LJ (/) > DC Z (0 DC K ffl j| U) b. LJ MECHA 1881 K LJ Z D O Z E u m RADUATI s* UJ W o w LJ I CVJ h- CM O - UJ igj LJ 5 Q z ^ < d o U H 5 z UJ -j UJ Z LJ 0) LJ I E fr THE AUTHOR elg etu'cates tfji's Booh TO ALBERT J. SCIIERZER, C.E., ItJ k, these are implements of his art which never become antiquated, and can never fall into dis- use. Those herein contained which are original will, 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 PREFACE. vii 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. SIIUNK. , N.J. ABBREVIATIONS. + Increased by. Diminished by. X Multiplied by. -f- Divided by. = Equal to. *. * Since, or seeing that. .*. Hence, or therefore. I Indicates the quotient of one divided by the other of the quantities it connects, called sometimes the ratio of the quan- tities. ; : Indicates an equality of ratios, and connects equal ratios in a proportion. Thus, a : b : : c : d indicates that a-S-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 6) -r- (a -\- b) requires that the product of a and b shall be divided by their sum. A 2 . A small secondary figure annexed thus to an expression is called its exponent. 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 2 = A X A ; A 3 = A X A X A, which is called the cube, or third power, of A. V This is called the square root sign: it signifies that the square root of the quantity covered by it is to be taken. V If preceded by a small secondary figure, called the index, as in the marginal figure, it indicates that the cube root of the quantity covered by it shall be taken ; and so on. \j 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 ABB11EYIA TIONS. as outcropping ledge, water-table of building, or other perma- nent 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, oil which the eleva- tion is marked with kiel. T. P. Turning-point : usually marked in the field-book. P. I. 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 a curve. P. T. Point of tangent: the stake-mark at the end of a curve. 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. P. R. C. Point 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. H. I. Height of instrument : elevation of the level above the datum or zero plane. II. W. High water. L. "W. Low water. TABLE OF CONTENTS. PAGE LOGARITHMS. I. Definitions and principles 3 II. Manner of using the tables 4 To find the logarithm of any number . . 4 To find the number corresponding to a given logarithm 5 Multiplication by means of logarithms . . 5 Division by means of logarithms ... 6 To raise a number to any power by means of logarithms 7 To extract roots by means of logarithms . . 7 PLANE TRIGONOMETRY. III. Definitions 11 IV. Natural sines 12 V. Logarithmic sines, &c 13 VI. General propositions ...... 15 VII. Solution of plane triangles 16 VIII. Right-angled plane triangles .... 18 ADJUSTMENT AND USE OF INSTRUMENTS. IX. General remarks on adjustment . . . .23 X. The level 24 To bring the intersection of the cross-hairs into the optical axis of the telescope ... 24 To bring the level bubble parallel with the tele- scope axis 25 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 ... 25 XI. Levelling 26 Correction for the earth's curvature and refrac- tion 28 To find differences in elevation by means of the barometer 29 Heights by the thermometer '. .... 29 xi Xii TABLE OF CONTENTS. PAGE XII. Setting slope stakes 30 XIII. Vertical curves . . 36 XIV. The transit 40 To adjust the level tubes 40 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 . . 40 To adjust the needle 41 XV. Miscellaneous 42 The vernier 42 To read an angle 43 To re-magnetize a needle 44 To replace cross-hairs 44 To fix a true meridian 44 PROPOSITIONS AND PROBLEMS RELATING TO THE CIRCLE. XVI. Propositions relating to the circle ... 49 XVII. Circular curves on railroads .... 50 XVIII. To find the radius, the apex distance, the length, the degree, &c., of a curve ... 52 Given the intersection angle I and radius R, to find the tangent T 52 Given the intersection angle I and tangent T, to find the radius R 53 Given the intersection angle I and chord AB= C, connecting the tangent points, to find the radius R 54 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 external secant S, or the tangent T 54 Given C, V, S, or T, of any curve, and D, the degree of curvature, to find the intersection angle I 55 Given the intersection angle I and deflection angle D, to find the length of the curve . . 55 Given any radius R and chord C, to find the de- flection angle D 57 Given any radius R and chord C, to find the de- flection distance d 57 Given any radius R and chord C, to find the tangential angle T 57 Given any radius R and chord C, to find the tangential distance t 58 TABLE OF CONTENTS. Xiii PAGE XIX. Ordinates 58 Given any radius R and chord C, to find the middle ordinate M 58 Given the radius R, chord C, and middle ordi- nate M, to find any other ordinate ... 59 Ordinates of a 1 curve, chord 100-feet . . 60 TRACING CURVES AND TURNING OBSTACLES IN THE FIELD. XX. To trace a curve on the ground with the chain only .63 XXI. To trace a curve on the ground with transit and 100-feet chain 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 ... 79 XXV. The curve-protractor and the projecting of loca- tions 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, x, may merge in said straight line . . . . 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 . . . . .89 PROBLEMS IN FIELD LOCATION. XXVL How to proceed when the P. C. is inaccessible . 93 XXVII. How to proceed when the P. C. C. is inaccessible, 95 XXVIII. To shift a P. C. so that the curve shall termi- nate in a given tangent 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 97 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 [V TABLE OF CONTENTS. 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 102 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 . 103 XXXIV. To locate a Y 103 XXXV. To locate a tangent to a curve from an outside fixed point 107 XXXVI. To substitute a curve of given radius for a tan- gent connecting two curves .... 108 LXXVII. To run a tangent to two curves already located . 109 RACK PROBLEMS. XXVIII. Reversed curves 115 XXXIX. To connect two parallel tangents by a reversed curve having equal radii 115 XL. To connect two parallel tangents by a reversed curve having unequal radii . . . .117 XLI. A reversed curve having unequal angles . . 119 XLII. A reversed curve between fixed points . . 120 XLIII. To connect two divergent tangents by a reversed curve 123 XLIV. To shift a P. R. C. so that the terminal tangent shall merge in a given tangent . . . .125 XLV. To pass a curve through a fixed point, the angle of intersection being given .... 127 XL VI. Frogs and switches 129 To find the radius of a turnout curve, the frog angles, and the distances from the toe of switch to the frog points 129 To find the angle of switch-rail with main track . 130 To find the distance from toe of switch to point of main frog 130 To find the radius of outer rail of turnout curve, 131 To find the main frog angle, the radius of outer rail being known 131 To find the angle of the middle frog in the case of a double turnout 131 To find the distance from toe of switch to point of middle frog ....... 131 Turnout tables ... . 135, 136 TABLE OF CONTENTS. X PAG XLVII. To locate a turnout lo XLVIII. Crossings on straight lines 13 XLIX. Crossings on curves 13 L. Elevation of the outer rail on curves . . . 14 Table for same 14 LI. Trackmen's table of curves and spring of rails . 14 Explanation of same 14 LIST OF TABLES. I. Time of meridian passage of North Star above the pole for the year 1870, and on 14 II. Time of extreme elongation of North Star for the year 1870, lat. 40, and on 1C III. Azimuths of the North Star, and their natural tan- gents 1* IV. Roods and perches in decimal parts of an acre . . K V. Decimals of an acre in one chain length of 100 feet, and of various widths 1" VI. Acres, roods, and pert lies in square feet . . .If VII. Circular arcs to radius of 1 1' VIII. Feet in decimals of a mile II IX. Inches reduced to decimal parts of a foot . . . It X. Radii and their logarithms, middle ordinates, and deflection distances If XI. Squares, cubes, &c., of numbers from 1 to 1042 . . 1( XII. Logarithms of numbers, 1 to 1000 11 XIII. Logarithmic sines, cosines, tangents, and cotan- gents 1! XIV. Natural sines and tangents . . . . . . & XVI. Chords, versed sines, external secants, and tangents of a one-degree curve 2( XV. Slopes for topography 3] XVII. Rise per mile of various grades 3] LOGAEITHMS. I. -II. LOGAEITHMS. 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. 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 power. 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 and 1 ; that is, it is equal to 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 O's that immediately follow the decimal point. The characteristic alone, in this case, is negative, the man- 3 4 MANNER OF USING THE TABLES. tissa being always positive. This is indicated by writing the negative sign over the characteristic : thus, 2.380211 is equiva- lent to 2 + .380211. 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 instead 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. 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 O'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 O's which immediately follow the decimal point will be one less than the number of units in the charac- teristic. Thus the number corresponding to the logarithm 5.233568 is 171225.296; that corresponding to the logarithm 2.233568 is .0171225. 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. Example. Find the continued product of 3.902, 597.16, and 0.0314728. Operation. Log. 3.902. . . 0.591287 Log. 597.16 . . . 2.776091 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. DIVISION 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. Example 1. Divide 24163 by 4567. Operation. Log. 24163 . . . 4.383151 Log. 4567 . . . 3.659631 0.723520 = log. 5.29078, the quotient. Example 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 TABLES. 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. Example. 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. Example. Find the 5th power of 9. Operation. Log. 9 0.954243 5 4.771215 = 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, aiid it will be the root required. 8 MANNER OF USING THE TABLES. Example. 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. Example. 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 TEIGO1STOMETEY. III.-VIII. PLANE TRIGONOMETRY. in. 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 arc 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 T" ^ ? ^ r' , ', ": thus the ex- pression 7 22' 33" is read, 7 degrees, 22 minutes, and 33 sec- onds. 3. The complement of an angle is the dif- ference between that angle and a right angle. 4. The supplement of an angle is the difference between that angle and two right angles. 11 12 NAT URAL SINKS, 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 arc enclosing it, to the diameter, through the other extrem- ity. Thus P M is the sine of the angle M O A. The cosine of an angle is the sine of the complement of the angle. Thus NM = O P is the cosine of the angle M O A. The tanyent 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 O T the secant of the angle MO 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 BT', BN, and OT' 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 ail arc 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- mic sines. LOGARITHMIC SINES, 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 FUNCTIONS 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, tang., or cotang., as the case may be; the number there found is the logarithm sought. Thus, log. sin. 19 55' .... 9.532812 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, cosine, 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' .... 10.111884 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. Example. Find the logarithmic sine of 40 26' 28 ;/ . 12 NATURAL SINE 8, 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 arc enclosing it, to the diameter, through the other extrem- ity. Thus P M is the sine of the angle M O A. 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 O T the secant of the angle MO 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 denned. Thus BT', BN, and OT' 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 fuiictioii of au arc 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- mic sines. LOGARITHMIC SINES, 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 FUNCTIONS 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, tang., or cotang., as the case may be; the number there found is the logarithm sought. Thus, log. sin. 19 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, cosine, 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 IS' . . . . 10.111884 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. Example. Find the logarithmic sine of 40 26' 28". 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. Example 1. Given angle A = 24 26'; angle = 36 43'; side 6 = 137.6: to find side a. _ u As sin. B . . log. 9.776598 Is to sin. A . log. 9.616616 So is b . . . log. 2.138618 11.755234, sum of 2d and 3d terms, To a, 95.2 . . log. 1.978636 less 1st term. SOLUTION OF PLANE TRIANGLES. 17, Example 2. Given, sides a and &, as above, and angle A; to find angle B. As side a (a. c.) log. 8.021364 Is to sin. A ....... log. 9.616616 So is side b log. 2.138618 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 180; 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. Example. Given side a = 95.2, side 6 = 137.6, and the included angle c = H8 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 34J ; . log. 9.771447 To tang. I their difference = 6 08$' . . . log. 9.031830 Adding half the difference to half the sum, 30 34^ + 6 OS}' =36 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 TRIANGLES. CASE TIT. 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 Method. 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. RIGHT-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 hypothenuse is equal to the sum of the squares of the remaining 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 RIGHT-ANGLED PLANE TRIANGLES. 19 become sines, tangents, cosines, or the like, and write upon them the proper 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 ABC, and calling the hypothenuse a radius, we shall have, c = a sin. C -f- R ; hence sin. C = Re -f- a. b = a cos. C -r- R; hence cos. C = Rb -f- a. Then, assuming the side b to be radius, we shall have, c = b tang. C -f- R; hence tang. C = Re -j- 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. ADJUSTMENT 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 rini 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 THE LEVEL. X. 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 000 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 moans 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-hairs may 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 BRING THE LEVEL BUBBLE PARALLEL WITH THE TELE- 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. C. 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 RIGHT 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 colhmation 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, 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. LEVELLING. 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 -j-200. Set the level at 13, 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 0, 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, -f 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. BM 200.00 B M on W. Oak. 2 22 202! 22 40 ft. N. of Sta. O. . sM 193.8 1 m 1.9 200.3 2 , 0.81 201.41 . . 2 64 204.05 3 g Si 7 206! 3 4 , 3.2 200.8 5 10.36 193.69 0. Iii levelling where great exactness is necessary, the rod at turning-points should be read to thousandths, and the reading checked by J,he leveller. Before taking it down, after clamp- 28 LEVELLING. 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 rodnian 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 writers practice has been to turn to hundredths 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 rodnian. 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 X D- if D is in "stations," and A = 0.6 X D - if D is in miles. Thus the correction for 10 stations would be LEVELLING. 29 .0215; for 50 stations, 0.5375; for 10 miles, 60 feet, and a spire or tree top apparently level with the instrument at that dis- tance would really be GO feet above it. Transposing the equa- tion we have D = Y/ A-f-0.6. In this form it is applicable to the determination of distances at sea. The Peak of Teneriffe, for example, 16,QOO feet high, should be just visible from the sea-level at a distance = ^10000-4-0.0 = 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, L; that at the upper stand, U. Then, D = [ (L U) -i- (L + U) I X 55000. Example. L = 26.64; U = 20.82. Then, L U= 5.82 .... log. 0.764923 = 47.46 .... lo. 1.676328 0.1226 Diff. 1.088595 And 0.1226 X 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 thai case, having found the difference as above, add ^1^ 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 55. When the upper thermometer reads highest, for "subtract" say "add," and vice versa in the foregoing rule. 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- enced, 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 -f T 2 . Example. T = 212 208 = 4. H = (513 X 4) -f 4- = 2068 feet. 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 "H 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 f^et road-bed, 9 feet cut, and slope of 1 to 1, the distance out would be 10 -f- 9 -f- 4 = 23 feet, as shown in the annexed diagram. 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 Example 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. -------- 26.0 ------------ *| 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 steop, 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 Jill 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 should not be discouraged if he misses the mark rather widely in his early trials. 32 SETTING SLOPE 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 tlie 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 -f- 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. i). Suppose him, by either method, or even by random guess, to think that 10 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 -f- 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 secolid 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 STAKES. 33 observation, will place a slope stake on the second or third trial, without conscious effort of mind. 11. Next, suppose the level at 13, 25.5 feet ahove 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 U.5 = 16 feet, corresponding to a distance out for the side stake of 10 -f- 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 -f- 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, practice 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.YDS 258 50 + 5.8 iO + 12.0 + 16.0 11.0 + 18.0 28.0 Example No. 2. 13. In the annexed figure, representing an embankment 14 feet wide on top, with side slopes of H to 1, the first thing to attract attention is that the instrument is 1 foot below grade, and that, therefore, 1.0 is to be added to all rods, in order to Imd the height of embankment above the points at which rods are takon. 14. Consider the down-hill side. The engineer, with the ground ill view, and with the height of embankment at the 34 SETTING SLOPE STAKES. 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 embankment slope. The slope being 1 to 1, or f, the hori- zontal reach of 23 feet corresponds to a vertical height of f of 23 = 15.3 feet;- and, since the instrument is 1 foot he- 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 -J- 6 = 25 feet, where, on trial, the rod proves to be 10 feet, instead of 11 feet, corresponding to an embankment height of 10 -f- 1 = 11 feet, and to a distance out of 7 -f 11 + 5.5 = 23.5 feet. Approximating thus, by shorter and shorter steps, he jfinally reaches the point sought. I 15. The process in fixing the upper slope stake is similar to that used in fixing the lower one in Example No. 1. The iseveral 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 hook as fol- lows: STA. Dis. LEFT. CENTRE RIGHT. AREA. C.Ycs. 140 62 9.4 22. G 6.3 3.2 12.7 Example No. 3. 17. Here is a case, partly in rock excavation, slope \ to 1; partly in embankment, slope 1^ 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 G.5 feet above grade ; centre cutting 2.5 feet. Find first, with a G.5 feet rod, the grade point, to left of centre line, which prmcs to be 2.5 foot out. Note it, and set a stake there marked "grade." Note also the change of ground 5.5 SETTING SLOPE STAKES. 85 feet out and 10.0 0.5 == 3.5 feet below grade. Then set the lower slope stake as in Example No. 2, observing that in this 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 snch 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 -7-4) for slope; total, 14.0 feet. The trial rod, however, at that distance, instead of reading 0, reads 6 feet, indicating a cut 22.5 0.0 = 10.5 feet deep, and a distance out correspond- ing thereto of 9.0 + (16.5 -f- 4) = 13.1 feet. Trying again at this distance out, the rod reads 7.6 instead of feet, requiring a further movement towards the centre line of (7.0 6) -4- 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: STA. Dis. LEFT. (,'ENTKE RIGHT. AKEA. C.Yus. 0.0 ^ 328 40 6.9- 2.5 i -3.5 f + 2.5 4-15.0 18.3 12.8 5.5 J VERTICAL CURVES. XIII. VERTICAL CURVES. DIAGRAM GIVING THE ORDINATES OF A PARABOLA AT IN- TERVALS OF ^V TO THE SPAN THE MIDDLE OKDINATE BEING UNITY. F ^x ^ ^^" X^ r-f ^~~ 1 > "S r-^ / + .t 7 e 9 Xie * a o .* 4 .C 5 . Q ' 7 5 ' 8 3 '- '12 II 1 o o 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 BF is equal to. the versed sine FD. Referring to the above diagram, ordinates 4 and 8 will be seen to correspond to the ordinates between chord AC and the curve in this instance, which ordinates therefore will be equal to the middle ordinatc FD multiplied by 0.89 and 0.55 respectively. Adding these multiples to the grade elevation at A, the elevations of the intermediate points selected will be ascertained. VERTICAL CURVES, 87 Example 1. Elevation at A = -f 94.0; A B = + 1 in 100; B C = 1 in 100; AD, D C, each = 150 feet or 1.5 stations of 100 feet each. Hence BD == 1.5; and FD = 0.75 feet. Ordinate 8 = 0.75 X 0.55 = 0.41. Ordinate 4 = 0.75 X 0.89 = 0.67. Elevation of grade at 8 8 = 94.0 -f- 0.41 = 94.41. Elevation of grade at 4 4 = 94.0 + 0.67 = 94.67. Elevation of grade at D = 94.0 + 0.75 = 94.75. Example 2. Elevation at A = + 94.0. A B = -f 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 II = 1 X 2 0.4 X 2 = 2.0 0.8 = 1.2 feet. Ascent from A to 8 = 0.15 per 50 feet. C along chord AC = CH -=- 8 = 1.2 -f- BE = BD I CH = 2 0.6 = 1.4. .'. FE = 1.4-f-2 = 0.7. Ordinates at 9 9 = 0.7 X 0.44 = 0.31. Ordinates at 6 6 = 0.7 X 0.75 = 0.52. Ordinates at 3 3 = 0.7 X 0.94 = 0.66. Mid-ordinate = 0.70. The elevations then along the chord AC, ascending at the rate of 0.15 per 50 feet, will be : A963 0369 C 94.0 94.15 94.30 94.45 94.60 94.75 94.90 95.05 95.20 38 VEK TIC A L CUR YES. to which add the ordinates just found: 0.0 0.31 0.52 O.C6 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 Example 3. Elevation at A = + 94.0 ; A B = -f 1 in 100 ; B C, A II, level. AD, BC, each 200 feet divided into 50-feet spaces, the points of division corresponding therefore to ordinates 3, 6, and 9 of the ordinate diagram C H = 1 X 2 = 2 feet. Ascent from A to C along chord A C = C H -f- 8 = 0.25 per 50 feet. BE = B I) & C II = 1 foot. .VFE = 1 -f-2 = 0.5. Ordinates 9 9 = 0.5 X 0.44 = 0.22. Ordinates 6 6 = 0.5 X 0.75 = 0.37. Ordinates 3 3 = 0.5 X 0.94 = 0.47. Mid.ordinale = = 0.50. The elevations then along the chord A C, ascending at the rate of 0.25 per 50 feet, will be: A9 6 3 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.C 94.47 94.87 95.22 95.5 95.72 95.87 95.97 96.0 VER TJCAL CUR VES. 39. Example 4. Elevation at A = + 04.0; AB = 1 in 100; BC, AH, level; AD, BC, each 150 feet, divided into 50-feet spaces, the points of division corresponding therefore to ordinates 8 and 4 of the initial diagram C II = 1 X 1.5 = 1.5. Descent from A to C along chord CH-f-6 = 0.25. E B = D B D E = 1.5 0.75 = 0.75 .-. FE = 0.75-1-2 = 0.375 Ordinates 8 8 = 0.375 X 55 = 0.21 Ordinates 4 4 = 0.375 X 89 = 0.33 Mid ordiuate = 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 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 0.33 0.37 0.33 0.21 0.0 And the grade elevations on the curve will be: 94.0 93.54 93.17 92.88 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 lilted 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 LEVEL TUBES. 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, AND MARK BACKSIGHT AND FORE- SIGHT POINTS IN THE SAME STRAIGHT LINE. 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 capstan 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 a church spire or chimney, on which point direct the cross-hair, and then, 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, atone 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 slay it in position by means of a small woodcif 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 MIS CEL LA NEO US. XV. MISCELLANEOUS. THE VEHXIER. 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 edge into ten equal 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, thl first mark of the latter, or ver- nier, 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 0.217 = six feet, two-tenths^ one-hundredth, and seven- thousandths. 3. The annexed figure represents the transit MIS CEL LANEO US. 43 vernier, together with a part of the graduated circle. The vernier is a double one, for con- venience in reading angles right or Jeft. 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 with some mark on the limb. Add the number of that 44 MISCELLANEOUS. 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 tlto 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 P'lX A TRUE MERIDIAN. By equal shadows of the sun. On level ground or ice, set up a pole. Two or three hours MISCELLANEOUS. 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 hours 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 means 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. Jjy observation of the North Star at its extreme elonga- tion. 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 a/imuth, 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 vernier instrument, the azimuth can be laid off directly, in degrees and minutes. PROPOSITIONS AND PROBLEMS RELATING TO THE CIRCLE. XVI. - XIX. PROPOSITIONS AND PROBLEMS RELATING TO THE CIRCLE. XVI. PROPOSITIONS RELATING TO THE CIRCLE. The following propositions, demonstrable by simple processes of geometrical reasoning, may be regarded as axiomatic. 1. In any circle a tangent is perpendicular to radius at the tangent point. Thus, 13 1 is perpendicular to B C. 41) 50 PROPOSITIONS RELATING TO TUE CIRCLE. 2. Tangents drawn to a circle from the same point are equal. Thus, I B = I E. . 3. The angle DIE, at the intersection of tangents, is equal to the central angle BO E, included between radii to the tan- gent points. 4. If a chord BE connect the tangent points, the angles I B E, I E B, are equal : each of them is equal to half of the central angle B C E, or of the intersection angle DIE. 5. Any angle, BCE, at the centre, subtended by the chord BE, is double the angle BFE, at the circumference, on the same side of the chord B E. 6. Acute angles at the circumference, subtended by equal chords, are equal. 7. An acute angle, KFH, between a tangent and a chord, is called a tanyential angle, and is equal to the peripheral angle LFH subtended by an equal chord; each is equal to half the central angles FCH, or HCL, 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. 9. U F K be made equal to F H, and H N be made equal to HL, H K is called the tangential distance, and LN the deflec- tion distance. 10. The exterior angle E HX 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. CIRCULAR CURVES ON RAILROADS. 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 tin; centre; its radius would be 5,729.6 feet, usually reckoned 5,730 feet. The CHICULAR CUR \'ES- O2T RAILKOADS. SI chain 100 feet long being the unit generally adopted by Ameri- can engineers for field measurements, any circular arc having that radius, of 5,7-30 feet, is called a one-degree curve, for the reason that one chain is equivalent to au 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 1,910 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 arc 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 d iff era 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 OG', 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 1, 2, 3, &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 Hie. degree of curvature, or central angle per 100 feet chord, varies. 52 TO FIND TDE RADIUS OF A CURVE. XVIII. TO FIND THE RADIUS, THE APEX DISTANCE, THE LENGTH, THE DEGREE, ETC., OF A CURVE. 1. Let EB, A O be two straight lines intersecting at E. Lay off equal distances, EA, EB; erect perpendiculars at A and B, meeting at G,.and con- n^ct A B, E G. From the centre G, with radius G A, draw the curve A H B. The point E will be the P. I. ; A and B, tangent points; E A, E B, the tan- gents, or apex distances, which denote by T; EH, the external secant, or S ; H N, the versed sine; or V. Let the long chord AB, connecting the tan- gent points, be called C, Call the deflection angle to a and G A or GB, the radius, R. chord of 100 feet D, as before. 2. By XVI. 3 and 4, angle E EBA=AGE o. GIVEN THE INTERSECTION ANGLE I AND RADIUS R, TO 9 FIND THE TANGENT T. T = R X tan. tr I. Example. R = 1,910.1, I = 35 2-1'. Then T = R tan. i I = 1,910.1 X 0.3191 = C09.5. 4. Measure from the P. I. equal distances, E M, E F, along the tangents. Measure, also, MF and EK, the distance from E to the middle point of MF. Then, by reason of similarity in the triangles MEK, EAG, M K : E K : : A G : A E : : R : T .-. T = RXEK-j-MK. TO FIND THE RADIUS OF A CURVE. 53 Example. Let MK= 190.5, EK = 60.8, R = 1910.1. Then R= 1910.1 .... 3.281056 EK= 60.8 .... 1.783904 MK = 190.5 ( a. c.) . . . 7.720105 T = 609.6 .... 2.785065 5. If 100-feet chords be used, find the tangent in Table XYI. 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. 6. GIVEN THE INTERSECTION ANGLE I AND TANGENT T, TO FIND RADIUS R. Transposing the equation in (3), R=T+ tan. I = T X cot. I, Example. T= 609.6, 1=35 24' R = T cot. i / = 609.6 X 3.1334 = 1910.1. By a like transposition of the equation in (4), 7. If 100-feet chords be used, find in Table XVI. the ^angent 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., 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 CURVE. 8. GIVEN THE INTERSECTION ANGLE I AND CHORD A B C, CONNECTING THE TANGENT POINTS, TO FIND RADIUS li. A N = i A n = i C; A G N = i I. AG = AN-^ sin. AG N ; i.e. y E = $ C -i- sin. $ I. Example. I = 35 24', (7=1161.4. Then I? = -J G f -=- sin. $!, = 580.7 -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, arid 5,730 divided by 3 makes the radius R = 1,910 feet. 10. 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 Y, THE EXTERNAL SECANT S, OK 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. Example. A 4 curve, 1 = 50 16'; to find the several functions above named. Table XVI. gives the designated functions of a 1 curve as follows: C 4,867.3, V 542.4, S 599.3, T 2,688.2. Dividing by 4 the degree of curvature, w r e have for the corresponding func- tions of a 4 curve as follows: C 1,216.8, V 135.6, S 149.8, T 672.0. RADII, 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. Example 1. Given T = 515, D = 5; to find I. Then T X D = 2,575, which corresponds in Table XVI. to 48 24' = I. Example 2. Given C = 1,656, D = 3; to find I. Then C X D = 4,968, which corresponds in Table XVI. to 51 23' = I. 12. GIVEN C, V, S, OR T, OF ANY CURVE, AND THE INTER- SECTION ANGLE I, TO P'IND THE DEGREE OF CURVATURE D. Take from the proper column of Table XVI. the number corresponding to the given an^le 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 = 22 26'; to find D. The Tan. corresponding to I in Table XVI. is 1,136.3. Then 1,136.3 -f- 587 = 1.935 = 1 56' = D. Example 2. Given S = 64, I = 30 25', to find D. The Ex. 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 CURVE. 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- venient method of effecting the division is, first, to reduce the RADII, DEFLECTION ANGLES, ETC. minutes, if any, in I to decimals of a degree; then divide by the degree of curvature. Example 1. I = 20 40', I) = 3. 20 40' is equivalent to 20.67 degrees. Dividing by 3, we have 6.80 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 lengthy were 50 feet eacii, 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 1 and D to minutes; then divide the former by the latter. Example 2. I == 3()Q 22', D = 2 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 who 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. 17. GIVEN ANY DEFLECTION AN- GLE D, AND CHORD C, TO FIND RADIUS R. FB -I- sin. L B = B L ; i.e., Example. Let C=100 feet, D = Then Z? = | C -f- sin. $ .0349 = 1432.7. = 50 -=- If the chords are 100 feet long, as is usual in railroad prac- tice, radius may.be found with sufficient accuracy by dividing RADII, DEFLECTION ANGLES, ETC. 57 5,730, Hie radius of a 1 curve, by the deflection Bangle, or de- gree of curvature. Thus, in the foregoing example, 5,730 -f- 4 = 1,432.5. 18. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE DE- FLECTION ANGLE D. From the preceding equation and example: Sin f D = i C -7- K = 50 -T- 1,432.7 = .0349 xin 2 = D 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 HAB in the figure being made equal to D, and HA = B A, B II will be the deflection distance. Draw A K to the middle point of II B. Then d = HB = 2 KB = 2 AB X sin KAB = 2 C X sin |D. Example, Let R = 1,146 feet, C = 100 feet. By (18) D will be "found = 5. Then ( I = 2 C X sin I D = 200 X .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. Thus, in the foregoing example, 10,000 -j- 1,140 = 8.72. 21. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE TAN- GENTIAL ANGLE T. The angle T is equal to \ U by construction; for mode of determining it, see preceding section (18). 58 ORDINATES. 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 13 A E in the figure being made equal to T, and AE = A 13, BE will be the tangential- distance. Draw AX to the middle point of BE. Then EB = 2 XB = 2 AB X sin N AB = 2 C X m Example. Let H = 1,146 feet, C = 100 feet. By sect. 1, T will be found = 2 30'. Then t = 2 C X sin \ T = 200 X .0218 = 4.06 feet. 23. In ordinary railroad practice the tangential distance may be considered equal to half the deflection distance. XIX. OKDIXATES. 1. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE MID- DLE ORDINATE M. G- 'L Iii the annexed figure, II X = M, H G = R, A B = C. ORDINATKS. 59 Example. = 819, C =100; to find the middle ordinate, M. M = 819 V67076l^^2500 = 1.53. 2. Angle HAN=MHGB; HGB = AGB, .'. HAN = A G B. UN = AN X tan. HAN; i.e., M = C X tan. D; D being the central angle subtended by the chord. Example. D = 7, C = 100; to find M, the middle ordinate. M = i C X tan. D = 50 X 0.03055 = 1.528. 3. GIVEN THE RADIUS R, CHORD C, AND MIDDLE ORDINATE M, TO FIND ANY OTHER ORDINATE E K = M', DISTANT d FROM N, THE MIDDLE POINT OF THE CHORD. KL = NG;NK=GL;EK = EL NG. E L = G E 2 N K 2 = R 2 1 O iO O OOlCOr-lO ?H M ?0 00 ri f^ -r < o TURNING OBSTACLES TO VISION IN TANGENT. 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. XXII. TURNING OBSTACLES TO VISION IN TANGENT. 1. Draw CF parallel to AB. Let lines BC, CE, FG, cut these parallels at equal inclinations. Call this angle I. Then B C = CE = FG. BE = A B- BD + DE = 2BD. But BD = BC cos. I, .-. BE = 2 BC cos. I. EG = C F. cos. I. Example. 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. B S to B, and deflect 20 right. Vernier will now read 10 right, and telescope will be in line C E. 'Hake C E = 200 feet. Move to E. See that vernier still reads 10 right. B S to C, arid turn 10 left. Ver- nier will now read zero, and telescope will be in line E G, or tangent A B prolonged. Distance BE = 2 B C cos. 1 = 2 (200 cos. 10) =400 X .985 = 394 feet. Then E = 24.50 + 394, = stake 28.44 on tangent A B prolonged. 72 TURNING OBSTACLES TO VISION IN TANGENT. If a parallel line C F were run, a deflection of 10 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 BHE will be equilateral, and BE = B H = H E. If the parallel D C or DF be run, BE = BD -f DC, and BG = BD + DF. 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. In 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 F G H rectangular d i s - tances A E, B F, I / i LJj ranging clear of A Bjf the obstacle. 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. 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 A D B, BDE, AB = BE, and may be measured on the ground. 2. Or, fix the point E, arid 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 B E, C B : B D : : B D : B E, Example. Suppose B D to be 60 feet, and B C 40 feet. Then B D 2 -h B C = 3600 -+- 40 = 90 feet = BE. 3. Or, with the instrument at D, measure the angle BDE. Then B E = B D tan. BDE. Example. B D = 120 feet, angle B D E = 54 40'. B D tan. B D E = 120 X 1.41 = 169.2 feet = B E. 4. Or, without an instrument, lay off any convenient lines B F or C H. Mark the middle point D. Line out H G, parallel to AB. Mark on it the point G in range with D and E. Then GF = BE, orGH = CE. 5. Should the use of a right angle be inconvenient, turn any angle E B D = x, measure B D about equal by estimation to B E, if the ground permits, and set a point D. Move to D, and measure angle 15 D E => y. 74 TURNING OBSTACLES TO MEASUREMENT IN TANGENT. Then the angle BED, or z, 180 (cc -}-?/), and, by trigonom- etry, sin. z : sin. y \\ BD:BE, .'. B E = B D sin. y 4- sin. z. Example. Let x = 44 02', y = 71 48', B D = 300 feet. Then z = 180 (x + y) = 180 (44 02' -f 71 48') = 180 115 50 ; = 64 10'. B E = B D sin. y 4- sin. z = 300 sin. 71 48' ~ sin. 64 10' = 300 X .95 -f- .90 = 316.6 feet. The calculation by logarithms would be as follows: Log. 300 2.477121 9.977711 Sum 12.454832 Log. sin. 64 10' 9.954274 Log. 316.6. Diff. . 2.500558 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 C D 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 DE, opposite angle x 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 C D E = 180 y = 180 71 48' = 108 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 ODE, therefore, CD + DE or 651.7 : CD DE or 188.3 :: tan. 108.12 -t- 2 or tan. 54 06' : tan. 54 06' X 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 Tair?2145', Diff.. , 9.601136 TURNING OBSTACLES 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 2V. 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 FG. Cross to E, B S to D, and deflect the angle z = 64 KX into the line of the tangent A B prolonged. SUGGESTIONS AS TO FIELD-WORK AND LOCATION -PROJECTS. XXIV. -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 hoth 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 perf unctory 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 oily 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 or more axemen. 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. 79 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. SUGGES TIONS CONCERNING FJEL D - WORK. 81 Always begin grade-lines at the summit, and work down. For such service, carry habitually a slip of profile paper, say six inches 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. FORM FOB SURVEY RECORD. STA. Dis. DEFLEC. COURSE M.C. ELEVA. SLOPE. REMARKS. FORM FOR LOCATION RECORD. w i * h B " 2 Z STATION DISTANC COURSE. a i w | GRADIE GRADE. VARIATI SLOPE. REMARKS. 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. 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 82 SUGGESTIONS CONCERNING 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 chainmaii carries the flag 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 transitmaii 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. Yery 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. THE TRANSITMAN will be expected to keep his instru- ment in adjustment, and to be quick and accurate in its manip- ulation. 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 sometim.es warp a little out of line. Never omit 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 20. At the beginning of a curve, let the rear chainman know the phis 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 regardetj as, a maximum. SUGGESTIONS CONCERNING 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 transitman will aid in the making of maps. 5. THE LEVELLED 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 thus 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 r*n eye to abrupt irregularities in the ground, arid give plus elevations when necessary. lie 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 v.'ith 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. 84 THE CURVE-PROTRACTOR. 7. THE SLOPEMAN will give backsights, and take the cross slopes for one hundred feet on each side of the line at everj station. 8. 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 wil hand it in daily to the transitman for record. As each succes- sive chain is stretched, the rear chainman calls out the numbei of the stake it is stretched from: this assures the selection o] the right number for the stake ahead. 9. ONE AXEMAX will be detailed to make stakes, another tr mark and drive them. Additional axemen may be employee 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, witt radii of 400 feet to an inch. After some years' use in his owr practice, the contrivance was transmitted by the writer to tin well-known firm of James W. Queen & Co., mathematical- instrument makers, New York and Philadelphia, by whom ii is now manufactured. It greatly facilitates the projecting oj lines and solution of field-problems on location. It enables the engineer, for example, by a short, graphical process ant rapid inspection, to find the curve which shall close an angk between tangents, or terminate a compound curve, and pass al the same time through some fixed intermediate point, withoul liability to the errors, and free from the loss of time, in velvet 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. THIS CURVE -PROTRACTOR. 85 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 C; thence by quarter-degrees to 4; and thence by ten-minute differences to 2^. This covers one side of the sheet, the scale being 400 feet to an inch. Xow 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 delicate stress. Practise beforehand on a separate slip, for the right intensity of stroke. Engrave 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. It is perfect for its use; durable, flexible, spotlessly transparent, not liable to warp or change dimensions withwchanges 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. 