 4°. | Then R = + c + sin. , D = 50+ { * .0349 = 1432.7. If the chords are 100 feet long, as is usual in railroad prac- tice, radius may be found with sufficient accuracy by dividing RAD II, DEFLECTION ANGLES, ETC. 57 5,730, the radius of a 1° curve, by the deflection angle, or de- gree of curvature. . Thus, in the foregoing example, 5,730 –– 4 = 1,432.5. 18. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE DE- FLECTION ANGLE ID. IFrom the preceding equation and example: — Sin D = } C+R = 50+ 1,432.7 =.0349 = sin 2° = D ..". D = 49. ,” 19. GIVEN ANY RADIUS R, AND CHORD C, To FIND THE DE- FLECTION DISTANCE d. First find the deflection angle by above method (18). Then, angle H A B in the figure being made equal to D, and H.A. = B.A, B II will be the deflection distance. Draw. A K to the middle point of III3. - - Then d = H B = 2 KIB = 2 A. B. × sin, K.A. B = 2 C × sin # D. - -- Fæample. Let R = 1,146 feet, C = 100 feet. By (18) D will be found = 5°. Then d = 2 C X sin 3 D = 200 × .0436 = 8.72 feet. 20. If the chords are 100 feet long, as is usual in field meas- urement, divide the constant number 10,000 by the radius in feet: the quotient will be the deflection distance. The deflec- tion distance with radius of 10,000 feet and chord of 100 feet is one foot: this rule is based upon the principle that deflection distances, the chord length being fixed, will vary inversely as the radii. 4. Thus, in the foregoing example, 10,000+ 1,146 = 8.72. 21. GIVEN ANY RADIUS R, AND CHORD C, TO FIND TIIE TAN- GENTIAL ANGLE. T. The angle T is equal to # D by construction; for mode of determining it, see preceding Section (18). 58 *. OR DINATES. 22. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE TAN- GENTIAL DISTANCE i. First find the tangential angle, as above directed. Then, angle BA E in the figure being made equal to T, and AIE = A B, B E will be the tangential distance. Draw AN to the middle point of B E. & Then t = EIB = 2 N B = 2 A. B. × sim, N A D = 2 C X sin, # T. e - - Ea:(limple. Let R = 1,146 feet, C = 100 feet. I3y sect. 1, T will be found = 2° 30'. Then t=2 c x sin T =200 × .0218 = 4.36 feet. 23. In ordinary railroad practice the tangential distance may be considered equal to half the deflection distance. XIX. oRDINATES. 1. GIVEN ANY RADIUS R, AND CHIORD C, To FIND THE MID- DLE ORDINATE M. _º N - N K E % |/ G L In the annexed figure, HN = M, H G = R, AB = C. N G = A/A Gº-AN3=A/Rº-3 C3; H N = H G —N G, i.e., M = R — A'Rº-3 C3, - - ORD INA TES. < 5 9 Ea:ample. - R = 819, C = 100; to find the middle ordinate, M. M = 819 – A/670761–2500 = 1.53. 2. Angle H AN = } IIG B; H G B = A A G B, ... H AN = # A G B. II N = A N × lan. HAN; i.e., M = } C X tan... }. D.; D being the central angle subtended by the chord. Ea:qmple. D = 7°, C = 100; to find M, the middle ordinate. M = } C x tan. ; D = 50 × 0.03055 = 1.528. 3. GIVEN THE RADIUS R, CHORD C, AND MIDDLE ORDINATE M, TO FIND ANY OTIIER ORDINATE E K = M/, DISTANT d FIROM N, TLIE MIDI) LE POINT OF THE CHORD. K L = N G ; N K = G. L.; E K = E L — N G. E L=A/GE2–N K*=A/R3-d?; N G (1) = A/Rº-3 C3. & ------------ Then E K– M = A/R3–d?– A/R2–; C3. 4. It is a property of the parabola, that ordinates vary as the products of their abscissas. This property may be assigned to the circle in cases where the arc encloses a small angle. Applying it here we have — H N : E K :: A N × N B : A K × K.B. Call any segments A K, KB, of the chord, a and b. Then M. : M! :: + C* : ab, ... M = M × 4 at + C2. Eacample. M = 1.528, C = 100, a = 60, b = 40; to find M!. M = 1.328 x 900+10000–1528 × 0.96–1467. 5. Multiply the corresponding ordinate of a 1° curve from the annexed table by the degree of curvature: the product will be the ordinate sought. 60 - . . ORDINA TES. ORDINATES OF A 19 CURVE, CHORD 100 FEET. DISTANCES OF THE ORDINATES FROM THE END OF THE 100-FEET Choiti). Middle | . s - Feet. Feet. Feet. | Foet. Feet. Feet. Feet. | Foet. Feet. | Feet. 50 || 45 40 35 30 25 20 15 10 5 LENGTIIS OF THE ORIDINATES IN FEET. 198 | 183 |.164 || 140 | .111 | .078 || 041 .218 .216 .209 IEa:(limple. What is the ordinate of a 69 curve, 30 feet from the end of the 100-feet chord? - - The corresponding tabular ordinate of a 1° curve is .183; which, multiplied by 6, gives 1.098, the required ordinale. 6. A quick way of laying off ordinates on the ground, and one sufficiently exact for the field, is, after fixing the point H. by means of the middle ordinate H N, to stretch a line from H to A, and make the middle ordinate F O = # II N.; then from F to A and F to H, making the middle ordinates = #FO; and SO OI). - ‘/7. A good track-layer will seldom require points at shorter intervals than 25 feet. TRACING OURVES - AND TURNING OBSTACLES IN THE FIELD, $ xx.-XXIII. TRACING CURVES AND TURNING OPSTA-CIES IN TEIE FIETD. XX. TO TRACE A CURVE ON THE GROUND WITH THE CHAIN. ONLY. 1. This is best taught by an example. M IEacample. Trom a point B, 18 feet in advance of A, on tangent A B, to trace a curve of 367 feet radius to the right, with chords 66 feet long, and consuming an angle of 34° 27'. 4. 63 64 TO TRACE A CURITE O N TIVE GROUND. 2. First, dividing half the unit chord, or 33 feet, by the radius, 367 feet (XVIII., 18), we have 0.09-i- for the sine of the tangential angle, corresponding to an angle of 5° 10': the de- flection angle, therefore, is 10° 20'. The tangential distance corresponding to the angle 5° 10', and chord 66 feet, is equal (XVIII., 22) to twice the chord multiplied by the sine of half the tangential angle, = 132 × 0.04507 = 5.95 feet. The deflec- tion distance (XVIII., 19) is equal to twice the chord multi- plied by the sine of half the deflection angle, = 132 × 0.09–H = 11.88, say 11.9 feet. • * - 3. To find the length of the curve (XVIII., 13): Divide the total central angle by the degree of curvature. The central angle, 34° 27', is equivalent to 2067 minutes; dividing by 620, the number of minutes in the deflection angle, we have 3.33, the number of chord lengths in the curve, - 3% chains = 220 feet. p - If A be a stake numbered 2, then the point of curvature, B, will be 2.18, and the point of laugºut, I, will fall at 2.18 + 3.22 = stake 5.40. 4. To determine the tangential distance C P, to the first stake on the curve, either of two methods may be used:— First, The sine of any tangential angle is equal to half the chord which limits the angle on one side divided by radius. The limiting chord B C in this instance is equal to 66 — 18 = 48 feet; half of 48, therefore, or 24 feet, divided by radius, 367 feet, gives 0.0654, the sine of 3° 45' = tangential angle P B C. The sine of half this angle multiplied by twice the given chord = 0.0327 × 96 = 3.14 feet, the tangential distance CP. 5. Secondly, C P may be found as follows, assuming that - the functions of small angles vary directly as the angles themselves, and vice versa. Let B F be a portion of the curve. Make the tangent B E equal to the 2^B N chord BF, and strike the A * chord B C, and strike the arc C. P. Prolong B C to D. EF may be taken as the tangen- tial distance due to the whole chord B F, and PC the tangen- arc E. F. Draw the Sub- tial distance due to the sub-chord B.C. To TRACE A GUI, VE ON THE GROUND. 65 Then P C : E D :: B C : B D or B.F.; and, by the foregoing supposition, E D : E F : : B C : B F. Combining these propor- tions, and cancelling E D, we have P C : E F : : B Cº : B F2 . . PC = EFX (B C –– B F)2. In words, the tangential distance for a sub-chord is to that for a whole chord as the square of the sub-chord is to the Square of the whole chord. The same is true of deflection dis- tall CeS. - 6. In the example we are treating, the tangential distance for the whole chord of 66 feet has been found to be 5.95 feet; that for 48 feet, therefore, is 5.95 × 48% -- 66% = 5.95 × 0.528 = 3.14, as before. - Stretch the 48 feet of chain from B to P, in prolongation of tangent A B, and mark the point P; then step aside, and stretch from B to C, making the distance P C = 3.14 feet: C will be a Stake On the curve. 7. Next, run out the whole chain length from C to O in the range B.C. To find O D, suppose the line N C T to be drawn tangent to the curve at C. Then N D may be considered the tangential distance due to the whole chord, = 5.95, as above determined. The angle O C N = T C B = P B C (XVI., 4); and (5) O N.: N D :: B C : C D ... O N = N D X B C –– C D ; i.e., O D =N D + O N = N D + [N D X (B C+C D)] =5.95 × [1+ (48 ––66)] = 5.95 × 1.727 = 10.27. - - 8. The point N may be fixed otherwise by laying off BT = CP, and running out the chain length CN in the range C.T. The point D on the curve may then be fixed by making ND equal to 5.95 feet, the tangential distance. Next run out the chain to M, in the range CD; make ME equal to the deflection distance, 11.9 feet, and fix the point E. The points C, D, and E will be stakes 3, 4, and 5 on the curve. 9. To set the point of tangent, F, at stake 5.40, prolong the chord line D E for 40 feet to L, and suppose V E to be drawn tangent to the curve at E. Then the angle L E V is equal to the tangential angle of the curve; and the sub-tangential dis- tance L V is to the whole tangential distance due to the 66– feet chord, as the sub-chord is to the whole chord (5); i.e., L V = 5.95 × 40 —– 66 = 3.6 feet. - By the method illustrated in (6), the distance FW will be 66 TO TRACE A CUR FE ON THE GROUND. equal to 5.95 × 40” -- 662 = 5.95 × 0.367 = 2.18 feet. With the distance L F = 3.6 –– 2.18 = 5.78 feet, thus obtained, and the sub-chord E F = 40 feet, the point of tangent F may be established. 10. Next, set off U E = F W = 2.18 feet, and lay out FK in prolongation of the range U F.; F K will be in the line of the terminal tangent. . g 11. This analysis has been somewhat minute and detailed, in order that the subject may be thoroughly understood. An instrument for measuring angles should always be used in rail- road service: it greatly simplifies and abridges the labor of tracing field-curves, and gives more exact results. But occa- sions sometimes rise, in miscellaneous practice, when strict accuracy is not required, and the clain only can be had: the young engineer should qualify against Such occasions. XXI. TO TRACE A CURWE ON THE GROUND WITH TRANSIT AND 100–FEET CHAIN. 1. This, also, is best taught by an example. Let it be a general rule, in locating, to fix the intersection of tangents, and to set the tangent points, or the P. C. at least, from the P. I. There are exceptional conditions, as a steep hillside, timber or broken ground, a very long arc, unimpor- tance of exact conformity to the project, and the like, which warrant its omission; but where these conditions do not obtain or are not prohibitory, and a Snug fit is desirable, time will usually be saved by fixing the P.I. It often proves serviceable as a reference point during construction: On the location, it gives confidence in the work and an assurance of safe progress, which are well worth a little painstaking beforehand. - 2. Having established the P. I., and found the intersection angle to measure, say, 66° 45', the first step is to find the apex distances so called, or tangent lengths IP, IF. These are each equal to R X tan. # I. If a 7° 30' curve be prescribed to close the angle, R X, tam. # I = 764 × 0.659 = 503 feet. TO TRACE A GUR VE ON THE GROUND. 67 Or, referring to Ta- ble XVI., the tangent corresponding to 669 45' is found by inter- polation to be 3774.6; dividing by 7.5, the rate of curvature in degrees and decimals, we have for the apex distance 503 feet, as above. 3. Before disturbing the instrument, which is presumed to stand in the range of the term in a l tangent, measure I F, = 503 feet, and set the P. T. at F. Then direct the telescope to the last point fixed on the ini- tial range A B, meas- ure I B, - 503 feet, and set the P. C. at B. Move to B. 4. Suppose the P. C. to have fallen at a stake 2.50. In order to find the length of the curve, divide the intersection angle by the degree of curva- ture, having first re- duced the minutes in each to hundreths of a degree by multiplying by 10 and dividing the product by 6. Thus the intersection angle becomes 66.759, and the degree of curva- ture 7.5°: dividing the 6S TO TRA C E A CUR VIE ON THYE GROUND. former by the latter, we have 890 feet for the length of the CUll"We. Or, the intersection angle 66° 45' is equivalent to 4005', and the degree of curvature 7° 30' is equivalent to 450': dividing the former by the latter, we have 890 feet for the length of the curve, as before. 5. Adding 8.90 to 2.50, the number of the P. C., the P. T. is found to fall at stake 11.40. Let the rear chainman make a note of this, that there may be no mistake in the terminal plus. 6. Next, to determine the proper deflections from the line of tangent at B, bear in mind that the deflection for a whole chain is half the degree of curvature ; and that, in field-curves of more than 300 feet radius, the deflections for sub-chords, or plusses, may, without material error, be held to vary directly as the sub-chords themselves; that is to say, the sub-deflec- tions due to 30, 60, and 80 feet, for instance, will be, to the deflection due to 100 feet, as 30, 60, and 80 are to 100. .7. Thus, in the example, 7° 30′ being the degree of curva- ture, one-half of this, or 3° 45', will be the deflection due to a chord of 100 feet; and ºr of this, or a deflection of 1° 524 from the line of tangent at B, will fix stake 3, 50 feet distant On the Cllrve. 8. The following is a simple rule for finding sub-deflec- tions: — - - Multiply the sub-chord in feet by the rate of curvature in degrees and decimals: three-tenths of the product will be the sub-deflection in minutes. Thus, in the example, 50 × 7.5 = 375, and 375 × 0.3 = 112.5/ = 1° 524', as before. 9. Having set stake 3, stakes 4 and 5 will be fixed by succes- sive deflections of 3° 45'. In establishing stake 5, the index will read, 1952} + 3° 45' + 3° 45' = 9° 22} = angle C B 5. 10. Suppose the instrument moved to 5. See that the ver- nier has not been disturbed, backsight to B, and deflect 9° 22}/ right; i.e., double the index angle. The index will now read, 180 45/ = the angle I CD; and the telescope will be directed along the line CD, tangent to the curve at 5, for the reason that the angle B 5C has been made equal to the angle C B 5 (XVI. 4). Proceed with successive deflections of 3° 45' from this tan- gent, and set stakes 6, 7, 8, and 9, at intervals of 100 feet. 11. Suppose the instrument moved to 9. In fixing this TO TRACE A CUAE VE O W THE GROUND. 69 stake, the index will read, 18° 45' + 4 times the constant angle 3° 45', - 189 45' + 159 = angle I CD + angle D 59, -33° 45'. In order to place the telescope in the line DE, tangent to the curve at 9, it is now necessary to turn an angle to the right, from backsight to 5, equal to D 9 5 = D 59 = 15°; i.e., the vernier should be moved from 33° 45' to 33° 45' + 15° = 48° 45'. The telescope will then be in tangent at 9. 12. A simple rule for finding the index angle which shall place the telescope in tangent at alry point on the curve is as follows: — Prom double the indea: angle which fiaced the given point, Sub- tract the indé2, reading in tangent at the last turning-point : the Yemainder will be the required indea: angle. Thus the index angle which established stake 9 was 33° 45/. Double this angle will be 67° 30'; subtracting 18° 45', the reading in tangent at the last turning-point, we have 48° 45', the required index angle, as before. * The reasons for the rule will be obvious from an examina- tion of the figure. 13. Being in tangent, then, at 9, and the index reading 48° 45', a deflection of 3° 45' will fix 10: a further deflection of 39 45' will fix 11, and the index will stand at 48° 45' -– 7° 30' = 560 15/. - 14. To find the deflection corresponding to the sub-chord 11 F, - 40 feet: by the foregoing rule (8), the degree of curva- ture, 7.5, multiplied by 40, the length of the sub-chord in feet, gives a product of 300, three-tenths of which amount to 90 minutes = 1° 30'. Adding 1° 30' to 56° 15', makes the index angle 57° 45' to fix the P. T. at 11.40. 15. Move to the P. T. at 11.40, see that the vernier has not been disturbed, and backsight to 9. By the foregoing rule (12), double the index angle, 57° 45', less the angle in tangent at 9, the last turning-point, 48° 45', - 115° 30' — 48° 45', - 66° 45', - the index angle in tangent at the P. T., - the total angle consumed by the curve. The work thus proves itself. 16. The preceding example would appear in the field-book as follows: — - 70 TO TRACE A OUR VE ON THE GROUND. • •u •© ®� �& �*ZI "GIZOI „ȘI ‘N "GI ZOO „GI’NI ŽĒ，95·e œ09 | G) 07’IL •- …• •/ 9† • 1,9A 08 a IOŤ*II • ！• ©«…» «…»/ 9I 999/ 97 48º ſa”OI … ->• •757-SŤ || … 08.39 | 4 gŤ „g | ··· | © ’6 • •· · ·• •a gï 488/ 9（† 48• •*$ ----→ ·• • •A 00 008/ 9 ſ 48• •"1. • •● ●«A ，/ 9I 99%A ºgŤ 48�, �*9 • •• •ZgĒĢī/ 08 4，3 || / 9Ť 48·· | G) ºg ºſas "I ‘ā ſąoog g0g ºsſGI xođv·© ®· · ·ÁZŐ 46/ gŤ 48• •*# *… ···€.}“J, *a[ 0ț¢”II|-• •• •„Ř18 ºg/ Ģf .800ſºg: + '/gŤ 499 4ųā ĶI 9A InſO /08 „1.• ‘O ‘ā’ 09”，• •• •• •“ȘI ÁZg „I |'''I /#Zg „I| 09G) Ogºz. ●• •• •• • •● ●& •09’， *sxºv IIŁyſ n°C, ‘W*GIS?IDOO“ĐNVI,*XEICINI*„IGIGI Į. “JOSICI*WĀĞŞ; *~~wº sº wrº"-- ~~ * TUR WING O BASTA OLES TO WISION IN TAWG ENT. 71 17. This mode of running curves secures a record of each step in the proceeding; so that, if any error occurs, it can readily be detected. At each turning-point, the number in the “tangent” column must correspond with the central angle due to the length of curve to that point; and at the P. T. that number must correspond with the total central angle. The work can thus be checked with facility during its progress, and checks itself at the end. 18. The young transitman is recommended, to rule blanks after the pattern given, and exercise himself thoroughly in computing the parts, and recording the field-notes of various curves assumed at will: drawings are not necessary. X. XXII. TURNING OBSTACLES TO VISION IN TANGENT. 1. Draw C F parallel to A. B. Let lines B C, CE, FG, cut these parallels at equal C F inclinations. Call this _* angle I. Then B C = _T \ C E = F G. B E = A-5 4. E. G BD + DE = 2 B D. But BD = BC cos. I, ... B E = 2 B C cos. I. E G = CF. B G = E G +B E = CF – 2 B C cos. I. IEacample. Suppose B to be a stake 24.50 on the tangent A B, and that a deflection left of 10° be made there for 200 feet to a point C. Set transit at C, vernier reading 10° left. BS to B, and deflect 20° right. Wernier will now read 100 right, and telescope will be in line C E. Make C E = 200 feet. Move to E. See that vernier still reads 100 right. BS to C, and turn 10° left. Ver- nier will now read zero, and telescope will be in line E G, or tangent A B prolonged. Distance B E = 2 B C cos. I = 2 (200 cos. 100) = 400 × .985 =394 feet. Then E = 24.50 + 394, - stake 28.44 on tangent A B prolonged. 72 TURNING OBSTACLES TO VISIOW IN TANG ENT. - If a parallel line C F were run, a deflection of 100 right would be made at each of the points C and F. If C F were 250 feet, then B G would be = 250 + 394 = 644 feet, and the point G would fall at stake 30.94 on tangent A B prolonged. 2. If angle I = 60°, the other conditions of above method being observed, triangle B H E will be equilateral, and B E = B H = HE. If the parallel D C or D F be run, B E = B D + DC, and B G = BD + D F. For field work see last example. 3. In turning obstacles by either of these methods, the angles should be measured with extreme niceness. Handle the instrument lightly, to avoid jarring the vernier; and, if possible, observe well-defined distant objects in the several short ranges, that the lines of foresight and backsight may accurately coincide. III locating, the following method is preferable to those given above, and should always be used on long tangents. - 4. Having established points A and B on the centre line, the farther apart the better within limits of distinct vision, set off the equal e - rectangular d i S- . f tances A E, B F, | ranging clear of A º the obst a cle. Place the transit at E or F, fix points G and H on the forward range, and, rectangularly to these points, establish others on the forward range of the centre line at C and D. The offset distances should be measured very carefully with the rod, or with a steel tape if they exceed in length the pocket rule which every engineer should have about him. G | H C D **** gº TURNING OBSTACLES TO MEASUREMENT IN TANGENT, 73 XXIII. TURNING OBSTACLES TO MEASUREMENT IN TANGENT. 1. Fix a point on tangent A B prolonged at E. Lay off at B a perpendicular of any convenient length. Move the instru- ment to D, make the angle B D A = B D E, and mark the point of intersection A. By reason of symmetry in the triangles ADB, BD E, AB = BE, and may be measured on the ground. 2. Or, fix the point E, and lay off the perpendicular BD as before. Move to D, direct the telescope to E, turn a right angle E D C, mark the point of intersection C, and measure C.B. Then, by reason of simi- larity in the triangles C B D, D BE, C B : B D : : B D : B E, .*. B E = B D 2 —— BC. G Example. Suppose B D to be 60 feet, and B C 40 feet. Then B D.” —— B C = 3600 —— 40 = 90 feet = B E. & 3. Or, with the instrument at D, measure the angle B D E. Then B E = B D tan. B D E. Eacample. BD = 120 feet, angle B DE = 54° 40'. BD tam. BDE = 120 × 1.41 = 169.2 feet = B E. - 4. Or, without an instrument, lay off any convenient lines BF a 4 ºf -er C H. Mark the middle point * D. Line out H. G., parallel to - A B. Mark on it the point G. in range with D and E. Then G F = BE, or G H = CE. 5. Should the use of a right angle be inconvenient, turn any - angle E B D = a, measure BD about equal by estimation to B E, if the ground permits, and set a point D. Move to D, and measure angle B D E = y. 74 TURNING OBSTACLES To MEAS UREMENT IN TA:NG ENT, Then the angle BED, or 2, = 180— (a + y), and, by trigonom- etry, sin. 2 : sin. y : : B D : BE, . . B E = B D sin, y -- sin. 2. Eacample. Let a = 44°02', y = 71° 48', BD = 300 feet, Then z = 1809 — (a + y) = 1800 – (440 02 + 71° 48') = 1809 – 115° 50' = 64° 10'. BE = BD sin. y -- sin. z = 300 sin. 71° 48' -- sim. 64° 10' = 300 × .95 ––.90 = 316.6 feet. The calculation by logarithms would be as follows:– Log. 800 . . . . . . . . . . . . . 2,477121 Log. sin. 71° 48' . . . . . . . . 9.97.77.11 Sum . . . . . . . . . . . 12,454832 Log. sin. 64° 10' . . . . . . . . 9.954274 Tog. 316.6. Diff. . . . . . . . . 2.500.558 If E is invisible from B, extend the line D B towards C, until a line C E clears the obstacle. The point E must then be established by intersection of the sides CE, DE, in triangle CD E. Supposing the extension B C to have been 120 feet, the side CD will be 420 feet, the angle y 71° 48'; and, by a calculation similar to the above, the side D E, opposite angle at in the lesser triangle, identical with DE in the larger one, will be found to be 231.7 feet. The sum of the angles at the base C E of the triangle CD E = 1800 — y = 1800 – 710 48' = 1080 12'. By trigonometry, two sides and the included angle being known in any plane triangle, the sum of the known sides is to their difference as the tangent of the half sum of angles at base is to the tangent of half their difference. In triangle CD E, therefore, CD + DE or 651.7 : CD — D E or 188.3 :: tan. 108.12 —– 2 or tam. 54° 06' : tam. 54° 06/ × 188.3 —- 651.7 = . .399 = tan. 21° 45', - half the difference of the angles at the base. Log. 188.3. . . . . . . . . . . . 2.274850 Tan. 54° 06' . . . . . . . . . . 0.140334 Sum . . . . . . . . . . . 2.415184 Log. 651.7 . . . . . . . . . . . 2.814048 Tan. 21° 45'. Diff. . . . . . . . . 9.601136 TURNING OBSTAGLES TO MEASUREMENT IN TANGENT. 75 The angle at C, being evidently the lesser of the two angles at the base, is equal to the half sum of these angles decreased by their half difference, - 54° 06' — 21° 45′ = 32° 21/. Set the transit, then, at C, foresight to D, deflect 32° 21' left, and fix in that range two points F and G, between which a cord may be stretched, and as nearly as can be judged on opposite sides of E. Move to D, foresight to C, deflect 71° 48' right, and establish a point E at intersection with F.G. Cross to E, BS to D, and deflect the angle z = 64° 10' into the lin of the tangent A B prolonged. . . SUGGESTIONS AS TO FIELD-WORK AND LOCATION – PROJECTS. YKXIV. — XXV. . SUGGESTIONS AS TO FIELD-WORK AND LOCATION – PROJECTS, XXIV. SUGGESTIONS CONCERNING FIELD–WORK. 1. THE CHIEF ENGINEER, after conference with his em- ployers in regard to the character of the work contemplated, and its general route, should, before organizing field-corps, go over the ground in both directions, and, aided by the best attainable maps, qualify himself by actual observation to in- struct his assistants as to the conduct of the survey. Equipped with hand-level, pocket-compass, and in rough regions with the aneroid, he can often not only prescribe lines for examina- tion, but indicate the gradients to be tried, thus saving a vast amount of random labor and needless expense. Such thorough preliminary exploration is due both to himself and his princi- pals: it is too often omitted, or done with a perfunctory rush. In broken topography, no maps, notes, or information derived from others can supply the want of personal acquaintance with the ground itself. He must indispensably make that acquaint- ance, in order to project an intelligent location, — a work which should rarely be delegated; being capital service, it comes Within the special function of the chief engineer, and only the necessary distribution of labor attending a great charge should relieve him from its direct performance. - 2. A FIELD-Corps in settled regions generally consists of One senior assistant or chief of corps, one transitman, one leveller, one rodman, two chainmen, one slopeman, and two Ol' In Ore 3, Xe]]]CI). § - The following notes in regard to the allotment of duties and the conduct of work may be acceptable. They are copied from the writer’s memoranda for the guidance of his field- parties, with the addition of some detail, and practical hints here and there, to aid the inexperienced. 7Q 80 SUGGESTIONS CONCERNING FIELD – WORK. 3. THE SENIOR ASSISTANT will receive instructions from the principal assistant in charge, or the chief engineer, and will act exclusively under his direction. He will be held responsible for the good conduct of the corps, and for the rapid, exact, and economical performance of the work. Indecent or blasphemous outcries in the field should be prohibited. The writer's various travel by land and sea has brought him acquainted with many cultivated, estimable, energetic, profane fellows, but not one in whom swearing was a grace; nor has he ever seen an instance where it forwarded work. Those considerate of others’ pride and self-respect will generally find that a good leader makes good followers. The senior assistant is empowered to appoint and dismiss employes below the rank of rodman, and will report any inefficiency or neglect of duty in the ranks above to his chief. - He will pay the authorized expenses of the corps for Sup- plies, repairs, transportation, and Subsistence, taking duplicate vouchers. Accommodations should be sought near the work. When not thus obtainable, transportation to and from the field is to be regarded as a measure of economy for the com- pany, compensating the expense incurred by Saving time and labor. - He will superintend field operations in person, keeping in advance of the transit to direct and expedite the work, and establish the turning-points. On preliminary Surveys, the axe should be little used; and on alternative locations, or such as may be subject to revision, trees over four inches in diameter need rarely be felled. He should be patient with sensitive landholders. He will find exercise for that amiable virtue, also, with the field vis- itors who so often spare time from useful toil to tell him he is on the wrong line, and to show him where the right one is. Note for record the kind and quality of material to be moved, observing quarries, wells, or other indications for the purpose; the timber and rock in the country traversed, with a view to their use in construction, and the widths of passage to be pro- vided for streams, together with the character of their banks and beds. Note the names of residents in the immediate vicinity of the work on survey; and, on location, cause the property-lines to be observed and recorded also when convenient. SUGGESTIONS CONCERNING FIELD-WORK. 81: Always begin grade-lines at the summit, and work down. For such service, carry habitually a slip of profile paper, say six inclies wide and two feet long. Rule the proposed grade- line On it, assume a Summit cut, mark the stations, and start down. When at fault, the elevation can be spotted on the profile, which will show at a glance, without any calculation, how you stand in relation to grade. The work of each day should be compiled and recorded in the evening, that no delay may result from the loss or deface- ment of a field-book. º º TOIRM FOR STUTRVEY IRECORD. STA. | DIs. | DEFLEC. |Courts E| M. C. ELEVA. | SLOPE. IöEMARKS. |-- FORM FOR LOCATION RECORD. &EMARKS. | ; | i ſ i i | | On location, check the transitman's calculation of the length of each curve and the fractional deflections. - The senior assistant must be qualified to locate a line accu- rately on the ground from the project furnished him. Lateral deviations exceeding five feet on ten-degree slopes, three feet on fifteen-degree slopes, and two feet on twenty-degree slopes, will be considered errors requiring correction. Measurements to the experimental line should be made and noted frequently, in order not only to check the field-work, but that the line may by means of them be laid down on the map. s The senior assistant will supply himself with drawing in- struments, colors, brushes, and the like personal furniture of an engineer. He will take care also that the stationery, field- books, instruments, and other articles of outfit supplied by the company, are not misused. His field equipment should always include a hand-level and a pocket-compass: to these may be S2 S UGGESTIONS CONCER WIWG FIELD – WORK. added a straight, round staff, five or six feet long, steel pointed; it will be found exceedingly useful. If without a topographer, he should make sketches of irregu- lar ground, of streams, buildings, roads, and the like, to help. in compiling the map. - - In hilly or wooded districts, the front chainman carries the ſlag on survey, and is at the head of the line. In open, plain country, work is greatly forwarded by detaching an axeman with flag, to accompany the senior in advance, and set turning- points for the transit. The transitman follows as rapidly as possible, and the chainmen come after, lining in their stakes by the eye from point to point. The whole force is thus kept pretty steadily in motion. On wide plains, a set of chain-pins may be used, and Survey- stakes placed five hundred or a thousand feet asunder. Very often stakes at intervals of two hundred feet are sufficient, the levels being taken every hundred. Location stakes are put in every hundred feet. - 4. TIII, TRANSTTAIAN will be expected to keep his instru- ment in adjustment, and to be quick and accurate in its manip- illation. It is not needful to plant it as if for eternity. On the contrary, it should be set gently, the legs thrust but slightly into the ground, and the screws worked without straining. On long tangents it is a good plan to reverse the instrument at each new point, putting the north and South ends forward alternately. Small errors in adjustment are thus balanced in some measure. Select also, in such a case, some distant object in range, when practicable, to run by. ' The telescope, in Wind or sun, will sometimes warp a little out of line. Never onlit to note both the calculated and magnetic bear- ings of the lines on survey, and of the tangents on location. Guard against the error of reading deflections or bearings from the wrong ten mark; as, for instance, 34 instead of 26. At the beginning of a curve, let the rear chainman know the plus of the P. T. Tell the front chainman the degree of curve, and instruct him how, by multiplying 1.75 by the degree, he can find the distance of each full station from the range of the last two. A quick fellow will soon pick this up, and become wonderfully skilful in practice. Thus accomplished, he is a check on wrong deflections. * In running curves, a tangential angle of fifteen degrees from one point should seldom be exceeded: twenty degrees is to be regarded as a maximum. SUGGESTIONS CONCER WING FIELD – WORK. 83 Carry a pocket-compass, and observe with it the magnetic bearings of streams and roads crossed. * Record daily each day’s run; fill out the distance column, transcribe the chain-book, and, on location, record the apex distances also in the column of remarks. On Survey, do not erase from the field-book the notes of abandoned lines. Simply cancel, and mark them “aban- doned,” in such manner that they may still be legible. When required by the senior assistant, the transitiman will aid in the making of maps. 5. TIIE LEVELLER must be familiar with the adjustments of his instrument, keep it in order, and handle it rapidly. On Survey, establish and mark benches at half-mile intervals; On location, four to the mile when practicable. - Note the surface elevations, the depths, and the flood heights, Of all considerable, streams crossed. . Take a rod in the beds of Small streams. Six hundred feet each way should be regarded as the maxi- mum sweep of the level. Carry a hand-level, and thuis Save the time required to peg across narrow hollows, or over heights which can be turned With the instrument. - The leveller should record his work, and make up the profile daily. - - - - - *. 6. THE RODMAN will give his intermediates close by the sta- tions, observing the number of each one as a check on the chainmen, and calling it out to the leveller. He should have an eye to abrupt irregularities in the ground, and give plus elevations when necessary. He will keep note of bench-marks and turning-pegs, describ- ing the latter occasionally with reference to the nearest stake, that the levels may be taken up speedily in case of a revision of the line. When unaccompanied by an axeman, the rodman is equipped with belt and hatchet. Sometimes he is furnished also with a steel pin for turning on. The pin has a ring through the head, by which it may be hung to a spring hook in the belt. The rodman will assist the leveller at record and profiles, and transcribe the slope-book daily. - If stakes of survey are set at intervals of two hundred feet, give rods every hundred feet, as nearly as the midway points can be guessed. - •r * 84 THE COTR WE-PROTP2A CTOR. 7. Tiri. SLOPEMAN will give backsights, and take the cross slopes for one hundred feet on each side of the line at every station. S. THE REAR CHAINMAN will carry a book in which to note the turning-points, the crossings of roads, streams, swamps, woodland, and, when convenient, property lines also. He will hand it in daily to the transitman for record. As each succes- sive chain is stretched, the rear chainman calls out the number of the stake it is stretched from ; this assures the selection of the right number for the stake ahead. 9. ONE AXEMAN will be detailed to make stakes, another to mark and drive them. Additional axemen may be employed at the discretion of the senior assistant, as the work requires them. Wanton destruction of timber, fences, growing crops, or other property, should not be allowed. Axemen must be careful, in passing through the country, to do as little damage as possible, - XXV. THE CURVE-PROTRACTOR, AND THE PROJECTING OF LOCATIONS. - 1. The curve-protractor is simply an eight-inch, semi-circu- lar horn protractor, upon which a series of twenty-three curves, from half a degree up to eight degrees, is finely engraved, with radii of 400 feet to an inch. After some years' use in his own practice, the contrivance was transmitted by the writer to the well-known firm of James W. Queen & Co., mathematical- instrument makers, New York and Philadelphia, by whom it is now manufactured. It greatly facilitates the projecting of lines and solution of field-problems on location. It enables the engineer, for example, by a short, graphical process and rapid inspection, to find the curve which shall close an angle between tangents, or terminate a compound curve, and pass at the same time through some fixed intermediate point, without liability to the errors, and free from the loss of time, involved in a tedious calculation. Other applications, such as the nice adjustment of line among buildings, on precipitous steeps, and the like, will suggest themselves to the experienced reader. TIME OUR PIE –PRO TRACTOR. S5 2. For office use, the Writer prefers a home-made curve- protractor of isinglass, prepared as follows: Take a thin, clear sheet, say six by ten inches, free from bubbles and cracks. Block it securely on the drawing-table with thumb-tacks, set- ting the shanks close against the edge of the sheet, but not piercing it, and the heads lapping its edge. . From a centre, midway of one of the long sides and near its margin, strike the curves from 12° or less, varying outwards by half-degrees to, 6°; thence by quarter-degrees to 4°; and thence by ten-minute differences to 24°. This covers one side of the sheet, the scale being 400 feet to an inch. Now release the sheet, turn it over, and on its other face strike the remaining curves," down to ten minutes, from centres on the table, in the reverse direc- tion, so that they shall cross the first series at a large angle. Space them about three-eighths of an inch asunder at the mid- dle. Use a needle-point centre for the first series, to avoid boring a large hole in the sheet. Add also, on that face, two radial lines drawn towards the corners. Score the fractional curves very lightly, the full figure curves a little deeper, but all of them with steadiness and (lelicate stress. Practise beforehand on a separate slip, for the ‘right intensity of stroke. Ingrave the numbers with a stiff steel point on the opposite side of the sheet to that upon which the corresponding curve is traced. Bring the work out by rubbing it with India ink. If preferred, the flat curves on the reverse side may be colored with carmine. Duplicate protractors will be found useful in projecting compound curves. Clip off the four corners, re- enforce the edges with a narrow ribbon of tracing-linen, folded over them and glued fast, and the article is complete. Ht is perfect for its use; durable, flexible, spotlessly transparent, not liable to warp or change dimensions with changes in the temperature or moisture of the air, and, withal, takes and pre- serves a visible line, thin as the gossamer. 3. To experienced locating engineers, the curve-protractor needs no wordy commendation. Contrasted with the incon- venient appliances of the old method, - cardboard, veneer, glass, or dividers, – its advantages will be manifest. A few hints as to the manner of using it may be in place. 4. First of all, let the experimental line approximate to the probable line of location; and, upon that base, construct a contour map, with reference to which special observations should be made in the field, and the chaining done with care, 2' * 86 THE GUE VE-PROTRACTOR. Extreme accuracy in the contours need not be attempted. Note the courses of streams, ravines, and ridges, the average slopes at frequent intervals, and, on irregular ground, make illustrative sketches to aid in utilizing the other notes. Prac- tice gradually teaches how to observe critical points intelli- gently, and to record them briefly. In valleys or plains, where the location indicated is made up of long tangents and easy curves, little detail is required; but on bluffy, tortuous ground, With unavoidable divides to overcome, and long reaches of maximum gradient to be fitted, the method by contours is not only the simplest and clearest way of compiling necessary information, but is an aid to the engineer in projecting the right line, which no substitute can fully replace. 5. The writer is forced by the strong constraint of experi- ence to differ on this subject with Mr. Trautwine. The dif- ference, however, is a permissible one, and implies no lack of grateful respect for that veteran engineer, whose books are our handy-books, and to whose genius we are all debtors. - 6. Iſaving made the map, with ten-foot contours, suppose, for example, that a continuous gradient five miles long is to be located. Spread the dividers to 500 feet by the scale, start at the foot of the ascent, and step up, complying with the general trend of the ground, to the summit. This needful preliminary gives about the distange you have to work on, which cannot in many cases be derived from the experimental line directly. The profile furnishes the height to be overcome; and you are thus prepared to assume a summit cut, and determine the gradient. Having adopted one, say, of 66 feet per mile, observe that this rises five feet in 400 feet. Spread the dividers, then, to 400 feet by scale, and stand one leg on or Inear the summit, at a point corresponding to a five or ten unit in the elevation of the gradient. That is to say, if the grade elevation at the summit be 362, for instance, stand the leg of the dividers a little beyond or a little short of the summit, at a point where the grade elevation is 365 or 360. Thence, exer- cising good judgment to conform in a general way to what the location ought to be, and to make no angular indirections which cannot be closed with the maximum curvature, step forward down the incline. Name each step mentally as it is made, 355, 350, 345, 340, &c., and spot at the same time with a pencil- point the contour or half space, directly opposite, correspond- * THE CURVE-PROTRACTOR. 87 ing to it in elevation. Connect the pencil-marks with a faint dotted line. - - - - 7. Were the ground a straight, regular hillside, the steps would be made directly from contour to half-space, thence to the next contour below, and the dotted line would mark Out a tangent conforming exactly to the ground surface. On devious slopes, rounding within the limit of the sharpest permissible curve, the same exact conformity could be obtained, if desired, and a grade-line laid down which should require the least possible expense in building. On irregular, winding ground, an approximation only to the dotted line can be made: it is nevertheless a guide to go by; and, the more nearly the loca- tion project approaches it, the lighter will the work of con- struction be. The dotted line, in short, is analogous to a . profile; and the engineer can prescribe his cuts and fills with reference to it, by means of curve or tangent, just as On the profile he does the same by means of grade-lines. A fairly correct map will enable him to construct a profile from the project, and to amend its errors without the trouble and ex- pense of tentative field-work. The writer’s habitual practice has been to base his preliminary estimates on a profile thus deduced from the map; and he recommends the practice to others. They will be surprised to observe the likeness between such a profile, tolerably well done, and that of the subsequent location. - 8. It is a good custom, and one which cannot prudently be neglected where long reaches of maximum gradient are en- countered, to “slacken grade ’’ on the curves. In making this adjustment, the contour map is exceedingly useful. An ap- proximate project is first required, in order to determine the -curvature, and, from that, the varying gradient. The location can then be laid down on the map with satisfactory precision. Opinions differ as to the right allowance per degree of curva- ture, and no experiments seem to have been made from which to deduce an authoritative rule. Some say 0.025 per degree per 100 feet; others, 0.05; others, variously between the two. Probably 0.05 is the safer rate. This amounts to 2.64 feet on a mile of continuous one-degree curve, and makes a nine-degree curve, about the curve of double resistance at ordinary passen- ger speeds. ! 9. In projecting iocations, the better way generally is to strike the curves first. - i '88 - TIME OUR VIE – PROTRACTOR. 10. The following tables may be of assistance. It was need- ful, calculating them at all, to calculate them right; but of course such exactness as the figures would seem to indicate is unattainable in practice. 11. TABLE SHOWING THE DISTANCE, D, IN FEET, AT WHICH A. STRAIGHT LINE MUST PASS FROM THE NIE AREST • POINT OF ANY CURVE, STRUCK witH It ADIUS r, IN ORDER THAT A TERMINAL BIRANCEI IIAVING RAIDIUS R=2 r, AND CONSUMING A GIVEN ANGLE, a, MAY MERGE IN SAID STRAIGLIT LINE. D = (R— r) × (1 — cos. ac). DEGREE OF THE MAIN CURVE. ANGLE 2° 3° 4° 5° 6° 7° 8° 9° 10° 3C. - I). 2° 1.72 | 1.15 0.86 0.69 0.57 0.49 0.43| 0.38 || 0.34 3° 4.01 || 2.67 || 2.00 | 1.60 | 1.34 | 1.15 | 1.00 0.89 || 0.80 4° 6.88 || 4.58 || 3.44 || 2.75 2.29 | 1.96 | 1.72 | 1.53 | 1.37 5° 10.89 || 7.29 || 5.44 4.35 | 3.63 || 3.11 || 2.72 2.42 2.18 6° 15.76 10.50 || 7.88 6.30 5.25 || 4.50 | 3.94 || 3.50 3.15 7° | 21.49 || 14.32 || 10.74 | 8.59 || 7.16 || 6.14 || 5.37 || 4.77 4.30 8° 28.36 | 18.91 || 14.18 || 11.35 | 9.45 8.10 || 7.09 || 6.30 || 5.67 9° 35.24 || 23.49 || 17.62 || 14.09 | I].75 | 10.07 || 8.81 || 7.83 || 7.05 10° | 43.55 | 29.13 21.77 17.42 14.52 | 12,44 10.89 9.68 || 8.71 THE GUE VE-PROTRACTOR. ' © 89 If R = 1; r, use half the tabular distance; if R = 3 r, use twice the tabular distance; if R = 4 r, use three times the tabular distance, and SO On. 12. TABLE SHOWING THE DISTANCE, d, IN FEET, AT WHICH CURVES OF CONTRARY FLEXURE MUST BE PLACED ASUNDER IN ORDER THAT THE CONNECTING TANGENT, T, MAY BE 300 FEET LONG. - à DEGREE OF CURVE. à tº P O O § | 1 || 2 | 3° | 4 || 5° 6° 7° 8° 9° 10° $ sº º - § ſº º ; ; º (7. ſº ſº ſº 1° 3.9 |5.24 5.02 | 6.29 6.35 | 668 6.86 7.00 7.08 || 7.18 1° 2° | . . 7.84 9.43 || 10.38 || 11.20 | 11.70 || 12.20 | 12.55 | 12.80 13.06 || 2° 3° . . . . . . 11.77 13.43 || 14.64 | 15.68 16.45 17.69 17.61 | 18.05 3° 4° . . . . . . . . . 15.65 || 17.39 || 18.76 | 19.90 20.82 21.64 22.31 4° 5° . . . . . * - ... 19.54 || 21.22 22.76 || 24.01 || 25.07 || 25.97 5° 6° . . . . . - - * * ... 23.32 25.20 26.70 || 28.00 || 29.13 || 6° 79 e * * * ... 27.25 || 29.01 || 30.58 || 31.93 || 7° 8° - - - - - - - * * - - * - • * 31.05 || 32.82 || 34.41 8° 9° * - I tº º - - * * - - - - - - * @ 34.82 || 36.31 9° 10° & º & & * * * * * * tº º - - -> ... ' | 38.56 || 10 . Eacamples. A 70 and 49 should be 19.9 feet asunder; a 5° and 99 should be 25.07 feet asunder. As approximations, for a connecting tangent 400 feet long, take twice the tabular distance: for a connecting tangent 200 feet long, take half the tabular distance. PROBLEMS IN FIELD LOCATION. xxvi-xxxvii. PROBLEMS IN FIELD LOCATION. XXVI. HOW TO PROCEED WHEN THE P. C. Is INACCEs. SIBLE. 1. Suppose, for example, a pro- jected 5° curve, beginning at stake 24.20, or B in the diagram. FIRST METHOD. — At any point A, which we will assume to be Stake 23.40, set up the transit. Let it be judged that stake 27, marked D in the diagram, must fall on ac- cessible ground. Then the distance B D, around the curve, is 280 feet, corresponding to an angle E B D of 7° at the circumference, or an angle of 149 at the centre. The chord of a 1° curve consuming this angle, by Table XVI., is 1,396.6 feet; that of - - a 5° curve, B D in the figure, is one-fifth of this, or 279.3 feet. In the triangle A B D are thus known the sides A B, BD, and the sum of the angles at A and D, which sum is equal to the angle E B D. - Hence, by trigonometry, - As the sum of the sides given =359.3 A. C. . . . . 7.444543 Is to their difference. = 199.3 . . . . . . . 2.299507 So is tan. ; sum of angles at base = 3° 30' . . . . 8.786486 To tan. ; their difference = 1° 56} . . . . 8,530536 Adding half the difference to half the sum, the larger angle, A, is found to be 5° 26%/; subtracting half the difference from half the sum, the smaller angle, D, is found to be 19334/. The 94 HOW TO PROCEED WHEN THE P. O. IS INA COESSIBLE. length of the side A D may be found in like manner by trigo- nometrical proportion; or, perhaps more simply, thus: — BD X mat. cos. D = D F = 279.2. BA X mat. cos. A = A F = 79.6. A F + F D = A D = 358.8. We are now prepared, from our point A, to deflect the angle 5° 264 R, and lay out the line A D to the point D on the curve. Moving the instrument to that point, and backsighting to A, a deflection of 1° 33}| R places the telescope on line D B; a fur- ther deflection of 7° places it in tangent at D, and the curve may thence be traced in both directions. 2. SECOND METHOD. — Having, as in the first method, judged that stake 27, marked D, must fall on accessible ground, and thus determined the central angle subtended by the arc BD, refer to Table XVI. for the tangent of a 19 curve, corré- sponding to 14°, the given angle. It proves to be 703.5 feet. One-fifth of this, 140.7 feet, is the tangent or apex distance, BC, of a 50 curve, which may be measured on the ground. Moving the instrument to C, turning 14° R, and laying off the line C D = B C, the point D on the curve is ascertained. - 3. The preceding methods are manifestly applicable to the ends also of curves, as well as the beginnings. A case not unfrequent in practice may be added in conclusion of the subject. . . Suppose a 2° curve terminating at C, in marsh or stream not measurable directly. Let C fall, at stake 32.20. At any con- venient point A, say stake 29, place the transit with telescope in tangent. The arc AC, - 320 feet, includes an angle of 6°24'. The tangent of a 19 curve corre- sponding to this angle in Table XVI. is 320.34 feet; that of a 2° curve is therefore H60.2 = A.B. Move to B, deflect Go 24, R into the range of the terminal tangent, and ſix E on the opposite shore. Fix also D, and note the angle EBD. Move to E. Measure the angle DEB, and the distance D.E. The tri- angle BED may then be solved. If B E is found to be 670 feet, CE = 670 – 160.2 = 509.8, and stake E = 32.20 + 509,8, = say 37.30. - t - - - - - - - - - - - HOW TO PROCEED TVIIEN TIME P. C. C. IS INACCESSIBLE, 95 XXVII. | HOW TO PROCEED WHEN THE P. C. C. IS INAC- CESSIBLE, - 1. Suppose a 4° curve, A B, compounding at B into a 69 curve B C. FIRST METHOD. — Place the transit at any point A, Say Stake 34. . Let the pro- / D \ posed P. C. C. fall at stake 36.25. Assume that we wish to reach C, on the second - Q curve, by means of the - straight line A. D. C. The arc A B, covering 225 feet O of a 49 curve, subtends an angle of 9°. A D is half the chord of twice this angle. - By Table XVI., the chord of 189 on a 19 curve is 1,792.7 feet. That of a 49 curve is therefore 448.2 feet, half of which = 224.1, = A D. The versin. of 189 on a 19 curve, by the same table, is 70.54 feet; one-fourth of which, or 17.635, is the versin. BD, corresponding to the same angle on a 49 curve. In order to find what angle on the 69 curve this versin. B D, = 17.635 feet, corresponds to, multiply it by 6, and seek the product, 105.81, in Table XVI., where it is found, nearly enough for field-practice, opposite the angle 22° 04'. The chord of that angle, on a 1° curve, is seen at the same time in the adjoining column to be 2,193.2 feet; on a 69 curve it is therefore 365.5 feet, one-half of which, = 182.75 feet, = D C, and one-half of 22° 04' = 11° 02', - the angle covered by the arc BC. Thus are found the angle at A = 9°, the angle at C = 11°02', and the distance AC = 224.1 + 182.75, − 406.85 feet. The angle 11° 02/ corresponds to a length of 1.84 feet on the 69 curve; C, therefore, falls at stake 36.25 + 1.84 = 38.09. With these data, the field-work is obvious. 2. SECOND METHOD. — Having reached the point A, and determined the arc AB = 0°, as above, find in Table XVI. the tangent 450.95 feet, corresponding to the given arc, one-fourth 96 TO SHIFT A P. C. of which = 112.7 feet, - tan. A E for the 49 curve. Move to B, deflect 9° R; range out the line E F, made up of EIB = A E = 112.7 feet, and B F any convenient distance, say 90 feet. This 90 feet is the assumed tangent of some unknown angle on the 69 curve. To find the angle, multiply 90 by 6, and seek the product, 540, in the tan. column of Table XVI., where it is found opposite 10° 46'. By moving then to F, deflecting 100 46' R, and measuring FC = 90 feet, the point C is fixed on the second curve. - 3. Should unexpected obstacles be met in carrying out either of these plans, the triangles A G C or E G F may be solved, and the point C fixed by means of the lines A G, G C. 4. The application of the foregoing methods to ourning obstacles on simple curves needs no special instance. XXVIII. TO SHIFT A T. C. SO THAT THE CURVE SHALL * TERMINATE IN A GIVEN TANGENT. 1. Suppose a 3° curve A B to 4. have been located, containing an angle of 44° 26', and ending in tangent B E: required, that it shall end in tangent IDF, parallel to B E. It is plain, from the diagram, that if the curve and its initial tangent be moved forward, like the blade of a skate, until the terminal tangent merges in D F, the P. T. will have traversed the line BD, equal and parallel to A.C. If, there- fore, on the ground at B, the angle E B D, equal to the whole angle consumed by the curve, in this case 44° 26', be laid off to the right, and the distance B D to the range of the proposed terminal tangent be measured, the equal distance A. C., from the original to the required P. C., is thus directly ascertained. Should such direct measurement be impracticable, range out the tangent B E, and, at any convenient point, measure the distance from it square across to the proposed terminal tan- gent DF, say 56 feet. Then in the right triangle B E D, mak- | | A - C ing BD radius, we have given the angle at B = 44° 26', and TO SUBSTITUTE A CUR VE OF DIFFERENT RADIUS. 97 the sine E D = 56 feet. Hence, by trigonometry, E D –- mat. sin. 44° 26', or 56 -- 0.7, − B D = 80 feet, - distance A C along the initial tangent, from the erroneous to the correct P. C. 2. This problem occurs more frequently than any other in the field; and the young engineer should have it by heart, that the distance square across between terminal tangents, divided by the natural sine of the total angle turned, will give him the distance he is to advance or recede with his P. C. to make a fit. 3. Excepting on precarious rocky steeps, city streets, or like exact confines, to strike within two feet of any point desig- nated in the project, may be considered striking the mark. Astronomical nicety, whether with transit or level, in an ordi- nary railroad location, is mere waste of time. 4. The observant reader will not fail to perceive that the foregoing rule applies to systems of curves, or to compound lines also, the angle E B D being the angle included between the initial and terminal tangents, let what flexures or indirec- tions soever have been interposed; and that, if the angle re- ferred to be either 180° or 360°, adjustment by shift of P. C. is impracticable. In those cases, a change of radius becomes necessary. XXIX. TO SUBSTITUTE FOR A CURVE ALREADY LOCATED, ONE OF DIFFERENT RADIUS, BEGINNING AT THE SAME POINT, CONTAINING THE SAME ANGLE, AND ENDING IN A FIXED TERMINAL TANGENT. 1. Suppose the 4° curve A B, containing an angle of 32° 20', to have been located, and that it is required to substitute for it an- other curve A C, which shall end in a parallel tangent CF, 60 feet to the right. FIRST M ET II O D. — Findl the – 1ength of the long chord A. C., a . A B + B C. Referring to Table.XVI., the chord of a 19 curve for 32°20' is seen to be 3,190.8 feet; that of a 49 curve, there- 98 To PIND THE POINT AT WHICH To doMPotºwd. fore, - 797.7 feet, say 798 feet, - AB. To find B C, solve the triangle BDC, observing that the angle D B C = BAI = one-half of the central angle 32° 20', - 16° 10', and that DC = 60 feet. Then D C -- nat. sin. 16° 10' = 60 +.278 = say 216 feet, + B C. Hence A C = A B + B C = 798 –– 216 = 1,014 feet. Having thus found the length of chord A C, the radius and rate of curvature may be deduced as in X. w . ..? Or, dividing the tabular chord of 32° 20' by chord A C = 1,014, the degree of the required curve is ascertained directly to be 3.15, equivalent to 3° 09'. . . . ** 2. SECOND METHoD.—Find the apex distance AH, - A I + IH. The tabular tangent of 32° 20' divided by 4 gives A I = 415 feet. In the triangle KD C, the side D C -- mat, sin. K = 60 -- nat. sin. 32° 20' = 112 feet = KC = IH. Then A H = AI + I H = 415 + 112 = 527 feet; and the tabular tangent 1,661 -- 527 gives 3.15, equivalent to 3° 09, the degree of the required curve AC, as before, . . . 2. XXX. HAVING LOCATED A CURVE A B C, TO FIND THE . POINT B AT WHICH TO COMPOUND INTO ANOTHER CURVE OF GIVEN RADIUS, WHICH SHALL END IN TANGENT E F, PARALLEL TO THE TERMINAL TANGENT OF THE ORIGINAL CURVE, AND A GIVE DISTANCE FROM IT. - 1. To find B, the angle BIC must be found. Call the given distance between tangents D; the larger radius, R; the smaller one, r, the required angle, a. Then, referring to the figure, observe that in the triangle I H K, I H being ra- dius, IK is the cosine a, i.e., IK -- IH = nat. cosine a. But I H = R — r, I K = 'I C — KC, and K C = KF or H E + FC, = r + D; i.e., IK = R — r — D. Hence mat. cosine a = R — r — D, + R — r = 1 – D –- (R – r)]. - TO SHIFT A. P. C. C. ' - 99 ° .The same reasoning would apply if A B E were the curve first located, and a terminal curve of larger radius required to . be put in. 2. We have, then, the following general rule for such cases: Divide the perpendicular distance between terminal tangents by the difference of the radii, and subtract the quotient from unity; the remainder is the natural cosine of the angle of re- treat along the located curve to the required P. C. C. Example. 3. A 3° curve on the ground, to find the P. C. C. of a 50 curve striking 27 feet to the right. Here D = 27; R — 1 = 1,910 — 1,146, = 764; D –– R — r = 27 —— 764, - .03534; and 1 — . .03534 = .96466 = mat. cosine 15° 17'. We must go back, therefore, 509 feet on the 3° curve, to compound into the 50 curve. Had the 5° curve been located first, we must have gone back 306 feet to begin the 39 curve which should strike 27 feet to the left. In either case, time might be saved by moving directly from E to C, or the reverse, and spotting in the curve : backwards. To do this, we have in the right triangle FEC, the angle E = half of 15° 19, = 7° 384, and the side FC = 27 feet. Then E C = 27 —— mat. sin. 7° 384', = 203 feet; and if E were stake 54.20 On the 59 curve, B would fall at Stake 54.20 — 3.06, = 51.14; and C, the P. T. of the 3° curve, at 51.14 + 5.09, = stake 56.23. XXXI. TO SHIFT A. P. C. C., SO THAT THE TERMINAL , BRANCH OF THE CURVE SHALL END IN A GIVEN TANGENT. FIRST CASE: the terminal branch having the shorter radius. 1. Suppose the compound curve A C N located, and that it is required to fix a new P. C. C. at B, from which the terminal branch B M shall merge in tangent ML, a given distance from N O. To fix B, the central angle B H M of the new terminal branch must be found, and substituted for CIN. Call the longer radius R; the shorter one, r; the dis- 100 - TO SHIFT A. P. C. C. :: * : tance asunder of the terminal tangents, D; the central angle, CIN, - IE K, of the located terminal branch, b ; and the central angle, B HIM, = H E F, to be substituted for it, a. In the right triangle, EIR, EIK = E I cos. IE K = (R — r) cos. b. In the right triangle HFE, IE F = E H cos. IIB. F = (R – r) COS. Ol. - Also, FK = LO = D, since each is equal to r — K L. Then E F = E K — F K; i.e., (R— r) cos. a = (R— r) cos. b -— D. Hence mat. cosime a = mat. cosime b — [D –– (R—r)]. Were the curve B M located, and the curve C N to be substi- tuted for it, — that is to say, were a given and b required, - we should have, by transposition, mat. cos. b = nat. cos. a + [D -- (R — r)] IEacample. A 39, compounding into a 5° curve at C, which consumes an angle CIN, + 30° 22', and ends in a tangent, N (), which is found, by measurement of LO, to be 34 feet too far to the left. Here, D = 34, R = 1,910, r = 1,146, b = 30° 22'; and, by the solution, mat. cos. a = nat. cos. 30° 22' — (34 –– 11,910 — 1,146]) = 0.8628 — (34 -- 764). 34 . . . . . . log. 1.531479 764 . . . . . . log. 2.883093 . .0445 . . . . . . log. 2.648386 Then 0.8628 – 0.0445 = 0.8183 = cos. 35° 05", = angle a ; a — b = IB II M – C IN = I2 EC = the angle of retreat from the orroneous P. C. C. = 35° 05' — 30° 22' = 4° 43', equivalent to 157 - feet, on the 39 curve, from C to B. 2. SECONI) . CASE: the terminal tºº? - branch having the longer radius. Let B N represent the terminal branch located with central angle IKO = b, and suppose it required to determine the new arc CM, with central angle IE F = a. Call the longer radius R, the shorter one r, the distance L N between tangents, D. In the To SIIIFT A P. C. C. 101 right triangle IR O, KO = K.I, cos. IK O = (R – r) cos. b. In the right triangle FIE, E F = E I, cos. IE F = (R – r) cos. a. Also, E H = L N = D, since each is equal to R — K L. Then E F = E H + H F = E H + KO; i.e., (R – r) cos. a = (R—r) cos. b + D. Hence mat. cos. d = mat. cos. b + |D -- (R — r)]. f Were the curve CM located, and the curve B N to be sub- stituted for it, that is to say, were a given and b required, we should have, by transposition, nat. cos. b = nat, cos. a- |D -- (R —r)]. • - Iºacample. A 5° compounding into a 3° curve at B, which consumes an angle of 44° 20', and terminates at N, 28 feet too far to the left. Here D = 28, R = 1,910, r = 1,146, b = 44° 20; and, by the solution, mat. cos. a = mat. cos. 44° 20' + (28 –– 764). The mat. cos. 44° 20' = 0.69883; 28 + 764 = log. 1.447158 — log. 2.883093 = log. 2.564065, corresponding to the decimal 0.03665, which, being added to mat. cos. 44° 20', gives 0.73548, the mat. cos. 429 29%. Then B KN – C E M = 44° 20' — 42°29′ = 1° 51/= angle BIC, equivalent on a 59 curve to 37 feet, which therefore is the distance around the are from B, the erroneous P.C.C., to C, the correct one. 3. From these formulas the following general rule may be drawn: Divide the distance between terminal tangents by the difference of the radii, and call the quotient Q. Find the nat- ural cosine of the terminal arc already located, and call it C. The sum or the difference of Q and C will be the natural COsine of the terminal arc to be substituted for that already located. With radii in the order R, r, should the terminal inside outside With radii in the order r, R, should the terminal tangent Outside inside tangent located strike } } the proposed tangent; or, located strike | } the proposed tangent, — take the } SUITQ. difference } Of Q and C for the required cosine. S{ 102. TO FIND THE POINT AT WII ICII TO BEGIN A CURVE. XXXII. HAVING LOCATED A. TANGENT, A, B, INTERSECTING . A CURVE, C. D, FROM THE CONCAVE SIDE, TO FIND THE POINT E ON SAID CURVE AT WHICH TO BEGIN A CURVE OF GIVEN RADIUS WHICH SEIALL MERGE IN THE LOCATED TANGENT. 2^ - 1. Place the transit at the 2 \, . . . intersection point B. Set - points at equal distances therefrom in both directions on the curve already located, by means of which the direc- tion of a tangent to that curve at B may be fixed, and the angle FIBA measured. Call that angle a ; and, as shown in the figure, suppose the lo- cated curve to be prolonged in- to a terminal tangent, parallel with the newly located tan- - ... • gent A. B. Complete the dia- gram. Call the larger radius R; the proposed radius, r, the central angle of the proposed curve, a. Then, obviously, the line A G = R cos. a. It is also equal to (R – r) cos. a. -- r. That is to say, R cos. a = (R – I') cos. a + r. Hence cos. a. = (R. cos. a - r) -- (R – r); and a — a = angle B G E, sub- tended by the arc BE, from which the length of the are may be deduced, and the point E ascertained. - F. Ba'ample. DC, a 1° curve; angle a = 64° 32'; to connect with a 49 curve. Here cos. a = (5,730 × 0.43) — 1,433 -i- (5,730 – 1,433) = 0.24 = cos. 76° 06'; and a — a = 11° 34', equivalent to a distance from B around the 19 curve of 1,157 feet to E, the point at which to begin the 4° curve. TO LOCATE. A. Y. - 103. XXXIII. HAVING LOCATED A TANGENT, A B, INTERSECTING A CURVE, C. D, FROM THE CONVEX SIDE, TO FIND THE POINT E ON SAID CURVE AT WHICH TO º, BEGIN A CURVE OF GIVEN RADIUS WHICH SHALL MERGE IN THE LOCATED TANGENT. • *, 1. This problem is analo- r D gous to the preceding one. . . . . ~~ The preparatory steps are the same in both. Having found 2%. - * - ~ bº 2’ the angle a, however, it will T->g 2’ - - - º - & be manifest to the attentive Nº - sk - reader, that, in this case, R º -\e w cos. a = (R -- r) cos. a + r. 2– - * } Hence cos. a = (R. cos. a - r) G . C -- (R – ?). - - - - J Facample. - . . . 2. DC, a 19 curve; angle a = 64° 32': to connect with a 4°. curve. Here cos. a = (5,730 × 0.43) — 1,433–4–(5,730+ 1,433) = 0.1439 = cos. 81° 43'; and a — a = 17° 11', equivalent to a distance from B around the 19 curve of 1,718 feet to E, the point at which to start the 4° curve. XXXIV. TO LOCATE. A. Y. 1. The processes of the two former problems may be adopted. In this case the angle a vanishes, and the cos. a clearly is equal to (R — r) -- (R -- r). **. 