86 THE CURVE -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. Having 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 distance 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 near 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, &e., 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 w,ill 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 w r ill 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; arid 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 locations, tiie better way generally ic to strike the curves first. 88 THE CURVE -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 NEAREST POINT OF ANY CURVE, STRUCK WITH RADIUS r, IN ORDER THAT A TERMINAL BRANCH HAVING RADIUS K = 2 r, AND CONSUMING A GIVEN ANGLE, X, MAY MERGE IN SAID STRAIGHT LINE. D = (R r) X (1 cos. x). DEGREE OF THE MAIN CURVE. ANGLE 2 3 4 5 6 7 8 9 10 X. 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 11.75 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 CURVE -PROTRACTOR. 89 If R = \\ 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. DEGREE OF CURVE. DEGREE OF CURVE. DEGREE OF CURVE. 1 2 3 4 5 6 7 8 9 10 ', or 56 -f- 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 v/ith 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 AB, containing an angle of 32 20', to have been located, and that it is required to substitute for it an- other curve AC, which shall end in a parallel tangent C F, 60 feet to the right. FIRST M E T H o D. Find the length of the long chord A C, = A B -{- B C. Referring to Table XVI., the chord of a 1 curve for 32 20' is seen to be 3,190.8 feet; that of a 4 curve, there- 98 TO FIND THE POINT AT WHICH TO COMPOUND. fore, = 797.7 feet, say 798 feet, = A B. To find B C, solve the triangle BDC, observing that the angle DBC = B AI = one-half of the central angle 32 20', = 16 10', and that t> C = 00 feet. Then D C -f- nat. *in. 16 10' = 60 -=- .278 = say 216 feet, = B C. Hence AC = AB-fBC = 798 -f 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. 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, = AI + I H. The tabular tangent of 32 20' divided by 4 gives A I = 415 feet. In the triangle KDC, the side DC -i- nat. sin. K = 60 -f- nat. sin. 32 20V = 112 feet = KG = I H. Then AH = A I -f- IH = 415 + 112 = 527 feet; and the tabular tangent 1,661 -f- 527 gives 3.15, equivalent to 3 09', the degree of the required curve A C, as before. 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. 1. To find B, the angle BIG must be found. Call the given distance between tangents D; the larger radius, K; 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, I K is the cosine a; i.e., I K -f- 1 II = nat. cosine a. But I II = R r; I K = I C K C, and K C = K F or II E -f F C, = r -f- D; i.e., 1 K = R r D. Hence 'nat. cosine a = R r D, -i- K r = 1 11) -f- (H r)]. TO SHIFT A P. C. C. 99 The same reasoning would apply if A BE 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 5 curve striking 27 feet to the right. Here D = 27; R = 1,910 1.146, = 764; D ~ R r = 27 -7- 784, = .03534; and 1 .03534 = .96466 = nat. cosine 15 17'. We must go back, therefore, 509 feet on the 3 curve, to compound into the 5 curve. Had the 5 curve been located first, we must have gone back 306 feet to begin the 3 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 F E C, the angle E = half of 15 19', = 7 38|', and the side^FC = 27 feet. Then E C = 27 -f- nat. sin. 7 38*', = 203 feet ; and if E were stake 54.20 on the 5 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 ACN located, and that it is required to fix a new P. C. C. at B, from which the terminal branch BM shall merge in tangent M L, a given distance from NO. 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, = IEK, of the located terminal branch, b; and the central angle, B H M, = HE F, to be substituted for it, . In the right triangle, EIK, EK = EI cos. I E K = (R r) cos. b. In the right triangle H F E, E F = E II cos. II E F = (II r) cos. a. Also, F K = L O = D, since each is equal to r K L. ThenE F = E K FK; i.e., (R r) cos. a= (K r) cos. b D. Hence nat. cosine a = nat. cosine b [D-f- (R r)j. 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, nat. cos. b = nat. cos. a -f- [D-MR-r)] Example. A 3, compounding into a 5 curve at C, which consumes an angle CIN, = 30 22', and ends in a tangent, NO, which is found, by measurement of L O, 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, nat. cos. a == nat. cos. 30 22' (34 -f- [1,910 1,146]) = 0.8628 (34 -4- 704). 34 log. 1.531479 764 , lo^;. 2.883093 .0445 log. 2.648386 Then 0.8628 0.0445 = 0.8.183 = cos. 35 05', = angle a ; a 6 = BUM C IN = EEC = the angle of retreat from the erroneous P. C. C. = 35 05' - 30 22' = 4 43', equivalent to LVT feet, on the 3 curve, from C'to B. 2. SECOND CASE: the terminal branch having the longer radius. Let B N represent the terminal branch located with central angle I K O = b, and suppose it required to determine the new arc CM, with central angle I E F = a. Call the longer radius R, the shorter one r; the distance L N between tangents, D. In the TO SHIFT A P. C. .C. 1 yi k right triangle IKO, KO = KI, cofc IK/0) = (R>- V); 233. <>. In the right triangle FIE, EF = ET, cos. 1F.F =' (R r) cos. a. Also, E II = L N = D, since each is equal to R K L. Then E F = E II + H F = E II -f K O; i.e., (R r) cox. =(R r) cos. b + D. Hence nat. cos. a = nat. cos. b -f iD-j-(R-r)]. Were the curve C M located, and the curve B N to be sub- stituted for it, that is to say, were a given and 6 required, we should have, by transposition, nat. cos. b = nat. cos. a [D-MR-r)]. Example. 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, nat. cos. a = nat. cos. 44 20' -|- (28 -f- 764). The nat. cos. 44 20'_= 0.69883; 28 -f- 764 = log. 1.447158 log. 2.883093 = log. 2.564065, corresponding to the decimal 0.03665, which, being added to nat. cos. 44 20 ; , gives 0.73548, the nat. cos. 42 29'. Then B K N C E M = 44 20' 42 29' = 1 51' = angle B I C, equivalent on a 5 curve to 37 feet, which therefore is the distance around the arc 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 tangent located strike ] ( the proposed tangent; or, ' OUtSlQG ' with radii in the order r, R, should the terminal tangent located strike | . , > the proposed tangent, take the i difference I of Q and for the re ^ uired cosine ' 100 TO SHIFT A P. C. C. tance asunder of the terminal tangents, D ; the central angle, C I N, = I E K, of the located terminal branch, b ; and the central angle, B H M, = HEF, to be substituted for it, . In the right triangle, EIK, EK = EI cos. I E K = (R r) cos. b. In the right triangle H F E, E F = E II cos. II E F = (II r) cos. a. Also, F K = L O = D, since each is equal to r K L. ThenE F = E K FK; i.e., (R r) cos. a=(R-r) cos. b D. Hence nat. cosine a = nat. cosine b [D -f- (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, nat. cos. b = nat. cos. a -f- ID-MR r)] Example. A 3, compounding into a 5 curve at C, which consumes an angle CIN, = 30 22', and ends in a tangent, NO, which is found, by measurement of L O, 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, nat. cos. a = nat. cos. 30 22' (34 -f- [1,910 1,146]) = 0.8628 (34 -f- 704). 34 log. 1.531479 764 , IK. 2.883093 .0445 log. 2.648386 Then 0.8628 0.0445 = 0.8183 = cos. 35 05', = angle a ; a 6 = BIIM CIN= BEC = the angle of retreat from the erroneous P. C. C. = 35 05' - 30 22' = 4 43', equivalent to 157 feet, on the 3 curve, from C'to B. 2. SECOND CASE: the terminal branch having the longer radius. Let BN represent the terminal branch located with central angle I K O = 6, and suppose it required to determine the new arc CM, with central angle I E F = a. Call the longer radius R, the shorter one r; the distance L N between tangents, D. In the TO SHIFT A P. C. .;<7. : ; v '. '.' 1(,jl v right triangle IKO, KO = KI, coS. IK/C) = (B r). 35.s. \ In the right triangle FIE, EF = El, cos. IF, F =' (R r) cos. a. Also, E H = L N = D, since each is equal to R K L. Then EF = EH + HF = EH + KO; i.e., (R r) cos. rt = '(R r) cos. b + D. Hence nat. cos. a = nat. cos. b -f [D--(R-r)]. Were the curve C M 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 P-MR-r)]. Example. 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, nat. cos. a = nat. cos. 44 20' -(- (28 -f- 764). The nat. cos. 44 20'_= 0.69883; 28 -f- 764 = log. 1.447158 log. 2.S83093 = log. 2.564065, corresponding to the decimal 0.03665, which, being added to nat. cos. 44 20', gives 0.73548, the nat. cos. 42 29'. Then B K N C E M = 44 20' 42 29' = 1 51' = angle BIG, equivalent on a 5 curve to 37 feet, which therefore is the distance around the arc 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 tangent located strike j the proposed tangent; or, with radii in the order r, R, should the terminal tangent located strike < . , > the proposed tangent, take the i difference I of Q and C for the re( l uired cosine - 104 TO LOCATE A Y. son that each is equal to D E. Make G F = R -4- r, the diame- ter of a semicircle. Said semicircle touches tangent 13 A at D, its middle point; and D E being perpendicular to G F, we have by geometry GE:DE::DE:EF; i.e., GE X EF, or R X r, = D E 2 . Hence D E = B D = U A = y/R X r = R tan. i x, and we are thus enabled to fix the points E and A. 3. In the two foregoing problems, the angle consumed by curve E A is = 180 x. Example. BE, a 2^ curve located; BA, a tangent: to complete the Y with a 6 curve, E A. By the first method, cos. x = (R ) -f- (R -f r) = (2,292 955) H- (2,292 -f 955) = 1,337 -f- 3,247 = log. 3.126131 - log. 3.511482 = 1.014649, which corresponds to log. cos. 9.614649, or to the decimal number 0.4118, indicating in either case the angle 65 41' = x. DE = BD = DA = R tan. $x = 2,292 X 0.6455 = 1,479.4. DE may be found also by reference to Table XVI., where the tangent of a 1 curve for 65 41' is seen to be 3,698.6. Dividing this number by 2|, we have 1,479.4, as above. Or, by the second method, D E X r = \/2 188860 = 1 ,479.4. 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- vex to the Y, construct the figure as in margin, and reason thus: In the triangle EGF, formed by lines connect- ing the curve-centres, the sides are respectively equal to the sums of the contiguous radii: the angles may therefore be found as in Case III., Trigonometry. Lines drawn bisecting the central angles of the severaJ 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. Example. Given B A, a 3, and B C, a 4 curve: to complete the Y with a 5 curve, C A. E F = 1,910 + 1,146 = 3,056. GF = 1,433 + 1,146 = 2,579. E G = 1,910 -|- 1,433 = 3,343. Then, by Case III., Trigonometry, As EG, 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. . . 2.678518 To diff. of segments of E G, 804 2.905276 Adding half the difference to half the sum of the segments of the base EG, 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 B Dor DA. 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 5 curve to be 72 12'. Arc B A, = 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 4 curve. Points being thus fixed at A and C, curve C A can be laid on the ground. 5. // curve B A is concave to the Y, the radii being given, construct the figure as follows: First draw the triangle GFE, 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 EB, EA, and GF. Then, comparing triangles GBD, GCD, the angles at G are equal by construction ; the angles at 13 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 CF = FA, and DA = DC. 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. 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- ness of the curves. It remains only to solve the triangle G F E, seeing that from its angles the required central angles can be obtained. Example. B A, a 1, B C, a 6 curve, located : to complete the Y with an 8 curve, C A. TO LOCATE A TANGENT TO A CURVE. 107 111 triangle G F E, E F = 5,730 717 = 5,013. E G = 5,730 955 = 4,775. G F - 955 + 717 = 1,672. Then, by Case III., Trigonometry, AsEF . . . 5,013 .... log. (a. c.) 6.299902 Is to E G -f G F, 6,447 .... log. . . 3.809358 SoisEG GF, 3,103 . . . . log. . . 3.491782 To cliff, seg. 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 -4- 4,775 = log. 3.653405 3.678973 = 9.974432 = cos. 19 28' = angle E. Cos. G F E = the shorter segment divided by GF = 511 -j- 1,672 = log. 2.708421 log. 3.223236 = 9.485185 = cos. 72 12' = angle G F E. The central angle, B G C, of the 6 curve, is equal to 180 F G E = the sum of the angles at E and F = 72 12' -f- 19 28' = 91 40', making the arc B C = 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 = 180 72 12' = 107 48', equivalent, on an 8 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. 108 TO SUBSTITUTE A CURVE. Example. A B = 1,500 feet ; D H B, a 4 curve ; angle F B D = 20 13'. First, the angle FED, between a tangent and a chord, is equal to half the central angle subtended by the same chord. Angle D C B, therefore, = 40 26'. By Table XVI., the chord of 40 26', for a 1 curve, = 3,960.2 feet; for a 4 curve, it is, say, 990 feet = D B ; and D I = I B = 495 feet. The versin. II I is, in like manner, found to be 88.25 feet. Deducting this from the radius of the 4 curve, we have I C = 1,344.4 feet. Then IC-i-IA = tan. I AC; i.e., 1,344.4 -r- (495 -f 1,500) = 0.674 = tan. 33 59' = angle I A C. Next, by geometry, the proposed tangent A E =\/A D X A B = \/2,490 X 1,500 = 1,932.6 ; and E C -f- 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. Example. 1. A B, a 4 curve; BC = 774 feet; CD, a 6 curve: to put in the 1 curve, E F. Sketch the figure as in margin, HK being parallel and equal to BC. Then KG = BG BK or CH = 1,433 955 = 478 feet; KH -7- GK = 774 H- 478 = 1.62 = tan. 58 19' = angle KGH; and KIT -r- sin. 58 19' = 774 -f- 0.851 = 909.6 feet = G H. TO RUN A TANGENT TO TWO CURVES. 109 In the triangle GUI we have then the sides given; namely G H = say, 910 feet, II I = 5,730 955 = 4,775 feet, and GI = 5,730 1,433 = 4,297 feet : to find the angles. Under Case 3, Trigonometry (in.), ill : IG + GH :: IG G H : I L L II ; i.e., 4,775 : 5,207 : : 3,387 : 3,093, the differ- ence of the segments into which the base III is divided by a perpendicular from G. Adding half the difference of the seg- ments thus found to half their sum, the longer segment, I L, is found to be 4,234 feet; subtracting half the difference from half the sum, the shorter segment, LH, is found to be 541 feet. Then H L -r- II G = 541 ^ 910 = 0.5945 = cos. 53 31' = angle GUI. In like manner, dividing IL by IG, we find the angle G I II to be 9 49'. The sum of these angles = angle E G II = G3 20', for the reason that each is equal to 180 II G I. Finally, E G H K G H = 63 20' 58 19' = 5 01' = angle E G B, equivalent to a distance from B of 125 feet around the 4 curve to the P. C. C. at E; and GIH EGB = 9 49' 5 01' = 4 48' = angle CHF, equivalent to a distance from C of 80 feet around the 6 curve to the P. C. C. atF. XXXVII. TO RUN A TANGENT TO TWO CURVES ALREADY LOCATED. 1. If one curve be visible from the other, or if both be visible from some inter- M mediate point, mark them on the ground with stakes at short intervals. The points M or L in the range of the required tangent may then be fixed by one or two trial settings of the transit, and the line put in. 110 TO RUN A TANGENT TO TWO CURVES. 2. Should obstacles prohibit this plan, measure any con- venient line, FG or BCD, from one to the other curve, and, completing the traverse A F G E or A 13 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 II and I ascer- tained as in the following example: Example. F B, a 4 curve; G D, a-G curve. X. S. E. W. A B, N. 20 E., 1,433 feet . . 1,340.6 490.0 B C, East, 3,570 feet . . 3,570.0 CD, X. 34 E., 1,800 feet . . 1,492.2 1,006.2 DE, X. 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' =- bearing AE; and 4,391 -f- sin. 50 20' = 5,704 feet = distance AE. Also, EK -^- AE = (1,433 -f 955) -!- 5,704 = xin. 24 45' = angle E AK; and angle AEK = 90 00' 24 45' = 65 10'. Hence the bearing of AK or III is X. 75 05' E., and that of AHorIE,N. 14 55' W. Since AB bears X. 20 E., the angle II AB = 20 00' -f 14 55' = 340 55^ equivalent to a distance of 873 feet from B around the 4 curve to the required P. T. at II; and, since DE bears N. 45 00' W., the angle IE D = 45 00' 14 55' = 30 05', equivalent to a distance of 501 feet from D around the 6 curve to the required P". C. at I. 3. Should the curves turn in the same 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. But 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 curve, 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 TANGENT TO TWO CURVES, 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. G. Suppose, for illustration, that a trial tangent, bearing by needle X. 54 30' E., is run out from stake 24.80 of a 4 curve, intending to touch a G, but is found to cut it. Suppose fur- ther that the objective curve was laid down and numbered in the direction of approach towards the 4 curve; that its P. C. is stake 25.10, and the magnetic bearing of its initial tan- gent S. 30 30' \V. The angle, then, between the bearing of the trial tangent and that of the initial tangent of the G curve, is 24, corresponding to a distance of 400 feet on the latter curve. At stake -25.10 + 4.0 = 20.10, therefore, a tangent to the G 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 87 feet, the nat. tan. of the angle of error is 0.0348 = tan. 2. By the method in (4), this calls for a shift of the P. T. 50 feet ahead on the 4 curve, making the new P. T. 24.80 -f 0.50 = stake 25.30, and ad- vances the P. T. of the G 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 2, equivalent to- a deviation of 1 feet scant in half a mile from the line corrected by the method in (4), and agreeing exactly with the correction determined by the method in (2). TRACK PROBLEMS. XXX VIII. -LI. TRACK PROBLEMS. XXXVIII. REVERSED CURVES. 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 PARALLEL TANGENTS BY A REVERSED CURVE HAVING EQUAL RADII. 1. The radius R, and the perpendicular distance D, between 110 TO CONNECT TWO PARALLEL TANGENTS. Draw the tangents, radii, and curves, fixing the P. R. C. midway of D. Draw the chords G I, I E, the line B F perpendicular to G I, and the line E H in prolongation of radius CE to an intersec- tion with B H passed through centre B parallel to tangents. That 1 falls midway of D, follows from the necessary sym- metry of the figure; and G IE must be a straight line, because the radii B I, 1C, perpendicular to a common tangent at the same point, form a straight line, to which the chords G I, IE, are equally inclined. C H-^ C B = cos. A; but C H = 2 R D, and C B = 2 R. .-. cos. A = (2 II D)-|-2R. B H = B C sin. A = 2 R sin. A ; GF = RMi. | A; GE = 4GF. . . G E = 4 R sin. $ A, and GIorIE = 2R sin. I A. Observe, that, in the right triangles G K E and B G F, the angles at G and B are each equal to | A: hence the triangles are similar. Example. R = 800 feet, D = 24 feet. To find angle A. Cos. A = (2 R D) -r- 2 R = 1,576 -f- 1,600 = 0.985 = nat. cos. 9 56'. BH may then be found =.2 R sin. A = 1,600 X 0.1725 = 276 feet, and laid 'off from the P. C. at G to K, the point E being fixed by a right angle from K. Or GE may be found = 4 R sin. i A = 3,200 X 0.866 = 277.1 feet, and laid off from G to E, the point I being fixed 138.5 feet from G, and angle K GE made equal to half of A = 4 58'. 2. The distances G K and D given, to find R. In triangle GKE, KE = D. D -T- GK = tan. A; D -j- sin. i A = GE; and GE -f- sin. $ A = 4 R. Or, having found GE, we have from the congruity of trian- gles GKE, BEG, D : GE :: i GEorGF : R. TO CONNECT TWO PARALLEL TANGENTS. 117 Example. = 300 feet, D = 28 feet. D-f-GK . . . . = Tan. \ D -r- sin. $ A . . . = GE G E -f- .sin. -J- A . . . Log. 28 Log. 300 A . 5 20' . Log. 28 Sin. 5 20' . . 301.24 . Sin. 5 20' = 4 R . . 3,241 .'. R = 810.2. 1.447158 2.477121 8.970037 1.447158 8.968249 2.478909 8.968249 3.510660 XL. TO CONNECT TWO PARALLEL TANGENTS BY A REVERSED CURVE HAVING UNEQUAL RADII. 1. Given the perpendicular distance, D, between two paral- lel tangents, and the unequal radii, R and r, of a reversed curve, to find the central angles, A, the chords, and the straight reach, G K, of the curve. 118 TO CONNECT TWO PARALLEL TANGENTS. Cos. A = C H -f- B C ; but C H = (R + r) D, and B C = K + r. .'. Cos. A= (K'-fr D)H- (R -f r). The straight reach GK = BH=(K-|-r) sin. A. The sum of the chords G E = G K -=- cos. | A. GI = 2Rsm. i A. IE = 2 r sin. i A = GE GI. Example. D = 28, E = 955, r = 574. Cos. A = (R -f r D) (R + r) = 1,501 ~ 1,529. 1,501 . . . log. 3.176381 1,529 . . . log. 3.184407 Cos. A, 10 59' ..... 9.991974 GK= (R + r) sin, A. R + r, 1,529 . . . log. 3.184407 Sin. A, 10 59' ... log. 9.279948 GK = 291.3 ..... 2.464355 G E = G K cos. -J A. GK, 291.3 ... log. 2.464355 Cos. i A, 5 29i' . . . log. 9.998014 GE = 292.6 ..... 2.466341 2 R, 1,910 . . . log. 3".281033 Sin. i A, 5 29^ ... log. 8.980916 G I = 182.8 ..... 2.261949 IE = GE GI = 292.6 182.8 = 109.8. 2. The distances GK and D, and one of the unequal radii, R, given, to find the other radius, r, and the central angles, A. REVERSED CURVE WITH UNEQUAL ANGLES. 119 Example. G K = 422, D = 30, R = 2,292. Tan. | A = D -f- G K. D = 30 . . . log. 1.477121 GK = 422 . . . log. 2.625312 Tan. i A, 4 04' . . . . . 8.851809 .-. A = 8 08'. G E = D -r- sin. i A. D = 30 . . . log. 1.477121 Sin. i A, 4 04' ... log. 8.850751 GE = 423 2.626370 GI = 2Rsw. i A. 2 R = 4,584 ... log. 3.661245 Sin. | A, 4 04' . . . log. 8.850751 GI = 325.1 2.511996 GE G I = 423 325 = 98 = IE. r = $ tO i ^ OS K> 0*-^. *; 10 Co W tO Or 4^^S ^iobb Ma Ra Mi Mi ce il frog dist. outer rail. frog dist. frog angle Switch-ra Do. angle Ma Ra Mi Mi fro out fro dis . outer rai . frog dist . frog angl witc Do. a -rai gle, 21 i! rt> ** p-p-H I* cP? 13* lll -S.5. i to -J CO , CO p. t) ' to en o aStf <3> GO tO CO O> CD 05 ^'-*Qi 4x CO -* M CO O5 O o ^j b lt> Main frog Rad. oute Mid. frog Mid. frog S* *P* otOCO CoV? 5 ? 3 *0000 y, CO 01 -1 00 , cob -ico : to co be o bo bi is OJCOO. F ^ ?' o< M O CO--1 ' 4- Oi 00 ^ fcS^S ^COOO ^^oo ;^i ^0,5. -^-00 i x; - 1 I <= M ?fe? ffek?! ^. os Oi *> I Xoo co -T fe *- 1- Jo to O OS H-" o, OT O Xbbi % tc o CC^i- r- TURNOUT TABLE. :s o S O 3 d I CM V I C ( z I p? O CO GO -1 1 2P| ^ICQO *O * < o ^ e 3 s s -oqv , cocOi-cv cocpov 0> TH GO 00 c-i oo v, -f CO CO o CO O O ( frog dist. uter rail. frog dist. rog angle witch-rail, L'gth. 12 ft. Ang.400'. Rad. 171.9. dist. r rail. dist. frog angle og dist. ter rail. g dist. g angle iiliili4 o ^-o 5 r^ 5 frog oute frog -;_; TO LOCATE A TURNOUT. 137 XL VII. TO LOCATE A TURNOUT. 1. Let the heavy parallels in the figure represent the rails of the main track. 2. Stick a pin or drive a spike at A, the toe of switch, at a distance from the gauge side of the main-track rail equal to the throw of the switch-rail. Lay off the distances AC and AB (if a double turnout), taken from the foregoing tables, and place the frogs C and B, or mark those points. Stretch the cord from A to B, and from B to C. Mark the middle points of those stretches at H and P. Catch the cord at H with your forefinger, and pull it outwards until your finger, at E, lines with the switch-rail, and also with the right gauge side of frog B. Stick a pin at L, half-way between H and E. Let the cord spring in against L, so that it shall stretch straight from A to L, and from L to B. Opposite the middle points, V, of those stretches, stick pins on the outside at a distance from the cord equal to one-quarter of H L. In like manner, catch the cord at P, the point midway between B and 0; stretch it to F, in line with the gauge sides of the frogs; a,nd stick a pin at I, half-way between P and F. 138 TO LOCATE A TURNOUT. 3. 'Next lay off the proposed line of the near rail of the side track, X D. Mark the point G on that line where the range of the proper gauge side of frog C strikes it. Measure C G. Set off G D, equal to C G, along the side-track line, and drive a pin at D. Stretch the cord from C to D. Mark the middle point of it at K, and drive a pin at N, half-way between K and G. Stretch the cord from C to N, and from N to D. Stick pins outside the middle points, M and O, of those stretches at a distance from those points, M and O, equal to one-quarter of KN. 4. These three sets of pins will fix the line of one rail of the turnout. The corresponding rail of a double turnout can be laid off from them, if required, by symmetrical measurements. 5. In the case of a single turnout, stretch the cord from the toe of switch, as above, to the point of frog, located by the foregoing tables ; catch it at the middle, and pull it outwards to a point in range with the line of the switch-rail in one direction, and the gauge side of frog in the other direction. Half-way between that point and the middle of the cord, when straight, stick a pin. Measure that half-way distance, and divide it by 4; call the quotient the "quarter-distance." Stretch the cord from the pin just set to the toe of switch in one direction, and to the point of frog in the other. Outside the middle points of these short stretches, lay off the " quarter- distance," as above found, and stick two other pins. These three pins will sufficiently mark the line of the outer rail of the turnout. 6. The same methods will apply in practice to turnouts from curves. In the latter case, the distance C G is to be calculated as follows : Multiply the distance Y D, between the nearest rails of the parallel tracks, by the number of the frog, taken from the fore- going table. Thus, on the full gauge, with a space between tracks of 7 feet and a No. frog, the distance C G would be 7 X 6 == 42 feet. Lay off C G, in range of the gauge side of the frog, and stick a pin at G. Measure out G D, equal to C G, and set another pin at D, making D Y the proper distance be- tween tracks. Then stretch the cord from D to C, and pro- ceed to stake off the curve C N D, as above directed. CROSSINGS ON STRAIGHT LINES. 139 XLVIII. CROSSINGS ON STRAIGHT LINES. 1. Having located frogs B and C by the preceding methods, stretch the cord any convenient distance, C D, in the range of the outer gauge side of the frog C. Set off E F parallel to C D, and distant the gauge-width from it. The intersection of said parallel at F with the near rail of the side track marks the spot for point of side-track frog ; the curve F G, thence to toe of switch, corresponds to AC on the main track, and may be staked out in like manner. XLIX. CROSSINGS ON CURVES. 1. Having located frogs B and C by the preceding methods, set off the width of gauge, CD, from point of frog C, and square to its outer gauge side. Stick a pin at D. 2. Next calculate the distance D E to the point of side-track frog as follows: Subtract the gauge of track from the dis- 140 CROSSINGS ON CURVES. tance, HI, between the gauge sides of the nearest rails of the main and side tracks; multiply the remainder by the number of the frog, taken from foregoing tables. The product will be the distance from D to the point of side-track frog at E. ELEVATION OF THE OUTER RAIL ON CURVES. 141 3. Suppose, for example, the gauge sides of the nearest rails of the main and side tracks are 6 feet 6 inches asunder; gauge of track, 4 feet 8 inches; frog, a No. 9. Reducing inches to decimals, we have then the distance between tracks 6.5 feet, less the gauge, 4.7 feet, = 1.8 feet; and 1.8 multiplied by 9, the number of the frog, gives 16.2 feet for the distance D E. The proper spring will be given to rail DE on the ground; and curve E G, from frog to toe of side-track switch, will be staked off as directed in the section on turnouts. L. ELEVATION OF THE OUTER RAIL ON CURVES. 1. Great precision in this adjustment is unattainable, owing to differences in the speed of trains and to the cost of track- maintenance, if it were attempted. The annexed table will be found convenient in practice. It has been calculated by the following simple rule: 2. Divide the speed in miles per hour by 10; multiply the square of the quotient by the degree of curve. The product is the elevation in sixteenths of an inch for full gauge of 4 feet 8| inches. Take two-thirds of it for 3-feet narrow gauge. 3. Molesworth gives the following formula for determining the elevation of the outer rail with any gauge: V = greatest velocity of trains in feet per second. G = gauge of railway in feet. C = length of chord whose middle ordinate will give the required elevation. Then C = \ V A modification of this formula gives the following approximate rules : To fix the elevation of the outer rail on the standard gauge of 4 feet 8| inches, multiply the speed of trains in miles per hour by 5, and divide 142 ELEVATION OF THE OUTER RAIL ON CURVES. the product by 3. This will give the length of tape, C, to stretch on the gauge side of the outer rail ; and the distance, e, from the middle of the tape to the gauge side of the rail, will be the proper elevation. For gauge of one metre, 3.28 feet, make C equal to one and one-third times the speed of trains in miles per hour. For 3-feet gauge, make C equal to one and one-fourth times the speed of trains in miles per hour. TABLE OF ELEVATIONS FOR OUTER RAIL ON CURVES. SPEED IN MILES PER HOUR. 10 20 30 40 H II II i fl b ELEVATION OF OUTER RAIL IN INCHES AND FRACTIONS. : O o ll o |d 1 d 1 &e | SO C T I o H W R J 9 c3 Is? * . CS-IC* 3 Jsr li H H Qj W 5 fi fr* fc w "* eo "* t^ w ** ^ M a 1 1-16 1-4 3-16 9-16 3-8 i 11-16 1 2 1-8 .. 1-2 5-16 1 1-8 3-4 2 1 5-16 2 3 3-16 1-8 3-4 1-2 1 11-16 11-8 3 2 3 4 1-4 1 11-16 21-4 1 1-2 4 2 11-16 4" 5 5-16 1 1-4 13-16 2 13-16 1 7-8 5 35-16 5 6 C 3-8 1-4 1 1-2 1 33-8 21-4 6 4 6 7 7-16 1 3-4 3-16 3 15-16 25-8 7 4 11-16 7 8 1-2 2 5-16 41-2 3 8 5 5-16 8 9 9-16 3-8 21-4 1-2 51-16 33-8 9 6 9 10 5-8 . . 2 1-2 11-16 5 10-16 33-4 10 611-16 10 11 11-16 23-4 13-16 (i 3-16 4 1-8 11 7 5-16 11 12 3-4 1-2 3 2 6 3-4 4 1-2 12 8 12 13 13-16 31-4 23-16 75-16 47-8 13 811-16 13 14 7-8 31-2 2 5-16 7 14-16 5 1-4 14 95-16 14 15 15-16 5-S 33-4 21-2 87-16 5 5-8 15 10 15 16 1 4 2 11-16 9 6 16 10 11-16 16 TRAQKMEN*8 TABLE OF CURVES. 143 LI. DEGREE OF CURVE. OOCC^OC.^COtOMO^OC-.CgC.^WtOM ||1| to*, ik to w to to to MM*M**Bm-. 100 Feet. MIDDLE ORDINATES IN INCHES AND FRACTIONS, THE LENGTH OF CHORD BEING tO 4- i' tO 4-4- tO 4- tO 4-4- tO 4- tO 50 Feet. WtOMMOCO<000-^05WOi4^*KWtOlOr- OO 1-0 OO 00 lO GO 4- tO OO 4- lO 4- GO 4- do 4- GO g ? to 1 5- "332S---SSSE- ^010> g 0105 003 20 Feet. sss |.*.-*. s gS 3S- *- 1 LENGTH OF CHORD IN FEET AND INCHES, WITH A MIDDLE ORDINATE EQUAL TO GAUGE OP TRACK. SS2SSSJ P ' 85 1 B-i-SoBS^o..B^So? DEGREE OP CURVE. SS8-J!'S!>S-.'?S81.!'! i 10890 T 7 4628.25 2 87120 2 21780 18 4900.50 3 130680 3 32670 ^9 5!7 2 -75 4 174240 4 435 6 20 5445-oo 5 217800 21 5717.75 6 261360 304920 Perches. Square Feet. 22 23 5989-50 6261.75 8 348480 24 6534.00 9 392040 i 272.25 25 6806.25 10 435600 2 544-5 26 7078.50 ii 479160 3 816.75 27 7350-75 12 522720 4 1089.00 28 7623.00 13 566280 1361.25 29 7895-25 *4 609840 6 ^S.so 30 8167.50 15 653400 7 I905-75 31 8439-75 16 696960 8 2178.00 32 8712.00 17 740520 9 2450.25 33 8984.25 18 783080 10 2722.50 34 9256.50 *9 827640 ii 2994-75 35 9528.75 20 871200 12 3267.00 36 9801.00 21 916760 13 3539-25 37 10073.25 22 960320 H 3811.50 38 10345-50 15 4083.75 39 10617.75 16 4356.00 40 10890.00 TABLE VII. CIRCULAR ARCS TO RADIUS OF 1. DEGREES. ' MINUTES. SECONDS. i .01745329 / i .00029089 // .00000485 2 .03490658 2 .00058178 2 .00000970 3 .05235988 3 .00087266 3 .00001454 4 .06981317 4 .00116355 4 .00001939 5 .08726646 .00145444 5 .00002424 6 .10471975 6 00174533 6 .00002909 7 .12217305 7 .00203622 7 .00003394 8 9 .13962634 .15707963 8 9 .00232711 .00261799 8 9 .00003878 .00004363 TABLES. 153 TABLE VIII. FEET IN DECIMALS OF A MILE. Feet. Dec in als of a Mile. I o I 8 9 3 9 4 2 o 3 7 8 7 9 8 3 o 5 6 8 i 8 2 4 o 7 5 7 5 7 6 5 o 9 4 6 9 7 6 o I i 3 6 3 6 4 7 o I 3 2 5 7 5 8 8 0.0 I 5 I 5 i 5 2 9 o I 7 O 4 5 4 6 TABLE TX. INCHES REDUCED TO DECIMAL PARTS OF A FOOT. In. O I 2 3 4 5 6 7 8 9 10 ii In. .OOOO 0833 .1667 .2500 3333 .4167 .5000 5833 .6667 .7500 8333 .9167 re .0052.0855 .1719 2552 .3385 .4219 5052 5885 .6719 7552 8385 .9219 A | .0104.0938 .1771 .2604 3438 .4271 .5104 5938 .6771 .7604 8438 .9271 ^ 1 3 6 .01561-0990 .1823 .2656 3490 4323 5156 599 .6823 .7656 .8490 9323 T 3 g i .0208 .1042 .1875 .2708 3542 4375 .5208 .6042 .6875 .7708 .8542 9375 \ 1 5 6 .026oj. 1094 .1927 .2760 3594 .4427 .5260 .6094 .6927 .7760 8594 9427 T 5 6 | .0313 .II46j.I 9 79 .2813 .3646 4479 5313 .6146 .6979 7813 .8646 9479 -| 7 16 0365 .1198 .2031 .2865 .3698 4531 5365 .6198 .7865 .8698 9531 A i .0417 .1250 .2083 .2917 3750 .45831.5417 .6250 .7083 .7917 .8750 9583 T 9 6 .0469 .1302 2135 .2969 .3802 4635! -54 6 9 .6302 7^35 .7969 .8802 9635 IT 1 .0521 1354 .2188 .3021 3854 .4688 5521 6 354 .7188 .8021 .8854 .9688 I H 0573 .1406 .2240 373 .3906 .4740 5573 .6406 .7240 8073 8906 974 u I .0625 .1458 .2292 3125 .3958 .4792 5625 .6458 .7292 .8125 8958 9792 1 1 3 1 15 .0677 .1510 2344 3177 .4010 .4844 5677 .6510 7344 .8177 .9010 9844 \l 7 8 .0729 .1563 .2396^3229 .4063 .4846 5729 6563 .7396 .8229 .9063 9896 1 8 ii .0781 .1615 .2448 .3281 43 4948 578i .6615 .7448 .8281 .9115 9948 ii TABLE X. RADII AND THEIR LOGARITHMS, MIDDLE ORDI- NATES, AND DEFLECTION DISTANCES. 156 RADII AND THEIR LOGARITHMS. : Degree of ! Curve. Radius. Logarithm of Radius. Arithmetical Comple- ment. Middle Ordinate, Chord 100 Feet. Deflec- tion Dis- tance. Tangen- tial Dis- tance. 3 5 ' 68754.9 4-837304 5.162696 .018 I 45 73 ! 10 34377-5 4.536274 5.463726 .036 .291 MS i 15 22918.3 4.360182 5.639818 55 436 .218 | 20 17188.8 4.235246 5-764754 73 582 .291 25 I 375 I -o 4-138335 5.861665 .091 .727 '364 3 11459.2 4.059154 5.940846 .109 873 436 35 9822.2 3.992209 6.007791 .127 i .02 59 46 8594.4 3-9342I5 6.065785 145 1. 16 582 45 7639-5 3.883066 6.116934 .164 I- 3 T 654 50 6875-5 3-837304 6.162696 .182 1.45 .727 55 6250.5 3.795914 6 . 204086 .200 i. 60 .800 1 5729.6 3.758128 6.241872 .218 T -75 873 5 5288.9 3-723365 6.276635 236 1.89 945 10 4911.1 3.691179 6.308821 255 2.04 1.02 15 4583-7 3.661216 6.338784 2 73 2.18 1.09 20 3-633I95 6.366805 .291 2-33 1.16 25 4044.5 3.606864 6.393136 309 2.47 1.24 3 3819.8 3.582041 6.417959 327 2.62 1.31 35 3618.8 3-558565 6-44M35 345 2.76 1.38 40 3437-9 3-536293 6.463707 364 2.91 45 6.484894 .382 3-5 I -53 50 3125.4 3.494906 6.505094 .400 3-20 i. 60 55 2989.5 3-475599 6.524401 .418 3-34 1.67 2 o 2864.9 3-457"4 6.542886 -436 3-49 1.74 5 2750-3 3-43938o 6.560620 455 3-64 1.82 10 2644.6 3.422359 6.577641 473 3-78 1.89 15 2546.6 3.405961 6.594039 .491 3-93 1.96 20 2455-7 3-390I75 6.609825 509 4.07 2.04 25 2371.0 3-374932 6.625068 527 4.22 2. II 3 2292.0 3.360215 6-639785 545 4-36 2.18 35 2218. i 3.345982 6.654018 564 4-5 1 2.25 40 2148.8 3-332196 6.667804 -582 4-65 2-33 45 2083.7 3-318835 6.681165 .600 4.80 2.40 50 2022.4 3.305867 6.694133 .618 4-94 2-47 55 1964.6 3.293274 6.706726 -636 5-09 2-54 3 o 1910.1 3.281056 6.718944 655 5-23 2.62 5 1858.5 3.269163 6.730837 673 5-38 2.69 10 i 809 . 6 3.257584 6.742416 .691 5-53 2.76 15 1763.2 3.246301 6-753699 .709 5-6 7 - 2.84 20 1719.1 3.235301 6.764699 .727 5-82 2.91 25 1677.2 3.224585 6.775415 745 5.96 2.98 3 1637.3 3.214129 6.785871 764 6.11 3-05 35 1599.2 3.203902 6.796098 .782 6.25 40 1562.9 6.806069 .800 6.40 3-20 45 1528.2 3. 184180 6.815819 .818 6-54 3-27 50 1495.0 3.174641 6.825359 836 6.69 3-34 55 1463.2 3.165303 6.834607 855 6.83 3-42 4 o 1432.7 3-i56i55 6.843845 .873 6.98 3-49 5 3.147212 6.852788 .891 7.12 3-56 10 1 375-4 3-138429 6.861571 .909 7-27 3-63 15 1348.4 3.129819 6.870181 9 2 7 7.42 3-7 1 20 1322.5 3-121395 6.878605 945 7-56 3.78 2 5 1297.6 6.886859 .964 7.71 3-85 30 1273.6 3-105033 6.894967 .982 7-85 3-93 35 1250.4 3.097048 6.902952 I. 00 8.00 4.00 40 1228.1 3-089233 6.910767 i. 02 8.14 4.07 RADII AXD THEIR LOGARITHMS. 157 Degree of Curve. Radius. Logarithm of Radius. Arithmetical Comple- ment. Middle Ordinate, Chord 100 Feet. Deflec- tion Dis- tance. Tangen- tial Dis- tance. | i o / 4 45 206.6 3-081563 6.918437 1.04 8.29 4.14 5 185.8 3.074011 6.925989. 05 8-43 4.22 55 l6 5-7 3.066587 6.933413 07 8.58 4.29 5 o 146.3 3.059299 6.940701 09 8.72 36 5 i 2 7-5 3.052117 6.947883 .11 8.87 43 10 109.3 3.045050 6.954950 J3 9.01 5i 15 091.7 3.038103 6.961897 15 9.16 -58 20 074.7 3.031287 6.968713 .16 9-3 -6 5 2 5 058.2 3.024568 6-975432 .18 9-45 .72 30 042.1 3.017910 6.982090 .20 9.60 .80 35 026.6 5.011401 6.988599 .22 9-74 87 40 on. 5 3.004967 6.995033 .24 9.89 94 45 996.9 2.998652 7.001348 2 5 IO.O 5.02 5 982.6 2.992377 7.007623 2 7 10.2 5-9 55 968.8 2.986234 7-013766 .2 9 10.3 5-i6 G o 955-4 2.980185 7.019815 31 10-5 5-23 5 94 2 -3 2.974189 7.025811 33 10.6 5-3i 10 929.6 2.968296 7.031704 35 10.8 5.38 15 917.2 2.962464 7.037536 3 10.9 5-45 20 5- r 2.956697 7-043303 ' -38 1 1 .0 S-S 2 25 3-4 2.951046 7.048954 .40 II. 2 5-6o 3 2.0 2.945469 7-05453I .42 "3 5-67 35 870.8 2.939918 7.060082 44 n-5 5-74 40 859.9 2.934448 7-065552 45 n .6 5-8i 45 849-3 2.929061 7.070939 47 ii. 8 5-89 5 839.0 2.923762 7.076238 49 11.9 5.- 96 55 828.9 2.918502 7.082498 5 1 12. I 6.03 7 o 819.0 2.913284 7.086716 53 12.2 6. io 5 809.4 2.908163 7.091837 55 12.3 6.18 10 800.0 2.903090 7.096910 56 12.5 6.25 1 5 790.8 2.898067 7- IOI 933 -58 12.6 6.32 20 781.8 2.893096 7.106904 .60 12.8 6-39 2 5 773- 1 2.888236 7.111764 .62 12.9 6-47 30 7 6 4-5 2-883377 7.116623 .64 13-1 6-54 35 756-I 2.878579 7.121421 .65 13.2 6.61 40 747-9 2.873844 7.126156 .67 13-4 6.68 45 739-9 2.869173 7.130827 .69 13-5 6.76 5 732.0 2.864511 7-I35489 7t !3-7 6.83 55 724-3 2.859918 7 . i 40082 73 13-8 6.90 8 o 716.8 2-855398 7.144602 75 14.0 6.98 5 709.4 2.850891 7.149109 76 14.1 7-5 io 702.2 2.846461 7-^53539 .78 14.2 7.12 15 695.1 2.842047 7- I 57953 .80 14.4 7.19 20 688.2 2-837715 7:162285 .82 14-5 7.27 25 681.3 2-833338 7.166662 .84 14.7 7-34 3 674.7 2.82QI I t 7.170889 .85 14-8 7.41 35 668.1 2.82 4 84I 7- I 79*59 .87 15.0 7.48 40 661.7 2.820661 7- I 79339 .89 15-1 7-56 j 45 655- '4 2.816506 7.183494 .91 15-3 7-63 5 649-3 2.8l2 44 5 7- I 8 7555 93 i5-4 7.70 55 643.2 2.808346 7-181654 95 iS-5 7-77 1 9 o 637-3 2.804344 7.195656 .96 5-7 7-85 5 631.4 2.800305 7.199695 .98 15.8 7-92 10 625.7 2.796366 7.203634 .00 16.0 7-99 J 5 620. i 2.792462 7-2U7538 .02 16.1 8.06 RADII AND THEIR LOGARITHMS. Degree of Curve. Radius. Logarithm of Radius. Arithmetical Comple- ment. Middle Ordinate, Chord loo Feet. Deflec- tion Dis- tance. Tangen- tial 'Dis- tance. ?j 2 ' 614.6 2.788593 7.211407 2.04 16.3 8.14 25 609.1 2.784689 7- 2T 53 11 2.06 16.4 8.21 30 603.8 2.780893 7.219107 2.07 16.6 8.28 35 598.6 2.777137 7.222863 2.09 16.7 8-35 40 593-4 2.773348 7.226652 2. II 16.8 8.43 45 588.4 2-769673 7.230327 2.13 17.0 8.50 5 583-4 2.765966 7-234I34 2.1 5 17.1 8-57 55 578-5 2.762303 7-237697 2.l6 17-3 8.64 1O o 573-7 2.758685 7- 2 4 I 3i5 2.l8 17.4 8.72 10 564-3 2.751510 7.248490 2.22 17.7 8.86 20 555-2 2-744449 7-25555I 2.26 18.0 9.00 30 54 6 -4 2-7375" 7.262489 2.29 18.3 9.15 4 537-9 2.730702 7.269298 2-33 18.6 9-30 5 529-7 2.724030 7.275970 2.36 18.9 9-44 11 521-7 2.717421 7.282579 2.40 19.2 9-58 IO 5 x 3-9 2.710879 7.289121 2.44 19-5 9 -73 20 506.4 2.704494 7-295506 2-47 19.7 9.87 3 499.1 2.698188 7.301812 2-5 1 20. o IO.O 40 492.0 2.691965 7-308035 2-55 20.3 10.2 5 485.1 2.685831 7.314169 2.58 20.6 10-3 12 o 478-3 2.679700 7.320300 2.62 20.9 10.4 IO 471.8 2-673758 7.326242 2.66 21 .2 10.6 20 465-5 2.667920 7.332080 2.69 21-5 10.7 3 459-3 2 . 662006 7-3379 4 2-73 21.8 10.9 40 453-3 2.656386 7-3436I4 2.77 22.1 II. 5 447-4 2.650696 7-349304 2.80 22.4 II .2 13 o 441.7 2.645127 7-354873 2.84 22.6 "3 10 436.1 2.639586 7.360414 2.88 22-9 20 43 -7 2-634175 7.365825 2.91 23.2 ii. 6 30 425-4 2.628797 7.371203 2-95 23-5 11.7 40 420.2 2.623456 7.376544 2.98 23-8 11.9 50 415.2 2.618257 7-38I743 3-02 24.1 12. 14 o 410.3 2.6I3I02 7.386898 3.06 24-4 12.2 IO 45-5 2.607991 7.392009 3.09 24.7 I2. 3 20 400.8 2.602928 7-397072 3-13 25.0 12-5 3 396.2 2.597914 7 . 402086 3- T 7 25.2 12.6 40 39 r -7 2092954 7.407046 3-20 25-5 12.8 50 387-3 2.588047 7-4"953 3-24 25-8 12.9 15 o 383-1 2-5 8 33 12 7.416688 3-28 26.1 13.0 10 378-9 2-578525 7-42I475 3-3 1 26. 4 I 3 .2 20 374-8 2.573800 7.426200 3-35 26.7 13-3 30 370.8 2.569140 7.430860 3-39 27.0 13-5 40 366.9 2.564548 7-435452 3-42 27-3 I 3 .6 50 363-0 2.559907 7.440093 3-46 27-5 13-8 16 o 359-3 2 -555457 7-444543 3-50 2 7 .8 13-9 IO 355-6 2.550962 7.449038 3-53 28.1 I4.I 20 352-o 2-546543 7-453457 3-57 28. 4 14.2 30 348.4 2 . 542078 7.457922 3.61 28.7 14-3 40 345- sJTJ 2.537819 7.462181 3-64 29.0 M-5 50 34'- 6 2-5335'8 7.466482 3-68 29-3 14.6 | 17 o 338.3 2.529302 7.470698 3-72 29.6 14-8 10 i 335-0 1 2.525045 7-474955 3.75 ,. 14.9 RADII ANT) 7777-77? LOGARITHMS. 159 ! Degree of 1 ; Curve. Radius. Logarithm of Radius. Arithmetical Comple- ment. Middle Ordinate, Chord loo Feet. Deflec- tion Dis- tance. Tangen- tial Dis- tance. o / 17 20 33'.8 .520876 7.479124 3-79 30.1 J5- 1 3 328.7 .516800 7.483200 3.82 3-4 15.2 40 325-6 .512684 7.487316 3-86 3-7 J 5-4 5 322.6 .508664 J'WSS^ '3-9 31.0 J5-5 18 o 319.6 .504607 7-495393 3-93 3 T -3 15-6 10 316.7 .500648 7-499352 3-97 31-6 15-8 20 3*3-9 .496791 7.503209 4.01 3i-9 15-9 30 311.1 .492900 7.507100 4.04 32.1 16.1 40 308.3 .488974 7.511026 4-08 3 2 -4 16.2 50 305-6 485153 7-5I4847 4.12 32-7 16.4 19 o 302.9 .481299 7.518701 4- I 5 33- 16.5 10 300.3 477555 7.522445 4.19 33-3 16.6 20 297.8 473925 7.526075 4- 2 3 33-6 16.8 3 295.2 .470116 7.529884* 4.26 33-9 16.9 40 292.8 .466571 7-5334 2 9 4-30 34- 2 17.1 50 290.3 .462847 7-537I53 4-34 34-4 17.2 3O o 287.9 2.459242 7-540758 4-37 34-7 17.4 TABLE XI. SQUARES, CUBES, ETC., OF NUMBERS FROM 1 TO 1042. TABLE SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMBERS No. Squares. Cubes. Sqimre Roots. Cube Roots. Reciprocals. 1 1 1 1-0000000 1-0000000 100000000 2 4 8 1-4142136 1-2599-210 500000000 3 9 27 1-7320508 1-4422496 333333333 4 10 64 2-0000000 1-5874011 250000000 5 25 125 2-2300f>80 1-7099759 200000000 6 36 216 2-44948.-I7 1 '8 171206 1666061567 7 49 343 2-0457513 1-9129312 142857143 8 64 512- 2-8284271 2-0000000 125000000 B 81 729 3-0000000 2-0800837 111111111 10 100 1000 3-1622777 2-1544347 100000000 II Kl 1331 3-3100248 2-2239801 090909091 It 144 1728 3-4041010 2-2894286 083333333 13 169 2197 3-005.").-) 13 2-3513347 070923077 14 190 2744 3-7410574 2-4101422 071428571 15 225 3375 3-8729833 2-4002121 006006607 10 256 4090 4-0000000 2-5198421 002500000 17 289 4913 4-1231056 2-5712816 0588-235-29 18 3-24 5832 4"24-2(>407 2-62J7414 05555555(5 19 3til 6859 4-3588989 2-0084016 052031579 20 400 8000 4-4721300 2-7144177 050000000 2! 441 9261 4-5825757 2-7589243 047(519048 22 484 10048 4-0904158 2-8020393 045454545 23 529 12167 4-7958315 2-8438070 043478261 24 570 13824 4-8989795 2-8844991 041000007 25 625 15625 5-0000000 2-9240177 040000000 20 670 17576 5-0990195 2-9024900 038401538 27 729 19683 5-190152-1 3-0000000 037037037 28 784 21952 5-2915020 3-0305889 035714286 29 841 24389 5-3851648 3-07-23108 034482759 30 900 27000 5-4772250 3-107-23-25 033333333 31 901 29791 5-5077G44 3-1413800 032-258005 32 1024 32768 5-6568542 3-1748021 031250000 33 1089 35937 5-744.10-20 3-2075343 030303030 34 1150 39304 5-8309519 3-2390118 029411705 35 1225 42875 5-9100798 3-2710003 028571429 3G 1296 46656 6-0000000 3-3019-272 027777778 37 1309 50053 0-08-270-25 3-33-2-2-218 0270270-27 38 1444 54872 6-1044140 3-3019754 026315789 39 1521 59319 6-2449980 3-3912114 025041026 40 1000 64000 6-3245553 3-4199519 025000000 41 1081 68921 6-4031-242 3-4482172 024390244 42 1704 74088 6-480741)7 3-4700-260 0238095-24 43 1849 7W507 6'5.")74385 3-5033981 023255814 44 1930 85184 6-0332490 3-5303483 02-2727273 45 2025 91125 6-7082039 3-550893:1 0222-2-2-2-22 46 2116 97336 6-7823300 3-5830479 0-21739130 47 2209 103823 6-8550540 3-0088201 021270000 48 2304 110592 6-9282032 3-0342411 0-20833333 49 2401 117049 7-0001)1100 3-0593057 020408103 50 2500 125000 7-0710678 3-0840314 020000000 162 SQUARES, CUBES, ETC.. OF NUMBERS. 163 Squares, Square Rn Reciprocals. 2001 132051 7-1414284 3-7084298 019607843 ' 2:04 140008 7-211 1026 3-7325111 019230709 8806 148877 7-2801099 3-7562858 018867925 2ffi 157404 7-3484092 3-7797631 018518519 3025 100375 7-4101985 3-80-29525 OI818181H 3130 175010 74833148 3-8258624 017857143 3249 185193 7-5498344 3-8485011 01754:5860 3364 195112 7-0157731 3-H708766 017241379 3481 2(15379 7-08 11 457 3-8D29905 016949153 3680 210000 7-7459667 3-9148670 01(>06f;667 3721 2-20981 7-8102497 3-93J4972 016393443 3844 238328 7-8740079 3-9578915 01612903^ 3909 250047 7-9372539 3-9790571 015873016 4096 202144 8-0000000 4-0000000 156251 MM) 4395 274025 8-0622577 4-0207256 01 538401 5 4356 28749G 8-1240384 4-0412401 015151515 4489 3007(3 8-1853528 4-0615480 011925373 4624 314432 8-2402113 4-0816551 014705832 47G1 328509 8-3066239 4-1015661 014492754 4900 343000 8-3060003 4-1212853 0142K>714 5041 357911 8-4201498 4-1408178 014084517 5184 373248 8-4852814 4-1001076 013888889 533) 389017 8-5440037 4-1793390 013698630 5476 405224 86023253 . 4-1983304 013513514 50-25 421875 8-6002540 4-2171033 013333333 5776 43897G 8-7177979 4-2358236 013157895 stay 450533 8-7749044 4-25432 10 012987013 6084 474552 8-8317009 4-2720586 012820513 6241 493039 8-8881944 4-2908404 0121)58228 6400 512000 89442719 4-3088695 012500000 6561 531441 9-0000000 4-3207487 012.-M5679 6724 551308 9-0553851 43444815 012195122 6889 571787 9-1104336 4-3(320707 012048193 7056 592704 9-1651514 43795191 011904762 7-225 614125 9-21! (544 5 4-3908296 011764706 7396 63CM50 9-2736185 4-4140049 011627907 7569 658503 9-3273791 4-4310476 011494253 7,44 081472 9-3808315 4-44791302 011303036 7921 704909 9-4339811 4-4647451 011235955 8100 729000 94868330 44814047 Oil 111 111 8281 753571 9-5393920 4-4979114 01 09890 II 84G4 778(588 95916030 45143574 OI08695G5 swy 804357 9-6436508 4-5306549 010752088 8836 830584 9-6953597 4-546H359 010038298 9025 857375 9-7407943 4-502!H)26 010520316 92 1G 88473G 9-7979590 4-5788570 010416607 9409 912073 9-8488578 45947009 010309273 9604 941192 9-8994949 4-6104303 010204082 9801 970299 9-9498744 4-0260050 010101010 10000 1000000 100000000 4-6415888 01001)00110 JU201 1030301 100498756 4-6570095 00990091)0 10404 1061208 100995049 4-0723287 00!)8()3:22 10G09 1092727 10-1488916 4-0875482 009708738 1081G 1124804 10-1980390 4-7020094 oojGir>:j85 11025 1157625 10-2469508 47176940 009523810 11236 119101G 10-2956301 4-7326235 009433902 11449 1225043 HKW40804 4-7474594 009345794 11(U)4 1259712 10-39-23048 4-7622032 00;)25!)259 11881 1295029 10-4403065 4-77ti8502 009174312 12100 1331000 10-4880885 4-7914199 009090909 12321 13G7031 10-5356538 4-8058995 009009009 12544 1404928 10-5830052 4-8202845 00-S928571 104 SQUARES, CUBES, ETC., OF NUMBERS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 113 12769 1442897 106301458 4-834.5881 00884! (558 114 12U.96 1481544 10K770783 4-8488076 QBH77IU9P 115 13225 1520875 10-7238053 4-86-2944 : 2 OOH6950.V2 JIG 134.56 1560896 107703296 4-87699!):) 008020690 117 13689 1601613 108166538 4-89097:*-2 008547009 118 13<)-24 1643032 10-8627^05 9048681 OOS474576 119 14161 1685159 109087121 9186847 008403361 ]20 14100 1728000 10-9544512 -.(M-24-24-2 008333333 121 14641 1771561 11 -0000000 9460874 008-264463 122 14834 1815848 11-0453610 9596757 008196721 123 15129 1860867 1] 0905365 9731898 008 130081 J24 15376 1906624 11 1355287 9866310 0080(545 16 125 15625 1953125 11-1803399 5-0000000 00*000000 126 15876 2000376 1I-2-2497-2-2 5-013297!) 007936508 127 16129 2048383 11-2694277 5-0265-257 007874016 128 16384 2097152 11-3137085 5-0396842 00781-2500 129 16641 2146689 11-3578167 50527743 007751938 130 16900 2197001) 11-4017543 5-0657970 007692308 131 17161 2248091 11-4455231 5-0787531 007633588 132 17424 2299968 11-489I-2.-3 5-0916434 007575758 133 17689 2352637 11-53-25626 5-1044687 007518797 134 17956 2406104 11-5758369 5 117-2-209 007462687 135 18225 2460375 11-6189500 5-1299-278 007407407 136 18496 2515456 H-6619038 5-14-2563-2 0073.V2941 137 18769 2571353 11-7040999 5-1551367 007299-270 138 19044 5628072 ]]-747:401 5-1676493 (KI7-240K77 139 19321 2685619 11-7898261 5-1801015 007194-245 140 19600 2744000 118321596 5- J 924941 00714-2857 141 19881 2803221 11-8743421 5-2048279 00709-2199 142 20164 2863288 11-9163753 5-2171034 00704-2-254 143 20449 2924207 11 -9582607 5-2293215 006993007 144 20736 2985984 ] 2-0000000 5-24148-28 006944444 145 21025 3048625 1-2-0415946 5-2535879 00089655-2 140 21316 3112136 1-2 0830460 5-2656374 006849315 147 21609 3176523 12-1243557 5-2776321 006802721 148 2191)4 3241792 12 1655251 5-2895725 006756757 149 22201 ' 3307949 122065556 53014592 006711409 150 22500 3375000 12-2474487 5-3132928 00666! 5fi(!7 151 22801 3442951 12-2882057 5-3250740 0006'2-25I7 152 23104 351 1808 123288280 5-3368033 006578947 153 23409 3581577 ] 2-36931 69 5-3484812 000535948 154 23716 36522(54 12-4096736 5-3601084 00649351 H> 155 24025 3723875 12-449H996 5-3716854 006451613 156 24336 3796416 12-4899960 5-3832126 006410-250 157 24649 3869893 l-2-5-29'.)64l 5-3946907 0003094-27 158 24964 3944312 125698051 5-4061-20-2 00(5329114 159 25281 4019679 12-6095202 5-4175015 00(5-289308 160 25600 4096000 1-26491106 54288352 1)06-250000 161 25921 4173281 1-2-6^85775 5-4401218 006-211180 102 26244 4251528 12-72792-21 5-451 36 18 00617-2840 163 26569 4330747 12-7671453 54625556 006134969 164 26896 4410944 12-8062485 5-4737037 006097561 165 27225 4492125 12-8452326 5-4848066 006060606 166 27556 4574296 ] 2-8840987 5-4958647 00(5024096 167 27889 4657463 12-9228480 5-5068784 OO.YJS8024 168 28224 4741632 129614814 5-5178484 00595-2381 169 28561 4826809 13-0000000 5-5287748 005917160 170 28900 4913000 13-0:{84048 5-5396583 005882:553 171 29241 5000211 130766968 5-55049IH 005817(153 172 29.584 5088448 131148770 5-561-2978 005813953 173 299'29 5177717 13-15-2<)404 5-5720546 0057H0347 174 30276 5268024 13-19(19060 5-582770-2 005747126 SQUARES, CUBES, ETC., OF NUMBERS. 105 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 175 30025 5359375 13-2287560 55934447 005714286 170 30976 5451770 13-2664992 5-6040787 005081818 177 31329 5545233 13-3041347 5-6146724 0050497 !R 178 31084 5639752 133416641 56252203 0056I797R 179 32041 5735339 13-3790882 5-6357408 00558055)2 180 32400 5832000 13-4164079 50402102 005555556 181 32761 5929741 13-4536240 5-0500528 005524862 182 33124 0028568 13-4907376 5-0670511 005494505 183 334 H9 6128487 135277493 5-07741 14 005464481 184 33H50 0229504 13-5646(500 5-6877340 005434783 185 34225 0331025 13-0014705 5-6980192 005405405 186 34590 0434856 13-038 18] 7 5-7082675 005376344 187 34969 0539203 13-0747943 5-7184791 005347594 18S 35344 6644072 13-7113092 5-7280543 005319149 189 35721 6751209 13-7477271 5-7387936 005291005 ]!K) 36100 ' 6859000 13-7840488 5-7488971 005263 1 58 191 36481 6967871 13-8202750 5-7589052 005235602 192 36864 7077888 13-8504065 5-7 >S9982 005208333 193 37249 7189017 13-8924440 5-7789906 005181347 194 37036 7301384 13-9283883 5-7889004 005154039 195 38025 7414875 13-9642400 5-7988900 0051 '28205 190 38410 7529536 14 -0000000 5-8087857 005102041 197 38809 7645373 14-0350088 5-8180479 005070142 198 39204 77(52392 14-0712473 5-8284767 0050.50505 199 39001 7880599 H'1067360 5-8382725 005025126 200 40000 8000000 H'1421356 58480355 005000000 20 J 40401 8120601 14'17744G9 5-8577600 004975124 202 40804 8242408 14-2120704 5-8074043 004950495 2o:i 41209 8365427 14-2478008 5-8771307 004926108 201 41010 8489004 14-2828509 58807653 004901961 20.") 42025 8GJ5125 H'317821 1 5-8903085 004878049 306 42430 8741810 14-3527001 5-9059406 004854369 207 42849 8809743 143874946 5-9154817 004830918 208 43264 8998912 14-4222051 5-9249921 0048070!)2 209 43081 9129329 14-4508323 5-9344721 00478408!) 210 44100 9201000 14-4913707 5-9439220 004761905 2J1 44521 9393931 14-5258390 5-9533418 004739336 212 44944 9528128 14-5002198 5-9627320 004716981 21.3 45369 .9663597 14-5945195 5-9720926 . -004 094836 214 45790 9800344 14-0287388 5-9814240 004672897 215 40225 9938375 14-6028783 5-9907264 004651163 210 46650 10077696 14-6969385 6-0000000 004029630 217 47089 10218313 147309199 6-0092450 004608295 218 47524 10360232 14-7648231 6-0184617 004587156 219 47961 10503459 14-7986486 6-0276502 004566210 220 48400 10648000 148323970 6-0368107 004545455 221 48841 10793861 14-8660687 60459435 004524887 2-22 49284 10941048 14-8996644 6-0550489 004504505 2-23 49729 11089567 14-9331845 6-0641270 004484305 224 50176 1J 239424 14-9666295 6-0731779 004404286 225 50025 1 1390625 15-OuOOOOO 6-0822020 004444444 226 51070 11543176 15-0332904 6-0911994 004424779 227 51529 ]J 097083 15-0065192 6-1001702 004405286 228 51984 1 1852352 150996689 6-1091147 004385905 22') 52441 12008989 15-1327460 6-1180332 00430(5812 230 52900 12167000 15- 1057509 6-1269257 004347826 231 53361 12320391 15- J 986842 6-1357924 0043-29004 23-2 53824 12487168 152315402 6-1446337 004310345 233 54289 121549337 15-2643375 6-1534495 004291845 234 54756 12812904 15-2970585 6-1622401 004273504 335 55225 12977875 15-3297097 6-1710058 004255319 230 55090 13144256 15-3622915 6-1797466 004237288 166 SQUAKES, CUBES, ETC., OF NUMBERS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocal*. 237 56169 13312053 15-3948043 6-1884628 004219409 238 50(544 13481272 15-4272486 6-1971544 004201681 2:39 57121 13651919 15-4596248 6-2058218 004184100 240 57600 13824000 15-4919334 6-2144(55;) 004 10(5607 241 58081 13997521 1 5- 524 J 747 6-2230843 004141(378 212 585(54 141724^ 15-5563492 6-2316797 0041322:!! 213 59049 14348907 1 5-58*1573 6-2402515 004115226 244 59536 1452(5784 15- (5204994 6-24879D3 00409^3(51 245 60025 14706125 15-6524758 6-2573248 004081633 246 60516 1488693(5 15-6843871 6-2(i582(56 0040(55041 247 61009 15069223 157102336 6-2743054 004048583 248 61504 15252992 15-7480157 6-2827613 004032258 249 62001 15438249 15-7797338 6-2911946 00401(50(54 250 62500 15625000 15-8113883 6-299(5053 IKMOOOOOO 251 63001 15813251 15-8429795 6-307HK35 0039840(54 25-2 63504 16003008 15-874.5079 6-31(535915 00390H254 253 64009 .16 194277 15-9059737 6-3247035 0039525(59 254 64516 163870(54 15-9373775 6-333025(5 003937008 255 65025 16581375 15-9687194 6-3413257 0039215(59 256 65536 16777216 16-0000000 6-3496042 00390(5250 257 66049 1697451)3 16-0312195 6-3578611 003891051 258 66564 17173512 JO-0623784 6-3660968 0038759(59 259 67081 17373979 16-0934709 6-3743111 0038(51004 260 67(500 17576000 16-1245155 6-3825043 00384(5154 261 68121 17779581 16-15.54944 6-39067(55 003831418 262 68644 17984728 16-18(54141 6-3988279 00381(5794 203 69169 18191447 162172747 6-40(59585 003H02281 264 69696 18399744 16-24807(58 6-4150(587 003787879 265 70225 18609625 16-2788206 6-4231583 003773585 266 70756 18821096 16-3095064 6-431227(5 003759398 267 71289 19034163 1(5-3401346 6-4392767 003745318 268 71824 19248832 16-3707055 6-4473057 003731343 2(59 72361 19465109 16-4012195 6-4553148 003717472 270 721)00 19(583000 16-4316767 6-4(533041 003703704 271 73441 19902511 16-4620776 6-4712736 003(590037 272 73!)84 20123(548 16-4924225 6-4792236 003676471 273 74529 2034(5417 16-5227116 6-4871.541 003663004 274 75076 20570824 16-5529454 6-4950653 003649(535 275 75625 20796875 16-5831240 6-5029572 003(531)3(54 276 76176 2102457(5 16-6132477 6-5108300 003(523188 277 76729 21253933 16-6433170 6-5186839 003610108 278 77284 21484952 16-673332) 6-5265189 00359712-2 279 77841 21717639 16-7032931 8" 53 ^ 1351 003584229 280 78400 2195-2000 16-7332005 6-5421326 003571429 281 78961 22188041 16-7630546 6-5499116 003558719 282 79524 22425768 16-7928556 6-5576722 003541)099 283 80089 22(565187 16-822(5038 6-5654144 0035335(59 284 80656 22906304 16-8522995 6-5731385 003522127 285 81225 23149125 168819430 6-5808443 003508772 286 81796 23393656 1(5-91 15345 6-5885323 003496503 287 82369 23(539903 16-9410743 6-59(52023 003484321 288 82944 23887872 16-9705627 6-6038545 003472222 289 83521 241375(59 17-0000000 6-6114890 0034(50208 290 84100 24389000 7-0293804 6-6191060 003448276 291 84(581 24(542171 7-0587221 6'62(i7054 00343(542(5 292 85264 24897088 7-0880075 6-6342874 003424058 293 85849 25153757 7-1172428 6-6418522 0034129(59 294 86436 25412184 7-1464282 6- (5493998 003401361 295 87025 7-1755(540 6-6509302 003389831 296 87616 35934339 17-204(1505 6-6644437 00337H378 297 88209 2(51981(73 17-2'n(>879 66719403 0033C7003 298 88804 26463592 17-2626765 6-6794200 0033a5705 SQUARES, CUBES, ETC., OF NUMBERS. 167 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 2.13 89101 26730899 17-2916165 6-6868831 003344482 300 90000 27000000 17-3205081 6-6943295 003333333 331 90001 27270901 17-3493516 6-7017593 003322259 302 91234 27.343608 17-3781472 6-7091729 00331 1258 3!>3 91809 27318127 17-4368952 6-7 1(55700 003301330 304 92416 2S094464 17-4355958 6-72.59508 003289474 305 93025 28372625 17-4642492 6-7313155 003278689 3015 93G36 28G52G16 17-4928557 6-7386641 0032(57974 307 94249 28934443 17-5214155 6-7459967 003257329 338 94804 2D218112 17-5499288 6-7533134 003246753 309 95481 29503G29 17-5783958 6-7606143 0032,36246 310 961 00 29791000 17-6068169 6-7678995 003225806 311 9G721 30080231 17-6351921 6-7751690 003215434 312 97344 30371328 17-6635217 6-7824229 003205128 311 97969 30664297 17-6918060 6-7896613 003194888 314 98596 30959144 17-7200451 6 7968844 003184713 3J5 99225 31255875- 17-7482393 6-8040921 003174603 316 99856 31554496 17-7763883 6-8112847 003164557 31? 100489 31855013 17-8044938 6-8184620 0031.54574 318 101124 32157432 7-8325545 G-8250242 003144654 319 101761 32461759 7-8605711 6-8327714 003134796 320 102400 327G8000 7-8885438 6-8399037 003125000 321 103041 33076161 7-9164729 6-8470213 003115265 3-2-2 103684 33386248 7-9443584 6-8541240 003105590 3-23 104329 33698267 7-9722008 6-8612120 003095975 3-24 104976 34012224 18-0000000 6-8682855 00308G420 325 105625 34328125 18-0277564 6-8753413 003076923 323 154449 60698457 19 8242276 7-3348295 0025445-29 394 155236 61162984 19 84!)4:532 7-3310369 002538071 395 156025 61629875 19 8746009 7-3372339 002531(546 396 156816 62099136 19-8997487 7-3434205 002525253 397 157609 62570773 19-9248588 7-3495966 002518892 398 158404 63044792 199499373 7-3557624 002512563 L99 159201 63521199 19 9749844 7-3619178 002506266 400 IWOOO 64000(100 200000000 7-3680(530 002500000 401 160801 64481201 20 0249844 7-3741979 002493706 402 161604 04964808 200499377 7-3803-227 00-24875(52 403 162409 65450827 20 0748599 7-3864:173 00-2481390 404 163216 65939264 20 09975 J 2 7-3925418 002475248 405 164025 66430125 201246118 7-3986363 0024(59136 406 164836 66923416 201494417 7-4047206 002403054 407 165649 67419143 20 1742410 7-4107950 002457002 408 166464 67917312 201990099 7-4168595 002450980 409 167281 68417929 20 2-2:57484 7-4229142 00-2444988 410 168100 68921000 20 2484567 7-4289589 00-2439024 411 168921 69426531 202731349 7-4349938 002433090 412 - 169744 6-28 202977831 74410189 00-24-27184 413 170569 70444997 203224014 74470342 002121308 4J4 171396 70957944 203469899 74530399 002415459 415 172225 ! 71473375 20-3715488 7 4590:i;>9 002409639 416 173056 71991296 20-3960781 7-4650223 002403846 417 173889 72511713 20-4205779 7-4709991 00-23 ( .8082 418 174724 73034632 20-4451 >4H3 7-4709004 002392344 419 175561 73560059 204694895 7-4829242 002:186035 420 176400 74088000 20-4939015 7-4888724 602380952 421 177241 74618461 20-5182845 7-4948113 002375297 422 178084 75151448 20V-426386 7-5007406 002369668 SQUARES, CUBES, ETC., OF NUMBERS. 1C9 Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 178929 75686967 20 5669638 7-5066607 002364066 179776 76225024 20-5912603 7-5125715 002358491 180625 76765625 20-6155281 7-5184730 002352941 181476 77308776 20-6397674 7-5243652 002347410 182329 77854483 20-6639783 7-5302482 002341920 18: J 184 78402752 20-6881609 7-5361221 082336449 184041 78953589 20-7123152 7-5419867 002331002 184900 79507000 211-7364414 7-5478423 002325581 185761 80062991 207605395 7-5536888 00232018(5 186624 80621568 207846097 7-5595263 002314815 187489 81182737 20-8086520 7-5653548 0023094(59 188356 81746504 20-8326667 75711743 002304147 189225 82312875 208566536 7-5769849 002298s.)! 190096 82881856 208806130 7-5827865 002293578 190969 83453453 209045450 7-5885793 002288330 191844 84027672 20-9284495 7-5943(533 002283105 192721 84604519 209523268 7-6001385 002277904 193600 85184000 20-9761770 7-6059049 002272727 194481 85766121 21-0000000 7-6116626 002267574 1 95364 86350888 21-0237960 7-6174116 002262443 196249 86938307 210475652 7-6231519 0022.17336 197136 87528384 210713075 7-628P837 0022.rJ-.232 198025 88121125 210950231 7-6346067 0(12247191 198916 88716536 21-1187121 7-6403213 002242152 199809 89314623 21-1423745 7-6460272 002237136 200704 89915392 21-1660105 7-6517247 002232143 201601 90518849 21-1896201 7-6574138 002227171 202500 91125000 21 2132034 7-6630943 002222222 203401 91733851 212367606 7-6687665 002217295 204304 92345408 21-2602916 7-6744303 002212389 205209 92959677 21-2837967 7-6800857 002207506 206116 93576664 21-3072758 7-6857328 002202643 207025 94196375 21-3307290 7-6913717 002197802 207936 94818816 21-3541565. 7-6970023 002192982 208849 95443993 21-3775583 77026246 002188184 209764 96071912 21-4009346 7-7082388 0021834015 210681 96702579 214242853 7-7188448 002178649 211600 97336000 214476106 7-7194426 002173913 212521 97972181 21-4709106 7 7250325 002169197 213444 9P61112S 21-4941853 77306141 0021 64502 214369 99252847 21-5174348 7-7361877 002159827 215296 99897344 21-540C592 7-7417532 002155172 216225 100544625 21-5638587 7-7473109 002150538 217156 101J94696 21-5870331 7-7528606 0021451)23 218089 101847563 21-6101828 7-7584023 002141328 219024 102503232 21-6333077 7-7639361 002136752 219961 103161709 21-6564078 7-7694620 002132196 220900 103823000 21-6794834 7-7749801 -002127660 221841 104487111 21-7025344 7-7804904 00212:5142 222784 223729 105154048 105823817 21-7485632 7-7859928 7 7914875 002118644 '002114165 224676 10C.496424 217715411 7-7969745 002109705 225625 107171875 21-7944947 7-8024538 002105263 226576 107850176 218174242 7-8079254 002100840 227529 108531333 218403297 78133892 0020964H6 228484 109215352 21 8632111 7-8188456 002092050 229441 109902239 21-8860686 7-8242942 002087(583 230400 110592000 21 91 189023 78297353 002083333 2313151 1112841)41 219317122 7-8351688 002079002 232324 11I!)*01()8 21-9544984 78405949 002074689 2332^9 11 2678587 21-9772610 7 8460134 002070393 234256 113379904 22 0000000 78514244 002066116 170 SQUARES, CUBES, ETC., OF XUMBERS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 485 233225 1 1 4084 1 25 ' 2202271 55 7-8568-281 002061856 480 23G196 114791256 28-0454077 7-862-2-242 002057613 487 2371153 115501303 2-2 OIW0765 78676130' 002053388 488 238144 116214272 22-0907220 7-8729944 002049180 483 23J121 110930169 22 U 33444 7-8783684 00-2044990 490 24J100 117649000 22 1359436 7 8837352 002040816 491 241081 118370771 22-1585198 7-8890946 002030660 49-2 2420G4 119095488 22-1810730 7-8944468 002(1325-20 4U3 243049 119823157 22 2036033 7-8997917 0020-28398 494 24403; > 120553784 222261108 7-905 12H4 002024-291 495 245025 121287375 222485953 79104599 00-20-20202 496 246016 122023936 222710575 79157832 002016129 497 247009 122763473 22-2934968 79210994 002012072 498 248004 123505992 223159136 7-9264085 00-200803-2 499 249001 124251499 223383079 7-9317104 002004008 500 25iK)00 125000000 223606798 79370053 002000000 501 251001 125751501 223830293 79422931 00199(5008 502 252004 126506008 224053565 7-9475739 Oi)19il-2032 503 253009 127263527 22-4276615 79528477 001988072 504 254016 128024064 224499443 7-9581144 001984127 503 255025 128787625 224722051 7-9633743 001980198 506 256036 123554216 22 4J44438 7-9686271 0019702^5 507 257049 130323843 225166605 7-9738731 001972387 508 258064 131096512 225388553 7-9791122 001908504 503 259081 131872229 225610283 7-9843444 001964637 510 260100 132651000 225831796 7-9895637 001960785 511 261121 133432831 22-6053091 7-9947883 001956947 512 262144 134217728 22-6274170 8-0000000 001953125 513 263169 135005697 22-6495033 8-0052049 001949318 514 264196 135796744 22-6715631 80104032 001945525 513 265225 136590875 226936114 80155946 001941748 516 266256 137383096 227153334 80207794 001937984 517 267289 138188413 227376341 8 0259574 001934236 518 208324 133991832 22-7596134 80311287 001930502 519 269361 139798359 22-7815715 80362935 001926782 520 270400 1401308000 228035085 80414515 001923077 521 271411 141420761 228-254244 80466030 001919386 522 272484 142236648 228473193 80517479 001915709 523 273529 143055667 228691933 8-0568862 001912046 524 274576 143877824 228910463 80620180 001903397 525 275625 144703125 22 9128785 80671432 001904762 526 276676 145531576 22 9346899 80722620 001901141 527 277729 146363183 22-9564806 80773743 001897533 523 278784 147197952 22-9782506 80824800 001893939 529 270341 148035389 230000000 80875794 001890359 530 233900 148877001 230217289 8-0926723 001H80792 531 281331 149721291 230434372 8-09775P9 001883239 532 283024 150568763 230651252 8- 1023390 001879699 533 284089 151419437 23 0867928 8-1079128 001876173 531 285156 152273304 23-1084400 8-1129803 001872059 535 283225 153130375 23 1300670 8-1 1804 H 001869150 53o 287238 153393656 23 151G738 8-1230902 001865672 537 2383G9 154354153 23-1732605 fi- 1281 447 001H62197 538 289444 155720872 231948270 8-1331870 001858736 539 290521 15G590819 23 2163735 8- 1382230 00 J 855288 54 3 291600 157464000 23 2379001 8 1432529 001851852 541 292681 1:8349421 21 2594007 8 14H27F5 001843429 542 293764 159220088 2'] 2-^8335 8-1532939 001845018 543 294849 169103007 2'5 3023694 8 1583051 00184 JH21 544 295936 160989184 2 < V323837f> 8 103310-2 001Kl 4J -j:!5 545 297025 ' 161878625 23 3-152 '51 8-K ; 83092 001H34Mia 541. 298113 1G277133G 23 3666429 81733020 001831503 SQUARES, CUBES, ETC., OF NUMBERS. 171 " Scuares. Cubes. Square Roots. Cube Roots, Reciprccals. 47 299209 1636(57323 23-3830311 8-1782888 001828154 48 300304 164566592 23-4093998 8-1832695 001824818 49 301401 165469149 23-4307490 8-1882441 * -001821494 so 302500 166375000 23-4520788 8-1932127 00181818-2 11 303601 167284151 23-4733892 8-1981753 001814H82 V2 304704 168196608 23-4946802 8-2031319 001811594 53 305809 169112377 23-5159520 8-2080825 001808318 54 306916 170031464 23-5372046 8-2130271 001805054 55 308025 170953875 23-5584380 8-2179657 001801802 56 309136 171879616 23-5796522 8-2228985 001798561 r>7 310249 172808693 23-6008474 8-2278254 001795332 58 311364 173741112 23-6220236 8-2327463 001792115 59 312481 174676879 23-6431808 8-2376614 00(788901) (50 313600 175616000 23-6643191 8-2425706 001785714 61 314721 176558481 23-6854386 8-2474740 001782.531 r,2 315844 177504328 23-7065392 8-2523715 00 77J359 53 316969 178453547 23-7276210 8-2572633 001776199 (54 318096 179406144 23-7486842 8-2621492 001773050 (15 319225 180362125 23-7697286 8-2670294 001769912 66 320356 181321496 23-7907545 8-2719039 001766784 87 321489 182284263 23-8117618 8-2767726 001763668 68 322624 183250432 23-8327506 8-2816355 0017(50563 69 323761 184220009 23-8537209 8-2864928 001757461) 70 324900 185 J 93000 23-8746728 8-2913444 001754386 71 326041 186169411 23-8956063 8-2961903 0-,) 175 13 13 7-2 327184 187149248 23-9165215 8-3010304 00174d253 715 328329 188132517 23-9374184 8-3058651 00 17452 Jl 74 329476 1891 19224 23-9582971 8-3106941 OOJ7421M) 7.1 330625 I9010'.)375 23-9791576 8-3155175 00 1 73 J 130 7ti 331776 1911021)76 24-0000000 8-3203353 001730111 77 332929 192100033 24-0208243 8-3251475 001733102 78 334084 193100552 24-0416306 8-3299542 001730104 79 33.1241 194104539 24-0624188 8-3347553 001727116 HO 336400 195112000 24-0831891 8-3395509 001724138 81 337561 196122941 24-1039416 8-3443410 001721170 83 338724 11)7137368 24-1246762 8-3491256 001718213 .H3 339889 198155287 24-1453929 8-3539047 001715266 84 341056 199176704 24-1660919 8-3580784 001712329 85 342225 200201625 24-1807732 8-3634466 00.703402 >8t; 343390 201230056 24-2074369 8-3682095 001706485 >87 3445151) 202262003 24-2280829 8-3729668 001703578 >88 345744 2032J7472 24-2487113 8-3777188 001700680 >8<) 346921 204336469 24-2693222 8-3824653 0016117793 )tl!) 348100 205379000 24-2899156 8-3872065 001694915 >!I2 350464 207474688 24-3310501 8-3966729 001681)189 393 351649 208527857 24-3515913 8-4013981 001686341 >9i 352836 209584584 24-3721152 8-4061180 OJKi835')2 ill.> 354025 210644875 24-3926218 8-4108326 00168,0672 V.M; 355216 211708736 24-4131112 8-4155419 001677652 3!)7 356409 212776173 24-4335834 8-4202460 001675042 >i)8 357604 213847192 24-4540385 8-4249448 001672241 39'J 358801 214921799 24-4744765 8-4296383 001669449 tilMJ 360000 216000000 24-4948974 "8-4343267 00166(5667 Jill 361201 217081801 24-5153013 8-4390098 0016G38J4 BU2 3(52404 218167208 24-5356883 8-4436877 001661130 >;)3 363609 211)256227 24-5560583 8-4483605 001058.T75 504 364816 220348864 24-5764115 8-4530281 001655621) G05 300025 221445125 24-5967478 8-4576906 001652893 500 367236 22-25451)16 24-6170673 8-4623479 001650165 507 368449 223648543 24-0373700 8-4670001 001647446 60d 369664 224755712 24-6576560 8-4716471 001644737 172 SQl'AKES, CUBES, ETC., OF NUMBERS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 609 370881 225806529 24-0779254 8-4702892 001042036 till) 372100 220981 (MM) 24-09X1781 8-4809201 001039344 Gil 373321 228099)31 24-7184142 8-4855579 001031)001 613 374544 229220-128 25-73803:58 8-4901848 00 10331)87 613 375765) 23104635)7 24-7588308 8-49-18065 0016:11321 C,I4 3705)96 231475544 24-775)0234 8-45)514233 001028004 615 378225 232008375 24-7991935 8-5040350 00 1 0201 U 6 (il(i 379456 233744890 24-815)3473 8-5080417 001023377 017 380089 234885113 24-8394847 8-5132435 001620740 Gis 381924 23!il)2lM)3i 24-8590058 8-5178403 001(518123 619 383101 237170059 24-8797106 8-5224331 001615509 620 384400 238328(100 24-85)97992 8-5270189 001612903 021 38nd41 239483061 24-9198716 5-53 10009 001010300 022 38U8H4 240041848 24-9399278 85301780 001007717 023 388129 24 J 804 307 24-951)9079 8-5407501 00100513(5 624 38i)37G 242970024 24-9791)920 &5453173 001602504 625 390625 244140025 25-0000000 8-5498797 001000000 020 391876 2453 1 4376 23-0 101)920 8-5544372 001597444 027 393129 240491883 25-035)9081 8-558985)9 001594896 628 394384 247073152 25-059.I282 8-5035377 001592357 629 3S5>41 248858189 25-0798724 8-5080807 0015811 182.', (iiill 390901) 250047000 25-0998008 8-5720189 001587302 631 398101 251239591 25-1197134 8-5771523 001584786 032 3911424 252435908 25-135)0102 8-5810809 001582278 633 400089 253030137 25-155)4913 8-5802047 001579779 G34 401956 254840104 25 179356G 8-5907238 001577287 635 403225 250047875 25 1992003 8-5952380 001574803 636 404490 257259456 25-215)0404 8-5997476 001572327 G37 405709 258474853 25-2388.->89 8-0042525 0015(59859 638 407044 2591)94072 252580019 8-6887528 001507398 639 408321 2605)17119 25-2784493 8-0132480 001504945 640 409000 202)44000 25-2982213 8-0177388 0015(52500 641 410881 203374721 25-3179778 8-0222248 001500062 642 4J2I04 2(54009288 25-3377189 8-02070(53 001557032 643 413449 205847707 2.V 3574447 8-0311830 001555210 644 414736 2670895)84 25-3771551 8-0356551 001552795 645 41GI25 268336125 25-3908502 8-G40122!) 001550388 64ii 417316 369583136 25-4105302 8-0445855 001547988 647 418009 270840023 25-4301947 8-6490437 001545595 648 419904 272097792 25-4558441 8-0534974 001543210 649 421201 273359449 25-4754784 8-0579405 001540832 650 422500 274025000 25-4950976 8-6(523911 ooi :>:!:< -102 651 423801 2758:14451 25-5147016 8-o:;o83io 001536098 652 425104 277167808 25-53429(fi 8-6712(505 (KH533742 653 420409 278445077 25-5538047 8-6736974 00153135)4 654 427716 279720204 25-5734237 8-0801237 0015291 152 655 429025 281011375 25-5929678 8-0845450 00152(5718 656 430336 282300416 25-0124909 8-6889030 00152435)0 657 431039 283593393 25-0320112 8- 65 133755) 001522070 658 432904 284890312 2.VG5I5107 8-( 5! 177843 001519757 659 434281 28fil91179 25-07095)53 8-7021882 00151/451 660 435000 287490000 250904052 8-7005877 001515152 6G1 4361)21 288804781 25-7099203 8-7109827 001512859 602 438244 290117528 25-725)3007 8-7153734 001510574 GG3 439509 25)1434247 25-7487804 8-7197590 00150825)6 604 440896 292754944 25-7081975 87241414 00150(5024 COS 442225 25)4079025 25-78755)35) 8-7285187 001503755) 606 44355G 25)5408290 25-8005)758 8-7328918 001501502 607 444889 25)0740903 25-82(53431 8-7372004 001499250 608 440224 29*077632 25-8450900 8-7410240 001497000 609 447501 299)18309 25-865* 134 3 8-7459840 .001494708 670 448900 3007G3000 25'8843582 8-7503401 001492537 SQUARES, CUBES, ETC., OF NUMBERS. No. ' Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 671 450241 302111711 25-9036677 8-7546913 001490313 672 451584 3034(54448 25- i (229628 8-7590383 001488095 673 452929 304821217 259422435 8-7633809 001485H84 674 454276 306182024 25-9615100 8-7677192 0014836HO 675 455625 307546875 25-9807621 8-7720532 00148U81 676 456976 308915776 26-0000000 8-7763830 OON7W90 677 458329 310288733 26-0192237 8-7807084 001477i()5 678 459684 311665752 26-0384331 8-7850296 001474926 679 401041 313046839 26-0576284 8-7893466 001472754 680 4G2400 314432000 26-0768096 8-7936593 001470.V8 681 463761 315821241 26-0959767 8-7979679 00 14(58429 682 465124 317214568 26-1151297 8-8022721 001466276 683 466489 318611987 26-1342687 8-8065722 OOH6412!) 684 467856 320013504 26-1533937 8-8108681 00 146 1! 188 685 469225 321419125 26-1725047 8-8151598 00145DH.14 686 470596 322828856 26-1916017 8-8194474 0014577-2(5 687 471969 324242703 26-2106848 8-8237307 001455604 688 473344 325660672 26-2237541 8-8280099 001453488 689 474721 327082769 26-2488095 8-8322850 001451379 690 476100 328509000 26-2678511 8-8365559 001449275 691 477481 329939371 26-2868789 8-8408227 001447178 692 478864 331373888 20-3058929 8-8450854 001445087 693 480249 332812557 26-3248932 8-8493440 001443001 694 481636 334255384 26-3438797 8-8535985 001440922 695 483025 335702375 26-3628527 8-8578489 001438849 696 484416 337153536 26-3818119 8-8620952 001436782 697 485809 338608873 26-4007576 8-8663375 001434720 698 487204 340068392 26-4196896 8-8705757 001432665 699 488601 341532099 2(5-4386081 8-8748099 001430615 700 490000 343000000 26-4575131 8-8790400 001428571 701 491401 344472101 26-4764046 8-8832661 001426534 702 492804 345948408 26-4952826 8-8874882 001424501 703 494209 347428927 26-5141472 8-8917063 001422475 704 495616 348913664 26-5329983 8-8959204 001420455 705 497025 350402625 26-5518361 8-9001304 001418440 706 498436 351895816 26-5706605 8-9043366 OOI4Ki4:U 707 499849 353393243 26-589471(5 8-9085387 001414427 708 501264 354894912 26-6082694 8-9127369 001412429 709 502681 356400829 26-6270539 8-9169311 001410437 710 504100 357911000 26-6458252 8-9211214 001408451 711 505521 359425431 2(5-6645833 89253078 001406470 712 506944 360944128 26-6833281 8-9294902 001404494 713 508369 362467097 26-7020598 8-9336687 001402525 714 509796 363994344 2(5-7207784 8-9378433 001400560 7J5 511225 365525875 26-7394839 8-9420140 001398601 716 512656 36706169(5 26-75817(13 8-94(51809 001396648 717 514089 368601813 26-7768557 8-9503438 001394700 718 515524 370146232 26-7955220 8-9545029 001392758 719 516961 371694959 26-8141754 8-9586581 001390821 720 518400 373248000 26-8328157 8-9628095 - -001388889 721 519841 374805361 26-8514432 8-9669570 001386963 722 521284 371)367048 26-8700577 8-9711007 001385042 723 522729 377933067 2(5-8886593 8-9752406 001383126 724 524176 379503424 2(5-9072481 8-9793766 001381215 725 525625 381078125 26-9258240 8-9835089 001379310 726 527076 382657176 26-9443872 8-987(5373 001377410 727 528529 384240583 26-9629375 8-9917620 001375516 728 529984 385828352 26-9814751 8-9958899 001373626 729 531441 387420489 27-0000000 9-0000000 001371742 730 532900 389017000 27-0185122 9-0041134 0013698(53 731 534361 390617891 27-0370117 9-0082229 001367989 732 5358:11 392223168 27-0554985 9-0123288 001366120 174 SQUARES, CUBES, ETC., OF NUMBERS. No. Squares. Cubes. Square Roo(s. Cube Roots. Reciprocals. 733 537289 393832837 270739727 90164309 ODI364256 734 538756 395446904 270924344 9 0205293 0013(52398 735 540225 3117065375 27-1108834 9-024(5239 0013(50544 730 541696 398(588256 27-1293199 90287149 .00135869R 737 543169 400315553 271477439 90328021 CJ1356852 738 544644 401947272 271661554 9-03(58857 001355014 739 546121 403583419 27-1845544 9 0409655 001353180 740 547600 405224000 272029410 90450419 001351351 741 549081 40(5869021 272213152 90491142 00 1341)528 742 550564 408518488 272396769 9-0531831 001347709 743 552049 410172407 27-2580263 9 0572482 001345895 744 553536 41 1830784 27-2763634 9 0013098 001344086 745 555025 413493625 27-2;)46S81 9 0653677 001342282 746 556516 415160936 273130006 9-0694220 001340483 747 558009 416832723 273313007 9 0734726 001338688 748 559504 418508992 273495887 90775197 00133681)8 749 561001 420189749 27-3678644 90815631 0013.35113 750 562500 421875000 27-3861279 9 0856030 001333333 751 564001 423564751 27-4043792 9-0896392 00133 1558 752 565504 425259008 274226184 9-0936719 001329787 753 567009 425957777 274408455 9 0977010 001328021 754 568516 428661064 27-451)0604 91017265 001326260 755 570025 430368875 274772633 9-1057485 001324503 756 571536 432081216 27-4954542 9-1097669 001322751 757 573049 433798093 275136330 9-1137818 001321004 758 574564 435519512 27-5317998 9 1177931 001319261 759 576081 437245479 275499546 9 1218010 001317523 760 577600 438076000 27-5680975 91258053 001315789 761 579121 440711081 2758(52284 9 1298061 001314060 762 580644 442450728 27 6043475 9 1338034 001312336 763 582169 444194947 276224546 91377971 001310616 764 583696 445943744 27-6405499 91417874 001308901 765 585225 4471)97125 27-6586334 - 91457742 001307190 766 58675(5 449455096 27-(57<57<)r>0 9-1497576 001305483 767 588289 4512J7663 27-6947648 9 1537375 001303781 768 589824 4.12984832 27-7128129 91577139 001302083 769 591361 454756609 277308492 9 1616869 001300390 770 592900 456533000 27-7488739 9 1656565 001298701 771 594441 458314011 277668868 9-1696225 0012970 17 772 595984 460099648 27-7848880 9 1735852 00129.1337 773 597529 461889917 27 8028775 91775445 00129:>-W51 774 ' 599076 463(584824 278208555 9 1815003 00 129 1990 775 6001525 465484375 278388218 91854527 00 1290323 776 602176 4672H8576 2785677(5(5 9-181)4018 001288660 777 603729 4691)97433 278747197 9 1933474 001287001 778 605284 470910952 27-8926514 9 1972897 001285347 779 606841 472729139 27-9105715 9 2012286 001283697 780 608400 474552000 271)284801 92051641 00 12820:> 1 781 609961 476379541 27-9463772 9 209*962 001281)410 782 611524 478211768 27-9IV42629 92130250 001278772 783 61308!) 480048687 279821372 92161)50.-) 001277139 784 614656 481890304 28-0000000 1) 22087215 001275510 785 616255 483736(525 280178515 92247914 00127:85 786 61771)6 48558765(5 28-0356915 922S7o8 001272265 787 619369 487443403 28-0535203 92326189 001270(548 788 620944 489303872 28(1713377 9 2365277 001269036 789 622521 4911(59069 28-0891438 9-2404333 0012(57427 790 624 100 41)30391 MH) 28-10(59386 92443355 0012(55823 791 625681 41)41113671 28-12-17222 92482.344 001261223 792 627624 49(5793088 28 1424946 9-2521300 00126-J6-J6 793 628849 498)577257 28-1(5(12557 925(50224 001261034 794 630436 500566184 28 1780056 92599114 001259446 SQUARES, CUBES, ETC., OF NUMBERS. 175 No. Squares. Square Roots. Cube Roots. Reciprocals. 795 632025 502459875 28-1957444 92037973 001257862 790 6330 l(i 504358336 282134720 9-2(57071)8 001256281 797 635209 500201573 28-2311884 92715592 001254705 798 636804 508169592 28-2488938 92754352 001253133 799 638401 510082399 28-2005*81 92793081 001251364 800 640000 5! 2000000 282842712 92831777 001250000 801 641601 513922401 283019434 9-2870444 001248439 80-2 643204 515849008 283196045 9-2909072 001246883 803 64481)9 517781627 283372546 9294:071 001245330 804 640416. 519718464 283548938 92986239 001243781 805 648.125 521660125 283725219 93024775 001242236 800 649036 523006616 28 3901391 93063278 001240695 807 651249 52.5557943 284077454 93101750 -001239157 808 652804 527514112 28 4253408 9-3140190 001237024 809 654481 529475129 284429253 9-3178599 001236094 810 65(5100 531441000 284(ii)4989 93216975 001234568 811 657721 533411731 28-4780017 93255320 00123304G 812 659344 535387328 28 4956137 9-3293634 001231527 813 6609(59 537367797 285131549 9-3331916 001230012 814 662596 539353144 285306852 9-3370167 001228501 815 604225 541343375 28-5482048 9-3408386 001220994 810 665856 543338496 285657137 9-344G575 001225490 817 607489 545338513 285832119 9-3484731 001223990 8J8 609124 547343432 "280000993 9-3522857 001222494 819 670761 549353259 286181760 93560952 001221001 820 672400 531368000 28 6350421 93599016 001219512 821 674041 553387601 28-6530976 9-3637049 001218027 822 675684 555412248 286705424 93675051 001216545 823 877329 557441767 286879766 93713022 001215067 824 678976 559470224 28-7054002 93750963 001213592 825 680625 501515625 2-7228132 9-3788873 001212121 82(5 682276 563559976 287402157 9-3826752 001210654 827 683929 5(55609283 287576077 9-3864600 001209190 828 685584 567663552 28-7749891 9-3902419 001207729 829 687241 5(59722789 28-7923601 93940206 001206273 830 688900 571787000 28-8097206 9-3977964 001204819 831 690561 573856191 288270706 9-4015091 001203369 832 692224 575930368 28-8444102 9-4053387 001201923 833 693889 578009537 28-8617394 9-4091054 001200480 834 695556 580093704 28-8790582 9-4128690 001199041 835 697225 582182875 288963666 9-4166297 001197605 836 69881)6 584277056 289136646 9-4203873 001196172 837 700569 586376253 28 9309523 94241420 001194743 838 702244 588480472 289482297 9-4278936 001193317 839 703921 590589719 239654967 9-4310423 001191895 840 705600 592704000 28 9827535 9-4353880 001190476 841 707281 594823321 29-0000000 9-4391307 001189061 842 708904 596947688 290.172363 9-4428704 001187648 843 710649 599'.)77107 290344023 9-4466072 001186240 844 712336 60121 1584 290516781 94503410 001184834 845 714025 603351125 2!) 0088837 94540719 001 183432 846 715716 60549.1730 29 0800791 9-4577999 001182033 847 717409 607(545423 291032044 94615249 001180638 848 719104 60!KM)1'.2 29 1204396 94652470 001179245 849 7208ri 61 1900049 29 1370046 9 4089661 001177856 85!) 722500 614125000 29 J547595 9 4720824 001176471 851 724201 610295051 291719043 947(539.">7 001175088 852 7251)04 618470208 29 185)0390 94801001 001 173709 853 727009 621)650477 292001037 94838136 001 172333 854 729316 622835H04 292232784 94875182 001170960 855 731025 025026375 292403830 94912200 001169591 856 732736 627222016 29-2574777 94949188 0011G8224 170 Squares. Cube*. Square Roots. Cube Hoots. Reciprocals. 734449 629422793 29-2745(523 9-498B147 001 100H11 736164 631028712 29-2SM6370 9-5023078 (101105501 737881 633839779 29 3087018 9-5059980 OOI104J44 739600 630056000 29- 32575(56 9-509(5854 0011027!)! 741321 638277381 29-3428015 9-5133699 00110i440 743044 640503928 29-3598365 95170515 001 1 000! 13 744769 1542735647 29-3708010 9-5207303 001158749 746496 644972544 29-3938709 !r 5244003 001 157407 748225 647214025 29-510H823 9-5280794 001 150009 749956 C49401890 29-4278779 9-5317497 001154734 751689 651714303 29-4448037 y- 5354 172 001153403 753424 653972032 29-4618397 9-5390818 001152074 755161 656234909 29-4788059 9-5427437 00115U748 750900 658503000 29-4957(524 9-5464027 001 149425 758641 6C07703J 1 29-5127091 9-55IJ05S9 00 1J 48 106 7003H4 6C3054848 29-5296461 95537123 001140789 702129 665338617 29-54(55734 9-5573630 001145475 76387(5 667627024 29-5(534910 9-5610108 001144105 765625 669921875 29-5803989 9-56-16559 001142857 767376 672221376 29-5972972 9-5682782 001141553 769129 674526133 29-61 4 J 858 9-5719377 001140251 770884 676836152 29-6310648 9 5755745 001138952 772641 679151439 29-6479342 9-5792085 001137656 774400 681472000 29 6047939 9-5828397 001130364 776161 683797841 290810442 9 5864682 001135074 777924 686128968 29-6984848 9-5900939 001133787 779689 688465387 29-7153159 9-5937169 001132503 781456 690807104 29-7321375 9-5973373 001131222 783225 693154125 29-7489496 9-6009548 001 129944 78491)6 695506456 29-7657521 9-6045690 001128668 786769 697804103 29-7825452 9-6081817 001127396 788544 700227073 29-7993289 96117911 001126126 790321 702595369 29-8161030 9-6153977 001 124859 792100 704969000 29-8328678 9-6190017 001123596 793881 707347971 298496231 9-6226030 001122334 795664 709732288 29-8663690 9-6262016 001121076 797449 712121957 29-8831056 9 6297975 001119821 799236 714516984 29-8998328 90333907 001118568 801025 716917375 29-9165506 9-0369812 001117818 802816 719323136 29-9332591 9-C405690 001116071 804609 721734273 29-9499583 9-0441542 001114827 806404 724150792 29-9666481 9-6477367 001113586 808201 726572699 29-9833287 9-6513166 001112347 810000 729000000 30-0000000 96548938 001 111 111 811801 731432701 30-0166621 9-8584084 001 109878 813604 733870808 30-0333148 9-0620403 001108647 815409 736314327 30-0499584 9-6050096 001107420 817216 738763264 30-0065928 9-0691762 001106195 819025 741217025 30-0832179 9-6727403 001 104972 82ii836 743077416 30-0998339 9-6763017 001103753 822649 746142043 30-1164407 9-6798604 001 102536 824464 748613312 30-1330383 9-0834166 001101322 826281 751089429 30-1496209 9-086'. (701 001100110 828100 753571000 30-1002003 9-0905211 001098901 829921 756058031 30-1827705 9-6940694 001097695 831744 758550528 30- 1993377 9-0976151 001090491 833569 761048497 30-2158899 9-7011583 001095290 83.13! >6 703551944 302324329 9-7040! H9 001094092 837225 70006(1875 30-2489(569 9-7082369 00 1(192896 839056 708575296 30-2654919 9-7117723 001091703 840889 771095213 30-2820079 9-7153051 0010!i()5I3 842724 773620U32 30-2985148 9-7188354 001089325 SQUARES, CUBES, ETC., OF NUMBERS. 177 Squares. Square R-oots. Reciprocals. 