2. Another solution of the Y problem is as follows: — - Draw the tangent E D in- tersecting the tangent B.A. Then is BD = D.A, for the rea- 104 - TO LOCATE. A. Y. Son that each is equal to D E. Make G F = R + r, the diame- ter of a semicircle. Said semicircle touches tangent B A at D, its middle point; and D E being perpendicular to G F, we have by geometry G E : D E : : D E : EF; i.e., G E X EF, or R × r, - DE”. Hence D E = B D = D A = WR × 1 = R tam. 3 a., and we are thus enabled to fix the points E and A. r 3. In the two foregoing problems, the angle consumed by curve E A is - 180° — a. - - Ea:ample. I3 E, a 24° curve lecated; BA, a tangent: to complete the Y with a 6° curve, E.A. - - By the first method, cos. a = (R – I') —- (R –– r) = (2,292 — 955) + (2,292 –– 955) = 1,337 –– 3,247 = log. 3.126131 — log. 3.511482 = 1.614649, which corresponds to log. cos. 9.614649, or to the decimal number 0.4118, indicating in either case the angle 65° 41' = a. D E = B D = D A = R tan. , a = 2,292 × 0.6455 = 1,479.4. DE may be found also by reference to 'I'able XVI., where the tangent of a 19 curve for 65° 41' is seen to be 3,698.6. Dividing this number by 23, we have 1,479.4, as above. Or, by the second method, - D E= A/RXT = A/21SSS60= 1,470.4. - Having thus the means- of fixing points E, D, and A, the curve E A can be laid down. 4. If B A is curved con- vea, to the Y, construct the figure as in margin, and reason thus : — - In the triangle E G F, formed by lines connect- ing the curve-centres, the sides are respectively equal to the sums of the contiguous r a dii : the C -º angles may therefore be | - found as in Case III., Trigonometry. - Lines drawn bisecting the central angles of the severa TO LOCATE. A. Y. 105 curves will pass through the points of intersection of the tan- gents to those curves severally. But lines so drawn in this case bisect also the angles of a triangle, and, demonstrably by geometry, meet in one point equidistant from the three sides of the triangle. That point, therefore, must be a com- mon P. I. for all the curves, and that equidistance the “tan- gent ’’ length common to them all. IEacample. * Given B.A., a 39, and B C, a 49 curve: to complete the Y with a 5° curve, CA. E F = 1,910 –- 1,146 = 3,056. G F = 1,433 + 1,146 = 2,579. E G = 1,910 – 1,433 = 3,343. Then, by Case III., Trigonometry, gº-ºº. As E. G., 3,343 . . . . log. (a. c.) 6.475864 Is to E F + G F, 5,635 . . . . log. . . 3.750894 So is E F – G F, 477 . . . . log. . . 2.6785.18 To diff. of segments of E G, 804. . . . . 2.905276 Adding half the difference to half the sum of the segments of the base E G, we shall have the greater of them; i.e., (3,343 – 804) -- 2 = 2,073.5, which is the cos. E, E F being radius. Hence 2,073.5 -- 3,056 = log. 3.316704 — log. 3.485153 = 9.831551 = cos. 47° 16' = E. By Table XVI., the tangent of a 1° curve corresponding to this angle is 2,507.3: that of a 3° curve, therefore, is 835.8 = the common tangent BD or D.A. Multiplying the common tangent by 4, we shall find opposite the product in Table XVI, the central angle of the 4° curve to be 60° 32'; multiplying it by 5, we find, in like manner, the central angle of the 59 curve to be 72°12'. Arc BA, = 47° 16', is equivalent to 1,575 feet on the 3° curve; arc B C, = 60° 32', is equivalent to 1,513 feet on the 49 curve. Points being thus fixed at A and C, curve C A can be laid on the ground. - 5. If curve BA is concave to the Y, the radii being given, construct the figure as follows: — First draw the triangle G. FE, the sides of which are obvi- ously derived from the given radii. Prolong the sides E G and E F indefinitely. Bisect the exterior angles at G and F with . 106. TO LOCATE. A. Y. lines meeting at D; and from D let fall perpendiculars on EP, EA, and G F. Then, comparing triangles G B D, G CD, the angles at G are equal by construction; the angles at B and C, are right angles, the side G.D. common. Hence the triangles are equal in all their parts: B G = G C, and B D = D C. By: like reasoning, it appears that C F = FA, and D A = D C. : The point D being equidistant from the right lines E B, EA, which limit angle E, a line bisecting that angle will strike point D. - E. - 6. It may be remarked, therefore, that lines bisecting the vertical angle and the exterior angles contained between the base and the prolongation of the sides of any triangle, will meet in a point equidistant from the base and the said prolon- gations. We thus have in the figure all the conditions for fit- mess of the curves. It remains only to solve the triangle G FE, seeing that from its angles the required central angles can be obtained. - - Example. B.A., a 19, B C, a 6° curve, located: to complete the Y with an 8° curve, O.A. - . . . %. . . . . . . . . . . . . . . . . .” To LoCATE A TANGENT TO A GURVE. 107 In triangle G FE,- . ... E F = 5,730–717 = 5,013. - E G = 5,730 – 955 = 4,775. . .* G F = 955–H 717 = 1,672. ... Then, by Case III., Trigonometry, – As E F . . . 5,013 . . . . log, (a, c.) 6.299902 . . Is to E G +G F, 6,447 . . . . log. . . 3.809858 So is E G – G F, 3,103 . . . . . log. . . .3491782 . " To diff. ség of base, 3,991 . . . log. ... 3,601042 - - The longer segment, therefore, is 4,502; the shorter, 511. Cos. E = the longer segment divided by E G = 4,502-5-4,775 = log. 3.653405 —3.678973 = 9.974432 = cos. 19°28′ = angle E. Cos. G. FE = the shorter segment divided by G F = 511 + 1,672 = log. 2,708421 — log. 3.223236 = 9.485185 = cos. 72° 12' = angle G F E. º * The central angle, B G C, of the 69 curve, is equal to 180– FG E = the sum of the angles at E and F = 72°12' + 19° 28! = 91° 40', making, the arc BC = 1,528 feet. The arc B.A., equivalent to 19°28' of a 1° curve, = 1,947 feet. Points C and A being thus ascertained, curve A C may be located. It will consume an angle = 1809 – 72°12' = 107°48', equivalent, on an 89 curve, to 1,347.5 feet. - - - - - % XXXV. To LOCATE: A TANGENT TO A CURVE FROM AN OUTSIDE FIXED POINT. . . \, 1. If the ground is open, and the curve can be seen from the fixed point, it may be marked by stakes or poles at short inter- vals, and the tangent laid off without more ado. ... - . 2. Suppose, however, that on. cumbered ground a trial tan- gent, A B, has been run out, intersecting the curve at B: it is required then to find the angle BAE, in order that the true tangent. A E may be laid down. \-' x ... Fºy, " . . . . ." - a ... ; ; * * * '-- - - “..” * , t , a • * * * * ** 108 To substitute 4 our VE. Facample. A B = 1,500 feet; D H B, a 49 curve; angle FBI) = 20° 13'. First, the angle FBD, between a tangent and a chord, is equal to half the central angle subtended by the same chord. Angle D C B, therefore, H 40° 26'. By Table XVI., the chord of 40° 26', for a 19 curve, = 3,960.2 feet; for a 49 curve, it is, say, 990 feet = D B; and DI = IB = 495 feet. The versin. HI is, in like manner, found to be 88.25 feet. Deducting this from the radius of the 49 curve, we have I C = 1,344.4 feet. Then IC -- IA = tan. IAC; i.e., 1,344.4 + (495 – 1,500) = 0.674 = tan. 33° 59' = angle I.A. C. Next, by geometry, the proposed tangent A E = A/AIDXAB =W2,490 × 1,500 = 1,932.6; and E C --A E=tan. E A C = 1,432.69––1,932.6=0,7413=tan. 36°33' = angle E A C. Then E A C–I A C =36°33' —33° 59' =2° 34' = angle B A E, the angle required, which can accordingly be laid off from the fixed point A, and the tangent located. XXXVI. To SUBSTITUTE A CURVE OF GIVEN RADIUS FOR A TANGENT CONNECTING TWO CURVES. - . . . . Eacample, 1. A B, a 49 curve; B C = 774 feet; CD, a 60 curve: to put in the 19 curve, EF. , , . * Sketch the figure as in margin, H K being parallel and equal to BC. Then K G = B G –(BK or CH)= 1,433 – 955 = 478 feet; KH -- G K = 774-i-478 = 1.62 = tan. 58° 19 = angle KGH; and K II -- sin. 58° 19 = 774-- 0.851 = 909.6 feet = G. H. . - TO PUW A TANGENT TO TWO CUP V.E.S. 109 In the triangle GHI we have then the sides given; namely G H = say, 910 feet, HI = 5,730 – 955 = 4,775 feet, and G. I = 5,730 — 1,433 = 4,297 feet: to - find the angles. - Under Case 3, Trigonometry (III.), III : I G + G H :: I G — G H : IL – L H; i.e., 4,775 : 5,207 :: 3,387 : 3,693, the differ- ence of the segments into which the base III is divided by a perpendicular from G. Adding lıalf the difference of the seg- ments thus found to half their sum, the longer segment, IL, is found to be 4,234 feet ; subtracting half the difference from half the sum, the shorter segment, L HI, is found to be 541 feet. . . Then H L –– H G = 541 –– 910 = 0.5945 = cos. 53° 31' = angle G. H. I. In like manner, dividing IL by I.G, we find the angle GIH to be 9° 49'. The sum of these angles = angle B G II = 63° 20', for the reason that each is equal to 180 — H G.I. Finally, E G H – K G H = 63° 20' — 58°19' = 50 01/ = angle E G B, equivalent to a distance from B of 125 feet around the 49 curve to the P. C. C. at E; and G IIH – E G B = 9° 49' — 5° 01' = 4° 48' = angle CIHF, equivalent to a distance from C of 80 feet around the 69 curve to the P. C. C. at F. w - XXXVII. TO RUN A TANGENT TO TWO CURVES ALREADY LOCATED. - ſº - 1. If one curve be visible a’.º \ from the other, or if both _2~ Y \p be visible from some inter- M † =#= |-2'----M mediate point, mark them F \, "ſe/ -----e. on the ground with stakes \ // w w at S h or t intervals. The %—------ points M or L in the range A t of the required tangent may then be fixed by one or two trial settings of the transit, and the line put in. w 110 TO PUN A TANG ENT TO TWO CUP VES: 2. Should obstacles, prohibit this plan, measure any con- venient line, FG or B C D, from one to the other curve, and, completing the traverse A. F. G E or A, B C D E, determine thence the bearings and distances asunder of the centres A and E. The right triangle A E K, in which E K = the sum of the radii, may then be solved, and the points H and I ascer- tained as in the following example:– * * \ Example. FB, a 49 curve; G D, a 6° curve. N. S. E. W. A B, N. 20° E., 1,433 feet . . 1,346.6. * . 490.0° - I3 C, East, 3,570 feet . . . — 3,570.0. cº-º C D, N. 34° E., 1,800 feet" . . 1,492.2 1,006.2 -- D E, N. 45° W., 955 feet . . 675.2 &- 675.2 3,514.0 5,066.2 675.2 Total northing, 3,514 feet; total easting, 4,391 feet. Then 4,391 + 3,514 – 1.2496 – tan. 50° 20' – boaring A. E.; and 4,391 –– sin. 50° 20' = 5,704 feet = distance A. E. Also, E K -- A E = (1,433 + 955) —— 5,704 = sin. 24°45' = angle E. A. K., and angle A E K = 90° 00' — 24° 45' = 65° 15'. IIence the bearing of AIK or III is N. 75° 05' E., and that of AH or IE, N. 14° 55' W. - l & • - Since A B bears N. 20° E., the angle H AB = 20° 00' + 149 55' = 34° 55', equivalent to a distance of 873 feet from B around the 49 curve to the required P. T. at H; and, since D E bears N. 450 00, W., the angle IIE D = 450 00 – 14° 55' = 30° 05, equivalent to a distance of 501 feet from D around the 69 curve to the required P. C. at I. . 3. Should the curves turn in the samé direction, the side E.K. of the triangle A E K is equal to the difference of the radii instead of their sum. In other respects, the method exemplified will apply to that case also. 4. The preceding solution may be useful as an exercise. ISut the problem is one of rare occurrence, and the conditions must be extraordinary which prevent a close approximation, at least, to the true line in the field. The better way in actual practice, then, is to run out a trial tangent as nearly right as possible. If it errs by passing outside the objective curvé, close with a compound (XXIX.); if that error be inadmissible, or if it errs by cutting the objective curve, measure the miss, and divide it by the length of the trial tangent. The quotient TO RUN A TANG ENT TO TWO CUR FES. 111 will be the natural tangent of the angle of retreat or advance on the first curve required to make the tangent fit. 5. A still closer adjustment, would be, after determining the angle approximately as above, to find the “tangents” corre- sponding to it for the two curves in Table XVI. Subtract the sum of these tangents from the length of the trial line, if it cuts the objective curve; add the sum, if it passes outside. With the number thus found, divide the measured amount of error for the tangent of the angle of retreat or advance, as the case may be. - 6. Suppose, for illustration, that a trial tangent, bearing by needle N. 54° 30' E., is run out from stake 24.80 of a 49 curve, intending to touch a 6°, but is found to cut it. Suppose fur- ther that the objective 69 curve was laid down and numbered in the direction of approach towards the 49 curve; that its P. C. is stake 25.10, and the magnetic bearing of its initial tan- gent S. 30° 30' W. The angle, then, between the bearing of the trial tangent and that of the initial tangent of the 69 curve, is 24°, corresponding to a distance of 400 feet on the latter curve. At stake 25.10 + 4.0 = 29.10, therefore, a tangent to the 6° curve would be parallel to the trial tangent. Go forward on the trial tangent, accordingly, to a point opposite 29.10, and measure the distance square across to that plus on the 6° curve. Assuming the trial tangent to be 2,500 feet long, and the amount of the miss to be S7 feet, the mat. tan. of the angle of error is 0.0348 = tam. 29. By the method in (4), this calls for a shift of the P. T. 50 feet allead on the 49 curve, making the new P. T. 24.80 + 0.50 = stake 25.30, and ad- vances the P. T. of the 69 curve to stake 29.43 of that numera- tion. The method in (5), applied to this case, brings the angle of error 2°02', instead of 29, equivalent to a deviation of 13 feet scant in half a mile from the line corrected by the method in (4), and agreeing exactly with the correction determined by T the method in (2). ** TRACK PROBLEMs. XXXVIII.-LI. TRA C K P R O B L E M S. XXXVIII. REVERSED CURWES. The following problems will be useful in laying off turnouts, the adjustment of tracks near stations or shops, and the like: but reversed curves should never be used on the main line between stations, where they are both objectionable and unne- cessary. Ground which allows any permissible location at all will allow straight reaches of at least two hundred or three hundred feet between curves of contrary flexure; and in every case it is Worth the Small additional Outlay to make Such a location. XXXIX. TO CONNECT TWO PARALLEI, TANGENTS BY A REVERSED CURVE HAVING EQUAL RADII. 1. The radius R, and the perpendicular distance D, between the tangents given. - 2C 2 : A i 2. § Aé ſ zº 2' 7] 7 2 7473 0 -7 i 192 LOGAEITIIMS OF WUMBERS. No. 0 1. 2 3 4 || 5 6 y S 9 | Diff. 760|880814|880871 1880028.18809851881042;8810001881.156188121318812711881328; 57 1| 1385| 1442| I499| 1556|| 1613 1670| 1727| 1784, 1841 1898 57 2] 1955, 2012| 2069| 2126| 2183. 2240| 2297| 2354| 2411] 2468| 57 3| 2525|| 2581| 2638 2695 2752| 2809| 2866| 2923| 2980|| 3037. 57 4| 3093| 3150| 3207| 3264; 3321|| 3377| 3434|| 3491 3548| 3605| 57 5| 366|| 3718| 3775|| 3832| 3888| 3945| 4002| 4059| 4115|| 4172| 57 6| 4229| 4285| 4342. 4399| 4455| 4512| 4569| 4625| 4682; 4739| 57 7| 4795| 4852; 4909| 4965 5022 5078| 5135|| 5192 5248; 5.305| 57 8| 5361| 5418| 5474| 5531|| 5587| 5644. 5700| 5757| 5813| 5870|. 57 9| 5926|| 5983. 6039| 6096 6152. 6209| 6265; 6321| 6378| 6434|| 56 770|886401 |886547|886604|886660}886716|886773|886829|886885886942886998| 56 l| 70.54|| 7111 || 7167| 7223 7280| 7336|| 7392|| 7449| 7505| 7561 56 2 7617| 7674; 773)| 7786|| 7842|| 7898 7955| 801 || 8067| 8123| 56 3| 8179| 8236|| 8292| 8348| 8404 846()| 85.16|| 8573| 8629| 8685| 56 4| 8741| 87.97 8853| 8909| 8965| 9021| 9077| 9134| 9190| 9246|| 56 5| 9302| 9358) 94.14 70| 9526|| 9582| 9633| 9694 9750| 9806| 56 6| 9862) 9918| 99.74|890030|8900861890 141|89()197|890253|890309|890365| 56 7|8904'21 |890477|890.533 0589| 0645| 0700 0756} 0812| 0868| 0924 56 8| 0980|| 1035| 1091| 1147| 1203| 1259| 1314|| 1370| 1426|| 1482| 56 9| 1537| 1593] 1649 1705| 1760| 1815| 1872, 1928|| 1983| 2039| 56 780|892095|892.150802206|892.262|892317|89237389.2429|892484892.540|892.595| 56 1. 2651 || 2707 762| 2818| 2873| 2929| 2985| 3040|| 3096|| 3151 56 2: 3207| 3262. 3318|| 3373| 3429 3484 || 3540|| 3595, 3651| 3706. 56 3| 3762 3817| 3873| 3928|| 3984| 4039 4094| 4150. 4205| 4261 55 4|| 4316 4371 4427| 4482| 4538|| 4593 4648. 4704| 4759| 4814. 55 5| 4870| 4925| 4980| 5036|| 5091 || 5146; 5201| 5257| 5312|| 5367| 55 6|| 54.23 5418| 5533 boč8| 5644. 5699| 5754 5809| 5864. 5920ſ 55 7| 5975|| 6030| 6085| 6140|| 6195| 6251 6306| 6361| 6416| 6471 55 8|| 6526|| 6581 | 6636|| 6692 6747| 6802| 6857| 6912| 6967| 7022. 55 9| 7077|| 7 132 7187| 7242| 7297| 7352 7407| 7462| 7517| 7572| 55 790897627|897682,897737|897792897847897902|897957|898012;898067|898.122 55 I 8176|| 8231 8286|| 8341| 8396|| 8451 8506| 856I 8615| 8670 55 2| 8725, 8780 88.35| 8890| 8944 8999| 90.54|| 9,109| 9164 9218 55 3| 9273 9328| 9383| 9437| 9492 9547| 9602| 9656| 971 || 9766|| 55 4| 982|| 9875|| 9930| 9985||900039|900094|900149|900203900258|900312|| 55 5||900367|9004:22; 900476|900531|| 0586|| 0640 0095| 0740|| 0804| 0859| 55 6|| 0913| ()968|| 1022, 1077; 1131 | I 186 1240|| 1295| 1349; 1404. 55 7| 1458. 1513| 1567| 1622, 1676|| 1731 || 1785| 184()| 1894| 1948 54 8| 2003| 2057| 2112| 2166] 2221. 2275] 23:29] 2384] 2438| 2492 54 9| 2547. 2601| 2655 710 764] 2818] 2873| 2927| 2981| 3036 54 800|903090|903.144|903109,903.253|903307|90336||903416|903470 |903524|903578 54 1. 3633, 3687| 3741 || 3795| 3849| 3904|| 3958) 4012| 4066|| 4120. 54 2| 417 || 4229| 4283| 4337| 4391 || 4445. 4499 || 4553| 4607 || 4661 54 3| 4716; 4770|| 4824|| 4878; 4932| 4986|| 5040|| 5094, 5148} 5202, 54 4| 5256 5310|| 5364|| 5418| 5472 5526|| 5580: 5634, 5688 5742. 54 5: 5796; 5850, 5904 5958| 6012| 6066 6119 6173 6227| 6281 54 6| 6335| 6389| 6443. 6497| 6551, 6604| 6658| 6719| 6766|| 6820, 54 7| 6874| 6927. 6981| 7035| 7089 7143| 7106 7250| 7304| 7358 54 8| 7411 || 7465| 7519| 7573| 7626|| 7680 7734|| 7787| 7841 || 7895| 54 9| 7949, 8002) 8056; 8110| 8163; 82.17| 8270|| 8324 8378; 843 I 54 810|908485||908539|908592}908646}908699|908753|908807908860908014 |908967| 54 I 9021; 9074| 9128| 9181| 0235 9289| 9349| 9396|| 9449| 9503 54 2| 9556| 9610| 9663| 97.16|| 9770| 9823| 98.77| 9930| 9984|9100.37| 53 3|9100.91 |910144|910197910251 |910304|910.358|910411|9,10464|910518) 057 || 53 4| 0624| 0678; 0731 ()784| 0838 08:) I | 0944|| 0998| 1051 | 1104| 53 5| 1158. 1211| 1264|| 1317| 1371 || 1424|| 1477|| 1530| 1584| 1637 53 6|| 1690|| 1743| 1797] 1850|| 1903|| 1956| 2009| 2063| 2 | 16| 2169 53 7| 2222 2275|| 2328|| 2381 2435| 2488| 2541 2594; 2647; 2700|| 53 8: 2753| 2806| 2859 2913. 2966: 3019. 3072 3195| 3178| 3231i 53 9| 3284| 3337| 3390) 3443 3496 3549| 3602| 3655| 3708| 3761| 53 No. 0 I 2 3 4 || 5 || 6 7 8 || 9 || Diff. LOGARITIIMS OF NUMBERS. No. 82 0. 0. 913814 4343 4872 5400 5927 6454 6980 7506 8030 8555 H 91.3867 4396 4925 5453 598() 840 85 8 7 9 19078 960] 920123 0645 II66 1686 2206 2725 3244 3762 924279 4796 53I2 ° 5828 6342 6857 7370 7883 8396 8908 9294.19 934498 5003 5507 60II 65H4 7016 75.18 8039 8520 9020 939519 940 () 18 {}5:6 -8 919130 9653 920176 0697 1218 I738 2258 2777 3296 3814 92433] 4848 5364 5879 6394 6908 7422 7935 8447 8959 929470 998.I 930491 1000 1509 2017 2524 303 I 3.538 4044 934549 5054 6061 939569 949068 §§§ 2 913920 4449 4977 5505 6(333 6559 7085 761 1 8135 8659 919183 97()6 920228 0749 1270 1790 231() 2829 3348 3865 924,383 4899 5415 .593] 6445 6959 7473 7986 8498 9010 92952; 930032 05:42 |(}51 1560 2068 2575 3082 3589 4094 93.4599 4 93.96:19 940] [8 06H6 III.4 I6il 2107 2603 3099 3593 4088 2 3 913973 45()2 5030 5558 6085 6612 7138 7663 8188 8712 919235 3917 924434 4951 92.9572 93(3083 0592 934650 5154 5658 6.162 6665 71.67 7668 8169 867() 9170 939(569 940 168 0666 4. 9 (40.26 4555 5083 561 I 6138 6664 7190 7716 8240 8764 919.287 9810 920332 0853 1374 1894 929623 930.134 0643 I 153 I661 2169 2677 3183 3690 4195 934700 5205 87.20 9.220 939.719 940.218 07:6 5 914079 4608 51.36 5664 6 (91 6717 7243 7768 8.293 8816 910340 98.62 0906 1426 1946 2466 2985 929674 930.185 0694 1204 1712 2220 2727 3234 3740 4246 93475.1 355 9270 939769 94(;267 ()765 920384|9 6 Q}4 132 4660 5189 5716 6243 6770 7295 7820 8345 8869 919392 9297.25 930236 0745 1254 I763 2271 2778 3285 3791 4296 934801 5306 5809 6313 6815 7317 78.19 8320 8820 9.320 939819 940317 08:15 1313 I809 2306 280i 32:37 379 | 4285 6 | 7 914.184 3 9967 924641 5157 5673 6.188 6702 7216 934852 5356 5860 6363 6865 939869 940367 | | 8 9] 423? 4786 5294 5822 6349 924693 5209 57.25 624() 6754 7268 7781 8.293 8805 93.17 92.98.27 930.338 0847 1356 1865 2372 2879 3386 3892 4397 934902 939918 940417 09:15 1412 I909 2405 2001 3396 3890 4384 i 8 §§6 Q19549 2674 3.192 371() 49.28 92.4744 5261 5776 6291 1915 2423 29.30 3437 3943 4448 93.4953 5457 5960 6463 93.9968 940467 0964 1462 1958 2455 2950 3445 3939 4433 | 9 194 LOGARITIIMS OF NUMBERS. O 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || Diff. 944.4831944532}944581 |944631 19446801944729|944779}9448.281944877|944927| 49. 4976|| 5025 5074| 5124| 5173| 5222. 5272 5321| 5370| 5419' 49 5469| 5518, 5567| 5616| 5665 5715 5764| 5813| 5862. 5912| 49 6059: 6108| 6157| 6207| 6256} 6305| 6354|| 6403|| 49 645S 650 || 6551| 6600| 6649, 6698; 6747 796|| 6845| 6894| 49 6943| 6992; 7041 7090| 7140|| 7189| 7238; 7287| 7336|| 7385| 49 7434 7483| 7532 7581| 7630| 7679| 7728. 7777. , 7826| 7875|| 49 7924 || 7973| 8022 8070| 8] 19| 8168| 8217| 8266 8315| 8364 49 8413 8462| 851 1| 8560| 8609| 8657 8706| 8755| '8804| 8853| 49 8902 | 8951; 8999 9048| 9097 || 9146|| 9,195| 9244|| 9292 934}| 49 949390|949439|949488|949536}949585;949634|949683|94973] |949780|9498.29| 49 98.78 9926 9975||950024.9500.73|950.121|950.17()|950219|950267|9503.16|| 49 950365||950414|950.462 0511 || 0560|| 0608; 0657 0706| 0754 0803|| 49 0851 || 0900| 0949| 0997.| 1046; 1095| II.43| 1192] 1240. I289| 49 1338|| 1386|| 1435. 1483| 1532| 1580|| 1629, 1677|| 1796|| 1775|| 49 1823, 1872; 1920, 1969| 2017| 2066 2114| 2163| 2211 226()| 48 2308; 2356 2405] 2453| 2502) 2550) 2599| 2647. 2696| 2744|| 48 2792| 2841 2889. 2938. 2986|| 3034, 3083| 3131|| 3180|| 3228| 48 3276. 3325|| 3373| 3421 || 3470 3518, 3566|| 3615 3663 3711 || 48 3760 3808) 3856. 3905| 3953. 4001 || 4049| 4098) 4,146|| 4194| 48 95.4243|954.29.1954339|954387|954435,954484|954532.954580|954628|954677 48 4725, 4773| 4821. 4869| 4918, 4966|| 5014 || 5062| 51.10|| 5158|| 48 5207| 5.255, 5303| 5351. 5399| 5447| 5495| 5543 5592 5640|| 48 5688 5736' 5784. 5832 5880 5928|| 5976|| 6024| 6072 6120 48 6168| 6216 6265| 6313| 6361| 6409| 6457| 6505 6553| 6601| 48 6640) 6607| BT45 6703 fiğdſ, 6888| 6036|| 6081| 7032| 7080| 48 7128, 7.176|| 7224| 7272| 7320|| 7368| 7416| 7464 7512| 7559| 48 7607 || 7655| 7703; 7751. 7799| 7847| 7894. 7942, 7990, 8038|| 48 8086|| 8134; 8181| 8229| 8277| 8325| 8373| 8421 | 8468| 85.16|| 48 8564| 8612, 8659| 8707| 8755| 8803| 8850. 8898| 8946 S994| 48 959041|959089,959137959185/959232|959280959328|959375||959423|95947.1 || 48 9518| 9566) 9614. 9661 | 9709| 9757| 9804. 9852| 9000| 99.47| 48 9995||960049.960090,960.138|960185.960233|960.281 |960328|960376'960423| 48 96047] | 0518; 0566 0613| 0661 0709| 0756|| 0804 (851; 0899| 48 0946 0994| 1041| 1089| 1136|| 1184 1231 1279| 1326|| 1374| 48 1421, 1469] 1516, 1563| 1611 | 1658. 1706| 1753, 1801 | 1848| 47 1895| 1943| 1990; 2038 2085| 2132| 2180 2227| 2275] 23:22| 47 2369| 2417| 2464] 2511 2559 . 2606. 2653| 2701 || 2748| 2795, 47 2843; 2890; 2937; 2985| 3032: 3079| 3126|| 3174; 3221 3268| 47 33.16|| 3363. 3410 3457| 3504| 3532 3599 3646. 36.93| 3741| 47 963788|963835||963882,963929|963977|9640249640719641 18|964.165||964.212| 47 4260| 4307| 4354 4401 4448. 4495 4542} 4590| 4637; 4684| 47 778 4825 72| 4919| 4966. 5013. 5061| 5108|| 5155| 47 5202| 5249, 5296 5343| 5399| 5437 5484. 5531|| 5578|s 5625) 47 5672, 57.19| 5766 5813| 5860|| 5907| 5954 6001| 6048. 6095| 47 6189| 6236 6283| 6329| 6376| 6423| 647()| 65|17| 6564| 47 6611| 6658 6705 6752 6799) 6845 6892| 6939| 6986| 7033| 47 7080 7127 7.173| 7220 72.7| 7314|| 7361| 7408 7454. 7501 47 7548| 7595| 7642 7688 7735| 7782 7829| 7875|| 7922 7969|| 47 80.16|| 8062 8109| 8156| 8203 8249| 8296 8343| 8390. 8436|| 47 968483|968530|968576|968623968670|968716|968763|968810|968856|968903|| 47 8950. 8996. 9043, 9090. 9136|| 9183| 92.29 9276 93.23| 9369| 47 9416| 9463| 9509| 9556; 9602; 9649. 96.95, 9742| 9789| 9835; 47 9882| 9928| 9975||97002 ||970068|970] 14|970.161 |970207|970.254|970300|| 47 97.0347|970393;970440 0486 || 0533| 0579| 0626|| 0672 0719 || 0765| 46 08.12 0858 0904 0951| 0997 || 1044|| 1090 l 137| 1183| 1229| 46 1276|| 13:22| 1369| 1415 1461 | 1508; 1554|| 1601 | 1647; 1693| 46 I740|| 1786, 1832, 1879| 1925, 1971| 2018. 2064| 21 10 2157| 46 2203| 2249| 2295] 2342 2388| 2434; 2481] 2527; 2573| 2619) 46 2666] 2712) 2758. 2804, 2851] 2897. 2943| 2989 3035) 3082 46 N 5 9 6 l 6 () l () 89 9 0 9 1 -9. 64 l7 423 l 9 3 N Qe o 1 i < | 3 || 4 || 5 || 6 || 7 || 8 || 9 | Did I, O GAR ITEIMS OF WUMBERS. No. | 0 940;973.128 077724 8181 8637 9093 95.48 1 973}74 36.36 4097 4558 50.18 5478 5937 6.306 6854 7312 Q77769 8226 8683 980003 0.458 0912 1366 1819 982271 969 () 9 | 7219 7666 8113 8559 9005 9450 9895 8|990339 9| 0783 980|991:226 1669 4757 5196 9956.35 6074 6512 990 986772 980049 0503 0957 1411 1864 982.316 2769 322() * | 2 97.3220 3682 4143 977815 8272 8728 9|84 96.39 980094 0549 1003 1456 I909 | 3 97.3266 8 7403 977861 8317 1954 98.2362 2814 3265 3716 4167 4617 986861 64 7.309 77.56 8202 8648 9094 9539 9983 9 3|990428 0871 99.1315 I758 2200 2642 3083 3524 3965 4405 4845 5284 082407 2859 3310 3762 4212 4662 5112 5561 6010 6458 986906 7.353 7800 8247 8693 91.38 583 900028 ()472 09]6 99.1359 I802 2244 2686 3127 3568 4009 4449 4889 5328 99.5767 6205 6643 7080 7517 7954 8390 8826 926] 9696 | 4 | 97.3313 774 4235 7449 77906 8363 88.19 9275 97.30 980185 0640 I093 1547 2000 9824.52 2904 3356 4257 4707 5157 5606 6055 6503 986.951 7398 7845 829 | 8.737 9183 96.28 990()72 0516 0960 99.1403 1846 2288 730 3172 3613 4053 4493 4933 5372 3807. 5 973359 3820 4281 4742 5202 5662 612I 6579 7037. 7495 97.7952 8409 8865 93.21 97.76 980231 0685 I 139 1592 2045 982497 2949 4752 5202 565? 6100 6548 986996 7443 7890 8336 8782 9227 96.79 99.0117 056] 1004 99.1448 I890 2333 4097 4537 4977 5416 995854 º: 6731 7 168 7605 804] 8477 8913 9348 9783 6 973405 3866 4327 4788 5248 707 97.7998 8454 8911 9366 98.21 98U2/{j 730 II84 I637 2090 98.2543 2994 3446 3897 4347 797 5247 56.96 6144 6593 98.7040 7488 7934 8381 8826 9272 97.17 990 16ſ 0605 1049 991492 I935 2377 28.19 326() 3701 4141 4581 502I 5460 995898 6337 6774 7212 7648 8085 859] 8956]. 9392 9826 7 973451 39.13 4374 4834 5294 5753 6212 667] 7 129 7586 9780.43 850ſ) 8956 9412 9867 980.322 0776 I229 1683 2135 98.2588 () 98.7085 7532 991536 1979 5065 5504 995942 638() 8 973497 3959 4420 978089 8546 9002 94.57 9912 980367 0821 1275 1728 2181 98.2633 3085 3536 3987 4437 987130 7577 8024 8470 8916 9361 9806 4229 4669 5108 5547 99.5986 6424 6862 6304 6763 7220 7678 978135 8591 9047 9503 9958 1320 1773 2226 982678 3130 990294 0.738 1182 99.1625 559] 996030| 6468 6906 7343 7779 8216 8652 9087 9522 99.57 N | 3 5 | 6 9 TABLE XIII. LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. - N. B. —THE minutes in the left-hand column of each page, increasing downwards, belong to the degrees at the top-; and those increasing upwards, in the right-hand column, belong to the degrees below. - g 198 (0 Degree.) LOGARITIIMIC SINES, COSIVES, ETO. M. Sine | D. Cosine D. Tang. I D. | Cotang. | () || 0-000000 I0-000000 0.000000 Infiniſe. 60 | | 6’463726 501717 000000 00 | 6’463726 50.1717 ||3:536274 59 2 764756 293485 000000 0ſ) 764756 29.3483 235.244 58 3 940847 | 208231 000000 00 940847 208231 059.153 57 4 || 7-065786 I (51517 00000() 00 7.065786 I61517 | 12.93.4214 56 5 J62696 131968 000000 00 162696 131969 837304 55 6 24 1877 I | 1575 9-999999 0.1 24 1878 I | 1578 758122 54 7 308824 96653 99.9999 0.1 308.825 99653 69] 175 53 8 366816 85.254 999999 01 366817 852.54 633 183 52 9 417968 76263 999999 01 4.17970 76263 58.2030 51 10 46.3725 68988. 999998 01 463727 68.988 536273 .50 II || 7-5051H8 6298] 9-999998 01 || 7-505120 6298; 12.404880 49 I2 542906 57936 999997 01 54.2909 57.933 457(991 - || 48 13 577668 53641 999997 0] 577672 53642 422328 47 I4 609853 49938 99.9996 01 609857 499.39 3:)() |43 46 15 639816 467 14 99.9996 ()] 630820 46715 36() 180 45 16 667845 43881 99.9995 01 667849 43882 332/51 44 I? 694 IT3 41372 99.9995 01 694 IT 9 41373 305821 43 18 7 18997 39 135 999994 ()1 719003 39136 280997 42 I9 742477 37127 999993 01 742484 37128 257516 41 : 20 |, 764754 || 35315 999993 01 || 764761 35.136 235239 40 21 7-785943 33672 9-909992 01 || 7-785951 33673 12.214049 39 22 806146 32]75 99999.1 01 806,155 32176 193845 38 23 82545.1 30805 999990 01 825460 30806 I74540 37 24 843934 29547 99.9989 02 843944 29549 156056 36 25 86 (662 28388 999988 ()2 86.1674 28390 I38326 35 26 878695 27.317 999988 02 878708 273.18 121292 34 7 89.5085 $263:23–- |. -999987 - || 02 || - 895099 . . 26325 104901 33 28 910879 25.399 999986 02 910894 25401 089106 32 29 926]. 19 24538 99.9985 02 926 134 24540 07:3866 31 30 940842 23733 99.9983 02 940858 23735 059 142 30 31 7.955082 22980 9.999982 || 02 || 7-955100 22981 12-044900 29 . 32 || 96.887() 22273 99.998 I ()2 || '968889 22275 03 || || 1 28 3. 982233 21608 999980 02 982253 21610 0.17747 27 34 995198 20981 99.9979 02 995219 20983 || 0()4781 26 35 | 8:007787 20390 99.9977 02 || 8:007809 20392 || II-99.2191 25 36 020021 1983 I 99.9976 02 020045 19833 979955 24 37 03.1919 193()2 99.9975 02 031945 I9305 968055 23 38 0435()] I880] 99.9973 02 04:35.27 18803 956473 22 39 05478] I8325 99.9979 02 054809 I8327 945.191 21 40 065776 I787.2 999971 02 065806 17874 934.194 20 41 || 8-076500 17441 9.999969 02 |8-076531 I7444 11-923469 19 42 086965 17031 99.9968 02 086997 I7034 91.3003 18 43 097 183 I6639 99.9966 02 097217 I6642 902783 17 44 107167 16265 99.9964 03 107.202 16268 892797 16 45 I 16926 15908 99.9963 03 I 16963 15910 88.3037 15 46 1264.71 15566 99.996.1 03 126510 15568 873490 14 47 1358]() 15238 99.9959 3 135851 15241 864 149 13 48 I44953 J4924 999958 ſ;3 144996 J4927 85.5004 12 49 J53907 I4692 99.9956 03 153952 J4627 846048 | 1 50 I62681 14333 99.9954 03 1627.27 14336 837.273 10 51 | 8-IT 1280 14054 || 9999952 03 || 8-171328 14057 11-828672 9 52 79713 I3786 99.9950 03 179763 13790 820237 8 53 187985 13529 99.9948 03 188036 13532 81 1964 7 54 196102 13280 99.9946 ()3 1961.56 I3284 803844 6 55 204070 I3()4 I 99.9944 03 204 126 13044 795874 5 56 21 1895 1281() 999942 04 21 j953 12814 788047 4 7 219581 I2587 99.9940 04 2 1964 [. 19590 780359 3 58 227 134 I2372 99.9938 04 227 195 I2376 772805 2 59 234557 12164 99.9936 04 234621 12168 765379 1 60 24.1855 l 1963 99.9934 04 24.1921 II967 758079 0 | Cosine l Sine | Cotang. | Tang. | M. 89 Degrees, 10G. ARITIIMIC, SINES, COSINES, ETC. (1 Degree.) 199 Tang. I 88 Degrees. M. Sine | D. Cosine | D. D. | Cotang. 0 8:24.1855 11063 |9-999934 04 || 8-241921 I 1967 II-758079 60 1 2496.33 I 1768 99.9932 04 249102 I lºº? 750898 59 2 256094 I 1580 999929 04 256165 | 1584 7438.35 58 3 263042 11398 900927 Q4 26.3] 15 ] I 402 736885 57 4 269881 11221 999925 04 269956 11225 7300.44 56 5 276614 11()50 999922 04 276691 11054 723309 55 6 283243 10883 99.9920 04 283323 I0887 716677 54 7 289773 10721 9999 ||8 04 289856 107.26 710}44 53 8 296207 10565 99.99.15 04 296292 I()570 703708 52 9 302546 10413 9999.13 04 302634 10418 697.366 51 10 308794 10266 9999.10 04 308.884 I0270 691116 50 11 8-314954 | 101.22 ||9-999907. 04 || 8-315046 | 101.26 11.684954 49 12 321027 99.82 999905 04 32] 122 99.87 678878 48 I3 327016 9847 99990.2 04 327 I I4 9851 672886 47 14 332924 9714 999899 05 333025 97.19 666.975 46 15 338753 9586 99.9897 05 338856 9590 661 || 44 45 I6 344504 946() 9998.94 05 344610 9465 655390 44 I7 350,181 9338 99.9891 05 350.289 9.343 649711 43 18 355.783 92.19 999888 05 355895 9224 644105 42 19 361315 9.103 999885 05 361430 9108 638570 41 20 | 366777 8990 999882 05 366895 8995 633105 40 21 8:37:2171 8880 ||9-999879 05 || 8:37.2292 8885 11-627708 39 22 377499 8772 99.9876 05 377622 8777 622378 38 23 382762 8667 99.9873 05 382889 8672 617] II 37 24 7962 8564 99.9870 05 388092 8570 61 1908 36 25 393101 8464 999867 05 393234 8470 606766 35 26 398ITQ 8366 999864 05 3983.15 8371 60.1685 34 27 403199 271 999861 05 403.338 8276 59666.2 33 28 408161 8177 99.9858 05 408,304 8182 59 1696. 32 29 413068 8086 99.9854 05 413213 8091 586787 31 30 417919? 7996 99.9851 06 4.18068 8002 581932 30 31 8-49.2717 7909 || 9-999848 06 || 8-422869 7914 |II-577131 29 32 427462 7823 999844 06 4276.18 7830 572382 28 33 432I56 7740 99.984] 06 432315 7745 567685 27 34 436800 7657 999838 06 436962 7663 56.3038 26 35 44.1394 7577 999834 06 44.1560 7583 558440 25 36 445941 7499 99983] 06 446110 7505 553890 24 37 450440 7422 9998.27 06 450613 7428 549387 23 38 454893 7346 9998.23 06 455070 7352 544930 22 39 459301 7273 9998:20 06 459481 7279 540519 21 40 463665 7200 9998I6 06 463849 7206 536151 + 20 41 8.467985 7129 9.999812 06 || 8-468,172 7135 |II-531828 19 42 472263 7060 999809 06 4724.54 7066 527546 18 43 76498 6991 999805 06 476693 6998 523.307 17 44 480693 6924 999801 06 || 480892 6931 519108 16 45 484848 6859 99.9797 07 485050 6865 514950 15 46 488963 794 99.9793 07 489.170 6801 510830 14 47 493040 673H 99.9790 07 493250 6.738 506750 13 48 497078 6669 99.9786 07 497.293 6676 502707 12 49 501080 6608 99.9782 07 50.1298. 6615 4987()2 11 50 505045 6548 999778 07 505267. 6555 494733 10 51 | 8:508974 6489 |9-999774 07 || 8-500200 6496 |II-490800 9 52 512867 6431 999769 07 513098 6439 486.902 8 53 516726 6375 999765 07 516961 6382 483()39 7 54 520551 6319 999761 07 520700 6326 4792] () 6 . 55 524343 6264 99.9757 ()7 524586 6272 475414 5 56 528102 62] I 989753 ()7 528349 6218 47 1651 4 57 531828 6158 99.9748 ()7 532080 6165 467920 3 58 535523 6106 99.9744 07 535779 61.13 464221 2 59 539 186 6055 99.9740 ()7 539.447 6062 460553 I 60 542819 6004 999735 07 543084 6012 4569 16 () | Cosine | | Sine | | Cotang. I | Tang, M. 200 (2 Degrees.) ZOGARITIIMIC, SINES, COSINES, ETC. M. Sine | D. | Cosine | D. Tang. D. Cotang. 