919 844501 770151559 303150128 9-7223031 001088139 920 840400 778088000 30-3315018 9-72.18883 001086057 921 848241 781220001 30-3470818 9-7204109 001085776 922 850084 783777448 30-3644529 9-7320309 001084599 923 85 J 929 78(i330407 303809151 9-73(34484 - -001083423 924 853770 7888891)24 30-3073083 9-7300034 001082251 925 855025 791453125 30-4 13.S 127 9-7434758 001081081 020 85747G 794022776 30-4302481 9-7400857 001079014 927 851)329 790597983 30-4400747 9-7504930 001078749 92^ 801184 799178752 30-4030924 9-7530979 001077586 921' 803041 801765089 30-4795013 9-7575002 001070426 930 804900 804357000 30-4959014 97010001 001075209 931 800761 80G954491 30-5122-)2G 97644974 001074114 932 808024 809557508 30-5280750 9-70791)22 001072901 933 870489 812100237 30-5450487 9-7714845 601071811 934 872350 814780504 30-5014136 9-7740743 001070064 935 874225 817400375 30-5777097 9-7784616 001069519 93G 870096 820025856 30-5941171 9-7819466 001068376 937 877909 822050953 '30-0104557 9-7854288 001067236 938 879844 8252D3G72 30-G207857 9-7889087 001066098 939 881721 827930019 30-043] 069 9-7923801 001064903 940 883000 830584000 30-0594 194 9-7958611 001003830 941 885481 833237021 30-6757233 9-7993336 001002099 942 887364 835890888 30-6920185 9-8028036 001061571 943 889249 838501807 30-7083051 9-8002711 001060445 944 891136 841232384 30-7245830 9-8097302 001059322 945 893025 843908025 30-7408523 9-8131989 001058201 94G 894916 840590536 30-7571130 9-8100591 001057082 947 890809 849278123 30-7733051 9-8201109 001055966 948 898704 851971392 30-7890086 9-8235723 001054852 949 900001 854070349 30-8058436 9-8270252 001053741 950 902500 857375000 30-8220700 9-8304757 001052032 951 904401 800085351 30-8382879 9-8330238 001051525 952 900304 862801408 30-8.544972 9-8373005 001050420 9515 908209 863523177 30-8700981 9-8408127 001049318 954 910116 808250004 30-8808904 9-8442536 001048218 955 912025 870983875 30-9030743 9-8470920 001047120 956 913936 873722816 30-9192497 9-8511280 001040025 957 915849 870407493 30-9354166 9-8545G17 001044932 958 917704 879217912 30-9515751 9-8579020 001043841 959 910081 881974079 30-9077251 9-8014218 001042753 900 921 GOO 884730000 30-9838008 9-8048483 001041007 981 923521 887503081 31-0000000 9-8682724 001040583 962 925444 890277128 310101248 9-8716941 001039501 9G3 9273G9 893050347 31-0322413 9-8751135 001038422 904 929296 895841344 31-0483494 9-8785305 001037344 905 931225 898032125 31-0644491 9-8819451 001030209 900 933156 901428096 31-0805405 9-8853574 001035197 907 935089 904231003 3l-090C)236 9-8887073 001034126 90S 937024 907030232 31 1120984 9-8021749 001033058 909 938961 909853209 31-1287(548 9-8955801 001031992 970 940900 912073000 31 144H230 9-8080830 001030928 971 942841 9 154! (HO 11 31 1(508729 9-91)23835 001029866 972 944784 918330048 31 1769145 9-9057817 001028807 973 94(5729 921107317 31 1920479 9-909177(5 001027749 974 948070 924010424 31 -'2080731 9-9125712 00102<)G94 975 95(Hi25 926850375 3l 2240000 9-9 159024 00 1025*541 970 952576 929714176 31 2409087 9-9193513 001024590 977 954529 932574833 31-25G0992 9-9227379 001023541 978 95154H4 935441352 312720915 9-92H1222 001022495 979 958441 938313739 31-2889757 9-9205042 001021150 980 960400 941192000 3] 3040517 99328839 001020408 178 SQUARES, CUBES, ETC., OF NUMBERS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 981 962361 944076141 31-3209195 9-9362613 001019168 982 964324 946966168 31-33*58792 9-931N5363 001018330 983 966289 9498621187 31-3528308 9-9430092 0111017294 984 968256 952763904 31-3687743 9-9463797 001016260 985 970225 955671625 31-3847097 9-9497479 001015228 986 1)72190 958585256 31.4006369 9-9531138 001014199 987 974169 961504803 31-4165561 9-956-1775 001013171 988 976144 964430272 31-4324673 9-9598389 001012146 989 978121 967361669 31-4483704 9-9*531981 001011122 990 980100 970299000 31-4l542li54 996(55549 001010101 991 982081 973242271 31-4801525 9-9699005 001009082 992 9840(54 976191488 31-4960315 9-9732619 001008065 993 986049 9791 4i 5(557 31-5 119025 9-9766120 -0(11007049 !)94 988036 982107784 31-5277655 9-979;).~>95) 00100(5036 995 990025 985074875 31-54W206 9-9833(155 001005025 996 992016 988047936 31-55(4<>77 9-986488 001004016 997 994009 991026973 31-5753068 9'9899'.MM) 001003009 ' 998 i; >t 5004 994011 992 31-591 J380 9-9933289 001002004 999 998001 997002999 31-6069613 9-!>966(;56 001001001 MM) 1000000 1000000000 31-62277(>6 10-0000000 001000000 1(M)1 1000201 1003003001 3 1 -6:^5840 10-0033222 0009990010 1002 1004004 1006012008 31-6543S36 10-0066J582 0009980040 1003 1006009 1009027027 31.6701752 10-(M)99899 0009970090 1004 1008016 1012048IM54 31-6859590 ] 0-0 133 155 0009960159 JU05 10J0025 10 15075 125 31-7017349 ]()() 16(5389 00(19950249 1000 1012036 1018108216 31-7175030 100199601 0009940358 1007 1014049 1021147343 31-7332633 10-0232791 0009930487 1008 XI16064 1024 192.) 12 31-7490157 10-0265958 00095)20635 1009156 1 105507304 32-1558704 10-1120726 0005X57 11 80 1035 1071225 1108717875 321714159 10-1153314 000966)836 1031i 1073296 1111934656 32-1869539 10-1185882 0009652510 1037 1075369 11 151 57653 32-2024844 10-1218428 0009(543202 1038 1077444 1 1 I8386H72 32-2180074 10-1250953 0009633911 1039 1079521 1121622319 32 -23:15229 10-1283457 00096-24 639 1040 1081600 1 134864000 322490310 10 -1315941 0009015385 1041 1083681 1 1281 1 1921 32- 21545316 10-1348403 0009606148 1042 1085764 1131366088 32-2800248 10-1380845 0009596929 TABLE XII. LOGARITHMS OF NUMBERS FROM 1 TO 10000. TABLE, CONTAINING THE LOGARITHMS OP NUMBERS FROM 1 TO 10,000. NUMBERS FROM I TO 100 AND THEIR LOGARITHMS, WITH THEIR INDICES. No. Logarithm. No. Logarithm. No. Logarithm. No. Logarithm. No. Logarithm. 1 o-oooooo 21 1-322219 41 1-612784 61 1-785330 81 1-908485 2 0- 301030 22 1-342423 42 1-623249 03 1-792392 82 1-913814 3 0-477121 23 1-361728 43 1-033468 S3 1-799341 83 1-919078 4 0-G020GO 24 1-380211 44 1-643453 64 1-800180 84 1-924279 5 0-0981)70 25 1-397940 45 1-653213 65 1-812913 85 1-929419 6 0-778151 26 1-414973 46 1-662758 66 1819544 86 1-934498 7 0-845098 27 1-431364 47 1-072098 67 1-820075 87 1-939519 8 0-903090 28 1-447158 48 1-681241 68 1-832509 88 1-944483 9 0-954243 29 1-462398 49 1-690196 69 1-838849 89 1-949390 10 i-oooaoo 30 1-477121 50 1-698970 70 1-845098 90 1-954243 11 1-041393 31 1-491362 51 1-707570 71 1-851258 91 1-959041 1-079181 32 1-505150 52 1-716003 72 1-857332 92 1-963788 13 1-113943 33 1-518514 53 1-724276 73 1-863323 93 1-968483 14 1146128 34 1-531479 54 1-732394 74 1-F69232 94 1-973128 15 1-176091 35 1-54400H 55 1-740363 75 1.875061 95 1-977724 16 1-204120 36 1-556303 56 1-748188 76 1-880814 96 1-982271 17 1-230449 37 1-568202 57 1-735875 77 1-886491 97 1 -1)86772 18 1-255273 38 1-579784 58 1-763428 78 1-892095 98 1-991226 19 1-278754 39 1-591065 59 1-770852 79 1-897627 99 1-995635 20 1-301030 40 1-602060 60 1-778151 80 1-9U8090 100 2-000000 NOTE. In the following part of the Table, the Indices are omitted, as *hey can be very easily supplied. Thus, the index of the logarithm of every integer number, consisting only of one number, is 0; of two figures, 1; of three figures, 2; of four figures, 3: being always a unit less than the number of figures contained in the integer number. The index to the logarithm of every number above 100, in the following part of the Table, is omitted; yet. in the operation, it must be prefixed, according to this remark : so that the logarithm of 600 is 2-77815, and that of 60UO is 377815, and so of the rest. 180 LOGARITHMS OF NUMBERS. 181 Na | | I 3 | 3 { 4 5 | 6 7 | 8 | 9 | Diff. 100 (MXKKK) 000434 1000868100 HOI 1001734 0021(56 002598 (K)3029 003461 0038511 1432 1 Bl 4751 5181 5(509 j (5038 646(5 (1894 7321 7748 8 174 '428 2 8(500 902(5 9451 987(5 10 1030* i 010724 01 1 147 011570 011993 012415 424 3 012837 013259 013(180 014 id!) i . 4521 4940 53(10 5779 (5197 6(516 420 4 7033 7451 78(58 8284 i 8700 91K5 9532 9947 020361 020775:416 5 02 11 HI) 0-2 1(503 022016 022428 022841 02325-2 (!2.'{()(i4 024075 4486 48961412 (i 5308 5715 6123 (5533 (5942 7350 7757 81(54 8571 8!)78 408 7 9384 978S) 030195 030(500 031004 031408 031812 03-2-21(5 032(519 03:50-21 404 8 0334-24 033826 4227 4628 5029 5430 58:50 6230 6(529 70281400 9 7426 7825 8223 8(520 91)17 9414 9S11 040207 040602 040998 397 1JO 041393 041787 042182 04257(5 0429(59 0433(52 043755 044148 044540 044932 393 1 53-23 5714 6105 6495 GH85 7275 7(5(5! 8053 8442 8831) 390 2 9218 0606 9993 050380 0507(56 051153 051538 051924 052309 052(594 386 3 053078 0531(53 05384(5 4230 4613 4996 53781 57601 SJ42 6524 383 4 0905 i 728(5 7066 8046 8426 8805 9185 95(53 9942 0(50320 379 5 060098 061075 061452 061821) 06220(5 062582 062i!o8 063:533 063709 4083:376 6 4458 4832 520(5 5580 5953 0326 GG99 7071 7443 7815373 7 8180 8557 81)28 9298 9(5(58 070038 070407 07077(5 071145 071514 370 8 07 188-2 072-250 072(517 072985 073352 3718i 4085 4451 481(5 5182366 9 5547 5912 6276 6640 7004 7368 7731 8094 8457 8819:363 120 079 18 J 079543 079904 0802(56 080626 ' 080987 1 081347 ! 081707 082067 082426 360 1 082785 083144 083503 38(51 4-211) 4576 4934 5291 5(547 (5004 ! 357 2 63(50 15710 7071 7426 7781 8136 8490 8845 9198 9552 355 3 4 9905 090258 0934-22 3772 090(511 4122 090963 4471 09J315 091(567 4820] 5169 092018 5518 092:170 586(5 092721 6215 093071 352 (15(52 349 5 GUI!) 7257 7(504 7951 8298 8(544 8990 9335 9(581 100026346 6 100371 100715 101059 101403 10 1747; 102091 102434 102777 103119 3462 343 7 3804 4J4(5 4487 4828 51(59 5510 5851 6191 6531 6871 341 8 7210 7549 7H88 8227 8565 8903 9241 9571) 991(5 110253 338 9L110SBO 110926 111203 111599 1119341112270 ( 112605 112940 113275 3609 335 130! 113943 [14-271 114(511 114944 115278 115611 115943 116276 11(5608 116940333 1 7271 7(i()3 7934 82(55 85 206556 271 1 6826 7096 7365 76:14 7904 8173 8441 8710 8979 9247 2ii: o 9515 9783 210051 210319 210586 210853 211121 211388 211654 211921 267 3 212188 212454 2720 2986 3252 3518 3783 4049 4314 4579 2iK5 4 4844 5109 5373 5(538 5902 6166 6430 6694 6957 7221 264 5 7484 7747 8010 8273 8536 8798 90(50 9323 9585 9846 262 6 220108 220370 220631 220892 221153 221414 221675 22193(5 222196 222456 261 7 2716 2976 3236 3496 3755 4015 4274 4533 4792 5051 259 8 5309 5568 5826 6084 6342 6600 6858 7115 7372 7(530 258 9 7887 8144 8400 8657 8913 9170 9426 9U82 9938 230193 256 170 23J449 230704 230960 231215 231470 231724 231979 232234 232488 232742 255 I 2996 3250 3504 3757 4011 4264 4517 4770 5023 5276 253 2 5528 5781 6033 6285 6537 6789 7041 7292 7544 7795 252 3 8040 8297 8548 8799 9049 9299 9550 9800 240050 240300 250 4 240549 240799 241048 241297 24ir>46 241795 242044 242293 2541 2790 249 5 3038 3286 3534 3782 40311 4277 4525 4772 5019 5266 248 6 5513 5759 600(5 6252 (5499 6745 6991 7237 7482 7728 246 7 7973 8219 8464 8709 8954 9198 9443 9687 9932 250176 245 8 250420 250664 250908 251151 251395 251638 251881 252125 252368 2610 243 9 2853 3096 3338 3580 3822 4064 4306 4548 4790 5031 242 18,) 255273 255514 255755 255996 256237 256477 256718 256958 257198 257439 241 J 7679 7918 8158 8398 8637 8877 9116 9355 9594 9833 239 2 260071 260310 260548 260787 261025 261263 261501 261739 261!>76 262214 238 3 2451 2688 2925 3162 3399 3636 3873 4109 4346 4582 237 4 4818 5054 5290 5525 5761 5996 6232 64(57 6702 6937 235 5 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 234 8 9513 974(5 9980 270213 270446 270679 270912 271144 271377 271609 233 7 271842 272074 272306 2538 2770 3001 3233 34(54 3696 3927 232 8 4158 4389 4620 4850 5081 5311 5542 5772 6002 6232 230 '.) 6462 6692 6921 7151 7380 7609 7838 8067 8296 8525 229 190 278754 278982 27921 1 279439 279667 279895 280123 280351 280578 280806 228 1 281033 281261 281488 281715 281942 282169 9396 2622 2849 3075 227 2 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 226 2) 5557 5782 6007 6232 6456 6681 6905 7130 7354 7578 225 4 7802 8026 8249 8473 8696 8920 9143 9366 95S9 9812 223 5 290035 290257 290480 290702 290925 291147 291369 291591 291813 292034 222 6 2256 2478 2699 2920 3141 3363 3584 3804 4025 424(5 221 7 4466 4(587 4907 5127 5347 5567 5787 6007 622(5 6446 220 8 66(55 6884 7104 7323 7542 7761 7979 8198 8416 8635 219 y 8853 9071 9289 9507 9725 9943 300161 300378 300595 300813 218 20.) 301030 301247 3014(54 301681 301898 302114 302331 302547 302764 302980 217 1 3196 3412 3(528 3844 4059 4275 44J1 4706 4921 513li 216 2 5351 5.-j66 5781 5996 6211 6425 6(539 6854 7068 7282 j 2 15 3 749*5 7710 7924 8137 8351 8564 8778 8991 9204 9417213 4 91)30 9843 310056 3102158 310481 31069:5 31090(5 311118 311330 311542212 5 31 1754 3119(5(5 2177 2389 2(500 2812 3023 3234 3445 3656 211 8 3rt67 4078 4289 4499 4710 4920 5130 5340 5551 57(50 210 7 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 209 8 80(53 8272 8481 8689 88i)8 911(6 9314 9522 9730 9938 208 y 32014(5 320354 320562 320769 320977 321184 321391 321598 321805 322012 207 210 32221!! 32242(5 322633 322839 323046 323252 323458 323665 323871 324077 206 1 4282 4488 4694 4899 5105 5310 5516 5721 592(5 6131 205 2 633(5 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 3 8IJ80 8583 8787 8991 9194 9398 9801 9805 330008 330211 203 4 330414 330(517 330819 331022 331225 331427 331630 331832 2034 $O6 202 5 2438 2(540 2842 3044 3246 3447 3649 3850 4051 4253 202 9 4454 4655 4856 5057 5257 5458 5658 5859 6059 6260 201 7 64(50 66(50 68(50 7060 7260 7459 7(559 7858 8058 82.">7 200 8 8456 84556 8855 9054 9253 9451 9650 9849 340047 340246! 199 9 340444 340642 340841 341039 341237 341435 341632 341830 2028 2225| 198 No. 1 O I 3 I 8 I LOGARITHMS OF NUMBERS. 183 No. | | 1 [ 3 | 3 | 4 5 ! 6 | 7 i 8 I 9 | D. 220 342423 U2020 .'{42817 343014 M3212 34340!) 343(50(5 J43802 343999 344196 197 J 4392 458i) 4785 4981 5178 5374 5570 57(50 5902 0157 196 o 6353 0549 0744 0939 7135 7330 7525 7720 7915 81 10 195 3 8305 8500 8(594 8889 9083 St-278 9472 9(5(515 9*00 350054 194 4 350248 350442 3501530 J50829 151023 351210 351410 151603 351796 1989 193 5 2183 2:575 2508 2701 2954 3147 3339 3532 3724 3916 193 6 41 OH 4301 4493 4085 4870 50(58 5-200 5452 5043 5834 192 7 6026 8217 (5408 6599 6790 (5981 7172 7303 7554 7744 191 8 7935 8125 8316 8500 8(596 8886 9076 9-2(5(5 9451) 9646 190 9 9835 360025 360215 360404 360593 360783 360972 301101 361350 361539 189 530 3(517-28 361917 302105 302294 362482 302(571 362859 3G3048 363536 363424 188 1 3619 3800 3988 4170 43(53 4551 4739 4920 5113 5301 188 2 5488 5(575 5802 6049 0230 0423 61510 0790 0983 7169 187 3 735(5 7542 7729 7915 8101 8287 8473 8(559 8845 9030 186 4 921(5 9401 9587 9772 9958 370143 370328 370513 370(598 370883 185 5 371008 371253 371437 371(522 371800 1991 2175 2300 2544 2728 184 6 2912 3090 3280 34(54 3047 3831 4015 4198 4382 4565 184 7 4748 4932 5115 5298 5481 5(504 5846 6029 6212 6394 183 8 6577 6759 0942 7124 7300 7488 7670 7852 8034 8216 182 9 8398 8580 8701 8943 9124 9300 9487 9068 9849 380030 181 240 380211 380392 380573 380754 380934 381115 381290 381476 381656 381837 181 1 2017 2197 2377 2557 2737 2917 3097 3277 3456 303(5 180 2 3815 3995 4174 4353 4533 4712 4891 5070 5249 5428 179 3 5(500 5785 5904 0142 6321 0499 0077 6856 7034 7212 178 4 7390 7508 7740 71C3 8101 8279 8450 8034 8811 8989 178 5 9100 9343 9520 9(598 9875 390051 3*0228 390405 390582 390759 177 (I 390935 391112 391288 3914(54 391041 1817 1993 2109 2345 2521 176 7 2097 2873 3048 3224 3400 3575 3751 3926 4101 4277 176 8 4452 46S7 4802 4977 5152 532(5 5501 5676 5850 0025 175 9 0199 0374 0548 6722 0896 7071 7245 7419 7592 7788 174 25(1 397940 398114 398287 398401 398034 398808 398981 399154 399328 399501 173 1 9074 9847 400020 400192 400305 400538 400711 400883 401050 401-228 173 2 401401 401573 1745 1917 2089 2201 2433 2005 2777 2949 172 3 3121 3292 3404 3035 3807 3978 4149 4320 4492 4(563 171 4 4834 5005 5176 5340 5517 5088 5858 6029 6199 6370 171 5 0540 0710 0881 7051 7221 7391 7501 7731 7901 8070 170 6 8240 8410 8579 8749 8918 9087 9257 9420 9595 9764 169 7 9933 410102 410271 410440 4 10(509 410777 41094(5 111114 411283 411451 109 8 411020 1788 1950 2124 2293 2461 2029 2790 2904 3132 108 i 4201 4357 4513 4(509 4825 49<-l 5137 5293 5449 156 91 56041 5760 5915 6071 0220 6382 6537 0(592 6848 7003,155 7|8|9 184 LOGARITHMS OF NUMBERS. Na|0| 1|J8|3|4:|5|G|7|8|9 2801447158 4473134474(58 447623 447778 447933 4480881448-242 448397 !4.-'5.V2 I."-.') 1 8706 88(51 9:>15 9170 93-24 9478 8033 9787 41 4500115 154 2 1 4302-10 450403 450557 45371 1 450885 451018 451172 45132:; 451479 1633 154 3 1788 1940 2093 2-247 240.) 2553 2706 2859 3012 3165 153 4 3318 3471 3o24 3777 3930 4082 4235 4387 4540 4692 .13 5 484.1 4937 5150 5302 5454 5:506 575t3 5iUO 60G2 6214 -, ) 6 6360 6518 6(570 68-21 6;i73 7125 7276 7428 7579 7731 5-3 7 7882 8033 8184 8336 8487 8638 8781) 8940 9091 9242 51 8 93J-2 9543 9694 9845 9.)1)5 460146 46.12.16 460447 46051(7 460748 151 S 4608 J8 46 1048 461198 461348 461499 1649 1799 1948 201)8 2248 15'J 290 4623;!8 4G2548 462697 52?47 4C.29J7 403146 463296 463445 463504 463744 150 3 3893 4042 4191 4340 4490 4(539 4788 493(5 5085 5234 149 2 5383 553-2 51580 582J 51)77 6126 6274 (54-23 6571 6719 149 3 6868 701(i 7164 7312 74(50 7(508 7756 7901 8052 8200 148 4 8347 8495 8643 8790 8938 9085 9233 9380 9527 9675 148 5 98-2-2 9969 470116 470'263 470410 470557 470704 470851 47091)8 471145 147 6 471292 471438 1585 1732 1878 2025 2171 2318 2464 3610 146 ? 2756 2903 3049 3195 3341 3487 *633 3779 3 ; )-25 4071 146 8 4216 4362 4508 4(553 4799 4944 5090 5235 5381 552(5 146 9 5671 5816 5J62 6107 6252 6397 6542 6687 6832 697(5 145 300 477121 47726;; 477411 477555 477700 477844 477989 478133 478278 478422 14.1 1 85(56 8711 8855 8999 9143 9287 9431 9575 9719 9863 144 2 480007 480151 48J2:>4 480438 480582 480725 480869 481012 481156 4812;)!) 144 3 1443 1586 1729 1872 2016 2159 2302 2445 .2588 2731 143 4 2874 3016 3159 3302 3445 3587 3730 3872 4015 4157 143 5 4300 4442 4585 4727 4869 5011 5153 5295 5437 5579 142 G 57 - 2 5863 6005 6147 6281) 6430 6579 6714 6855 6997 142 7 7138 7280 7421 7563 7704 7845 7986 8127 82(59 8410 141 8 8551 8692 8833 8i)74 9114 9255 939(5 9537 9677 98 18 141 9 9958 490099 490239 490380 4905-20 490661 490801 490941 491081 491222 140 310 4913G2 49150-2 491642 491782 491922 492062 41)2201 492341 492481 492(521 140 1 27(50 2900 3040 3179 3319 3458 3597 3737 3876 4015 139 2 4155 4-294 4433 4572 4711 48.10 4989 5128 52(57 540(5 139 3 5544 5683 5822 5960 (501)9 6238 6376 6515 6(553 6791 139 4 6930 7068 7206 7344 74H3 7621 7759 7897 8035 8173 138 5 8311 8448 8586 87-24 8862 899!) 9137 9275 9412 9550 138 6 9687 98-24 99(52 50001);) 500-236 500374 500511 500(548 500785 500922 137 7 501051) 501 19S 501333 1470 1607 1744 1880 2017 2154 221)1 137 8 24-27 2564 2700 2837 2973 3109 3246 3382 3518 3(555 136 9 3791 3927 4063 4199 4335 4471 4(507 4743 4878 5014 136 320 5051.50 505286 505421 505557 505693 505828 505964 506099 506234 506370 136 j 6505 6640 6776 6911 7046 7181 731(5 7451 758!i 7721 13.1 2 7856 7991 8126 82(50 8395 8530 8664 8799 8934 9068 135 3 9-203 9337 9471 UBU6 9740 9874 510009 510143 510277 510411 134 4 510545 511X579 510813 510947 511081 511215 1349 1482 1616 1750 134 5 1883 2017 2151 2284 2418 2551 2(584 2818 2951 3084 133 6 3218 3351 3484 3617 3750 3883 4016 4149 4282 4415 133 7 4548 4li81 4813 4946 5079 5211 5344 547(5 5609 5741 133 8 5874 GO! 16 6139 6-271 6403 6535 6(5(58 6800 6932 7064 132 9 7196 7328 7460 7592 7724 7855 7987 3119 8251 8382 132 330 5185141518646 518777 518909 519040 519171 519303 519434 519566 519697 131 JU 98281 9959 5200.X) 120221 520353 520484 520615 520745 52087(5 521007 131 2 521138 521-269 14 X) 1530 1(561 1792 1922 2053 2183 2314 131 3 2-444 2575 2705 2835 29156 3096 3226 3356 3486 3(516 130 4 3746 387(5 4();)6 4136 4266 4396 4536 4656 4785 4915 130 5 5045 5174 5304 5434 5563 5693 5822 5951 6081 6210 129 6 6339 6469 65!>8 6727 (585'j 6985 7114 7243 7372 7501 129 7 7630 7759 788S 8016 8145 8274 8402 853 1 86(50 8788 129 8 8917 9045 9174 930-2 9430 9559 9(587 9815 9943 530072 ! 128 9 530200 530328 530456 530584 53071-2 530840 530968 531096 531223 1351| 128 No. I O I 1 .1 3 | -1 | 5 | J 7 LOGARITHMS OF NUMBERS. No. | | 1 3 | 6 | 7 j 9 | Dig 34 J 53l-17i> 531607 531734 531802:531990 532117 532245 532372 532500 532627 128 1 2754 2882 3009 313(5 3204 3391 3518 3(545 3772 3899 127 o 4026 4153 4280 4407 4534 4661 4787 4914 5041 5167 J27 3 5294 5421 5547 5674 5800 5927 6053 6180 6306 6432 126 4 6558, 6(185 6811 69' J7 7063 7189 7315 7441 7567 7693 126 5 7819 7945 81)71 8197 832-2 8448 8574 8699 8825 8951 126 G 90761 0-20-2 9327 9452 9578 9703 9829 9954 540079 540204 1-25 540329 540455 540580 540705 540830 540955 541080 541205 1330 1454 1-25 8 1579 1704 1829 1953 2078 2203 2327 2452 2576 2701 1-25 9 9823 2:)50 3074 3199 3323 3447 3571 3696 3820 3944 124 350 544008 544192 544316 544 MO 544564 544688 544812 544936 545060 545183 1-24 1 5307 5431 5555 5678 5862 5925 6049 6172 6296 6419 124 2 0543 6666 6789 69 1 3 7036 7159 7282 7405 7529 7652 123 3 7898 8021 8144 8267 8389 8512 8635 8758 8-881 123 4 !K)()3 9126 9249 9371 9494 9616 9739 9861 9984 550106 123 5 550228 550351 550473 550595 550717 550840 550962 551084 551206 1328 122 C 1450 157-2 1694 1816 1938 2060 2181 2303 2425 2547 122 7 26G8 2790 2911 3033 3155 3276 3398 3519 3640 3762 121 8 3883 4004 4126 4247 4368 4489 4(510 4731 4852 4973 121 9 5094 5215 5336 5457 5578 5699 5820 5940 6061 6182 121 3GO 556303 556423 556544 556664 556785 556905 557026 557146 557267 557387 120 ] 7507 7027 7748 7868 7988 8108 8228 8349 8469 8589 1-20 2 8709 8829 8948 9068 9188 9308 9428 9548 9667 9787 120 3 9907 5600-26 560 140 560265 560385 560504 560(524 560743 560803 5(50982 119 4 561J01 1221 1340 1459 1578 1698 1817 1936 2055 2174 119 5 2-293 24121 2531 2C-50 2769 2887 - 3006 3125 3244 3362 119 6 3481 3600 3718 3837 3955 4074 4192 4311 4429 4548 119 7 4660 4784 4903 502 1 5139 5257 5376 5494 5612 5730 118 8 5848 5966 6084 6-21 !-2 (53-20 6437 6555 6073 6791 6909 118 9 7020 7144 7262 7379 7497 7614 7732 7849 7967 8084 118 370 568202 568319 568436 568554 568671 5G8788 568905 569023 569140 569257 117 ] 93:4 9491 9I508 9725 9842 9959 57007(5 570193 570309 570426 117 2 570543 570660 570776 570893 571010 571126 1243 1359 1476 1592 117 3 1709 1825 1942 2058 2174 2291 2407 2523 2639 2755 116 4 2872 2988 3104 3220 333(5 3452 3568 3684 3800 3915 116 5 4031 4147 4263 4379 4494 4(510 4726 4841 4957 5072 116 6 5188 5303 5419 5.134 5650 5765 5880 5996 6111 (5-226 115 7 6341 6457 6572 6G87 6802 6917 7032 7147 7262 7377 115 8 7492 7007 7722 78315 7951 8000 8181 8-295 8410 8525 115 9 8639 8754 8868 8983 9097 9212 932(5 9441 9555 9669 114 380 570784 579898 580012 580120 580241 580355 580469 580583 580697 580811 114 1 580925 581039 1153 12(57 1381 1495 1608 1722 1836 1950 114 2 3063 2177 2291 2404 2518 2631 2745 2858 2972 3085 114 3 3199 3312 3426 3539 3(552 3765 3879 3992 4105 4218 113 4 4331 4444 4557 4670 4783 489(5 5009 5122 5235 5348 113 5 5461 5574 51)86 5799 5912 6024 6137 6250 6362 (5475 113 6 6587 6700 6812 6925 7037 714J 7262 7374 7486 7599 112 7 7711 7823 7935 8047 8160 8272 8384 8496 8608 8720 112 ci 8833 8944 905(5 9167 9279 9391 9503 9615 9726 9838 112 9 9950 590061 590173 590284 590396 590507 59061'J 590730 590842 590953 112 390 591065 591176 591287 591399 591510 591621 :91732 591843 591 955 ' 592066 111 J 2177 2288 2399 2510 2621 2732 2843 2954 3064 3175 111 3-286 3397 3508 3618 3729 3840 3950 4061 4171 4282,111 3 4393 4503 4(514 4724 4834 4945 5055 5165 5276 5386 110 4 5496 5606 57-17 5827 5937 (5047 6157 6267 6377 6487 110 5 6597 6707 6817 6927 7037 7146 725(5 73(56 74761 758(5110 6 7(595 7H05 7914 8024 8134 8243 8353 84(52 8572 868L 1 10 7 8791 89001 9009 91 19 9228 9337 9446 955(5 9665 9774 109 8 9883 9992161)0101 600210 (00319 600428 (500537 600(546 600755 61)0864 H'9 9 60097 3 | 60 1082 j 1191 1299 141,8 1517J 1625| 17:54 18431 1951 109 No. | O | I 7 I 8 186 S OF NUMBERS. No.|0|l|3|3|4-|5 G 7 | 8 | 9 TDK 400 6C20i50 0021(59 G02277 802386 002494 602603 002711 602819 602928 60303!! 108 1 3144 3253 3361 34(59 3577 3G8G 3794 3932 4010 4118 108 2 4226 4334 4442 4550 4(558 47G6 4874 4982 5089 5197 108 3 5305 5413 5521 5628 573(5 5844 5951 6059 GJGG 6274 103 4 6381 6489 6590 6704 6811 6919 7026 7133 7211 7348 107 5 7455 7562 7G69 7777 7884 7991 8098 8-205 8312 8419 107 6 85261 8G33 8740 8847 8954 9061 91G7 9274 93dl 9488 107 7 9594! 9701 9808 9914 610021 610128 610-234 610341 610447 610554 107 8 010060 G107G7 610873 G10979 108G 1192 1298 1-105 1511 1G17 106 9 1723 1829 1936 2042 2148 2254 2360 2466 2572 2678 106 410 1 G12784 3842 612890 3947 612996 4053 613102 613207 4159 4264 6133131613419 4370 4475 613525 4581 G13G30 4686 613736 4792 106 106 4897 5003 5108 5213 5319 5424 5529 5(534 574!) 5845 105 3 5950 6055 6160 6265 6370 G47G 6581 CG86 6790 6895 105 4 7000 7105 7210 7315 742!) 7525 76-29 7734 7839 7943 105 5 8048 8153 8257 8362! 8460 8571 8G76 8780 8884 8989 105 6 9093 9198 9302 9406 9511 9615 9719 9824 9928 620032 104 J 1 62013(5 620240 620344 620448 623552 620656 620700 6208G4 6209G8 1072 104 8 1176 1280 138-1 1488 1592 1695 1799 1903 2007 2110 104 9 2214 2318 2421 2525 2628 . 2732 2835 2939 3042 3146 104 420 6232-19 623353 623456 623559 623663 623766 623869 G23973 624070 624179 103 1 4282 4385 4488 4591 4695 4798 4901 5004 5107 5210 103 2 5312 5415 5518 5621 5724 5827 5929 6032 6133 6-238 103 3 6340 6443 6546 6648 6751 6853 6956 7058 716! 72(53 103 4 73(56 74C8 7571 7673 7775 7878 7980 8082 8185 8287 102 5 8389 8491 8593 8695 8797 8900 9002 9104 920G 9308 102 6 9410 9512 9613 9715 9817 9919 630921 630123 630224 G3032G 102 7 630428 630530 630(531 630733 630835 630936 1038 1139 1241 1I54-2 102 8 1444 1545 1(547 1748 1849 1951 2352 2153 2255 2356 101 9 2457 2559 2GGO 2701 2862 2903 3004 3165 32(56 3367 101 430 633468 633569 G33G70 633771 633872 633973 634074 634175 634276 G34376 101 1 4477 4578 4679 4779 4883 4981 5081 5182 5283 5383 101 2 5434 5584 5685 5785 5886 598(5 6087 6187 6287 6388 100 3 6488 6588 6688 6789 6889 6989 7089 7189 7290 7390 100 4 7490 7590 76UO 7791) 7893 7990 8090 8190 8290 8389 103 5 8489 8589 8689 8789 8888 8988 9088 9188 9287 9387 100 C 9486 9586 9686 9785 9885 9984 G4(K)84 640183 640283 (540382 99 640481 640581 (540680 640779 640879 640978 1077 1177 127(5 1375 93 8 1474 1573 1G72 1771 1871 1970 21)69 2168 2267 2366 99 9 2465 2563 2662 2761 2860 2959 3058 3156 3255 3354 93 440 643453 643551 643650 643749 643847 643946 644044 644143 614242 G44340 98 1 4439 4537 4(536 4734 4832 4931 5029 5127 522!i 53-24 98 c 5422 5521 5619 5717 5815 5913 6011 6110 6208 (5306 95 3 6404 6532 6600 6698 679(5 6894 6992 7089 7187 7283 98 4 7383 7481 7579 7676 7774 7872 7989 81)67 81(55 82(52 98 8360 8458 8555 8053 8750 8848 8945 9043 9140 9237 97 ( 9335 9432 9530 9327 9724 9821 91)19 6500 1C 650113 97 7 650308 (550405 653502 650599 650(59(5 650793 050890 0987 1084 ' 1181 97 8 1278 1375 1472 1569 1(566 17(52 185< 195( 2053 2150 97 i 2246 2343 2440 2536 2633 2730 2826 2923 3019 3116 97 450 653213 653309 653405 653502 653598 G53G95 G53791 653888 653984 654080 90 1 4177 4273 4361 4465 45(52 4658 4754 4851 4946 5042 96 c 5133 5235 5331 5427 5523 5(519 5715 58K 59! H 8003 96 5 60D8 (5194 G290 G38G 6482 6577 6673 6769 6864 6960 96 i 7056 7152 7247 7343 7438 7534 7629 775 782 791C 96 i 8011 8107 8202 8298 8393 8488 8584 8679 8774 8870 95 ( 8965 9001 9155 9250 9346 9441 9536 9G31 972C 9821 95 ' 9:uo 660011 (5G010< 660201 66()2<)G 660391 660486 660581 660(57* GG0771 95 8 G60865 0960 1055 1150 1245 1339 1434 1529 1623 17J8I 95 ( 1813 1907 2002 2U6 2191 226 2ItO 2475 2)<9 2663 1 95 No. | 1 | | 3 1 4- | 5 fi "l 7 I * | I.,IE LOGARITHMS OF NUMBERS. 187 No. | 1 I 3 [ | 7 I 8 | 9 |D.ff. 400 662758 , C62852, 662947 663041 (563135 663230 1| 37011 3795 3889 1 3983 4078 417-3 563324 663418 42(56 4360 863512 663(507) 94 4454 4548! 94 2 48421 4736 4830! 49-24 - 5018 5112 5206 5299 5393 5487| 94 3 5581 5675 5769 58(52 5956 6050 (5143 6237 6331! 6424 94 4 65 J 8 6612 6705 6799 6892 698(5 7079 7173 7-2f>i 7360' 94 5 7453 7546 7640 7733 7886 71*30 8013 8106 8199 8293 93 6 838(5 8479 8572 8665 8759 8852 8945 9038 9131 9224 93 7 9317 9410 9503 9596 9689 9782 9875 9967 670060 670153 93 8 9 670-246 1173 570339 1265 670431 1358 670.124 1451 70617 1543 670710 1636 670802 670895 1728 1821 0988 1913 1080 2005 93 93 470 372098 672190 672283 672375 672467 672560 672652672744 672836 672929 92 1 30-21 3113 3205 3297 3390 3482 3574 3666 3758 3850 92 2 394-2 4034 4126 4218 4310 4402 4494 4586 4677 4769 92 3 4861 4953 5045 5137 5228 5320 5412 5503 5595 5687 92 4 5778 5870 5962 6053 6145 6236 6328 6419 6511 6602 92 5 G694 6785 6876 6968 7059 7151 7242 7333 7424 7516 91 6 7607 7698 7789 7881 7972 8063 8154 8245 8336 8427 91 7 8518 8609 8700 8791 8882 8973 9064 9155 9246 9337 91 8 9428 9519 9610 9700 9791 9882 9973 680063 680154 680245 91 9 680336 680426 680517 680607 680698 680789 680879 0970 1060 1151 91 480 681241 681332 681422 681513 681603 681693 681784 681874 681964 682055 90 1 2145 2235 2326 2416 2506 2596 2686 2777 2867 2957 90 2 3047 3137 3227 33 J 7 3407 3497 3587 3(577 3767 3857 90 3 3947 4037 4127 4217 4307 4396 4486 4576 4666 4756 90 4 4845 4935 5025 5114 5204 5294 5383 5473 5563 5652 90 5 5742 5831 5921 6010 6100 6189 6279 6368 6458 6547 89 6 6636 6726 6815 6904 6994 7083 7172 7261 7351 7440 89 7 7529 7618 7707 7796 7886 7975 8064 8153 8242 8331 89 8 8420 8509 8598 8687 8776 8865 8953 9042 9131 9220 89 9 9309 9398 9486 9575 9664 9753 9841 9930 690019 690107 89 490 690196 690285 690373 6904(52 690550 690639 690728 690816 690905 690993 89 1 1081 1170 1258 1347 14:55 1524 1612 1700 1789 1877 88 2 1905 2053 2142 2230 2318 240(5 2494 2583 2(571 2759 88 3 2847 2935 3023 3111 3199 3287 3375 3463 3551 3639 88 4 3727 3815 3903 3991 4078 4166 4254 4342 4430 4517 88 5 4605 4693 4781 4868 4956 5044 5131 5219 5307 5394 88 6 5482 5569 5657 5744 5832 5919 6007 6094 6182 62(59 87 7 6356 6444 6531 6618 6706 (5793 6880 6968 7055 7142 87 8 722U 7317 7404 7491 7578 7665 7752 7839 7926 8014 87 9 8101 8188 8275 8302 8449 8535 8622 8709 8796 8883 87 500 698970 699057 699144 699231 699317 699404 699491 699578 699664 699751 87 ] 9838 9924 700011 70009& 700184 70027 1 700358 700444 700531 700617 87 2 700704 700790 0877 0963 1050 1136 1-2-22 1309 1395 1482 86 2 1568 11)54 1741 1827 1913 1999 2086 2172 2258 2344 86 4 431 2517 2603 2689 O**7j 2861 2947 3033 3119 3205 86 5 'J291 3377 3463 3549 3635 3721 3807 3893 3979 4065 86 6 4151 4236 4322 4408 4494 4579 4665 4751 4837 4922 86 7 5008 5094 5179 5265 5350 5436 5522 5(507 5693 5778 86 8 5864 5949 6035 6120 6206 6291 6376 6462 6547 6153'. 85 9 6718 6803 6888 6974 7059 7144 7229 7315 7400 7485 85 510 707570 707655 707740 70782* 707911 707996 708081 708166 708251 708336 85 J 8421 8500 8591 867( 8761 8846 8931 9015 9100 9185 85 S 9270 9355 9440 9524 9609 9694 9779 9863 9948 710033 85 3 710117 7102th. 710287 710371 710456 710540 710625 7107JO 710794 0879 85 4 096: 1048 1132 121" 1301 1385 1470 1554 1639 1723 84 5 1807 189V. 1976 2060 2144 222 1 2313 2397 2481 2566 84 6 2650 27:54 2818 290- 2986 3070 3154 j 3238 332; 340" 84 7 3491 3575 3(559 3742 382 3910 39941 40781 4JC.2 4iJ4r 84 8 4330 4414 4497 4581 4665 4749 4833 4916 5000 5084 84 9j 5167 5251 5335 5418 55U2| 5586 5669 5753 5836 5920 84 No. 6 | | 8 9 | Diff 188 LOGARITHMS OF NUMBERS. No. | 3 f 3 | 5 | 6 | 7 | 8 f 9 fniit 520 716003 716087 710170 716254 716337 716421 716504 716588 71(5(571 71(5 1 6838 6921 7004 7088 7171 7254 7338 7421 7:504 o 7671 7754 7837 7920 8003 808(5 8169 8253 8336 8- 3 8502 8585 8668 8751 8834 8917 9000 9083 91(55 9 4 9331 9414 9497 9580 96(53 9745 9828 9911 9994 7-20 5 720159 720242 720325 720407 720490 720573 720(555 720738 72082 11 6 098(5 1068 1151 1-233 1310 1398 1481 15(53 164(5 1 7 1811 1893 1975 2058 2140 2222 2305 2387 2409 t> 8 2)534 2716 2798 2881 29(53 3045 3127 3209 3291 3 9 3450 3538 3620 3702 3784 3860 3948 4030 4112 4 530 724270 724358 724440 724522 724604 724685 724767 724849 72493 1 725 1 5095 5176 5258 5340 5422 5503 5585 5(5(57 5748 5 2 5912 5993 6075 6156 6238 6320 6401 6483 (55(54 li 3 6727 6809 6890 6972 7053 7134 7216 7297 7379 7. 4 7541 7(523 7704 7785 786(5 7948 802S 8110 8191 8- 5 8354 8435 8516 8597 8678 8759 8841 8922 9003 9 6 91(55 924(5 9327 9408 9489 9570 9651 9732 9813 9 7 9974 730055 730136 730217 73D298 730378 730459 730540 730(521 730 8 730782 0863 0944 1024 1105 11815 126(5 1347 1428 1 9 158J 1669 1750 1830 1911 1991 2072 2152 2233 o 540 732394 732474 732555 732635 732715 732796 73287(5 732956 733037 733 1 3197 3273 3358 3438 3518 3598 3679 375! t 3839 3 ( 2 3993 4079 4160 4240 4320 4400 4480 45!50 4(540 4' 3 4800 4880 4960 5040 5120 5200 5279 5359 5439 4 5593 5679 5759 5^3 J 5918 5998 (5078 6157 0237 o: 5 6397 6476 655(5 0(535 6715 (5795 0874 (5954 7034 7 6 7193 7272 7352 7431 7511 7590 7(570 7749 7829 75 7 7987 8067 8140 8225 8305 8384 8463 8543 8022 8' 8 8781 8860 8939 9018 9097 9177 9250 9335 9414 9 9 9572 9651 9731 9810 9889 9968 740047 740120 740205 740- 550 740363 740442 740521 740600" 740678 740757 740830 740915 740994 74 1 ( I 1152 1230 1309 13K8 1467 1540 1024 1703 1782 u 2 1939 2018 2090 2175 2254 2332 2411 2489 25(58 21 3 2725 2804 2882 29(51 3039 3118 3190 3275 3353 3- 4 3510 3588 3667 3745 3823 3902 3980 40.58 4130 4- 5 4293 4371 4449 4528 46015 4(584 4702 4840 4919 4' 6 5075 5153 5231 5309 5387 54(55 5543 5021 5699 5' 7 5855 5933 6011 0089 6167 6245 (5323 0401 0479 (>. 8 (5(534 6712 6790 0808 6945 7023 7101 7179 725(5 7; 9 7412 7489 7567 7(545 7722 7800 7878 7955 8033 * 560 748188 74826(5 748343 748421 748498 748570 748(553 748731 748808 748? ] 8963 9040 9118 9195 9272 9350 9127 9504 9.582 9' 2 973(5 9814 9891 99(58 750045 750123 750200 750277 750354 750 3 751)508 75058(5 750(5(53 750740 0817 0894 0971 1048 1125 1- 4 1279 1350 1433 1510 1587 1004 1741 1818 1895 1' 5 2048 2125 220-2 2279 23511 2433 2509 2580 2663 2 6 281(5 2893 2970 3047 3123 3200 3277 3353 3430 3. 7 3583 3660 3736 3813 3889 3900 4042 4119 4195 4 8 434^ 4425 4501 4578 4(554 4730 4807 4883 4960 5 9 5112 5189 5205 5341 5417 5494 5570 5040 5722 5 570 755875 755951 750027 750103 756180 750250 750332 756408 756484 750 I 6030 6712 6788 08154 6940 7016 7092 7108 7244 7. 2 7390 7472 7548 7(524 7700 7775 7851 7927 8003 8 3 8155 8230 8300 8382 8458 8533 8009 8085 8761 8f 4 8912 8988 9003 9139 9214 9290 930(5 9441 9517 9 5 90(58 9743 9819 9894 9970 760045 70.) 121 76019(5 760272 700: 6 760422 760498 760573 7001149 700724 0799 0875 0950 1025 1 7 1170 1251 1321) 1402 1477 1552 1(527 1702 1778 If 8 1928 2003 2078 2153 2228 2303 2378 2453 2529 21 9 2679 2754 2829 2904 2978 3053 3128 3203 3278 3: No. | | J4 8 9 LOGARITHMS OF NUMBERS. 189 Nu. | 1' . 1 | | 3 4 | 5 | 6 7 | 8 | 9 | Diff. 580 763428 763503 7(53578 763(553 763727] 763802 763877 763952 764027 764101 75 1 417(5 4251 4326 4400 4475 4550 4(124 4699 4774 4848 75 o 4923 495W 5072 5147 5221 5296 5370 5445 5520 5594 75 3 5069 5743 5818 5892 5966 6041 6115 6190 6264 6338 74 4 6413 6487 6562 6636 6710 6785 6859 6933 7007 7082 74 5 7156 7230 7304 7379 7453 7527 7601 7675 7749 7823 74 G 7898 7972 8046 8120 8194 8268 8342 8416 849(1 8564 74 7 8638 8712 8786 8860 8934 9008 9082 9156 9230 9303 74 8 9377 9451 9525 9599 9673 9746 9820 9894 9968 770042 74 9 770115 770189 770263 770336 770410 770484 770557 770631 770705 0778 74 590 770852 770926 770999 771073 771146 771220 771293 771367 771440 771514 74 1 1587 1661 1734 1808 1881 1955 2028 2102 2175 2248 73 2 2322 23!)5 2468 2542 2615 2688 2762 2835 2908 2981 73 3 3055 3128 3201 3274 3348 3421 3494 3567 3640 3713 73 ''4 3786 3860 3933 4006 4079 4152 4225 4298 4371 4444 73 5 4517 4590 4663 4736 4809 4882 4955 5028 5100 5173 73 6 524(5 5319 5392 5405 5538 5610 5683 5756 5829 5902 73 7 5974 6047 6120 6193 6265 6338 6411 6483 6556 6629 73 8 6701 6774 6846 6919 6992 7064 7137 7209 7282 7354 73 9 7427 7499 7572 7644 7717 7789 7862 7934 8006 8079, 72 600 778151 778224 778296 778368 778441 778513 778585 778658 778730 778802 72 I 8874 8947 9019 9091 9163 9236 9308 9380 9452 9524 72 2 9590 9(569 9741 9813 9885 9957 780029 780101 780173 780245 72 3 780317 780389 780461 780533 780605 780677 0749 0821 0893 0965 72 4 1037 1109 1181 1253 1324 1396 1468 1540 1612 1684 72 5 1755 1827 1899 197] 2042 2114 2186 2258 2329 2401 72 C 2473 2544 2(516 2688 . 2759 2831 2902 2974 304(5 3117 72 7 3189 3260 3332 3403 3475 3546 3618 3689 3761 3832 71 8 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546 71 9 4617 4689 4760 4831 4902 4974 5045 5116 5187 5259 71 610 785330 785401 785472 785543 785615 785686 785757 783828 785899 785970 71 1 6041 6112 6183 62.54 6325 6396 64G7 6538 6609 6680 71 2 6751 6822 6893 6964 7035 7106 7177 7248 7319 7390 71 3 7460 7531 7602 7673 7744 7815 7885 7956 8027 8098 71 4 8168 8239 8310 8381 8451 8522 8593 8(363 8734 8804 71 5 8875 8946 9016 9:;R7 9157 9228 9299 93(59 9440 9510 71 6 9581 9051 9722 9792 9863 9933 790004 790074 790144 790215 70 7 79025 790356 790426 79049G 790567 790637 0707 0778 0848 0918 70 8 0988 1059 1129 1199 1269 1340 1410 1480 1550 1620 70 9 JC91 1761 1831 1901 1971 2041 2111 2181 2252 2322 70 620 792392 792462 792532 792602 792672 792742 792812 792882 792952 793022 70 1 3092 3102 3231 3301 3371 3441 351 1 3581 3651 3721 70 2 3799 3860 3930 4000 4070 4139 42.19 4279 4349 4418 70 3 4488 4558 4627 4697 4767 4836 49t)!5 4976 5045 5115 70 4 5185 5254 5324 5393 5463 5532 5602 5(572 5741 5811 70 5 5880 5949 6019 6088 6158 6227 6297 6366 643(5 6505 69 6 6574 6644 6713 6782 6852 6921 6990 7060 7129 7198 69 7 72158 7337 7406 7475 7545 7614 7C83 7752 7S2J 7890 69 8 7960 8029 8098 8167 8236 8305 8374 8443 8513 8582 69 9 8651 8720 8789 8858 8927 8996 9065 9134 9203 9272 69 630 1 791)341 300029 799409 800098 799478 800167 799547 800236 799616 800305 799685 30037* 799754 800442 799823 800511 799892 800580 799961 800(548 69 69 2 0717 0786 0854 0923 0992 1061 1123 1198 1266 1335 69 3 1404 1172 1541 1609 1678 1747 1H15 18H4 1952 2D21 69 4 2089 2158 2226 2295 2363 2432 2500 25(58 2637 2705 68 5 2774 28-12 2910 2979 3047 3116 3184 32.V2 3321 3389 68 C 3457 3525 3594 36R2 3730 3798 3bfi7 3935 4003 4071 68 7 4139 4208 4276 4344 4412 4480 45-18 4616 4685 4753 68 8 4821 4889 4957 5(125 5093 5161 ! 52.9 5297 5365 5433 68 9 5501 5569 5637 5705 57731 5841| 5908 5970 (1044 6112 68 No. | | 1 j 3 | I 5 I 6 | 7 9 I Drff. 190 LOGARITHMS OF NUMBERS. I 1 | 5 | 6 8 | 9 640; 800 180 806218 8063*6 1 306384] 806451] 80651 9 800587 80(5655 806*23 S00790 08 j 6858 6926 6994 706l| 712! 7197 72(54 7332 7400 7467; 68 t 7535 7003 7<5?( 7738! 78()( 7873 7941 8008 807(5 8143 68 J 8211 8279 834f P414 8481 854! 861(5 8t;*4 875! 8818 67 4 8886 8953 9021 9088 9l5f 9223 9290 9338 9425 9492 67 j 9560 9(527 9(194 9762 9829 9896 9904 810031 810098 810165 67 6 8 J 0233 810300 8103(57 810434 810501 81050! 810(i3t 0703 0770 0837 (57 7 0904 0971 J039 1101 1173 1240 1307 1374 1441 1508 (57 8 1575 1042 1709 1776 1843 19'0 1977 2044 2111 2178 67 9 2245 2312 2379 2445 2512 2579 264f 2713 2780 2847 67 650 812913 812980 813047 813114 813181 813247 813314 813381 813448 813514 67 J 3581 3648 3714 3781 3848 3914 3981 4048 4114 4181 (57 2 4248 4314 4381 4447 4514 4581 4647 4714 4780 4847 (57 2 4913 4980 5046 5113 5179 5246 531- 5378 5445 5511 66 4 5578 5644 5711 5777 5843 5910 5976 6042 6109 6173 66 3 ' 6241 6308 6374 6440 6506 6573 6631 6705 6771 6838 66 6 G'J()4 0970 7036 7102 7169 7235 7301 7367 7433 7499 66 7505 7631 7698 7764 7830 7896 7962 8028 8094 8160 66 8 8236 8292 8358 8424 8490 8556 81522 8688 8754 882( 66 9 8885 8951 9017 9083 9149 9215 9281 9346 9412 9478 66 660 819544 819610 819676 819741 819807 819873 819939 820004 820070 8201 3f 66 1 820-201 820267 820333 t>2399 820464 820530 820595 0061 0727 0:92 66 2 0858 0924 0989 1055 1120 1186 1251 1317 1382 1448 66 3 1514 1579 1645 1710 1775 1841 1906 1972 2037 2103 C3 4 2168 2233 2-299 2364 2430 2495 2560 2626 2691 2756 65 5 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 65 6 3474 3539 3605 3670 3735 3800 3805 3930 3998 4001 65 7 4126 4191 425G 4321 438(5 4451 4516 4581 4646 4711 65 8 4776 4841 4906 4971 5036 5101 516(5 5231 5296 5361 65 9 5426 5491 5556 5621 5686 5751 5815 5880 5945 6010 65 670 826075 826140 826204 826269 826334 826399 82(5464 826528 826593 826658 65 1 6723 6787 6852 6917 6981 70-16 7111 7175 7240 7305 65 2 7369 7434 7499 7563 7628 7692 7757 7821 7886 7951 65 3 8015 8080 8144 8209 8273 8338 8-102 8467 8531 8595 64 4 8060 8724 8789 8853 8918 8982 9046 9111 9175 9239 64 5 9304 9368 6432 9497 9561 91525 9li!K) 9754 9818 9882 64 9947 83001 1 830075 830139 830204 830268 830332 830396 830460 830525 64 7 830589 0653 0717 0781 0845 0909 0973 1037 1102 1166 64 8 1230 1294 1358 1422 I486 1550 1614 1678 1742 1806 64 9 1870 1934 1998 2062 2126 2189 2253 2317 2381 2445 64 680 832509 832573 832637 832700 832704 832828 832892 832956 833020 833083 64 3147 3211 3275 3338 3402 3466 3530 3593 3657 3721 64 2 3784 3848 3912 3975 4039 4103 4166 4230 4294 4357 64 3 4421 4484 4548 4011 40,75 4739 4802 4866 4929 4993 64 4 5050 5120 5183 5247 5310 5373 5437 5500 55(5-1 5(527 63 5 5691 5754 5817 5881 5944 6007 6071 6134 6197 6261 63 6 6324 6337 6451 6514 6577 6(541 6704 6767 6830 6894 63 7 6957 7020 7083 7146 7210 7273 7336 7399 7462 7525 63 8 7388 7652 7715 7778 7841 7904 7967 8030 8093 8156 63 9 82J9 8282 8345 8408 8471 8534 8597 8660 8723 8786 (53 690 838849 838912 838975 839038 839101 839164 83922? 839289 839352 839415 63 1 9478 9541 9(504 96(5? 9729 9792 9855 9918 9981 840043 63 2 8401 )6 840169 840232 840294 840357 840420 840482 8411545 S401508 0(571 63 3 0733 079(5 0859 0021 0984 104(5 1109 1172 1234 121)7 63 4 1359 1422 14H5 1547 1610 1(572 1735 1797 18(50 1922 63 5 1985 2047 2110 2172 2235 2297 23(50 2422 2484 2517 (52 6 2609 2072 2734 2796 2859 2921 29H3 3046 3108 3170 (52 7 3233 3295 3357 3420 3482 3544 3(50(5 3869 3731 3793 (52 8 3855 3!) 18 39^0 4042 4104 41(56 4229 4291 4353 4415 (52 9 4477! 4539 4601 46154 4726 4788 4850 4912 4974 5036 62 No, | 3 I 4: J 5 D>tf. LOGARITHMS OF NUMBERS. 191 No. | | 1 3 I 4 6 | 7 | 8 | 9 |Diff. 700 84509R 845160 845222 845284 845346 845408 84547 845532 845594 84.5656 62 57J8 5780 5842 5904 5966 6028 ' 6090 6151 6213 6275 62 E 6337 6399 6461 6523 6585 6640 6708 6770 6832 6894 62 j 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 62 ^ 7573 7634 7696 7758 7819 7881 7943 8004 8066 8128 62 8189 8251 8312 8374 8435 849" 8559 8620 8(582 8743 62 ( 8805 8866 8928 8989 9051 9112 9174 9235 9297 93.58 61 9419 9481 9542 9604 9665 9726 9788 9849 9911 9972 61 8 85003:5 850095 R50156 850217 850279 850340 85040 8504C.-2 850524 850585 61 9 0646 0707 0769 0830 0891 0952 1014 1075 1136 1197 61 710 851258 851320 851381 851442 851503 R51564 851625 851686 851747 851809 61 1 1870 1931 19!t2 203 2114 2175 2236 2297 2358 2419 61 2 2480 2541 2602 2(563 2724 27R5 2H4f 2907 2968 3029 61 2 3090 3150 3211 3272 3333 3394 345. 3516 3577 3637 61 4 3698 3759 3820 3881 3941 4002 4063 4124 4i&5 4245 61 5 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 61 6 4913 4974 5034 5095 5156 5216 5277 5337 5398 5459 61 5519 5580 5640 5701 5761 5822 5882 5943 6003 6064 61 8 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 60 9 6729 6789 6850 6910 6970 7031 7091 7152 7212 7272 60 720 857332 857393 85745' 857513 857574 857634 857694 857755 857815 857875 60 1 7935 7995 8056 8U( 8176 823f 8297 8357 8417 8477 60 2 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 60 3 9138 9198 9258 9318 9379 9439 9499 9559 9619 9K79 60 4 9739 9799 9859 95* lh 9978 860038 860098 860158 860218 860278 60 5 860338 860398 P60458 860518 860578 0637 0697 0757 0817 0877 60 6 0937 0996 1056 1116 1176 1236 1295 1355 1415 1475 60 1534 1594 16.54 1714 1773 1833 1893 1952 2012 2072 60 8 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 60 y 2728 2787 2847 2906 29B6 3025 3085 3144 3204 3263 60 730 863323 863382 (753442 863501 863561 863620 863680 863739 863799 863858 59 J 3917 3977 4036 4096 4155 4214 4274 4333 4392 4452 59 2 4511 4570 4630 4fi) 4748 4808 486" 492( 4985 5045 59 3 5104 5163 5222 5282 5341 5400 5459 5519 5578 5637 59 4 5696 5755 5814 5874 5933 5992 6051 6110 6169 6228 59 5 G287 6346 6405 6465 6524 6583 6642 6701 6760 6819 59 6 6878 6937 6996 7055 7114 7173 723L 7291 7350 7409 59 7 7467 7526 7585 7644 7703 7762 7821 7K*0 7831 7998 59 8 8056 8115 8174 8233 8292 8350 8409 8468 8527 8586 59 9 8644 8703 8762 8821 8879 8938 8997 9056 9114 9173 59 740 869232 869290 869349 869408 869466 869525 869584 869642 869701 869760 59 1 9818 9877 9935 9994 870053 870111 870170 870228 870287 870345 59 370404 870462 870521 870579 0638 0696 0755 0813 0872 0930 58 3 0989 1047 1106 1164 1223 1281 1339 1398 14.56 1515 58 4 1573 1631 1690 1748 1806 1865 1923 1981 2040 2098 58 5 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 58 6 2739 2797 2855 913 2972 3030 3088 3146 3204 32(52 58 7 3321 3379 3437 3495 3553 3611 3669 3727 3785 3844 58 8 3902 39*50 4018 4076 4134 4192 4250 4308 4366 4424 58 9 4482 4540 4598 4656 4714 4772 4830 4888 4945 5003 58 750 875061 875119 875177 875235 875293 875351 875409 875466 875524 875582 58 J 5640 5698 5756 5813 5871 5929 5987 6045 6102 6160 58 2 6218 6276 6333 6391 0449 6507 6564 6622 6(580 6737 58 3 6795 6853 6'JK) 6968 7026 7083 7141 7199 7256 7314 58 4 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 58 5 7947 8004 8062 8119 8177 8234 8292 8349 8407 8464 57 6 8522 8579 8637 86JM 8752 8809 8866 8924 8981 9039 57 7 9096 9153 9211 9268 9!J25 9383 9440 9497 9555 9612 57 8 9069 9726 9784 9841 9898 9950 880013 880070 #0127 3801KJ 57 9880-.M2 88C299 880356 880413 880471 880528 0585 0642 0699 0756 i 57 fc. I o 192 LOGARITHMS OF NUMBERS. No I I 1 I 3 I 3 I 4: I 5 | G | 7 | 8 9 |DiS 760 880814 880871 880. >28 880985 881042 881099 881150 881213 881271 881328 57 1 1385 1442 149!) 155*6 1613 1670 1727 1784 1841 1898 57 2 1955 2012 2069 2120 2183 2240 2297 2354 2411 24(58 57 2 2525 2581 2(538 2(595 2752 2809 28(5(5 2923 2080 3037 57 4 3093 3150 3207 3-2(54 3321 3377 3434 3491 3548 3605 57 *j 3661 3718 3775 3832 3888 3945 4002 40.19 4115 4172 57 6 422 4285 4342 4399 4455 4512 45(59 4625 4(582 4739 57 7 4?!).-> 4852 4909 4905 5022 5078 5135 5192 5248 5305 57 8 5301 5418 5474 5531 5587 5644 5700 5757 5813 5870 57 9 5926 5983 6039 609(5 6152 6209 6265 6321 (5378 6434 56 770 886491 886547 886004 880(5(50 886716 886773 886829 886885 880942 886998 50 1 7054 7111 7167 72'VJ 7880 733(5 7392 7449 7505 7561 56 2 7617 7674 773i ) 7780 7842 7898 7955 8011 8067 8123 50 3 8179 8236 8292 8348 8404 8460 8510 8573 8629 86a r > 515 4 8741 8797 8853 8909 8965 9021 9077 9134 9190 924(5 56 5 9302 9358 9414 9470 9526 9582 9(538 9094 9750 9800 56 6 98G2 9918 9974 890030 890086 890141 890197 890253 890309 890305 56 7 890421 890477 890533 0589 9(545 0700 0756 0812 08(58 0924 56 8 0980 1035 1091 1147 1203 1259 1314 1370 1426 1482 56 9 1537 1593 1649 1705 1760 18115 1872 1928 1983 2039 56 780 892095 892150 892206 892202 892317 892373 892429 892484 892540 892595 56 1 2051 2707 2762 2818 2873 2929 2985 3040 309(5 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 4048 4704 4759 4814 55 5 4870 4925 4980 503(5 5091 514(5 5201 5257 5312 5367 55 6 5423 5478 5533 5588 5644 5699 5754 5809 5804 5920 55 7 5975 6030 6085 6140 6195 6251 6300 6301 6416 6471 55 8 6526 6581 6636 6692 6747 6802 6857 6912 6907 7022 55 9 7077 7132 7187 7242 7297 7352 7407 7462 7517 7572 55 790 897627 897682 897737 897792 897847 897902 897957 898012 898007 898122 55 1 81 76 8231 8286 8341 8396 8451 8506 8561 8(515 8(570 55 2 8725 8780 8835 &890 8944 8999 9054 9109 91(54 9218 55 3 9273 9328 9383 9437 9492 9547 9602 9656 971 1 9766 55 4 9821 9875 9930 9985 90003:) 900094 900149 900203 900258 900312 55 5 90031)7 900422 900476 900531 0580 0040 0695 0749 0804 0859 55 6 0913 09(38 1022 1077 1131 1180 1240 1295 1349 1404 55 7 1458 1513 1567 1622 1676 1731 1785 1840 1894 1948 54 8 2003 2057 2112 2160 2221 2275 2329 2384 2438 ' 2492 54 9 2547 2601 2655 2710 2764 2818 2873 2927 2981 3036 54 800 903090 903144 903199 903253 903307 903301 903416 903470 903524 903578 54 1 3633 3687 3741 3795 3849 3904 39.58 4012 4060 4120 54 2 4174 4229 4283 4337 4391 4445 4499 4553 4607 4001 54 3 4716 4770 4824 4878 4932 498(5 5040 5094 5148 5202 54 4 5256 5310 5364 5418 5472 552(5 5580 5634 5688 5742 54 5 5796 5850 5904 5958 6012 6066 6119 6173 6227 6281 54 6 6335 6389 6443 6497 6551 6(504 6058 6712 6760 6820 54 7 6874 6927 6981 7035 7089 7143 7196 7250 7304 7358 54 g 7411 7465 7519 7573 7(52(5 7680 7734 7787 7841 7895 54 9 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 54 810 908485 908539 J08592 90804(5 908099 908753 908807 908860 908914 908967 54 1 9021 9074 9128 9181 9335 9289 934-' 9390 9449 9503 54 2 8356 9610 9663 9716 9770 9823 9877 9930 9984 910037 53 3 9J0091 910144 J10197 910251 910304 910358 9101 1 1 9104(54 910518 0.-.71 53 4 0024 0(578 0731 07H4 0838 0891 0944 0998 1051 1104 53 5 1158 1211 1264 1317 1371 1424 1177 1530 1584 1037 53 6 1690 1743 1797 1850 1903 1950 2009 2003 2110 2109 53 7 2222 2275 2328 2381 2435 2488 2541 2594 2047 2700 53 8 27531 2806 2859 2913 2906 3019 3072 3125 3178 3231 53 9 32841 3337 3390 3443 3496 3549 3(502 3055 3708 3761 53 No. I | 1 | 3 5 | 6 | 7 I 8 | LOGARITHMS OF NUMBERS. 193 | 2 | 3 | 4: | 5 I 7 I 8 | 9 820 913814 913867 913920 913973 914026 914079 914132 914184 914237 914290 1 4343 4396 4449 4502 4555 4(508 4(5(50 4713 4766 4819 2 4872 4925 4977 5030 5083 5136 5189 5241 5294 5347 3 54(H) 5453 5505 5558 5611 5664 5716 57(59 5822 5875 4 5927 5980 6033 (5085 6138 6191 (5243 6296 6349 6401 5 6454 6507 6559 6612 6664 6717 6770 6822 6875 6927 6 6980 7033 7085 7138 7190 7243 7295 7348 7400 7453 7 750G 7558 7611 7663 771(5 7768 7820 7873 7925 7978 8 8030 8083 8135 8188 8240 8293 8345 8397 8450 8502 9 8555 8607 8659 8712 8764 8816 8869 8921 8973 9026 830 919078 919130 919183 919235 919287 919340 919392 919444 919496 918549 1 9001 9653 97061 9758 98 10 9862 9914 9967 920019 920071 2 920123 920176 920228 ,920280 920332 920384 92043(5 920489 0541 0593 3 OG45 0697 0749 0801 0853 0906 0958 1010 1062 1114 4 116G 1218 1270 13-2^2 1374 1426 1478 1530 1582 1634 5 I:t2 7581 7630 7679 7728 7777 7826 7875 49 7 7924 7973 8022 8070 8119 8168 8217 8260 8315 8364 49 8 8413 8462 8511 8560 8609 8657 8700 8755 8804 8853 49 9 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 49 890 1 949390 9878 949439 9926 949488 9975 949530 950024 949585 950073 949634 950121 949083 950170 949731 950219 949780 9502(57 949829 95031(5 49 49 2 9503(55 950414 950402 0511 0560 0608 0657 0706 0754 0803 49 3 0851 0900 0949 0997 1046 1095 1143 1192 1240 1289 49 4 1338 1386 1435 1483 1532 1580 1629 1677 1726 1775 '49 s 1823 1872 1920 1969 2017 2066 2114 2103 2211 22(50 48 f, 2308 2356 2405 453 2502 2550 2599 2047 2696 2744 43 7 2792 2841 2889 2938 2980 3034 3083 3131 3180 3228 48 8 3276 3325 3373 3421 3470 3518 3566 3015 36(53 3711 48 9 3760 3808 3856 3905 3953 4001 4049 4098 4146 4194 48 900 954243 954291 954339 954387 954435 954484 954532 954580 954628 954677 48 1 4725 4773 4821 4869 491S 4966 5014 5002 51 10 5158 48 2 5207 5255 5303 5351 5399 5447 5495 5543 5592 5640 48 3 5688 5736 5784 5832 5880 5928 5976 6024 6072 6120 48 4 6168 6216 0205 6313 0301 6409 6457 6505 6553 6601 48 5 6049 6(i97 6745 6793 6840 6888 6936 6984 7032 7080 48 6 7128 7176 7224 7272 7320 736K 7416 7464 7512 7559 48 7 7007 7655 7703 7751 77-J'J 7847 7894 7942 7990 80:18 48 8 8086 8134 8181 8229 8277 8325 8373 8421 8408 8516 48 9 8564 8012 8059 8707 8755 8803 8850 8898 8940 8994 48 9 JO 959041 959089 959137 959185 959232 959280 959328 959375 959423 959471 48 ] 9518 9500 9014 9061 9709 9757 9804 9852 9900 9947 48 2 9995 960042 900090 960138 91)0185 960233 9602HJ 96032H 960370 900423 48 3 900471 0518 0500 0013 0601 0709 0750 0804 0851 0899 48 4 0940 OUM 1041 10S9 1130 1184 1231 1279 1320 1374 48 5 1421 1469 1516 1563 1011 1058 1700 1753 1801 1848 47 c 1895 1943 1990 2038 2085 2132 2180 2227 2275 2322 47 7 2369 2417 2404 2511 2559 2000 2053 2701 2748 2795 47 8 2843 2890 2937 2985 3032 3079 3120 3174 3221 32(58 47 i) 3316 3363 3410 3457 3504 3552 3599 3646 3093 3741 47 920 963788 963835 963882 963929 963977 964024 964071 964118 964165 964212 47 ] 42tK) 4307 4354 4401 4448 4495 4542 4590 4037 4084 47 4731 4778 4825 4872 4919 4960 5013 5001 5108 51.55 47 3 5202 5249 5296 5343 5390 5437 5484 5531 5578 5025 47 4 5072 5719 5766 5813 5860 5907 5954 6001 604H 6095 47 5 C142 6189 6236 G2S3 632J 6376 6423 6470 6517 (55(54 47 6 6611 6658 6705 6752 6799 6845 6892 6939 698(5 7033 47 7 7080 7127 7173 7220 7207 7314 7361 7408 7454 7501 47 8 7548 7595 7042 7688 7735 7782 7829 7875 7922 7909 47 9 8016 8062 8109 8156 8203 8249 8296 8343 8390 8430 47 930 j 968483 8950 968530 8996 968576 9043 968623 9090 968670 9136 968716 9183 968763 9229 968810 9276 968856 9323 968903 9369 47 47 9416 9403 9509 9556 9002 9(549 9Ji!)5 9742 9789 9835 47 3 9882 9928 9975 970021 970008 9701 14 970161 J70207 970254 970300 47 4 970347 970393 970440 0480 0533 0579 0020 0072 0719 0765 46 5 U8J2 0858 0904 0951 0997 1044 1090 1137 1183 1229 46 6 1276 1322 1309 1415 1461 1508 1554 1(50] 1647 IfiJM 46 7 1740 1786 1832 1879 1925 1971 2018 2004 2110 2157 46 8 2203 2249 2295 2342 2388 2434 2481 2527 2573 2619 46 9 2666 2712 2758 2804 2851 2897 29431 2989 3035 3082 46 No.) | 1 i 13|4|5|6|7|8|&|I>* LOGARITHMS OF NUMBERS. 195 Na| 6 | 3 | | 7 940 973128,973174 )73220 )73206 973313 9733,39 973405 973451 973497 973543 46 J 3590 3636 368-2 3728 3774 3820 3866 3913 3959 4005 46 2 4051 4097 4143 4189 4235 4281 4327 4374 4420 4466 46 3 4512 4558 4604 4650 4696 4742 4788 4834 4880 4926 46 4 4972 5018 5064 5110 515C 5202 5248 5294 5340 5386 46 5 5432 5478 5524 5570 5616 5662 5707 5753 5799 5845 46 6 5891 5937 5983 6029 (5075 6121 6167 6212 6258 (5304 46 7 6350 6396 6442 6488 6533 6579 6625 6671 6717 6763 46 8 6808 6854 6900 6946 6992 7037 7083 7129 7175 7220 46 9 7206 7312 7358 7403 7449 7495 7541 7586 7632 7678 46 950 977724 977769 977815 977861 977906 977952 977998 978043 978089 J78135 46 J 8181 8226 8272 8317 8363 8409 84.34 8500 8546 8591 46 2 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 46 3 9093 9138 9184 9230 9275 9321 9366 9412 9457 9503 46 4 9548 9594 9639 9(>85 9730 9776 9821 9867 9912 9958 46 5 980003 980049 )80094 980140 (80185 980231 980276 980322 980367 980412 45 6 0458 0503 0549 0594 0640 0685 0730 0776 0821 0867 45 7 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 45 8 1366 1411 1456 1501 1547 1592 1637 1683 1728 1773 45 9 1819 1864 1909 1954 2000 2045 2090 2135 2181 2226 45 960 982271 982316 982362 J82407 982452 982497 982543 982588 982633 982678 45 1 2723 2769 2814 2859 2904 2949 2994 3040 3085 3130 45 2 3175 3220 3265 3310 3356 3401 3446 3491 3536 358 J 45 3 3026 3671 3716 3762 3807 3852 3897 3942 3987 4032 45 4 4077 4122 4167 4212 4257 4302 4347 4392 4437 4482 45 5 4387 4572 4617 4662 4707 4752 4797 4842 4887 4932 45 G 4977 5022 5067 5112 5157 5202 5247 5292 5337 5382 45 7 5426 5471 5516 5561 5606 5651 5696 5741 . 5786 5830 45 8 5875 5920 5965 6010 6055 6100 6144 6189 6234 6279 45 9 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 45 970 986772 986817 986861 986906 986951 986996 987040 987085 987130 987175 45 1 7219 7264 7309 7353 7398 7443 7488 7532 7577 7622 45 2 7666 7711 7756 7800 7845 7890 7934 7979 8024 8068 45 3 8113 8157 8202 8247 8291 8336 8381 8425 8470 8514 45 4 8559 8604 8048 8693 8737 8782 8826 8871 8916 8960 45 5 9005 9049 9094 9138 9183 9227 9272 9316 9361 9405 45 c 9450 9494 9539 9583 9628 9672 9717 9761 9806 9850 44 7 9895 9939 9983 990028 9900/2 990117 990161 990206 990250 090294 44 81990339 990383 990428 0472 0516 0561 0005 0650 0694 0738 44 9 0783 0827 0871 0916 0960 1004 1049 109*3 1137 1182 44 980 991226 991270 991315 991359 991403 991448 991492 991536 991580 991625 44 . I 1669 1713 1758 1802 1846 1890 1935 1979 2023 2067 44 2 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 44 3 2554 2598 2642 2686 2730 2774 2819 2863 2907 2951 44 4 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 44 5 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 44 6 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 44 7 4317 4361 4405 4449 4493 4537 4.381 4625 4669 4713 44 8 4757 4801 4H45 4889 4933 4977 5021 5065 5108 5152 44 9 5196 5240 50*4 5328 5372 5416 5460 5504 5547 5591 44 990 995635 995679 995723 995767 995811 995854 995898 995942 995986 996030 44 J 6074 6117 6161 6205 6249 6293 6337 6380 6424 6468 44 2 6512 6555 6599 6643 6687 6731 6774 6818 6802 6906 44 3 6949 6993 7037 7080 7124 7168 7255 7299 7343 44 4 7386 7430 7474 7517 7561 7605 7648 7698 . 7736 7779 44 5 7823 7867 7910 71)54 7998 8041 8085 8129 8172 8216 44 6 8259 8303 8347 8390 8434 8477 8521 8564 8608 8652 44 7 8695 8739 8782 8826 8869 8913 8956 9000 9043 9087 44 8 9131 9174 9218 9261 9305 9348 939-2 9435 9479 9522 44 S 9565 9609 9652 9696 9739 9783 9826 9870 9913 9957 43 Mu I O | 1 [ 3 5 I 6 8 I 9 |i>ift 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. 198 (0 Degree.) LOGARITHMIC SINES, COSINES, ETC. M. | Sine | D. Cosine | D. | Tnng. D. Cotanff. o-oooooo 10-000000 o-oooooo Infinite. 1 6-4637-20 501717 000000 00 6-41537-26 501717 1 3-536-274 2 7(54756 293485 000000 00 76475(5 293483 235-244 3 940H47 2 18231 0(10(1(10 00 940847 208231 059153 4 7-065786 101517 00(1000 00 7-065786 1(51517 12-934214 5 ] 6269(5 131968 000000 00 162*596 1319(59 837304 6 241877 111575 9-999999 01 241878 111578 75*1-2-2 7 308824 96<>53 999999 or 308825 99653 691175 8 366816 85254 999999 01 366817 85-254 6331*3 9 417968 76263 999999 01 417970 76-263 582030 10 463725 68988 999998 01 463727 68988 536-273 11 7-505118 62981 9-999998 01 7-505120 6298J 12-494*80 12 542906 57936 999997 01 542909 57933 457091 13 577(5(58 53641 999997 01 577672 53(542 4-223-28 14 609853 49938 999996 01 609857 49939 390143 15 639816 46714 999996 01 639820 46715 3(50180 16 667845 43881 999995 01 6(57849 43882 33-2151 17 694173 41372 999995 01 694179 41373 305821 18 718997 39135 999994 0! 719003 39136 2*0997 19 742477 37127 999993 01 742484 37128 257516 20 764754 35315 999993 01 764761 35136 235-239 21 7-785943 33672 9-999992 01 7-785951 33673 12-214049 22 806146 32175 999991 01 806 J 55 32176 193845 23 825451 30805 999990 01 825460 30806 174540 24 843934 29547 999989 02 843944 29549 15(5056 25 8616(52 28388 999988 02 861674 28390 13*32(5 2(5 878(595 27317 999988 02 878708 27318 121292 27 895085 2(53-23 999987 02 895099 20325 104901 28 910879 25399 9! 19986 02 910894 25401 0*9106 29 926119 24538 999985 02 9S6IU4 24540 0738(56 30 940842 23733 999983 02 940858 23735 059142 31 7-955082 22980 9-999982 02 7-955100 22981 12-044900 32 9(58870 22-273 99998J 02 9(58889 22273 031111 33 982233 21(508 999980 02 982253 21(510 017747 34 995 198 20981 999979 02 995219 '20! 1*3 004781 35 8-007787 20390 999977 02 8-007*09 20392 11-99-2191 36 020021 19831 999976 02 020045 19833 979955 37 031919 19302 999975 02 03 lit 15 19305 9(58055 38 043501 18801 999973 02 043527 1*803 95(5473 39 054781 18325 999972 02 054809 18327 945191 40 065776 17872 999971 02 065806 17874 934194 41 8-07(5500 17441 9-999969 02 8-070531 17444 11-923469 42 0869(55 17031 999968 02 08(5997 17034 913003 43 097183 1(5(539 999986 02 097-217 16642 90-27*3 44 1071(57 162(55 9999(54 03 107-202 16-2(58 892797 45 1J6926 15908 999903 03 1 169(53 15910 8*3037 46 J 26471 155(56 999961 03 126510 155(58 873490 47 135*10 15238 999959 0.) 135*51 15-241 864149 48 1441153 14924 999958 03 14499(5 149-27 855004 49 153907 14622 99995(5 03 153952 14(527 846048 50 162681 14333 999954 03 162727 J4336 837273 51 8-171280 ' 14054 9-999952 03 8-171328 14057 11-828672 52 179713 13786 999950 03 1797(53 13790 820237 53 187985 13529 999948 03 18*036 13532 811964 54 196102 13280 999946 03 196156 13284 803844 55 204070 13041 999944 03 2041-2(5 13044 795874 56 211*95 12810 999942 04 '211953 1-2*14 7HH047 57 219581 12587 999940 04 219641 12590 7*0359 58 2-27134 1-237-2 999938 04 227195 1-237(5 772805 59 234557 12164 99993(5 04 234621 1-21(18 765379 60 241855 11963 999934 04 241921 1 1967 75*079 | Cosine Sine | Cotang. | ) Tang. | M. 89 Degrees. LOGARITHMIC SINES, COSINES, ETC. (1 Degree.) 199 M. Sine D. Cosine D. | Tang. D. Golan*. | 8-241855 11963 9-999934 04 8-241921 11967 11-758079 1 249033 1 1768 999932 04 249102 11772 750898 2 256094 11580 999!29 04 256165 11584 743835 3 203042 11398 999927 04 2f)31 15 11402 736885 4 269881 11221 999925 04 269956 11225 730044 5 276614 11050 999922 04 276691 11054 723309 6 233243 10883 999920 04 283323 10887 716677 289773 10721 999918 04 289856 10726 710144 8 296207 10565 999915 04 296292 10570 703708 9 302546 10413 999913 04 302634 10418 697366 ]<> 308794 10266 999910 04 308884 10270 691116 11 8-314954 10122 9-999907 04 8-315046 10126 11-684954 19 321027 9982 999905 04 321122 9987 678878 13 327016 9847 999902 04 327114 9851 672886 14 332924 9714 999899 05 333025 9719 666975 IS 338753 9586 999897 05 338856 9590 661144 16 344504 9460 999894 05 344610 9465 655390 17 350181 9338 999891 05 350289 9343 649711 18 355783 9219 999888 05 355895 9224 644105 19 361315 9103 999885 05 361430 9108 638570 20 366777 8990 999882 05 366895 8995 633105 21 8-372171 8880 9-999879 05 8-372292 8885 11-627708 22 377499 8772 999876 05 377622 8777 622378 23 382762 8667 999873 05 382889 8672 617111 24 387962 8564 999870 05 388092 8570 611908 SS 393101 8464 999867 05 393234 8470 606766 26 398179 8366 999864 05 398315 8371 601685 27 403199 8271 999861 05 403338 8276 596662 28 408161 8177 999858 05 408304 8182 591696 29 413068 8086 999854 05 413213 8091 586787 30 417919 7996 999851 06 418068 8002 581932 31 8-422717 7909 9-999848 06 8-422869 7914 11-577131 32 427462 7823 999844 06 427618 7830 572382 33 432156 7740 999841 06 432315 7745 567685 34 436800 7657 999838 06 436962 7663 563038 35 441394 7577 999834 06 441560 7583 558440 3f> 445941 7499 999831 06 446110 7505 553890 37 450440 7422 999827 06 450613 7428 549387 38 454893 7346 999823 06 455070 7352 544930 39 459301 7273 999820 06 459481 7279 540519 40 463665 7200 999816 06 463849 7206 536151 41 8-467985 7129 9-999812 06 8-468172 7135 11-531828 42 472263 7060 999809 06 472454 7066 527546 43 476498 6991 999805 06 476693 6998 523307 44 480693 6924 999801 06 480892 6931 519108 45 484848 6859 999797 07 485050 6865 514950 41) 488963 6794 999793 07 489170 6801 510830 47 493040 6731 999790 07 493250 6738 506750 4H 497078 6669 999786 07 497293 6676 502707 49 501080 6608 999782 07 501298 6615 498702 50 505045 6548 999778 07 505267 6555 494733 51 8-508974 6489 9-999774 07 8-509200 6496 11-490800 52 512867 6431 999769 07 513098 6439 486902 53 516726 6375 999765 07 516961 6382 483039 54 520551 6319 999761 07 520790 6326 479210 55 524343 6264 999757 07 524586 6272 475414 50 528102 6211 989753 07 528349 6218 471651 57 531828 6158 999748 07 532080 6165 467920 58 535523 6106 999744 07 535779 6113 464221 59 539186 6055 999740 07 539447 6062 460553 GO 542819 ' 6004 999735 07 543084 6012 456916 | Cosine | | BUM Cotang. i I Tang. 88 Degrees. 200 (2 Degrees.) LOGARlTmilC SINES, COSINES, ETC. M. | Sine D. | Cosine D. | Tan?. | D. Cotan ? . | 8-542819 6004 9-999735 07 8-543084 6012 11 45(i916 (H) 1 54G422 5955 999731 07 546691 5962 453309 59 g 549995 5906 999796 07 550268 5914 449732 58 3 55X539 5858 91)9722 08 553817 5866 446183 57 4 557054 5811 999717 08 557336 5819 442664 56 5 560540 57G5 999713 08 56082S 5773 439172 55 6 563999 5719 999708 08 564291 5727 435709 54 7 567431 5674 990704 08 567727 5682 432273 53 8 570836 5630 999699 08 571137 5638 428863 52 9 574214 5587 999694 08 574520 5595 425480 51 W 5/ <566 5544 999689 08 577877 5552 422123 50 11 8-580892 5OJ2 9-999685 08 8-581208 5510 11-418792 49 12 584193 5460 999680 08 584514 5468 .415486 48 13 587469 5419 939075 08 587795 5427 412205 47 14 590721 5379 999070 08 591051 5387 408949 46 15 593948 5339 999665 08 594283 5347 4057J7 45 16 597152 5300 999660 08 597492 5308 402508 44 17 600332 5261 999655 08 600077 5270 399323 43 18 603489 5223 999650 08 603839 5232 396361 42 19 606623 5186 999645 09 606978 5194 393022 41 20 609734 5149 999640 09 610094 5158 389906 40 21 8-612823 5112 9-999635 09 8-613189 5121 11-386811 3D 2 615891 5076 999629 09 616262 5085 383738 38 23 618937 5041 999624 09 619313 5{50 380687 37 24 621962 5006 999619 09 622343 5015 377(557 36 25 624965 4972 999614 09 625352 4981 374648 35 26 627948 4938 999608 09 628340 4947 371660 34 27 630911 4904 999603 09 631308 4913 368C92 33 28 633854 4871 999597 09 634256 4880 365741 32 29 (536776 4839 999592 09 6371*4 4848 362816 31 3') 639680 4806 999586 09 640093 4816 359907 30 31 8 642563 4775 9-999581 09 8-642982 4784 11-357018 29 32 645428 4743 999575 09 645853 4753 354147 28 33 648274 4712 999570 09 648704 4722 351296 27 34 651102 4682 999564 09 651537 4691 348163 26 35 65391 1 4652 999558 10 654:152 4661 345(548 25 36 656702 4622 999553 10 657149 4631 342851 24 37 659475 4592 999547 10 659928 4602 340072 23 38 662230 4563 999541 10 662689 4573 33731 1 22 39 664968 4535 . 999535 10 665433 4544 334567 21 40 667689 4506 999529 10 668160 4526 331840 20 41 8-670393 4479 9-999524 10 8-670870- 4488 11-329130 19 42 673080 4451 999518 10 673563 4461 326437 18 43 675751 4424 999512 10 676239 4434 323761 17 44 678405 4397 999506 10 678900 4417 32 J 100 16 45 681043 4370 999500 10 681544 4380 318456 15 40 683665 4344 999493 10 684172 4354 315838 14 47 686272 4318 999487 10 686784 4328 313216 13 48 688863 4292 999481 10 689381 4303 310619 12 49 691438 4267 999475 10 691963 4277 308037 11 50 693998 4242 999469 10 694529 4252 3(15471 JO 51 8-696543 4217 9-999463 11 8-697081 4228 11-302919 9 52 699073 4192 999456 11 699617 4203 300383 8 53 701589 4168 999450 11 702139 4J79 2978(51 7 54 704090 4144 999443 11 704646 4155 295354 6 55 706577 4121 999437 11 707140 4132 2928(50 5 56 709049 4097 999431 11 709618 4108 290382 4 57 711507 4074 999424 11 712083 4085 287917 3 58 713952 4051 999418 11 714534 4062 285465 2 59 71(5383 4029 9! (9411 11 7l(i972 4040 2H3028 ] 60 ' 718800 4006 999404 11 719396 4017 280604 | Cosine 1 I 87 Degrees. Cola.,-. I Tang. LOGARITHMIC SINES, COSINES, ETC. (3 Degrees.) 201 M. I Sine | D. | Cosine | D. | Tang. | D. | Cotnnp. | 1) 8-718800 4006 9-999404 11 8-719396 4017 11-280004 60 1 721204 3984 999398 11 721806 3995 ' 278194 59 o 723.11)5 3962 999391 11 724204 3974 275796 58 3 7251)72 3941 999384 11 720588 3952 273412 57 4 723337 3919 999378 11 728959 3930 271041 56 5 731)033 3898 999371 11 731317 3909 268083 55 6 733027 3877 999304 12 733063 3889 2G6337 54 7 735354 3857 999357 12 735996 3868 264004 53 8 737GG7 3836 999350 12 738317 3848 201083 52 9 73'J9:;9 3816 999343 12 740026 3827 259374 51 10 7422-59 3796 999336 12 742922 3807 257078 50 Jl 8-744536 3776 9-9993-29 12 8-745207 3787 11.254793 49 12 746802 3756 999322 12 747479 3768 252521 48 13 749055 3737 999315 12 749740 3749 250200 47 14 751297 3717 999308 12 751989 3729 248011 46 15 753528 3098 999301 12 754227 3710 245773 45 16 755747 3079 999294 12 750453 3092 243547 44 17 757955 3001 999286 12 758668 3073 241332 43 18 700151 3642 999279 12 760872 3655 239128 42 19 702337 3024 999272 12 763065 3636 236935 41 20 764511 3606 9992G5 12 765246 3618 234754 40 21 8-700075 3588 9-999257 12 8-71.7417 3600 11-232583 39 22 708828 3570 999250 13 709578 3583 230422 38 23 770970 3553 999242 13 771727 3565 228273 37 24 773101 3535 999235 13 773866 3548 226134 36 25 775223 3518 999227 13 775995 3531 224005 35 26 777333 3501 999220 13 778114 3514 221886 34 27 779434 3484 999212 13 780222 3497 219778 33 28 781524 3407 999205 13 782320 3480 217680 32 29 783605 3451 999197 13 784408 3404 215592 31 30 735G75 3431 999189 13 786486 3447 213514 30 31 8-787736 3418 9-999181 13 8-788554 3431 11-211440 29 32 789787 3402 999174 13 7900 J 3 M14 209387 28 33 791828 3386 999166 13 7926G2 3399 207338 27 34 793859 3370 999158 13 794701 3383 205299 26 35 795881 3354 999150 13 790731 3368 203209 25 3(5 797894 3339 999142 13 798752 3352 201248 24 37 799897 3323 999134 13 800703 3337 199237 23 38 801892 3308 999126 13 802765 3322 197235 22 39 803876 3293 999118 13 804758 3307 195242 21 40 805852 3278 999110 13 800742 3292 193258 20 41 8-807819 3263 9-999102 13 8-808717 3278 11-191283 19 42 809777 3249 999094 14 810083 3202 189317 18 43 811726 3234 999086 14 812641 3248 187359 17 44 813007 3219 999077 14 814589 3233 185411 16 45 815599 3205 999069 14 816529 3219 183471 15 46 817522 3191 999061 14 818461 3205 1H1539 14 47 819436 3177 999053 14 820384 3191 179010 13 48 821343 3103 999044 14 822298 3177 177702 12 49 823240 3149 999036 14 824205 3163 175795 11 50 825130 3135 999027 14 826103 3150 173897 10 51 8-827011 3122 9-999019 14 8-827992 3136 11-172008 9 52 828884 3108 999010 14 829874 3123 170120 8 53 830749 3095 999002 14 831748 3110 108252 7 54 832007 3082 998993 14 833013 3096 166387 6 55 834450 3009 998984 14 835471 3083 164529 5 56 830297 3050 998970 14 837321 3070 162(579 4 57 838130 3043 998967 15 839103 3057 100837 3 58 839956 3030 998958 15 840998 3045 159002 2 59 841774 3017 998950 15 842825 3032 157175 1 60 843585 3000 998941 15 844644 3019 1 155356 | Cosine J Tang. | M. 80 Degrees. 202 (4 Degrees.) LOGARITHMIC SIXES, COSINES* ETC. M. | Sine D. | Cosine I). Tan. 1 D. Cotang. U 8-843585 3005 9-998941 15 8-844044 3D 19 11-155356 60 1 845387 2992 998932 15 84G455 3007 153545 59 a 847183 2980 998923 15 848260 2395 151740 58 3 848971 29G7 998914 15 850057 2982 149943 57 4 850751 2955 998905 15 851846 21)70 148154 56 5 852525 21)43 91)8896 15 853028 21)58 140372 55 () 854291 2631 998887 15 855403 21)46 144597 54 7 850049 2919 998878 15 857171 2935 J 42829 53 8 857801 2D07 9988G9 15 858932 21)23 1410(58 52 9 859546 28 JO 998860 15 8G3G8G 2!) 11 139314 51 10 861283 2884 998851 15 802433 2330 137507 50 11 8-863014 2873 9-998841 15 8-804173 2888 11135827 49 1-2 864738 2801 998832 15 605906 2877 134094 48 13 860455 2850 998823 16 837032 2300 1323G8 47 14 8681 05 2839 998813 1G 8(39351 2854 130649 46 15 869HG8 2828 998804 1G 8710G4 2343 128936 45 16 871505 2817 998795 1G 872770 2832 127230 44 37 873255 2800 9D8785 1G 8744G9 2821 125531 43 18 874938 2795 99877G 1G 870102 2811 123838 42 n 87GG15 278(5 998766 10 877849 2800 122151 41 20 878285 2773 998757 16 879529 2789 120471 40 21 8-879945) 2703 9-998747 10 8-881202 2779 11-118798 39 2-2 881007 2752 998738 10 882809 2708 117131 38 215 883258 2742 998728 10 884530 2758 115470 37 24 884903 2731 998718 10 880 185 2747 113815 36 2.1 88(3542 2721 998708 1G 887833 2737 112107 35 28 888174 2711 998699 16 889476 2727 1 10524 34 27 889801 2700 998G89 16 891112 2717 108888 33 28 891421 2G90 998079 16 892742 2707 107258 32 29 893035 2G80 998G69 17 894306 2697 105634 31 30 894643 2G70 998G59 17 895984 2087 10401G 30 31 8-896246 2000 9-998G49 17 8-897596 2677 11-102404 29 32 897842 2051 998639 17 899203 2GG7 100797 28 33 899432 2041 998G29 900803 2058 099197 27 34 901017 2031 998019 7 902398 2048 097G02 26 35 902596 2622 998G09 7 903987 2038 090013 25 36 904 1G9 2012 998599 905570 2029 094430 24 37 905736 2G03 998589 7 907147 2020 092853 23 38 907297 2593 998578 7 908719 2010 091281 22 30 908853 2584 9985G8 7 910285 2001 089715 l 40 910404 2575 998558 17 911846 2592 088154 20 41 8-911949 25GG 9-998548 17 8-913401 2583 11-080599 19 42 913488 255G 998537 17 914951 2574 0850 4 9 18 43 915022 2547 998527 17 91(3495 2505 083505 17 44 910550 2538 9985 1C tiB 918034 2550 081906 16 45 918073 2529 998500 18 9195G8 2547 080432 15 46 919591 2520 998495 18 921096 2538 078904 14 47 921103 2512 998485 18 922019 2530 077381 13 48 922010 2503 998474 18 924130 2521 0758(34 12 4!) 924112 2494 9984G4 18 925649 25 J 2 074351 11 50 925G09 2480 998453 18 92715G 2503 072844 10 51 8-927100 2477 9-998442 18 8-928058 2495 11-071342 9 52 928587 24(59 998431 18 930155 248G 069845 8 53 930008 2400 998421 18 931047 2478 068353 7 54 931544 2452 998410 18 933134 2470 OG68GG 6 5.") 933015 2443 998399 18 934G1G 24G1 005384 5 56 934481 2435 998388 18 930093 2453 003907 4 57 935942 2427 998377 18 9375G5 2445 OG2435 3 58 937398 2419 9983GG 18 939032 2437 000908 2 59 938850 2411 998355 18 940494 2430 059500 1 GO 94029G 2403 998344 18 941952 2421 058048 | Cosine Sine | | Coding. 85 Degrees | Tang. | 1C, LOGARITHMIC SINES, COSINES, ETC. (5 Degrees.) 203 M. Sine D. Cosine | D. | Tang. L>. Cotanj?. ! 8-940296 2403 9-998344 19 8-941952 2421 11 058048 60 1 1)4)738 2394 9! 18333 19 943404 2413 05651 K> 59 2 943174 2387 l)>)83-.>2 19 944852 2405 055148 58 3 944006 2379 99831 1 19 946295 2397 053705 57 4 94GOS4 2371 998300 19 947734 2390 052206 56 5 947456 2363 998289 19 949168 2382 050832 55 6 948874 2355 998277 19 950597 2374 049403 54 7 950287 2348 998266 19 952021 2366 047!79 53 8 951696 2340 998255 19 953441 2360 046559 52 9 953100 2332 998243 19 95485(5 2351 045144 51 10 954499 23-25 998232 19 951)267 2344 043733 53 11 8-9558D4 2317 9-998220 19 8-957674 2337 11042326 49 12 957284 2310 998209 19 959075 2329 040925 48 13 958670 2302 91)8197 19 960473 2323 039527 47 14 960052 2295 9118186 19 961866 2314 038134 46 15 961429 2288 998174 19 963255 2307 036745 45 16 962801 2280 998163 19 964639 2300 035361 44 n 964170 2273 998151 19 966019 2293 033981 43 J8 965534 2266 998139 20 967394 2286 032606 42 19 966893 2259 998128 20 968766 2279 031234 41 20 968249 2252 998116 20 970133 2271 029867 40 21 8-969600 2244 9-99H104 20 8-971496 2265 11028504 39 93 970947 2238 998092 20 972855 2257 027145 38 23 972289 2231 998080 20 974209 2251 025791 37 24 973628 2224 998068 20 975560 2244 024440 36 25 974962 . 2217 998056 20 976906 2237 023094 35 2(i 976293 2210 998044 20 978248 2230 021752 34 27 977619 2203 998032 20 979586 2223 020414 33 28 978941 2197 998020 20 980921 2217 019079 32 29 980259 2190 998008 20 982251 2210 017749 31 30 981573 2183 997996 20 983577 2204 016423 30 31 8-982883 2177 9-997984 20 8-984899 2197 11-015101 29 32 984189 2170 997972 20 98(5217 2191 013783 28 33 985491 2163 997959 20 987532 2184 012468 27 34 986789 2157 997947 20 988842 2178 011158 26 33 988083 2150 997935 21 990149 2171 009851 25 36 989374 2144 997922 21 991451 2165 008549 24 37 990660 2138 997910 21 992750 2158 007250 23 38 991943 2131 997897 21 994045 2152 005955 22 39 993222 2125 997885 21 995337 2146 004063 21 40 994497 2119 997872 21 996624 2140 003376 20 41 8-995768 2112 9-997860 21 8-997908 2134 11-002092 19 42 997036 2106 997847 21 9991 SB 2127 000812 ]8 43 998299 2100 997835 21 9-000465 2121 10-999535 n 44 999560 2094 997822 21 001738 2115 998262 16 45 9-000816 2087 997809 21 003007 2109 996993 15 46 002069 2082 997797 21 004272 2103 995728 14 47 003318 2076 997784 21 005534 2097 994466 13 48 004563 2070 997771 21 006792 2091 993208 12 49 0058(15 2064 997758 21 008047 2085 991953 11 50 007041 2058 997745 21 009298 2080 990702 10 51 J-008278 2052 9-997732 21 9-010546 2074 10-989454 9 52 009510 2046 997719 21 011790 2068 988210 8 53 1 010737 2040 997706 21 ' 013031 2062 986969 7 54 01 1962 2034 997693 22 014268 2056 985732 6 55 013182 2029 997680 22 015502 2051 984498 5 5C 014400 2023 997667 22 016732 2045 983268 4 57 015613 2017 997654 22 017959 2040 982041 3 58 016824 2012 997641 22 019183 2033 980817 2 59 j 018031 2006 997628 22 020403 2028 979597 1 60 ' 019235 2000 997614 22 021620 2023 978380 | Cosmo | | Sine | | Cotang. ; Tang. M. 84 Degrees. 204 (6 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. | Sine D. | Tang. | D. Cotang. 1 9-019235 020435 2000 1995 9-997(514 997(501 22 22 9-021620 022834 2023 2017 10-978380 9771(56 2 021032 1989 997588 22 024044 2011 975956 3 022825 1984 997574 22 025251 2006 974749 4 024016 1978 9975(51 22 020455 2000 973545 5 025203 1973 997547 22 027055 1995 972345 6 026386 1967 997534 23 028852 1990 971148 7 027567 1962 997520 23 030046 1985 9(59954 8 028744 1957 997507 23 031237 1979 908703 9 029918 1951 997493 23 032425 1974 967575 10 031089 1947 997480 23 033609 1969 906391 11 9-032257 1941 9-097466 23 9-034791 1964 10-965209 12 033421 1936 997452 23 035909 19,58 9(54031 13 034 582 1930 997439 23 037144 1953 962856 14 035741 1925 997425 23 038316 1948 9(51(584 15 036896 1920 997411 23 039485 1943 960515 16 038048 11115 997397 23 040(551 1938 959349 17 039197 1910 997383 23 041813 1933 958187 18 0411342 1905 997309 23 042973 1928 957027 19 041485 1899 997355 23 044130 1923 955870 20 042(525 1894 997341 23 045284 1918 954716 21 9 043762 1889 9-997327 24 9-046434 1913 10-9535(16 22 044895 1884 997313 24 047582 1908 952418 23 04(5026 1879 997299 24 048727 1903 951273 24 047154 1875 997285 24 049869 1898 950131 25 048279 1870 997271 24 051008 1893 948992 26 049400 1865 997257 24 052144 1889 947856 27 050519 1860 997242 24 053277 1884 940723 28 051635 1855 997228 24 054407 1879 945593 29 052749 1850 997214 24 055535 1874 944465 30 053859 1845 997199 24 056659 1870 943341 31 054966 1841 9-997185 24 9-057781 1865 10-942219 32 05(5071 1836 997170 24 058900 1869 941100 33 057172 1831 997156 24 00001(5 1855 939984 34 058271 1827 997141 24 061130 1851 938870 35 059367 1822 997127 24 002240 1846 937700 36 0604(50 1817 997112 24 0(53348 1842 93(5652 37 061551 1813 997098 24 004453 1837 935547 38 0(52(539 1808 997083 25 065556 1833 934444 39 0(53724 1804 997068 25 06(5655 1828 833345 40 0(54806 1799 997053 25 067752 1824 932248 41 9-065885 1794 9997039 25 9-068846 1819 10-931154 42 060902 1790 997024 25 009938 1815 9300(52 43 068036 1786 997009 25 07)027 1810 928973 44 069107 1781 996994 25 072113 1806 927887 45 070176 1777 996979 25 073197 1802 926803 46 071242 1772 996964 25 074278 1797 925722 47 072306 1768 99(5949 25 075356 1793 924644 48 073306 1763 99(5934 25 070432 1789 9235(58 49 074424 1759 99(5919 25 077.505 1784 922495 50 075480 1755 996904 25 078576 1780 921424 51 9-070533 1750 9-996889 25 9-079044 1776 10-920356 52 077583 1746 99(5874 25 080710 1772 919290 53 078631 1742 996858 25 081773 1767 918227 54 079676 1738 996843 25 082833 763 917167 55 080719 1733 990828 25 083891 759 916109 56 081759 1729 996812 26 084947 755 915053 57 082797 1725 990797 26 080000 751 914000 58 083832 1721 996782 26 087050 747 912950 59 0848(54 1717 99(5766 26 088098 743 911902 60 085894 1713 996751 26 089144 1738 910856 Cosine I | Cotang. | I Tang. ( M. & Degrees. LOGARITHMIC SINES, COSINES, ETC. (7 Degrees.) 205 M. Sine D. Cosine | D. | Tan?. D. | Coianff. \ IP 9-085894 J713 9-996751 26 9-089144 1738 10-910856 60 1 086922 1709 996735 26 090187 1734 909813 59 2 087947 1704 996720 2(5 091228 1730 908772 58 3 088970 1700 99(5704 26 0922(56 1727 907734 57 4 089990 1696 996(588 2(5 093302 1722 906698 56 5 0910(18 1692 996673 26 094336 1719 9056(54 55 6 092024 1(588 99(5657 26 0953H7 1715 904633 54 7 093037 1684 996641 26 09(5395 17JI 903605 53 8 094047 1680 996(525 06 097422 1707 902578 52 9 095050 HJ76 996(510 26 098446 1703 901554 51 JO 09G062 1673 996594 26 099468 1699 900532 50 11 9-097065 1668 9-996578 27 9-100487 1695 10-899513 49 12 098006 1685 996562 27 101504 1691 898496 48 13 099065 11561 99(55-1(5 27 102519 1887 897481 47 14 100062 1657 99(5530 27 103532 1684 89(54(58 46 15 10)056 1633 996514 27 104542 1(580 895458 45 16 1020-48 1649 99(5498 27 105550 1676 894450 44 17 103037 1645 996482 27 106556 1672 893444 43 18 104023 1641 996-1(55 27 107559 1(569 89244 1 42 19 105010 1638 996449 27 1085(50 1665 891440 41 20 105992 1634 996433 27 109559 1661 890441 40 21 9-106973 1630 9-996417 27 9-110556 1658 10-889444 39 22 107951 1627 996400 27 111551 1654 888449 38 23 108927 1623 99(5384 27 1 12543 1650 887457 37 24 109901 1619 996368 27 113533 16-1(5 886467 36 25 1J0873 1616 996351 27 114521 1643 885479 35 26 111842 1612 99(5335 27 115507 1639 884493 34 27 112809 1608 99(5318 27 116491 1636 883509 33 28 113774 1605 996302 28 117472 1632 882528 32 29 114737 1601 99(5285 28 118452 1629 881548 31 30 115698 1597 996269 28 119429 1625 880571 30 31 9-116656 1594 9-996252 28 9-120404 1622 10-879596 29 32 117613 1590 996235 28 121377 1618 878623 28 33 11 8567 1587 996219 28 122348 1615 877652 27 34 119519 15B3 996202 28 123317 1611 876683 26 35 120469 1580 996185 28 124284 1607 875716 25 36 121417 1576 996168 28 125249 1604 874751 24 37 122362 1573 996151 28 126211 1601 873789 23 38 123306 1569 996134 28 127172 1597 872828 22 39 124248 1566 996117 28 128130 1594 871870 21 40 125187 1562 996100 28 129087 1591 870913 20 41 9-126125 1559 9-996083 29 9-130041 1587 10-869959 19 42 127060 1556 996066 29 130994 1584 8(59006 18 43 127993 1552 99(5049 29 131944 1581 868056 17 44 128925 1549 996032 29 132893 1577 867107 16 45 129854 1545 996015 29 133839 1574 866161 15 46 130781 1542 995998 29 134784 1571 865216 14 47 131706 1539 995980 29 135726 1567 864274 13 48 132630 1535 995963 29 136667 1564 8(53333 12 49 133551 1532 995946 29 137605 1561 8(52395 11 50 134470 1529 995928 29 138542 1558 861458 10 51 9-135387 1525 9-995911 29 9-139476 1555 10-860524 . 9 52 1315303 1522 995894 29 140409 1551 859591 8 53 137216 1510 995876 29 141340 1548 858660 7 54 138128 1516 995859 29 142269 1545 857731 8 55 139037 1512 995841 29 143196 1542 856804 5 56 139944 1509 995823 29 144121 1539 855879 4 57 140850 1506 995806 29 145(144 1535 85495(5 3 58 141754 1503 995788 29 145966 1532 854034 2 59 142655 1500 995771 29 146885 1529 853115 60 1 143555 1496 995753 29 147803 1526 852197 1 I Sine Cotang-. | I Tang. | M. 82 Degrees. 206 (8 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. | Sme j Cosine | D. | | D. | 9-143555 1496 9-995753 30 9-147803 1526 10.852197 60 1 144453 1493 91)5735 30 148718 1523 851 282 59 2 145349 1490 9957.17 30 149(532 1520 850368 58 3 146243 1487 995699 30 150544 1517 849456 57 4 147136 1484 995(581 30 151454 1514 848546 56 5 148026 1481 9956(54 30 1523(53 1511 847637 55 6 148915 1478 995646 30 153269 1508 846731 54 7 141)802 1475 995628 30 154174 1505 845826 53 8 150(586 1472 995(510 30 155077 1502 844923 52 D 151569 1469 995591 30 155978 1499 844022 51 10 1 52451 1466 995573 30 156877 1496 843123 50 11 9-153330 1463 9-995555 30 9-157775 1493 10-842225 49 12 154208 1460 995537 30 158671 1490 841329 48 13 155083 1457 995519 30 159565 1487 840435 47 14 155957 1!54 995501 31 160457 1484 839543 46 15 156830 1451 99548-2 31 161347 1481 838653 45 16 157700 1448 995464 31 162236 1479 837764 44 17 158569 1445 99544G 31 1(53123 1476 836877 43 18 159435 1442 995427 31 164008 1473 835992 42 19 160301 1439 995409 31 164892 1470 835108 41 20 161164 1436 995390 31 165774 1467 834226 40 21 9-162025 1433 9-995372 31 9' 166654 1464 10-833346 39 22 162885 1430 995353 31 167532 1461 832468 38 23 163743 1427 995334 31 168409 1458 831591 37 24 164600 1424 995316 31 169284 1455 830716 36 25 165454 1422 995297 31 170157 1453 829843 35 26 J (56307 1419 995278 31 171029 1450 828971 34 27 167159 1416 995260 31 171899 1447 828101 33 28 168008 1413 995241 32 172767 1444 827233 32 2!) 16f<856 1410 995222 32 173634 1442 826366 31 30 169702 1407 995203 32 174499 1439 825501 30 31 9-170547 1405 9-995184 32 9-1753(52 1436 10-824638 29 32 71389 1402 995165 32 17G234 1433 823776 28 33 72230 1399 995146 32 177084 1431 822916 27 34 73070 1396 995127 32 177942 1428 822058 26 35 73908 1394 995108 32 178799 1425 821201 25 36 74744 1391 995089 32 179655 1423 820345 24 37 75578 1388 995070 32 180508 1420 819492 23 38 764 1 1 1386 995051 32 181360 1417 818640 22 39 77242 1383 995032 32 182211 1415 817789 21 40 178072 1380 995013 32 183059 1412 81(5941 20 41 9-178900 1377 9-994993 32 9-183907 1409 10-816093 19 42 179726 1374 994974 32 184752 1407 815248 18 43 180551 1372 994955 32 185597 1404 814403 17 44 181374 1369 991935 32 18(5439 1402 8135(51 16 45 182196 1366 9J4916 33 187280 1399 812720 15 46 183016 1364 99189(3 33 188120 139(5 811880 14 47 183834 1361 994877 33 1889.58 1393 811042 13 48 184651 1359 994857 33 189794 1391 81(1206 12 49 185466 135(5 994838 33 190629 1389 809371 11 50 186280 1353 994818 33 191462 1386 808538 10 51 9-187092 1351 9-994798 33 9- 192294 1384 10-807706 9 52 187903 1348 994779 33 193124 1381 80687(5 8 53 188712 1346 994759 33 193953 1379 80(5047 7 54 189519 1343 9111739 33 194780 1376 805220 6 55 190325 1341 99171 9 33 195606 1374 804394 5 56 191130 1338 994700 33 196430 1371 803570 4 57 191933 1336 994(580 33 197253 1369 802747 3 58 192734 1333 994660 33 198074 13(5(5 801926 2 59 193534 1330 994640 33 198894 1364 801106 1 60 194332 1328 994(520 33 199713 1361 ' 800287 Cosme Sine 1 | Cotang. 81 Decrees. | Tang. LOGARITHMIC SINES, COSINES, ETC. (9 Degrees.) 207 } Sine | D. | Cosine D. Tan?. | D. Cotansr. | 9-194332 1328 9-994(520 33 [9-199713 1361 10-800287 60 195129 1326 994000 33 200529 1359 799471 59 195925 1323 !!!)4580 33 201345 1356 798055 58 196719 1321 9945(50 34 202159 J354 797841 57 197511 1318 994540 34 202971 1352 797029 56 198302 1316 994519 34 203782 1349 790218 55 199091 1313 994499 34 204592 1347 795408 54 199879 1311 994479 34 205400 1345 794(500 53 200(500 1308 994459 34 20(5207 1342 793793 52 201451 1300 994438 34 207013 1340 792987 51 202234 1304 994418 34 207817 1338 792183 50 0-203017 1301 9-994397 34 9-208619 1335 10-791:581 49 203797 129!) 994377 34 209420 1333 790580 48 204577 1290 994357 34 210220 1331 789780 47 205354 1294 994330 34 211018 1328 788982 46 200 I'M 1292 994316 34 211815 1326 788185 45 206906 1289 994295 34 212011 1324 787389 44 207(579 1287 994274 35 213405 1321- 780595 43 208452 1285 994254 35 214198 1319 785802 42 209222 1282 994233 35 214989 1317 785011 41 209992 1280 994212 35 215780 1315 784220 40 9-210700 1278 9-994191 35 9-21(5508 1312 10-783432 39 211526 1275 994171 35 217350 1310 782044 38 212291 1273 994150 35 218142 1308 781858 37 213055 1271 994129 35 218926 1305 781074 36 213818 1208 994108 . 35 219710 1303 780290 35 214579 1266 994087 35 220492 1301 779508 34 215338 1264 994006 35 221272 1299 778728 33 216097 1201 994)45 35 222052 1297 777948 32 21(5854 1259 994024 35 222830 1294 777170 31' 217609 1257 994003 35 223606 1292 770394 30 9-218363 1255 9-993981 35 9-224382 1290 10-775018 29 219116 1253 993900 35 225150 1288 774844 28 219868 1250 993939 35 225929 1286 774071 27 220018 1248 993918 35 226700 1284 773300 26 221367 1240 993890 30 227471 1281 772529 25 222115 1244 993875 30 228239 1279 771701 24 222861 1242 993854 30 229007 1277 770993 23 223006 1239 993832 36 229773 1275 770227 22 224349 1237 993811 36 230539 1273 7(59401 21 225092 1235 993789 36 231302 1271 768(598 20 9-225833 1233 9-993708 36 9-232065 1269 10-707935 19 220573 1231 993740 36 232820 1207 707174 18 227311 1228 993725 I6 233586 1265 760414 17 228048 1226 993703 36 234345 1262 7(55055 16 228784 1224 993081 36 235103 1200 704897 15 229518 1222 993000 3(5 235859 1258 704141 11 230252 1220 993038 36 230(514 1250 763386 13 230984 1218 993010 36 2373(58 1254 762032 12 231714 1210 993594 37 238120 1252 701880 11 232444 1214 993572 37 238872 1250 701128 10 9-233172 1212 9-993550 37 9-239022 1248 10-700378 9 233899 1209 993528 37 240371 1240 759029 3 234025 1207 993500 37 241118 1244 758882 7 235349 1205 993484 37 241805 1242 758135 6 236073 1203 993402 37 242010 1240 75739(1 5 23(5795 1201 993440 37 243354 1238 750046 4 237515 1199 993418 37 244097 1230 755903 3 238235 1197 993396 37 244839 1234 755101 1 238953 1195 993374 37 245579 12" 754421 1 239670 1193 993351 37 246319 1230 753681 | Coswe | | Coiang. 80 Degrees. | Tang. I M. 208 (10 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. | Sine J D- Cosine 1 D. Tan*. D. 1 Cotang, 1 9--239670 1193 9-993351 37 9246319 1230 10-7;j%81 60 1 '24U:J86 1191 9933-29 37 247057 1228 752943 59 2 241101 1189 993307 37 247794 2-26 75-221 Hi 58 3 241814 1187 993285 37 248530 -224 751470 57 4 242526 1185 99321 a 37 249204 2-2-2 750736 56 5 243237 J l;<\ 993-240 37 249998 2-20 75001)2 55 6 431)47 1181 993-217 38 250730 -218 74SW70 54 7 244656 1179 993195 38 251401 -217 748539 53 8 245363 1177 993172 38 '252191 -215 747809 52 9 240069 1175 993149 38 2529-20 213 7470HO 51 10 246775 1173 993127 38 253648 '211 746352 50 11 9-247478 1171 9-993104 38 9-254374 1209 10-745'.i26 49 12 248181 1169 993081 38 255100 1-207 744900 48 13 248883 1167 993059 38 255824 1205 744 176 47 14 249583 1165 993036 38 256547 1203 743453 46 15 250282 1163 993013 38 257-269 1-201 74-2731 45 16 250980 1161 99-2990 38 257990 1-200 742010 44 17 251677 1159 99-2967 38 258710 JJ98 741290 43 18 252373 1158 99-2J44 38 259429 1196 740571 42 19 253067 1156 992921 38 260146 1194 739854 41 20 253761 1154 99-2898 38 260863 119-2 739137 40 21 9-254453 1152 9-992875 38 9-261578 1190 10-738422 39 22 255144 1J50 992852 38 262-292 1189 737708 38 23 255834 1148 99-2829 39 263005 na7 730995 37 24 256523 1146 992806 39 263717 1185 730283 36 25 257211 1144 992783 39 204428 1183 735572 35 26 257898 1142 992759 39 205138 1181 734802 34 27 258583 1141 99-2736 39 205847 1179 734153 33 28 259208 1139 992713 39 '26u555 1178 733445 32 29 259951 1137 992690 39 207201 1176 73-2739 31 30 260633 1135 992o66 39 267907 1174 732033 30 31 9-261314 1133 9-992643 39 9-268671 117-2 10-731329 29 32 261994 1131 992619 39 209375 1170 730625 28 33 26-2673 1130 99-2596 39 270077 1169 7299'23 27 34 263351 1128 992572 39 270779 1167 729221 26 35 264027 1126 992549 39 271479 1165 7285-21 25 36 264703 1124 992525 39 27-2178 1164 727822 24 37 265377 1122 992501 39 272876 1162 7271-24 23 38 266051 1130 992478 40 '273573 1160 726427 22 3!) 