0 8°54′2819 6004 9-999735 07 || 8'54.3084 6012 | 11 .456916 60 1. 546492 5955 9997.31 07 546(591 5962 453309. 59 2 549.995 5906 9997.26 07 550268 5914 4497.32 58 3 553539 5858 99.9722 08 553817 5866 446183 57 4 557054 5811 Q997.17 08 557.336 5819 , 442664 56 5 560540 5765 9997 13 08 560823 5773 439172 55 6 563999 5719 99.9708 08 56429 i 57.27 435709 54 7 56743} 5074 99.0704 08 567727 5682 432273 53 8 570836 5630 999699 08 71137 5638 428863 52 9 57.4914 5587 - 9996.94 08 574520 5595 425480 51 I () 577566 5544 99.9689 08 577877 5552 422123 50 I 1 || 8-580892 5.02 9.990685 08 || 8:58.1208 5510 || 11-418792 49 12 584193 5460 999680 08 584.514 5468 415486 48 I3 587469 5419 999675 08 587795 5427 412205 47 14 5907:21 537 9996.70 || - 08 591051 5387 408949 46 15 593948 5339 999665 08 594283 5347 405717 45 I6 597 152 5300 999660 08 597492 5308 402508 44 I7 600332 5261 999655 08 600677 5270 399323 43 18 603489 5223 999650 08 603839 5232 396.161 42 19 606623 5186 999645 09 606978 5.194 39.3022 41 20 609734 5149 999640 09 610094 5:58 38.9906 40 21 8-612823 5112 9.999635 09 |8-613189 5:21 |II.386811 39 22 615891 5076 999629 09 || 616262 5085 3837.38 . 38 23 618937 504] 999624 09 619.313 5(;50 380087 37 24 62.1962 5006 9996.19 09 63.23.43 5()15 377657 36 25 624965 4072 9996]4 09 625.352 4981 37.4648 35 26 627948 4038 QQQ6ſº (10 698?40 4047 371600 | {}4 27, 630911 4904 999603 09 631308 49]3 368692 33 28 633854 4871 990597 09 634.256 4880 365744 32 29 636776 4839 999592 09 637184 4848 362816 31 3ſ) 639680 4806 990586 09 640003 4816 359907 30 31 8:642563 4775 9.999581 09 || 8-642082 4784 || 11-357018 29 32 (545428 4743 99.95.75 09 645853 475.3 354147 28 33 648.274 4712 909570 09 648704 4722 35;Q96 27 34 65] 102 4682 999.564 09 65.1537 4691 348463 26. 35 6539]] 4652 999558 I0 654352 4661. 345648 25 36 656702 4622 900553 I0 657.149 4631 34285.1 24 37 6594.75 4502 900547 10 659928 4602 34007:2 23 38 662230 4563 99.9541 10 662689 4573 3.37311 22 30 664968 4535 990535 I0 665433 4544 334567 21 40 667689 4506 99.9529 10 668160 4526 33 1840 20 41 8:670393 4479 9.999524 10 || 8:670870 4488 || 11-320,130 19 42 ($73080 4451 999518 10 673563 4461 326437 18 43 67575] 4424 909512 10 76239 4434 323761 17 44 678405 4397 990506 10 678000 44.17 32II()0 16 45 681043 4370 999.500 10 68.1544 4380 3.18456 15 46 683665 4344 999493 10 68417.2 4354 315828 14 47 6862.72 4318 909487 10 686784 4328 313216 13 48 688863 4292 999481 10 689381 4303 310619 12 49 691438 4267 99.94.75 I0 691963 4277 308037 11 50 693998 4242 999.469 I0 694529 4252 305471 J0 51 |8-696543 4217 | 9.999463 I1 8-697081 4228 11-302019 52 699073 4.192 99.9456 11 6996.17 4203 300.383 53 701589 4,168 99.9450 11 702139 4179 297861 54 704090 4,144 909443 11 704646 4155 295.354 55 706577 4121 990437 11 707 140 4132 292.860 56 700049 4097 999.431 I 1 709618 4 IO8 200382 57 7I1507 4074 999424 11 712083 4085 2879.17 58 713952 4051 909418 11 714534 4062 285.465 59 716383 4029 9994 II 11 716972 4040 283028 l 60 7:8800 4006 999.404 11 7 19396 4017 280604 | Cosine l | Sine | Cotang, | Tang. M. 87 Degrees. LOGARITIIMIC, SIVES, COSINES, ETC. (3 Degrees.) 201 M. Sine ! D. : Cosine | D. Tang. D. 1 Cotang. I 0 |8-718800 4006 || 9.999404 I1 8-7 19396 4017 | 11-280604 i 60 l 721204 3984 99.9398 11 721806 3995 278.194 || 59 2 723595 3962 99.9391 11 724204 3974 275796 58 3 725972 3941 999384 11 726.588 3952 2734 12 57 4 72S337 3919 999378 11 728959 3930 27 1041 56 5 730688 3898 99.9371 11 | 731317 3909 268683 || 55 6 733027 3877 99.9364 12 733663 3889 266337 || 54 7 735,354 3857 99.9357 12 7.35996 3868 264004 || 53 8 737667 3836 999.350 12 738317 3848 26.1683 || 52 9 739960 3816 999343 12 740626 3827 259374 || 51 I0 742259 3796 999336 12 742922 3807 25.7078 50 11 || 8-744536 3776 9-999329 12 || 8-745207 3787 | 11.254793 || 49 12 746802 3756 999322 12 747479 3768 252521 || 48 13 740055 3737 99.9315 12 749740 3749 250260 47 14 751907 3717 999.308 12 |. 751989 3729 248011 || 46 15 '753528 3698 999301 12 754.227 3710 245773 || 45 I6 755747 3679 999.294 I2 756453 3692 243547 || 44 17 757955 3661 999286 12 758668 3673 24.1332 || 43 18 760,151 3642 999279 12 760872 3655 239128 || 42 19 762337 3624 999272 12 76.3065 36.36 236935 || 41 20 764511 3606 999265 I2 765246 3618 234.754 || 40 21 ; 8.766675 3588 9-999257 12 || 8.7074.17 3600 || 11.232583 || 39 22 7688.28 3570 99.0250 13 76,057 3583 2304.22 || 38 23 770970 3,553 999.242 13 771727 3565 228273 || 37 24 773.101 3535 900235 I3 77.3866 3548 226134 36 25 775223 35.18 99.0227 I3 775995 3531 224005 || 35 26 777333 3501 999.220 I3 778114 3514 221886 || 34 27 779434 3484 9992.12 13 780.222 3497 219778 || 33 28 78.1524 3467 990.205 13 782320 3480 217680 || 32 29 78.3605 3451 999.197 13 784408 3464 215592 || 31 30 785675 3431 99.9189 I3 786486 3447 213514 || 30 31 | 8.787736 3418 9-999181 I3 || 8-788554 3431 | 1.1.211446 29 32 789787 3402 099.174 I3 790613 3414 209387 28 33 791828 3386 999166 13 79.2662 3399 207338 27 34 793859 337 999,158 I3 794.701 3383 205.209 || 26 35 79.5881 3354 999150 13 796731 3368 203269 || 25 36 797894 3339 999.142 13 798752 3352 201248 || 24 37 799897 3323 999.134 13 800763 3337 1992.37 23 38 801892 3308 999.126 13 802765 3322 I97235 22 39 803876 3203 999]18 13 804758 3307 195249 || 21 40 805852 3278 999.110 13 806742 3292 193258 20 41 8-80.7819 3263 || 9.099/02 13 |8-8087.17 || 327 II.19.1283 19 42 809777 3249 99.9094 14 810683 3262 180317 | 18 43 811726 3234 99.9086 14 812641 3248 187359 || 17 44 813667 3219 900077 14 81.4589 3233 1854.11 || 16 45 8.15599 3205 990069 14 816529 32.19 183471 || 15 46 817522 3.191 990061 14 818461 3205 181539 || 14 47 | . 810436 3177 9900.53 14 820384 3.191 1796.16 || 13 48 821343 3.163 999044 14 822.298 3177 I77702 || 12 49 823240 3149 999.036 14 824205 316.3 I75795 11 50 825,130 3135 990027 14 826103 3150 173897 || 10 51 8-827011 3122 || 9-900019 14 || 8-827992 3.136 11-172008 9 52 828884 3.108 999010 14 829874 3.123 I70I26 8 53 830749 3695 900002 14 831748 31.10 168252 7 54 832607 3082 99.8003 14 833613 3096 j66387 6 55 834456 3069 998.984 14 835471 3083 164529 5 56 836297 3056 998.076 14 837321 3070 I62679 4 57 838.130 3043 99.8967 15 830163 3057 160837 3 58 839956 3030 99S958 15 840998 3045 I59002 2 59 84.1774 30.17 998.050 15 84.2825 3032 1571.75 1 60 843585 3000 99.8941 15 844644 3019 155356 0 | Cosme J | Sine -l | Cotang. ". | Tang. M. 86 Degrees. 202 (4 Degrees.) LOGARITIIMIC, SINES, COSINES, ETC. M. Sine | D. Cosine | D. Tang. D. Cotang. I - 0 || 8-843585 3005 || 9-998.941 15 8-844644 3019 || II* 155356 || 60 I 845387 2992 998.932 I5 846455 3007 I53545 59 2 847 183 2980 99.8923 15 848260 2005 I517.40 58 3 848971 2967 998.914 15 8500.57 2982 149943 57 4 850751 2955 998.905 15 851846 2970 148154 56 5 85.2525 24)43 908896 15 85.3628 2958 ] 46372 55 6 854291 2931 998887 I5 8.55403 2.946 144597 || 54 7 856049 29:19 998878 15 857171 2935 142829 || 53 8 857801 2007 998869 15 8589.32 2923 14 ()68 52 9 859546 2896 998860 15 860686 29] I I39.314 || 51 J0 861283 2884 99885.1 15 862.433 2900 137567 50 11 8-8630.14 2873 || 9-908841 15 |8-864.173 2888 || 11-135827 49 12 864738 2861 998832 15 865906 2877 I34(){}4 || 48 13 866455 2850 908823 16 867632 2866 132368 47 14 868165 2839 99.8813 16 869351 2854 130649 46 15 869868 2828 99.8804 I6 871064 284.3 IQ8936 || 45 16 87.1565 2817 998795 16 87.2770 2832 197230 44 J7 873255 2806 908785 I6 874469 2821 125531 || 43 18 874938 2795 998.776 16 876.162 2811 123838 42 19 87(5615 2786 998.766 I6 877849 2800 122151 41 20 78.285 773 99.8757 16 79.529 789 190471 || 40 21 | 8-879949 763 || 9-908747 I6 | 8-881902 277 II"] 18798 || 39 22 881607 2752 998.738 16 882869 2768 I 17131 38 23 883258 2742 998.728 16 884530 2758 II54.70 || 37 24 884903 2731 99.8718 16 886.185 747 113815 36 25 886542 2721 998.708 16 8878.33 2737 112167 || 35 . 20 888174 27I1 998699 16 889476 27.27 I 10524 34 2 / §§§§U1 2700 9900ſ]9 10 801 II? Q717 108888 || 33 28 89.1491 2690 998679 j6 892742 2707 I07258. 32 29 89.3035 2680 998669 17 894366 2697 105634 || 31 30 894643 2670 998659 17 805984 2687 104016 || 30 31 || 8.896246 2660 9-998649 17 | 8.897506 2677 II* 102404 || 29 32 897842 2651 998630 . 17 899.203 2667 100707 || 28 33 899.432 2641 9986.29 17 900803 2658 ()9919.7 | 27 34 901017 2631 998619 17 902398 2648 097602 || 26 35 902596 2622 998609 17 90.3987 2038 096013 || 25 36 904I69 2612 998.599 17 905570 2629 094430 24 37 905736 2603 908580 17 907 147 2620 09.2853 || 23 38 90.7297 2593 998.578 J7 9087.19 26.10 09 1281 - || 22 39 908853 2584 998568 17 910285 2601 0897 15 21 40 910.404 2575 998558 17 91 1846 2592 088154 || 20 41 || 8-91 1949 2566 9.998548 17 | 8-913401 2583 || 11-086599 || 19 42 913488 2556 998.537 17 9I4951 2574 085049 | 18 43 9.15022 2547 9085.27 17 916495 2565 0.83505 || 17 44 9.16550 2538 99851C 19 918034 2556 08.1966 || 16 45 918073 2529 998.506 18 919568 2547 080432 15 46 919591 2520 998495 18 921().96 2538 078904 || 14 47 921103 2512 99.8485 18 92.2619 25.30 077381 13 48 92261() 2503 99.8474 18 924,136 2521 075864 12 49 924 119 2494 998.464 18 9.25649 2512 074351 | 11 50 925609 2486 99.8453 18 927.156 2503 0.72844 || 10 51 || 8-927 100 2477 9.998.442 18 8-928658 2495 11-071349 9 52 92.8587 2469 99.8431 || 18 930.155 2486 069845 8 53 930068 2460 99.8421 18 93.1647 2478 068353 7 54 93.1544 2452 998.410 18 933134 2470 066866 6 55 933015 2443 998399 18 934616 2461 065.384 5 56 934481 2435 998.388 18 936093 2453 063907 4 57 935.942 2427 998377 18 937565 2445 062435 3 58 93.7398 2419 998366 18 939.032 2437 060968 2 59 938850 24II 998.355 18 940494 2430 059506 1. 60 940296 2403 998.344 18 94.1952 24.21 058048 0 | Cosine l | Sine | | Cotang. | Tang. | M. 85 Degrees LoGARITHMIC SINES, CoSINES, ETC (5 Degrees.) 203 ~, 84 Degrees. M. Sine | D. | Cosine | D. Tang. D. ( Cotang. I 0 8-940.296 2403 |9.998344 I9 |8-94.1952 24.21 11:058048 60 I 94 1738 2394 - 99.8333 19 943404 24 13 056596 59 2 943.174 2387 998322 19 944852 2405 0.55148 58 3 9446(96 2379 998.311 19 946295 2397 053735 57 4 946034 2371 998.300 19 947734 2390 0.52266 56 5 947456 2363 998.289 19 949 168 2.382 050832 55 6 948874 2355 998.277 19 95()597 2374 049403 54 7 950287 2348 998266 19 952021 2366 047979 53 8 951696 2340 998.255 }9 95.3441 2360 046559 52 9 95.3100 2332 998.243 19 95.4856 2351 045144 51 10 95.4499 2325 998.232 19 956.267 2344 0437.33 50 I 1 || 8.955894 2317 | 9.998.220 19 |8-957674 2337 11.042326 49 12 95.7284 2310 998.209 I9 959075 2329 040925 - || 48 13 958670 2302 | . 998197 19 96()473 2323 0395.27 47 14 96.0053 2295 -998.186 19 96.1866 2314 0.38134 46 15 961429 2288 998.174 19 96.3255 2307 036745 45 16 96.2801 2280 998.163 19 964639 2300 035.361 44 17 964 170 2273 998.151 I9 966019 2.293 033981 43 18 965534 2266 998.139 20 96.7394 2286 03:2606 42 19 966893 2259 998.128 20 968766 2279 031234 41 20 968249 2252 998116 20 97() 133 2271 0.29867 40 21 8.969600 2244 9.998104 20 | 8,971496 2265 | 11:028504 39 22 970947 2238 99.8092 20 97.2855 2257 027145 38 23 97.2289 2231 99.8080 20 974.209 2251 0.25791 37 24 97.3628 2224 99.8068 20 97.5560 2244 024440 36 25 974962 2217 998056 20 97.6906 2237 023094 35 26 976.293 2210 998044 20 978248 2230 0.21752 34 27 9776#9 2203 998.032 20 97.9586 2223 020414 33 28 97894] 2197 99.8020 20 980921 2217 0.19079 32 29 Q80259 2190 99.8008 20 982251 2210 0.17749 31 30 98.1573 2}83 997996 20 98.3577 2204 0.16423 30 31 || 8-982883 2177 || 9-997984 20 | 8'984899 2197 11-015101 29 32 984].89 2170 99.7972 20 || 986.217 2191 ()13783 28 33 985.491 2163 997.959 20 987532 2184 0.12468 27 34 || 986789 2157 997.947 20 988842 21.78 01 || H58 26 35 988083 - || 2150 99.7935 21 990 149 2] 71 || ()09851 25 36 989374 2I44 907022 21 991451 2165 008549 24 37 99.0660 2138 997910 21 992750 2158 007250 23 38 99.1943 2I3] 997897 21 99.4045 215.2 005.955 22 39 993222 2125 99.7885 2] 995.337 2146 004663 21 40 994497 2119 997872 21 99.6624 2140 003376 20 41 8-99.5768 2112 || 9-997860 21 |8-99.7908 2134 || 11:002092 19 42 99.7036 2106 99.7847 21 999 IS8 2127 000812 I8 43 998.299 2100 997.835 21 |9-000465 2121 ||10°9995.35 17 44 999560 2094 997822 21 00.1738 2115 998262 I6 45 ||9-000816 2087 997809 21 003007 2109 99.6993 15 46 002069. 2082 90.7797 21 004:272 2103 995.798 14 47 003318 2076 997784 21 ()055.34 2097 994466 13 48 004563 2070 997771 21 006792 2091 9932()8 12 49 00:5805 2064 997758 21 008047 2085 99.1953 | 1 50 007044 2058 99.7745 21 009298 2080 990702 || 10 51 8:008278 2052 |9-997732 21 9:01.0546 2074 || 10-989.454 9 52 0.095.10 2046 997719 21 0.11790 2068 988210 8 53 01.0737 2040 997706 21 0.13031 2062 986969 7 54 01 1962 2034 997693 22 0.14268 2056 985732 6 55 0.13182 2029 997680 22 0.15502 2051 984.498 5 56 0.14400 2023 997667 22 0.16732 2045 983268 '4 57 015613 2017 997654 22 017959 2040 98.2041 3 58 || 0 16824 2012 997641 22 ()19183 2033 980817 2 59 0.18031 2006 997628 22 020403 2028 795.97 I 60 ! 019235 2000 997614 22 021620 2023 97.8380 0 | Cosine | Sine | | Cotang. •l Tang. M. 204 (6 Degrees.) Loganrººſic SINES, CoSINES, ETC. M. Sine | D. Cosine | D. Tang. D. Cotang. I 0 9-019235 2000 |9-997614 22 ||9-021620 2023 10-978380 60 1 0204:35 1995 997601 22 ()22834 2017 977166 59 2 021632 I989 997.588 22 024044 2011 975.956 58 3 02:28:25 1984 997574 22 {}25251 2006 97.4749 57 4 ()24() || 6 I978 997561 22 0.26455 2000 97.3545 56 5 0.25%)3 1973 99.7547 22 0.27655 1995 972345 55 6 026:386 1967 997.534 23 028852 I990 97] 148 54 7 0.27567 1962 997.520 23 03(){}46 1985 969954 53 , 8 028744. 1957 99.75()7 23 ()3]237 1979 968763 52 9 0.299 18 I951 997.493 23 03:24:25 J974 967575 51 10 03 ſ ()89 1947 997.480 23 033609 1969 966391 50 11 |9-032257 1941 9-997466 23 ||9-0347.91 1964 || 10-965209 49 12 03:34.21 1936 99.7452 23 03:5969 1958 964().31 48 13 034582 J930 99.7439 23 037 || 44 1953 962856 47 14 0.35741 1925 997.425 23 ()38.316 I948 96.1684 46 15 0.36896 I920 99.74 I 23 ()39485 1943 960515 45 16 0.38()48 J915 997.397 23 0.40651 1938 959349 44 17 0.39) 197 j9 () 99.7383 23 04 I813 1933 958].87 43 18 ()4()342 1905 997.369 23 042973 1928 957027 42 . I9 ()4}485 1899 997355 23 044 130 1923 955870 41 20 (94.2625 I894 997.341 23 045284 I918 954,716 40 21 || 0 ()43762 1889 9-997327 || 24 ||9-046434 1913 |10-95.3566 39 22 044895 1884 997.313 24 047582 1908 95.2418 38 23 ()46()26 1879 9972.99 24 0487.27 I903 95 1273 37 24 047154 1875 9972.85 24 0.49869 1898 95013] 36 25 | 048279 1870 997:271 24 05:1008 I893 948992 35 26 0494.00 1865 997257 || 24 052] 44 1889 947856 || 34 2/ | U}{}}) IJ 1860 997242 $24 ()53'277 1884 9467.23 33 28 051635 I855 997228 24 054407 1879 945.593 32 29 05:2749 I850 997214 24 055.535 1874 944.465 31 30 053859 1845 997 199 24 56659 1870 943.34.1 30 31 054966 I841 9-997.185 24 9-057781 1865 | 10:942219 29 32 056071 1836 907170 24 058900 I869 941 100 28 33 057172 1831 997 156 24 0600 16 1855 93.99.84 || 27 34 058271 1827 997 141 24 061130 I851 93887() 26 35 059.367 1822 997127 24 062240 1846 937760 25 36 '06()460 1817 997I12 24 063348 I842 936652 24 37 061551 1813 907098 24 064453 I837 1935,547 23 38 || 06:2639 1808 907083 25 065556 1833 934444 |' 22 39 063724 1804 99.7068 25 066655 1828 83.3345 21 40 ()64806 1799 907053 25 067752 1824 932248 || 20 4] 9:(165885 I794 || 9:907039 25 | 9-068846 1819 10-931154 I9 42 066962 1790 99.7024 25 ()69938 1815 930062 18 43 068()36 j786 997.009 25 07 1027 1810 928973 J7 44 069107 I781 996994 25 072] I3 - 1806 927887 16 45 || 07() 176 1777 996.979 25 0.73197 1802 926803 15 46 || 07 1242 1772 996964. 25 074278 I797 92.5722 14 47 || 07:2306 I768 996.949 25 07:5356 I793 924644 13 48 || 073366 I763 996934 25 076432 I789 923568 12 49 || 074424, I759 996919 25 0775()5 I784 92.2495 11 50 || 07548() 1755 996.904. 25 078576 1780 921424 10 51 9-076533 1750 || 9,996889 25 |9-070644 1776 ||10920356 9 52 || 077583 1746 99.6874 25 08()710 1772 91929() 8 53 ()786.31 I742 996858 25 08:1773 I767 9.18227 7 54 || 0796.76 1738 996843 25 08:2833 1763 917.167 6 55 || 080719 I733 996828 25 083891 1759 9.16109 5 56 081759 1729 996812 26 0.84947 1755 9 15053 4 57 0827)7 I725 996797 26 086000 1751 9.14000 3 58 || 083832 1721 996782 26 087()50 1747 912950 2 59 || 084864 1717 996766 26 088098 I743 9] 1902 I 60 H 085894 I713 996751 26 0891.44 I738 910856 0 * Cosine A | . Sine | | Cotang. Tang. UM. 83 Degrees. LOGARITIIMIC SINES, COSINES, ETC. (7 Degrees.) 205 82 Degrees. M. j. Sine D. Cosine | D. Tang. D. Cotang. 0 ||9-08589; I7]3 9.996751 26 9-089.j44 1738 || 10-910856 60 I {}86922 I709 Q967:35 26 090187 I734 9098.13 59 2 087947 I704 9967:20 26 09 [228 I730 908772 58 3 088070 1700 9967()4 26 09.2266 1727 907734 57 4 080990 I696 996688 26 0933()2 1722 906698 56 5 09 1008 J692 996673 26 094336 1719 905664 55 6 09.2024 1688 99.6657. 26 095367 I715 904633 54 7 093037 I684 99.664] 26 096395 I7 || 1 903605 53 8 094()47 1680 906625 26 ()97492 I707 902578 52 9 095056 I (576 99.661() 26 ()984.46 17()3 90.1554 51 10 096()62 1673 996594 26 099.468 I699 900532 50 11 || 9-097065 1668 || 9,996578 27 | 9' 100487 1695 || 10-8995.13 49 12 098066 1665 996562 27 I() 1504 I691 898496 48 13 090065 I (56 | 996546 27 102.519 }687 897.481 47 14 J{}{}{}62 1657 996530 27 103532 I684 896468 46 15 I () ()56 1653 9965 14 27 1()4542 I680 89.5458 45 I6 1020.48 1649 996498 27 1()5550 1676 894.450 44 17 103037 1645 996482 27 106.556 1672 893444 43 18 104025 I(541 996.465 27 107559 1669 892441 42 19 + I()5010 J638 9964.49 27 J08560 1665 89.1440 41 20 105992 1634 996433 27 109559 1661 89.0441 40 21 || 9' 106973 I630 ||9-9964.17 27 9' 110556 1658 10.8894.44 39 22 107951 1627 9964(jū 27 II I551 1654 888449 38 23 108927 1623 996.384 27 I 12543 1650 887457 37 24 I(\99() I. 1619 996368 27 I 13533 1646 886467 36 25 I 10873 I616 996.351 27 II 4521 1643 885479 35 26 II 1842 1612 996.335 27 I 15507 1639 884493 34 27 II 2809 J608 996318 27 I 16491 I636 88.3509 33 28 II3774 1605 9963()2 28 I 17472 I632 882528 32 29 I 14.737 1601 996285 28 I 18452 1629 881548 3] 30 | 15698 1597 996269 28 119.429 1625 880571 30 31 || 9 || 16656 1594 || 0.996252 28 9' 120404 I622 || 10-879596 29 32 1176 J3 1590 996235 28 121377 1618 78623 28 33 I}8567 1587 9962.19 28 122348 I (315 877652 27 34 IJ9519 I583 996202 28 1233}7 16|| 1 76683 26 35 120460 1580 996.185 28 I24284 1607 875716 25 36 J2 (4.17 1576 996}68 28 125249 I604 874751 24 . 37 I22362 I573 996.151 28 1962] I 1601 873789 23 38 123.306 1569 906134 28 127172 1597 872828 22 39 1242.48 1566 996] 17 28 128130 1594 871870 21 40 I25187 1562 99.6100 28 120087 1591 87()913 20 41 9' 126125 1559 || 9-006083 20 | 9' 130041 1587 | 10:869959 19 42 I27060 1556 996006 29 130994 1584 869006 18 43 127903 1552 906049 29 131944 1581 868056 17 44 |28925 1549 996(332 $20 13289.3 1577 867 I()7 16 45 120854 1545 996015 . $20 1338.39 I57 86616.1 15 46 I30781 I542 995998 29 134784 1571 865216 14 47 131706 1539 99.5980 20 1357.26 1567 864974 13 48 1326.30 1535 995963 29 136667 1564 863333 12 49 133551 1532 995946 29 I37605 1561 862395 11 50 I34470 1529 995928 29 138542 I558 86.1458 10 51 9-135387 1525 | 9.9959] 1 29 9' 139476 I555 ..] HC)'860524 9 52 I36303 152:2 995894 29 I4()409 1551 85959 | 8 53 137916 1519) 995876 29 141340 1548 858660 7 54 I38 198 1516 995859 29 142269 1545 857731 6 55 139037 1512 99584 I 29 143196 I542 856804 5 56 139944 509 9958.23 29 144 121 1539 855879 4 57 140850 1506 995806 29 I45044 1535 854.956 3 58 141754 1503 995788 29 145966 1532 854034 2 59 142655 1500 99.577 I 20 146885 1529 853II; | I 60 143555 1496 99.5753 29 147803 I526 85.2197 0 i Cosine | Sine | | Cotang. | | Tang. | M. 206 (8 Degrees.) LOGARITIIMIC SINES, COSINES, ETC. ‘M. Sine | D. Cosine D. Tang. D. Cotang. I 0 | 9°143555 - || 1496 ||9-99.5753 30 || 9 |47803 1526 IIO,852197 60 I 144453 1493 995.735 30 I48718 I523 85.1282 59 2 145349 j490 995.717 30 149632 1520 850368 58 3 || 146243 }487 995699 30 150544 1517 849456 || 57 4 1471.36 1484 995081 30 151454 1514 8485.46 56 5 148026 I481 995664 30 152363 1511 847637 55 6 148915 1478 995646 30 153269 1508 846731 54 7 1498)2 1475 995628 30 154174 1505 845826 || 53 8 150686 J472 995610 30 155077 1502 844923 52 9 151569 I-469 995591 30 155978 ]499 84.4022 51 10 | 152451 1466 99.5573 30 156877 1496 843,123 || 50 11 9. 153330 1463 |9-99.5555 30 9. 157775 1493 10.842225 49 12 154208 1460 995537 30 I58671 1490 841329 48 13 155083 I457 995519 30 I59565 1487 840435 47 14 I55957 1454 99.5501 31 160457 1484 839.543 46 15 J5683() 1451 995482 31 j61347 1481 838653 45 16 157700 I448 995.464 31 162236 I479 837764 || 44 I7 158569 1445 99.5446 31 163.123 1476 836877 43 18 159435 I442 995427 31 I64008 J473 835992 42 I9 160301 1439 995409 31 164892 1470 835.108 i.41 20 161164 1436 995.390 31 16577 1467 834226 40 21 || 9' 162025 || 1433 || 9'995.372 || 31 9' 166654 || 1464 10.833346 || 39 22 | 162885 1430 995.353 31 J67532 1461 832468 || 38 23 163743 1427 995334 31 I68409 1458 83.1591 37 24 I64600 1424 995.316 31 169284 1455 830716 36 35 165454 1422. 995997 31 I?() 157 1453 82984.3 35 Q6 Iſj6307 14 IQ Q05:278 31 171ſº |,450 $28071 34 27 167159 1446 995:260 31 171899 1447 828101 || 33 28 168008 1413 995241 32 I72767 1444 827233 32 29 I68856 141() 995.222 32 |. 173634 1442 826366 31 30 1697 (J2 . 1407 995203 32 1744.99 1439. 825501 30 31 9-170547 1405 || 9°995.184 32 || 9, 175362 1436 || 10-82.4638 29 32 I7 1380 1402 995165 32 176224 1433 823776 28 33 1722.30 1399 9951.46 32 I77()84 1431 822916 2 34 173070 1396 9951.27 32 177942 1428 822058 26 35 I73908 1394 995108 32 l?8799 1425 8212() I 25 36 174744 1391 995089 32 179655 1423 820.345 24 37 175578 1388 995070 32 180508 1420 819492 || 23 38 176411 I386 995051 32 181360 1417 8.18640 22 39 177242 1383 995032 32 182211 1415 817789 21 40 || 178072 1380 995013 32 183059 1412 816941 | 20 41 9° 178900 1377 9°994.993 32 || 9-183907 I409 || 10-816093 || 19 42 || 1797.26 || 1374 994974 || 32 | 184752 || 1407 815248 || 18 43 180551 I372 99.4955 32 185597 1404 814403 17 44 181374 1369 99.4935 32 1864:39 1402 813561 I6 45 182196 1366 99.4916 33 187280 1399 812720 15 46 183016 1364 99.4896 33 188120 1396 8] 1880 14 47 183834 1361 994877 33 188958 I393 811042 13 48 18465 1359 94)4857 33 1897.94 1391 810206 I2 49 185466 1356 99.4838 33 190629 1389 809371 11 50 186280 1353 994818 33 191462 I386 8085.38 10 51 || 9, 187092 1351 9-994798 33 || 9, 192294 1384 || 10.807706 9 52 187903 1348 99.4779 33 193124 1381 806876 8 53 | 188712 1346 99.4759 33 193953 1379 806047 7 54 | 189519 1343 994.730 33 194780 1376 805220 6 55 | 1903.25 1341 99.4719 33 195606 1374 8()4394 5 56 I91130. 1338 994.700 33 1964.30 T37I 8()3570 4 57 | 191933 1336 994680 33 197253 I369 802747 3 58 192734 1333 99.466() 33 198()74 1366 801926 2 59 || 193534 1330 99.4640 33 1988.94 1364 80.1106 I 60 l 194332 1328 99.4620 33 199713 1361 800287 0 1. Cosine | Sine | | Cotang. I | Tang. | M. 81 Degrees. LOGARITHMIC SINES, COSINES, ETC. (9 Degrees.) 207 80 Degrees. M. : Sine D. | Cosine | D. Tang. D. Cotang. { 0 9-194332 I328 9-994620 33 || 9, 199713 1361 |10-800287 60 I 195199 1326 99.4600 33 200529 1359 799471 || 59 2 195925 I323 99.4580 33 "| 201345 1356 798655 58 3 : 196719 1321 99.4560 34 2021.59 I354 79.7841 57 4 || 197511 I318 99.4540 34 202971 I352 797(329 56 5 | 198302 1316 9945.19 34 203782 1349 796218 55 6 1990.91 1313 994499 34 204592 1347 795.408 54 7 | 199879 1311 994479 34 205400 I345 794600 || 53 8 200666 I308 994459 34 206207 1342 793793 52 9 | 201451 I306 994438 34 207013. 1340 79.2987 || 51 10 || 202234 1304 9944.18 34 207817 1338 79.2183 || 50 I1 ; 9:203017 J301 | 9.994397 34 ||9-208619 I335 |10-70.1381 - || 49 12 || 203797 1299 99.4377 34 209420 1333 790580 || 48 I3 i 204577 1296 994.357 34 210220 1331 789780 47 14 || 205354 1294 99.4336 34 21 1018 1328 788.982 46 I5 206131 I292 99.4316 34 211815 1326 788185 45 I6 || 206906 1289 994.295 - 34 212611 1324 787389 || 44 17 | 2076.79 I287 99.4274 35 213405 1321 786595 || 43 18 2084.52 1285 99.4254 35 214198 1319 785802 || 42 19 || 209222 1282 994233 35 214989 1317 785011 || 41 20 209992 1280 994212 35 215780 1315 784220 | 40 21 || 9:210760 1278 |9.994.191 35 | 9:216568 1312 |10.783432 39 22 || 21 1526 1975 994ITI 35 217356 1310 78.2644 38 23 212291 I273 99.4150 35 218142 1308 781858 || 37 24 || 213055 1271 994.129 35 218926 1305 78.1074 || 36 25 || 213818 1268 994.108 35 219710 1303 780290 35 26 214579 1266 99.4087 35 2204.92 1301 779508 34 27 215338 1264 99.4066 - || 35 221272 1299 7787.28 33 28 216097 I961 994045 35 222052 I297 777948 32 29 216854 1259 994024 35 22.2830 1294 777.170 31 30 || 217609 I257 994003 35 223606 1292 776394 30 31 ||9-2}8363 1255 |9-993.981 35 | 9.224.382 1290 ||10-775618 29 32 219,116 I253 903960 35 225,156 1288 774844 28 33 || 219868 1250 993939 35 225929 1286 774071 27 34 220618 1248 90.39.18 35 226700 1284 773300 26 35 221367 1246 993896 36 2274.71 1281 772529 || 25 36 2221.15 I244 993875 36 228239 1279 77.1761 24 37 || 2:22861 1242 993854 36 229007 1277 770993 || 23 38 22.3606 I239 90.3832 36 229773 1275 770.227 22 39 224349 1237 993811 36 230539 1273 769461 21 40 || 225092. 1235 993789 36 23.1302 I271 768698 || 20 . 41 ||9-2258.33 I233 |9.99.3768 36 9-232065 1269 |10-767935 I9 42 226573 1231 993746 36 232826 1267 767174 18 43 2273.11 1228 90.3725 36 23.3586 1265. 76{j414 17 44 || 2:28048 1226 99.3703 36 234.345 1262 7(55655 16 45 228784 1224 993681 36 235103 1260 764897 15 46 || 229518 I222 993660 36 235859 1258 76414 I | 1 47 || 230.252 1220 993638 36 236614 1256 763386 13 48 || 230984 I218 9936.16 36 237368 1254 762632 12 49 || 231714 1216 99.3594 37 238.120 1252 76.1880 11 50 232444 1214 993:572 37 238872 1250 761128 10 51 9-233.172 1212 9-993550 37 9-239.622 1248 ||10-76()378 9 52 || 233899 .1209 99.3528 37 940371 1246 7596.29 8 53 234625 1207 99.3506 37 241 118 1244 758882 7 54 || 235349 1205 993484 37 241865 1242 758135 6 55 || 236073 1203 993462 37 242610 1240 757.390 5 56 236795 1901 993440 37 243354 1238 756646 4 57 2375.15 1199. 99.34.18 37 244097 1236 755903 3 58 2382.35 I 197 90.3396 37 244839 1934 755161 2 59 || 2:3895.3 1195. 99.3374 37 245579 1232 754421 I 60 239670 1193 993.351 37 246319 1230 75.3681 0 | Cosine |. Sine | | Cotang. | | Tang, M. 208 (10 Degrees.) LOGARITIIMIC, SINES, COSINES, ETC. | Cosine | | Sune | | Cotang. . . | Tang. JM. I. Sine : D. l Cosine D. ( Tang. D. 1 Cotang. I 0 | 9:239670 | 193 l 9'993.351 37 9:246,319 1230 | 10-75°3681 60 I 240386 1191. 993.329 37 247()57 I928 752943 || 59 2 || 24.1101 1189 993.307 37 247794 1226 752206 || 58 3 241814 I RS7 993285 37 24853() 1224 75 [470 57 4 24:25:26 1185 993262 37 249.264 I222 750736 56 5 24,3237 1 183 993240 37 249998 1220 750002 || 55 6 2433)47 I 181 9932.17 38 250730 1218 74927() 54 7 244656 1179 993 195 38 251461 1217 748539 53 8 245.363. 1177 99.3172 38 252191 1215 747800 52 9 246(969 1175 | 99.3149 38 2529:20 1213 747080 51 10 240775 1173 99.3127 38 25.3648 1211 746352 50 II 9:247478 117H | 9-993 104 38 || 9:25.4374 1909 || 10-745526 49 12 248.181 1169 993081 38 255 [()() 1207 744900 48 13 248883 1167 993059 38 255824 1205 744 176 47 14 249583 II65 993036 38 256547 1203 743453 || 46 15 250282 1163 993013 38 257.269 12i}! 74:2731 45 16 250980 I 161 99.2990 38 257990 I2()() 74:2010 44 17 251677 II59 99.2967 38 2587 IU 1198 74.1290 43 18 252373 1158 99.2944 38 259429 1196 740571 42 19 253067 1156 99.2921 38 260/46 1194 739854 4 | 20 253761 Ilā4 992898 38 260863 I 192 7391.37 40 21 | 9:254453 1152 9-902875 38 9-261578 1190 |10.738422 || 39 22 || 255144 1150 992852 38 262292 I 189 737708 38 23 255834 1148 992829 || 39 203005 || 1187 | 736995 || 37 24 256523 1146 992806 39 2ü37 17 1185 736283 36 25 257211 II.44 992783 39 2544:28 1183 73557: 35 26 257898 I 142 992759 39 265138 1181 734862 || 34 27 258583 II4] 992736 39 255847 1179 734153 33 28 259268 II39 9927.13 39 26ü555 1178 733445 32 29 259951 II37 99.2690 39 257.251 1176 73:2739 31 30 260633 1135 99.2566 39 57:}67 1174 732033 || 30 31 ||9-261314 II.33 ||9-992643 39 9-268671 1172 |10'731329 29 32 261994 II31 99.2619 39 26.9375 1170 73U625 28 33 26.2673 1130 99.2596 39 27(){J77 1169 729923 || 27 34 263351 1128 992572 39 27U779 I [67 729221 26 35 | .264027 II26 99.2549 39 271479 1165 7285:21 25 36 264703 II:24 902525 39 272,178 'I I64. 727822 24 37 265377 1122 99.2501 39 27.2876 1162 727124 || 23 38 || 266051 1120 99.2478 40 273573 1160 726427 22 30 266723 1119 99.2454 40 274,269 1158 725731 21 40 267395 III.7 99.2430 40 274964 1157 725036 20 41 ||9-268065 II 15 9-992406 40 || 9-275658 II55 10-724342 | 19 42 268734 1113 992.382 40 276351 1153 7:23649 I8 43 | 269402 IłII 99.2359 40 277043 1151 722957 I7 44 270069 1110 992,335 40 277734 1150 722266 I6 45 || 270735 II08 99.2311 40 2784.24 1148 721576 15 46 || 27.1400 1106 99.2287 40 2791.13 I 147 720887 | 1.4 47 272064 1105 99.2263 40 270801 1145 7.20199 13 48 || 272726 I103 992239 40 280488 1143 719512 | 12 49 || 27.3388 II.01 992214 40 281174 l 14 | 7 18826 I 1 50 | 274049 1099 992.190 40 28.1858 1140 718142 || 10 51. 9:274768 1098 || 9°992.166 40 | 9-282542 IlºS 10-717458 9 52 27.5367 1096 99.2142 40 283225 II36 716775 8 53 276024 1094 992] 17 41 283907 1135 716093 7 54 || 276681 1092 992)93 41 284588 1133 715412 6 55 || 277337 109] 99.2069 41 285268 1131 714732 5 56 || 277991 I689 99.2044 41 285947 1130 714053 4 57 | 278644 I087 992020 41 286624 11:28 7 13:376 3 58 279297 I()86 99.1996 41 287.301 I 126 712699 2 59 || 279948 1084 99.1971 41 287977 11:25 71.2023 I .60 l 280599 IU82 99.1947 41 288652 1123 711348 0. | M. 79 Degrees. zo&ARITILIrg. SINES, CoSINES, ETC (11 Degrees) 209 78 Degrees, M. Sine | D. 1 Cosine D. | Tang. D. 1 Cotang. I 0 9-280.59%) 1082 ||9-99.1947 41 9:288652 1123 || 10-711348 60 I 28.1248 1081 99.1922 41 289.326 II22 710674 59 2 || 281897 1079 99.1897 41 289999 1120 .71()001 58 3 || 332544 ió77 | 351873 || 4i ºf iii.3 || 70.j | # 4 || 283190 1076 99.1848 41 291342 II 17 708658 56 5 || 283836 1074 99.1823 41 292013 1115 707987 || 55 6 || 284480 I()72 99.1799 41 292682 1114 707318 54 T | 285.194 I07I 99.1774 42 293350 III2 706650 53 8 || 285766 1069 99.1749 42 2940.17 IIII 705983 || 52 9 || 286408 1067 991724 42 294684 1109 705316 || 51 10 287048 1066 99.1699 42 295.349 II07 704651 50 II 9.287687 1064 ||9-991674 42 (9,296013 1106 || 10-703987 || 49 12 || 288.326 1063 991649 42 296677 1104 703323 || 48 13 || 288964 I061 991624 42 297339 I 103 T02661 47 . I4 289600 1059 991599 . 42 298001 1 101 70 1999 || 46 I5 290236 1058 991574 42 298662 1100 701.338 45 J6 || 200870 1056 99.1549 42 200322 1098 700678 || 44 I7 || 29 1504 I(\54 991524 42 299.980 1096 700020 || 43 18 2921.37 I053 991498 42 300638 I095 699362 42 I9 || 292768 1051 991473 42 30.1295 1093 698705 || 41 20 | 293399 1050 99.1448 42 30.1951 1092 698049 || 40 21 9-29.4029 1048 || 9'991422 42 9:302607 1090 10-697.393 || 39 22 || 294658 1046 99.1397 42 303261 1089 696.739 || 38 23 295286 1045 99.1372 43 303914 1087 696086 || 37 24 295913 1043 99.1346 43 304567 I0S6 69.5433 36 25 | 296539 1042 99.1321 || 43 305.218 1084 694782 || 35 26 297164 1040 99.1295 43 305869 1083 694131 || 34 27 297788 1039 99.1270 43 306519 || 1081 693481 || 33 28 2984 12 I037 99.1244 43 307 168 1080 692832 || 32 29 || 299034 1036 99.1218 43 307815 1078 692185 31 30 299655 1034 991 193 43 308463 1077 69.1537 || 30 31 9°300276 I032 ||9-99.1167 43 9-309.109 1075 10-690891 || 29 32 || 300895 1031 991 141 43 309754 1074 690246 || 28 33 30 1514 I029 99.11.15 43 3.10398 1073 68.9602 || 27 34 || 302132 1028 99.1090 43 31.1042 1071 688958 26 35 || 30.2748 1026 99.1064 43 3I 1685 1070 6883.15 25 36 || 303364 1025 99.1038 43 312327 1068 687673 24 37 30.3979 I023 99}012 43 312967 1067 687033 23 38 304593 1022 990.986 43. 313608 1065 686,392 || 22 39 305207 1020 99.0960 43 314247 1064 685753 || 21 40 3058.19 I019 99.0934 44 3I4885 1062. 6851 15 20. 41 9:306430 I017 | 9,990908 44 || 9 315523 1061 |10-684477 | 19 42 30704] 1016 99.0882 44 316159 1060 683841 I8 43 307650 1014 99.0855 44 3.16795 1058 683.205 || 17 44 308259 1013 990829 44 317430 I057 6825.70 | 16 45 || 308867 10II 900803 44 3.18064 1055 681936 15 4(5 309474 1010 990777 44 3]8697 1054 681303 || 14 47 310080 1008 990.750 44 319.329 1053 680671 13 48 || 310685 1007 99{)724 44 3.1996.1 1051 680039 || 12 49 || 311289 1005 990697 44 320592 1050 679408 II 50 3] 1893 1004 99067.1 44 3.21222 1048 678778 10 51 ||9-312495 1003 || 9.990644 44 9-321851 1047 | 10:678:149 9 52 313097 1001 9006R8 44 322479 I045 67752I 8 53 313698 1000 990591 44 3231()6 1044 676894 7 54 || 314297 998 990565 44 323.733 1043 676267 6 55 || 3.14897 997 990538 || 44 324358 1041 675642 5 56 || 315495 996 990511 45 32.4983 1040 6750.17 4 57 || 316092 994 99()485 45 325607 1039 674303 3 58 || 316689 993 990458 45 3262:31 1037 6.73769 2 59 || 31.7284 '991 990431 45 326853 1036. 673147 I 60 ! 317879 99() 990404 45 327475 1035 672525 0 ° | Cosine | L. Sine | | Cotang. | Tang. I Me 210 (12 Degrees.) LOGARITHMIC SINES, GOSINES, ETC. M. : Sine | D. Cosine | D. Tang- || D. | Cotang. ' 0 9-317879 990 9-990404 45 9-327474 1035 | 10-672526 60 l 318473 988 99()378 45 328095 1033 671905 59 2 3}9066 987 99.0351 45 3287.15 1032 671285 58 3 319658 986 99.0324 45 329334 1030 670666 57 4 320249 984 990.297 45 32995.3 1029 670047 56 5 320840. 