266723 1119 992454 40 2M209 1158 725731 21 40 267395 1117 992430 40 274964 1157 7'25036 20 41 9-268065 1J15 9-992406 40 9-275658 1155 10-724342 19 42 268734 1113 992382 40 276351 1153 72;>049 18 43 269402 1111 992359 40 277043 1151 722957 17 44 270069 1110 992335 40 277734 1150 72226U 16 45 270735 1108 99-2311 40 278424 1148 721576 15 46 271400 1106 992287 40 279113 1147 720887 14 47 272064 1105 99:2263 40 '279801 1115 720199 13 48 272720 1103 992239 40 2c04d8 1143 719512 12 49 273388 1J01 99-2214 40 2rilJ74 1141 7188'26 11 50 274049 1099 992190 40 281858 1140 718142 10 51 J-274708 1098 9-99211)6 40 9-282542 1138 10-717458 9 52 275367 1096 J92142 40 283!2'25 1136 710775 8 53 270024 1094 992117 41 '283907 1135 716093 7 54 270681 1092 992093 41 284588 1133 715412 6 55 277337 1091 99-2009 41 28.V208 1131 714732 5 56 277991 1089 992044 41 285947 1130 714053 4 57 278644 1087 992020 41 iMS024 ll;J8 713376 3 58 279297 108(5 991996 41 '287301 1121) 712699 2 59 279948 1084 991971 41 2871*77 1125 712023 .60 280599 1082 991947 41 288652 . 1123 711348 | Cusine | 8,ne | | Cotang. I Tang. 79 Degrees. LOGARITHMIC SINES, COSINES, ETC. (11 Degrees.) 209 M. | Sine | D. | Cosine Tang. Cotang. I 9 -28059! 1082 9-991947 41 9-28S652 1123 10-711348 60 1 281248 1081 991922 41 2H!):WI5 1122 710674 59 2 281897 1079 991897 41 289999 1120 710001 58 3 282,544 1077 991873 41 290671 1118 709329 57 4 283190 1076 991848 41 291342 1117 708658 56 5 283836 1074 991823 41 292013 1115 707987 55 6 284480 1072 991799 41 292682 1114 707318 54 7 2S5124 1071 991774 42 293350 1112 706650 53 8 285766 1069 991749 42 294017 1111 705983 52 9 28(i408 1067 991724 42 294684 1109 705316 51 10 287048 1066 991699 42 295349 1107 704651 50 11 9-237687 1064 9-991674 42 9-296013 1106 10-703987 49 IS 288326 1063 991649 42 296677 1104 703323 48 13 288964 1061 991624 42 297339 1103 702661 47 14 289600 1059 991599 42 298001 1101 701999 46 15 290236 1058 991574 42 298662 1100 701338 45 JO 290870 1056 991549 42 299322 1098 700678 44 17 291504 1054 991524 42 299980 1096 700020 43 If 292137 1053 991498 42 300(538 1095 699362 42 19 2927(58 1051 991473 42 301295 1093 698705 41 20 293399 1050 991448 42 301951 1092 698049 40 21 9-294029 1048 9-991422 42 9-302607 1090 10-697393 39 82 2946.58 1046 991397 42 303261 1089 696739 38 2:5 295286 1045 991372 43 303914 1087 696086 37 24 295913 1043 991346 43 304567 1086 695433 :> 2r, 29(5539 1042 '991321 43 305218 1084 694782 35 26 297164 1040 991295 43 305869 1083 694131 34 27 297788 1039 991270 43 306519 1081 693481 33 28 298412 1037 991244 43 307168 1080 692832 32 89 299034 1036 991218 43 307815 1078 692185 31 30 299055 1034 991193 43 308463 1077 691537 30 31 9-300276 1032 9-991167 43 9-309109 1075 10-690891 29 32 300895 1031 991141 43 309754 1074 690246 28 33 301514 1029 991115 43 310398 1073 689602 27 34- 302132 1028 991090 43 311042 1071 688958 26 35 302748 1026 991064 43 311685 1070 688315 25 36 303364 1025 991038 43 312327 1068 687673 24 37 303979 1023 991012 43 312967 1067 687033 23 38 304593 1022 990986 43 313608 1065 68(5392 22 39 305207 1020 990960 43 314247 1064 685753 21 40 305819 1019 990934 44 314885 1062 685115 20 4! 9-306430 1017 9-990908 44 9315523 1061 10-684477 19 42 307041 1016 990882 44 316159 1060 683841 18 43 307650 1014 990855 44 316795 1058 683205 17 44 308259 1013 990829 44 317430 1057 682570 16 45 308867 1011 990803 44 318064 1055 681936 15 4fl 309474 1010 990777 44 318697 1054 681303 14 47 310080 1008 990750 44 319329 1053 680671 13 48 310685 1007 990724 44 319961 1051 680039 12 4!) 311289 1005 990697 44 320592 1050 679408 11 50 311893 1004 990671 44 321222 1048 678778 10 51 9-312495 1003 9-990644 44 9-321851 1047 10-678149 9 52 313097 1001 9!K)618 44 322479 1045 677521 8 53 313698 1000 990591 44 323106 1044 676894 7 54 314297 998 090565 44 323733 1043 676267 6 5.1 314897 997 990538 44 324358 1041 675642 5 56 315495 996 990511 45 334983 1040 675017 4 57 316092 994 990485 45 325(507 1039 674393 3 58 316689 993 990458 45 326231 1037 673769 o 59 317284 991 990431 45 326853 1036 673147 1 60 ' 317879 990 990404 45 ' 327475 1035 672525 Couite | Sine | I Cotang. | 78 Degrees. I Tang. | M. 210 (12 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. | Bine | D. Cone | D. | Tang. | D. Cotang. 9-317879 990 9-990404 45 9-327474 1035 10-672526 (50 1 318473 988 990378 45 328095 1033 671905 59 2 319066 987 990351 45 328715 1032 G71285 SB 3 319(558 986 990324 45 329334 1030 670(JG6 57 4 320249 984 990297 45 321)953 1029 670047 5(5 5 320840 983 990270 45 330570 1028 661)430 55 6 321430 982 990243 45 331187 1026 6C8813 54 7 322019 980 991,215 45 331803 1025 668197 53 8 322607 979 990188 45 332418 1024 667582 52 y 323194 977 91)0101 45 333033 1023 666967 51 10 323780 976 990134 45 333046 1021 660354 50 n 9-324366 975 9-990107 - 46 9-334259 1020 10-665741 49 12 324!,'50 973 990075) 46 33487] 1019 665129 48 13 3-2.3534 972 99U052 46 335482 1017 664518 47 14 3261 17 970 990025 46 336093 1016 6631)0*7 46 15 320700 969 981)997 46 336702 1015 - 663298 45 16 327281 968 989970 46 337311 1013 662689 44 17 327862 966 989942 46 337919 1012 662081 43 18 328442 965 989915 46 338527 1011 661473 4-> 19 329021 964 989887 46" 339133 1010 660867 41 20 329599 962 989860 46 239739 1008 660361 40 21 9-330176 961 9-989832 46 9-340344 1007 10-65!: 056 39 22 330753 960 989804 46 340948 1006 651)1152 38 23 331329 958 989777 46 341552 1004 058448 37 24 331903 957 989749 47 342155 1003 657845 3(5 25 332478 956 989721 47 342757 1002 657243 35 26 333051 954 989693 47 343358 1000 656642 34 27 333G24 953 989065 47 343958 999 656042 33 28 334195 952 989637 47 344558 998 655442 32 29 334766 950 989609 47 345157 997 654843 31 30 335337 949 989582 47 345755 996 654245 30 31 9-335906 948 9-981)553 47 9-340353 994 10-653647 29 32 336475 946 989525 47 346949 9i)3 653051 28 33 337043 945 989497 47 347545 992 652455 2? 34 337610 944 981)469 47 348141 991 651859 26 35 338176 943 989441 47 348735 91)0 651265 25 36 338742 941 989413 47 349329 988 650671 24 37 339306 940 989384 47 349922 987 650078 23 38 339871 939 989356 47 350514 986 649486 22 39 340434 937 98!)3'28 47 351106 985 64881)4 21 40 340996 936 989300 47 351697 983 648303 20 41 9-341558 935 9-989271 47 9-352287 982 10647713 19 42 342119 934 989243 47 352876 981 647124 18 43 342679 932 989214 47 353465 980 646535 17 44 343239 931 989186 47 354053 979 645947 16 45 343797 930 989157 47 354640 977 645360 15 46 344355 929 989128 48 355227 976 644773 14 47 344912 927 989100 48 355813 975 644187 13 48 345469 926 989071 48 356398 974 643602 12 49 346024 925 989042 48 35(5982 973 643018 11 50 346579 924 989014 48 357566 971 ,642434 10 51 9-347134 922 9-988985 48 9-358149 970 10-641851 9 52 347(587 921 9HH956 48 358731 969 6412(19 53 348240 920 988927 48 351(313 968 640687 7 54 348792 9J9 988898 48 359893 i>r,7 040107 6 55 349343 917 988869 48 360474 966 63952G 5 56 349893 916 988H40 48 361053 965 638947 4 57 350443 915 988811 49 36J63-J 963 638368 3 58 350992 914 988782 49 3IW210 962 637700 2 59 351540 913 988753 49 302787 961 637213 1 60 352088 911 988724 49 363364 960 636636 Cosine I I Coung. | Tang. 77 Degrees. LOGARITHMIC SINES, COSINES, ETC. (13 Degrees.) 211 | Sine Cosine I D. I Tan<*. Cotang. | 9-352088 911 9-988724 49 9-303364 960 10-030636 (K) 1 352035 910 988095 49 303940 959 030000 59 2 353181 909 988060 49 364515 958 035485 58 3 353720 908 988(536 49 365090 957 634910 57 4 354271 907 988607 49 305004 955 (534336 56 5 354815 905 988578 49 306237 954 633703 55 6 355358 904 988548 49 360810 953 033190 54 7 355901 903 988519 49 367382 952 632618 53 8 356443 902 988489 49 367953 951 632047 52 9 35(1984 901 9884(50 49 368524 950 631476 51 10 357524 899 988430 49 3(59094 949 630906 50 11 9-358064 898 9-988401 49 9-309663 948 10-630337 49 12 358603 897 988371 49 370232 946 0297(58 48 13 359141 896 988342 49 370799 945 629201 47 14 359078 895 988312 50 371367 944 028033 46 15 360215 893 988282 50 371933 943 (5281)67 45 16 360752 892 988252 50 372499 942 627501 44 17 361287 891 988223 50 373004 941 626936 43 18 361822 890 988193 50 373629 940 620371 42 19 362356 889 988163 50 374193 939 025807 41 20 3(52889 888 988133 50 374756 938 625244 40 21 9-363422 881 9-988103 50 9-375319 937 10-624681 39 22 303954 885 988073 50 375881 935 624119 38 2:5 364485 884 988043 50 376442 934 623558 37 24 365016 883 988013 50 377003 933 622997 36 25 365540 882 987983 50 377563 932 G22437 35 26 306075 881 987953 50 378122 931 621878 34 27 366604 880 987922 50 378081 930 621319 33 28 367131 879 9878!)2 50 379239 929 620761 32 29 367(559 877 987862 50 379797 928 620203 31 30 368185 876 987832 51 38(1354 927 G 19646 30 31 9-368711 875 9-987H01 51 9-3809 JO 926 10-019090 29 32 369236 874 987771 51 381466 925 618534 28 33 369701 873 987740 51 382020 924 617980 27 34 370285 872 987710 51 382575 923 617425 26 35 370808 871 987679 51 383129 922 610871 25 3(5 371330 870 987649 51 :*83682 921 616318 24 37 371852 8(59 987618 51 384234 920 615766 23 38 372373 807 987588 51 384786 919 615214 22 3<; 5 44-2535 730 98-2000 01 459875 7i>0 540125 55 6 442973 729 982624 01 400349 790 539051 51 7 443410 728 982587 61 400823 7f J 9 539177 53 8 443847 727 983551 01 401297 788 538703 52 9 4-14284 727 98-2514 01 401770 788 538230 51 10 444720 720 98-2477 61 462342 787 537758 50 11 9-445155 725 9-982441 61 9-402714 780 10-537286 49 12 445590 724 982404 01 403180 785 530814 48 13 44G025 723 982307 01 403058 785 530342 47 14 446459 723 982331 61 404120 784 535871 40 15 446893 722 982294 61 464599 783 535401 45 1G 447320 721 982257 61 405009 783 534931 44 17 447759 720 982220 02 405539 782 534401 43 18 448191 720 982183 02 400008 781 533992 12 19 448023 719 982140 62 400470 780 533524 41 20 449054 718 982109 62 400945 780 533055 40 21 9-449485 717 9-982072 62 9-407413 779 /0-532587 39 22 449915 710 982035 62 407880 778 532120 38 23 450345 716 981993 62 408347 778 531053 37 24 450775 715 981901 62 4G8814 777 531180 30 25 451204 714 981924 62 409280 770 530720 35 26 451032 713 981880 62 409740 775 530254 34 27 452000 713 981849 62 470211 775 529789 33 28 452488 "712 981812 62 470G7G 774 529324 32 29 452915 711 981774 62 471141 773 528859 31 30 453342 710 981737 62 471005 773 528395 30 31 9-453708 710 9-981099 63 9-472008 772 10-527932 29 32 454194 709 981002 63 472532 771 527408 28 33 454019 708 981025 63 4721);!.-) 771 527005 27 34 455044 707 981587 63 473457 770 520543 20 35 455409 707 981549 63 473919 709 520081 25 36 455893 700 981512 63 474381 7(59 525019 24 37 450310 705 981474 63 474842 708 525158 23 38 450739 704 981430 63 475303 707 524097 22 39 457102 704 981399 63 475703 707 524237 21 40 457584 703 981301 63 470223 700 523777 20 41 9-458006 702 9-981323 63 9-470083 783 10-523317 19 42 458427 701 981235 63 477142 7(55 522858 18 43 458848 701 981247 63 477001 704 522399 17 44 459208 700 981209 63 478059 703 521941 10 45 459(588 699 981171 63 478517 7(53 521483 15 46 400108 698 981133 64 478975 702 521025 14 47 400527 698 981095 64 479432 701 520508 13 48 400940 697 981057 64 479889 701 520111 12 49 401304 690 981019 64 480345 700 519055 11 50 401782 095 980981 64 480801 759 519199 10 51 9-402199 695 9-980942 64 9-481257 759 10-518743 9 52 402010 094 980904 64 481712 758 5J8288 8 53 403032 693 9808(56 64 482(07 757 517833 7 54 403448 093 980827 64 482021 757 517379 55 4G3804 092 980789 64 483075 750 510925 5 56 404279 091 980750 64 483529 755 510471 4 57 404094 090 980712 04 483982 755 510018 3 58 405108 090 980073 04 48441(5 754 515505 2 59 405522 089 980035 04 484887 753 515113 1 60 405935 688 980590 64 485339 753 514001 I Cosii 1 Cotang. Tang. 73 Degrees LOGARITHMIC SINES, COSINES, ETC. (17 Degrees.) 215 J*. / __Sme |D. | Cosine | D. | Tang. | D. 9-405935 688 9-9S<)5% 64 9-4*5339 755 0-5H661 M) 1 406348 688 980558 64 485791 752 514209 r )8 3 467173 686 980480 65 48GC93 751 513307 ->7 4 467585 685 980442 65 487143 750 512857 r >l> 5 407996 685 980403 65 487593 749 512407 r >5 r> 468407 684 980304 65 488043 749 511957 54 408817 683 980325 65 488402 748 511508 53 8 4119227 683 9802S6 65 488941 747 511059 52 9 469637 682 980247 65 489390 747 5 ICG 10 r >l 10 47004G C81 986208 65 489838 746 510162 50 11 9-470455 680 9-980169 65 9-490286 746 10-51,9714 41) 1-2 470803 680 980130 65 490733 745 509267 48 13 471271 679 980091 65 491180 744 51*8820 47 14 471G79 678 980052 65 491627 744 508373 46 15 472086 678 980012 65 492073 743 507927 45 16 472492 677 979973 65 492519 743 507481 44 1? 472898 67G 979934 66 4929G5 742 507035 43 18 473304 676 979895 66 493410 741 506590 42 19 4737 JO 675 979855 66 49:3854 740 506146 41 20 474115 674 979816 66 494299 740 505701 4!) 21 9-474519 674 9-979776 66 9-494743 740 10-505257 3!l 22 474923 673 979737 66 495186 739 504814 38 23 475327 672 979697 66 495630 738 504370 37 24 475730 672 979058 66 496073 737 5(i3927 36 25 47G133 671 979GI8 C6 49G515 737 503485 35 26 476536 670 979579 66 496957 736 503043 34 27 470938 669 979539 66 497399 736 502601 33 28 477340 6(19 979499 66 497841 735 502159 32 29 477741 668 979459 66 498282 734 501718 31 30 478142 6G7 979420 66 498722 734 501278 30 3J 9-478542 667 9-979380 66 9-499163 733 10-5CC837 29 32 478942 666 979340 GG 499CC3 733 500397 28 33 479:^42 6(55 979300 67 50CC42 732 499958 27 34 479741 6(15 9792HO 67 500481 731 499519 26 35 480140 604 979220 67 50QS20 731 499(J80 25 36 480539 663 979180 67 601 359 730 498041 24 37 480937 663 979140 67 501797 730 498203 23 38 48 J 334 662 979100 C7 502235 729 4977(15 22 3D 481231 661 979059 67 5C2072 728 497328 21 40 482128 661 979019 67 503109 728 496891 2(J 41 9-48252.') GOO 9-978979 67 9-503546 727 10-496454 19 42 482921 659 978939 67 503982 727 49G018 Id 43 483316 659 978898 67 504418 726 495582 17 44 483712 658 97858 67 504854 725 495146 If 45 484107 657 978,817 67 505289 725 494711 i" 46 484501 657 978777 67 505724 724 494276 n 47 484895 656 978736 67 500159 724 49.'i841 i: 48 485289 655 978696 68 506593 723 493407 IS 49 485682 655 978G55 68 507027 722 492973 11 50 48G075 654 978615 68 507460 722 492540 11 51 9-480407 653 9-978574 68 9-507893 721 10-492107 < 52 4815800 653 978533 68 508326 721 491674 1 53 487251 652 978493 68 508759 720 491241 54 487(i43 651 978452 68 509191 719 49G809 < 55 488034 651 978411 68 509622 719 490378 5(i 488424 650 978370 68 510054 718 489946 , 57 48814 650 97S529 68 510485 718 489515 58 489204 649 978288 68 510916 717 4891 !84 59 489593 648 978247 68 511340 716 488654 60 489982 648 9782C6 68 511776 716 488224 I .Cosme | I Colang. | I Ta,, S . 72 Degrees 216 (18 Degrees.) LOGARITHMIC SINES, COSINES, ETC. I P. I Tang. | D.' | Cn (1 9*489982 648 9-978206 68 9-51 1776 716 10-48*224 00 1 490371 648 978165 68 512206 716 487794 o9 2 490759 647 978124 68 512635 715 4873(i5 58 3 491147 646 978083 69 5130(34 714 486936 r ,7 4 491535 646 978042 69 513493 714 486507 M 5 491922 645 978001 69 513921 713 486079 i>:> 6 492308 644 977JJ59 69 514349 713 485051 54 7 492G95 644 977918 69 514777 712 485223 53 8 493081 643 977877 69 515-204 712 48471)6 52 9 493466 642 977835 69 515631 711 484309 :.i 10 493851 642 977794 69 51G057 710 483943 50 11 9-494236 641 9-977752 69 9-510484 710 10-483516 49 IS 494021 641 977711 69 510910 709 483090 48 13 495005 640 977669 69 517335 709 482005 47 14 495388 639 977G28 69 517761 708 482239 4(i 15 495772 639 977586 69 518185 708 481815 45 10 496154 638 977544 70 518610 707 481390 44 17 496537 637 977503 70 519034 706 480966 43 18 49G919 637 9774(51 70 519458 706 480542 4-2 19 497301 636 977419 70 519882 705 480118 41 20 497682 636 977377 70 520305 705 479095 40 21 9-498064 635 9-977335 70 9-520728 704 10-479272 39 22 498444 634 977293 70 521151 703 478849 38 23 498825 634 977251 70 521573 703 478427 37 24 499204 633 977209 70 521995 703 478005 3(5 25 499584 632 977167 70 522417 702 477583 35 2(i 499963 632 977125 70 5228:18 702 477162 34 27 500342 631 977083 70 523259 701 476741 33 28 500721 631 977041 70 523680 701 470320 32 29 501099 630 970999 70 524100 700 475900 31 30 501476 629 970957 70 524520 699 475480 30 31 9-50 1H54 G29 9-97f914 70 9-524939 699 10-475001 29 3-2 502231 628 970872 71 525359 698 474041 28 33 502446 71 529535 693 470405 18 43 506354 622 970404 71 52J950 693 470050 17 44 506727 621 970301 71 530306 692 469034 JO 45 507099 620 976318 71 530781 691 409219 J5 46 507471 620 976275 71 531196 691 468804 11 47 507H43 619 97G232 72 531611 690 468389 13 48 508214 619 976189 72 532025 690 407975 1-2 49 508585 618 976146 72 53-2439 689 467561 11 50 508956 618 976103 72 532853 689 407147 10 51 9-509326 617 9-976060 72 9-533266 688 10-406734 9 52 509696 616 976017 72 533079 688 460321 8 53 510065 616 975974 72 5340U2 687 405908 7 54 510434 615 975930 72 534504 687 465496 55 510803 615 975887 72 534916 686 405084 5 5fi 511172 614 975844 72 535328 686 404672 4 57 511540 613 975800 72 535739 685 404-261 3 58 511907 613 975757 72 530150 685 403850 2 59 512275 612 975714 72 536561 684 403439 GO 512642 612 975670 72 536972 684 463028 Colang. I Tang. U Decrees. LOGARITHMIC SINES, COSINES, ETC. (19 Degrees.) 217 Sine ! D. | Cosine D. Tulip;. D. \ Cotan?. 9- -.12642 612 9-975670 73 9-336972 684 10-463028 60 513009 611 975(527 73 537382 (583 46.2618 59 513375 611 975583 73 5377! 12 183 462208 58 513741 610 975539 73 538902 682 461798 57 514107 609 975496 73 5386 1J 682 4(51389 50 514472 609 975452 73 539020 681 4(50980 55 514837 608 975408 73 539429 681 4(50571 54 515302 608 975G5 73 539P37 680 460163 53 515506 607 9/5321 73 54(3245 680 459755 52 515930 607 975277 73 540G53 679 459347 51 516294 606 975233 73 541061 679 458939 50 9-516657 605 9-975189 73 9-541468 678 10-458532 49 517020 605 975145 73 541875 678 458125 48 517382 604 975101 73 542281 677 457719 47 517745 604 975057 73 542G88 677 457312 46 518107 603 975013 73 543094 676 456906 45 5184G8 603 974969 74 543499 676 456501 44 518829 602 974925 74 543905 675 456095 43 519190 601 974880 74 544310 675 455690 42 519551 601 974836 74 544715 674 455285 41 519911 600 9747J2 74 545119 674 454881 40 9-533271 600 9-974748 74 9-545524 673 10-454476 39 520G31 599 974703 74 545928 673 454072 33 520990 599 974659 74 54G331 672 453(569 37 521349 598 974614 74 54G735 672 453265 36 521707 593 974570 74 547138 671 432862 35 52206G 597 974523 74 547540 671 4524(50 34 522424 596 974481 74 547943 670 452057 33 522781 596 974436 74 548345 670 451655 32 523,138 595 974391 74 548747 669 431253 31 523495 595 974347 75 549149 669 450851 30 9-523852 594 9-974302 75 9-543550 668 10-450450 29 524308 594 974257 75 54-W51 6G8 45i)049 28 524504 593 974212 75 550352 667 44JG48 27 524920 593 974 J 67 75 550752 667 449248 26 525275 5U2 974122 75 551152 666 448848 25 525G30 591 974077 75 551552 666 448448 24 525984 591 974032 75 551932 665 448048 23 52G339 590 973987 75 552351 665 447649 22 52GG93 590 973942 75 552750 665 447250 21 527046 589 973897 75 553149 664 446851 20 9-527400 589 9-973852 75 9-553548 664 10-446452 19 527753 588 973807 75 553946 663 446054 18 528105 588 973761 75 554344 663 445656 17 528458 587 973716 76 554741 6(52 445259 16 528810 587 973671 76 555139 662 444861 15 52'J161 586 973625 76 555.536 661 444464 14 529513 586 973580 76 553933 661 4440(57 13 529864 585 973535 76 55(5329 660 443671 12 530215 585 973489 76 536725 660 443275 11 530565 584 973444 76 557121 659 442879 10 9530915 584 9-973398 76 9-557517 659 10-442483 9 531265 583 973352 76 557913 659 442087 8 531614 582 973307 76 558308 658 441692 7 531963 582 973261 76 558702 658 441298 6 532312 581 973215 76 559097 657 440903 5 532661 581 9731(59 76 539491 657 440509 4 533009 580 973124 76 559885 656 440J 15 3 533357 580 973078 76 560279 656 439721 2 533704 579 973032 77 560673 655 439327 ] .53-1052 578 972986 77 561066 655 438934 | Cotiae | Cotang. I 1 Tang. | M. 70 Dtgreea. 218 (20 Degrees.) LOGARITHMIC SINES, COSINES, ETC. Sine | D. | Cosine | D. | Tan7 4 535-1:18 576 972802 77 562636 653 4373(54 56 5 535783 576 972755 77 563028 653 430972 53 6 53(5129 575 972709 77 563419 652 43(5581 54 7 536474 574 972063 77 563811 652 43(5189 53 8 53(3818 574 972617 77 5(54-202 651 435798 53 9 5371G3 573 972570 77 564 .V. 12 651 435408 r>l in 537507 573 972524 77 564983 650 43.~>017 50 11 9-537851 572 9-972478 77 9-565373 650 10-4346-27 49 12 538194 572 97-2431 78 5657(53 649 434-237 48 13 538538 571 972385 78 5(56153 (549 433847 47 14 5388^0 571 972338 78 560542 649 433458 46 15 539-223 570 972291 78 566932 648 433068 45 16 539565 570 972-245 78 567320 648 432680 44 17 539907 569 972198 78 567709 647 432291 43 18 540249 569 972151 78 568098 (547 431902 4-2 19 540590 508 972105 78 568486 646 431514 41 20 540931 568 972058 78 568873 646 431127 40 21 9-54 J272 567 9-972011 78 9-5692(51 645 10-430739 39 22 541613 567 971964 78 5(51)648 645 430352 38 23 541953 566 971917 78 570035 645 4-2911(55 37 24 542-21I3 566 971870 78 570422 644 429578 3(3 25 54263-2 565 971823 78 570809 (544 421)191 35 26 542!71 565 971776 78 571 11)5 643 428805 34 27 543310 564 971729 79 571581 643 428419 33 28 543649 564 971682 79 5711)67 642 428033 3-3 29 543987 563 971635 79 572352 (542 427648 31 30 544325 563 971588 79 572738 642 427262 30 31 9-544663 562 9-971f>40 79 9-573123 641 10-42(5877 2!) 32 545000 562 971493 79 573.-)07 641 4-264113 28 33 545338 561 971446 79 573892 640 426108 27 34 545(574 561 971398 79 574276 640 425724 -26 35 546011 560 971351 79 574660 639 42:>34() 25 36 54(i347 560 971303 79 575044 639 424956 -24 37 546683 559 971256 79 575427 639 4-24:>73 -23 38 547019 559 971208 79 575810 638 424190 -2-2 39 547354 558 971161 79 576193 638 4-23H07 21 40 547689 558 971113 79 576576 637 423424 20 41 9-548024 557 9-971066 80 9-576958 637 10-423041 IS) 42 548359 557 971018 80 577341 638 422659 18 43 548693 556 970970 80 5777-23 636 422277 17 44 549027 556 970922 80 578104 636 421896 16 45 541)260 555 970H74 80 578486 635 421514 15 46 549093 555 970827 80 578867 (535 421133 14 47 550026 554 970779 80 579248 e:J4 420752 13 48 550359 5.54 "70731 80 579629 634 42U371 1-2 49 550692 553 970(583 80 580009 634 419991 11 50 551024 553 970635 80 580389 633 419611 10 51 9-551356 552 9-970~>86 80 9-580769 633 10-419231 9 52 551687 552 9705:58 80 581149 632 41 HH51 8 53 552018 552 9704!M) 80 581528 632 418472 7 54 552349 551 97044-2 80 581907 632 4181)93 6 55 552680 551 970394 80 582286 631 4I77I4 5 56 553010 550 970345 81 582(565 631 41733$ 4 57 553341 550 970297 81 583043 630 41(5957 3 58 553070 549 '.(70-249 81 5H34-22 cno 41C>578 I 59 554000 549 9702IX) 81 583800 629 410200 1 60 554329 548 970152 81 584177 629 415823 | Cosine | | Sine | Couuig. 1 Tang. | Dejreea LOGARITHMIC SINES, COSINES, ETC. (21 Degrees.) 219 M. | Sine D. | Cosine | D. ~ Tan?. D. Cotnng. 9-554329 543 9-970152 1 81 9-584177 G29 10-415823 60 554! 558 548 970103 81 584555 629 415445 59 2 554987 547 9"0055 81 584932 628 415008 58 3 555315 547 970006 81 585309 628 414(591 57 4 555643 546 9(59957 81 58508G 627 414314 56 5 555971 546 960309 81 580002 627' 413938 55 6 556299 545 9(59860 81 58043D 627 412561 54 7 556(526 545 969811 81 586815 G2G 413185 53 8 55(i953 544 9G9762 81 587190 626 41^310 52 9 557-280 544 969714 81 587506 625 412434 51 10 557606 543 909G65 81 587941 625 412059 50 !1 9 557932 543 9-969616 82 9-588316 625 10-411084 49 12 558358 543 9695(57 82 588691 624 411209 48 13 558583 542 969518 82 5B90G6 624 41C934 17 14 558909 542 969469 82 589-140 623 41C560 46 15 559-234 541 969420 82 589814 623 410186 45 16 559558 541 909370 82 5901 88 623 4C9812 44 17 559883 540 969321 82 590562 622 409438 43 18 560207 540 969272 82 590935 622 409065 42 1 562468 536 968926 83 593542 619 400458 35 2ti 562790 536 908877 83 593914 618 400086 34 27 563112 536 908827 83 594285 618 405715 33 88 563433 535 908777 83 594656 - 618 405344 32 39 5(53755 535 968728 83 595027 617 404973 31 30 5(54075 534 968678 83 595398 617 4046C2 30 31 9-5(54396 534 9-968628 83 9-595768 617 10-404232 29 3-2 5(54716 533 908578 83 590138 616 403802 28 33 5(55036 533 908528 83 590508 616 403492 27 34 565356 532 968479 83 590878 616 403122 26 35 5(55676 532 94931 88 621852 591 378148 17 44 587085 502 964879 88 622207 590 377793 16 45 587386 502 964826 88 622561 590 377439 15 40 587688 501 9(54773 88 622915 590 377085 14 47 587989 501 964719 88 623269 589 376731 13 48 588289 501 964060 89 623(523 589 376377 12 49 588.390 500 964613 89 623976 589 376024 11 50 588890 500 964560 89 624330 588 375670 10 51 9-589190 499 9-964507 89 9-624683 588 10-375317 9 52 589489 499 964454 89 (525036 588 374964 8 53 5S97H9 499 964400 89 625388 587 374612 7 54 590088 498 964347 89 625741 587 374259 6 53 590387 498 964294 89 62(5(193 587 373907 5 50 590086 497 9(54240 89 626445 586 373555 4 57 590984 497 964187 89 626797 586 373203 3 58 591282 497 904133 89 627149 586 372851 2 59 591580 496 964080 89 627501 585 372499 1 f>0 591878 496 9(54026 89 627852 585 372148 | Cosine | Sine | | Cotang. 67 Degrees. 1 Tang. I LOGARITHMIC SINES, COSINES, ETC. (23 Degrees.) 21 | Sine. | 1). Cosine D. Tan?. D. | Cotang'. | 9-5!) 1878 496 9-964026 89 9-627852 585 10-372148 60 592176 495 903972 89 628203 585 371797 59 51)2473 495 963919 89 6285.54 585 371446 58 592770 495 963865 90 628905 584 371095 57 593067 494 963811 90 629255 584 370745 56 593363 494 963757 90 629606 583 370394 55 593659 493 963704 90 629956 583 370044 54 593955 493 9636.50 90 630306 583 369694 53 594251 493 963596 90 630656 583 369344 52 594547 492 963542 90 631005 582 368995 51 594842 492 963488 90 631355 582 368645 50 9-595137 491 9-963434 90 9-631704 ' 582 10-368296 49 595432 491 963379 90 632053 581 367947 48 595727 491 963325 90 632401 581 367599 47 596021 490 963271 90 632750 581 367250 46 596315 490 963217 90 633098 580 366902 45 596609 489 963163 90 633447 580 366553 44 596903 489 963108 91 633795 580 366205 43 597196 489 963054 91 634143 579 365857 42 597490 488 962999 91 634490 579 365510 41 597783 488 962945 91 634838 579 365162 40 9-598075 487 9-962890 91 9-635185 578 10-364815 39 598368 487 962836 91 635532 578 364468 38 598660 487 962781 91 635879 578 364121 37 598952 486 962727 91 636226 577 363774 36 599244 486 962672 91 636572 577 363428 35 599536 485 962617 91 636919 577 363081 34 599827 485 962562 91 637265 577 362735 33 600118 485 962508 91 637611 576 362389 32 600409 484 962453 91 637956 576 362044 31 600700 484 962398 92 638302 576 361698 30 9-600990 484 9-962343 92 9-638647 575 10-361353 29 601280 483 902288 92 638992 575 361008 28 601570 483 962233 92 639337 575 360663 27 601860 482 962178 92 639682 574 360318 26 602150 482 962123 92 640027 574 359973 25 602439 482 962067 92 640371 574 359629 24 602728 481 962012 92 640716 573 359284 23 603017 481 961957 92 641060 573 35H940 22 603305 481 961902 92 641404 573 358596 21 603594 480 961846 92 641747 572 358253 20 9-603882 480 9-961791 92 9-642091 572 10357909 19 604170 479 961735 92 642434 572 357566 18 604457 479 961680 92 642777 572 357223 17 604745 479 961624 93 643120 571 356880 16 605032 478 961569 93 643463 571 356537 15 605319 478 961513 93 643806 571 356194 14 605606 478 961458 93 644148 570 355852 13 605892 477 961402 93 644490 570 355510 12 606179 477 9G1 34G 93 644832 570 355168 11 606465 476 9612i)0 93 645174 569 354826 10 9-606751 476 9-961235 93 9-645516 569 10-354484 9 607036 476 961179 93 645857 569 354143 8 607322 475 961123 93 646199 569 353801 7 607607 475 961067 93 646540 568 353460 6 607892 474 961011 93 646881 568 353119 5 608177 474 960955 93 647222 568 352778 4 608461 474 960899 93 647562 567 352438 3 608745 473 960843 94 647903 567 352097 2 609029 473 960786 94 648243 567 351757 6093 13 473 900730 94 648583 566 351417 Cosine | t 66 Degrees. Cotang. Tang. 222 (24 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. Sine D. Cosine D. | Tang. D. CotP.ng. | 9-609313 473 9-960730 94 9-648583 566 10-351417 1 609597 472 9(50674 94 648923 566 351077 2 609880 472 960618 94 649263 566 350737 3 610164 47-2 960561 94 649(502 5(16 35031*8 4 610447 471 960505 94 649942 565 350058 5 6 1 0729 471 9(50448 94 650281 505 349719 6 611012 470 960392 94 650620 5K5 349380 7 611294 470 960335 94 650959 564 349041 8 611576 470 960279 94 651-297 5(i4 348703 9 611858 469 900222 94 65!ti36 51)4 3483(54 10 612140 469 960i65 94 651974 563 348026 11 9-612421 469 9-960109 95 9-65-2312 563 10-347688 12 6127i(2 41)8 960052 95 652650 5(53 347350 13 61-2983 4H8 959995 95 652988 563 347012 14 61*864 467 959938 95 653326 562 . 34(5074 15 61:1545 467 959882 95 653663 562 346337 16 613825 4b7 959825 95 654000 562 346000 17 6I4ID5 466 9597(58 95 654337 561 3456(53 18 614385 466 959711 95 654(574 561 345326 ]L) 614665 466 959654 95 655011 561 344!(89 20 614944 465 959596 95 655348 561 344652 21 9-615223 465 9-959539 95 9-655(584 560 10-344316 22 6i.Wo2 465 959482 95 65(5020 560 343980 23 615781 4ti4 959425 95 656356 5150 343644 24 616060 464 959:568 95 656692 559 343308 25 616338 464 959310 96 657028 559 34-2972 26 616616 463 959-253 96 657364 559 342636 27 616894 463 959195 96 657699 559 342301 28 617172 4(52 959138 96 658034 558 341966 29 617450 462 959081 96 658369 5.58 341631 30 617727 462 959023 96 658704 558 341-296 31 9-618004 461 9-958965 96 9-659039 558 10-340961 32 618281 461 958908 9(5 659373 557 340(127 33 618558 461 958850 96 659708 557 340-292 34 618834 460 958792 96 660042 557 339958 35 619110 460 958734 96 6(50376 557 339(524 36 619386 460 958677 96 660710 556 339290 37 619662 459 958619 96 661043 556 338957 38 619938 459 958561 96 6(51377 556 3386-23 39 620213 459 958503 97 661710 555 338-290 40 620488 458 958445 "97 662043 555 337957 41 9-620763 458 9-958387 97 9-662376 555 10-337624 42 621038 457 958329 97 6627(0 554 337291 43 62*1313 457 958271 97 663042 554 3369.58 44 621587 457 958213 97 663375 554 336625 45 621861 456 958 1.54 97 663707 554 33(5-293 46 622135 456 95809(5 97 664039 553 335961 47 622409 456 9583 335297 49 622956 455 957921 97 665035 553 334965 50 623229 455 957863 97 665366 552 334634 51 9-623502 454 9-957804 97 9-66")6!)7 552 10-334303 52 623774 454 957746 98 666! (29 552 333971 53 624047 454 957687 98 666360 551 333640 54 624319 453 957628 98 666691 551 333309 55 624591 453 957570 98 6671121 551 332979 56 624863 453 957511 98 667358 551 33-2(548 57 625135 452 957452 98 66768-2 550 332318 98 625406 453 957393 98 6(>80I3 550 331987 59 625677 45-2 957335 98 668343 550 331(557 60 625948 451 95727(5 98 6(58672 550 331328 | CttUM I Siae | Cotang. | 1 Tang-. 65 Degrees. LOGARITHMIC SINES, COSINES, ETC. (25 Degrees.) 223 M. Sine D. Cosine _D. | Tang. D. | Coiang. | 9-625948 451 9-957276 98 9-6681 i73 550 10-331327 ] 626219 451 957217 98 669002 549 330998 o 62JH90 451 957158 98 669332 549 330668 3 6267150 45J 957099 98 669661 549 330339 4 627 tO 450 957(140 98 669991 548 331)009 5 627:fK) 450 9569H1 98 670320 548 329080 8 627 j 70 449 956921 99 670649 548 329351 7 6-27*40 449 950862 99 670977 548 329023 8 628 j 09 449 956803 99 671306 547 328694 9 688378 448 956744 99 671034 547 328366 1U 628647 448 956684 99 67 J 963 547 328037 31 9 628916 447 9-956625 99 9-672291 547 10-327709 12 629 H5 447 950566 99 672619 546 327381 13 629453 447 956506 99 672947 546 327053 14 629721 448 956447 99 673274 546 326726 15 629989 446 956387 99 673002 546 326398 16 630257 446 951)327 99 673929 545 326071 17 630524 446 956263 99 674257 545 325743 18 630792 445 956208 100 674584 545 325416 19 63IK59 445 956148 JOO 674910 544 325090 98 63 J 326 445 956089 100 675237 544 324763 21 9- S3 1593 444 9-956029 100 9-675564 544 10-324436 22 631859 444 955969 100 675890 544 324110 23 632125 444 955909 JOO 676216 543 323784 24 632392 443 955849 100 676543 543 323457 25 632058 443 955789 JOO 676869 543 323131 26 632923 443 955729 JOO 677194 543 322806 27 633189 442 955669 100 677520 542 322480 28 633454 442 955609 100 677846 542 322154 29 B337I9 442 955548 100 678171 542 32IH29 30 633984 441 955488 100 678496 542 321504 31 19-634249 441 9-955428 101 9-678821 541 10-321179 32 634514 441) 9553(58 101 679146 541 320854 33 634778 440 955307 J01 679471 541 3205-29 34 635042 440 955247 101 679795 541 320205 35 6o5306 439 955186 101 680120 540 319880 36 635570 439 955126 101 680444 540 319556 37 6.1X34 439 955065 101 680768 540 319232 38 636097 438 955005 101 681092 540 318908 39 6363150 438 954944 J01 681416 539 318584 40 636623 438 954883 101 681740 539 318260 41 1 9-636886 42 637148 437 437 9-954823 954762 101 101 9-682063 682387 539 539 10-317037 317613 43 637411 437 954701 101 682710 538 317290 44 637673 437 9.54640 101 683033 538 316967 45 637935 436 954579 101 683356 538 316644 46 638197 436 954518 102 683679 538 316321 47 638458 436 954457 102 684001 537 315999 48 638720 435 954396 102 684324 537 315676 49 638981 435 954335 102 684646 537 315354 50 639242 435 954274 102 684968 537 315032 5] 9-630503 434 9-954213 102 9-685290 536 10-314710 52 639764 434 954152 102 685612 536 314388 53 641)024 434 954090 102 685934 536 314066 54 640284 433 954029 102 686255 536 313745 55 640544 433 9539 6641(53 405 947995 110 716168 514 283832 31 30 664406 404 947929 110 716477 514 283523 30 31 9-664(548 404 9-947863 110 9-716785 514 10-283215 . 29 32 664891 404 947797 110 717093 513 282907 28 33 665133 403 947731 110 717401 513 282599 27 34 665375 403 947665 110 717709 513 282291 26 35 665617 403 947(500 110 718017 513 281983 25 36 665859 402 947533 110 718325 513 281675 24 37 666100 402 947467 110 718633 512 281367 23 38 666342 402 947401 110 718940 512 281060 22 39 666583 402 947335 110 719248 512 280752 21 40 666824 401 947269 110 719555 512 280445 20 41 9-667065 401 9-947203 110 9-719862 512 10-280138 19 42 667305 401 947136 111 720169 511 279831 18 43 667546 401 947070 111 720476 511 27S524 17 44 667786 400 947004 111 720783 511 279217 16 45 668027 400 946937 111 721089 511 278911 15 46 668267 400 946871 111 721396 511 278604 14 47 668506 399 94r>804 111 721702 510 278298 13 48 668746 3<>9 9467.18 111 722009 510 277991 12 49 668986 399 9-) lit ,71 111 722315 510 277685 11 50 669225 399 946(504 111 722621 510 277379 10 51 9-69464 398 9'946.->38 iii 9-722927 510 10-277073 9 52 6:o:i 398 946471 111 723232 509 276768 8 53 6WW42 398 946404 111 723538 509 276462 7 54 - 67(1181 397 946337 111 723844 509 276156 6 55 670419 397 94R270 112 724149 509 275851 5 5(5 670H.T8 397 94(5203 112 724454 509 275546 4 57 67(l89fi 397 946136 112 724759 508 275241 3 & 67li:t4 396 94ttO9 112 725065 508 274935 2 59 671372 390 946002 112 725369 508 274631 1 (HI 671609 3% 945935 112 725674 508 274326 Coane Cotang. I Tang. 62 Degrees. 226 (28 Degrees.) LOGARITHMIC SIXES, COSItfES, ETC. M. | Sine D. f Cosine D. Tansr. | D. Cotan ? . ~60" 9-671609 396 9-9459:15 1J2 9-7251)74 508 10-274326 1 671847 395 :M58I>8 112 7-25979 508 2740-21 59 2 67-2084 3p5 945800 112 7-2(1-284 507 273716 58 3 672321 395 945733 112 726588 507 273412 57 4 67-2558 395 945666 112 726892 507 273108 56 5 672795 394 945598 112 727197 507 272803 .">."> 6 673i>32 394 945531 112 727503 5U7 272499 54 7 6732(58 394 945464 113 727805 506 272195 53 8 673505 394 945396 113 728109 50(5 27I8-.JI 52 9 673741 393 945328 1J3 728412 5(1(5 271588 51 10 673977 393 945261 113 728716 506 271284 50 11 9-674213 393 9-945193 113 9-729020 506 10-270980 49 B 674448 392 945125 113 729323 505 270677 48 33 674(584 392 945058 113 729026 505 270374 47 34 f>74!H'J 392 'J449UO 113 729929 505 270071 46 35 67 5 J 55 392 944922 113 730233 505 269767 45 16 675390 391 944854 113 730535 505 269465 44 37 675624 391 944786 113 730838 504 269162 43 38 675859 391 944718 113 731141 504 268859 42 39 676094 391 944(550 113 731444 504 2(58556 41 20 676328 390 944582 314 73J746 504 268254 40 21 9-676562 390 9-944514 314 9-732048 504 30-267952 39 22 676796 390 944446 314 73-2351 503 267649 38 23 677030 390 944377 114 732653 503 267347 37 24 677264 389 944309 314 732955 503 267045 36 25 677498 389 944241 114 733257 503 2(56743 35 24(533 28 33 679360 387 943693 115 735668 501 204332 27 34 679592 387 943624 315 735969 501 2(54031 26 35 679824 386 943555 115 736269 501 263731 25 36 680056 386 943486 315 736570 501 263430 24 37 680288 386 943417 135 736873 501 263129 23 38 680519 385 943348 315 737173 500 262829 22 39 680750 385 943279 315 737473 500 2C2529 21 40 680982 385 943210 115 737773 500 262229 20 41 9-681213 385 9-943141 315 9-738071 500 30-261929 19 42 681443 384 943072 315 438371 500 261629 18 43 681674 384 943003 315 738671 499 201329 17 44 681905 384 942934 115 738971 499 261029 16 45 682135 384 942864 315 739271 499 200729 15 4f> 682365 383 942795 316 739570 499 'JH0430 14 47 682595 383 942726 316 739870 499 2(50130 13 48 682825 383 942656 316 740169 499 259831 12 49 683055 383 942587 316 740468 498 259532 31 50 683284 382 942517 116 740767 498 259233 30 51 9-683514 382 9-942448 316 9-741066 498 10-258934 9 52 6R3743 382 942378 116 741365 498 258635 8 53 683972 382 942308 116 741664 498 258336 7 54 684201 381 942239 316 741962 4!I7 258038- 6 55 684430 381 942169 116 742261 497 257739 5 56 684658 381 942099 116 742559 497 257441 4 57 684887 380 942029 116 742858 497 257142 3 58 685115 380 941959 116 743156 497 25f>844 2 59 685343 380 941889 117 743454 497 2511546 3 60 685571 330 941819 117 743752 496 256248 I Cosine | I | Cotang. 61 Degrees. I Tang. | M. LOGARITHMIC SINES, COSINES, ETC. (20 Degrees.) 227 M. | Cosine | D. | Tang. | D. | Cotang. 9-tJH.")f)7 1 380 9-941819 117 9-743752 496 10-256248 1 685799 379 94 J 749 117 744050 496 255950 2 686027 379 941(i79 117 744348 496 255052 3 686254 379 941609 117 744645 496 255355 4 6864*2 379 941539 117 744943 496 255057 5 686709 378 9414(19 117 745240 496 254760 6 686936 378 941398 117 745538 495 254462 7 687 J 63 378 941328 117 745835 495 254165 8 687389 378 941258 117 746132 495 25H868 9 687616 377 941187 117 746429 495 253571 10 6*7843 377 941117 117 746726 495 253274 It 9-688069 377 9-941046 118 9-747023 494 10-252977 M 6*8295 377 940975 118 747319 494 252681 J3 68*521 376 940905 118 747616 494 252384 14 688747 376 940*34 118 747913 494 252087 15 688972 376 940763 118 748209 494 251791 16 68!) 198 376 940(593 118 748505 493 251495 n 689423 375 940622 118 748801 493 251199 J8 689648 375 940551 118 749097 493 250903 19 689*73 375 940480 118 749393 493 250607 20 690098 375 940409 118 749689 493 250311 21 9-690323 374 9-94(1338 118 9-7499*5 493 10-250015 22 690:>48 374 940267 118 750281 492 249719 23 6!)0772 374 940196 118 750576 492 249424 24 690996 374 940125 1 19 750*72 492 249128 23 6! 1 12-20 373 940054 119 7511(57 492 248833 2f)71 368 938619 120 757052 489 242948 46 695892 368 938547 120 757345 488 242(555 47 696113 368 938475 120 757038 488 2-12362 48 696334 367 938402 121 757931 488 242069 49 696554 387 93*330 121 75*224 488 241776 50 696775 367 938258 121 758517 488 241483 51 9-696995 367 9-938185 121 9-75*810 488 10-241190 52 697215 366 938113 121 759102 487 240898 53 697435 366 938040 121 759395 487 240605 54 697654 366 937967 121 759(587 4*7 240313 55 697874 366 937895 121 759979 487 240021 56 698094 365 937822 121 760272 4?7 239728 57 698313 365 937749 121 7605(54 487 239436 58 698532 365 937676 121 760856 .486 239] 44 59 698751 365 93"604 121 761148 486 238852 60 698970 364 937531 121 761439 486 23856] I Cosine | Sme | | Cotang. | 60 Degrees. I Tang. | M. 228 (30 Degrees.) LOGARITHMIC SIXES, COSINES, ETC. M. Sme D. Cosine D. Tang. D. | Cotanjy. | 9-698970 364 9-907531 121 9-761439 486 10--2385I51 60 1 699189 364 9374.58 J22 761731 486 238269 59 2 699407 364 937385 122 762023 486 237977 58 3 699626 364 937312 122 702314 486 2376*6 57 4 699844 363 937238 122 762(106 485 237394 56 5 700062 363 937165 199 762897 485 237103 55 6 700280 363 937092 122 7031 88 485 236812 54 7 700498 363 937019 122 763479 485 2365-21 53 8 700716 363 93(5946 122 763770 485 23(5230 ;>2 9 700933 362 93(5872 122 7(54061 485 235939 51 10 701151 362 93(5799 122 764352 484 235648 50 11 9-701368 362 9-936725 122 9-764(543 484 10-235357 49 12 701585 362 936652 123 7(54933 484 235067 48 13 701802 361 936578 123 765824 484 234776 47 14 702019 361 936505 123 765514 484 23448(5 46 15 702236 361 936431 123 7(55805 484 234195 45 16 702452 361 936357 123 76C.095 484 233905 44 17 702669 360 936284 123 7(5(5385 483 233615 43 18 702885 360 936210 1*3 7(5(5(575 483 233325 42 19 703101 360 936136 123 766U65 483 233035 41 20 703317 360 936062 123 767255 483 23-2745 40 21 9-703533 359 9-935988 123 9-767545 483 10-232455 39 22 703749 359 935914 123 767834 483 '23-2166 38 23 703964 359 935840 123 768124 482 231876 37 24 7(i41V9 359 935766 124 768413 482 231587 36 25 704395 359 935692 124 768703 482 231297 35 20 704610 358 935618 124 768992 482 231008 34 27 704825 358 935543 124 769281 482 230719 33 28 705040 358 935469 124 769570 482 230430 32 29 705254 358 935395 124 7698GO 481 230140 31 30 705469 357 935320 124 770148 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 33 706112 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 30 706753 356 934873 124 771880 480 228120 24 37 706967 356 934798 125 772168 480 227832 23 38 707180 355 934723 125 772457 480 227543 22 39 707393 355 934 (54 9 125 772745 480 227255 21 40 707606 355 934574 125 773033 480 226967 20 41 9-707819 355 9-934499 125 9-773321 480 10-226679 19 42 708032 354 934424 125 773608 479 22(5392 18 43 708245 354 934349 125 7738! )f> 479 22(5104 17 44 708458 354 934274 125 774184 479 2-25816 16 45 708670 354 934199 125 774471 479 225529 15 46 7(18882 353 934123 125 774759 479 225241 14 47 709094 353 934048 125 77504(5 479 2-24954 13 48 70930fl 353 933973 125 775333 479 224(567 12 49 709518 353 933898 126 775621 478 2-24379 11 50 709730 353 933822 126 775908 478 224092 10 51 9-709941 352 9-9:0747 . 126 9-776195 478 10-223805 9 52 710153 352 933671 126 77(5482 478 223518 8 53 710364 352 933596 126 776769 478 223231 7 54 710575 352 933520 126 777055 478 222945 6 55 710786 351 933445 126 777342 478 222658 5 56 71 01197 351 933369 126 7776-28 477 222372 4 57 711208 351 933293 126 777915 477 222085 3 58 711419 351 933217 126 778201 477 221799 o 511 711629 350 9:0141 196 778487 477 221512 1 GO 71 1839 350 9330(56 126 778774 477 2-2 J -2-26 I Sine | | Cotanf*. I 59 Degrees. J M. LOGARITHMIC SINES, COSINES, ETC. (31 Degrees.) 