983 990.27() 45 330570 1028 669430 , 55 6 321430 982 99t)243 45 33.1187 1026 668813 54 7 322019 980 9902}5 45 33.1803 1025 668197 53 8 3226()7 979 990.188 45 332418 IU24 667582 52 9 323.194 977 990 161 45 33.3033 1023 666967 51 10 323780 76 990.134 45 333646 1021 666354 50 11 || 9-324366 975 9-990 107 46 9-334259 1020 || 10.665741 49. 12 324950 973 990()79 46 334871 1019 665129 48 13 325534 972 9900.52 46 335482 I017 664518 47 14 326117 970 990025 46 336093 1016 6639(7 46 15 326700 969 989997 46 336702 1015 663298 45 I6 327281 968 989%)70 46 3373]] 1013 662680 44 17 327862 966 989.942 46 337919 1012 662081 43 18 328442 | x 965 98.99.15 46 338527 1011 661473 42 19 329021 964 9898.87 46 || 339]33 1010 660867 41 20 3295.99 962 989860 46 339.739 1008 660261 40 21 9-330.176 961 9-989832 46 9-340344 1007 || 10-659656 39 22 330753 96() 989804 46 340948 1006 (559()52 38 23 331329 958 989777 46 34 1552 1004 (558.448 37 24 331903 957 989749 47 342155 1003 657845 36 25 332478 956 98.9721 47 || 342757 1002 657243 35 26 333051 954 98.9693 47 343358 1000 656642 34 27 333624 953 989665 47 343958 999 656042 33 28 334}95 952 98.9637 47 344558 998 655442 32 29 334766 950 989609 47 345157 997 654843 31 30 335337 949 989582 47 345755 996 654245 30 31 || 9-335006 948 || 9-989553 47 9°346353 994 || 10-65.3647 29 32 336475 946 98.9525 47 346949 993 65.3051 28 33 337()43 945 98.9497 47 347545 992 652455 27 34 337610 944 989.469 47 348141 991 653859 26 35 338 176 943 98.944? 47 3487.35 990 65.1265 25 36 338742 941 98.9413 47 349329 988 65067.1 24 37 339306 940 98.9384 47 349922 987 650078 23 38 33987 | 939 989356 47 350514 986 649486 22 39 340434 937 98.9328 47 351 106 985 648894 21 40 340996 936 989.300 47 351697 983 648.303 20 4] | 934 1558 935 9-98.9271 47 | 9-352287 982 | 10 647713 19 42 3421 19 934 989.243 47 . 352876 981 647124 I8 43 342679 932 98.9214 47 353465 980 646535 17 44 34.3239 931 989 186 47 354053 979 645947 16 45 343797 930 98.9157 47 354640 977 645,360 15 46 344355 929 98.9128 48 355227 976 64.4773 14 47 344912 927 989,100 48 355813 75 644187 13 48 345469 926 989071 48 356398 974 643602 12 49 346024 925 98.9042 48 356982 97 64.3018 U1 50 3465.79 924 98.9014 48 357566 71 (54.2434 10 51 9°347134 922 9-988985 48 9°358].49 97() || 10-641851 9 52 347687 921 988.956 48 358731 969 64 1269 9 53 34824() 920 988927 48 3593.13 968 640087 7 54 348792 919 9888.98 48 3598.93 967 640ſ ()7 6 55 349.343 917 988869 48 360474 966 639526 5 56 34989.3 916 988840 48 361053 965 638.047 4 57 350443 915 9888] I 49 36.1632 963 638368 3 58 350992 914 988.782 49 36221() 962 637,700 2 59 351540 913 988753 49 36.2787 961 637213 1 60 352088 911 988724 49 363364 960 6366.36 0 | Cosine | | Sine | | Cotang. I | Tang. M. 77 Degrees. zo&ARITHiſto SINES, CoSINES, ETC. (18 Degrees) 211 76 Degrees. M. Sine D. Cosine D. : Tang. | D. Cotang. I 0 | 9-352088 911 9.988724 49 9-363364 960 Y()-6366.36 6%) 1 352635 910 988695 49 36.394() 959 63606() 59 2 35318] 909 988666 49 364515 958 635485 58 3 353726 908 9886.36 49 365090 957 634910 57 4 354271 907 9886()7 49 365664 955 63.4336 56 5 354815 905 988578 49 366237 954 633763 55 6 355.358 904 988548 49 366810 953 633190 54 7 3559()] 903 9885.19 49 367382 952 6326||8 53 8 356443 902 9.88489 49 367953 951 63.2047 52 9 356984 9()1 9.88460 49 368524 950 631476 51 10 357524 899 988.430 49 369094 949 630906 50 11 || 9-358064 898 ||9-988.4()] 49 9°369663 948 || 10-630337 49 12 358603 897 98837] 49 376232 946 629768 48 13 359,141 806 988342 49 70799 945 629201 47 14 || 359678 895 9883.12 50 37.1367. 944 6286.33 46 15 36()215 893 988.282 50 . 371933 943 628067 45 I6 360752 892 * 988259 50 372499 942 627501 44 I7 361287 891 988223 50 37.3064 941 626936 43 18 36 1822 890 98.8193 50 373629 940 626371 42 19 362356 889 988163 50 374193 939 625807 41 20 362880. 888 988,133 50 374756 938 625244 40 21 | 9-363422 887 | 9.988103 50 9-375.319 937 10-624681 39 22 363954 885 988073 50 375881 935 624] [9 38 23 364485 884 988043 50 376442 934 623558 37 24 365010 883 988013 50 37.7003 933 62.2997 36 25 36.5546 882 987983 50 377.563 932 622437 35. 26 366075 88.1 987953 50 378122 931 621878 34 27 366604 880 98.7022 50 378681 930 621319 33 28 367 131 870 987892 50 70239 929 620761 32 29 367659 877 987862 50 379797 928 620203 31 30 368 185 876 9878.32 51 380354 927 619646 30 31 || 9-368711 875 || 9.987801 51 9.380910 926 10-619090 29 32 369236 874 987771 51 381466 925 618534 28 33 369761 873 98.7740 51 382020 924 617980 27 34 370285 872 9877 10 51 38.2575 923 617425 26 35 370808 871 9876.79 51 , 383129 922 61687] 25 36 371330 870 987649 51 383682 921 616318 24 37 || 371852 869 987618 51 384234 920 615766 23 38 372373 867 987588 51 38.4786 919 615214 22 39 || 372894 866 987557 51 385337 918 614663 21 40 373414 865 987526 51 385888 917 6141 12 20 41 || 9°37.3933 864 || 9.987496 51 9.386438 915 10-613562 19 42 374452 863 987.465 51 386987 914 613013 18 43 374970 862 987434 51 387536 913 612464 17 44 75487 861 987.403 52 388084 912 61 1916 I6 45 76003 860 || 98.7372 52 388631 9 II 6] 1369 15 46 376519 859 98.734] 52 389,178 910 610822 j4 47 377035 858 987310 52 38.9724 909 610276 13 48 377549 857 98.7279 52 390270 908 6097.30 12 49 378063 856 987248 52 390815 907 609185 11 50 378577 854 987217 52 391360 906 608640 10 51 9-379089 853 9.987 186 52 9°391903 905 |10,608097 9 52 379601 .852 987 I55 52 392.447 904 607553 8 53 3801 || 3 851 987124 52 392089 903 607()11 7 54 380624 850 98.7092 52 393531 902 60(;469 6 55 381.134 849 98.7061 52 394()73 901 60,5927 5 56 38.1643 848 987030 52 394614 90ſ) 605.386 4 57 3821.52 847 986998 52 395154 899 604846 3 58 382661 846 986967 52 395694 898 6()4306 2 59 383.168 845 986936 52 396.233 897 603767 I 60 383675 ! .. 844 986904 52 39677 896 603229 0 | Cosime | Sine | | Cotang. | | Tang. | M. 212 (14 Degrees.) LOGARITHMIC SINES, COSINES, ETC. # M. Sine | D. Cosime D. Tang. I D. Cotang. I () 9-3836.75 844 9,986904 52 9-396771 896 106()3229 60 I 384.182 843 986873 53 307300 896 002691 59 2 384687 842 980.841 53 397846 895 G02154 58 3 385.192 841 986800 53 398.383. 894 60.1617 57 4 385697 840 986778 53 398,019 893 601(381 56 5 386201 839 986746 53 30945.5 892 60()545 55 6 386704 838 9867.14 53 309090 891 60()()1() 54 7 387207 837 986683 53 400.524 890 599476 53 8 387700 836 986651 53 40.1058 889 598.942 52 9 388210 835 986619 53 401591 888 508409 51 I0 3887] 1 834 986587 53 402124 887 59.7876 50 11 ||9-3892] 1 8.33 9,986555 53 |9-402656 886 10°597344 49 12 3807 || 1 832 986523 53 403187 885 596813 48 13 390210 831. 986.401 53 403718 884 596.282 47 14 390708 . 830 986459 53 404240 883 595.75l. 46 15 391206 828 986.427 53 404778 882 595.222 45 16 || 391703 827 986.305 53 405308 881 50.4692 || 44 17 302199 826 986363 54 405836 880 594164 43 18 302695 825 98.633 I 54 406364 879 593636 42 19 303.191 824 986.299 54 406892 878 593] 08 41 20 39.3685 823 986.266 54 407419 877 502581 40 21 |9-394179 822 9,986934 54 || 9:407945 876 10°592055 39 22 394.673 821 986202 54 40847] 875 59,1529 38 23 395.166 820 986160 54 408997 874 59,1003 37 24 395658 819 986137 54 40952] 874 5004.79 36 25 396150 818 9S6104 54 4](){}45 873 589.955 35 26 || 3000 11 817 Q$1}{\72 54 4T ſºft C) 872 589431 || 34 27 397 132 817 986(339 54 41 092 871 588908 33 28 397621 816 986007 54 4 | 1615 870 588385 32 29 - 3981 II 815 985974 54 4121.37 869 587863 31 30 398600 814 985942 54 412658 868 587.342 30 31 ||9-399088 813 9.985909 55 |9-413179 867 10-586821 29 32 39.95.75 812 985876 55 413609 - 866 586,301 28 33 400062 8|| 98.5843 55 414219 865 58578.1 27 34 400549 810 98.58] 1 55 4.14738 864 585262 26 35 4()}{}35 809 98.5778 55 415257 864 58.4743 25 36 401520 808 98.5745 55 415775 863 584225 24 37 || 402005 807 985,712 55 4.16203 862 5837(37 23 38 402489 806 985670 55 416810 861 583]90 22 39 402972 805 985646 55 4.17326 860 58.2674 21 40 403455 804 985613 55 4.17842 859 58.2158 20. 41 9.403938 803 9.985.580 55 | 9-4 18358 858 10-581642 19 42 404420 802 9855.47 55 4 18873 857 581127 I8 43 404001 801 98.5514 55 410387 856 580613 17 44 405382 800 98.5480 55 4 19901 855 580009 16 45 405862 700 985.447 55 4204.15 855 579585 15 46 | 40634.1 798 985-114 56 420927 854 5790.73 14 47 406820 797 98538() 56 42.1440 853 578560 I3 48 407.299 796 985347 56 42.1952 852 57.8048 12 49 407777 795 985314 56 422.463 851 77537 II 50 408254 794 985280 56 422974 850 577026 |. 10 51 |9-408731 794 9-985247 56 |9-423.484 849 10:576516 9 52 409907 793 985213 56 423993 848 . 576007 8 53 409682 792 98.5180 56 424503 848 575,497 7 54 410,157. 701 985.146 56 425011 847 574989 6. 55 410632 790 985] 13 56 425519 846 57.4481 5 56 411106 789 985079 |! : 56 426027 845 57.3973 4 57 4] 1579 788 985045 56 . 426534 844 57.3466 3 58 412052 787 985011 56 427041 843 572959 2 59 412524 786 984978 56 427547 843 572-153 1 60 412996 785 984944 56 428052 842 7 1948 0 | Cosine | Sine ! | Cotang. I | Tang. | M. 75 Degrees. LOGARITHMIC SINES, COSINES, ETC. (15 Degrees.) 74 Degrees, M. I. Sine | D. | Cosine I D. Tang. D. Cotang. I 0 9°412996 785 | 9.984944 57 | 9-428052 842 |10°57'1948 I 60 1 || 4 13467 784 9849 I() 57 428557 841 57.1443 59 2 || 4 J3938 783 984876 57 429062 840 570938 || 58 3 || 414408 783 984842 57 429566 839 57()434 || 57 4 || 414878 782 98.4808 57 430070 838 569030 56 5 § 4.15347 78I. 984?74 57 430573 838 569427 | 55 6 || 415815 780 984.740 57 431075 837 5689.25 || 54 | 7 || 416.283 779 98.4706 57 431577 836 568423 || 53 8 || 416751 778 98.4672 57 432079 835 56792} 52 9 || 417217 777 984637 57 | 432580 834 567420 || 51 T0 || 417684 776 98.4603 57 433()80 833 566920 | 50 II 9:418150 775 9.984569 57 9.433580 832 |10°566420 || 49 I2 4186H5 774 984535 57 434080 '832 565920 || 48 13 419079 773 984500 57 434.579 831 565421 || 47 14 || 419544 77 984.466 57 435.078 830 564922 || 46 I5 4:20007 772 984432 58 435576 829 564424 || 45 16 || 420470 771 98.4397 58 436073 828 563927 || 44 17 420933 770 98.4363 58 43657() 828 563430 || 43 18 421.395 769 98.4328 58 437067 827 562033 || 42 19 || 421857 768 984294 58 437563 826 562437 || 41 20 4223.18 767 984259 58 438059 825 561941 | 40 21 |9-422778 767 9.984224 58 ||9-438554 824 |10:561446 39 22 423238. 766 98.4190 58 439048 823 560952 || 38 23 423697 765 984 155 58 430543 823 560457 || 37 24 || 424,156 764 984 120 58 440036 822 559964 || 36 35 | 424615 763 984085 58 440529 821 559471 || 35 26 425073 762 98.4050 58 44.1022 820 558978 || 34 27 4255.30 761 98.40I5 58 441514 819 558486 || 33 28 || 425987 760 983981 58 44.2006 819 557994 || 32 29 || 426443 760 98.3946 58 442497 818 557503 || 31 30 || 426899 759 98.30] I 58 44.2988 817 557012 || 30 31 || 9.427354 758 ||9-083875 58 ||9-443479 816 |10°556521 29 32 || 427809 757 98.3840 59 443968 816 556032 28 33 || 428.263 756 983805 59 444458 815 555542 27 34 || 428717 755 983770 59 444947 814 555053 26 35 | 4291.70 754 983735 59 445435 813 554565 || 25 36 || 429623 753 983700 59 445923 812 554077 24 37 || 430075 752 983664 59 446411 812 553589 23 38 || 430527 752 983629 59 446898 811 553102 || 22 39 430978 75I 983594 59 447384 810 552616 21 40 || 43I429 750 98.3558 59 447870 809 552130 | 20 41- || 9,431879 749 || 9'983.523 59 9-4483.56 809 |10:551644 19 42 || 432329 749 983487 59 448841 808 551159 | 18 43 || 432778 748 983452 59 449326 807 550674 || 17 44 || 433226 747 98.3416 59 449810 806 55() 190 | 16 45 || 433675 746 98.3381 59 450294 806 549.706 15 46 || 434122 745 98.3345 59 450777 805 549223 14 47 434.569 744 98.3309 59 451260 804 548740 13 48 || 435016 744 983273 60 45.1743 803 548257 | 12 49 || 435462 743 9832.38 6() 452225 802 547775 11 50 435908 742 983202. 60 452706 802 547294 || 10 51 9-436353 74] 9'983I66 60 | 9.453187 801 ||10.5468.13 9 52 || 436798 740 98.3130 60 453668 800 54.6332 8 53 || 437242 740 98.3094 60 454148 799 54.5852 7 54 || 437686 739 98.3058 60 454628 799 545372 6 55 || 438129 738 98.3022 60 455107 798 544893 5 56 || 438572 737 98.9986 60 455586 797 544414 4 57 || 439014 736 98.2950 60. 456064 796 543936 3 , * 58 || 439.456 736 98.2914 - || 60 456542 796 543.458 2 59 || 439897 735 98.2878 60 457019 795 542981 l 60 || 440338 734 98.2842 60 45.7496 794 542504 0 | Cosme I | Sine | | Cotang. | Tang, | M. 214 (16 Degrees.) LOGARITHMIC, SINES, COSINES, ETC. *- M. : Sine D. Cosime D. Tang. D. Cotang. I 0 | 9-440338 734 9,982.842 60 9°45.7496 794 | 10°542504 60 ! 440778 733 98.2805 60 457973 793 54.2027 59 2 44.1218 732 . 98.2769 61 458449 703 54.1551 58 3 44.1658 731 98.2733 61 4589.25 792 54 1075 57 4 442096 731 98.2696 61 459400 701 540600 56 5 442535 730 98.2660 61 459875 790 540 125 55 6 442973 729 982624 61 460349 700 53965 [ . 5.4 7 443410 728 98.2587 61 460823 780 539ſ 77 53 8 || 443847 727 98.255.1 61 461297 . 788 538703 52 9 444284 727 98.25 14 61 46] 770 788 5382.30 51 10 444720 726 982477. 61 462242 787 5377.58 50 II | 9°445155 725 9.982441 6 I 9°46’27]4 786 || 10:537286 || 49 12 || 445590 724 982404 61 463186 785 536814 48 13 446025 723 98.2367 61 463658 785 536342 47 14 || 446459 723 98.2331 61 464 129 784 535871 46 15 || 446893 722 982204 61 464599 783 5354()1 45 16 447326 721 982257 61 465.069 783 534931 44 I7 4477.59 720 98.2220 62 465539 782 534461 43 I8 || 448191 720 982I83 62 466008 781 533992 42 19 448623 719 982] 46 62 466476 780 533524 41 20 || 449054 718 98.2109 62 466945 780 533()55 40 21 9-449485 717 9-98.2072 62 9-467413 779 || 10.532587 39 22 || 449915 716 98.2035 62 467880 77 532120 | 38 23 450.345 716 98.1998 62 468.347 77 53.1653 37 24 450775 715 98.1961 62 468814 777 531186 36 25 451204 714 98.1924 62 469.280 776 530720 || 35 26 || 451632 713 98.1886 62 469746 75 53()254 || 34 27 452060 ºl{} 90.1049 62 4702II 775 52)709 || 3:0 28 452488 712 98.1812 62 470676 77 529324 32 29 || 452915 7II 98.1774 62 47] 141 773 528859 || 31 30 453342 710 98.1737 62 471605 773 528395 || 30 31 9:453768 710 |9-98.1699 63 9:47.2068 77 10:52.7932 || 29 32 || 454,194 709 98.1662 63 472532 771 527468 || 28 33 || 454619 708 98.1625 63 472995 771 527005 || 27 34 || 455044 707 98.1587 63 473457 77() 526543 26 35 || 455469 707 98.1549 63 473919 769 526081 25 36 455893 706 981512 63 474381 769 525619 24 37 || 456.316 705 98.1474 63 474842 768 525,158 23 38 456739 704 98.1436 63 475.303 767 524697 || 22 39 || 457162 704 98.1399 63 475763 767 524237 21 40 || 457584. 703 98.1361 63 476.223 766 523777 | 20 41 9°458006 702 ||9-981323 63 9-476.683 765 | 10-5233.17 | 19 42 || 458427 701 98.1285 63 477142 765 522858 18 43 458848 701 981247 63 477601 764 522.399 17 44 || 459268 700 98.1209 63 478059 763 521941 I6 45 459688 699 981 ITI 63 4785.17 763 52I483 15 46 || 460.108 698 98II.33 64 478975 762 521025 l4 47 460527 698 981095 64 479432 761 520568 I3 48 || 460.946 697 98105.7 64 479889 761 52011} 12 49 46.1364 696 98.1019 64 480345 760 519655 11 50 461782 695 980981 64 480801 759 5.1919.9 10 51 9°46’2199 695 || 9°980942 64 9°481.257 759 |10:518743 9 52 462616 694 980904 64 481712 758 518288 8. 53 || 46.3032 693 980866 64 48.2167 757 517833 7 54 || 463448 693 980827 64 482621 757 5] 7.379 6 55 || 463864 692 980,789 64 48,3075 756 51(39.25 5 56 464279 691 98()750 64 483539 755 51 (547.] 4 57 || 464694 690 98.0712 64 483982 755 5160.18 3 58 || 4651.08 690 980673 64 484435 754 515565 2 59 || 465522 689 980635 64 484887 753 515] 13 I 60 465935 688 980596 64 485339 753 514661 0 | Cosine | Sine | | Cotang. | , Tang. M. 73 Degrees LOG ARITIIMIC, SINES, COSINES, ETC. 215 (17 Degrees.) M. Sine | D. Cosine | D. | Tang. D. Cotang. 0 | Q-465,035 688 9-980596 64 9-485,339 755 10-514661 60 l. 466348 688 98()558 64 485701 752 514209 || 59 2 466761 687 980519 65 '486242 751 513758 58 3 467 173 686 980480 65 486t,93 751 513307 || 57 4 467,585 685 98()442 65 487 143 750 512857 56 5 467996 685 980403 65 487593 749 512407 || 55 6 468407 684. 980,364 65 488043 749 511957 54 7 468817 683 980325 65 488492 748 511508 || 53 8 469227 683 . 980:286 65 48894.1 747 5] 1059 52 9 4(;9637 682 980.247 65 489,390 747 51(X610 || 51 1() 470046 681 98(3208 65 489838 746 510162 |\50 | 1 || 9-470455 680 9-980169 65 9-490286 746 10-509714. || 49 12 47(3863 680 980ſ 30 65 490733 745 509207 48 13 47 1971 67 980091 65 491180 744 508S20 47 14 47 1679 (378 98()052 65 491627 744 508373 46 15 472086 678 980ſ)12 65 402073 743 507927 45 16 4724.92 677 97.9973 65 492519 743 507481 || 44 I7 472898 76 9799.34 66 492965 742 507035 || 43 18 473304 676 9798.95 66 49.341() 74.1 506590 || 42 19 473710 675 97.985.5 66 493854 740 506,146 || 41 20 474,115 674 97.9816 66 494.299 740 505701 || 4Q 21 9-474519 674 9-97.9776 66 9°494743 740 10-505257 39 22 47-4923 673 97.9737 66 495186 739 504814 || 38 23 475.327 72 –979697 66 4956.30 738 5()4370 37 24 4757.30 672 97.9658 66 496073 737 503927 36 25 476133 671 97.9618 66 4965.15 737 50.3485 35 26 476,536 670 97.9579 66 496957 736 503043 || 34 27 476938 669 97.9539 66 497.399 736 502601 33 28 477.340 669 979499 66 497841 735 502159 - || 32 29 47774.1 668 97.9459 66 498.282 734 501.718 31 30 478||42 667 97.9420 66 4987.22 734 501278 || 30 31 9-478542 667 9-970380 66 9°400IG3 733 10-500.837 29 32 478942 666 979.340 66 || 499603 733 500.397 28 33 479342 665 979.300 67 506042 732 49.9958 27 34 479741 665 979260 67 500.481 731 499.519 26 35 480 140 664 979220 67 500020 731. 49.9080 || 25 36 48ſ)539 663 97.9180 67 601359 730 498641 24 37 480937 663 979140 67 50.1797 730 498.203 || 23 38 48.1334 662 97.91 (30 67 502235 729 497765 22 39 481731 661 97.9059 67 50267 728 497.328 21 40 48.2128 661 97.90.19 67 503109 728 496891 20 41 9-482525 660 9-978979 67 9°503546 727 10-496454 I9 42 482921 659 978039 67 503982 727 4960.18 I8 43 48,3316 659 78.898 67 504418 726 495.582 I? 44 48,3712 658 978858 67 50.4854 725 495/46 I6 45 484 107 657 978817 67 5t)5289 725 494711 I5 46 484501 657 978777 67 505724 724 494.276 14 47 484895 656 97.8736 67 50615.9 724 49.3844 13 48 485280 655 978696 68 506593 723 493.407 || 12 49 485682 655 978655 68 507(327 722 492973 11 50 486075 654 978615 68 507.460 722 49254() J() 51 9-486.467 65 9°978574 68 9-507803 721 10-492.I()7 9 52 || 486860 653 97853.3 68 508.326 7:21 491674 8 53 487.251 652 978493 68 508759 720 491241 7 54 487643 651 978452 68 50919.1 719 490809 6 55 488034 651 9784.11 68 509622 719 490.378 5 56 48.8494 650 978:370 68 510054 718 4899.46 4 57 48.8814 650 978.339 68 510485 718 4895.I.5 3 58 484}2()4 649 78.288 68 5109ſ 6 717 489()84 2 59 489593 648 978:247 68 511,346 7I6 488654 I 60 489982 648 978206 68 5] 1776 716 , 488224 0 | – Cosuve | Sine i | Cotang. . Tang. M. 72 Degrees 216 (18 Degrees.) LOGARITHMIC, SINES, COSIWES, ETO. M. Sine | D. Cosine | D. Tang. | D. | Cotang. –4 0 9:489982 648 9°9782.06 68 9:51 1776 716 || 10:488224 60 I 4903.71 648 978}65 68 512206 716 487794 || 59 2 490759 647 978124 68 5126.35 715 487.365 58 3 491.147 646 978083 69 513064 714 486936 57 4 49.1535 646 978()42 69 513493 714 486.507 56 5 49.1922 645 978001 69 513921 713 486079 55 6 492308 644 977359 69 514349 713 485651 54 7 492695 644 97.79;8 69 514777 712 485223 53 8 49.3081 643 977877 69 515204 712 484796 52 9 49.3466 642 977835 69 515631 711 48.4369 51 I0 493851 642 97.7794 69 5,16057 710 48394.3 50 11 9:494236 | 641 9-97.7752 | 69 9'516484 || 710 || 10:48.3516 || 49 12 49.4621 64 | 97.77.11 69 51(;910 709 48.3090 48 13 495005 640 77669 69 5.17335 7(39 48.2665 47 14 495.388 639 97.7628 69 5.17761 708 48.2239 46 15 495772 639 977586 69 518185 708 481815 45 16 496.154 638 977544 70 . 518610 707 48.1390 44 I7 496537 637 977503 7 519034 7(36 480966 43 18 4969.19 637 977461 70 5.19458 706 480542 42 19 497.301 636 977419 70 519882 7{}5 480118 || 41 20 497682 636 977377 70 520305 705 479695 40 21 || 9.498064 635 | 9-97.7335 70 9-520728 704 || 10-479272 39 22 498.444 63 977293 70 521151 703 478849 38 23 498825 634 977.251 70 52.1573 703 478427 37 24 499.204 633 977209 70 52 1995 703 478005 36 25 499584 632 977 167 70 5224.17 702 477583 35 26 499.963 632 977125 70 52.2838 702 477|62 34 27 500342 ū31 9 a USJ 'it) 5:23:259 701 476741 || 33 28 500791 631 977041 70 523680 701 476390 32 29 501099 630 76999 7 524 100 700 475900 31 30 50.1476 629 976.957 70 524520 699 475480 30 31 || 9°501854 629 9-07(.944 70 9°52.4939 699 || 10:475061 29 32 502231 628 976872 71 525359 698 474641 28 33 502(;07 G28 97.68:30 71 525778 698 474.222 27 34 502984 627 97.678 71 526 197 697 473803 26 35 50.3360 626 976745 7 1 52(5615 697 473,385 25 36 50.3735 ($26 976.702 71 52.7033 696 472967 24 37 504 II () 625 976(360 7 1 527451 696 472549 23 38 504485 6.25 97 CGIT 71 52.7868 695 472132 22 30 504860 624 976574 T} 528285 695 47 1715 21 40 505234 623 976532 71 528702 694 47 1298 20 41 || 9°505(;08 623 9-976489 71 9-520119 693 || 10-47(;881 I9 42 505981 622 {}^{j446 71 5295.35 693 47(9465 I8 43 506354 622 976404 71 523950 693 470ſ)50 17 44 506727 621 76.361 71 530,366 692 469634 16 45 507099 620 976318 71 530781 691 4692.19 15 46 507471 620 976275 . TI 531196 691 468804 14 47 507843 619 976232 72 531611 690 468.389 13 48 508214 619 976189 72 532625 690 46.7975 12 49 508585 618 97.6146 72 532439 689 467561 11 50 508956 618 976103 72 532853 689 467.147 10 51 || 9°509326 617 9-976060 72 9-533.266 688 || 10-466734 9 52 509696 616 976()IT 72 53.3679 688 466321 8 53 510065 616 97.5974 72 534092 687 465908 7 54 510434 615 97.5930 72 534504 687 465.496 - || 6 55 51(\803 615 975887 72 534916 | 686 465084 5 56 51] 172 614 975844 72 535328 686 46.4672 4 57 511540 6.13 975800 72 535.739 685 464261 3 58 5] 1907 6.13 97.5757 72 536150 685 46.3850 2 59 512275 612 97.5714 72 536561 684 403439 1 60 5}2642 612 975670 72 536.972 684 46.3028 0 Cosime I . Sine ! | Cotang. I | Tang. . . M. 71 Degrees. LOGARITHMIC, SINES, COSINES, ETG. (19 Degrees.) 217 70 Degrees. M. Sine . I D. Cosine | D. Tang. D. 1 Cotang. I {} | 9' 12642 612 9-97.5670 T3 || 9°536.972 684 10-463()28 60 T 513(){}9 611 975627 73 5.37.382 (583 462(; 18 || 59 2 513375 61] 975.583 73 537702 683 46.3208 || 58 3 51374.1 6]{) 97.5539 73 538202 - 682 46]798 || 57 4 514107 (;09 97.5496 73 53861J 682 461389 || 56 5 5.14472 609 97.5452 73 539020 681 460980 55 6 514837 608 97.5408 73 539429 68] 460571 || 54 7 515202 608 975'265 73 539837 680 460163 || 53 8 515566 607 915.321 73 546245 6S0 45.9755 52 9 515930 - || 607 975.277 73 54C653 6.79 459347 || 51 10 516294 606 97.5233 7 541061 6.79 458939 || 50 II 9'516657 605 || 9°975189 7 9-54.1468 678 || 10:458532 || 49 12 517020 605 975145 73 541875 67 458,195 || 48 13 5.17382 || 604 975101 73 542,281 77 4577 19 || 47 14 517745 604 975057 73 542688 77 457.312 || 46 15 5181.07 603 975013 73 543094 76 456906 || 45 16 518468 603 97.4969 74 543499 76 456501 || 44 I7 518829 602 974925 74 543905 675 456095 || 43 18 519.190 601 97.4880 74 544310 675 455690 || 42 I9 519551 601 97.4836 74 544715 674 455.285 41 20 51991] 600 97.4792 74 5451.19 674 454881 | 40 21 9:52:271 600 9.974748 74 || 9-545524 67. 10-454476 39 22 520631 599 974.703 74 545928 67 454()72 || 38 23 5.20900 599 • 97.4659 74 546,331 672 ,453669 || 37 24 521349 598 974614 74 . 5,46735 672 453265 36 25 521707 598 97.4570 7 547 138 671 452862 35 26 522066 597 97.4595 74 547540 671 452460 | 34 27 5224.24 596 97.4481 74 547943 670 452057 || 33 28 522781 596 974436 74 548345 670 451655 32 29 523138 595 974.391 74 548747 669 451.253 31 30 523495 595 974347 75 549149 669 450851 30 31 9:523852 594 9.974302 75 | 9-540550 668 || 10-450450 | 29 32 524288 594 97.42.57 75 54:3951 668 450049 28 33 524564 593 974.212 75 550352 667 449648 || 27 34 524020 593 97.4167 75 554)752 667 449248 || 26 35 525275 592 97.4122 75 55115.2 666 448848 || 25 36 525630 591 97.4077 75 551552 666 448448 || 24 37 525,984 591 97.4032 75 551952 665 448048 || 23 38 536339 590 97.3987 75 552351 665 447649 || 22 39 5:26093 590 97.3942 75 552750 665 447250 21 40 527046 589 973897 75 553149 664 446851 | 20 41 9°527400 589 9-973852 75 9-553548 664 || 10-446452 | 19 42 527753 588 97.3807 75 553946 663 446054 | 18 43 528105 588 97.3761 ºf 75 554.344 663 445656 || ||7 44 528458 587 97.37.16 76 554741 662 445.259 | 16 45 5288.10 587 97.367.1 76 555,139 662 444861 | 15 46 52916.1 586 97.3625 76 555536 661 444464 14 47 529513 586 97.3580 76 555933 661 444067 || 13 48 529864 585 97.3535 76 556,329 660 443671 12 49 530215 585 973489 76 556,725 660 443275 | II 50 530565 ' | 584 973444 76 557 121 659 442879 10 51 9-530915 584 || 9-973398 76 9°557517 659 || 10:442483 9 52 53.1265 583 973352 76 5579.13 659 442087 8 53 531614 582 973.307 76 558.308 658 44.1692 7 54 || 531963 582 97.3261 76 558702 658 44.1298 tº 55 532312 58] 97.3215 76 559097 657 440903 5 56 532661 58! 97.3169 76 550491 657 440509 4 57 5.33009 580 973124. 76 55.9885 656 4401 15 3 58 53.3357 580 973878 76 560279 656 439721 2 59 533704 579 97.3032 77 560673 655 430327 I 60 534052 578 97.2986 77 56.1066 655 438934 {} | Cosine | l Sine | Cotang. | Tang. M. 218 (20 Degrees) LoGARITIIMIG SINES, CoSINES, ETC. Af M. Sine D. | Cosime D. | Tang. T). Cotang. I () 9-5.34052 578 9.972.986 77 9.561066 655 | 10°438934 60 l 534.399 577 979.j40 7 561459 654 438541 59 2 534745 577 97.2894 77 561851 654 4381.49 58 3 535092 577 97.2848 77 56.2244 65.3 4377.56 57 4 535438 576 97.28()2 77 5626.36 653 437364 56 5 535.783 57 972755 77 563028 65.3 436979 55 6 536,129 75 97.2709 77 5634 [9 652 436581 54 7 536474 57 972663. 77 5638||1 652 436189 53 8 536818 574 97.261? 77 5642i;2 651 435798 52 9 537.163 573 97.2570 77 564592 651 435408 51 10 53.7507 73 972524 77 56.4983 650 4350I '7 50 II | 9°537851 572 9-972478 77 9.565373 650 10°434627 49 12 || 538.194 72 97.2431 78 565763 649 434237 48 13 5385.38 57.1 97.2385 78 566.153 649 433847 47 14 538880 57.1 972.338 78 566542 649 433458 46 I5 539223 570 97.2291 78 566932 648 433068 45 16 539565 570 97.224.5 78 567320 648 432680 44 I7 539907 569 97.2198 78 567700 647 432291 43 18 540249 569 97.2151 78 568098 647 431902 42 I9 540500 568 97.2105 78 568486 646 431514 || 41 20 540931 568 || 972058 78 568873 646 431127 40 2I 9-54.1272 567 9-97.2011 78 9-569.261 645 || 10°430739 || 39 22 541613 567 97 1964 78 5696.48 645 430352 38 23 541953 566 7 1917 78 70035 645 429965 37 24 542293 566 97.1870 78 570422 644 429'578 36 25 542632 565 97.1823 78 570809 644 42919 L 35 26 542971 565 97.1776 78 571 195 643 428805 34 27 543310 564 97.1799 7) 571581 643 42849 }} 28 543649 564 97 1682 79 57.1967 642 428033 32 29 543987 563 97.1635 79 572352 642 427648" | 31 30 544325 563 97.1588 79 57.2738 642 427.262 || 30 31 9°54.4663 562 9-97.1540 79 9°57'3123 641 || 10-426877 29 32 545000 562 971493 79 57.3507 64 I 426493 28 33 5.45338 561 971446 79 573892 640 426108 27 34 545674 561 971398 79 574276 640 425724 26 35 5460If 560 971351 79 574660 639 425.340 25 36 546347 560 97.1303 79 75044 639 424956 24 37 546683 559 71.256 79 575427 639 424573 23 38 547019 559 971208 79. 575810 638 424 [90 22 39 547354 558 971.161 79 576.1Q3 638 423807 2I 40 547689 558 ‘97III.3 79 576576 637 423424 20 41 9-548024 557 9-971066 80 9-576958 637 10-423041 I9 42 548359 557 71018 80 577341 636 422659 I8 43 548693 556 970970 80 ° 577723 636 422.277 I7 44 5400.27 556 970922 80 578104 636 421896 I6 45 5.49360 555 970874 80 578486 635 421514 15 46 549693 555 970827 80 578867 635 421133 14 47 550026 554 970779 80 579248 634 420752 13 48 550359 554 970731 80 796.29 634 420371 I? 49 550092 553 97.0683 80 580.009 634 4.19991 I I 50 551024 553 97()(335 80 580389 633 4.19611 IQ 51 ()-551356 552 9-970.586 80 9°580769 633 10-4}9231 9 52 551687 552 970.538 80 581,149 632 4.1885.1 8 53 5520 18 552 7()490 80 581528 632 4.18472 7 54 552349 551 970442 80 581907 632 4.18).93 6 55 55.2680 551 970394 80 582286 631 417714 5 56 553010 550 970345 81 58.2665 631 417.335 4 57 553341 550 97()297 . 81 58.3043 630 416957 3 58 5.53670 549 970249 81 58.3492 $30 4.16578 2 59 554000 549 970200 81 583800 629 4}{3200 1 60 554329 548 970.152 81 584.177 629 4.15823 0 Cosine ! | Sine \ | Cotang. Tang. | M. 69 Degrees. LOGARITHMIC, SINES, COSINES, ETC. (21 Degrees.) 219 M. Sine | D. Cosine | D. Tang. D. Cotang. I 0 9°554329 548 9-970 152 . 81 9-584177 629 10°415823 60 1 554658 548 97.01(33 81 584555 620 415445 59 2 554987 547 97.0055 81 584932 628 4.15068 58 3 555.315 547 97.0006 8L 585309 628 4 14691 57 4 555643 546 969957 81 585686 627 4.14314 56 5 555971 546 96990ſ) 81 586062 627 4]3938 55 6 556299 545 969860 81 586439 627 413561 54 7 556626 545 96.9811 81 586815 626 413185 53 8 556953 544 969769 81 587100 626 412310 52 9 557280 544 969714 81 587566 625 412434 51 10 557606 543 969665 81 S 587941 625 412059 50 #1 || 9 557932 543 9°969616 82 9-588.316 625 10°4 II (;84 49 12 558.258 543 969,567 82 588691 624 4] 1309 48 13 558583 542 96.9518 82 589066 624 416934 47 14 558909 542 96.9460 82 580.440 623 410500 . 46 15 559234 541 969420 82 589814 623 4]{}}86 45 16 559558 541 969370 82 590.188 623 40.9812 44 17 559883 540 969321 82 590562 622 409438 43 18 560207 540 969.272 82 590935 622 409065 42 19 560531 539 969.223 82 59.1308 622 408692 41 20 560855 539 969,173 82 591681 621. 408339 40 21 9.561 IT8 538 9-96.9124 82 9-502054 621 10-407.946 39 22 561501 538 969075 82 59.2426 620 407574 38 23 561824 537 96.9025 82 502798 620 407.202 37 24 56.2146 537 968.076 82 503170 619 406829 36 25 562468 536 968926 83 593542 619 406458 35 26 562790 536 968877 83 59.3914 618 406086 34 27 563] 12 536 968827 83 594285 618 405715 33 28 5634.33 535 968777 83 594656 618 405344 32 29 563755 535 968728 83 595027 617 404973 31 30 56407.5 534 968678 83 595398 617 404602 || 30 31, 9-564396 534 9-968628 83 0.595768 617 10.404232 29 32 5647 16 533 968578 83 596,138 616 403862 28 33 565036 533 968528 83 596508 616 403402 || 27 34 565356 532 9684.79 83 506878 616 403122 26 35 565676 532 968429 || 83 597247 615 402753 || 25 36 565995 531 968.379 83 597.616 615 402384 || 24 37 566314 531 968.329 83 507985 615 402015 23 38 566632 531 968278 83 598.354 614 401646 22 39 56695I 530 96S228 84 5987.22 614 401278 21 40 567269 530 968178 84 500091 6.13 400909 20 41 9°567587 529 || 9-968128 84 9°59.9459 613 10-400541 19 42 567904 529 968078 84 590827 6.13 400173 I8 43 568222 528 968027 84 600,194 612. 399806 17 44 568539 528 96.7977 84 600562 612 3994.38 16 45 568856 528 96.7027 84 600929 6II 399071 15 46 569 ITV2 527 967876 84 60.1296 611 . 398704 14 47 569488 527 967826 84 601662 611 39.8338 I3 48 569804 526 967775 84 602029 610 397.971 12 49 5701:20 526 96.7725 84 602395 || 610 397605 II. 50 70435 525 967674 84 . 60276}. 610 397239 10 51 || 9'570751 525 9.967624 84 9.603127 609 10:306873 9 52 571066 524 96.7573 84 603493 609 3.96507 8 53 57.1380 524 967522 85 603858 609 396142 7 54 571695 523 967471 85 604:223 608 395777 6 55 57.2009 523 967421 85 604588 608 395.412 5 56 57.2323 523 967:370 85 604953 607 395047 4 57 57.2636 522 96.7319 85 605.317 607 394683 3 58 572950 522 967.268 85 60.5682 607 3943.18 2 59 57.3263 521 967:21.7 85 606046 606 393054 I 60 57.3575 521 967166 85 606410 606 303590 O | Cosine | | Sine | | Cotang. I | Tang. | M. 68 Degrees. 220 (22 Degrees.) LOGARITIIMIC, SINES, COSINES, ETC. M. L. Sine | D. Cosine | D. | Tang. | D. 1 Cotang. 0 | 9°573575 521 9.967166 85 | 9-6064]() 606 || 10:39.3590 60 l 57.3888 520 967 115 85 606773 666 393227 59 2 574.200 520 96.7064 85 607 137 605 39.2863 || 58 3 574512 519 96.70I3 85 607500 605 302500 57 4 574824 519 966961 85 607863 604 39.2137 56 5 575136 519 9669.10 85 608225 604 39.1775 55 6 5754.47 518 966359 85 608588 604 3914 12 || 54 7 575758 518 966808 85 G08950 603 391050 || 53 8 576069 || 517 966756 86 609312 603 390688 52 9 576379 517 966705 86 609674 603 390326 51 10 576689 516 966653 86 610036 602 389964 || 50 I 1 || 9:576900 516 9-966602 86 9-610397 602 I0.389603 49 I9 57.7309 516 966550 86 610759 602 389.241 || 48 13 577618 515 9664.90 86 61] [20 601 388880 47 14 |. 577027 515 966.447 86 61 1480 601 388520 46 15 578236 514 966305 86 6] 1841 601 388,159 || 45 16 578545 514 966.344 86 61220I 600 387799 || 44 I7 578853 513 966.292 86 619.561 600 387439 || 43 I8 579.162 513 96634() 86 612921 600 387079 42 19 570-470 513 966.188 86 613281 599 386719 || 41 20 579777 512 966136 86 6.13641 599 386.359 || 40 21 9-580085 512 9 966085 87 9-614000 || 598 || 10:386000 || 39 22 58()392 511 9660.33 87 614359 598 385641 38 23 580699 5II 965981 87 614718 598 385282 37 24 58 1005 51 I 965928 87 615077 597 384923 || 36 25 581312 510 96.5876 87 615435 597 384565 35 26 581618 510 96.5824 87 615793 597 384207 34 27 581924 509 965772 87 616151 596 3838-49 33 28 582229 509 9657.20 87 * 616509 596 383491 32 29 582535 509 965668 87 616867 506 383.133 || 31 30 582840 508 9656.15 87 617224 595 382776 || 30 31 || 9'58.3145 || 508 9.965563 87 9-617582 || 595 || 10:3824|18 || 29 32 58.3449 507 9655] I 87 6.17939 595 382061 28 33 58,3754 507 96.5458 87 618295 594 38H 705 || 27 34 58.4058 506 965.406 87 618652 594 381348 26 35 584.361 506 965353 88 619008 594 380992 25 36 584665 506 965,301 88 619.364 593 380636 24 37 584968 505 965248 88 61972] 593 380.279 23 38 585272 505 965,195 '88 620{)76 503 379924 22 30 585574. 504 96.5143 88 620432. 502 379568 21 40 585877 504 96.5000 88 620787 502 3792.13 20 41 | 9'586.179 || 503 || 9.065037 || 88 || 9-621142 592 || 10-378858 19 42 586482 503 964984 88 62.1497 59] 378503 I8 43 586783 - 503 96493I 88 621852 591 378 I 48 17 44 587085 502 964879 88 622207 590 377.793 l6 45 587386 502 964826 88 622561 590 377439. 15 46 | . 587688 501 96477. 