229 M. I Sine D. | Cosine | D. | Trng. | D. | Colang. I o 9-711839 350 9-933066 126 9-778774 477 10-221226 60 712050 350 932990 127 7790(50 477 220940 59 2 712260 350 932914 127 779346 476 220654 58 3 712469 349 932838 127 779632 476 220368 57 4 712679 349 932762 127 779918 476 220082 56 5 712889 349 932685 127 7802<3 476 219797 55 6 713098 349 932609 127 780489 476 219511 54 7 713303 349 932533 127 780775 476 2111225 53 8 713517 348 932457 127 781060 476 218940 52 9 71 3726 348 932380 127 781346 475 218654 51 10 713935 348 932304 127 781631 475 2.18369 50 11 9-714144 348 9-932228 127 9-781916 475 10-218084 49 12 714352 347 932151 127 782201 475 217799 48 13 714561 347 932075 128 782486 ' 475 217514 47 14 714769 347 931998 128 782771 475 217229 46 15 714978 347 931921 128 783056 475 2-16944 45 Ifi 715186 347 931845 128 783341 475 216659 44 17 715394 346 9317(58 128 783626 474 2J6374 43 18 715(502 346 931691 128 783910 474 216090 42 19 715809 346 931(514 128 784195 474 215805 41 20 716017 346 931537 128 784479 474 215523 40 21 9-716224 345 9-931460 128 9-78474 474 10-215236 39 22 716432 345 931383 128 785048 474 214952 38 2:} 716639 345 931308 128 785332 473 214668 37 24 716846 345 931229 129 785616 473 214384 36 25 717053 345 931152 129 785900 473 214100 35 2(5 717259 344 931075 129 786184 473 213816 34 27 717466 344 930998 129 786468 473 213532 33 28 717673 344 930921 J29 78(5752 473 213248 32 29 717879 344 930843 129 787036 473 2129(54 31 30 718085 343 930766 129 787319 472 2J2<581 30 31 9-718291 343 9-930r>88 129 9-787603 472 10-212397 29 32 718497 343 93061 1 129 787886 472 212114 28 33 718703 343 93:1533 129 788170 472 211830 27 34 718909 343 930456 129 788453 472 211547 26 35 719114 342 930378 129 788736 472- 211264 25 30 719320 342 930300 130 789019 472 210981 24 37 719525 342 930223 130 789302 471 210698 23 38 719730 342 930145 130 789585 471 210415 22 39 719935 341 930067 130 789868 471 210132 21 40 720140 341 929989 130 790151 471 209849 20 41 9-720345 341 9-929911 130 9-790433 471 10-209567 19 42 720549 341 929833 130 790716 471 209234 18 43 720754 340 929755 130 790999 471 209001 17 44 720958 340 929677 130 791281 471 208719 J6 45 721162 340 929599 130 791563 470 208437 15 46 721366 340 929521 130 791846 470 208154 14 47 721570 340 9211442 130 792128 470 207872 13 48 721774 339 929364 131 792410 470 207590 12 49 721978 339 " 929286 131- 792692 470 207308 11 51) 722181 339 929207 131 792974 470 20702G 10 51 j-722385 339 9-929129 131 9-793256 470 10-206744 9 52 722588 339 929050 131 793538 4(59 206462 8 53 722791 338 928972 131 793819 469 206181 7 54 722994 338 928893 131 794101 469 205899 6 55 723197 338 928815 131 7941183 469 205G17 5 5(5 723400 338 928736 131 7(146(54 469 205336 4 57 "723603 337 928657 131 794945 469 205055 3 58 723805 337 928578 131 795227 469 204773 2 59 724007 337 928499 131 795508 468 204492 1 60 724210 337 928420 131 795789 468 204211 ' | Cosine | Sine | | Cotang. | 53 Decrees, Tang. ) M. 230 (32 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. | Sine Cosine | D. | Tang. | D. | Cota 9-724210 337 9-92H420 132 9-795789 468 10-204211 1 724412 337 928342 132 796070 468 203930 2 724614 336 928263 132 790351 468 203649 3 724816 336 928183 132 796632 468 203368 4 725017 336 928104 132 796913 468 203087 5 725219 336 928025 132 797194 468 202806 6 725420 335 927946 132 797475 468 202525 7 725622 335 927867 132 797755 468 202245 8 725823 335 927787 132 798038 467 201964 9 726024 335 927708 132 798316 467 201684 10 720225 335 927629 132 798596 467 201404 11 9-726426 334 9-927549 132 9-798877 467 10-201123 12 726626 334 927470 133 799157 467 200843 13 726827 334 927390 133 799437 467 200563 14 727027 334 927310 133 799717 467 200283 15 727228 334 927231 133 799997 466 200003 16 727428 333 92715% 133 800277 466 199723 17 727628 333 927071 133 800557 466 199443 18 727828 333 926991 133 800836 466 199164 19 728027 333 926911 133 801116 466 198884 20 728227 333 920831 133 801396 4(56 198604 21 9-728427 332 9-926751 133 9-801675 466 10-198325 22 728626 332 926671 133 8C1955 468 198045 23 728825 332 926591 133 802234 465 ] 97766 24 729IJ24 332 926511 134 802513 465 197487 25 729223 331 926431 134 802792 465 197208 26 729422 331 926351 134 803072 465 196928 27 729621 331 926270 134 803351 465 196649 28 729820 331 926190 134 803030 465 196370 29 730018 330 926110 134 803908 465 196092 30 730216 330 926029 134 804187 465 195813 31 9-730415 330 9-925949 134 9-804466 464 10-195534 32 730613 330 925868 134 804745 464 195255 33 730811 330 925788 134 805023 464 194977 34 731009 329 925707 134 805302 464 194698 35 731206 329 925626 134 805580 464 194420 3(5 731404 329 925545 135 805859 464 194141 37 731(502 329 925465 135 806137 464 193863 38 731799 329 925384 135 806415 463 193585 39 73 J 996 328 925303 135 806693 463 193307 40 732193 328 925222' 135 806971 463 193029 41 9-732390 328 9-925141 135 9-807249 463 10-192751 42 732587 328 925060 135 807527 463 192473 43 732784 328 924979 135 807805 463 190195 44 732980 327 924897 135 808083 463 101917 45 733177 327 924P16 135 808361 463 191639 46 733373 327 9247H5 136 808638 4C2 1913G2 47 733569 327 924654 136 808916 462 191084 48 733765 327 924572 136 809193 . 4C2 190807 49 733961 326 924491 136 809471 462 190529 50 734157 326 924409 136 809748 462 190252 51 9-734353 326 9-924328 136 9-810025 462 10-189975 52 73-1549 326 92-1246 136 810302 4C2 189698 53 734744 325 924164 136 810580 462 189420 54 734939 325 924083 136 810857 462 1P9143 55 735135 325 924001 136 811134 461 18P866 56 735330 325 5)23919 136 811410 461 18P590 57 735525 325 923837 136 811687 461 18P313 58 7?5719 324 9237.-).-) 137 811964 461 188036 59 735914 324 923673 137 812241 461 187759 <*) 736109 324 923591 137 812517 461 187483 | Cotan-. I Tta % . \ M. 67 Degrees. LOGARITHMIC SINES, COSINES, ETC. (33 Degrees.) 231 M. ' Sine } D. Cosine | D. Tang. D. | Cotang. 1 9-730109 324 9-923591 137 9-812517 461 10-187482 00 1 730303 324 923509 137 812794 401 187206 59 2 736498 324 923427 137 813070 461 180930 58 3 730692 323 923345 137 813347 460 186053 57 4 736886 323 923263 J37 813623 460 180377 56 5 737080 323 923181 137 813899 460 180101 55 6 737274 323 923098 137 814175 460 185825 54 7 737467 323 923016 137 814452 460 185548 53 8 737(501 322 922933 137 8147-28 460 185272 52 9 737855 322 922851 137 815004 460 184996 51 10 738048 322 922768 138 815279 460 184721 50 ]] 9-738241 322 9-922086 138 9-815555 459 10184445 49 12 738434 322 922603 138 815831 459 1H4109 48 13 738627 321 922520 138 816107 459 183893 47 14 738820 321 922438 138 816382 459 183618 46 15 739013 321 922355 138 816658 459 183342 45 16 739206 321 922272 138 816933 459 183007 44 17 739398 321 922189 138 817209 459 182791 43 18 739590 320 922 1G6 138 817484 459 182516 42 19 739783 320 9221,23 138 817759 459 182241 41 20 739975 320 921940 138 818035 458 181965 40 21 9-740167 320 9-921857 139 9-818310 458 10-181690 39 22 740359 320 921774 139 818585 458 181415 38 23 740550 319 921691 139 818800 458 181140 37 24 740742 319 92 J 607 139 819135 458 180865 36 25 740934 319 921524 139 819410 458 180590 35 20 741125 319 921441 139 819684 458 180316 34 27 741316 319 921357 139 819959 458 180041 33 28 74 J 508 318 921274 139 820234 458 179766 32 29 741699 318 921 190 139 820508 457 179492 31 30 741 889 318 921107 139 820783 457 179217 30 31 9-742080 318 9-921023 139 9-821057 457 10-178943 29 32 742271 318 920939 140 821332 457 178668 28 33 7424G2 317 92C856 140 8210(,6 457 178394 27 34 742052 317 920772 140 821880 457 178120 26 35 742842 317 920G88 140 822154 457 177846 25 30 743033 317 92G6G4 140 822429 457 177571 24 37 74:>223 317 92G520 140 822703 457 177297 23 38 743413 316 920436 140 822977 456 177023 22 39 743602 316 920352 140 823250 456 176750 21 40 743792 316 D202C8 140 823524 456 176476 20 41 9-743982 316 9-920184 140 9-823798 456 10-176202 19 42 744171 316 920099 140 824072 456 175928 18 43 744361 315 921015 140 824245 456 175055 17 44 744550 315 919931 141 824619 456 17o381 16 45 744739 315 919846 141 824feD3 456 175107 15 40 744928 315 919702 141 825JC6 456 174834 14 47 745117 315 9J9077 141 825439 455 174501 13 48 745306 314 919593 141 825713 455 174287 12 49 745494 314 919508 141 825986 455 174014 11 50 745683 314 919424 141 826259 455 173741 10 51 9-745871 3J4 9-919339 141 9-826532 455 10-173468 9 52 746059 314 919254 141 820805 455 173195 8 53 746248 313 919109 141 827078 455 172922 7 54 740436 313 919085 141 827351 455 172649 6 55 746624 313 919000 141 827624 455 172376 5 56 746812 313 9J8915 142 827897 454 172103 4 57 746999 313 918830 142 828170 454 171!:30 3 X 747187 312 918745 142 828442 454 171558 2 59 747374 312 918659 142 828715 454 171285 j 60 747562 312 918574 142 828987 454 171013 l I Coeine I Sine | Cotang. Tnng. 56 Degrees 232 (34 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. | Sine D. Cosine D. | Targ. f D. Golan*. 9-7475G2 312 9-918574 142 9-828987 454 10-171013 1 747749 312 i) 18489 142 829200 454 170740 2 747J36 312 918404 142 829532 454 170408 3 71dl23 311 918318 J42 82-.i*< 15 454 170195 4 746310 311 9 J 8233 142 Kin 177 454 169923 5 748497 311 918147 142 83u3-19 453 169651 746683 311 91riM>2 142 83(>021 453 10! (379 7 748870 311 917976 143 83urt93 453 109107 8 74JU56 310 917891 143 8311t>5 453 168835 9 74J243 310 9178u5 143 831437 453 16KVJ3 10 7494-29 3JO 917719 143 831709 453 168291 11 9-749615 310 9-917634 143 9-831981 453 10-168019 12 74J3J1 310 917548 143 832253 453 107747 13 749987 309 917462 143 832525 453 107475 14 750172 309 917376 143 832796 453 107204 15 750358 309 9J7290 143 833068 4f>2 100932 10 750543 309 9 J 7204 143 833339 452 - KiOOOl 17 750729 309 917118 144 833011 452 100389 18 750914 308 917032 144 833882 452 100118 19 751J99 308 916946 144 834154 452 105846 20 751284 308 916859 144 834425 452 105575 21 9-751469 308 9-910773 144 9-834096 452 10-105304 2-2 751G54 308 910087 144 834907 452 1 65033 23 751839 308 910000 144 835238 452 104702 24 752u23 307 916514 144 835509 452 104491 25 752-208 307 916427 144 835780 451 104220 26 752392 307 916341 144 830051 451 163949 27 752576 307 91i,254 144 830322 451 103078 28 752760 307 916167 145 830593 451 163407 29 752944 306 916081 145 830804 451 163136 30 753128 306 915994 145 837134 451 162866 31 9-753312 306 9-915907' 145 9-837405 451 10-162595 32 753495 306 915820 145 837675 451 162325 33 753679 306 915733 145 837946 451 102054 34 753862 305 915646 145 838216 451 101784 35 754046 305 915559 145 838487 450 101513 36 754229 305 915472 145 838757 450 101243 37 754412 305 915385 145 839027 450 100973 38 754595 305 915297 145 839297 450 100703 31) 754778 304 915210 145 839508 450 100432 40 754960 304 915123 146 839838 450 160162 41 9-755143 304 9-915035 146 9-840108 450 10-159892 4-2 755326 304 914948 146 840378 450 159622 43 755508 304 914860 146 840047 450 159353 44 755690 304 914773 146 840917 449 159083 45 755872 303 914685 146 841187 449 158813 46 756054 303 914598 146 841457 449 158543 47 756236 303 914510 146 841726 449 158274 48 756418 303 914422 146 841996 449 158004 49 756600 303 914334 146 842206 449 157734 90 756782 302 914246 147 842535 449 157465 51 9-756963 302 9-914158 147 9-842805 449 10157195 a 757144 302 914070 147 843074 449 150926 53 757326 302 913982 147 843343 449 150657 54 757307 302 913894 147 843612 449 156388 55 757688 301 913806 147 843882 448 156118" 56 757869 301 913718 147 844151 448 155849 57 758050 301 913630 147 844420 448 155580 58 758230 301 913541 147 844689 448 155311 r>u 758411 3*)1 91:5453 147 844958 448 15.-JU42 90 758591 301 913365 147 845227 448 154773 ! Cosine | | S.ne Cotang. I Tang. M. 55 Degrees. , LOGARITHMIC SINES, COSINES, ETC. (35 Degrees.) 233 Sine ( D. | Cosine 1 D. | Tang-. | D. | Golan?. | 9-758591 301 9-913365 147 9.845227 448 10-154773 (k) 1 758772 300 913276 147 845496 448 154504 59 2 758952 300 913187 148 845764 448 154236 58 3 759132 300 913099 148 846033 448 153967 57 4 75U312 300 913010 148 846302 448 153698 58 5 75U492 300 912922 148 846570 447 153430 5.) . 6 7.19G72 299 912d33 148 846839 447 153161 54 7 75985-2 299 912744 148 847107 447 152893 53 8 760031 299 912655 148 847376 447 152624 52 9 700211 299 912566 148 847644 447 152356 51 10 760390 299 912477 148 847913 447 152087 50 11 9-7G0569 298 9-912388 148 9.848181 447 10-151819 49 12 760748 298 D12299 149 848449 447 151551 48 13 760927 298 912210 149 848717 447 151283 47 14 761106 298 912121 W9 848986 447 151014 46 15 761-285 298 9121)31 149 84 25 763067 296 9imo 150 851931 446 148069 35 26 763245 296 911040 150 852199 446 147801 34 27 763422 296 910956 150 852466 446 147534 33 28 763600 295 910866 150 852733 445 147267 32 29 763777 295 91 0776 150 853001 445 1461)99 31 30 763954 295 910686 150 853268 445 146732 30 31 9-764131 295 9-910596 150 9-853535 445 10-146465 29 32 764308 295 910506 150 853802 445 146198 28 33 704485 294 910415 150 854069 445 145:>31 27 34 764662 294 910325 51 8543J6 445 145664 26 35 764838 294 910235 51 854603 445 145397 25 36 765015 294 910144 51 854870 445 145130 24 37 765191 294 910054 51 855137 445 144863 23 38 765367 294 909963 51 855404 445 144596 22 39 765544 293 909873 51 855671 444 144329 21 40 765720 293 909782 51 ' 855938 444 144062 20 41 9-765896 293 9-909691 51 9-856204 444 10-143796 19 42 766072 293 909601 51 856471 444 143529 18 43 766247 293 909510 51 856737 444 143263 17 44 766423 293 909419 151 8570;t4 444 142996 6 45 766598 292 909328 152 857270 444 142730 5 46 766774 292 90U237 152 857537 444 142463 4 47 766949 292 909146 152 857803 444 142197 13 48 767124 OIJO 909055 152 858069 444 141931 12 49 767300 292 908964 152 858336 444 141664 H 50 767475 291 908873 152 858602 443 141398 10 51 9-767649 291 9-908781 152 9-858863 443 10-141 1.72 9 52 767824 291 90H690 152 85U134 443 140866 8 53 767999 291 908599 152 859400 443 140600 7 54 768173 291 908507 152 859666 443 140334 6 55 768348 290 908416 153 859932 443 140068 5 56 768522 290 908324 153 860198 443 139802 4 57 768097 2!0 908233 153 860464 443 13<). r >36 3 58 768871 2!K) 908141 153 860730 443 139270 2 59 769045 2!H) 908049 153 860995 443 139005 1 00 769219 290 907958 153 861261 443 138739 Sine | I Cotang. | I Tang. 54 Degrees. 234 (36 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. | Sine | D. Cosine | D. | Tan?. | D. Cotang. j 9-769219- 290 9-907958 153 9-861261 443 10-138739 60 1 709393 289 907806 153 861527 443 138473 59 2 709566 289 907774 153 861792 442 138208 58 3 7G;)740 289 907082 253 862058 442 137942 57 4 709913 289 907590 153 862323 442 137077 50 5 770087 289 907498 153 802589 44t> 137411 55 6 771)200 288 907-406 153 882854 442 137146 54 7 770433 288 907314 154 803119 442 136881 53 8 770006 288 907222 154 8G3385 442 136015 59 9 ' 770779 288 907129 154 803050 442 136350 51 10 770952 288 907037 154 803915 442 130085 50 11 9-771125 288 9-906945 154 9-804180 442 10-135820 49 12 77121)8 287 906852 154 804445 442 135555 48 13 771470 287 900700 154 804710 442 135290 47 14 77 J 043 287 906607 154- 864975 441 135025 40 15 771815 287 900575 154 805240 441 134700 45 16 771987 287 906482 154 805.505 441 134495 44 17 772159 287 900389 155 805770 441 134230 43 18 772331 280 906290 155 800035 441 J33'..G5 42 19 772503 2F6 90(W04 155 860300 441 133100 41 20 772675 286 900J11 155 800564 441 133436 40 21 9'772847 28G 9-900018 155 9-866829 441 10-133171 39 22 773018 286 905925 155 867094 441 132906 38 23 773190 286 905832 155 367358 441 132642 37 24 773301 285 905739 155 867623 441 132377 30 25 773533 285 905645 155 8o7S87 441 132113 35 20 773704 285 905552 155 8(58152 440 131848 34 27 773875 285 905459 155 808416 44U 131584 33 28 77J04G 285 905366 156 80uoO 440 131320 32 99 774217 285 905272 156 808945 440 131055 :n 30 774388 284 905179 156 869209 440 13(1791 30 31 9-774558 284 9-905085 156 9-869473 440 10-130527 2<) 32 774729 284 904992 156 809737 440 J 30203 28 33 774899 284 904898 156 870001 440 12!l!l;9 87 34 775070 284 904804 156 870205 440 129735 28 35 775240 284 904711 156 870529 440 129471 25 31. 77:>410 283 904617 156 870793 440 129207 24 37 775580 283 904523 156 871057 440 128943 23 38 775750 283 904429 157 871321 440 128679 22 39 7751120 283 904335 157 871585 440 128415 21 40 776090 283 90424] 157 871849 439 128151 20 41 9-776259 283 9-904147 157 9-872112 439 10-127888 19 42 77G429 282 ' 904053 157 872376 439 127624 J8 43 770598 282 903959 157 872640 439 127360 17 44 776768 282 903864 157 872903 439 127097 10 45 776937 282 903770 157 873167 43!) 120833 15 46 777106 282 903676 157 873430 439 126570 14 47 777275 281 903581 157 873694 439 120306 13 48 777444 281 903487 157 873957 439 120043 12 49 777013 281 903392 158 874220 439 125780 11 50 777781 281 903298 158 874484 439 125516 II) 51 9-777950 281 9-903203 158 9-874747 439 10-125253 9 52 778119 281 903108 158 875010 439 124990 8 53 778287 280 9(0014 158 875273 438 124727 7 54 778455 280 902919 158 875536 438 124404 6 55 778(524 280 902824 158 875800 438 124200 5 50 778792 280 902729 158 876063 438 123937 4 57 77PSK50 280 902034 KB 876326 438 123(574 3 58 779128 280 902539 159 870589 438 123411 2 59 779295 279 902444 159 876851 438 123)49 1 60 779403 279 903349 159 8771 14 438 122886 | Cotang. I Tang. 53 Degrees. LOGARITHMIC SINES, COSINES, ETC. (37 Degrees.) 235 M. Sine I D. | Cosine 1 D. I Tan*. | D. | Cotang. 9-779463 279 9-902349 159 9-877114 438 10-122886 60 1 779631 279 902253 159 877377 438 122(523 59 2 779798 279 902158 159 877640 438 122360 58 3 779966 279 902063 159 877903 438 122097 57 4 780133 279 901967 159 878165 438 121835 56 5 780300 278 901872 159 878428 438 121572 55 6 780467 278 901776 159 878691 438 121309 54 7 780634 278 901681 159 878953 437 121047 53 R 780801 278 90 J 585 159 879216 437 120784 ;>2 780968 278 901490 159 879478 437 120522 51 10 781134 278 901394 160 879741 437 120259 50 ]1 9-781301 277 9-901298 160 9-880003 437 10-119997 49 is 781468 277 901202 160 880265 437 119735 48 13 781634 277 901106 160 880528 437 119472 47 14 781800 277 901010 160 880790 437 119210 46 15 781966 277 900914 160 881052 437 118948 45 10 782132 277 900818 160 881314 437 118686 44 IT 782298 276 900722 ]60 881573 437 1J8424 4;J 18 782464 276 900626 160 881839 437 118161 42 11) 782630 276 900529 160 882101 4H7 117899 41 20 782796 276 900433 161 882363 436 117637 40 21 9-782961 276 9-900337 161 9-882625 436 10-117375 3'J 23 783127 276 900240 161 882887 436 1171J3 38 S3 783292 275 900144 Ifil 883148 436 116852 37 24 783458 275 900047 Itil 883410 436 116590 36 So 783623 275 899951 361 8*3672 436 116328 35 96 783788 275 899854 J61 883934 436 116066 34 27 783953 275 899757 161 884196 436^ 115804 33 38 784118 275 899660 161 884457 436 115543 32 2!) 784282 274 899564 161 884719 436 115281 31 30 784447 274 89941)7 162 884980 436 115020 30 31 9-784612 274 9-899370 162 9-885242 436 10-114758 29 33 784776 274 899273 162 885503 436 114497 28 33 784941 274 899176 162 885765 436 114235 27 34 785105 274 899078 162 886026 43(5 113974 26 35 785269 273 898981 162 886288 436 113712 25 36 785433 273 898884 162 886549 435 113451 24 37 785597 273 898787 162 886810 435 1131;)0 23 38 785761 273 898689 162 887072 435 112928 22 39 785925 273 898592 162 887333 435 112667 21 40 786089 273 898494 163 887594 435 11240G 20 41 9-786252 272 9-898397 163 9-887855 435 10-112145 19 4-2 786416 272 898299 163 888116 435 11J8H4 18 43 786579 272 898202 163 888377 435 111623 17 44 786742 272 898104 163 888(539 435 111361 16 4."> 786906 272 898006 163 888'JOO 435 111100 15 4(1 787069 272 897908 163 889160 435 110840 14 47 787232 271 897810 163 889421 435 110579 13 48 787395 271 897712 163 889(182 435 110318 13 4!) ' 787557 271 897614 163 889943 435 110057 11 50 787720 271 897516 163 890204 434 109796 10 f>l 9-787883 271 9-897418 164 9-890465 434 10-109535 9 52 788045 271 897320 164 89(1725 434 109275 8 5:5 788208 271 8! (72-2-2 164- 890986 434 109014 7 54 788370 270 8971 -23 164 891247 434 108753 6 55 7*8532 270 897025 164 891507 434 108493 5 56 7HW94 270 PW926 164 891768 434 108232 4 57 7^'8856 270 896828 Ifi4 892028 434 1071)72 3 58 789018 270 896729 164 892289 434 1077 11 2 30 7891 80 270 896K3I 164 892549 434 107451 1 GO 789342 269 896532 164 892810 434 107190 | Cosine | | Sine I 52 Degrees. Cotang. I Tang. 236 (38 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. | Sine | D. | Cosine | D. | Tang. | Cotang. | u 9-789342 269 9-896532 164 9-892X10 434 10-107190 60 1 789504 269 896433 165 893070 434 106930 59 2 789665 269 896335 165 893331 434 Hi6GG9 58 3 789827 269 896236 165 893591 434 100409 57 4 789988 269 896137 165 893X51 434 106149 56 5 790149 269 896038 165 894JI1 434 105X89 55 6 790310 268 895939 165 894371 434 105029 54 7 790471 268 895840 165 894632 433 105368 53 8 790632 268 895741 165 894892 433 105108 52 9 7907 y3 2(58 895641 165 895152 433 104X48 51 JO 790954 268 895542 165 895412 433 104588 50 ]] 9-791115 268 9-895443 166 9-895672 433 10-104328 49 12 791275 2b7 895343 j(W 895932 433 1040(58 48 J3 79 J 436 267 895244 166 896192 433 103808 47 14 791 596 267 895145 5(5 896452 433 103548 46 15 791757 267 895U45 ]-i6 896712 433 1032X8 45 16 791917 867 894945 ]i>6 896971 433 103029 44 17 792077 267 81)4846 ]i>6 897231 433 1027(59 43 18 792237 266 894746 166 897491 433 102509 42 19 792397 266 81)4046 166 897751 433 102249 41 20 792557 266 89454(5 166 898010 433 101990 40 21 9-792716 266 9-894446 167 9-8982TO 433 10-101730 39 22 792876 266 894346 1(57 89X530 433 101470 38 23 793035 266 894246 167 898789 433 101211 37 24 793195 265 894146 167 899(1-19 432 100951 36 25 793354 265 84*4046 167 X99.'I(IS 432 j 00692 35 2G 793514 265 893946 167 899568 432 100432 34 27 793073 265 893846 167 X99827 432 ] 00 173 33 28 793832 265 893745 167 900086 432 099914 32 29 793991 265 893645 167 900346 432 099654 31 30 794150 264 893544 167 900605 432 099395 30 31 9-794308 264 9-893444 1C8 9-90(1864 432 10-099136 29 32 794467 264 893343 168 901124 432 0! 18876 28 33 794626 264 893243 168 901383 432 098017 34 794784 264 893142 168 901642 432 098358 96 35 794942 264 893041 108 901901 432 098099 25 30 795101 264 8929-10 108 902160 432 097840 24 37 795259 264 892X39 168 902419 432 097581 23 38 795417 263 892739 108 902679 432 097321 22 39 795575 263 892638 168 902938 432 097062 21 40 795733 263 892536 168 903197 431 096803 20 41 9-795891 263 9-892435 169 9-903455 431 10-096545 19 42 796049 263 892334 169 903714 431 096286 18 43 796206 263 892233 169 903973 431 09(5027 17 14 790364 262 892132 169 904232 431 095768 10 45 796521 262 892(130 169 904491 431 09550!) 15 46 79(5679 262 891929 1(59 904750 431 095250 14 47 79(5X36 262 891X27 109 905008 431 09-4 9! 12 13 48 796993 262 891726 109 905207 431 094733 12 49 797150 261 891624 ]09 905526 431 094474 11 50 797307 261 891523 170 905784 431 094216 10 51 9-797464 261 9-891421 70 9-906043 431 10-093957 9 52 797621 261 891319 70 900302 431 093098 8 53 797777 261 891217 70 906500 431 093440 7 54 791934 261 891115 70 906X19 431 093181 6 55 79X091 261 ' 891013 70 907077 431 0925)23 5 56 79X247 261 890911 70 907336 431 092664 4 57 798403 260 890809 70 907594 431 092406 3 58 79X5HO 2(50 890707 170 907X52 431 092148 2 SB 798716 2(50 890605 170 90X111 430 Oil 1X89 1 60 798872 260 890503 170 908369 430 091631 Conne | Sine | | Cotang. | Tang. 51 Degrees. LOGARITHMIC SINES, COSINES, ETC. (39 Degrees.) 237 M. | Sine I | D. | Tang. | I). | Coin 9-798872 260 9-890503 170 9-908309 430 10-091631 60 1 799028 260 890400 171 908G28 430 091372 59 2 799184 260 891)298 171 908886 430 091114 58 3 791)339 259 89J195 71 909144 430 090856 57 4 79949.1 259 890093 71 909402 430 090598 56 5 799(551 259 88.)9;)0 71 909660 430 090340 55 6 79980G o-,(j 889888 71 909918 430 090082 54 7 799MB 259 889785 71 910177 430 089823 53 8 800117 259 889682 71 910435 430 089565 52 9 800272 258 889579 71 910693 430 089307 51 10 800427 258 889477 k 71 910951 430 089049 50 11 9-800582 258 9-889374 72 9-911209 430 10-088791 49 12 800737 258 889271 72 91 1467 430 088533 48 13 800892 258 8891(58 72 91 1724 430 088276 47 14 801047 258 889064 72 911982 430 088018 46 IS 801201 258 888961 72 912240 430 0877(50 45 16 801 350 257 888858 72 912498 430 087502 44 17 801511 257 888755 72 912756 430 087244 43 18 801665 257 888651 72 913014 429 086986 42 .1!) 801819 257 888548 72 913271 429 086729 41 20 801973 257 888444 73 913529 429 086471 40 21 9-802128 257 9-888341 73 9-913787 429 10-086213 39 2-2 802282 256 888237 73 914044 429 085956 38 2i{ 802436 256 888134 73 914302 429 085(598 37 24 802589 256 888030 73 914560 429 085440 36 2.-, 802743 256 887926 73 91481*7 429 085183 35 220 428 081580 21 40 805039 254 886362 75 918677 428 081323 20 41 9-805191 254 9-886257 75 9-918934 428 10-081066 19 42 805343 253' 886152 75 919191 428 0808i)9 18 43 805495 253 88(5047 75 919448 428 080552 17 44 805647 253 885942 75 919705 428 080295 16 45 805799 253 885837 75 919962 428 080038 15 46 805951 253 885732 75 920219 428 079781 14 47 80(5103 253 885(527 75 92047(5 428 079524 13 4H 806254 253 885522 75 920733 428 0792(57 12 4!) 801 140(5 252 885416 75 920990 428 079010 11 50 806557 252 885311 76 921247 428 078753 10 51 9-80(5709 252 9-885205 76 9-921503 428 10'()78497 9 52 80(5860 252 885100 76 9217(50 428 078240 8 5:5 807011 252 884994 76 922017 428 077983 7 54 807163 252 884889 76 922274 428 077726 6 55 807314 252 884783 76 922530 428 077470 5 50 807465 251 884677 76 922787 428 077213 4 57 807615 251 884572 76 923044 428 076956 3 58 H07766 251 8844(56 76 923300 428 07(5700 2 59 807917 251 884360 176 923557 427 876443 1 GO 808067 251 884254 177 923813 427 076187 Cosine | I. 50 Decrees. Tang. 238 (40 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. Sine D: Cosine | D. | Tang. D. Can ff . | 9-808007 251 9-884254 177 9-923813 427 10-070187 1 8118218 251 884148 177 924070 427 075930 2 8083(58 251 884042 177 924327 427 075073 3 808519 250 883936 177 924583 427 0754J7 4 808609 250 , 883829 177 924840 427 0751(50 5 808819 250 883723 177 925096 427 074904 6 808909 250 883017 177 925352 427 074(548 7 8091 19 250 883510 177 925009 427 074391 8 809-209 250 883404 177 9258(55 427 074 J 35 9 8(19419 249 883297 78 926122 427 073878 JO 809509 249 883191 78 920378 427 073022 11 9-809718 249 9-883084 78 9-9200:14 427 10-073366 12 809868 249 8821*77 78 920890 427 073110 13 810017 249 882871 78 927147 427 07-2853 J4 810167 249 882764 78 927403 427 072597 15 810316 248 882657 78 927059 427 072341 16 810465 248 882550 78 927915 427 072085 17 8i0614 248 882443 78 928171 427 071829 18 810763 248 882336 79 928427 427 071573 19 810912 248 882229 79 928(583 427 071317 2U 81 J 061 248 882121 79 928940 427 071060 21 9-811210 248 9-882014 79 9-929196 427 10-070804 22 811358 247 88 J 907 79 929452 427 070548 23 811507 247 881799 79 929708 427 070292 24 81 1055 247 8*1092 79 929904 426 070036 25 811804 247 881584- 79 930220 426 009780 26 811952 247 881477 79 930475 426 069525 27 812100 247 881309 79 930731 426 0(59209 23 8 12248 247 881201 180 930987 426 069013 29 812396 246 881153 180 931243 426 008757 30 812544 246 881046 180 931499 426 068501 31 9-812092 246 9-880938 180 9-931755 426 10-068245 32 812H40 246 880830 180 932010 426 OG7990 33 812988 246 880722 180- 9322GG 426 OG7734 34 813135 246 880013 180 QOOjOO 426 CG7478 35 813283 246 880505 180 932778 426 OC7222 36 813430 245 880397 180 933033 426 06G9G7 37 813578 245 88(1289 181 933289 426 OGG711 38 813725 245 880180 181 933545 426 OGG455 39 813872 245 88W072 181 933800 426 06G200 40 814019 245 879963 181 934056 426 065944 41 9-814166 245 9-879855 181 9-934311 426 10-065689 42 814313 245 87'.)746 181 934507 426 065433 43 814400 244 879037 181 934823 426 065177 44 814(507 244 879529 181 935078 426 004922 45 814753 244 879420 181 935333 420 0(540(57 46 8I49IK) 244 87! 13 11 181 935589 426 0644 1 1 47 815046 244 879202 182 935844 426 064156 48 815193 244 879093 182 936100 4-2(5 0(53900 4!) HI 5339 244 878984 182 930355 4-26 003045 50 815485 243 878875 182 930(510 426 063390 51 9-815031 243 9-878766 182 9-930806 425 10-003134 52 815778 243 878056 182 937 121 425 002879 53 8 15! 124 243 878547 182 937376 425 00-2024 54 H| til nil) 243 878438 182 937(532 425 002308 55 810215 243 878328 182 937887 425 002 113 56 810301 243 8782 19 183 938142 425 001858 57 81 651 17 24-2 878109 183 938398 425 001002 58 810052 242 877999 183 938053 425 001347 59 816798 242 877890 183 938908 425 001092 60 810943 242 877780 183 939163 425 0(50837 I Cosine Sine | | Cotang. | 49 Degrees. Tang. I M. LOGARITHMIC SINES, COSINES, ETC. (41 Degrees.) 239 Sine [ D. | Tang. D. 1 Cotang. 9-816943 242 9 877780 183 9-939163 425 10-060837 60 1 817088 242 877670 183 9394 J 8 425 060582 59 2 817233 242 8775t>0 183 939673 425 060327 58 3 817379 242 877450 183 939928 425 060072 57 4 817524 241 877340 1H3 941JJ83 425 059F17 56 5 817668 241 877230 184 9404:i8 425 059562 55 6 817813 241 877120 184 940694 425 059306 54 7 817958 241 877010 184 940949 425 059051 53 8 818103 241 871)899 184 941204 425 058796 52 9 818247 241 876789 184 941458 425 058542 51 10 818392 241 876678 184 941714 425 058286 50 11 9-818536 240 9 876568 184 9-941968 425 10-058032 49 12 818681 240 8715457 184 942223 425 057777 48 is 818825 240 876347 184 942478 425 057522 47 14 818969 240 876236 185 942733 425 057267 46 J5 819113 240 876125 185 942988 425 057012 45 J6 819257 240 87(5014 185 943243 425 056757 44 17 819401 240 875904 185 943498 425 056502 43 J8 819545 239 875793 185 943752 425 056248 42 19 819689 239 875682 185 944007 425 055993 41 20 819832 239 875571 185 944262 425 055738 40 21 9-819976 239 9-875459 185 9-944517 425 10-055483 39 22 820120 239 875348 185 944771 424 055229 38 23 820263 239 875237 185 945026 424 054974 37 24 820406 239 875126 186 945281 424 054719 36 25 820550 238 875014 186 945535 424 054465 35 26 820693 238 874903 186 945790 424 054210 34 27 820836 238 874791 186 946045 424 053955 33 28 820979 238 874680 186 946299 424 053701 32 29 821122 238 874568 186 946554 424 053446 31 30 821265 238 874456 186 946808 424 053192 30 31 9-821407 238 9-874344 186 9-947063 424 10-052937 29 32 821550 238 874232 187 947318 424 052(582 28 33 82 1093 237 874121 187 947572 424 052428 27 34 821835 237 874009 187 947826 424 052174 2e 35 821977 237 873896 187 948081 424 051919 25 30 822120 237 873784 187 948336 424 051664 24 37 8222G2 237 873672 187 948590 424 051410 23 38 822404 237 8735CO 187 948844 424 051156 22 39 822546 237 873448 187 949099 424 050901 21 40 822688 236 873335 187 949353 424 050647 20 41 9-822830 236 9-873223 187 9-949607 424 10.050393 19 42 822972 236 873110 188 949862 424 050138 18 43 823114 236 872998 188 950116 424 049884 17 44 823255 236 872885 188 950370 424 049630 16 45 823397 236 872772 188 950625 424 049375 15 40 823539 236 872659 188 950879 424 049121 14 47 823680 235 872547 188 951133 424 0488(57 13 48 823821 235 872434 188 951388 424 048612 12 49 823963 235 872321 188 951642 424 048358 11 50 824104 235 872208 188 951896 424 048104 10 51 9-824245 235 9-872095 189 9-9521.50 424 10-047850 9 52 824:186 235 871981 189 952405 424 047595 3 53 82-4527 235 871 HIM 189 952659 424 047341 7 54 8246(>8 234 871755 189 952913 424 047087 6 55 824808 :J4 871641 189 9531(57 423 04H833 5 56 824949 2M 871528 189 953421 423 046579 4 57 825090 234 871414 189 953675 423 046325 3 58 825230 234 871301 189 953929 423 046071 2 59 825371 14 871187 189 954183 423 045817 60 825511 234 871073 190 954437 423 045563 Sine | | Cotang. 48 Degrees. I Tang. f 240 (42 Degrees.) LOGARITHMIC SINES, COSINES, ETC. | D. | Tang. D. | Cotang. 9-825511 234 9-871073 190 9-954437 423 10-045563 (50 1 825651 233 870960 190 954691 423 045309 59 2 8-25791 233 870846 190 954945 423 045055 58 3 8-25J31 233 870732 190 955200 423 044800 57 4 82<>G7l 233 870618 190 955454 423 044546 56 5 82(5211 233 870504 190 955707 423 044-293 55 6 826351 233 870390 1'JO 955S)1 423 044039 5-1 7 82(5491 233 870276 190 956215 423 043785 53 8 826631 233 870161 190 956469 423 043531 52 9 826770 232 870047 191 956723 423 043-277 51 10 826910 232 869933 191 95(5977 423 043023 50 11 9-827049 232 9-869818 191 9-957231 423 10-042769 4'.) 12 827 189 232 869704 19L 957485 423 042515 48 13 8273-28 232 869589 191 957739 423 04i>261 47 14 827467 232 8(59474 191 9579SJ3 423 042007 4(5 15 827606 232 869300 191 958246 423 041754 45 16 827745 232 869245 191 958500 423 041500 44 17 827884 231 869130 191 958754 423 041246 43 18 828023 231 8(59015 192 959008 423 040992 42 19 828162 231 8681)00 192 9592(52 423 0407:18 41 20 828301 231 868785 192 959516 423 040484 40 21 9-828439 231 9-86F670 192 9-959769 423 10-040231 39 22 828578 231 868555 192 960023 423 039977 3rt 23 828716 231 868440 192 960277 423 039723 37 24 828855 230 868324 192 960531 423 039469 3(5 25 828993 230 8(58209 192 960784 423 039216 35 26 829131 230 8(58093 192 961038 423 038962 34 27 829269 230 8(57978 193 961291 423 038709 33 28 829407 230 8678(52 193 961545 423 0-8455 32 29 829545 230 867747 193 961799 423 038201 31 30 829(583 230 867631 193 962052 423 037948 30 31 9-829821 229 9-867515 193 9-962306 423 10-037C94 29 32 829959 229 867399 193 962560 423 037440 28 33 830097 229 867283 193 9C2813 423 037187 27 34 830234 229 867167 193 963067 423 036933 26 35 830372 229 867051 193 963320 423 0366PO 25 36 830509 229 86(5935 194 963574 423 036426 24 37 &30646 229 866819 194 9C3827 423 036173 23 38 830784 229 8(i(i703 194 9f:4081 423 035919 22 39 830921 228 8(5(5586 194 964335 423 035665 21 40 831058 228 866470 194 964588 422 035412 20 41 9-831195 228 9-866353 194 9-964842 422 10-035158 19 42 831332 228 86(5237 194 965095 422 034905 18 43 831469 228 866120 194 965349 422 034651 17 44 831(506 228 866004 195 965(502 422 034398 16 45 831742 228 865887 195 965855 422 034145 15 46 831879 228 865770 195 966109 422 033891 14 47 832015 227 8G5C53 195 966362 422 033638 13 48 832152 227 805536 195 966616 422 033384 12 49 832288 227 805-1 19 195 96(5869 422 033131 11 50 832425 227 8(55302 195 9(571 '23 422 032877 10 51 9-832561 227 9'805IF5 195 9-967376 422 10-032624 9 52 832(597 227 865068 195 9(57(529 422 032371 8 53 832833 227 8649.50 195 9C.7W3 422 032117 7 54 832969 22f> 864833 196 968136 422 031864 6 55 833105 226 864716 196 968389 422 03 1C- 11 5 56 83:;24i 226 8(54598 196 9(58.643 422 031357 4 57 833377 226 8(54481 196 9(58896 422 031104 3 58 833512 226 864303 196 969149 422 030H51 2 59 83IH548 226 864245 196 969403 422 030597 1 60 833783 226 8(54127 196 969(556 422 030344 | Cotang. I Tang. | M. 47 Degrees. LOGARITHMIC SINLS, COSINES, ETC. (43 Degrees.) 241 Cos.nc | D. | | D. 9-833783 226 9-864127 196 9-969056 4-J2 10-030344 | 60 1 833919 225 864010 1% 969909 422 030091 59 2 834054 225 86381(2 197 970162 422 029838 58 3 834189 225 803774 197 970416 422 029584 57 4 834325 2-25 863656 197 970669 422 029331 56 5 834460 225 863538 197 970922 422 029078 55 6 834595 225 863419 11)7 971175 422 02H825 54 7 834730 225 803301 197 971429 422 028571 53 8 834865 225 863J83 197 971682 422 028318 52 9 834999 2-24 863064 197 971935 422 028065 51 10 835J34 224 8G2946 198 972188 422 027812 50 11 9-835269 224 9-862827 198 9-972441 422 10-027559 49 12 835403 224 862709 198 972694 422 027306 48 13 835538 224 862590 198 972948 422 027052 47 14 835672 224 862471 198 973201 422 026799 46 15 835807 224 862353 198 973454 422 026546 45 16 835941 224 862234 198 * 973707 422 026293 44 17 &36075 223 862115 198 973960 422 026040 43 18 836209 223 861996 198 974213 422 025787 42 19 836343 223 861877 198 974466 422 025534 41 20 836477 223 861758 199 974719 422 025281 40 21 9-836611 223 9-861638 199 9-974973 422 10-025027 39 22 836745 2-23 861519 199 975226 422 024774 38 23 836878 2-23 861400 199 975479 422 024521 37 24 837012 222 801280 199 975732 422 024268 36 25 837146 ooo 861161 199 975985 422 024015 35 26 837279 222 861041 199 976238 422 023762 34 27 837412 222 800i>22 199 976491 422 023509 33 28 837546 222 860802 199 976744 422 023256 32 i2i) 837679 222 860682 200 976997 422 023003 31 30 837812 222 800562 200 . 977250 422 022750 30 31 9-837945 222 9-860442 200 9-977503 422 10-022497 29 32 838078 221 800322 200 977756 422 022244 28 33 838211 221 860202 200 978009 422 021991 27 34 838344 221 860082 200 978202 422 021738 26 35 838477 221 859962 200 978515 422 021485 25 30 838010 221 859842 200 978768 422 021232 24 37 838742 221 859721 201 979021 422 020979 23 38 838875 221 859601 201 979274 422 020726 22 39 839007 221 859480 201 979527 422 020473 21 40 839140 220 859360 201 979780 422 020220 20 41 9-839272 220 9-859239 201 9-980033 422 10-019967 19 42 839404 220 859119 201 980280 422 019714 ' 18 43 839536 220 858998 201 980538 422 019462 17 44 839668 220 858877 201 980791 421 019209 16 45 839800 220 858756 202 981044 421 018956 15 46 839932 220 858635 202 981297 421 018703 14 47 840064 219 858514 202 981550 421 018450 13 48 840196 219 858393 202 981803 421 018197 12 49 840328 2J9 858272 202 9H2056 421 (117944 11 50 840459 219 858151 202 982309 421 017691 10 51 9-840591 219 9-858029 202 9-982562 421 10-017438 9 52 840722 219 857908 202 982814 421 017186 8 53 840854 219 857786 202 983067 421 016933 7 54 840985 219 857665 203 983320 421 010080 6 55 841116 218 857543 203 983573 421 016127 5 56 841247 218 857422 203 98.-J826 421 016174 4 57 841378 218 857300 203 984079 421 015921 3 58 841509 218 857178 203 984331 42i 015669 2 59 841640 218 857056 203 984584 421 015416 1 60 841771 218 856934 203 984837 421 015163- | I Cotang. 46 Degrees. J Tang. I 242 (44 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. Sine JD 1 Cosine D. Tan?. | D. Cotnn?. | 9-841771 218 9-850934 203 9-984837 421 10-015103 tin 1 8411)02 218 850812 203 985090 4-21 014910 59 2 84-2033 218 850090 204 985343 421 OJ4057 58 3 842 1G3 217 856508 204 985596 421 014404 57 4 842294 217 856446 204 985848 421 01 1152 5i> 5 8424-24 217 850323 204 980101 421 013899 55 (i 842555 217 850201 204 980354 421 013040 54 7 842085 217 850078 204 980007 4-21 01 3393 53 B 842815 217 8551)50 204 986800 421 013140 52 9 842940 217 855833 204 987112 421 012888 51 10 84307G 217 855711 205 987305 421 012035 50 11 9-843206 216 9-855588 205 9-987018 421 10-012382 4!> 12 843330 210 855405 205 987871 421 012129 48 13 843406 216 85534-2 205 9881-23 421 011877 47 14 84351)5 216 855219 205 988376 421 V) 11 024 46 15 8437-25 216 855090 205 988029 421 011371 45 16 843855 216 854973 ' 205 988882 421 011118 44 17 843984 216 854850 205 989134 421 010800 43 1H 844114 215 854727 200 989387 421 010013 42 19 844243 215 854603 206 989640 421 010300 41 20 844372 215 854480 200 989893 421 010107 40 21 9-844502 215 9-854356 206 9-990145 421 10-009855 3!) 22 844031 215 854233 206 990398 421 009002 an 2!1 844700 215 854109 206 990051 421 009349 37 24 844889 - 215 8531)80 206 91)01)03 421 009097 30 25 845018 215 853802 200 991156 421 008844 35 90 845147 215 853738 200 991409 421 008591 34 27 845270 214 853014 207 991002 421 008338 33 28 845405 214 853490 207 91)1914 421 008080 33 2J 845.333 214 8533(50 207 99-2107 421 007833 31 30 845002 214 853242 2J7 99-2420 421 007580 30 31 9-845790 214 9-853118 207 9-992072 421 10-007328 29 3-2 845919 214 852994 207 9:i2J-25 421 007075 28 3:$ 840047 214 852869 207 91)3178 421 000822 27 34 846175 214 852745 207 993430 421 000570 215 35 840304 214 852020 207 993(583 421 000317 25 3(i 840432 2J3 85-2490 208 99IW3G 421 000004 24 37 840500 213 852371 208 994189 421 005811 23 38 84IK588 213 852247 208 994441 421 005559 22 39 840816 213 852122 208 994694 421 005306 21 40 846944 213 851997 208 994947 421 005053 20 41 9-847071 213 9-851872 208 9-995199 421 10-004801 19 42 847199 213 851747 208 995452 421 004548 18 43 8473-27 213 851022 208 995705 421 004295 17 44 847454 212 851497 209 995957 421 004043 10 45 847582 212 851372 209 990210 421 003790 15 46 847709 212 851246 209 99>403 421 003537 14 47 847836 212 85 11 '21 209 990715 421 003285 13 48 847964 212 850996 209 990908 421 003032 12 41) 848091 212 850870 209 997221 421 002779 11 50 848218 212 850745 209 997473 421 002527 10 51 9-848345 212 9-850619 209 9-9977-20 421 10-002274 9 53 848472 211 850493 210 997979 421 009021 8 53 848599 211 85(1308 210 9982:11 421 0017G9 7 54 848720 21J 85024-2 210 998484 421 001516 6 55 848852 211 850110 210 91IH737 421 001203 5 50 848979 211 84991)0 210 91)8;81) 421 001011 4 57 8491 (Hi 211 849804 210 999242 421 000758 3 58 849232 211 849738 210 999495 421 (I0,;5!i5 2 .VJ 849:159 211 8490 1 1 210 999748 421 00(1253 1 (M) 849485 211 849485 210 jo-oooooo 421 000000 Cosuw Ccuug. I Tanp. | 4.5 De-Tees. TABLE XIV. NATUKAL SINES AND TANGENTS. 246 NATURAL SINES. / 16 17 a 1 18 19 20 21 22 28 / 275 C374 292 3717 309 0170 325 5C82 342 0201 358 3C79 374 C066 -390 731V 60. 1 9170 C499 2936 8432 2935 6395 8763 S9S9 59 a 276 1915 9280 5702 326 1182 5CG8 9110 375 1459 .391 2166 58 3 4761 93 2001 84G8 3932 8400 359 1825 4156 5343 57 4 7551 4842 310 1234 6681 o43 1133 4540 6852 8019 56 6 277 0352 7623 3999 9430 38C5 7254 9547 -392 OG95 1 55 6 3147 294 0403 67 C4 327 2179 6597 9968-8762243, 3371 54 7 5941 3183 9529 4928 9329 3602G82 40381 C047 53 S S73C 5963 311 2294 767G 3442060 5395; 76321 8722 52 y 278 1530 8743 5058 328.0424 4791 8108J 377 0327 393 1397 ( 51 IQ 4324 295 1522 7822 3172 7521 361 OS21 3021 4071 50 11 7118 4302 312Q586 5919 345 0252 3534 5714 6745 49 VI 9911 7081 3349 8666 2982 6246 8408 94191 48 13 279 2704 9859 6112 329 1413 5712 8958 3781101-3942093! 47 14 5497 2962638 8875 4160 8441 362 1G69 3794J 47661 46 la 8290 5416 313 1C3S 6906 3461171 4380 6486 7439! 45 16 280 1083 8194 4400 9653 3900 7091 9178 -395 01111 44 17 3875 297 0971 71GS 330 2398 6628 9802 3791870 2783! 43 IS 6667 3749 9925 5144 9357 363 2512 4562 5455 42 18 9459 6526 314 2681 7889 347 2085 5222 7253 8127 41 20 281 2251 9303 544> 331 0634 4812 7932 9944 3960798 40 21 5042 298 2079 8209 3379 7540 3640641 380 21,34 34G8 39 22 7833 4856 315 09G9 6123 348 02C7 3351 5324 6139 38 2;', 2820624 7632 3730 8867 2994 6059 8014 8809 37 24 3415 299 0408 6490 3321611 5720 8768 381 0704 397 1479 36 25 6205 3184 9250 4355 8447 365 1476 3393 4148 35 26 8995 5959 316 2010 7098 349 1173 4184 6082 6818 34 27 2831785 8734 4770 9841 3898 tS91 8770 9486 33 28 4575 300 1509 7529 3332584 6624 9599 382 1459 398 2155 32 29 7364 4284 317 0288 5326 9349 366 2306 4147 4823 31 GO 2840153 7058 3047 80G9 350 2074 5012 6834 7491 30 31 2942 9832 5805 3340810 4798 7719 9522 .3990158 29 32 5731 301 2GOC 8563 3552 7523 367 0425 383 2209 2825 28 83 8520 5380 318 1321 6293 351 0246 3130 4895 5492 27 34 285 1308 8153 4079 9034 2970 5836 7582 8158 26 35 409G 3020926 683C 3351775 5693 8541 3840268 400 0825 25 86 6884 3699 9593 45U 8416 368 1246 2953 3490 24 37 9671 6471 319 2350 725C 3521139 3950 5t39 6ft6 23 3s 2862458 9244 510C 9996 38G2 6654 8324 8821 22 39 5246 303 201C 7863 3362735 6584 935S 385 1008 4011486 21 40 8032 4788 3200619 5475 930C 369 2061 3C93 4150 20 41 287 0819 - 7559 3374 8214 853 202" 4765 6377 6814 19 42 3605 3040331 6130 337 0953 4748 7468 906C 9478 18 4:5 6391 3102 8885 3691 7469 370 0170 3861744 4022141 17 44 917" 5872 321 1640 6429 3540190 *.S* 1 4427 4804 16 45 288l96o 8G43 439f 916" 2910 5574 7110 7467 15 46 474 305 1413 7149 338 190o 5630 8276 9792 .4030129 14 47 753 4183 9903 464'. 8350 371 0977 387 2474 2791 13 48 289031 6953 3222657 7379 355 1070 3G78 5156 5453 12 49 310 9723 5411 339 Oil 3789 6379 7837 8114 11 50 588 306 249L 8164 285 650 907S 388 0518 4040775 10 51 867 5261 3230917 558 922 3721780 3199 3436 9 52 290 145 803C 367C 832 356194 447S 5880 C096 8 53 423 307 079 L 6422 340 106 4C6 717S 8560 8756 7 54 702 35C 9114 379 738 987* 389 124C 405 1416 6 55 980 6334 324192C 653 357 009 373 2577 391S 4075 5 5f, 291 258 910 4G7 926 281 527 659S 6734 4 57 537 308 186 74 341 200 553 797C 9277 9393 3 58 815 468 325018C 473 824 374 0671 -390 195 406 2051 2 59 292093 740 2931 746S 358 096J 3369 463 470S 1 60 309 017 568$ 342020 367 60661 731] 7366 / 73' 1 72 71 70 69 68 j 67 66 | ' NAT. COSINE. NATURAL SINES. 247 / 24 25 26 27 28 29 | 30 31 / 406 7300 4226183 4383711 4:,:; nw:> 469 4716 -484 8096 -500 0000 515 03S1 00 1 407 0024 8819 6326 4542497 7 2s 4 485 06401 2519 2874 59 2 2681 423 1455 8940 5 us si 9S52 3184 5037 5307 58 2 6337 40.0 439 1553 707'.' -470 2419 5727 7556 7859 57 4 7993 6725 4166 4550269 4986 8270-5010973 5160:551 56 5 408 O04'. 9:jt50 8779 2859 7;V,:; : -4SOOS12l 2591 2842 55 6 3305 424 l'.H>4 9392 544'.) -47 Id 11'.' :.">4 5107 5333 54 7 5980 4628 44020.