88 62.2015 590 377085 14 47 587089 501 964719 88 62.3269 589 376731 13 48 588.289 50I 964666 89 623623 589 376.377 | 12 49 588.500 500 , 964613 89 62.3976 589 376024 || 11 50 588890 500 96.4560 80 624330 588 3756.70 || 10 51 || 9°589190 499 || 9,964.507 89 || 9-624683 588 || 10-375.317 9 52 589489 499 964.454 89 625036 588 374964 8 53 589789 . 499 Ø64400 89 625.388 587 374612 7 54 590088 498 964347 80 695741 537 374950 6 55 590387 - 498 964204 80 626093 587 3730()7 5 56 590686 407 964340 89 626445 586 373555 4 57 590.984 497 964|87 89 626.707 586 37.3203 3 58 50.1282 497 Q64]33 89 627149 586 37285] 2 59 591580 496 96.4080 89 62750} 585 372409 I 6ſ) 591878 496 964026 89 627852 585 372148 0 | Cosìne | Sune | | Cotang, | | Tang. I .67 Degrees. * LOGARITIIMIC, SINES, COSINES, ETC. (23 Degrees) 221 Cosine * M. Sine | D. | D. | Tang. D. Cotang. : () 9°59'1878 496 || 9.964026 89 || 9-627852 585 | 10-372148 60 l 502176 495 96.3972 89 628203 585 37.1797 59 2 592473 495 96.3919 89 628554 585 37 1446 || 58 3 59.2770 495 963865 90 628905 584 37 1095 57 4 503067 494 9638] I 90 629255 584 370745 56 5 593363 494 963757 90 629606 583 370394 55 6 59.3659 493 963704 90 629956 583 370044 || 54. 7 593955 493 963650 90 63()306 583 369694 53 8 59425.1 493 96.3596 90 630656 583 369344 52 9 594547 492 96.3542 90 63]005 582 368995 51 Jū 594.842 492 963488 90 631355 || 582 368645 || 50 11 9°595137 491 9°963434 - || 90 9-631704 582 | 10-368296 || 49 12 59.5432 491 96.3379 90 63.2053 581 367947 || 48 I3 595.727 491 963.325 90 632401 581 367599 || 47 14 59.6021 400 963271 90 632750 581 367250 46 J5 596315 490 963217 90 633098 580 366902 || 45 16 596609 489 96.3163 90 633447 580 366553 || 44 I7 596903 489 963.108 91 633795 580 366205 ' || 43 18 597196 489 963054 91 634143 579 365857 || 42' 19 597490 488 96.2999 91 634490 579 365510 || 41 20 597.783 488 96.2945 91 634838 579 365162 | 40 21 | 9-508075 487 9-96.2890 91 9'635.185 578 i 10:364815 39 22 598368 487 96.2836 91 635.532 578 364468 38 23 598660. 487. 962781 91 635879 578 . 36412] 37 24 598952 486 962727 9] 636226 577 36.3774 36 25 599.244 486 96.2672 91 636572 577 363428 || 35 26 5995.36 || 485 962617 9] 636919 577 363081 34 27 5998.27 485 962562 - || 91 637265 577 362735 | 33 28 600118 485 96.2508 91 637611 576 362389 || 32 ’29 600409 484 96.2453 – 91 637.956 576 362044 || 31 30 600700. 484 962398 92 638302 576 361698 || 30 31 9'600990 484 || 9-962343 92 || 9-638647 575 10-361353 29 32 60 1280 483 962.288 92 638992 575 361008 || 28 33 60.1570 483 962233 92 639337 575 360663 27 34 60.1860 482 962]78 92 639682 574 360318 26 35 602150 482 96.2123 92 6400.27 574 359973 || 25 36 602439 482 96.2067 92 640371 57 3596.29 24 37 || –602728 481 96.2012 92 640716 573 359284 23 38 603017 481 96.1957 92 64 1060 573 358940 22 39 603305 481 96.1902 92 64 1.404 573 358596 || 21 40 603594 480 96.1846 92. 641747 572 358253 20 41 9.603882 480 || 9-96.1791 92 || 9-642091 572 | 10:357909 19 42 604 IT0 479 96.1735 92 642434 572 357566 18 43 604457 479 96.1680 92 642777 572 357.223 I7 44 604745 479 96.1624 93 643120 571 356880 16 45 605032 478 96.1569 93 643463 57.1 356537 15 46 605.319 478 96.1513 93 643806 571 356.194 14 47 605606 478 96.1458 93 644148 570 355852 | 13 48 605892 77 96.1402 93 644490 570 355510 || 12 49 606179 477 96.1346 93 644832 570 3551.68 11 50 606465 476 96.1290 93 645174 569 354826 || 10 51 || 9-60675I 476 9.961235 93 || 9-645516 569 || 10:354484 9 52 607.036 476 961 I?9 93 645857 569 354143 8 53 . 607322 475 961 123 93 646,199 569 . 353801 7 54 607607 475 96.1067 93 646540 568 353460 6 55 607892 474 96.1011 93 646881 568 353119 5 56 608.177 474 96()955 93 647922 568 352778 4 57 608461 474 96()899 93 647562 567 352.438 3 58 608745 473 960843 94 647903 567 352007 2 59 6()9()29 473 96()786 94 648.243 567 351757 1 60 6093.13 '473 960730 94 648583 566 351417 () | Cosine | | Sine | Cotang. I | Tang. | M. 66 Degrees. 222 (24 Degrees.) LOGARITHMIC SINES, COSINES, ETC. A. M. Sine | D. Cosine | D. Tang. | D. Cotang. I 0 9.6093.13 473 9.960730 94 9-648.583 566 || 10-351417 60 I 609597 472 960674 94 648923 566 35.1077 59 2 609880 472 96(36.18 94 649263 566 350737 58 3 610|64 472 96()561 94 649602 566 350398 57 4 610447 471 960.505 94 649942 565 3500.58 56 5 610729 471 96.0448 94 650281 565 349719 || 5 || 6 61] ()12 470 96()392 94 650620 565 349380 54 7 611294 47, 0 960335 94 650959 564 349041 53 8 6]]576 470 96()279 94 65 1297 564 3487 ()3 52 9 6] 1858 469 96().222 94 65 it;36 564 348364 51 10 || 612140 469 960 165 94 651974 563 348026 || 50 II 9-6.12421 469 9-960 109 95 9-652312 563 10-347688 49 12 6]27 02 468 96(){}52 95 652650 563 34.7350 || 48 13 6]2983 468 95.9995 95 652.988 563 347() 12 47 14 613264 467 95.9938 95 653326 562 346674 || 46 15 6] 3545 467 95.9882 95 65.3663 562 346337 45 J6 613825 467 95.9825 95 654000 562 346000 || 44 17 614 105 466 95.9768 95 654337 561 345663 || 43 18 614385 466 95.97 | 1 95 654674 561 345326 || 42 I9 614665 466 959654 95 655() | 1 561 344989 || 41 20 614944 465 95.9596 95 655.348 561 344652 40 21 9-615223 465 '9'959539 95 9-655684 560 || 10-344316 || 39 22 615502 465 959.482 95 65t;()20 560 343.980 38 23 615781 464 959.425 95 656356 56() 343644 37 24 616060 464 95.9368 95 656692 559 343.308 || 36 25 616338 464 959310 96 65.7028 559 342972 35 26 6]6616 463 959253 96 657364 559 34:26:36 || 34 27 616894 463 95.9195 96 ,657699 559 342301 33 28 617172 462 959 138 96 658034 558 341966 || 32 29 617450 462 959081 96 |. 658369 558 341631 31 30 617727 462 959023 96 658.704 558 341296 || 30 31 9-6.18004 461 9°958965 96 9-659039 558 || 10-340961 29 32 618281 461 958908 96 659:373 557 340627 | 28 33 618558 461 958.850 96 659708 557 340292 || 27 34 618834 460 958792 96 660042 557 339958 26 35 619110 460 958734 96 660376 557 339624 25 36 619386 460 958677 96 660710 556 339290 || 24 37 619.662 459 9586.19 96 661()43 556 338957 || 23 38 619938 459 958561 96 661377 556 338623 22. 39 620213 459 958503 97 661710 555 338290 21 40 62()488 458 958445 97 662043 555 337957 20 41 9-620763 458 9,958.387 97 9-662376 555 || 10:337624 19 42 621()38 457 958329 97 6627()9 554 337291 18 43 621313 457 958271 97 66.3042 554 336958 17 44 62.1587 457 958.213 97 663375 554 336625 I6 45 621861 456 958 154 7 66.37()7 554 336293 15 46 6221.35 456 958()96 97 664()39 553 335961 14 47 622409 456 958038 97 664371 553 335629 13 48 622682 455 957979 97 6647()3 553 335297 12 49 622956 455 957921 97 665().35 553 334965 11 50 6.23229 455 95.7863 97 665,366 552 334634 10 51 || 9.623.502 454 9-957804 97 9.665607 §52 || 10-334303 9 52 62.3774 454 957746 98 666029 552 339971 8 53 624047 454 95.7687 98 666.360 551 3.33640 7 54 624319 453 95.7628 98 666691 551 333.309 6 55 624591 453 95.7570 98 667()?1 551 332979 5 56 624863 453 95.75 i ! 98 667.352 551 33.2648 4 57 625,135 452 95.7452 98 667682 550 332318 3 58 625406 452 95.7393 98 668() 13 550 33 1987 2 59 625677 452 957335 98 668,343 550 33 1657 l 60 625948 451 95.7276 98 668672 550 331328 0 | Cosine ! Sine | | Cotang. I | Tang. | M. 65 Degrees. LOGAR ITHMIC, SINES, COSINES, ETC. (25 Degrees.) 223 | 64 Degrees. M. I. Sine | D. Cosine | D. Tang. D. 1 Cotang. Tú 19:625948 45L 9°957.276 98 || 9-668673 550 | 10°331327 60 l 6262 j9 451 957217 98 669()()2 549 330998 || 59 2 626490 451 957 158 98 669,332 549 330668 || 58 3 626760 450 957099 Q8 669661 549 3:30339 || 57 4 627.030 450 9:57t)40 98 66999 | 548 330009 || 56 5 6273)0 45() 956981 98 67t)320 548 3.29t;80 || 55 6 6275 70 449 956921 99 670649 548 329351 54 7 || 6278,40 449 956862 99 670977 548 329023 53 8 || 628 |{)9 449 956803 99 67 1306 547 328694 || 52 9 || 628.378 448 956744 99 7 |634 547 328.366 || 5 | 10 || 628647 448 956684 99 671963 547 328037 || 50 11 || 9 628916 447 9-956625 99 || 9-672291 547 10-327709 || 49 12 629 Jºj 447 956566 99 67.2619 546 327381 48 13 || 62.9453 447 9565t)6 99 672947 546 327053 47 14 || 62.9791 446 956.447 99 67.3274 546 325726 46 15 || 6’29.989 446 956.387 99 || 673602 546 326398 || 45 16 || 630257 446 956397 99 67:34,29 545 326()71 || 44 17 | 630.524 446 956968 99 674.257 545 3257.43 || 43 18 63()792 445 95t;208 || 100 674584 545 3254 ||6 || 4-2 I9 63|{}59 445 956 |48 | 00 6749 () 544 3.25090 4 | 20 | 631326 445 956089 lù0 75237 544 324763. 40 21 9-531593 444 || 9.956029 || 100 ||9-675564 544 10-324436 39 22 || 63|859 444 955969 100 675890 544 324] 10 || 38 23 632}25 444 955909 || 100 676.216 543 323784 || 37 24 b32392 443 955849 || 100 676543 543 323457 || 36 25 || 632658 4:43 955789 || 100 676869 543 323 131 || 35 26 632923 443 955729 100 677 194 543 322806 34 27 | #33 jS9 442 955669 100 677520 542 322480 || 33 28 || 633.454 442 955t;09 || 100 677846 542 322 154 || 32 29 6337 (9 442 955548 100 678 [71 542 321829 || 31 30 || $33984 441 955488 ° 100 678496 542 32.1504 || 30 31 ||9-634249 441 9-955.498 || 101 || 9-678821 541 #10-321 j79 || 29 32 6345 14 44() 955368 || 101 679.146 541 3.90854 || 28 33 634778 440 955307 || 101 679471 541 320529 || 27 34 6.35042 440 955247 | 101 679795 541 320205 || 26 35 | 63,5306 439 955186 || 101 680 120 540 3.19.880 || 25 36 || 63557t) 439 955 126 101 680444 540 319556 24 37 635834 439 955065 | 101 680768 540 3]{}232 23 38 636097 4.38 955005 || 101 68.1092 540 318908 || 22 39 636:350 4.38 954944 || 101 681416 539 318584 || 21 40 || 636623 4.38 954883 || 101 681740 539 318260 | 20 41 9°636886 437 9.954823 || 101 9.682063 539 10-3]7937 19 42 637 || 48 437 954762 || 101 68.2387 539 3.17613 I8 43 || 6374] 1 437 954701 101 6827.10 538 317290 17 44 637673 437 95.4640 || 101 68.3033 538 316967 J6 45 637935 436 95.4579 || 101 68.3356 538 316644 15 46 || 6381.97 436 95.4518 102 683679 538 3.16321, 14. 47 || 638.458 436 954457 || 102 68400] 537 315999 13 48 || 638720 435 95.4396 102 684324 537 3.15676 I2 49 || 638981 435 954,335 | 102 684646 537 315354 11 50 | 639249 435 954274 I02 684968 537 3.15032 10 51 9-630503 434 9°954.213 || 102 ||9-685290 536 10-314710 9 52 639754 434 954 I52 102 685512 536 314388 8 . 53 || 640ſ)24 434 95.4090 I02 685934 536 314066 7 54 || 640284 433 954ſ)29 || |{}2 686255 536 3 13745 6 55 || 64()544 433 95.3968 ; } {}2 686577 535 313423 5 56 640804 433 95.3906 } 02 686898 535 313102 4 57 64 |{}{;4 4.32 95.3845 } {}2 687219 535 312781 3 58 || 64 1324 4.32 953.783 102 687540 535 312.460 2 59 || 64.1584 432 95.3722 103 687861 534 312|39 l 60 64.1842 431 953660 " 103 688.182 534 31.1818 () | Cosine | | Sine | | Cotang, | | Tang. | M. 224 (26 Degrees.) LoGARITHMIC SINES, COSINES, ETC. * M. : Sine ( D. | Cosine D. Tang. . D. | Cotang. 0 9-641842 431 9.953660 103 || 9,688.182 534 || 10-31 1818 60 1 64:21()] 431 95.3599 1()3 6885(32 534 31 1498 59 2 642360 421 95.3537 103 688823 534 311 177 58 3 642618 430 95.3475. I()3 689143 533 3|{}857 57 4 || 64:2877 430 953413 1(;3 689463 533 310537 56 5 || 64.3135 430 953352 I (33 689.83 533 310217 | 55 6 | 643,393 430 95.3290 103 690103 533' 309897 || 54 7 || 643650 429 95.3228 103 69.0423 533 309577 || 53 8 || 643908 429 95.3166 103 690742 532 309.258 52 9 644 165 429 95.3104 103 69 [(362 532 308.938 5]. 10 || 644423 428 953042 103 691381 532 308619 || 50 11 || 9-644680 428 9-95.2980 104 ||9-691700 531 10-308300 || 49 12 || 64.4936 428 952018 104 692019 531 307981 || 48 13 || 645193 427 95.2855 104 , 692.338 531 307662 47 14 || 645450 427 952793 104 692656 531 307344 || 46 15 || 6457(36 427 952731 104 692075 531 307()25 || 45 16 || 645962 426 952CC0 104 693203 530 306.707 || 44 17 | 646218 426 952600 104 693612 530 306388 || 43 18 646474 426 95.2544 104 693930 530 306070 || 42 19 || 646729 425 95.2481 1C4 694248 530 305752 41 20 | 646984 425 95.2419 104 694566 529 305434 || 40 21 ||9-647240 425 9-952356 I04 || 9-694883 520 | 10-305117 | 39 22 || 647494 424 95.2294 IQ4 695201 529 304799 || 38 2 647749 424 9522.31 104 695518 529 304482 37 24 648004 424 95.2168 I65 695836 529 304164 || 36 25 648.258 424 952}06 I05 696.153 528 303847 || 35 26 6485 12 423 95.2043 105 69.6470 528 303530 || 34 27 | 648766 423 - || 951980 IG5 | 696787 528 303.213 || 33 28 || 649620 423 951917 105 697 103 528 302897 || 32 29 || 649.274 422 951854 105 697.420 527 302580 || 31 30 649527 422 951791 105 697736 527 302264 || 30 31 || 9-64978.1 422 9-951728 IG5 9-6980.53 527 | 10:30.1947 29 32 || 550034 492 951665 I(35 698369 527 301631 || 28 33 || 650287 421 951602 105 698685 526 30.1315 27 34 || 650539 421. 951539 I05 699001 526 300999 || 25 35 | 650792 421 951476 105 6993 16 526 300684 || 25 36 || 551044 420 951412 105 699632 526 300368 || 24 37 || 65||1297 420 951349 106 699.947 526 300053 23 38 || 65||1549 420 951286 I(\6 700263 525 2997.37 22 39 || 651800 419 951222 I(;6 700578 525 2994.22 || 21 40 # 652052 419 95] 159 106 700893 525 299.107 || 20 - 41 || 9-652304 419 9:951006 106 ||9-701208 524 |10-208792 19 42 || 652555 4.18 951032 106 701523 524 298.477 | 18 43 || 652806 418 950968 I06 701837 524 298.163 || 17 44 || 653057 418 950905 106 70215.2 524 297848 || 16 45 || 65.3308 418 950.841 106 702466 524 297534 || 15 46 || 653558 417 950778 106 702780 523 207220 14 47 || 653808 4.17 9507 14 106 703095 593 296.905 || 13 48 || 654059 417 950650 106 703409 523 296591 || 12 49 654309 416 950586 106 703723 523 296277 II 50 || 654558 416 950522 107 704036 522 295964 10 5] 654808 416 9:95()458 107 ||9-704350 522 || 10-295650 9 52 655058 416 950394 107 704663 522 295337 8 53 655.307 415 . 950330 I07 704977 522 295023 7 54 655556 415 950266 I07 705290 522 2947 10 6 55 || 655805 415 950:202 I07 705603 52I 294,397 5 56 656.054 414 9501.38 I07 7()59 |{} 52I 294(384 4 57 656.302 414 95007.4 107 706228 52I 29.3772 3 58 656551 414 9500 10 107 706541 521 293459 2 59 656799 413 9499.45 107 706854 521 2931.46 l 60 657047 413 94.9881 I()7 707166 520 29:2834 0 | Cosine l ! Sine | | Cotang. ! Tang. | M. 63 Degrees. LOGARITIIMIC SINES, COSIWES, ETC. (27 Degrees.) 225 & | Cotang. I 62 Degrees. M. Sine | D. Cosine D. Tang. D. 0 || 9,657047 413 9-94.9881 107 9-707166 520 || 10:29:2834 60 I 657995 413 949816 107 707.478 520 292522 || 59 2 657542 412 9497.52 " I ()7 707790 520 292210 || 58 3 657790 412 949688 108 || 708102 520 29]898 57 4 658037 412 9496.23 I08 7084I4 519 291586 || 56 5 658284 412 949.558 108 708726 519 29 1274 || 55 6 658531 4] I 949494 108 709037 519 290963 54 7 658.778 4] I 949.429 108 709349 5 19 290651 53 8 659025 4 : 1 949.364 || 108 709660 519 290340 || 52 9 659.271 410 949.300 108 70.9971 518 290029 5 L It) 659517 410 949235 I08 710282 518 2897.18 50 II 9.659763 410 || 9-94.9170 108 || 9-710593 518 10-289407 || 49 12 . 660009 409 949 105 108 710904 518 289096 48 13 660255 4()9 949040 108 . 7] 1215 5|8 288785 47 14 6605()1 409 948.975 I08 711525 5||7 2884.75 46 15 660746 409 948910 I08 7I1836 517 288.164 || 45 lö 660991 408 948845 108 712I46 517 287854 44 17 661236 408 948780 109 7 12456 5H7 287544 43 I 3 661481 408 9487 |5 109 712766 516 287234 42 19 | 661726 407 9486.50 109 7|13076 516 286924 || 41 20 66 1970 407 948584 I09 713386 516 286614 || 40 2] | 9-662214 407 || 9-9485.19 I09 || 9°713696 516 || 10:286304 || 39 22 | 662459 407 948.454 I09 714005 5I6 285995 || 38 23 662703 406 948.388 I09 714314 515 285686 37 24 662946 406 948.323 I09 714624 515 2853.76 || 36 25 663,190 406 948.257 109 714933 515 285067 35 26 663433 405 948.192 I09 715242 515 284758 || 34 27 66.3677 405 948126 I09 715551 5|| || 284449 || 33 28 6639.20 405 948060 I09 715860 514 284 （40 || 32 29 664163 405 947995 II0 716168 514 283832 31 30 664406 404 947929 II0 71647.7 514 283523 || 30 31 9-66.4648 404 || 9-94.7863 110 || 9-716785 514 || 10.283215 || 29 32 66489.1 404 947797 II.0 717.093 513° 282907 || 28 33 665133 403 947731 110 717401 513 282599 27 34 665375 403 94.7665 110 717709 513 282291 || 26 35 6656.17 403 7600 L10 718017 513 281983 25 36 665859 || 402 94.7533 II.0 7183.25 513 28.1675 || 24 37 666100 402 947467 I 10 718633 512 28H367 23 38 666342 402 947401 II0 718940 512 28.1060 22 39 666583 402 94.7335 I 10 719248 512 280752 || 21 40 | 666824 401 947269 110 || 719555 512 280,445 || 20 41 9-66.7065 401 || 9-947203' 110 || 9-719862 512 10-280.138 . .19 42 667305 401 947H36 III 720169. 5II 2798.31 I8 43 667546 401 947070 III 720476 511 2795.24 I7 44 667786 400 94.7004 III' 720783 511 279217 | #6 45 668027 400 946937 111 721089 511 278911 15 46 668.267 400 946871 III 721396 511 278604 || 14 47 668506 399 946804 111 7.21702 510 278298 || 13 48 668746 399 946.738 111 722009 510 277991 || 12 49 668986 399 94.6671 III 722315. 510 277685 11 50 669225 399 94.6604 III 722621 510 277.379 || 10 51 9-669464 398 || 9-946538. III | 9-722927 510 || 10.277073 9 52 6697 ()3 398 946471 III 723232 509 276768 8 53 669,942 398 946.404 III 723.538 509 276462 7 54 67() (81 397 946.337 I 11 723844 509 276.156 6 55 67.04 |9 397 946270 II2 7241.49 509 27585.1 5 56 6.0658 397 94.6203 112 7244.54 509 27.5546 4 57 7 (1896 397 946 136 || 112 724759 508 275.24H. 3 58 67 ||34 396 946(169 II2 T25065 508 27.4935 2 59 67.1372 396 946(MK2 I 12 725.369 508 27.4631 I 60 l 671609 396 945935 112 725674 508 274326 0 | Cosine ! | Sine { | Coung. I t Tang. M. 226 (28 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. Sine | D. Cosine D. Tang. D. 1 Cotang. 0 | 9-671609 396 || 9-945935 I J3 || 0-725574 508 || 10-274326 60 I 671847 395 345868 112 || 725979 508 27.4021 59 2 672084 395 945.800 | 12 726.284 507 273716 58 3 672.321 395 945733 1 12 7265.88 507 273412 57 4 672558 395 945666 I 12 726892 507 273.108 56 5 ($72.795 394 94.5598 | 12 727 197 507 272803 55 6 67.3032 394 94.553] I 12 727501 507 272499 54 7 67.3268 394 94.5464 J 13 727805 506 272195 53 8 673505 394 945396 I 13 728109 5i)6 97 1891 52 9 67374.1 393 945,328 | 13. 7284 12 506 27 1588 51 |{} 6.73977 393 945261 113 || 728716 506 27 1284 50 } | | 9-674213 393 || 9.945.193 ..] I 3 9-729020 506 || 10:270980 49 12 674448 392 945.125 J 13 729.323 505 270677 - || 48 I3 674684 392 945058 113 729626 505 270374 47 14 6749 19 392 944990 II3 729929 505 270071 46 15 675155 392 944922 113 730233 505 259767 45 16 675390 391 944854 113 730535 505 259465 44 I7 67.5624 391 944786 | 13 730838 504 269,162 43 18 675859 391 9447].8 I 13 73] [4] 504 268859 42 19 676(394 39] 9446.50 ]]3 73.1444 504 268556 41 20 676328 390 944582 114 731746 504 268254 | 40 21 || 9.676562 390 ||9-94.4514 114 9-732048 504 || 10-267952 39 22 676796 390 944446 114 73235.1 503 267649 38 23 77030 390 944377 I 14 732653 503 267347 37 24 677264 389 944309 I j4 732955 503 267045. 36 25 677498 389 944241 I 14 733257 503 266743 35 $26 67773] 389 944 17.2 114 733558 503 266442 34 27 677964 388 944}{}4 114 733860 502 266 j4() 33 28 678]97 388 944036 | 14 734.162 502 265838 32 29 678430 388 943967 114 734463 502 965537 31 30 678663 388 943899 114 734764 502 265236 30 3] | 9.678895 387 9-943830 II4 i 9-735066 502 || 10-264934 29 32 679.128 387 943761 I 14 735367 502 264.633 28 33 679360 387 943693 115 735668 501 964332 27 34 7.9592 387 943624 115 735969 501 264.031 26 35 679824 386 943555 115 736269 501 263731 25 36 | 680056 386 943486 115 T36570 501 96.3430 24 37 680288 386 943417 115 736871 501 263129 23 38 680519 385 943.348 115 737 171 500 262829 22 39 680750 385 943.279 115 73747] 500 262529 21 40 680982 • 385 943.210 115 737771 500 262229 20 41 9-68.1213 385 9-94.3141 I]5 || 9-738071 500 | 10.261929 |. 19 42 681443 384 943072 115 4383.71 500 261629 J8 43 681674 384 943003 II5 738671 499 26].329 I? 44 681905 384 94.2934 II5 738971 499 26) ()29 16 45 6821.35 384 942864 I 15 739271 499 260729 15 46 68.2365 383 942.795 J 16 739570 499 360430 14 47 682595 383 942.726 1}6 739870 499 260130 13 48 682825 383 94.2656 116 740169 499 2598.31 12 49 68.3055 383 94.2587 I 16 740468 498 250532 11 50 | 683284 382 94.2517 116 740767 498 259233 10 51 9'683514 382 9-942448 II6 9-74] ()66 498 || 10:258934 9 52 6837.43 382 942378 116 74 1365 498 258635 8 53- | 683972 382 942308 II6 741664 498 258.336 7 54 58420H 381 942.239 II6 74]962 497 258938 6 55 684430 381 942.169 II6 742.261 497 257739 5 56 684658 381 942099 116 742559 497 257441 4 57 684887 380 94209.9 | 16 742858 497 957 j49 3 58 685] 15 380 94 1959 I 16 743 156 497 256.844 2 59 685343 380 941889 117 | 743.454 497 256546 I 60 685571 380 941819 II? 743752 496 256248 0 | Cosine I Sine I' | Cotang. | | Tang. M. 61 Degrees. LOGARITHMIC SINES, COSINES, ETC. (29 Degrees.) 227. | Cotang. M. Sine | D. | Cosine D. Tang. | D. * 0 9-685571 380 9.94 1819 iI7 9-743752 496 || 10°256248 || 60 l 685799 379 94.1749 II 7 744050 496 255.950 || 59 2 | 686027 379 941679 117 744348 496 255652 58 3 | 686254 379 94]{509 J 17 744645 496 25.5355 || 57. 4 686482 379 94 |539 I 17 744943 496 255057 || 56 5 686709 378 94 |469 J I7 745240 496 254760 55 6 6869.36 378 94 1398 J 17 745.538 495 254462 54 7 | 687].63 78 9413:28 1.17 745835 495 254165 53 8 || 687389 378 94 1258 | | 7 746132 495 25.3868 || 52 9 | 6876 16 377 94.1187 I |7 74.6429 495 25.3571 51 |(} 68.7843 377 941 117 I 17 746726 495 25.3274 50 II | 9-688069 377 || 0-94]{)46 118 9.747023 494 || 10-252977 || 49 J2 688295 377 940975 J 18 747319 494 252681 || 48 J3 | 688521 376 94()!}{}5 I 18' 747616 494 252384 || 47 14 | 6887.47 376 940.834 118 747.913 494 252087 46 I5 | 6889.72 76 94()763 II8 748209 494 251791 || 45 I6 | 689198 76 94.0693 118 748505 493 251 495 || 44 J7 | 689423 375 94.0622 I 18 748801 493 25 | 199 || 43 18 || 68964.8 75 94055.1 118 749097 493 250903 || 42 19 | 6898.73 375 940480 118 749393 493 250607 || 41 20 | 6900.98 75 940409 118 749689 493 250311 || 40 21 |9-690.323 374 || 0-9403.38 118 || 9,749985 493 || 10-250015 || 39 22 || 690548 374 94()?67 118 75(;281 492 249719 || 38 23 || 690772 374 94() 196 118 750576 492 24.9424 || 37 24 || 690996 374 94()125 119 || 750872 492 249]28 || 36 25 || 6.91220 373 940ſ).54 119 75] 167 492 248833 || 35 26 || 6.91444 373 93.9982 119 751462 492 2485.38 34 27 69.1668 373 93991 1 | 19 75.1757 492 248243 || 33 28 || 6.91892 373 939840 119 752052 491 247948 || 32 29 || 6921 15 372 93.97(58 I 19 752347 491 247653 || 31 30 | 692339 372 93.9697 119 752642 491 247358 || 30 31 ||9-692562 372 | 9-9.396.25 119 || 9-752937 491 || 10-247063 29 32 || 692785 371 939554 119 753231 491 246769 || 28 33 603008 371 939482 I 19 753526 491 246474 27 34 || 693231 371 9394.10 119 753820 490 246.180 || 26 35 | 693453 371 93.9339 119 754II5 490 245885 25 36 6936.76 70 939.267 120 754.409 490 245591 || 24 37 || 693898 370 939.195 120 754.703 490 245297 || 23 38 694.120 370 939.123 I20 754997 490 245003 || 22 39 || 69.4342 370 93.9052 I20 755201 490 244709 || 21 40 || 694564 , 369 938980 12() 7.55585 489 244415 || 20 41 ||9-694786 369 |9.938008 120 9-755878 489 10.244.122 || 19 42 695007 369 938836 120 756172 489 243828 18 43 || 695229 369 938763 120 756465 489 243535 | 17 44 || 6.95450 368 938691 . 120 756759 489 24324] 16 45 || 6956.71 368 938619 Hy:0 \ , 757052 480 242048 || 15 46 || 695892 368 93.8547 120 757.345 488 242655 14 47. 6961.13 368 9.38475 120 757638 488 242362 || 13 48 || 696334 367 938.4()2 121 757.03] 488 242069 || 12 49 || 696554 367 938.33() 121 758224 488 241776 || 11 50 | 6.96775 367 938.258 121 7585.17 488 241483 10 51 ||9-696995 367 || 0-938.185 121 || 9-758810 488 || 10:241 190 9 52 | 697:215 366 938] I3 121 759 102 487 240898 8 53 || 697435 366 938040 191 759395 487 240605 7 54 || 697654 366 937.967 121 759687 487 240313 6 55 | 697874 366 937895 121 750979 487 24002I 5 56 || 698094 365 937822 J21 760272 487 2397.28 4 57 | 6983;3 365 93.749 121 760564 487 239436 3 58 || 698532 365 93.7676 121 760856 486 239].44 2 59 || 69875] 365 937 604 191 76] 148 486 238852 l 60 l 698970 364 93.7531 121 76.1439 486 23856] 0 | Cosime - Sine | Cotang. I | Tang, || M. 60 Degrees, 228 (30 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. I. Sine D. Cosine | D. Tang. D. 1 Cotang. () 9:698970 364 9.937531 121 || 0-761439 486 || 10-238561 60 I 699,189 364 937458 122 761731 486 238269 j9 2 (399.407 364 937385 122 762023 486 237977 58 3 699626 || 364 || 937.312 |. 122 |* 762314 || 486 237686 57 4 6998.44 363 937238 122 762606 485 23.7394 56 5 700062 363 937165 122 762897 485 237103 55 6 700280 363 937092 122 763.188 485 236812 54 7 700498 363 937019 122 763479 485 236521 53 8 700716 363 936.946 122 76.3770 485 236230 32 9 700933 362 9:36872 I22 764061 485 235939 51 10 701.151 || 362 936799 122 764352 484 235648 50 11 9-701368 362 9.936725 122 ||9-764643 484 || 10-235,357 49 12 701585 362 936652 . I23 764933 484 235067 48 13 701802 361 936578 123 765224 484 234776 47 14 702019 36] 936505 123 765514 484 234486 46 15 702236 361 936431 123 76.5805 484 234195 45 I6 702452 || 361 930.357 123 7(56095 484 233905 44 17 702669 360 936.284 123 766385 483 || 233615 43 I8 702885 360 936.210 1z3 766(375 483 233325 42 19 703101 360 936136 I23 766965 483 233035 41 20 703317 360 936062 I23 767255 483 232745 40 21 9-703533 359 |9.935988 123 ||9-767545 || 483 |10-232455 39 22 703749 359 935914 123 767834 483 232166 38 23 703964 359 || 93.5840 123 768124 482 231876 37 24 || 7041.9 359 935766 124 7684.13 482 231587 || 36 25 704395 359 935692 124 768703 482 || 231297 35 25 TÜ4(310 3.j8 935.018 I:24 7'08392 482 2,31UU8 34 27 704825 358 935543 124 769281 482 230719 33 28 705040 |. 358 935469 I24 769570 482 230430 32 29 705254 358 935395 124 769860 481 230140. 31 30 705469 357 935320 124 770,148 481 229852 30 31 9-705683 357 ||9-935246 124 || 9-770437 481 || 10-229563 29 32 705898 357 935171 124 770726 481 229274 28 3 706I12 357 935097 124 771015 481 228985 27 34 706326 356 935022 124 771303 481 228697 26 35 706539 356 934948 124 771592 481 228408 25 36 706753 356 934873 124 771880 480 2281.20 24 37 706967 356 934798 125 772.168 480 2278.32 23 38 707180 355 934723. 125 772457 480 227543 22 39 707393 355 934649 125 772745 480 227255 2I 40 707606 355 934574 125 "73033 480 226967 20 41 9.707819 355 9-934499 125 |9-773321 480 || 10-226679 19 42 708032 354 934424 I25 773608 479 226392 18 43 708245 354 934349 I25 |. 77.3896 479 2261()4 17 44 || 708458 354 934274 125 774.184 479 225816 16 45 || 708670 354 934 199 I25 774471 479 225.529 15 46 708882 353 934.123 195 774759 479 225241 14 47 700094 353 93.4048 125 775046 479 224954 13 48 709306 353 933973 125 775333 79 224667 12 49 709518 353 933898 126 775621 478 224379 11 50 7097.30 353 933822 126 77.5908 478 224092 10 51 || 9-70994.1 352 9.933747 I26 ||9-776195 478 || 10-223805 9 52 || 710153 352 933($71 126 776482 478 2235.18 8 53 710364 352 93.3506 126 776769 478 |. 223231 7 54 || 7105.75 352 93352() I26 777055 478 222945 6 55 || 710786 35I 933445 126 777342 478 222658 5 56 710997 351 933.369 I26 777628 477 222372 4 57 7II208 351 933293 126 777015 477 ° 222085 3 58 7I1419 351 03:3217 126 77820.1 477 221799 2 59 7I1629 350 933141 126 778487 477 221512 1 60 "I J339 350 ° 933066 126 778774 477 221226 0 | Cosine ! | Sine | | Cotang. I | Tang. J M. 59 Degrees, I.OGARITIIMIC, SINES, COSINES, ETC. (31 Degrees.) 229 | Cotang. I 58 Degrees, M. Sine | D. Cosine D. Trng, D. 0 |9-711839 350 9-93.3066 126 9-778774 477 | 10-221226 60° I 712050 350 93.2990 127 779060 || 477 220.940 59 2 712260 350 932914 - || 127 779.346 || 476 220654 || 58 3 71.2469 349 932838 | 127 779632 476 220368 || 57 4 712679 349 932762 | 127 779918 476 220082 56 5 7 12889 349 932685 # 127 780.203 476 219797 55 6 .713098 349 932609 || 127 780489 476 2195] 1 54 7 71330S 349 932533 | 127 78()775 476 219225 || 53 8 713517 348 || 932457 | 127 781060 76 218940 || 52 9 713726 348 || 932380 | 127 781346 475 218654 || 51 I0 713935 348 || 93.2304 || 127 || 781631 475 2.18369 50 11 ||9-714144 348 ||9-932228 || 127 9-781916 475 10-218084 49 12 714352 347 || 93.2151 | 127 782201 475 217790 48 13 714561 347 || 93.2075 128 78.2486 475 217514 47 14 || 714769 347 || 931998 128 782771 47 217229 46 I5 714978 347 || 93.1921 128 78.3056 475 2.16944 45 16 715186 347 || 93.1845 128 78.3341 475 216659 44 17 715394 346 || 93.1768 128 78.3626 474 216374 43 18 715602 346 || 931691 128 , 783910 474 216090 42 19 || 715809 346 931614 128 784-195 474 215805 41 20 716017 346 || 931537 128 784479 47 21552] 40 21 || 9:716224 345 ||9-93.1460 128 9.784764 474 ||10–215236 39 716432 345 || 93.1383 128 || 785048 474 2I4952 38 23 716639 345 | 931306 128 785332 473 214668 37 24 716846 345 931229 199 785616 473 214384 36 25 717053 345 # 93] 152 129. 785900 473 214,100 35 26 717259 344 93.1075 199 '786.184 473 213846 34 27 717466 344 930998 129 786468 473 213532 33 28 717673 344 930921 129 786752 473 213248 32 29 717870 344 930843 129 787.036 473 212964 31 30 718085 343 930.766 129 787319 472 212681 30 31 ||9-718291 343 |9930688 129 9.787603 || 472 |10212397 29 32 718497 343 || 93061 I 129 || 787886 472 212114 28 33 718703 343 930533 I20 788170 472 21 1830 27 34 7 18909 343 930456 129 788453 472 2] 1547 26 35 719114 342 930378 129 788736 472 2[1264 25 36 719320 342 930.300 130 789019 472 210981 24 37 719525 342 930.223 130 789.302 471 210698 23 38 7 19730 342 930.145 I30 789585 471 2104.15 22 39 7|19935 341 || 9300.67 I30 789868 471 210132 21 40 720,140 341 929989 i80 790.151 471 209849 20 41 9-720.345 341 ||9-929911 130 ||9-790433 471 10-200567 19 42 720549 34I | 9298.33 330. 790716 471 209.284 18 43 720754 340 || 9297.55 130 790999 471 209001 17 44 || 720958 340 92.9677 130 791281 471 2087.19 16 45 721162 340 || 929.599 J30 7.91563 70 208437 15 46 721366 340 929.521 130 791846 470 208154 14 47 721570 340 929.442 130 79.2128 470 207872 13 48 721774 339 929364 I31 79.2410 470 207590 12 49 721978 339 929.286 13] 79.2602 470 207308 II 50 722.181 339 929207 131 792974 470 207026 I0 51 | {}•722385 339 9-929129 I31 9-703256 470 ||10-206744 9' 52 722588 339 92.9050 131 793538 469 206462 8 . 53 || 722791 338 || 928.972 | 131 || 793819 469 206181 7 54 722994 338 928893 131 794.101 469 205899 6 55 | 723197 338 9288.15 134 794383 469 205617 | 5 56 723400 338 928.736 131 794664 469 205336 4 57 723603 337 9.28657 i31 794945 469 205055 3 58 723805 337 9.28578 13I 795.227 469 204773 2 59 724()07 337 928499 I31 795.508 468 204492 I 60 's 724210 337 928420 131 795789 468 204211 - 0 | Cosine | Sine | Cotang, Tang, M. 230 (32 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. : Sine D. : Cosine | D. ( Tang. D. Cotang. 0 97.24:210 337 ||9-928420 132 9-795789 468 || 10.20421] 60 } 724412 337 928.342 132 796070 468 203930 59 2 724.614 336 928263 132 796351 468 203649 58 3 724816 336 928.183 132 796632 468 203368 57 4 725()17 336 928.104 132 7969.13 468 203087 56 5 7252.19 336 92.8095 I32 797194 468 202806 55 6 T25420 335 927946 132 7.97475 468 202525 54 7 725622 335 927867 132 797755 468 2022.45 53 8 725823 335 9.27787 I32 798036 467 20 1964 52 9 726024 335 9.27708 I32 7983]6 467 201684 51 10 726225 335 92.7629 132 '798596 467 || 201404 50 11 || 9:726426 334 9-927549 132 ||9-798877 467 10-201123 - || 49 12 26626 334 927470 133 799157 467 200843 48 13 726827 334 92.7390 133 799437 467 200563 47 14 727(327 334 927.310 133 7.99717 467 200283 , || 46 15 727228 334 927231 133 799997 466 200003 45 I6 727428 333 927.151 133 800.277 466 1997:23 44 I7 727628 333 92.7071 133 800557 466 199443 43 I8 727828 333 926994 . 133 800836 466 199164 42 I9 728027 333 9269]I 133 80 1116 466 198884 4.1 20 728227 333 92683]. 133 801396 466 198604 40 21 ; 9.728427 332 9-926751 133 ||9-801675 466 10° 198325 39 22 728626 332 926671 133 80.1955 466 198045 38. 23 728825 332 926591 133 802234 465 I97766 37 24 729024 332 9265II 134 802513 465 197487 .36 25 729223 331 926431 134 || 802792 || 465 197208 || 35 96 | 73942? | 33i | 92635i 134 || 803072 465 I96928 34 27 72962] 331 92.6270 134 80335] 465 196649 33 28 7298.20 331 926,190 134 803630 465 196370 32 29 730018 330 926110 134 803008 465 196092 31 30 730216 330 926029 134 804,187. 465 I95813 30 31 9.7304.15 330 | 9-925949 I34 9.804466 464 10° 195534 29 32 730613 330 9.25868 134 804745 464 I95.255 28 33 730811 330 925788 134 805023 464 194977 27 34 73.1009 329 925.707 134 805302 464 194698 26 35 | 731206 329 925626 134 805580 464 194420 25 36 731404 329 9.255.45 135 805859 - 464 I94141 24 37 73.1602 329 925.465 I35 806137 464 193863 23 38 73.1799 329 925.384 135 | . 806415 463 I93585 22 39 73]996 328 925.303 135 806693 463 I93307 21 40 7321.93 328 925.222 135 806971 463 1930.29 20 41 9-782300 328 9.92514] 135 H 9-807249. 463 10, 192751 19 42 | 732587 || 328 925060 || 135 8075.27 | 403 I92473 18 43 732784 || 328 924979 || 135 807805 || 463 I92195 || 17 44 73.2980 || 327 | 924807 || 135 808083 463 191917 16 45 7.33177 327 924816 135 808361 463 191639 15 46 733373 327 924735 36 808638 462 191362 14 47 733569 327 924654 I36 808916 462 191084 13 48 733765 327 924572 136 800193 462 190807 12 49 733961 326 9.24491 I36 80947? 462 190529 11 50 734,157 326 924400 136 809748 462 190252 I0 51 || 0-734353 326 9-924.328 136 9-8100.25 462 10, 1890.75 9 52 '734549 326 924246 136 810302 462 1896.98 8 53 734744 325 924164. 136 810580 462 180420 7 54 734939 325 93.4083 136 810857 462 189143 6 55 735135 325 924001 I36 811134 461 188866 5 56 735.330 325 92.3919 . 136 811410 461 188590 4 57 7355.25 325 923837 136 81:1687 461 188313 3 58 735.719 324 92.3755 137 811964 461 188036 2 59 735914 324 923673 137 812241 461 187759 I (;0 736,109 324 923591 I37 81.2517 461 187483 s! 0 | Cosine | Sine ..] Cotang. | | ‘Tang. I M. 57 Degrees. LOGARITIIMIC SINES, COSINES, ETC. (33 Degrees.) 231 56 Degrees. M. 1 Sine ! D. Cosine D. Tang. D. | Cotang. I 0 9:736109 324 ||9-923591 137 ||9-812517 461 10, 187482 60 I 736,303 324 923509 I37 812794 461 187206 || 59 2 736498 324 923497 137 813070 461 186930 58 3 736692 323 92.3345 I37 813347 460 I86653 57 4 736886 32.3 923263 137 813623 460 186377 56 5 737()8() 323 92318] I37 813899 460 186101 55 6 737274 323 92.3098 I37 814175 460 185825 54 7 737467 323 923016 I37 814452 460 185548 - || 53 8 737661 322 922.933 I37 814728 460 185272 52 9 737855 322 92285.1 ..I.37 815()()4 460 184996 || 51 10 | 738048 322 92.2768 I38 815279 460 184721 50 II 9-738241 322 ||9-922686 138 9-815555 459 || 10:184445 49 12 738434 322 92.26()3 |38 8.15831 459 184169 || 48 13 738627 32I 922520 138 816107 459 I83893 47 14 738820 321 92.2438 138 816382 459 I83618 || 46 15 739013 321 922355 138 816658 459 183342 || 45 16 739206 321 92.2272 138 816933 459 183067 || 44 17 739398 321 922.189 138 817209 459 182791 43 18 739590 320 922IG6 138 817484 459 182516 || 42 I9 739783 320 92.2023 I38 8.17759 459 182241 41 20 739.975 320 92.1940 138 818035 458 I81965 | 40 21 9:740167 320 | 9-92.1857 139 ig-818310 458 || 10:181690 || 39 22 740359 320 92.177 139 818585 458 181415 || 38 23 740550 319 92.1691 139 818860 458 181140 37 24 740742 319 021607 l39 819]35 458 I80865 || 36 25 740934 319 92.1524 I39 819410 458 180590 35 26 74 IJ25 319 921441 139 819684 458 180316 || 34 27 74.1316 3.19 92I357 139 819959 458 180041 33 28 74.1508 3.18 921274 139 820234 458 I79766 32 29 74.1699 || 318 921 190 139 820508 457 I79492 31 30 74.1889 3.18 92] 107 139 $20.783 457 . . 179217 30 31 || 9,742080 318 9.92.1623 139 9-8.21057 45 10:178943 29 32 742.271 3.18 920939 140 82.1332 457 178668 28 33 74:2462 3I? 926856 J40 821606 457 178304 || 27 34 74.2652 317 920.772 140 821880 457 178120 26 35 742.842 317 020688 140 822154 457 |, 177846 || 25 36 743033 3I7 920604 140 822429 457 I77571 || 24 37 74.3223 317 |, 920.520 140 822703 - || 457 177297 23 38 743413 316 926436 140 822977 456, 177023 22 39 743602 316 920352 140 S23250 456 176750 21 40 743.792 316 926:268 140 823524 456 176476 | 20 41 || 9,743982 316 ||9-920184 140 ||9-823798 456 || 10-176202 || 19 42 744,17] 316 92.0099 140 824072 #56. I75928 18 43 744.361 315 920()ij 140 | , 8.24345 456 175655 17 44 7445.50 315 9.1993] 141, 824619 456 I73,381 16 45 744739 315 9.19846 141 82.4893 456 175107 15 46 744928 315 919762 I4]: 825166 456 174834 || 14 47 745,117 315 919677 141 825439 455 I74561 13 48 745306 3.14 9.19593 141 825713 455 174287 12 49 745494 314 919.508 141 825986 455 174014 11 50 745683 314 919424 141 -826259 455 173741 10 51 9.745871 314 ||9-919339 I41 || 0-826532 455 | 10-173468 9 52 746059 314 919.254 141 826805 455 173195 8 53 746248 313 919 169 141 82.7078 455 172922 7 54 74.6436 313 919085 141 827351 455 I72649 6 55 746624 3.13 919000 141 827624 455 172376 5 56 746812 3.13 918915 I42 827897 454 I72103 4 57 7.46999 313 918830 I42 828I70 454 I71830 3 59 747187 312 9.18745 142 828442 454 I71558 2 59 74.7374 312 9.18659 142 828715 454 171285 - || 1 60 747562 312 9.18574 142 82.8987 454 171013 0 i Cosine | ! Sine . . Cotang. | | Tang. M. 232 (34 Degrees.) LoGARITHMIC SINES, CoSINES, ETC. .” M. | Sine | D. . Cosine D. Targ. ſ D. 1 Cotang. I 0 9-747562 312 9-9.18574 142 9-82.8987 454 10° 171013 60 l 747749 312 918489 J42 829260 454 I70740 59 2 747s)36 312 918.404 14:2 8295.32 454 170468 58 3 || 748.123 311 918318 142 8.29805 454 1701.95 57 4 748,310 3.11 9.18233 142 83iſt)77 454 169923 56 5 748497 311 948.147 I42 83U349 453 169651 55 6 748683 311 918(362 142 83i)621 453 169379 54 7 | < 748870 311 917976 J43 83U893 453 169107 53 8 743.056 310 947891 143 83] 165 453 168835 52 9 743243 310 917855 143 834437 453 168563 51 10 | . T494.29 310 917.719 143 831709 453 16829] 50 11 || 9-74.9615 310 || 9-9.17634 143 9-83.198 453 || 10-168019 49 I2 749801 310 917548 143 832253 453 167747 48 13 749987 309 9.17462 143 832525 453 167475 || 47 14 750.172 309 .91.7376 143 832796 453 167204 || 46 lö 750358 309 947290 143 833068 452 166932 45 16 750543 309 917.204 143 833.339 452 166661 || 44 17 750.729 309 917.118 144 833611 . . 452 166389 || 43 18 , 750914 308 917032 144 833882 452 1661.18 42 19 75.1099 308 916.946 144 834,154 452 165846 41 20 75.1284 308 916859 144 834425 452 165575 | 40 21 | 9.751469 || 308 || 9.916773 144 9.834696 452 || 10-165304 || 39 22 || 751654 308 916687 144 834967 || 452 165033 || 38 23 75.1839 || 308 916600 144 835238 452 164762 || 37 24 752623 || 307. 916514 144 835509 || 452 164491 || 36 25 || 752208 || 307 916427 144 835780 || 451 164220 35 26 752392 307 9.16341 144 836051 451 !63949 || 34 27 | '152576 || 307 913254 144 836322 || 451 163678 || 33 28 || 752760 307 916167 145 836593 451 163407 || 32 29 || 752944 306 916081 145 836864 451 163136 || 31 30 || 753128 306 915994 145 || 837134 451 162866 || 30 31 || 9-753312 306 ||9-915907 145 || 9.837.405 451 10:162595 29 32 || 753495 306 915820 145 837675 451 162325 28. 33 75.3679 306 915733 145 837.946 451 162054 27 34 753862 305 9.15646 145 838.216 451 161784 26 35 | 754046 305 9.15559 145 838487 450 161513 25 36 754229 305 915.472 145 838757 450 161243 || 24 37 || 754412 305 9.15385 145 8390.27 450 160973 23 38 754595 305 9.15297 145 839297 450 160703 22 39 754778 304 9.15210 145 839568 || 450 160432 21 40 || 754960 304 915.123 146 839838 450 160162 20 41 || 9,755.143 304 ||9-915035 146 ||9-840.108 450 | 10:15.9892 || 19 42 || 755326 304 914948 146 840378 450 159622 | 18 43 755508 304 914860 146 840647 450 159353 || 17 44 || 755690 304 914773 146 840917 449 159083 16 45 || 755872 303 914685 146 841187 449 || 158813 15 46 756654 303 914538 146 84.1457 449 158543 14 47 || 756.236 303 914510 146 84.1726 449 158274 13 48 || 756418 303 9,14422 146 84.1996 449 158004 || 12 49 || 756600 303 914334 146 84.2266 449 157734 I 1 50 || 756782 302 914246 147 842535 449 157465 10 51 | 9-7.56963 302 || 9-9.14158 147 |9-849.805 449 || 10-157195 9 52 757144 302 914070 147 843074 449 156926 8 53 || 757326 302 9]3982 147 843343 449 156657 7 54 757507 302 913894 147 843612 449 156388 6 55 || 757688 301 913806 147 843882 448 156118 5 56 757869 301 913718 147 84415] 448 155849 4 57 || 758050 301 913630 147 84.4420 .448 155580 3 58 || 758230 301 91354I 147 844689 448 1553] 1 2 59 || - 7584.11 301 913453 147 84.4958 448 155042 i 60 ! 758591 301 913365 147 845227 448 15.4773 0 ! Cosine ! | Sine . . | Cotang. I | Tang. | M. 55 Degrees. LOG ARITIIMIC, SINES, COSINES, ETC. 233 (35 Degrees.) 54 Degrees. M. : Sine D. Cosine D. Tang. I D. Cotang. ) 0 | 9-758591 301 || 9°913365 147 9.8459.27 448 || 10-154773 60 l 758772 300 913.276 147 845496 448 154504 59 2 758952 300 913187 148 845764 448 I54236 53 3 750 132 300 91309.9 148 846033 448 153967 57 4 759312 300 913010 148 8463()2 448 I536.98 56 5 759492 300 912922 148 846570 447 15.3430 55 6 7596.72 299 9]2833 148 846839 447 153161 54 7 759852 299 912744 148 847 107 447 I52893 53 8 760031 299 912655 148 847376 447 152624 52 9 .7602.11 299 912566 148 S47644 447 152356 51 10 760390 299 912477 148 847.913 447 152087 50 11 || 9-760569 298 || 9-91.2388 148 || 9.848.181 447 || 10: 151819 49 12 760748 298 912.299 149 8484.49 447 151551 48 13 760027 298 912210 149 8487.17 447 151283 47 14 761106 298 912.121 149 848986 447 151014 46 15 76.1285 298 912931 149 849.254. 447 150746 45 16 761464 238 9; 1942 149 849522 447 150478 -| 44 17 76.1642 297 911853 149 84979 446 150210 43 I8 761821 . .297 9] 1763 149 850658 446 149942 42 . 19 761999 297 91.1674 149 850325 446 149675 |,41 20 .762177 297 911584 149 850593 446 149407 40 21 9-762356 297 || 9-91,1495 I49 || 9.850861 446 |S|0"149139 30 22 762534 296 91.1405 149 85] 129 446 148871 38 23 7627 12 296 911315 150 85.1396 446 148604 37 24 7.62889 296 91 1226 150 85.1664 446 148336 36 25 763067 296 911 136 150 85.1931 446 148069 35 26 763.245 296 9| 1046 150 85.2199 446 I47801 34 27 763422 296 910956 I50 85.2466 446 I47534 33 28 763600 295 910866 I50 8:52733 445 147267 32 29 76.3777 295 9,10776 150 853001 445 146999 31 30 763954 295 910686 150 853268 445 146732 30 31 || 9,764]31 295 9-910596 150' | 9-85.3535 445 10° 146465 || 29 32 764308 295 910506 150 85.3802 445 146198 28 33 764-485 294 910415 150 854.069 445 145931 27 34 764662 294 9}{}325 151 8543.3 445 145664 26 35 764838 294 910235 15i 85.4603 445 145397 25 36 765015 294 910144 151 854870 445 145130 24 37 765,191 294 910054 151 8551.37 445 . 144863 || 23 38 76.5367 294 90.9963 151 855.404 445 144596 22 39 76.5544 293 909873 151 855671 444 144329 21 40 765720 293 909782 151 855.938 444 144062 20 41 || 9-765896 293 || 9-909691 151 9°856.204 444 10° 143796 19 42 766072 203 90960| 151 856471 444 143529 18 43 766247 293 9095.10 I51 856.737 444 143263 || 17 44 766423 293 9094.19 151 857004 444 142996 6 45 766598 292 909328 152 $357.270 444 I42730 5 46 766774 292 909237 152 857537 444 1424.63 4 47 766949 292 909146 152 857803 444 142197 13 48 767 124 292 900055 152 858069 444 14 1931 12 49 767300 202 908964 I52 858.336 444 141664 11 50 767475 291 908873 152 858602 443 141398 |0 5] | 9,767649 || 291 || 9.908781 | 152 || 9-858868 || 443 10-141132 9 52 767824 291 908690 152 859134 443 140866 8 53 767999 291 9085.99 152 859.400 443 140600 7 54 768173. 291 90S507 152 859666 443 140334 6 55 768348 200 9084.16 I53 859932 443 140068 5 56 768522 290 908324 153 86() 198 443 139809 4 57 768697 290 908233 I53 860464 443 I39536 3 58 768871 290 908141 153 86()730 443 130270 2 59 760045 290 908{}49 153 860995 443 139005 I 60 l 769219 290 907958 153 861261 443 138739 0 | Cosine. } l Sine ! | Cotang. I | M. | Tang. 234 (36 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. I. Sine | D. Cosine | D. ( Tang. | D. 1 Cotang. 0 9-7692.19 290 || 9.907958 153 9-861261 443 | 10-138739 60 I 769393 289 907866 153 861527 443 138473 59 2 769566 289 907774 I53 86.1792 442 I38208 58 3 769740 289 90.7682 153 862058 442 137942 57 4 7699.13 289 907590 I53 86.2323 442 137677 56 5 770037 289 907498 153 86.2589 44.2 137411 55 6 770260 2 9{}74(36 I53 86.2854 442 1371.46 54 7 770433 288 907314 154 86.3] 19 442 136881 53 8 770606 288 907222 154 863.385 442 1366.15 52 9 770779 288 907129 154 863650 442 136350 51 10 770952 288 907037 154 863915 442 136085 50 II | 9.771 125 288 9-906945 154 9-864I80 442 | 10-135820 49 12 77 1298 287 906852 154 864.445 442 135555 48 I3 771470 287 906760 154 864710 442 135290 47 14 771643 287 906667 154 864975 || 441 I35025 46 15 771815 287 906575 154 865240 441 134760 45 16 |. 771987 287 906482. 154 865505 441 I34495 44 I7 772150 287 906389 155 86577 441 13423() 43 18 772331 286 906296 155 || 866035 441 J3.3965 42 I9 || 772503 286 906.204 I55 866.300 441 133700 41 20 772675 286 906] 11 155 866564 44.1 133436 40 21 9-772847 286 9-906018 155 9-866829 44.1 || 10-133171 39 22 77.3018 286 905925 155 | . 867094 441 132906 38 93 773.190 286 905832 155 S67358 441 132642 37 24 773.361 285 905.739 155 867623 441 I32377 36 25 || 7735.33 285 905645 155 8b7887 441 1321 13 || 35 26 77.3704 285 905552 155 || 868152 44() I31848 34 27 | 77.3875 285 905459 155 8084Ib 44U J.31584 || 33 28 774046 285 905.366 156 868UC0 440 131320 32 29 774217 285 905272 156 868945 440 J3 |(}55 31 30 774388 284 905179 156 869209 440 g30791 30 31 || 9.774558 284 9-905085 | . 156 9.869473 440 || 1ſ). I50527 29 32 774729 284 904992 156 869737 440 130263 28 33 774899 284 904898 156 87()001 440 J29099 27 34 775070 284 904804 156 87{}265 440 I?0735 26 35 775.240 284 9047II 156 870529 440. 1294.71 25 36 775410 283 9046.17 156 870793 440 1292()7 24 37 775580 , 283 904523 156 87I057 440 128943 23 38 775750 283 904429 157 87.1321 440 1286.79 22 39 775020 283 9()4335 157 87.1585 440 . 1284.15 21 40 776690 283 90424] 157 87.1849 439 128151 20 41 9-776250 283 9-904147 157 9-87.2112 439 10-127888 19 42 776429 282 904053 157 872376 4:39 127624 18 43 776598 282 903959 157 87.264() 4:39 127360 17 44 776768 282 903864 157 872903 439 127097 16 45 776937 282 903770 157 873167 439 126833 15 46 777106 282 903676 157 873430, 439 126570 |4 47 777275 281 903581 157 873694 439 I263(16 I3 48 777.444 281 903487 157 87.3957 439 126(43 12 49 777613 281 903392 158 74220 439 125780 | 1 50 777781 281 903.298 158 874484 439 I2551.6 10 51 9.777950 281 || 9.903203 158 9-874747 439 || 10-125253 9 52 778] 19 281 903] 08 . 158 87.5010 439 124990 8 53 778287 280 903(; 14 158 75273 438 124727 7 54 778455 280 9(329 IQ 158 875536 438 124464 6 55 778624 280 902824 158 875800 438 124200 5 56 778792 280 902729 158 876063 438 I23937 4 57 778960 280 902634 158 876326 438 193674 3 58 779.128 280 902530 159 876589 438 1234 II 2 59 770295 279 902444 159 876851 438 123149 l 60 779463 279 902349 159 877] 14 438 192886. 0 | Cosine | Sine | | Cotang. | Tang. M. 53 Degrees. 2 LOG ARITH MIC SINES, COSIWES, ETC. (37 Degrees.) '235 M. Sine | D. | Cosine | D. I Tang. D. Cotang. 0 | 9.779463 279 9.902349 159 || 0-877I14 438 10-122886 60 H 779631 79 902253 159 877377 438, 122623 59 2 779798 279 902158 159 877640 438 122360 58 3 779966 279 902063 159 877903 4.38 122097 57 4 780 133 79 90 j967 159 878.I65 438 121835 56 5 780300 278 90 1872 I59 78428 438 121572 55 6 780467 278 90ſ 776 159 878691 438 121309 54 7 780634 278 90]681 159 878.953 437 121047 53 8 780801 78 901585 159 879216 437 120784 52 9 780968 278 901490 J59 879478 437 120522. 51 10 78] 134 278 90 1394 160 879741 437 120259 50 II | 9-781301 277 | 9-90.1298 I60 0-880003 437 10-119997 49 12 78.1468 277 901202 160 880265 437 1 19735 48 I3 78.1634 277 901106 160 880528 437 119472 47 14 78.1800 277 901010 I60 880790 437 119210 46 15 78.1966 277 900914 160 88 i()52 .437 118948 45 16 782132 277 900818 160 881314 437 I 18686 44 I7 782298 27 900.722 J60 88.1576 437 I 18424 43 18 78.2464 276 900626 160 88.1839 437 118161 42 19 782630 76 900599 160 882] () L 437 I 17899 41 20 7827.96 27 900433 161 882363 436 1176.37 40 21 9-782961 76 9-000337 161 |9-882625 436 10.117375 39 22 7831.27 276 900240 161 882887 4.36 II 71.13 38 23 783.292 75 9001:44 ] [5] 883].48 436 116852 37 24 783458 275 900047 16] 88.341() 436 II6590 36 25 78.3623 275 899951 161 883672 436 116328 35 26 78.3788 75 899854 j61 883934 436 116066 34 7 783953 275 8997.57 161. 8841.96 436 115804 33 28 784] 18 75 899660 161 884.457 - || 436 115543 32 29 784282 274 8995[j4 J61 8847 19 436 I 15281 31 30 784447 74 899.467 162 884980 436 I 15020 30 31 || 9,784612 74 ||9-899370 162 |\}:885242 436 10-114758 29 32 784776 274 899.273 162 8.85503 436 II.4497 28 33 78494.1 27 899 |76 I62 885765 436 114235 27 34 785) ()5 274 809078 I62 886()26 436 113974 26 35 785269 273 898981 162 886288 436 | 137:2 25 36 785433 273 898.884 162 886549 435 I 13451 24 37 785597 27 808787 162 886810 435 113190 23 38 785761 273 898689 I62 887()72 435 II.2928 22 39 785925 273 898592 162 887333 435 112667 21 40 786089 73 898494 163 887594 435 112406 20 41 || 9-786.252 272 9-898397 163 |9.887855 435 10-112145 19 42 786416 272 898.299 163 888] 16 435 Il 1884 18 43 78657 272. 898.202 163 , 888377 435 II 1623 17 44 786742 272 898104 163 8886.39 435 11||1361 16 45 786906 272 898006 163 888900 435 111100 15 46 787()69 272 || 897908 I63 |, 889.160 435 II ()840 14 47 787.232 27] 897810 163 8894.21 435. 110579 }3 48 787395 271 89.7712 163 889682 435 110318 12 49 787557 271 897614 163 88994.3 435 II0057 ll. 50 7877.20 271 897516 163 890.204 434 109796 | 10 51 9-787883 71 9,897.418 I64 9-890.465 434 10:109535 9 52 788045 271 897320 ]64 89()725 434 109275 8 53 788208 271 897.222 164 89()986 434 109014 7. 54 788370 7() 897 123 164 - 89 247. 434 108753 6 55 788532 270 | . 8970.25 164 89.1507 '434 Jſ)8493 5 56 788694 27 806026 J64 89.1768 434 J08232 4 57 78.8856 70 896828 164 892028 434 107972 3 58 7890 18 270 896799 J64 892.289 434 1077 Il 2 59 789 180 270 89663 | 164 892549 434 I ()7451 l 60 789342 269 896532 I64 892810 434 107190 0. | Cosime | Sine | | Cotang. | - Tang. | M. 52 Degrees. 236 (38 Degrees.) LOGARITHMIC SINES, COSIWES, ETC. M. Sine D. Cosine D. Tang. D. Cotang. 0 || 9-789342 269 9-896532 164 9.892810 434 || 10° 107190 t 60 I 789.504 269 896433 J65 893070 434 IU6930 59 , 2 789665 269 896.335 165 89.3331 434 106669 || 58 3 789827 269 896236 165 89.3591 434 106409 || 57 4 789988 269 896137 }65 89.3851 434 } 06.149 || 56 5 790 149 269 896038 I65 894] I l 434 105889 || 55 6 || 7903 |0 268 895939 165 894371 434 I05629 54 7 790471 268 895840 165 894632 433 I05368 53 8 790632 268 895741 I65 89.4892 433 I05108 52 9 790793 268 89.5641 165 895,152 433 I04848 || 51 10 || 790954 268 895542 165 895412 433 I04588 || 50 II 9-79] I 15 268 9-895443 166 9-895672 433 || 10-104328 || 49 12 7.91275 267 895.343 j66 895932 433 j04068 || 48 ]3 || 791436 267 895.244 166 896.192 433 103808 - || 47 I4 791.596 267 895 145 166 896452 433 103548 || 46 lă 79.1757 267 895()45 I j6 896712 433 I03288 || 45 16 791917 267 894.945 I fió 896971 433 I()3029 || 44 I7 792077 267 89.4846 | tit; 897.231 433 I02769 43 18 792237 266 894746 I66 897.491 433 1025()9 || 42 19 79.2397 266 89.4646 166 897751 433 102249 || 41 20 79.2557 266 8945.46 166 8980 10 433 101990 | 40 21 9-79.2716 266 9-894.446 167 9898270 433 10-101730 39 22 2792876 266 89.4346 167 898530 433 101470 38 23 793035 266 89.4246 I67 898789 433 101211 || 37 24 793.195 265 894,146 I67 ‘899(149 432 100.951 36 25 793354 265 89.4046 I67 894)3()8 4.32 i()0692 || 35 26 793514 265 893946 I67 894,568 4.32 Iſſhq92 || 34 2] '1936'73 265 89.3846 I67 899827 4.32 J001.73 || 33 28 793832 265 893745 167 900086 432 0999 || 4 || 32 29 793991 265 893645 167 900346 432 0.99654 || 31 30 794,150 264 893544 I67 900605 432 099395 : 30 31 || 9-794308 264 |9-893444 168 9.900.864 432 || 10-099.136 29 32 794467 264 89.3343 I68 901 124 432 ()!}8876 28 33 794626 264 893243 168 90.1383 432 ()98617 | 97 34 794784 264 893.142 I68 90.1642 432 098.358 26 5 794942 264 893041 I68 90.1901 432 098099 || 25 36 795101 264 8929.40 I68 902160 4.32 097840 || 24 37 795.259 264 89.2839 168 902419 4.32 ()97581 23 38 7954.17 263 892739 168 902679 432 097.321 22 39 7955.75 263 892638 I68 902938 432 097062 21 40 795.733 263 892536 I68 903 197 431 096803 || 20 41 || 9-795891 263 9.892435 169 |9-90.3455 431 || 10-0965.45 19 .42 796049 263 892334 169 903714 431. 096286 18 43 796.206 263 892233 || 169 90.3973 4:31, 006027 17 #4 796364 262 892; 32 169 9042.32 431 09:5768 16 45 796521 262 892030 169 90449 | 431 09:5509 15 46 7966.79 262 89 1929 169 9()4750 431 09:5250 14 47 796836 262 89.1827 169 905()08 431 094992 I3 48 796993 262 891726 I69 905.267 4:31 09:4733 12 49 797 150 261 89.1624 I69 905526 431 094474 11 50 797307 261 89.1523 I7() 905784 431 094216 10 51 | 9-797464 261 || 9-891421 I70 9.906043 431 10-093957 9 52 797621 261 891319 I70 906302 431 0.93698 8 53 797777 261 89.1217 I70 906560 431 093440 7 54 797.934 261 89 || 15 170 9068.19 431 093I81 6 55 798091 261 89 () 13 170 907077 431 09:2923 5 56 798247 261 89()911 I70 9()7336 431 09.2664 4 57 T984()3 260 89(809 I70 90.7594 431 09:2406 3 58 798560 260 89()707 170 - || 907852 431 09.2148 2 59 798716 260 89.0605 170 908 || 1 430 09 1889 1 60 798872 260 89.0503 170 908369 430 091631 0 | Cosine ! - Sine | Cotang. Tang. " M. 5l Degrees, LoGARITHMIC SINES, CoSINES, ETC. (30 Degrees.) 237 50 Degrees, M. Sine | D. Cosine I D. Tang. D. Cotang. ) () 9-798872 260 9-890503 170 9-908369 430 | 10°09'163} | 60 I 799628 260 800400 I71 008028 430 091.372 59 2 || 799,184 260 899.298 I71 908886 430 09] I 14 58 | 3 || 790339 259 || 89.9105 I71 90%)] 44 430 090856 || 57 4 || 799495 259 89.0003 171 900.402 430 000598 || 56 5 || 700051 259 889930 I71 909660 430 090340 || 55 6 || 799806 259 88.9888 I71 900018 430 090082 || 54 7 || 709962 259 880785 171 910.177 430 080823 53 8 || 800ſ IT 259 889682 I71 9,10435 430 089565 52 9 || 800272 258 889579 I71 910693 430 089307 || 51 10 || 800427 258 889477 I71 910951 430 080049 || 50 11 || 9'800.582 258 || 9,889374 I72 0.911209 430 10-08879 I 49 12 800737 258 889.271 I72 911467 430 088533 48 13 || 800892 258 889168 172 9] 1724 430 088276 47 I4 801047 258 889064 I72 9] 1982 430 0880 18 || 46 15 8() 120 I 258 88896i I72 912240 430 087760 || 45 16 || 801356 257 88.8858 I72 912.498 430 087502 || 44 17 | 80 I5]. I 257 888755 I72 Q12756 430 0.87244 43 18 80.1665 257 888651 I72 913014 429. 086986 42 19 || 801819 257 888548 172 913271 429 086729 || 41 20 | 80.1973 257. 888444 I? 91.3520 429 086471 || 40 21 || 9-802128 257 | 9-83834.1 173 |9-913787 429 || 10-086213 39 22 || 802282 256 888237 I73 914044 429 085956 || 38 23 || 8024.36 256 888134 I7 914302 429 . 08:5698 || 37 24 || 802580 256 888030 I73 914560 429 085440 || 36 25 | 802743 256 7926 I73 9,14817 429 0S5183 || 35 26 || 802897 256 887822 I73 915075 429 084925 || 34 27 | 803050 256 8877 IS 173, 915332 429 084668 33 28 803204 256 887614 I73 9 15590 429 0844 10 || 32 29 || 803357 255 887510 173 915847 429 0.84153 || 31 30 8035.11 255 887.406 174 916.104 429 08:3896 || 30 31 || 9-803664 255 |9-887302 I74 99.16362 429 || 10:083638 29 32 803817 255 887 198 174 916619 429 || 083381 28 33 803970 255 8870.93 I74 916877 429 083123 || 27 34 || 804123 255 886989 I74 9.17 134 429 082866 26 35 | 804276 254 886885 174 9]7391 429 082609 || 25 36 804428 254 886780 174 9.17648 429 0.82352 24 37 804581 254 886676 I74 9.17905 429 082095 || 23 38 || 804734 254 886571 I74 918,163 428 081837 || 22 39 804886 254 886466 174 918420 428 08.1580 || 21 40 805039 254 886.362 I75 9]8677 428 081323 20 41 9-805.191 254 9-886257 I75 9.918934 428 || 10-081066 19 42 805343 253 886152 I75 919 19.1 428 || 080809 18 43 || 805495 253 886047 I75 919448 428 080552 17 44 805647 253 885942 I75 919.705 428 080295 || 16 45 805799 253 885837 I75 919962 428 080038 || 15 46 || 805951 253 885732 I75 920219 428 079781 14 47 806103 253 885627 I75 920476 428 079524 13 48 || 806254 253 885522 I75 920733 428 070267 || 12 49 || 806406 252 8854 16 I75 920990 428 079010 11 50 | 806557 252 885311. I76 921.247 428 078753 10 51 9-806700 252 | 9-885205 176 9-92.1503 428 || 10'078497 9 806860 252 885,100 I76 92.1760 428 078240 8 53 |. 807011 252 884994 I76 922()17 428 077983 7 54 807163 252 884889 I76 Q22274 428 0777.26 '6 55 807314 252 884783 I76 9225.30 428 077470 5 56 807465 251 884677 176 922787 428 077213 4 57 8076.15 25I 884.572 I76 92.3()44 428 076956 3 58 807766 251 884.466 I76 {}23.300 428 076700 2 59 8079.17 251 884360 J76 923557 427 8)76443 I 60 l 808067 251 884254 177 92.3813 427 076.187 0 ! Cosine | | Sine ! | Cotang, | Tang. I bi, 238 (40 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. I. Sine , | D. Cosine | D. Tang. D. : Cotang. I 0 || 9-808067 251 9-884.254 177 |9-923813 427 10-076.187 60 I 808.218 251 884 148 177 924070. 427 075930 59 2 808.368 251 884()42 177 9:24:327 427 07:5673 58 3 8()8519 250 88.3936 177 924.583 4:27' 0754.17 57 4 808669 25() 88.3829 177 9:24840 4:27 75]60 56 5 8()88.19 250 88.3793 177 9:25.096 427 074904 55 6 808969 250 8836.17 177 9:25:352 427 07.4648 54 . 7 8()!} | 19 250 88.3510 177 9:25.609 427 74.391 53 8 8()9269 250 88.3404 177 925865 497 07:4135 52 9 8t)!}.419 249 88.3.297 J78 926.192 427 073878 51 l() | 809569 2.49 88.3.191 I78 926.378 427 07:3622 50 II 9-8097.18 249 || 9-88.3084 I78 |9-9266.34 427 | 10-073366 49 12 809868 249 88:2977 178 99.6890 427 07:3] 10 48 J3 8] ()()) 7 249 88:2871 I78 927] 47 427 07:2853 47 14 81() [67 249 882764 178 927.403 427 07:2597 46 lă 81.0316 248 882657 I78 927.659 4:27 07:234.1 45 J6 8:0465 248 88.2550 I78 9:27.915 4:27 07:2085 44 } 7 8'10614 248 882443 178 928.171 427 7 1829 43 I8 8|{}763 2.48 882336 179 928.427 427 07.1573 42 19 8|{}} {2 i. 248 8822:29 I79 92.8683 427 07 1317 41 20 81 1061 248 88:21:21 179 928940 427 07 1060 40 21 9-8] 1210 248 || 9,882014 179 |9-920196 427 | 10-070804 || 39 22 || 8 || 358 2.47 88.1907 I79 92.9452 427 070548 || 38 23 8] 1507 2.47 881799 179 |. 929708 427 070202 || 37 24 || 8 || |655 247 88.1692 179 929964 426 07.0036 || 36 25 || 8 || 1804 247 88.1584 179 930.220 426 069780 || 35 36 || 8 || |!)52 2.47 88.1477 179 9,3174 (5 4:26 Ü695.25 | 34 27 | 812100 247 881369 179 93(j731 4:26 069269 || 33 28 8iºq8 247 881261 180 930.987 426 069013 || 32 29 || 812396 246 88] 153 180 931243 426 0.68757 - 3I 30 | 812544 246 881046 180 93.1499 426 O68501 || 30 31 9-812602 246 || 0-880.038 180 || 0-031755 426 10-068245 20 32 || 812840 246 880830 180 93.2010 426 067990 || 28 33 || 812988 246 880722 180 93.2266 426 || 067734 27 34 813135 246 880613 180 93.2522 426 067.478 26 35 | 81.3283 246 880505 180 Q32778 426 || 007922 || 25 36 | 81.3430 245 880307 180 933033 426 066967 || 24 37 813578 245 880.289 181 933280 426 066711 23 38 813725 245 880 180 181 933.545 426 066455 22 39 813872 245 880072 181 933800 426 066200 21 40 || 814019 245 879963 181 934()56 426 06:5944 20 41 || 0-844 166 245 || 0-879855 181. 9,9343| 1 426 || 10-065689 19 42 814313 245 879746 181 934567 426 06:54:33 18 43 814460 244 870637 181 934893 426 065177 17 44 814607 |, 244 879529 181 935()78 426 ()64922 16 45 814753 244 79420 J81 935333 426 ()64667 15 46 81.4900 || 244 870311 181 935589 426 || 064411 14 47 815()46 244 874)2()2 182 93.5844 426 ()64156 13 48 815.1%)3 244 879.093 182 936) ()() 426 ()($3900 19 49 815,339 244 78984 182 936355 426 ()63645 | 1 50 815485 243 878875 I82 93(;610 426 06.3.390 10 51 9-815631 243 9-87S766 I82. 9-936866 425 || 10-063134 9 52 8||5778 243 878656 J82 {)37 || 2 | 425 062879 8 53 815024 243 878547 182 937:376 495 062624 7 54 || 8 |6069 243 784.38 . 182 9:37632 4:25 062368 6 55 8|{{2|5 24.3 878.328 182 9:37887 425 062) 13 5 56 816.361 943 878219 183 938142 425 06:1858 4 57 8 16507 242 878 109 J83 938.898 425 0616t)2 3 58 8|{}{352 242 87.7999 JS3 938653 425 061347 2 59 8|6798 242 87789ſ) 183 9:38908 425 06]()92 I 60 816943 242 87778) 183 939.163 425, 06()837 () I Cosine | Sine ! | Cotang, | | Tang. | M. 49 Degrees, LOGARITIIMIC, SINES, COSINES, ETC. (41 Degrees.) 239 Sine | D. Cosine | D. | Tang. | D. I Cotang. I M. O 9-816943 242 1 0877780 183 || 9-939,163 425 || 10-060837 60 I | 817088 242 877670 183 939418 4:25 060582 || 59 2 817:233 242 87.7560 183 939673 4:25 060327 58 3 || 817379 242 877.450 183 9:39.928 425 060072 7 4 817524 241 877.340 183 94t)183 4:25 0598.17 56 5 || 817668 241 77:230 184 94.04:38 425 059562 || 55 6 || 817813 24! 877 120 |, 184 940t;94 425 059.306 || 54 7 817958 241 877()10 184 940949 425 059051 53 8 || 81814)3 241 876899 I84 94.1204 425 058796 || 52 9 818247 241 876789 184 941458 425 058542 || 51 10 || 818392 241 876678 184 94.1714 425 058286 50 II 9-818536 240 || 9 876568 184 9-94.1968 425 || 10-058032 || 49 12 || 8.18681 240 876457 184 94.2223 425 057777 || 48 13 818825 240 876347 184 949478 425 057522 || 47 J4 || 818969 240 876236 185 94.2733 || 425 057267 46 I5 8191 13 240 876125 185 942.988 425 0570 12 || 45 16 || 819257 240 || 87.6014 185 943.243 425 05ſ,757 44. I7 || 81944)1 240 || 875904 185 943498 425. 056502 || 43 18 || 8 j9545 239 875793 185 943752 425 || 056248 || 42 19 || 819689 239 875682 185 944()()7 425 055993 || 41 20 | 819832 239 875571 185 || 944262 425 055738 || 40 21 |9-819976 . . 239 9.875459 I85 9-94.4517 425 || 10:055483 || 39 22 || 820120 239 875348 I85 944771 424 055229 || 38 23 820.263 2.39 875237 185 945026 424 054974 || 37 24 || 820406 239 875.126 186 945281 424 054719 || 36 25 || 820550 2.38 875014 || - 186 945.535 424 054465 35 26 820693 238 874903 186 94.5790 424 054210 || 34 27 820836 238 8747.91 186 946()45 424 . 053955 || 33 28 820979 238 874680 186 946.299 424 05:3701 || 32 29 821122 238 874568 I86 946554 424 053446 || 31 30 821265 238 874456 186 946808 424 053.192 || 30 31 || 9,821407 238 || 9-874344 186 ||9-947063 424 10-052937 || 29 32 821550 238 874232 187 947.318 424 052682 28 33 821693 237 874 121 187 94.7572 424 052428 27 34 82.1835 237 874009 187 94.7826 424 052174 26 35 | 82.1977 237. 873896 187 948()8] 424 05 1919 25 36 || 822120 237 873784 I87 948.336 424 051664 || 24 37 || 822262 237 873672 187 948590 424 051410 || 23 38 822404 237 873560 187 948844 494 05] 156 22 39 || 822546 237 873448 187 949()99 424 || 050901 || 21 40 || 822688 236 873335 187 949353 424 050647 || 20 41 9-822830 236 9-873.223 I87 9-949607 424 10.050393 || 19 42 822972 236 873.110 188 949862. 424 050.138 18 43 || 823] 14 236 || 872998 188 950] 16 424 049.884 || 17 44 || 823255 236 87.2885 I88 950370 424 049630 | 16 45 823397 236 87.2772 188 950625 424 04.9375 | 15 46 823539 236 87.2659 188 950879 494 || 049 121 || 14 47 || 823680 235 872547 188 951133 424 || 048867 13 48 || 82.3821 235 872434 188 95]388 424 048612 || 12 49 || 823963 235 872.321 188 951642 424 048.358 || 11 50 | 824104 235 87.2208 || 188 951896 424 048] ()4 10 51 |9-824245 ' || 235 | 9-872095 189 || 9,952.150 424 || 10-047850 9 52 | 894386 235 7 1981 I89 95.2405 || 424 0475.95 8 53 || 82,4527 235 87 1868 I89 95.2559 424 047341 7 54 || 824668 234 871755 I89 952013 424 047087 6 55 824808 234 871641 189 95.3]67 423 0468.33 5 56 824949 234 87.1528 189 95.3421 423 046579 || 4 57 825090 234 87] 414 189 95.3675 423. 046325 3 58 || 8252:30 234 87.1301 189 953929 423 ()46()71 2 59 825:371 234 87] 187 189 954 183 423 04:5817 I 60 | 8255] 1 234 87.1073 190 954437 423 045563 0 | Cosino l' | Sine I. | Cotang. I | Tang. ( M. 48 Degrees. 240 (42 Degrees.) LOGARITIIMIC, SINES, COSINES, ETC. * | D. M. : Sine | D. Cosine Tang. D. Cotang. I 0 | 9-8.255.11 234 9:871073 190 9.954437, 423 | 10-045563 60 1 | 825651 233 870960 I90 954691 423 045309 || 59 2 825791 233 870846 I90 954945 423 045055 || 58 | 3 || 825931 233 870732 190 9552(;0 423 044800 57 4 826071 233 87()618 190 955.454 423 0.44546 || 56 5 8262 II 233 870504 190 955707 423 044:293 || 55 6 || 826351 233 870.390 190 955961 423 044039 || 54 7 || 826491 233 87().276 I90 956215 423 04:3785 || 53 8 || 826631 233 870 161 190 956469 423 043531 |.52 9 || 826770 232 870047 I91 956723 423 043277 || 51 10 || 826910 232 869933 191 956977 423 043023 50 ll 9-827049 232 ||9-8698.18 191 9-957'231 423 |10-042769 || 49 I2 || 827 189 232 869704 I91 95.7485 493 042515 || 48 13 827328 232 869589 191 957739. 423 04:2:261 || 47 14 || 827467 232 869474 191 957993 423 042007 || 46 15 827606 232 869360 I91 958246 423 041754 || 45 I6 827745 232 869245 I91 958500 423 041500 || 44 17 | 827884 231 860130 191 958754 423 04:1246 || 43 18 || 828023 231 8690.15 I92 959008 423 040992 || 42 I9 828.162 231 868900 192 95.9962 423 040738 || 41. 20 | 828301 231 868785 192 959516 423 040484 || 40 21 9:828439 231 || 9-868670 192 9°959769 423 |10-040231 39 22 || 828578 231 868555 192 960(;23 423 039977 || 38 23 8287I6 231 868440 192 960.277 423 0.39723 37 24 828855 230 868324 192 960531 423 039469 || 36 25 828993 230 868209 192 960784 423 0392.16 35 £6 || 82013 I 200 80000:) I92 961000 423 038902 || 34 27 | 829269 230 86.7978 193 96.1291 423 0.38769 || 33 28 || 829407 230 867862 193 96.1545 423 038455 || 32 29 || 829545 230 867747 193 96.1799 423 C38201 || 31 30 829683 230 867631 193 962052 423 037048 30 31 9-8298.21 229 |9-8675I5 193 9-962306 423 || 10-037694 || 29 32 829959 229 867.399 193 96.2560 423 037440 28 33 || 830097 229 867283 193 96.2813 423 037187 27 34 830234 229 867 I67 I93 963067 423 036.933 26 35 | 830372 229 867051 I93 96.3320 423 0.36680 25 36 8305.09 229 866935 194 96.3574 423 036426 24 37 830646 229 8668.19 I94 .963827 423 036.173 || 23 38 830784 229 866703 194 96.4081 423 035919 22 39 830921 228 866586 194 964335 423 035665 [.21 40 | 831058 228 866470 I94 964.588 422 0.354.12 20 41 ||9-831.195 228 ||9-866353 194 || 9°964842 422 || 10-035158 19 42 | 83.1332 228 866237 194 96.5095 422 0.34905 || 18 43 || 83I469 228 866,120 194 965349 422 034651 || I7 44 || 83.1606 228 866004 195 965602 422 0.34398 || 16 45 831742 228 865887 195 965855 422 0.34145 15 46 || 83 1879 228 865770 I95 966109 422 033891 || 14 47 832015 227 865653 I95 966362 422 03:3638 || 13 48 || 832.15.2 227 865536 I95 966616 422 033384 12 49 || 832288 227 865419 195 966869 422 0.33131 || 11 50 | 832.425 227 865.302 195 967123 422 03:2877 || 10 51 9'832561 227 9'865.185 195 9-96.7376 422 10°03′2624 9 52 | 832697 227 865(168 I95 96.7629 422 03:2371 8 53 || 8.32833 227 864950 I95 967883 422 0.321 [7 7 54 || 83.2969 226. 864833 196 968136 422 031864 6 55 | 833105 226 864.716 196 968.389 422 0.31611 5 56 || 83,3241 226 86,4598 196 968643 422 031:357 4 57 833377 226 864481 196 968896 422 ()3 J ()4 3 58 || 833512 226 864363 196 969 149 422 030851 2 59 || 83.3648 226 8649.45 196 96.9403 422 030597 | I 60 833.783 226 864.127 196 969656 422 030344 0. Cosine | Sine | Cotang. I Tang, M. 47 Degrees. Z06 ARITHMIC, SINES, CoSINES, I.T.C. (48 Degrees.) 241, M. Sine. !: D. || Cosine D. t Tang. F. D. - Cotang. " 0 9:833.783 226 9-864. 127 196 9-969656 4:2 || 10-030344 60 I 83.3919 225 || - 864010 196 969909 422 0300.91 59 2 834054 225 863802 197 970.162 422 0.29838 58 3 834 189 225 863774 197 9704]6. 422 0.29584 57 4 834.325 225 863656 197 970669 422 0.293.31 56 5 834460 225 863.538 197 970922 422 0.29078 55 6 834595 225 863419 197 97II?5. 422 0288.25 54 7 8347.30 225 863301 I97 7 1429 422 02857 I | 53 8 834865 225 863183 197 71.682. 422 028318 52 9 8349.99 224 863064 197 97 1935 422 028065 5] 10 835134 224 862.946 I98 97.2188. 422 || 027812 50 |*; ; *; # | *; : *; I2 35403 || 224 862709 198 972694 422 || 027.306 || 48 13 835.538 224 862590 I98 97.2948 422 0.27052 47 14 835672 || 2:24. 862471 I98 97320i 422 0.26799 || 46 15 835807 224 862353 198 973454 422 026546 45 I6 835.941 || 224 862234. I98 97.3707 422. 0.26293 44 17 836075 223. 862115 198 973960 422. 026040 || 43 18 836209 223. 86.1996 198 97.4213 422 025787 42 19 836343 223 86.1877 198 974466. 422 0.255.34 41 20 836477 223 86.1758 199 974719 422 0.25281 40 21 9.836611 223 ||9-861638 199 9-974973 422 10.025027 39 22 836745 223 861519. 199 97.5226 422 0.24774 38 23 836878 223 86 [400 199 97.5479 422 024521 37 24 837()12 222. 861:280 I99 97.5732 422 02.4268 36 25 837146 222 86] 161 199 97.5985 422 0.240.15 35 26 837.279 222 86.1041 199 97.6238 422 023762 34 27 | 837412 222 860922 199 976491 422 023509 || 33 28 8375.46 222 860802 199 976744 422 023256 32 29 837679 222 860682 200 976997 422 023003 3} 30 83.7812 222. 860562 200 977:250 422 || 02:2750 30 31 9.837045 222 ||9-860442 | 200 9.977503 || 422 | 10:022497 29 32 838078 221 860322 200 97.7756 422 02:2244 28 33 83821 I 221 860202 200 978009 422 021991 27 34 838.344 || 2:21 860082 200 978:262 422 0.21738 26 35 | 838477 221 | 859962 200 978515 || 422 021485 25 36 8386.10 221 859842 200 978768 422 021232 24 37 838742 221 859724 201 97.9021 422. 020979 23 38 838875 221 8596(){ 201 979:274 422 020726 22 39 839007 || 221 859480 201 979527 422 020473 21 40 839140 220. 859360 201 97.9780 422 020220, 20 41 9:839272. 220 9-859239 201 9-980033 422 || 10-049967 I9 42 839.404 220 8591.19 201 || 980286 422 0.19714 I8 43 839536 220 | 858998 || 201 980538 422 0.19462 17 44 839668 || 2:20. 858877 201 980791 || 421 0.19209 I6 45 839800 220 858756 202 981044 || 421 018955 | 15 46 839932 220 85863.5 202 98.1297 421 018703 14 47 840064 219. 8585.14 202 98.1550 421 018450 13 48 840,196 249. 858.393 202 98.1803 421 {}18197 12 49 840328 219 858272 202 98.2056 421 (17944 11 50 840.459 219 858I51 202 982.309 424. 0.17691 10 51 9-840591 219 9-858029 || 202 || 9-982562 421 | 10-017438 9 52 840722 219 857908 202 98.2814 421 0.17186 8 53 840854 219 857786 202 98.3067 421 016.933 7 54 840985 219 857665 203 98.3320 421 0.16680 6 55 84III6 218 857543 203 98.3573 421 0.16427 5 56 84.1247 218 85.7422 203 98.3826 4:24 0.16174 4 57 84.1378 218 857.300 203 98.4079 421 0.1592, 3 58 84509 218 857.178 203 98.433; 42: 0.15669 2 59 84.1640 218 857056 203 984584 42.É. {}15416 I 60 841771 218 856934 203 984837 42i 0.15163 {} | Coaine ! | Sine | Cotang. . Tang. M. 46 Degrees. 242 (44 Degrees.) LOGARITIIMIC, SINES, COSINES, ETC. M. Sine I. D. Cosine D. I Tang. D. . . Cotang. {} | 9-841771 218 9.856934 203 9°984837 21 | 10:015163 | 60 ..] | 84.1902 218 856812 203 98.5090 || 4:21 0.14910 59 2 | 842033 218 856690 204 || 985343 421. ()14657 | 58 3 | 842163 217 856568 204 985596 421 0.14404 || 57 4 842294 217 | 856446 204 985848 421 0.14152 || 56 5 | 8424.24 217 85.6323 204 986101 421 0.13899 || 55 6 | 842555 217 856201 || 204 || 986.354 421 0.13646 || 54 7 | 842685 217 856,078 204 986607 421 013393 || 53. 8 || 84.2815 217 855.956 204 986860 421. 0.13140 52 9 || 84.2946 217 855833 || 204 987] 12 421 0.12888 || 51. 10 || 843076 217 855711 . . .205 987.365 421. 01:2635 | 50: . 11 ||9-843206 || 216 ||9-85.5588 || 205 || 9.987618 421 , |10-01.2382 || 49 12 | 84.3336 216 855465 205 98.7871 421" | 012129 || 48 13 | 843466 216 855342 205 988 123 421. 01:1877 || 47 14 | | 84.3595 216 855219 205 || 988376 - 421 W11624 || 46 15 # 843725 216 . , 855096 205 , 988629, , 421 01.[371 . . .45 16 -843855 216 || 854973. 205 988882 || 421" 0.11118 || 44, 17 | 843984 || 216 854850 205 989134 || 421. 0.10866 .43. 18, 844114 || 215 854727 206 989.387. 421 01.0613 || 42 19. 844243 || 215 || 85.4603 || 206 989640 || 421 () 10360 || 4 | 20 | 844372 || 215 | 854.480 206 989893 4:21 - || 010.107 40 21 |9-844502 || 215 ||9-854356 206 9-990145 || 421 10-009855 - || 39 22, 844631 215. 854233 || 206 99.0398 || 421 009602 || 38 23 844760 215 854109 206 900651 421 009349 37 24 | 844889 215 853986 206 99.0903 421 009097 || 36 25 | 845.018 215 85.3862 206 99 II56 421 0.08844 || 35 26 845147 215 85.3738 $206 99.1409 421 008591 || 34 27 | 845276 214 853614 207 991662 421 008338 33 28 845.405 || 214 853490 207 99.1914, 4:21 008086 32 29 | 845.533 214 853.36(5 207 99.2167 4:21 007833 31 30 | 845662 214 853242 207 99.2420 421 007580 || 30 31 9-845790 214 ||9-853.118 207 || 9-992672 421 | 10:007328 29 32 | 845919 2.14 '852934 207 902925 421 007075 28 33 | 846047 214 852869 207 99.8178 421 006822 27 34 846;75 214 852745 207 99.3430 421 006570 26 35 | 846304 214 85.2620 207 99.3683 421 ()06317 || 25 36 | 846432 213 85.2496 208 993.936 || 421 006064 24 37 | 846560 213 | 852371 208 994.189 421 00:5811 || 23 38 | 846688 213 852247 208 || , 994441 421 0.05559 22 39 || 846816 213 85.2122 208 994694 421 005306 || 21, 40 | 846944 213 || 85.1997 208 99.4947 421 005053 || 20 41 ||9-847071 213 ||9-851872 || 208 || 9995199 || 421 | 10:004801, 19 4? | 8471.99 213 85.1747 208 995452 421 004548 || 18 43 | 847327 213 851622 208 995705 421 004:295 17 44 | 847454 212 85.1497 |, 209 995957 421 004043 | 16 45 | 847582 212 85.1372 209 9962.10 421 003790 15 46 | 847709 212 || 85 1246 209 996463 421 003537 14 47 | 847836 212 851 121 209 , || 996715 421 003285 || 13. 48 847964 212 850.996 209 996968 || 421 0.03032 | 12 49 | 84.8091 212 850870 209 997221 421 002779 11 50 | 848.218 212 || 850745 909 99.7473 421 002527 | 10 51 9-848345 212 || 9.850619 209 || 9-0977.26 421 10.002274 9 52 | 848472 21] 850493 210 99.7979 4:21 002021 8 53 | 848599 211 850368 210 99.823] 421 00I '769 7 54 848726 21] 850242 210 99S484 421 00.1516 || 6 | 55 | 848852 21i 850.116 210 99.8737 424 00I 263 || 5 56 | 848979 211 849.990 210 998989 || 421 001011 || 4 57 | 849 106 2II 849864 210 999.242 421 0{}{}758 3 58 849232 211 | 849738 210 999495 421 000505 || 2 59 | 849359 211 8496.11 210 99.9748 421 000253 | 1 60 l 849485 21] 849,485 210 10:000000 421 000000 || 0 Cosine | Sine § | Cotaug. 1. | Tang. M. 45 Degrees, TABLE XIV. NATURAL SINES AND TANGENTs. ~ ºf ºf º-º/-4...R. 47-f ºf rºy a Oo | 1o 29 | 3o 40 - 50 60 '000 0000-0174524,0348995-0523360-0697565-087.1557|1045285 2009, ..., 7432,035.1992 (234.0709467 4:455 8178 5818 -018 0341|| 4809; 9169| 336S 7353|-105 1070 87.27 3249 7716-053 2074 6270-0880251 3963 001 1636 61581.0360623; 4979; 9171 3148| 6856 4544: 9066: 3530 7883-0712073 6046, 97.48 7453'-019 1974 6437°054 0788; 497.4 8943|-1062641 002 0362; 4883, 9344; 3693 WS7 6|-089 1840. 5533 3271 7791}-03'ſ 2251 6597-07207.77 4738|| 84.5 6180}-020 0699| 5158 9502: 3678 7635-107 1318 90S9 360S 8065-055 2406, 6580|-090 0532. 4210 003 1998 6516|-0380971 5311| 9481 3429| 7102 4907 94.24, 3878 8215|-073 23S2 6326 9994 78.15-0212332 67853-05611.10: 52S3 9223|-108 2885 •004 0724. 5241 ..., 9692. 4024 . , 8184.091 2119. 5777 3633 8140.039 2598 6928-074 10S5 5016 8669|| 6542}-0221057 5505 9832 3986 7913|-109 1560 9451 3.965| 8411|-057 2736 6SS7|-0920S09| 4452 005 2360 6873|-040 1318 5640; 9787 3706| 7343 5268 9781 4224 8544.075.2688 6602-110 0234 817.7 °023.2690 7131.058 1448 5589 9499| 3126 006 10S6 559Si.041 0037 4352 8489-093 2395 6017 3995 8506. 2944 - 7256-076.1300 5291. 8908|- 6904-024 1414 5850.059 0160, 4290 8187|-111 1799 98.13 4322 8757. 3064. 7190|-094 1083. 4689 007 2721 7230|-042 1663 5967.077 0091 3979 75S0|. 5630-025 0138|| 4569 8871 2991 6875-112 0471 8539 3046; 7475|-060 1775 5891 9771. 3361 •008 1448ſ 5954-0430382 4678 8791|-095.2666 6252 4357 : 3288 7.58%|-ſ)78, 1601 5562 0.142|. 7265|-026 1769 619.4|-061 0485 4591 8458|-113 2032 .0099.174 T 1677 gió0|| 3333 iſgil.096 iáš 1932 3083| 75853-044 2006. 6292-079 0391 4248 7812|- 5992'-027 0.493. 4912 9196| 3290 7144|-114 0702 8900 3401: 7818-062.2099 6190|-097 0039. 3592 •010 1809 6309:045 0724 5002, 9090 2934 6482 4718 92.16; 3630 7905-080 1989 5829| 9372|" 7627.028.2124 6536.0630808 4889 8724-115 2261 •011 ()535 5032, 94.42 711] 7788-098 1619) 5151 3444 7940:0462347 6614'081 0687| 4514 8040|- 6353,029.0847; 5253 9517 35S7 7408-116 0929 9261 3755: 8159|-0642420. 6486,099 0303 3818 •012 2170 666.2 °047 1065 5323 93.85 31.97 6707 5079 9570, 3970 8226|-082 2284 6092] 9596|. 76񊁅. 6876.065ii.29 51.83 8986|-117 2485 •013 0896 5385: 9781 4031 S082|-100 1881 5374 3805 8293-0482687 6934|-083 0.981 4775 826 6713:0311200 5592 .9836. 3880 7669|-118 1151 9622, 4108; 8498.066 27.39| 6778-101 0563 4040 •014 25.30 7015-049 1403 564 96.77 3457 6928 5439 9922 4308 8544|-084.2576 6351| 9816 8348-032 2830 7214-067 14:46, 5474 9245|-119 2704 '015 1256 5737 '050 0119 4349 8373|-102.2138 5593 4165 8644 3024 7251|-085 1271 5032. 848 7073'-033 1552 5929|-068 0153. 41.69 7925-120 1368 99.82; 4459 8835 3055| 7067|-103 0819| 4256 0.16.2890 7366-051. 1740 5957 9966 3712| 7144'. 5799, '034 0274 4645 8859-086 2864 6605-121 0031 8707 3181 7550.069 1761 57.62 9499. 2919 0.17 1616 6088 052 ()455 4663 8660-104 2392. 5806 4524 8995 3360 7565|-087 1557 5285| 8693 899 889 87o 869 859 849 839 7o •121 S693 •122 1581 4468 7355 “123 02:41 31.28 6015 89.01 '124 1788 4674 7560 ‘125 04:46 2 7156 •1360038 2919 NAT. CoSINE, . .1460830 3o •T39 1731 - , 7492 •140.0372 3252 •1420531 ‘3410 ‘6289 '9168 •143 2047 4926 *7805 •144,0684 3562 '6440 93.19 •145.2197 '5075 7953 3708 ‘8094 •148 0971 3848 67 24 9601 •149 2477 ‘5353 . . 8230 *150 1100 3981 . . . 9733 1512608 5484 , , 8359 *152 1234 4109 , 6984 . . 9858 -153 2733 5607 8482 “154 1356 4230 7104 997S “155 2851 5725 8598 “156 1472 ;4345 “810 : 4612| 6857| * WATURAL SINES. 90 10o -1564345]:1736482 7218 . , 9346 •157 0091.374,2211 2963| 5075 '583t| 7939 870SI-175 0803 2}-158.1581 3667 4453. 6531 7326|| 9395 • 159 0.1971-1762258 3069|| 5121 5940 7984 '8812||177.0847 160 1683 3710 4555 , 6573 ... 7426|| 9435 1610297|1782298 3167| 5160 ºl.$9.3% ‘8909-179'0884 *1621779| 3746 4650 6607 7520) 9469 -1630390-1802330 3260 5.191 6129| 8052 '8999-181,0913 -1641868] 3774 4738. 6635 "7607| 9495 *1650476]-1822355 3345. 5215 6214 8075 9082-1830935 1661951] 3795 4819 6654 . . .7687| 9514 *167 0556-1842373 3423 5232 6291 (8091 ... 9159||1850949 ‘1682026, 3808 4894. 6666 7761| 9524 ‘169,0628-1862382 3495] 5240 6362 8098 9228-187 0956 170.2095 : 3813 4961 6670 7828 9528 '171 0694-188 2385 3560 5241 64.25 8098 9291-1890.954 *172 2156 3811 5022 6667 7887. 9523 '173 0752-190 2379 3617| 5234 _6482 '8090 :800 || 790 11o 1908090 •191. 0945 3801 6656 95.10 •1922365 5220 807 •193 092S $37.82 '6636 ... 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COSINE, 550 - -819 1520 3189 4S56 6523 81.89 9784 •8221440 3096 4751 6405 8059 97.12 '8231364 3015 • 4666 6316 7965 9614 •824 1262 569 -8290376 2002 3628 3541 •832 1.15 2768 4380 7643| . . .9236|| •837-0827 2418 4009 5598 7187 |. 87.75 |-838 0363 NATURAL SINES. 570 •8386706. - 8290. 98.73 '839.1455. 3037 4618 6199 7778 9357 •840 0930 2513 4090 5666 , 72.11 . " 8S16 , 841 0390 1963 3914 5477 7039 8600 •844 016? 1720 3279 4838 6395 7952 •848 04S1 320 589 •848 0481 2022 3562 : 5102 66.41 8179 97.17 '849 1254 2790 4325 |, •851 1167 i 2693 42.19 | 8538 •854 0051 1564 3077 4588 9664 •856 II68 2671 417.3 5674 590 •857 1673 3171 . 4668 6164 , , 7660 $5sº 8523 •860 0007 5673 7134 8595 ‘865 0055 1514 2973 4430 5887 7344 8799 ‘866 0254 309 NAT. COSINE, 600 •866 0254 1708 77.56 91.96 •869 0636 •870 0691 2124 3557 4989 6420 ! • 7851 928.1 •871 0710 2138 3566 4993 6419 784-1 9269 •872 0693 2116 3538 4960 ’874 0550 ; : | 61o •874 6197 7607 9016 •875 04:25, 1 , . 1832 #455. 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'5456 3936 3714|-901.0770 5082, '6626 5288 '5021 2031 "6297 7795 6639 6326 3292 75.11 '8963 7989 7632 , 4551 '8725 -916 01:30 9339 893 5810 9938 1297 886,0688 894 0240 7068|909 1150 $2402 2030 1542 '8325 2361 3627 3383 2844 9582 3572 A791 4730 4146|| 9020838 4781 5955 6075 5446 209 '5990 7118 7420 6746 3347 71.99 8279 87.65 8045 '4600 8406 9440 •887 0108 9344 5853 96.13 -917 0601 1451 | 8950641 7105 || '910'0819 1760 2793 1938 8356 2024 2919 || - 4134 3234 . 9606 3228 4077 ‘5475 4529 '903 0856 4432 5234 6815 5824 2105 5635 '639]. 8154 7118 3353 6837 7546 9492 84.11 4600 8038 8701 888.0830 9703 || 5847.| . , 9.238 9855 2166 .896 0994 7093 | "911 0438 -918-1009 3503 2285 8338 1637 2161 4838 3575 9582 2835 6172 4864 || 904 0825 403 4464 7506 6153 2068 5229 '5614 8839 #440 3310 '6425 67 C3 889 0171 8727 4551 7620 7912 1503 | 897 0014 5792 88.15 '9000 2834. 1299 7032| 912 0008 |'.919,0207. 4164 2584 '8271 1201 1353 5493 3868 9509 2393 2499 6822 5151 |'905 0746 3584 3644 81.49 6433 1983 '4775 4788 947 7715 32.19 5965 5931 890 0803 8996 4454 W154 7073 2128 |-898 0276 5688 8342 8215 3453 1555 6922 9529 9356 4777 2834 8154 || “913 0716 •920 0496 6100 4112 9386 1902 1635 7423 5389 || '906 0618 3087 2774 8744 6665 1848 4271 3912 891.0065 -7940 3078 5455 5049 270 269 - || 250 240 239 67o 1920-5049 C185 7320 8455 9589 51] .9210722 1854 2986 4116 || - 5246 6375 7504 8632 9758 •922.0884 2010 3134 '4258 5381 6503 7624 8745 '9865 •023'0004 2102 3220 9805 •926'0902 2000 3096 4.192 5286 6380 65° *927 1839 2928 4016 || ". , 917.3 ‘929.0250; 1326; 240i 3475 4549 5622, 7765 8835 9905 || - '930,0974 2042 3109. 4176 5241 6306 7370 8434 '9496 •931 0558 1619 2679 '3739, 4797 (5855 6912 7909 9024 •932:0079 1133 2186 3238 4290 5340 6300 7439 8488 {}UU4 || NAT, COSINE, ſ 750 699 740 700 730 710 729 '933.5804 6846 9898 '9380906 1913 2920 : 1944 '939%; 0890 1849 2807 3764 4720 5675 '945,5186 , 6132 7078 8023 8968 991.I. 2736 3677 4616 6493 7430 8366 9301 •947 0236 * 1170 2103 3035 3966 4897 5827 67.56 7.684 8612 95.38 •948 0464 1389 2313 3237 4159 5081 •950.0629 1536 2443 3348 4253 5157 6061 .9460854. 1795 . 5555 . . '951 0565 1464 •956.3048 3898 4747. 5595 6443 98.25 •957 0669 NAT, CoSINE. •961 26.17 - 3418 42.19 5019 5818 6616 7413 8210 9005 9800 •9620594 1387 |. •965 9258 2 5 4 WATURAL SINES. C . . , 8432 1973 760 •970 2957 C0 4363 5065 77.59 9105 9777 111. 1789 | 2.458 3125. 3793 4.458 5124 5789 6453 7116 777s , 8.439 , 9100 0449 | 77o 78o 790 800 819 820 / •974 3701 -978 1476 •981 6272 '9848 078 [•9876883 -9902 681 | 60 . . 4355 2080 6826 || 582 ‘9877 338 '9903 085 59 5008 2684 738() •9849 086 792 . . 489 58 5660 3287. 7.933 589 i-9878 245 891 i 57 6311 |, .. 3889 8485 '9850 091. 697 '9904.293 56 , 6962 | 4490 9037 593 |1987.9 148 694 55 7612 5090. 9587 - '9851 093 ;- 599 || 9905095. 54 -8261 5689 '982 0137 ...,593 9880.048 || . 494; 53 * 8909 || | | 6288 0686: 9852092 | . . 497 || "... 893 || 52 9556 6886. 1234 590 || 945||9906290. 51 grä0203| 7483. 1781 -9853087: 9881392 || | 687 '50 ‘. . .0849 8079|... .2327 || ". 583 || 1838|9967,083, 39 ... 1494 |, . . .867.4%. 2873. 9854 079 |:9882284 || || || 478 |^48 ...2138|| 9268|, ... 34.17 | . 574 i..., 728 jº ; 47 ". .2781. 9862 || 3961|:985.5068|9883 172|{990.8266, '46 ** 3423 ||3790435, ºf 450+ ... à61 ... 615 ||...}}}|45 4065: ' ', 1047". 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'837 182 482 * 395 0 254 631 C 262 511 727 •9953 122 485 811 •9990 C98 341 539 •9935 058 403 715 , 99.1 419 567 wº 389 683 945 •9981 170 355 .497 593 719 , 962; 9969.173 348 482 573 .. 619 ‘9936047|19954.240 401 525 609 649 644 375 517 . 628 701 734 724 608 703 794 854 877 859 798 } - 692 •9937 029 -9955 070 9970 080 1-9982052 983 871 || 714| “, 355 '345 || 304 225 |-9991106 |, ..., 943| 736] ... 619 |*, *620 , 528 398 228|-9997.015|, .. 756. •9938 003 89 750 570 || 350 086 || 776 326|-9956.165: . , 972 j : 742 470 1561 .. 795 || 648] 437 |-9971193) 912 590 224| , 813| 969] . . 708]. . . , 413|-9983,082 || | | 709| ; 292, 3, . 831 •9939.290 . . . .978. 633 250 827 360 | . , 847 | 610 -995: 247 851 j : 118 944 |. 426 863 3 928. 515 -997.2069 585 |-9992 060 492 . , 878 . •9940 246 783 . , 286. 751 176 $56, 892 | 563|-9958 049 502 || , 917 | 290} . 620 | . . . .905 | 880 315 717 | -99.84 081 '404 " 683 917 •994.1 195 580 931. 245 517 745 || 928 844 -9973 145 40S 629 807 ,939 823 •9959 107 357 . 570 740 867 949 •9942136 370 969, 731 || 851 927 |. 958 448 631 780 891 || 960 986]" . 966|| , 760 892 900 -99.85 050 |-9993069|-9998 044 973 •9943 0.70 || -9960 152 | *ggſ 4 199 209 177 101 979 379 , 411 40S 367 284 157 | 985 688 || || | 669 615 524 390 213 989 | 996 2 822 680 495 267 w 9944 303 •996.1 1834 -99.75 028 835 | 600 321| 996 . 609. 438 ' ' ' 233 - 989 F : 704 374; ' ' ' '. 998 || , 914 693} , 437 -9986,143 806|. 426||1-0000000| •9945.219 , 947 . , 641 295 || 908 || " .. 477 000 || ºl 56"| 48"|. 38' 26"| 1s"| 0 || NAT, GOSINE, 256 NATURAL TANGENTS. 1A 09 •000 0000 2909 •002 0362 3271 6180 9089 •003 1998| 49 4907 7816 •004 0725] 36 3634 6542 9451 •005 2360 5269 8178 |:006 1087 3996 6905. 9814 •007 2723 5632]. 8541 Ö08 1450 4360 7269 •009 O1.78 3087 5996]: 8905 ; :010 1814 4724 96 '0142545 5.454 7453° 1o -0174551 746 0 -018 0370 3280 61.90 •020 0740 3650 6560 9470 -0212380 5291 8201 -0221111 4021 6932 9842 023. 27.53 8948 •026 1859 4770 7681 8059 :029 0970 3882 •0320086 2998 8822 :033 1734 4646 5910;" -0340471 3383. 6295. 9208; 835 7558. 29 3o 49 •0349208-0524078-0699268. •035:2120 6995-0702191 5033 9912 5115 7945-053 2829) 8038 w; 5746-0710961 3771 8663. 3885 6683-054 1581 6809 9596 4498 9733 037:2500' .7416.0722657 5422-055 0333 5581 8335 3251 . , 8505 -0381248. 6169.073 1430 4161| 9087| 4354- 7074.056 2005 7279 º 4923|0749.2% •0392901 7841, 312 5814-057 0759 6053. 8728. 3678 8979 -040 1641; 6596-075 1904 4555 9515 4829 7469.058 2434 7755 :04.10383 5352.076 0680 3296, 8271 3606 62.10.059 1190 6532 9124, 4109 9458 '0422038; 7029 .077 2384 4; 2,9948 #1 1836,060 2867|..., $23, 043 0181 b/87-078 1184 3695 8706 4090 6609-0611626 7017 º: 4546 9944 -0442438 ..., 7466-079.2871 5353-0620386 5798 8268. 3306 87.26 4:045 is: 226,080 1653 4097 .9147 4581 1012-063.206. ..., 7509 9927; 4988:08.10437 0462842; 7908 3365 5757'0640829 6293 8673; 3750 .9221 047 1588; 6671-0822150 4503 ..., 9592 5078 7419.065 2513 8007 0480334 5435.0836936 3250; ..., 8356 3865 6166-0661278] 3794 9082, 4199 9723 -049 1997: 7121-084.2653 #;"; ; - 65 8512 050 0746, 5887.085 1442 3662, ... 8809 4372. 6578-068 1732 7302 9495; 4654.086 0233 0512411 7577; 3163 §º 0.499. 6094-10. º 3.422 9025 .052 1161; 6345,087 ig56 gº 9268| 4887. 87o 860 | 850 7818 •088 0749 -0893ſ; 5408|- 813 -100 107 5| 9 | 1012824 ‘1021641 45 59. 087 4887 1. 8 3681. 6612". 9544 6 8341 -090 1273 .093 0606 3540|. 096.2890 5826 8763 •097. 1699 4635 7572 .098 0509 3446 4 6 6383 9320 -099 2257 5.194 1 4009 6947 886 3 5763 702 8 O 7.520 4|-113 0517 6o -105 1042 -108 0462 3405 6348 9291 8||109 2234 5178|. 3463 9356 ‘1142303 5:250 8197 *115 1144 4092 7039 998.7 -1162936 5884 8832 -117 1781 4730 7679 t 6 -1180628 6528 9478 -119 2428 5378 6410| 3578. 7o -1227846| 3|-123 0798 3752 6705 965S -1242612 5566 8520 125 1474 4.429 7384 -126 0339 3294 -131 (607 3566 6525 9484 '1322444 5404 8364 •133 1324 4285 7246 •134 0207 '135 2053 50.15 :137 2793 8329 •120 1279 4230 7182, -121 0133 3085 6036; 8.988 1|122 1941 4893 7846 839 5757 8721 •138 1685 4650 76.15 -139 0580 3545 6510 9476 -140 2442 3408) . 82° 8168| 6129 8091|| | NA.T. COTANs . AWA TURAA, TANGENTS. 80 .90 •140 540S}-158 3844 8375 6826 141 1342; 9809 4308-159 27.91 727 6 5774 •142 02:43, 8757 3211-160 1740 6179| 4724 9147| 7708 143 2115-1610692 5084, 3677 8053| 666.2 144 1022 9647 3991|-1622632- 6961 5618 9931 8603 145 2901-163 1590}. 5872. 4576 88.42| 7563 •146 1S13-164 0550 4784} 3537 77.56 6525 •147 07:27; 9513 3699-165 2501 6672, 5489 96.44 8.478 1482617-1661467 5590; 4.456 8563 T 146 •149 1536-167 0436 4510| 3426 7484; 64.17 •150 045S] 9.407 3.433|-168 2398 6408. 5390 93.83 8381 151 2358-1691373 5333. 4366 8309|| 7358 *1521285-1700351 4262. 3344 7238, 6.338 | 1530215 . , 9331 3192-1712325i. 6170. 5320 9147| 8314 T54 2125|-172 1309:- 5103| 4304 8082 7300 • 155 1061-173 0296 4040: 3292 7019 6288 9998, '9285 •1562978-174.2282 5958) 5279 8939, 8277 ‘157 1919-175 1275 4900; 4273 78SI 727 158 0863|-1760271 3844; 32 810 809 NAT. COTAN. 100 | 11o 12o 130 140 150 .1763270-1943803-212 5566 2398682.249.3289.231949? 6269 6822 8606 -231. 1746 6370-268 2610 9269 984.1]-213 1647 481.1 9460 5728 '177 2269,195 2861 4688 7876°250 2551 8S47 5270 5881 7730 -232 0941 56.42|-269 1967 8270 89.01|-214 0772, 4007 734 5087 •178 1271-196 1922 3814; 7073-251 1826 8207 4273| 4943 6857-233 0140 4919-270 1328 727.4 7964 9900 3207 8012 4449 •179 0276-1970980|-215 2944 6274|-252 1106 7571 3279 4008 5988 93.42 4200}•271 0694 6281 7031 90.32-234 2410 7294 3817 9284-1980053-216 2077 5479|-2530389 6940 1802287 3076 5122 8548 3484-2720064 5291 6100 8167|-235 1617 6580 3.188 8295 912.4-217 1213 4687 96.76 6313 1811299|-199 2148 4259 7758|-2542773 9438 430 5172 7306.236 0829 5870-2732564 7308 8197|-218 0353 3900 8968 5690 -182 0313|-200 1222 3400 6971-255 2066 8817 3319 4248 64481.237 00:44 5165°2741945 6324 727.4 9490 31.16 8264 5072 9330-201 0300!'.219 2544 6189)-256 1363 8201 •1832.337 3327 5593 9262 4463-275 1330] : 5343 6354 8643.238 2336 7564 4459 8350 9381}*220 1692 5410|-257 0664 7589 •184 1358-2022409 4742 8485 3766-276 0719 4365 5437 7793-239 1560 6868 3850 7373 8465-221 0844 4635 9970 6981 •185 0382-2031494 3.895 7711|-258.30731-277 01.13 3390 4523 6947|-240 0788 6176 3245 6399 7552 9999 3864 9280 6378 9.409-204 0582-222.3051 6942|-259 2384 9512 •1862418 361.2 6104|-241 0019 5488-278.2646 5428 6643 9157 3097 8593 5780 8439 9674|-223 2211 6176 °260 1699 89.15 *1871449|-205 2705 5265 9255 4805-279 2050 4460 5737 8319.242 2334 7911 518 74.71 8769|-224 1374 5414-261 1018 8322 •1880483|-206 1801 4429 8494 4126-280 1459 3495 4834 7485|-2431575 7234 4597 6507 7867-2250541 4656.262 O342 77.35 9520|-207 0900 3597 7737 3451,281 0873 189 2533 3934 6654.2440819 6560 4012 5546 6968 9711 3902 9670 7152 8559-208 0003-226 2769 6984|-2632780-282 0292 190 1573 3038 5S27-245 0068 5891 34.32 4587 6073 8885 3151 9002 6573 7602 9109|-227 1944 6236-264. 2114 97.15 •191 0617.209 2145 5003 9320 5226-283 2857 3632 5181 8063-246 2405 8339 5999 6648 8218-228 1123 5491-265 1452 9143 9664.210 1255 4.184 8577 4566-284 2286 •1922680 4.293 7244-24; 1663 7680 5430 5696 7331|-229 0306 4750°2660794 8575 8713-211 0369 3367 783 3909 -285 1720 193 1731 3407 6429-248 0925 7025 4866 4748 6446 9492. , 4018-2670141 8012 7766 94.86|-230 2555 7102 3257-286.1159 1940.784.2122525 5618-249 0.191 6374 4306 3803 5566 8682 3280 9492 7454 790 789 77o 769 750 749 2 5 8 AVATURAL TANGENTS. 8 •290 2114 160 •286 7.454 •287 0602 37 51 6900 •288 00:50 320i 6352 9503 “289 2655 5808 961 5269 8423 •291. 1578 473 7890 •292 1047 4205 7363 •293 0521 3680 6839 9999 •294 3160 632} 94$3 •295 2645 5808 897.1 •299 0634 3803 6973 “300 014.4 3315 6486 96.58|- •301 2831 6004 91.78|. •302 2352 5527 8703. -303 1879 5055 8232 -304 1410|. 4588 7767 •305 0946|. 4126 7307 730 I7o 180 190 200 •305 7307 -324.9197|-344 3276/.363.970% '306 0488}:325 24.13 6530}-364. 299'ſ 3670 5630 9785 629. 6852 8848}*345 3040 9588. •307 0034-326 2060 62961-365 2S85!” 32.18 5284 9553 6182 6402 8504:34628.10 9.480 9586'327 IT24 6068|-366 2779 •3082771 4944 93.27. 79 5957 8165-347 2586 ° 93.79 9143-328 1387 5846}.367 2680 •309. 2330 4610 9107 5981 5517 7833-348 2368 9284 8705-329 IQ50 5630-3682587 -310 1893 4281 8S93 589(; 5083 7505.349 2156 919.5 82725-330 0731 54.20(.369 2500 •311 1462 3957 8685 5800 4653 7 1844-350 1950 9112 7845-331 0411 5216|-370 2420 -312 1036 3639 8483 572S 4229 6868|-351 1750 9036 7422}-3320097 5018|-371. 234( -313 0616 3327 8287 5650. 3S10 6557|-352 1556 896 7005 9788 4826-372 2278 -314 0200-333.3020) 8096 559ſ) 3396 625. •353 1368 8903 6593 94.85 4640}.373 2217|. 7901-334 2719 7912 5532 5|-315 2988 5953-354 1186 8847 6186 9.188 4460|-374. 21631. 93.85:335 2424 7734 5479 •316 2585 5660-355 1010 8797 5785 8896 28G|-375 2115 8986:3362134 7562 5433 -317 218.7 5372-3560S40 8753 5389 86.10 4118-376 2073. 8591]:337 1850 7397. 5394 •318 1794 5000-357 0676 8716 4998 8330 3956-377. 203S 8202}.338 1571 7237 5361 •319 1407 4813.358 0518 8685 4613 8056 3801|-378 2010 7819; 339 1299 7083 533 320 1025 4543.359 0367 8661 4232 7787 3651-379 1988 7440ſ-340 1032 6936 5315 321 0649 4278-360 0222 8644 3858 T524 3508-380 1973 7067.341 0771 6795 5302 3220278 4019-361 0082 8633 3489 7267 3371 |-381 1964. 6700-342 0516 6660 29C 9912 3765 99.49 86.29 323 3125 7015|-3623240|-3821962 6338|-343 0.266 6531 5296 9552 35.18 98.23 863 3242766 6770-3633115|-383 1967 5981}-344 0023 6408 530 91.97 3.276 97.02 8640 720 71o 700 690 210 220 •383 8640-404 0262 '384 1978. 3646 5317| 7031 8656|-405 04.17 385 1996, 3804 5337 7191 867.9°406 0579 -386 2021] 3968 5364 T 358 87.08°407 0748 -387 2053. 4139 539S| 7531 874.4°40S 0924 -388 2091 4318 5439| 7713 8787 -400 1108 -389 2136: 4504 5486| 7901 8837 -410 1299 •390 2189 4697 554.1. 8097 8894-411 1497 •391 2247 4898 5002 8300 7| 8957|-412 1703 •392.2313| 5106 ñ67() 851() 9027-413 1915 393 2386 5321 745) 8728 9105|-414 2.136 394 2465, 5544 5827 | 8953 9189|-415 2363 395 2552 5774 5910| 91.86 92SG|-416 2598 396 2645 6012 601||| 94.26 9378||417 2841 •397 2746 6257 61.14| 96.73 94.83°418 3091 '3982853 6509 6224| 9928 9595|-419 3348. •399 296S 6769 6341|-4200190 97.15| 3613|. •400 3089 7036 6465-421 0460 9841. 3885| 401 3218 7311 6596]-4220738 99.74. 4165- 4023354|| 7594 6734°423 1023 403 0115| 4453. 3496 7884 6879-424, 1316 404 0262. 4748|- 689 67o 30 '424 4748 8182 •425 1610. 5051 8487 *426 1924 5361 8800 •427 2239 5680 9121 •428 2.563 6005 9449 •429 2894 6339 9785 '430 3232 6680 .431 0129 3579 7030 •4320481 3933 7386 •433 0840 4295 # e 208 434 4665 8124 •435 1583 5043 8504 •436 1966 5429 8893 •4372357 2 9289 •438 2756 6224 9693 •439 3163 3 NAT, COTAN. NATURAL TANGENTS. 259 26o 46106634823127 āşşı 7014 7119°483 0601 240 | 250 27o 445.22871-466.30771-487 7326.509 5254 § 6618; 488 0927 8919 9260-167 0161 4530|| 510 2585 ºf 462747 3705 8133 6252 6236 ..., 7250.489.1737 99.19 97.26'468 0796 5343-511 3588 •447 331; 4342| 8949| 7259 708; 7890-490 2557 -512 0930 448 0200'4691439 61.66 4602 3693 498: 9775 8275 7187; 8539°491 3386-513 1950 4490682:470 2090 G997 5625 41.78 5643°4920610 9302 7675 9196 4224.1-5.14 2.980 4501173i'471 2751 7838 6658 4672 6306'493 1454'-515 0338 8171 9863 5071 4019 451 1672; 472 3420 8689| 7702 5173 6978, '494 2308}-516 1385 8676-473 0.538 5928 5069 •452 21.79 4098 ... 9549 8755 5683. "7659.495317.1-5.17 2441 9188!-474 1222 6794 § '453 2694 4785}'496 0418 98.18 6201 8349 4043:518 3508 97.09]-475 1914 7669: 7199|- •454 3218 5481'497 1297-519 0891 6728 90.48 4925 458 •455 0.238°476 2616 8554 827S. 37.50 6185*498 2185}.520 1974 7263 97.55 5816 5671 $56 0776|-477 3326 9449 9368|"5 4290 6899, '499 3082.521 3067 išjāqīsāī; iſ “à. 457 1322 4046-5000352.5220468 4839 7621 3989 4170 835.71°479 1197 7627 7S74|+5 458,1877 4774-501 1266.5231578 5397 8352 4906 5284 8918; 480 1932 - 85.47 8990 '459 2439 5512-502 2189}-524. 2698 5962: ... 9093 5832 6407|- 9486481.26; 94.76%-525 0.117 460.391]: #508312|| 3829 6537 9842 768 7541|| 6168 ‘504 04:15:526 1255 4063 4969 3; 8685 *505 1363 - 527 2402 *462 06:49: 4189 4179, 7778 7710; .484 1368 4631243. 4959 4776; 8552 83101'485 2145 '464 1845; 5739 5382, 9334 :465.2457 6528 5996:487 0126 9536 37.26 466 3077 7326 65° | 64° 8919:486 2931-5080607-530 2178 5015i 6120 8668' 9839 '506.2322 528 3560 5977. , 7281 96.33°529 100.4 ‘507 3290; 4727 6948; 8452 4267 i 5906 7929 9634 •509 1591 °531 3364 5254. 7094 630 || 622 NAT, COTAN. . 280 290 300 31o / ‘531 7094-554 3091 577 3503-600 8606' 60 ‘532 0826; 6894 7382-601.2566 59 4559-555 0698 '578.1262 6527 58 8293 4504, 5144.6020490 57 ‘5332029. 8311, 9027| 4454 56 57.65|-556 2119.579.2912, 8419 55 9503 5929 6797]-6032386 54 ‘534 3242, 97.39 °580 0684, 6354 53 698, 55735;}| #13||6040323 52 ‘585.0728, 1864, 8162 4294, 51 . 4465-5581179.58.12353 8266 50 8208 4994 6245-605 2240, 49 ‘5361953 88.11:582 Q139 , 6215. 48 5699|-559:2629 4034-606 0192, 47 9446. 6449| 7930. 4170 46 -537 3194,5600269.583 1828. , 8149, 45 6943 409|| 5126.607 2130 ºf ‘538 0694| 7914. 9627| 6112' 43 4445|-561 1738, '584 3528-608 º 42 8198| 5564] 7431| 4080; 41 •539 1952, 9391}*585 1335| 8067| 40 5707-562.3219 5241-609 2054, 39 9464 # 91.48 6043, 33 540.3221-563.0819-5863056-610.0034; 37 j" iſ ºil ſº gº 541 0740 8543°587 0876 8019. 35 #oil-564:3378 4788-61.1%iſ 34 8263| 6213. 8702. 6011 33 542 2027|-565 0050-588.2616|-612 0008; 32 791. 3888 6533 4007; 31. 9557| 7728-589 0450 8008] 30 543 3324|-566.1568 4369|-613 2010, 29 7092 5410 8289. 6013; 28 544 OS62 9254-590 2211|-614 0018; 27 4632°567 3098| 61.34 4024, 26 8404 69.44''591 0058. 8032. 25 545 21.77|-568 0701] 3984|-615 2041; 24 5951, 4639| 7910. 6052. 23 97.27| 84881-592 1839-616 0064; 22 '546.3508:5692339| 576 4077. 21 7281| 6191| 9699| 8092. 20 547 1060-570 0045-5933632-617 2108; 19 484.0) 3899 7565. 6126, 18 8621 7755-594.1501.6180145||17 548 2404|-571 1612, 543 4166 16- 6188 5471) 9375] 8188 15 997 9331-595 3314|-619 2211| 14 "549 3759|-572,3192| 7255 6236, 13 7547| 7054°596 11961-620 0263. 12 '550 1335-573 0918, 5140|| 4291; 11 5.125| 4783| 9084| 8320i 10 89.16|| 8649-597 3030-621. 2351 9 -551 2708]-574.2516. 6978 6383| 8 6502 6385-598 0926-622 0417| 7 •552 0297|-575 0255| 48.77| 4452 6 4093 4126 8828; 8.488; 5 7890| 7999.599 2781-623 2527| 4 '553.1688-576 1873| 67.35|| 6566 3 5488 5748-600 0691,.624 0607 2 9288 96.25 4648}. 4650; 1 '554.3091,577 3503 - 8606 . 8694 0 61o 600 599 589 / NATURAL. TANGENTS. 5 •631 3598 76 320 •624 8694 -6252739 6786 '626 0834 4884 8935 -627.3388 7042 •628 1098 5155 9214 •629 3274 7336 •630 1399 5464 9530 67 •632 1738 5810 9883 •633 3959 • 8035 •634. 21.13 619.3 .635 0274 4357 8441 •636 2527 fió14 •637 0703 4793 8885 •638 2978 7073 -639 1169 . •664 3984 -645 2797 6918 •646 1041 5165 929t) •647 34.17 7546 '648 1676 5808 6 •667 3374 ‘5 7580. j}-668 1786 9 5995). •669 0205 50 44.17|' 33o l 2 – 6490 4774 '652 3064 7211 '653 1360. 5511 9663 •654 3817 7.972 '655.2129 6287 •656 0447 \ 4609 87.72 •657 2937 2^ 7103 •658 1271 • 5441 •659 3785 •660 2136 6313 4673 8856 *6623040 7225 '663. 1413 5601 792 8178 •665 2373 6570 4969 9171 8630 •670 2845 •671 1280 97.21 8169 •673 2396 6624;" •674 0854 •649 40', 6. 821. '650 2350). '651 0631j. 8918. 96.12. 7960. 661093. "666 0769. 7061}. 5500/. •672 3944. 569 5085.” '6'78 3243 7492 •679 1741 5993). •680 0246 4501 87.58 '6813016 7276 •682 1537 5801 •683 0066 •6872810 7.093 •688 1379|. 5666 99.55). •689 4246 8538 •690 2832. 7128 •691. 1425 5725 730 •707 1664 602SH-7: ‘708 0395 6( •718 1319 350 '700 2075, 360 726 5425 98.71 $1.730 1041 ‘704 116. 5515 9869|| 7 05 4224 85813-7 •7062940 4763 91.33 ‘709 3504 7878 •710 2252 43 66.30 -711 1009 539. 9772 -712 4157 8 854. .713 2931 7163698 8 1 0 0 •717 2505 911 5729 •719 0141 4554 8970 720 3387 7806 .7212227 6650 '722 1075 5502 9930 51 ‘723 4.361 8793 97 2|-724.3227 7663 •725 2101 6540 4|| 726 0982 1 .” 5501 9963 '735 0210 4691 917.4 •736.3660 8147 737 2636 71.27 '738 1670 6115 739 061] 511(, 961] '740 41.13 86.18 ſ. ‘741 3124 7633|- |-742 2143 6055]- 43 1170 7 5686). ‘744 0204 4724 9246 '7453770 8296 •746 2824 73.54 •747 1886 6420 -748 0956 5494 •749 0033 45.75 9119 • 750 3665 8212 •751 2762 7314 •752 1867 6423° '753 0981 5541 539 SI-755 3799 370 ‘753 5541 '754 0102 4666 9232 8369 •756 2941 7514 '757 2090 6668 •758 1248 5829 ‘759 0413 4999|" 9587 8769 •762 2557 7157 •763 1759 6363 •764 0969 4800 ‘766 4031 8649 9589 •772 4233 88 S •773 3526 •774. 2827 •775 21:37 ‘776 1455 6H18 '779 4135 •781 2S56 ‘781 2856 •782 22:29 7S3 1611 6305]- ‘784 1002 5700ſ. •785 0400 •760 4177}. 5577 •765 0188]. 9414- -767 3270. 8176. 7481. 6795. -777 0782. 5448 •778 01.17|. 4788 94.60). 529 380 7542 6919 5103 9808 •786 4515. •790 2248 6975 •791 1703 6434 •792 1167 •800 1963. 6736 78|-801 1511. 288 8 9|822.3840 390 •S09 7840 •810 2658 7.478 •811 2300 7124 '814 12SO 6118 97.64 •8]9 4625 9488 •820 4354 9222 •821 4093 965 -82. 3364 8251 •825 3140 31 dº •834 1547 6481 '835 1418 6357 983-836 1298 6242 '837 1.188 61.36 '838 1087 6041 '839 0996 500 NAT. COTAN. NATURAL TANGENTS. 261 49 50 51 52 53 54}. 56}. 57 58. 59 60 400 595.) 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CHORDS, VERSED SINES, EXTERNAL SECANTS, AND TANGENTS OF A ONE-DEGREE CURWE. The angles of the table are the intersection angles, I, equal to the total central angle included between the tangent points. •K To find the corresponding func- . tion for any other curve, divide / the tabular number by the de- gree of curvature. The unit chord is assumed to be one hundred feet long. By using radius of 5,730 feet, the chord column of the table can be made serviceable for plotting. 269 270 CIIoIRDS, VERSED SINES, ExTERNAL SECAMTS. 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Izz · 89914 · 9984zț7 Oły8 • Io99† • £ Izz9 · 96SrZ-9664offyoț»o • 9oy9./. • ZyIz1° zgºr€ ±994off;" 38I • 96#92, º II 22;9 • 969 r.£ • ț786/3€38gº zoºg9 • 947 Izoº 19918* 198438 9€.6 • 476 #9oº 6ozz€ • #69. I6 • 18.6/,9€.9€.A, º 6689$ • £; Iz:9 · 6991† • 699498 #8Z • I6ț793° 9ozzz • 869 I9 • 6z 6/#8#89 · 9689z • I? Iz/* 9991o · Zºg/#8 zºS · 99999 • țyozz;o • z69 r.I • Zz6€.z€z€# • £6€9o • 6€Iz - || 9 · ŻGGI· 9 ·#994zº £º • Sgºg£ º zozz6 • o69 I9-ſz64 © e o . $500 CLARK (D. K.). Fuel; its Combustion and Economy. Embracing portions of the well-known works of C. WYE WILLIAMS, “Combustion of Coal and the Prevention of Smoke,” and of T. T. PRIDEAUx’s work on “The Economy of Fuel.” With extensive additions on Recent Practice in the Combustion and Economy of Fuel, Coal, Coke, Wood, Peat, Petroleum, &c. 12mo, cloth . . ge e ge o A Manual of Rules, Tables, and Data for Mechanical Engineers. Large 8vo. 1,012 pages. Illustrated. 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