>t 2685 5895 7024 7824 53 8 8615 7262 4615 456 0027 5250 S43'i; -502 0140 5170:314 52 9 409 1269 9895 7227 3216 7815 487 0977 1 2655 2S04 51 10 302:; 425 2.V2S 9838 5804 472 0380 3517 5170 529.". 50 11 ; 6577 5161 441 2 Us 8392 2944 6057 7X-, 7782 49 12 92: jo 13 -4101SS:5 7793 4260425 50:>! 7668 457 0979 356(1 5508 8071 8597 -503 0199 488 1136 2713 518 0270 2758 48 47 14 4536 3056 442 027 S 6153 473 0634 3674 5227 5246 46 16 7189 56S7 2SS7 8789 3197 6212 7740 77:;.", 45 16 9841 831 S 5496 458 1325 5759 8750-5040252 5190219 44 17 4112192 4270949 8104 8910 321 489 1288 2765 2705 43 IS 5L44 8579 4430712 6496 47408S2 3825 5270 5191 42 1'J 7795 62'. is 8319 9080 3443 6361 77SS 7070 41 2o 412 0445 8S38 5927 459 1665 6004 8897 505 0298 520 0161 40 21 S096 428 1407 8534 42 is 8564 490 1433 2809 2040 39 22 5745 4095 4441140 6832 475 1124 3968 5:519 5130 38 23 8395 6723 8746 9415 8683 6503 7828 7613 37 24 4131 OH 9351 6352 460 199S 0242 9038 50603:5* 521 00'. Mi 36 25 3693 429V.I7U, 8957 4680 8801 491 1572 2840 2579 35 B8 6342 46061-445 1562 7162 476 1359 4105 63ftf 5001 34 27 8990 72:;:; 416? 9744 3917 8638 7863 7541 33 28 414 io:;s 9859 6771 461 2325 6474 9171 507 0370 5220024 32 23 4285 4302485 9375 4906 9031 492 1704 2877 250.- 31 88 6932 5111 446 1978 74SC, 477 15SS 4286 5384 4981 30 81 9579 7738 4.") si 462 0066 4144 6707 789 7401 29 32 415 2226 431 OSiil 7184 2646 6700 9298 508 0390 994* 28 S3 4S72 2986 978fl 6228 '.)_'.,: 4931829 2901 r>2:>.22 27 31 7517 5610 447 2:iss 7804 478 1810 4359 540< 4'.Hi: 26 "<> 416 0103 82:; 4 4991 463 0382 4864 6889 79U 73S1 25 86 2808 432 0857 7591 '2960 691! 9419 5090414 9S5! 24 SI 5453 3481 448 0192 6538 9472 494 194H 2.1 is W42:;:;< X S8 8097 6103 2792 8115 479 202i 447' 5421 481: 22 80 417 0741 87 2' i 5892 464 0692 457'. 7005 7924 729! 21 40 8385 433 134S 7992 3269 7131 9532J-5100426 97 20 41 ' 6023 3970 449 <>f.:il 6845 90s: 4952060 292s 525224' 19 42 8671 6591 319! 8420 480 2235 4587 542: 4717 IS 4:; 41S i:!i:! 9212 5789 465 0990 478b 7113 793i 7191 17 41 3956 434 1832 83S7 3571 7:i: '.7 9639 5110431 9661 16 4:. 6597 4463 4500-.IS4 6145 9888 496 2105 2.i:il 52021:;: 15 46 923 ) 7072 35S2 8719 481 2438 40',R) 5431 401: 14 47 419 1S.SO 6179 4661293 4987 7215 799 7osr 13 4S 4521 4352311 8775 3806 7537 9740 512 042! 955^ 12 4'. 7101 4::: > 451 1372 6439 482 0086 497 2204 2927 527 2031 11 M 9301 7548 3967 9012 2684 4787 542f 450: 10 61 420 2441 436 0106 6563 467 1584 5182 7310 792 697: 9 52 50SO 2784 9158 4156 779 9833 5130420 941: 8 53 77T.I 5401 452 1 75." 6727 483 0277 -49S 2:455 2916 528 1914 7 54 421 o:;:,s 8018 4347 9298 2S24I 4877 541: 438; 6 55 2996 437 0034 6941 4681809 5370 7399 7908 C.S.V 5 56 5634 8251 953J 44:;:* 7916 9320 514 0404 9322 4 57 8272 5866 453 2128 7009 4840462-4992441 2894 529 179( 3 58 422ir.ii):) 84S2 4721 9578 3007 i 4961 639! 425* 2 59 3546 4381097 7313 4692147 r.552 1 7481 7887 072' 1 BO 61 S3 3711 9905 47 10 8096 -500 0000 5150381 919: / 65 6i 63 02 61 60 59 58 / NAT. COSINE. 246 NATURAL SINES. 10 \1 \ 18 19 2C 21 1 22 28 ' .275 C374 292 3717 309 0170 325 5082 342 0201 358 3679 3746066-3907311 60 9170 C499 2936 8432 2935 6395 8763 9989 59 276 19U5 92SO 5702 326 1182 5CC8 9110 3751459.3912066 58 4701 293 2061 84C8 3932 8400 359 1125 4156 5313 57 755i 4842 3101234 6681 343 1133 4540 6852 8019 56 "277 0352 7023 3999 9430 3805 7254 9547 -392 0095' 55 3147 294 0403 67 C4 327 2179 6597 9968 -376 2243 3371 54 5941 3183 9529 4928 9329 36021821 4938 C047 53 8731 5963 311 221)4 7670 3442000 5395; 7032 8722; 52 278 1530 8743 5058 m0424 4791 810S|-377 0327 3931397' 51 4324 295 1522 .7822 3172 7521 361 021 3021 4071 50 7118 4302 3120586 5919 345 0252 3534 5714 6745 49 9911 7081 3349 8666 2982 6246 8408 9419 48 279 2704 9859 6112 329 1413 5712 8958 378 110L-394 2093 47 5497 2962638 8875 4160 8441 362 1069 37941 4766 46 8290 5416 313 1G3N 6906 3461171 4380 64S6| 7439 45 280 10S3 8194 4400 9653 3900 7091 9178-3950111 44 3875 297 0971 71C3 330 2398 6628 9802 379 1870 2783 43 6667 3749 9925 5144 9357 363 2512 4562 5455 42 9459 6526 3142681 7889 347 2085 5222 7253 8127 41 281 2251 9303 544- c 331 0634 4812 7932 9944 -396 0798 40 5042 298 2079 8209 3379 7540 3640641 380 2034 3468 39 7833 4856 315 0909 6123 348 0207 3351 5324 6139 38 282 0624 7632 3730 8867 2994 6059 8014 8809 37 3415 2990408 6490 3321011 5720 8768 381 0704 397 1479 36 6205 3184 9250 4355 8447 365 1476 3393 4148 35 8995 5959 316 2010 7098 349 1173 4184 6082 6818 34 283178;; 8734 4770 9841 3898 CS91 8770 9486 33 4575 300 1509 7529 333 2584 6624 9599 382 1459 398 2155 32 7364 4284 317 0288 6326 9349 366 2306 4147 4823 31 2840153 7058 3047 8069 350 2074 5012 6834 7491 30 2942 9832 5805 3340810 4798 7719 9522 .3990158 29 5731 301 2COO 8563 3552 7523 367 0425 383 2209 2825 28 8520 5380 318 1321 6293 351 0240 3130 4895 5492 27 285 1308 8153 4079 9034 2970 5836 7582 8158 26 4096 3020926 683C 335 1775 5693 8541 3840268 400 0825 25 6884 3699 9593 4511 8416 368 1246 2953 349(1 24 9671 6471 319 2350 725C 3521139 3950 5039 ofce 23 2862458 9244 51 OC 9996 3862 6654 8324 8821 22 524C 303 201G 7863 336 2735 6584 9358 385 1008 401 148t 21 8032 4788 320 0619 5475 9300 369 2061 3093 4150 20 287 0819 - 7559 3374 8214 3532027 4765 6377 C814 19 3605 3040331 6130 337 0953 4748 7468 906C 9478 18 6391 3102 8885 3691 7469 370 0170 386 1744 402 2141 17 9177 5872 321 1C4C 6429 3540190 2872 4427 4804 16 288 1963 8G43 439f 9167 2911 5574 7110 7467 15 4748 305 1413 7141 338 1905 5630 8276 9792 4030129 14 7533 4183 9903 4642 8350 371 0977 387 2474 2791 13 289 0318 6953 3222657 7379 355 107t 307 S 5150 5453 12 3103 9723 5411 3390110 3781 6379 7837 8114 11 5887 306 2491. 8164 2852 6508 9079 3880518 404 0775 10 8671 5201 3230917 5589 922f 3721780 3199 343C 9 2901450 8030 3670 8325 3561944 4479 5880 096 8 4239 307 0798 642L 340 1061 4C62 7179 8560 875C 7 702L 3566 9174 3796 738( 9878 389 1240 405 1416 6 980^ 6334 3241926 6531 357 0097 373 2577 3919 4075 5 291 258F 910'. 4678 9265 2814 5275 659S 6734 4 537 308 1869 7429 341 2000 5531 7973 9277 9393 3 815S 463C 325 01SC 4734 8248 374 0671 ,-3901955 406 2051 2 292 093 7403 2931 7468 358 0964 3369 403S 470Q 1 37r 309 017U 5682 3420201 3679 60661 7311 736C 7o 72 71 70 69 68 i 67 66 / NAT. COSINE. NATURAL SINES. 247 24 25 i 26 27 28 29 30 81 / 406 7366 407 0024 422 6183 8819 4383711 6326 4531)905 -469 4716 -484 8096' -500 0000 4542497: 7284-48506401 2519 515 0381 2874 ro 59 2681 423 1455 8940 50881 9852 3184J 5037 53<'.7 58 5337 4090 439 1553 7079 -470 2419 57271 7556 7859 57 7993 6725 4K)6 455 0263 49Si 8270-5010073 5160351 50 408 0049 9360 6779 2859 75531-486 0812] 2591 2842 f.5 3305 424 1994 9392 5449I-4T10119 3354 5107 5333 r,t 5900 4628 440 2004 8038 2685 58951 7024 7824 53 8615 7282 46JL5 456 0627 5250 843,; 5020140 5170314 52 409 1269 9895 7227 3216 7815 487 0977 2655 2804 61 3920 425 2528 9838 5804 472 0380 3517 5170 529.". 50 6577 5161 441 2 44S 8392 2944 6057 7685 7782 49 9230 7793 5059. 457 0979 5508 8597 503 0199 518 0270 48 4101883 4260425 7668 3560 8071 488 1136 2713 2758 47 4536 3050 4420278 6153 473 0634 3674 5227 5240 46 71S9 56S7 2887 8739 3197 6212 7740 7733 45 9841 8318 54'.K1 458 1325 5759 8750 504 0252 519 0219 44 411 2492 4270949 8104 3910 8.321 489 1288 2765 2705 43 5144 3579 4430712 6496 474 0882 3825 5276 5191 42 7795 6208 3319 9080 3443 6361 7788 7676 41 412 0445 8838 5927 459 1665 6004 8897 505 0298 520 0161 40 3096 428 1467 8534 4248 8564 490 1433 2809 2(34(1 39 5745 4095 4441140 6832 475 1124 3968 5319 5130 38 8395 6723 3740 9415 3683 6503 7828 7613 37 4131041 9351 6352 460 1998 6242 9038 506 0338 521 0096 30 3693 429 1979 8957 4580 8801 491 1572 2846 2579 35 6342 4liOi)l -445 1562 7162 476 1359 4105 5355 5061 34 8990 7233 4107 9744 3917 6638 7 863 7543 33 414 1638 9859 6771 461 2325 6474 9171 507 0370 522 0024 32 42S5 4302485 9375 4906 9031 492 1704 2877 2505 31 6932 5111 446 1978 7486 477 1588 4236 5384 4986 30 9579 773ii 4581 462 0066 4144 6767 7890 7461 29 415 2226 431 0361 7184 2646 6700 9298 508 0390 994f 28 4872 2386 9786 5225 9255 493 1829 2901 523 2424 27 7517 5610 447 2388 7804 478 1810 4359 540* 4903 26 416 0163 8234 4990 463 0382 4364 6889 7910 7381 25 280S 432 0857 7591 2960 6919 9419 509 0414 9859 24 5453 3481 448 0192 5538 9472 494 1948 2918 524 233f 23 8037 6103 2792 8115 479 2020 4476 5421 481: 22 417 0741 8720 5392 464 0692 4579 7005 7924 7291 21 3385 433 1348 7992 3269 7131 9532! -510 042f 97 6f 20 0025 3970 449 0591 5845 9683 495 2060 292s 5252-241 19 8671 6591 3190 8420 480 2235 4587 5421 4717 IS 418 1313 9212 5789 465 0996 4786 7113 7930 7191 17 3956 434 1832 8387 3571 7337 9639 5110431 966," 16 6597 4453 450 0984 6145 9888 496 2165 2931 526 213; 15 9231 7072 3582 8719 481 243S 4690 5431 46i: 14 419 1880 9692 6179 4661293 4987 7215 793( 7085 13 4521 4352311 8775 3866 7537 9740 512 0429 9558 12 7161 4930 451 1372 6439 482 0086 497 2264 2927 527 2030 11 9301 7548 3967 9012 2634 4787 5425 450? 10 420 2441 436 0166 6563 467 1584 5182 7310 7923 697: 9 5080 2784 9158 4156 7730 9833 513 0420 944! 8 7719 5401 4521753 6727 483 0277 498 2355 2916 528 1914 7 421 0358 8018 4347 9298 2824 4877 5413 438: 6 2996 437 0031 6941 4681869 5370 7399 7908 686J 5 5634 325 1 9535 4439 7916 9920 514 0404 932i 4 8272 5800 453 2128 7009 4840462-4992441 289{ 529 1791 3 4220909 8482 4721 9578 3007 4961 5393 425,^ 2 3546 4381097 7313 469 2147 6552 7481 7887 67 2f 1 6183 3711 9905 4716 8096 -500 0000 515 0381 9193 65 | 64 G3 62 61 00 59 58 / NAT. COSINE. 248 NATURAL SINES. / 32 | 33 34 35 36 37 38 39 t 529 9193-544 6390 5591929 573 5764 587 7853 601 8150 615 6615 -629 3204 00 1 2 3 5301659 8830 4125-545 1269 6591 3707 4340 6751 916-2 8147 574 0529 2911 5880206 2558 4910 602.0473 2795 5117 8907i 5464 6161198! 7724 3489 9983 59 58 57 4 9057 ' 6145 560 1572 5292 7262 7439 5780/630 2242 56 5 531 1521 8583 3981 7672 9613 97 CO 80691 4500 56 6 3986 546 1020 6390 575 0053 589 1964 603 2080 617 0359 6758 54 7 6450 3456 879S 2432 4314 4400 2648 9015 53 8 8913 5892 561 1206 4811 6663 6719 4936 631 1272 52 9 5321370 8328 3614 7190 9012 9038 7224 3528 51 10 3839 547 0763 6021 9568 590 1361 .604 1356 9511 5784 50 11 6301 3198 8428 576.1946 3709 3674 618 1798 8039 49 12 8763 5632 5620834 4323 6057 5991 4084 632 0293 4S 18 5331224 8066 3239 6700 8404 8308 6370 2547 47 14 3685 548 0499 5645 9076 591 0750 605 0624 8655 4800 46 15 6145 2932 8049 577 1452 3096 2940 619 0939 7053 45 16 8605 5365 563 0453 3827 5442 5255 3224 9301; 44 17 5341065 7797 2857 6202 7787 7570 5507 633 1557 43 IS 352.3 5490228 5260 8576 5920132 9884 7790 3809 42 19 5982 2659 7663 578 0950 2476 606 2198 620 0073 6059 41 20 8440 5090 5640066 3323 4819 4511 2355 8310 40 21 5350898 7520 2467 5696 7163 6824 4636 634 0559 39 2-2 3355 9950 4869 8069 9505 9136 6917 2808 00 23 5812 550 2379 727C 579 0440 593 1847 607 1447 9198 5057 37 24 8268 4807 9670 2812 4189 3758 621 1478 7305 36 25 5360724 7236 565 2070 5183 6530 6069 3757 9553 35 2(j 3179 9663 44(59 7553 8871 8379 6036 6351800 34 27 5634 551 2091 6868 9923 594 1211 608 0689 8314 4046 33 28 8089 4518 9267 580 2292 3550 2998 6220592 6292 32 29 537 0543 6944 566.1665 4661 5889 5306 2870 8537 31 30 2996 9370 4062 7030 8228 7614 5146 6360782 30 31 5449 5521795 6459 9397 595 0566 9922 7423 3026 29 32 7902 4220 8S56 581 1765 2904 609 2229 9698 5270 2S 33 538 0354 6645 567 1252 4132 5241 4535 623 1974 7513 27 34 2806 9069 3648 6498 7577 6841 4248 9756 2(5 35 5257 553 1492 6043 8864 9913 9147 6522 637 1998 '25 30 770S 3915 8437 582 1230 5962249 610 1452 8796 4240 37 5390158 6338 568 0832 3595 4584 3756 624 1069 6481 23 38 2608 8760 3225 5959 6918 6060 3342 8721 22 39 5058 5541182 5619 8323 9262 8363 5614 638 0961 40 7507 3603 8011 5830687 597 1586 611 0666 7885 3201 20 41 9955 6024 5690403 3050 3919 2969 625 0156 5440 19 42 540 2403 8444 2795 5412 6251 5270 2427 7678 IS 43 4851 5550864 5187 7774 8583 7572 4696 9916 17 44 7298 3283 7577 584 0136J -598 0915 9873 6966 639 2153 16 45 9745 5702 9968 2497 3246 612 2173 9235 4390 15 4<> 541 2191 8121 570 2357 4857 5577 4473 6261503 6626 14 47 4637 5560539 474" 7217 7906 6772 3771 8862 13 4S 7082 2956 7136 9577 599 0236 9071 6038 640 1097 12 49 9527 5373 9524 585 1986J 2565 613 1369 8305 3332 11 50 542 1971 7790 571 1912 4294| 4893 "3G6: 627 0571 5566 10 51 4415 557 0206 4299 6652 7221 5964 2837 7790 9 52 6S59 2621 6686 901 9549 8260 5102 641 0032 8 53 9302 5036 9073 586 1367, -600 1876 614 055f 7366 2264 7 54 543 1744 7451 572 1459 3724! 4202 2852 9631 449C 6 55 4187 9865 3844 6080| 6528 5147 628 1S94 672S 5 66 6628 558 2279 6229 8435; 8854 7442 4157 8958 4 57 9069 4<>92 8614 5870790-6011179 9736 6420 6J21189 ,'! 58 5441510 7105 5730998 3145! 3503 615 2029 8682 3418 2 59 3351 9517 3381 5499 5827 4322 0943 5647 1 60 6390 559 1929 5764 7853 8150 6615 629 3204 7876 / 57 56 55 54 53 52. 51 50 / NAT. COSINE. NATURAL SINES. 249 40 41 42 43 44 45 46 47 / 6427876 6560590 669 1306 681 9984 -694 6584 707 1068 193398 731 3537 60 b430104 2785 34()fc 68221111 8676 3124 6418 5521 59 2332 4980 5628 4237 -695 0767 5180 7438 7503 58 4559 7174 7789 6363 285'8 7236 9457 9486 57 .6785 9367 9948 8489 4949 9291 201476 "32 1467 56 9011 657 15CO 670 2108 6830613 7039 708 1345 3494 3449 55 6441236 3752 4266 2738 9128 3398 5511 5429 54 3461 5944 6424 4861 69C1217 5451 7528 7409 53 5685 8135 8682 6984 2306 7504 9544 9388 )2 7909 6580326 671 0739 9107 5392 9556 21 1559 733 1367 51 645 0132 2516 2895 684 1229 '7479 7091 07 3574 3345 50 2355 4706 5051 3350 9565J 3657 5589 5322 49 4577 6895 7206 5471 697 1651 5707 7C02 7299 48 6798 9083 9361 7591 3736 7757 615 9275 47 9019 0591271 6721515 9711 58211 9806 -221C28 734 1250 46 6461240 3458 3668 685 1830 79051-7101854 3640 3225 45 3460 5G45 5821 3948 9988J 3901 5651 5199 44 5679 7831 7973 6066 698 2071 6948 7C61 7173 43 7898 6600017 6730125 8184 4153 7995 9671 914C 42 647 0116 2202 2276 686 0300 62341-711 0041 "23 1681 735 111? 41 2334 ^4386 4427 2416 8315 2086 3690 3C90 40 4551 6570 6577 4532 699 0396 4130 5698 5001 39 6767 8754 8727 6647 2476 6174 7705 7032 38 8984 661 0936 674 0876 8761 4555 8218 9712 9002 37 648 1199 3119 3024 C87 0875 6633i-7120260 7241719 736 0971 36 3414 5300 5172 2988 8711 2303 3724 2940 35 6628 7482 7319 5101 7000789 4344 5729 4908 34 7842 9662 9466 7213 2806 6385 7734 6875 33 6490056 662 1S42 6751612 9325 4942! 8426 9738 8842 32 2268 4022 3757 688 1435 7018 -713 0465 725 1741 737 0808 31 4480 6200 5902 3546 9093J 2504 3744 2773 30 6C92 8379 8046 5655 701 1167 4543 5746 4788 29 8903 663 OC57 6760190 7765 3241 6581 7747 6703 28 650 1114 2734 2333 9873 5314! 8618 9748 8666 27 3324 4910 4476 689 1981 7387 1-714 0655 7261748 738 OP.29 26 5533 7087 6618 4089 9459 2C91 3748 2592 25 7742 9262 8760 61% 7021531 4727 5747 4553 24 9951 6641437 677 0901 8302 3601 6762 7745 6515 23 651 2158 3612 3041 690 0407 5072 8796 9743 8475 22 4366 5785 6181 2512 7741 715 0830 727 1740 739 0435 21 6572 7959 7320 4617 9811 2863 3736 2394 20 8778 665 013 9459 6721 7031879 489o 5732 4353 19 6520984 2304 678 159" 8824 3947 692- 7728 6311 18 3189 4475 3734 691 0927 6014 895Q 9722 82C8 17 5394 6646 5871 3029 8081 716098 728 1716 740 022o 16 7598 881- 800- 5131 704 0147 301 3710 218 15 980 666 098" 679014," T282 2213 504 5703 413 14 653 200 31 5f 2278 9332 4278 707 769o 609 13 420 5325 4413 692 1432 6342 910 968 804 12 640 7493 6547 3531 8406 717-1134 729 167 741 000 11 860 966 68 5630 705 0469 316 3668 195 10 654 081 667 182 680 OS! 7728 2532 618 5657 390 9 301 399 294 9825 4594 721 7646 585 8 520 616 507 693 1922 6655 9238 9635 780 7 740 832 720 4018 871 718126 730 1623 975 6 960 668 OJ9 933 6114 7060776 328 3610 742170 5 655 180 m 681 146 820S 283J 531 5597 866 4 400- 481 3599 694 0304 4894 7333 758S 560 3 619 698 572 239 6953 935 956S 755 2 839 9144 785 449 9011-719137 731 155? 950 1. 6560591 669 130 998 6584 707 106S 339 3537 ,'743 144 49 48 47 46 45 44 43 42 NAT. COSINE NATURAL SINES. 48 49 50 51 52 53 54 713 1448 : -754 7090 766 0444 -777 1460 -788 0108 798 6355 809 0170 3394 9004 2314 3290 1898 8105 . 1879 5340 -7550911 4183 : 5120 < 3088 9S55 3588 7285 : 2818 6051 ; 6949 5477 7991004 5296 9229 i 4724 7918 8777 7266 3352 7004 7441173 6030 9785 ''7780(504 9054 5100 8710 3115 8535 7671052 2431 -7890841 6847 8100410 5058 j -756 0439 3517 4258 2627 8593 2122 6999 2342 5382 1 6084 4413 800 0338 8820 8941 4240 7246 1 7909 6198 2083 5530 745 0881 6148 9110 9733' 7983 3827 7234 2821 8050 70S 0973 -7791557 9707 5571 8930 4760 9951 2835 3380 790 1550 7314 811 0038 6099 757 1851 4697 5202 3333 9056 2339 8G36 3751 6558 7024 5115 801 0797 4040 7460574 5650 8418 | 8845 6896 2538 5740 2510 7548 769 0278 ; -780 0005 8676 4278 7439 4446 9440 2137 1 2485 791 0456 6018 9137 6382 758 1343 3990 4304 2235 7756 812083*5 8317 3240 5853 6123 4014 9495 2532 747 0251 5136 7710 7940 5792 802 1232 4229 2184 7031 9507 9757 7569 2969 5925 4117 8926 770 1423 ; '781 1574 9345 4705 7620 6049 759 0820 3278 3390 7921121 6440 9314 7981 2713 5132 5205 289(1 8175 813 1008 9912 4000 6986 1 7019 4071 9909 2701 748 1842 6498 8840 8833 6445 803 1042 4393 3772 8389 771 0692 : '782 0046 8218 3375 6084 5701 7600280 2544 2459 9990 5107 7775 7029 2170 4395 4270 793 1762 6838 9466 9557 4060 6246 60S2 3533 8569 8141155 749 1484 5949 8096 7892 5304 8040299 2844 3411 7837 9945 9702 7074 2028 4532 5337 9724 7721794 783 1511 8843 3750 6220 7202 7611011 3042 3320 7940611 5484 7900 9187 3497 5489 5127 2379 7211 951)3 7501111 5383 7336 6J35 4140 8938 815 1278 3034 7268 9182 8741 5913 805 0004 2903 4957 9152 7731027 j '784 0547 7078 2389 4047 6879 762 1036 2872 2352 9444 4113 6330 8800 2919 4716 4157 795 1208 5S37 8013 751 0721 4802 6559 5961 2972 7560 9695 2641 6683 8402 7764 4735 9283 8161370 4561 8564 7740244 9566 6497 8061005 3050 6480 7630445 2086 785 1368 8259 2720 4730 8398 2325 3926 3109 796 0020 4446 6410 752 0316 4204 5767 4970 1780 6166 8094 2233 6082 7606 G770 3540 7885 9772 4149 7960 9415 8509 5299 9603 817 1449 6005 9838 775 1283 7860367 7058 807 1321 3125 V980 7641714 3121 2165 8816 3038 4801 9894 3590 4957 3963 797 0572 4754 6470 7531808 5465 6794 5759 2: ;_*.> 6470 8151 3721 7340 8029 7555 4084 8185 9824 5034 9214 776 0464 9350 5839 9899 818 1497 7546 765 1087 2298 787 1145 7594 808 1012 3109 9457 2900 4132 2939 9347 3325 4841 754-1368 4832 5905 4732 7981100 5037 0512 3278 6704 7797 6524 2853 6749 8182 5187 8574 9629 8316 4004 8400 9852 7096 766 0444 777 1460 788 0108 0355 8090170 8191520 41 40 39 38 37 36 35 NAT. COSINE. NATURAL SINES. 251 55 56 57 58 59 60 61 / 819 1520 829 0376 838 6706 848 0481 857 1673 8660254 874 6197 60 3189 2002 8290 2022 3171 1708 7607 59 4850 3628 9873 3562 4668 3161 9016 58 6523 5252 839 1455 5102 6164 4614 875 0425 57 8189 6877 3037 6641 76CO C006 1832 56 9854 8500 4618 8179 9155 7517 3239 55 820 1519 830 0123 6199 9717 858 0649 8967 4645 54 3183 1745 7778 849 1254 2143 867 0417 6051 53 484(3 336(3 9357 2790 3635 1866 7455 52 6509 4987 840 0936 4325 5127 . 3314 8859 51 8170 6607 2513 5860 6619 4762 8760263 50 983:2 8226 4090 7394 8109 6209 1665 49 821 1492 9845 5066 8927 9599 7655 3067 18 3152 831 1463 7241 850 0459 859 1088 9100 4468 47 4811 3080 8816 1991 2576 868 0544 5868 46 6469 4696 841 0390 3522 4064 1988 7268 45 8127 6312 1963 5053 5551 3431 8666 44 9784 7927 8536 6582 7037 4874 877 0064 43 822 1440 9541 5108 8111 8523 6315 1462 42 3096 8321155 6679 9639 8600007 7756 2858 41 ! 4751 2768 8249 851 1167 1491 9196 4254 40 6405 4380 9819 2693 2975 869 0636 5649 ;;9 8059 5991 S42i:;ss 4219 4457 2074 7043 38 9712 7602 2356 5745 5939 3512 8437 :'-7 8231364 9212 4524 7269 7420 4949 9830 36 3015 833 0822 6091 8793 8901 6386 878 1222 35 . 4666 2430 7657 852 0316 861 0380 '.821 2613 84 6316 4038 9222 1839 1859 9256 4004 33 7965 5646 8430787 3360 3337 8700691 5394 32 9614 7252 2351 4881 4815 2124 6783 31 824 12(32 8858 3914 6402 6292 3557 8171 30 2909 834 0463 5477 7921 7768 4989 9559 29 4556 2068 7039 9440 9243 6420 879 0946 28 6202 3672 8600 8530958 8620717 7851 2332 27 7847 5275 844 0161 2475 2191 9281 3717 26 9491 6877 1720 3992 3664 871 0710 5102 25 825 1135 8479 3279 5508 5137 2138 6486 24 2778 835 0080 4838 7023 6608 3566 7869 23 4420 1680 6395 8538 8079 4993 9251 22 6062 3279 7952 8540051 9549 6419 880 0633 21 '7703 4878 9508 1564 8631019 7844 2014 20 9343 6476 845 1064 3077 2488 92(59 3394 V.) 826 0983 8074 2618 4588 3956 872 0693 4774 18 2622 9670 4172 6099 5423 2116 6152 17 4260 8361266 5726 7609 6889 3538 7530 16 5897 2862 7278 9119 8355 4960 8907 15 7534 4456 8830 855 0627 9820 6381 881 0284 14 9170 6050 8460381 2135 8641284 7801 1660 13 827 0806 7643 1932 3643 2748 9221 3035 12 2440 9236 3481 5149 4211 8730640 4409 11 4074 837 0827 5030 6655 5673 2058 5782 10 5708 2418 6579 8160 7134 3475 7155 9 7340 4009 8126 9664 8595 4891 8527 8 8971 5598 9673 856 1168 865 0055 6307 9898 7 8280603 7187 847 1219 2IJ71 1514 7722 8821269 6 2234 8775 2765 4173 2973 9137 2638 5 3864 838 0363 4309 5674 443f 8740550 4007 4 5493 1950 5853 7175 5887 1963 5376 3 7121 3536 7397 8675 7344 3375 6743 2 8749 5121 8939 857 0174 8799 4786 8110 1 829 0376 6706 848 0481 1673 8660254 6197 9476 34 33 82 31 30 29' 28 / NAT. COSINE. 252 NATURAL SINES. 62 63 04 65 06 67. 68 882 9476 .891 OOC5 ( -898 7940 9063078 913 5455 920 5049 927 1839 8830841 13S5 9215 4307 6C37 C185 2928 2206 2705 8990489 5535 7819 7320 4016 35C9 4024 1763 C7C2 9001 8455 5104 4933 5342 3035 79S9 91401S1 9589 G191 62-J5 CC59 4307 9215 13G1 .921 0722 7277 7656 7975 5578 907 0440 2540 1854 8363 9017 9291 6848 1665 3718 2986 9447 884 0377 8920C06 8117 2888 4895 4116 928 0531 1736 1920 9386 4111 6072 6246 1014 3095 3234 900 0654 5333 7247 6375 2696 4453 4546 1921 6554 8422 ' 7504 377S 5810 5858 3188 7775 9597 8632 4S58 71C6 7169 4453 8995 9150770 9758 5938 8522 8480 5718 9080214 1943 9220884 7017 9876 9789 6982 1432 3115 2010 8096 885 1230 8931098 8246 2C49 4286 3134 9173 2584 2406 9508 3866 6456 4258 9290250 3936 3714 901 0770 6082 6626 6381 1326 5288 6021 2031 6297 7795 6503 2401 6639 6326 3292 7511 89C3 7624 3475 7989 7632 4551 8725 9160130 8745 4549 9339 8936 5810 9938 12D7 9865 5C22 886 0688 8940240 7068 909 1150 24C2 92309S4 6094 2036 1542 8325 2561 3627 2102 7705 3383 2844 9582 3572 4791 3220 8835 4730 4146 9020838 4781 5955 4330 9905 6075 5446 2092 5990 7118 5452 9300974 7420 6746 3347 7199 8279 C567 2042 8765 8045 4600 8406 9440 7682 3109 887 0108 9344 5853 9613 917 OC01 8795 4176 1451 895 0041 7105 9100819 17 CO 9908 5241 2793 1938 8356 2024 2919 924 1020 C306 4134 3234 9606 3228 4077 2131 7370 5475 4529 903 0856 4432 6234 3242 8434 6815 5824 2105 6035 6391 4351 9496 8154 7118 3353 6837 7546 5460 931 0558 9492 8411 4600 8038 8701 6508 1019 888 0830 9703 5847 9238 9855 7676 2079 2166 .8960994 7093 9110438 918 1009 8782 3739 3503 2285 8338 1637 2161 9888 4797 4838 3575 9582 2835 3313 9250993 5855 6172 4864 9040825 4033 44G4 2097 6912 7506 6153 2068 5229 5614 3201 79C9 8839 7440 3310 6425 67C3 4303 9024 8890171 8727 4551 7620 7912 5405 9320079 1503 897 0014 5792 8815 90CO 6506 1133 2834 1299 7032 9120008 9190207 7C06 2186 4164 2584 8271 1201 1353 8706 3238 5493 3868 9509 2393 2499 9805 4290 6822 5151 905 0746 3584 3644 926 0902 5340 8149 6433 1983 4775 4788 2000 6390 9476 7715 3219 5965 5931 3096 7439 890 0803 899fi 4454 7154 7073 4192 8488 2128 898 0276 5688 8342 8215 5286 9535 3453 1555 6922 9529 9356 6380 933 0582 4777 2834 8154 9130716 920 0496 7474 1C28 6100 4112 9386 1902 1035 8566 2073 7423 5389 9060618 3087 2774 9658 3718 8744 6665 1848 4271 3912 927 0748 4701 891 0065 7940 3078 5455 5049 1839 5804 27 26 25 24 23 22 21 NAT. COSINE. NATURAL SINES. 253 69 70 71 72 73 74 75 933 5804 6846 939 6926 7921 945 5186 6132 951 0565 1464 9563048 3898 961 2617 3418 965 9258 9660011 7888 8914 1 7078 2361 4747 4219 0762 8928 9907 8023 3258 5595 5019 1513 9968 940 0899 89(38 4154 6443 5818 2263 934 1007 1891 9311 5050 7290 6616 3012 2045 288> 9460854 5944 8136 7413 3761 3082 3871 1795 6838 8981 8210 4508 4119 4860 2736 7731 9825 9005 5255 5154 5848 3677 8623 957 0669 9800 6001 6189 6835 4616 9514 1512 9620594 6746 7223 7822 5555 952 0404 2354 1387 7490 8257 8808 6493 1294 3195 2180 8234 9289 9793 7430 2183 4035 2972 8977 9350321 941 0777 8366 3071 4875 3762 9718 1352 1760 9301 3958 5714 4552 967 0459 2382 2743 1 -947 0236 4844 6552 5342 1200 3412 3724 1170 5730 7389 6130 1939 4440 4705 2103 6615 8225 6917 2678 5468 5686 3035 7499 9060 7704 3415 6495 6665 3966 8382 9895 8490 4152 7521 7644 4897 9264 958 0729 9275 4888 8547 8621 5827 9530146 1562 9630060 5624 9571 9598 ! 6756 1027 2394 0843 6358 936 0595 942 0575 7684 1907 3226 1626 7092 1618 1550 8612 2786 4056 2408 7825 2641 2525 9538 3664 4886 3189 8557 3662 3498 -94804(54 4542 5715 3969 9288 4683 4471 1389 5418 6543 4748 968 0018 5703 5444 2313 6294 7371 5527 0748 6722 6415 3237 7170 8197 6305 1476 7740 7386 4159 8044 9023 7081 2204 8758 8355 5081 8917 9848 7858 2931 9774 9324 6002 9790 9590672 8633 3658 937 0790 943 0293 6922 954 0662 1496 9407 4383 1806 1260 7842 1533 2318 964 0181 5108 2820 2227 8760 2403 3140 0954 5832 3833 3192 9678 3273 3961 1726 6555 4846 4157 9490595 4141 4781 2497 7277 5858 5122 1511 5009 5600 3268 7998 6869 6085 2426 5876 6418 4037 8719 7880 7048 3341 6743 7236 4806 9438 8889 8010 4255 7608 8053 5574 9690157 9898 8971 5168 8473 8869 6341 0875 938 0906 ' 9931 6080 933f 9684 7108 1593 1913 . -944 0890 6991 955 0199 0499 7873 2309 2920 ! 1849 7902 106- 960 1312 8638 3025 3925 i 2807 8812 1923 2125 940- 3740 4930 i 3764 9721 2784 2937 965 0165 4453 5934; 4720 -9500629 3643 3V48 0927 5167 6938 5675 i 1536 450- 4558 1689 5879 7940 6630 2443 5361 536 2440 6591 8942 7584 3348 6218 617 3209 7301 9943 8537 4253 7074 698 3968 8011 939 0943 9489 5157 7930 779 472 8720 1942 9450441 6061 878o 859 548 9428 2940 1391 6963 963 940 6240 970 0135 3938 2341 7865 956049 961 020 699 0842 4935 3290 8T6C 134 101 775 154S 5931 4238 9666 219 181 850 2253 6920 5186 951 0565 304 261 925 2957 20 19 18 17 16 15 14 NAT, COSINE. 254 NATURAL SINES. / 76 77 78 1 79 80 81 82 / 970 2957 974 3 7 01 978 1476 981 6272 9848 078 9876 883 9902 681 60 1 36 4355 2080 6826 582 -9877 338 9903 085 59 2 4363 5008 2684 7380 9849 086 1 792 489 58 3 5065 5660 3287 7933 589 j '9878 245 ' 891 57 4 5766 6311 ' 3889 8485 9850091 GJI7 9904 293 56 5 6466 6962 4490 9037 593 1 -9879 148 694 ' 55 6 7165 7612 5090 9587 -9851093 599 9905 095 54 7 7863 8261 5689 9820137 593 -9880048 494 53 8 8561 8909 6288 0686 9852092 497 893 52 9 9258 9556 6886 1234 590 945 9906 290 51 10 9953 975 0203 7483 1781 9853 087 9881 392 687 50 11 971 0649 0849 8079 2327 583 1 838 9907 083 49 12 1343 1494 8674 2873 9854079 -9882284 478 48 13 2036 2138 9268 3417 574 i 728 873 47 14 2729 2781 98(12 3961 9855 068 i -9883 172 9908 266 46 15 10 3421 4112 3423 4065 979 0455 1047 4504 5046 561 i 615 98560531-9884057 659 9909 051 45 44 17 4802 4706 1038 5587 544 498 442 43 IS 5491 5345 2228 6128 9857 035 939 832 42 19 6180 5985 2818 6668 524 -9885378 9910 221 41 20 6867 6623 3406 7206 9858 013 817 610 40 21 7554 7260 3994 7744 501 9886255 997 39 22 8240 7897 4581 8282 988 692 9911 384 38 215 8926 8533 5167 8818 9859 475 9887 128 770 37 24 9610 9168 5752 9353 960 564 9912 155 36 972 0294 9802 S337 9888 9860 445 998 540 35 26 0976 976 0435 6921 983 0422 929 9888 432 923 34 27 1658 1067 7504 0955 9861 412 865 9913 306 83 28 2339 1699 8086 1487 894 9889 297 688 32 29 3020 2330 8667 20191-9862375 728 9914 069 31 30 3699 . 2960 9247 2549 856 9890 159 449 30 31 4378 3589 9827 3079 9863336 588 828 29 32 5056 4217 980 0405 3608 815 9891 017 9915 206 28 S3 5733 4845 0983 4136 9864 293 445 584 27 34 6409 5472 1560 4663 770 872 961 26 35 7084 6098 2136 5189 9865 246 9892 208 9916337 25 36 7759 6723 2712 5715 722 723 712 24 37 8432 7347 3286 6239 9866 196 9893 148 9917 086 23 38 9105 7970 3860 6763 670 572 459 22 39 9777 8593 4433 7286 9867 143 994 832 21 40 973 0449 9215 5005 7808 615 9894416 9918 204 20 41 1119 9836 5576 8330 9868 087 838 574 19 42 1789 977 0456 6147 8850 557 9895258 944 18 43 2458 1075 6716 9370 9869 027 677 9919 314 17 44 i 3125 ' 1693 7285 9889 496 9896 096 682 16 45 3793 ; 2311 7853 984 0407 964 514 9920049 15 46 4458 2928 8420 0924 9870 431 931 416 14 47 6124 3544 1 8986 1441 897 9897 347 782 13 48 5789 4159 ; 9552 1956 9871 363 762 9921147 12 49 6453 4773) '981 0116 2471 827 9898 177 511 11 50 7116 5386 | 0680 2985 '-987 2 291 590 874 10 51 7778 5999 1243 3498 754 9899 003 9922 237 9 52 8439 6611 1805 4010 -9873216 415 599 8 53 9100 7222 2366 4521 678 826 959 7 54 9760 7832 2927 6032 9874 138 9900 237 9923 319 6 55 i -974 0419 8441 3486 5542 598 646 679 5 56 1077 9050 4045 6050 9875 057 9901 055 9924 037 4 57 1734 9658 4603 6558 514 462 394 3 5S 2390 9780265 61CO 7066 972 869 751 2 59 3046 0871 5716 7572 9876428 9902 275 9925107 1 60 3701 1476 1 6272 8078 883 681 462 13 12 11 10 8 7 ' NAT. COSINE. NATURAL SINES. 255 / 83 84 80 86 87 88 89 / 9925 462 9945 219 9961 947 9975 641 S86 295 993 3 8 992S271 625 948 237 486 C93 8C6 2 9 618 921 9964 195 433 631 788 900 M 10 965 9948 217 440 627 775 881 942 r ;0 11 9929310 513 685 821 919 974 984 49 12 655 807 929 9978015 9988 061 9995 OC6 9999 C25 48 13 999 9949 101 9965172 207 203 117 005 47 14 9930342 393 414 399 344 247 105 40 15 685 685 655 580 484 336 143 45 16 9931 026 976 895 779 623 424 181 44 17 367 9950 266 9966135 9C8 761 512 218 43 18 706 556 374 9979 156 99 599 254 42 19 9932045 844 612 343 9989 C35 184 289 41 20 384 9951 132 849 530 171 770 323 40 21 721 419 9967 085 716 306 854 357 39 22 9933057 705 321 900 440 9C7 389 m 23 393 990 555 9980 084 573 9996 020 421 37 24 728 9952 274 789 2(,7 706 101 452 86 25 9934 062 557 9968 022 450 837 '182 482 35 26 395 840 254 631 9C8 262 611 34 27 727 9953 122 485 811 9990 098 341 639 S3 28 9935 058 403 715 991 227 419 667 32 29 389 683 945 9981 170 355 497 693 31 30 719 962 9969 173 348 482 573 C19 no 31 9936047 9954 240 401 525 609 64Q . 644 29 32 375 517 628 701 734 724 6C8 28 33 703 794 854 877 859 79 692 21 34 9937 029 9955 070 9970 080 9982052 983 87 714 26 35 355 345 304 225 9991 106 94 736 25 36 679 620 528 398 228 9997 01 756 24 37 9938 003 893 750 570 350 08 776 23 38 326 9956165 972 742 470 15 795 22 39 648 437 9971 193 912 590 224 813 21 40 969 708 413 9983 082 709 29 831 20 41 9939 290 978 633 250 82" 36 847 19 42 1 610 9957 247 851 118 944 42 863 18 43 928 515 997C9 JM 9992 00 49 878 17 44 9940 246 783 286 751 176 65 892 16 45 563 9958 049 502 917 290 62 90 15 46 880 315 717 9984 OS1 404 68 917 14 47 9941195 580 931 245 617 74 92 13 48 510 844 9973 145 408 629 80 93 12 49 823 9959107 357 570 74 86 94 11 50 9942 136 370 569 731 86 927 95 10 61 448 631 780 R91 9C 98 96 9 52 760 892 990 9985 050 9993 06 9998044 97 8 53 9943 070 9960152 9974199 209 17 10 97 7 54 379 411 408 367 28 15 98 C 55 688 C69 615 524 39 21 98 5 66 996 926 821, 680 49 26 99 4 57 9944 303 9961 183 9975028 835 60 32 99 3 58 C09 438 233 989 70 37 99 2 69 914 693 437 9986143 80 42 1-000000 1 60 9945 219 947 641 295 90 47 00 C / 5 4 3 f)0 1 / NAT. COSINE. 258 NATURAL TANGENTS. / i 16 17 18 10 20 21 22 1 23 / 2867454 305 7307 324 9197 344 327 G 363 9701 383 8640 40-1 0262! -424 47 48 60 1 287 0602 J06 0488 325 241'i 6530 364 29JV, 384 1978 36461 8182 59 2 3751 3670 5630 9785 62i)L 6317 7031-4251611 58 3 6900 6852 8848 3453-040 858S-. 8650 405 0417 5051 57 4 288 0050 307 0034 326206t. 6296 3652886 385 1996 3804 8487 66 5 3201 3218 5284 9553 618- 5337 7191 426 1924 65 6 6352 6402 8504 3462810 94S< 8679 406 0579 5361 54 7 9503 9586 3271724 6068 366 27 7 1 386 2021 3968 8800 53 8 2892655 308 2771 444 9327 6079 5304 7358 427 2239 52 9 5808 5957 81C5 347 2580 937 f J 8708 4070748 5680 51 10 8961 9143 328 1387 5846 367 2680 387 2053 4139 9121 50 11 290 2114 30 21530 4610 9107 5981 5398 7531 428 2563 49 12 5269 5517 7833 348 2308 9284 8744 408 0924 6005 48 13 8423 8705 329 105t 5630 368 2581 388 2091 4318 9449 47 14 2911578 310 1893 4281 8893 58W 5439 7713 429 2894 46 15 4734 5083 7505 3492156 919C 87S7 4091108 6339 45 16 7890 8272 330 0731 5420 369 25(K: 389 2130 4504 9785 44 17 2921047 311 1462 3957 8685 5801 5480 7901 430 3232 43 18 4205 4653 7184 350 1950 911 -2 8837 410 1299 6680 42 19 7363 7845 3310411 5210 370 2420 390 2189 4697 4310129 41 20 293 0521 3121036 8639 8483 57 2? 5541 8097 3579 40 21 3C80 4229 6868 351 1750 9030 8894 411 1497 7030 39 22 6839 7422 3320097 5018 371 234( 391 2247 4898 4320481 38 20 9999 313 0616 3327 8287 5C50 5002 8300 3933 37 24 294 3160 3810 6557 3521556 8967 8957 4121703 7386 36 25 6321 7005 9788 4820 372 227 fr 392 2313 5100 433 0840 36 26 9483 314 0200 3333020 8030 6590 5670 8510 4295 84 27 295 2645 3396 6252 3 ,ol368 8903 9027 4131915 7751 33 28 5808 6593 9485 353 4640 373 2217 393 2380 5321 434 1208 32 29 8971 9790 3342719 7 ( J12 6632 5745 8728 466! 31 30 2962135 315 29SS 5953 3541186 8847 9105 4142130 8124 30 31 5299 6186 9188 4400 374216: 394 24C5 6544 435 1583 29 32 8464 9385 335 2424 7734 547! 5827 8953 5043 28 33 297 1630 31625S5 5660 355 1010 8797 9189 415 2363 8504 27 34 4796 5785 *89t 42SC 375 21K 395 2552 5774 436 196f 26 35 7962 8986 3362134 7562 54' ;: 5910 9180 5429 25 30 298 1129 317 2187 5371 8600840 87.'.: 928C 416 259S 8893 24 37 4297 5389 8610 4118 376 207: 396 2045 6012 437 235" 23 38 7465 8591 3371850 7307 539- 6011 9420. 5823 22' 89 2990634 318 1794 5090 357 0070 8716 937 417 2841 9289 21 40 3803 499S 8330 3950 377 203S 397 2740 6257 438 275 20 41 6973 8202 338 1571 7237 5361 6114 9673 6224 19 42 3000144 319 1407 481o 358 051? 8085 9483 418 3091 9693 18 43 3315 4613 8050 3801 3782010 398 285L 050'. 439 3163 17 44 6486 7819 339 1299 70S3 5335 622- 9928 6634 16 45 9658 320 1025 4543 3590367 8661 959C 419 334F 440010 15 46 301 2831 4232 7787 3651 379 1988 399 290f 0701 357 14 47 6004 7440 340 1031 6936 5315 6341 420 0190 705 13 4S 9178 321 0649 427? 360 0222 8044 9715 861! 441 052 12 49 302 2352 3858 7524 3508 380 197u 4003081 703C 400 11 50 5527 706" .341 0771 6795 5302 646f 421 04C( 747 10 51 8703 3220278 4010 361 OOSl 8633 9841 3885 442095 9 52 303 1879 3489 T267 3371 3811964 .401 3218 7311 413 8 5:5 505o 6700 342 051 f 6660 529f C590 4220738 7910 7 54 8232 991'. 376o 994f 8629 9974 4165 4431390 6 55 3041410 323312o 701, 3623240 382 1902 402 3354 7594 4S71 56 4588 6338 343 02Cf 6531 6294 6734 423 1023 835^ 4 57 7767 9552 351 9823 . 8631 403 Oil f 4453 4441834 3 58 305 094 324 2766 67 7C 363 311 f 383196" 34 9( 7884 5318 2 59 4126 5981 344 002, 6408 530. 687 424 1311 8802 1 60 7307 9197 327 9702 8040 404 02G1 4748 44.1 22S7 / 73 72 71 70 69 68 67 66 / NAT. COTAN. NATURAL TANGENTS. 259 24 25 26 1 27 28 29 30 31 I / 445 2287 466 3077 487 7326 509 5254 -531 7094 5543091 577 3503 600 8606 60 5773 6618' 488 0927 89191 -532 0826 6894 7382 601 2566 59 9260 467 0161 4530 510 25851 4559 555 0698 578 1262 6527 58 4462747 3705 8133 6252 8293 4504 5144 602 04<(0 57 6236 7250 -489 1737 9919 -533 2029 8311 9027 4454 56 9726 468 0796 5343 511 3588 5765 5562119-5792912 8419 55 447 3216 4342 8949 7259 9503 5929 6797 603 2386 54 6708 7890-490 2507 ! '512 0930 534 3242 9739 -580 0684 6354 53 448 0200 469 1439 6166 4602 6981 557 3551 4573 604 0323 52 3093 j 4988 9775 8275 535 0723 7364 8462 4294 51 7187 8539 -491 3386 -513 1950 4465 558 1179 581 2353 8266 50 449 0682 470 2090 6997 5025 8208 4994 6245 605 2240 4'J 4178: 5643-4920010 9302 536 19o3 8811 -582 0139 6215 48 7075 9196 4224 514 2980 5699 559 2629 4034 6060192 47 45011731-4712751 7838 4672| 6306'-4931454 6658 9446 5150338-5373194 6449 5600209 7930 583 1828 4170 8149 46 45 8171 9863 5071 4019 6943 4091 5720 607 2130 44 4511672-4723420 8689 5173 6978-4942308 7702 -538 0694 516 1385 4445 7914 561 1738 9627 584' 3528 6112 608 0095 43 4-2 8676-4730538| 5928 5069 8198 5564 7431 4080 41 452 2179 4098! 9549 8755 539 1952 9391 585 1335 8067 40 5683 7659/495 3171 517 2441 5707 5623219 FJ41 609 2054 39 9188 474 1222j 6794 6129 9464 7048 9148 6043 38 453 2094 4785 496 0418 981 3 >540 3221 5630879 5863050 610 0034 37 6J01 8349 4043 -518 3508 6^80 4710 6905 4026 30 9709 454 3218 475 1914 5481 766&] 7199 49712971-5190891 541 0740 4501 8543 -587 0876 564 2378 4788 8019 611 2014 35 34 6728 9048 4925 4584 8263 6213 8702 6011 33 455 0238 476 2616 8554! 8278 542 2027 5650050/5882616 612 0008 ' 32 3760 6185 498 2185 520 1974 5791 3888 6533 4007 31 7263 9755 5816 5671 9557 7728 589 0450 8008 30 4560776 477 3326 9449 9368 543 3324 566 1568 4369 613 2010 '29 4290 689J -499 3082 521 3067 7092 5410 8289 6013 28 7806 4780472 6717 6767 544 0862 9254 590 2211 6140018 27 457 1322 4046 500 0352 522 0468 4032 567 309 6134 4024 26 4839 7821 3989 4170 8404 6944 591 0058 8032 25 8357 i'479 1197 7627 7874 -545 2177 568 0791 3984 615 2041 458 1877 4774 501 1266 523 1578 5951 4639 7910 6052 23 5397 8352 4906 5284 9727 8488 592 1839 6160064 22 8918 480 1932 8547 8990 5463503 569 2339 5768 4077 21 4592439 5512 502 2189'-524 2098 7281 6191 9699 8092| 20 5962 9093 5832 6407 547 1060 570 0045 593 3032 617 2108 19 9486/481 2675 9476 525 0117 4840 3899 7565 6126 IS 460 3011 6258 5033121 3829 8621 7755 5941501 618 0145 17 6537 9842 6768 7541 548 2404 571 1612 5437 4166 16 401 0063 482 3427 504 0415 5261255 6188 547l| 9375 8188 15 3591 7014 4063 4969 9973 9331 595 3314 619 2211 14 7119/4830601 7713 8685 549 3759 5723192| 7255 6236 ]3 462 0649 4189 505 1363 527 2402 7547 7054 5961196 620 0263 12 4179 7778 5015 6120 5501335 5730918 5140 4291 11 7710 484 1368 8668 9839 5125 4783 9084 8320 10 463 1243! 4959 -506 2322 528 3560 8916 8649 597 3030 621 2351 1 9 4776 8552 5977 7281 551 2708 574 2516 6978 6383 8 8310 485 2145 9633 529 1004 6502 6385 598 0926 622 0417 7 464 1845 5739/507 3290 4727 552 0297 575 0255 4877 4452 6 5382 9334 6948 8452 4093 4126 8828 8488 5 8919 4862931-5080607 530 2178 7890 7999 5992781-6232527 4 465 2457 6528 4267 5906 553 1688 5761873 6735 6566 3 5996 487 0126 7929 9634 5488 5748 600 0691 624 0607 2 9536 3726-509 1591 -531 3304 9288 96251 4648 4650 1 4663077 7326 5254 7094 -554 3091 577 3503 8606 8694 n 65 64 63 62 61 60 | 59 68 | ' KAT. COTAN. 258 NATURAL TANGENTS. / i 10 17 18 19 20 O|o 22 1 23 / 2867454 305 7307 324 9197 3443276 363 9701 383 8640 40-1 0262! -424 47 48 60 1 287 0602 3060488 325 241;, 630 364 299', 384 1978 36461 8182 59 2 3751 3070 5630 9785 629L 317 7031 -425 1GK 58 3 6900 6852 8H4S 3453040 9588 8650 405 0417 5051 57 4 288 0050 307 0034 326206t. 6296 365 2886 385 1996 3804 8-VLS7 66 5 3201 3218 52S4 9553 6182 5337 7191 426 1924 55 6 6352 6402 8504 3462810 948(. 8679 4060579 5301 54 7 9503 9586 3271724 6068 366277^ 386 2021 3968 8800 53 8 289 2655 308 2771 4941 9327 6079 5364 7358 427 2239 52 9 5808 5957 8105 347 2586 9379 8708 407 0748 5680 51 10 8961 9143 323 1387 5840 367 2680 387 205C 4139 9121 50 11 2902114 301 2:530 4610 9107 5981 5398 7531 428 2563 49 12 5269 5517 7833 348 2368 9284 8744 4080924 6006 48 13 8423 8705 329 105t 5630 368 2587 388 2091 4318 9449 47 14 291 1578 310 1893 42S1 8893 5891 5439 771:; 429 2894 46 15 4734 5083 7505 349 2156 B19 87 S7 409 1108 6339 45 16 7890 8272 3300731 5420 369 2500 389 213( 4504 9785 44 17 292 1047 311 14G2 3957 86S5 5801 548C 7901 430 3232 43 18 4205 4653 7184 350 1950 9112 8837 410 1299 6680 42 19 7363 7845 3310411 521C 370 2420 390 2189 4697 4310129 41 20 2930521 3121036 3639 8483 5728 5541 8097 3579 40 21 3G80 4229 GSG8 351 1750 90o(i 8894 411 1497 7030 39 22 6839 7422 3320097 5018 371 234( 391 2247 4898 432 0481 38 2C 9999 313 061 6 oo27 8287 5C5( 5002 8300 3933 37 24 294 3160 3810 6557 3521556 8967 8957 412 1703 7386 36 25 6321 7005 9788 482G 372 2278 392 2313 6106 433 0840 36 26 9483 3140200 3333020 8006 5590 5670 8510 4295 34 27 295 2645 3396 6252 o.o 1368 8903 9027 413 1915 7751 33 28 5808 6593 9485 3j3 4(540 373 2217 393 238C 5321 434 1208 32 29 8971 9790 3342719 7912 6632 5745 8728 4666 31 30 296 2135 315 2988 5953 354 11SC 8847 910& 414 21 3f 8124 30 31 5299 6186 9188 4400 374 216L 394 24Cf 5541 435 1583 29 32 8464 9388 335 2424 7734 5479 5827 8953 5043 28 38 297 1C30 316 2585 5660 355 1010 8797 9189 415 2363 8504 27 34 4796 5785 ^896 42SC 375 21 U 395 2552 5774 4361966 26 35 7962 8986 3362134 7562 ' 5432 59K 9186 5429 25 36 298 1129 317 2187 537 - 350084-0 8753 928C 416 2598 889.3 24 37 4297 5389 8610 4118 3762073 390 2C45 6012 437 2357 23 38 74(i5 8591 337 1850 7387 5394 6011 942C 5823 22' 39 299 0634 318 1794 5090 357 0(576 8716 937,' 417 2841 9289 21 40 3803 4998 8330 395C 377 2038 397 274f 6257 438 2756 20 41 6973 8202 338 1571 7237 5361 6114 9673 6224 19 42 300 0144 319 1407 4813 358 0518 8685 9483 418 3091 9693 18 43 3315 4613 805C 3801 378 2010 398 285C 6609 439 3163 17 44 6486 7819 3391299 7083 5335 6224 9928 6634 16 45 9658 3201025 4543 3590367 8661 959t 419 .",34* 440 0105 15 46 301 2831 4232 7787 3G51 379 1988 399 29Cf 6769 3578 14 47 6004 7440 340 1032 6936 5315 6341 420 0190 7051 13 4S 9178 321 0649 427? 360 0222 8C44 9715 3613 441 052C 12 49 302 2352 3858 7524 3508 380 197o 40030Si 703G 4001 11 50 5527 7067 .341 0771 6795 5302 6465 421 04CO 7477 10 51 8703 3220278 4019 361 0082 8633 9841 3885 442 0954 9 52 303 1879 3489 7207 3371 3811964 .401 3218 7311 4132 8 5:j 5055 6700 342 051 f 6660 529( C59f 4220738 7910 7 54 8232 9912 37 994V 862r 9974 4165 4431390 6 55 3041410 3233125 7015 3623240 382 1902 4023354 7594 4871 5- 56 4588 6338 343 02C( 6531 6296 6734 423 1023 8352 4 57 7767 9552 3518 982." . 8c:n 403011f 4453 444 1834 3 58 305 094P 324 2766 (1771 3633115 383 1907 349< 7884 5318 2 59 4126 5981 344002. 6408 6303 687J 4241 31 f 8802 I 60 7307 9197 3276 9702 8640 404 02GL 4748 445 2287 / 73 72 71 70 69 68 67 66 / NAT. COTAN. NATURAL TANGENTS. 259 24 25 26 27 | 28 29 30 31 r 445 2287 466 3077 487 7326 509 5254! -531 7094 -554 3091 577 3503 6008606 60 5773 6618 4880927 8919-5320826 6894 7382- 601 2566 59 9200 467 0161 4530 510 2585 4559 5550698-5781262 6527 58 4462747 3705 8133 6252 8293 4504 5144 602 04K) 57 6236 7250 -489 1737 9919 533 2029 8311 9027 4454 56 9726>468 0796| 5343 '511 3588 5765 5562119-5792912 8419 55 447 3216 4342 8949 7259 9503 5929 6797 603 2386 54 6708 7890 -490 2557 5120930 534 3242 9739 580 0684 6354 53 448 0200 469 1439 6166 4602 69S1 557 3551 4573 604 0323 52 3693 4988 9775 8275 535 0723 7364 8462 4294 51 7187 8539 -491 33861-513 1950 4465 5581179-5812353 8266 50 449 0682 470 2090 6997 5025 8208 4994 6245 605 2240 4'J 4178 5643-4920610 9302 -536 1953 8811 582 0139 6215 48 7675 9196 4224 514 2980 5699 559 2629 4034 606 0192 47 4501173 471 2751 7838 6658 9440 6449 7930 4170 46 4672] 6306-4931454 8171! 9863! 5071 5150338-5373194 4019 6943 560 0269 4091 583 1828 5726 8149 607 2130 45 44 451 1672 472 3420 8689 7702-5380094 7914 9627 6112 43 5173 6978-4942308 5161385 4445 561 1738 584'352b 6080095 42 8676 4730538 5928 5069 8198 5564 7431 4080 41 452 2179J 409S! 9549 8755 539 1952 9391 585 1335 8067 40 5683 7659-495 3171 '517 2441 9188 -474 1222 6794| 6129 4532694 4785-4960418; 9Sl3 5707 9404 '540 3221 562 3219 7048 563 0879 5241 9148 5863050 609 20541 39 6043' 38 6100034' 37 6201 8349 4043 -518 3508 6J80 4710| 6965 4026' 36 9709 4751914 7669 7199 541 0740 8543 587 0876 8019 35 454 3218 5481 497 1297 519 0891 4501 564 2378 4788 611 2014 34 6728] 9048 4925 4584 8263 6213| 8702 6011 33 455 0238 476 2616 8554 8278 542 2027 565 0050 588 2616 612 0008 ' 32 3750 6185 498 2185 520 1974 5791 3888 6533 4007 31 7263 9755 5816 5671 9557 7728 589 0450 8008 30 4560776 477 3326 9449 9368 543 3324 566 1568J 4369 613 2010 '29 4290 689J -499 3082 521 3067 71KI2 5410 8289 6013 28 7806 4780472! 6717 6767 544 0862 9254 590 2211 6140018 27 457 1322 4046 -500 0352 522 04C8 4632 567 3098 6134 4024! 26 4839 7621 3989 4170 8404 6944 591 0058 8032 25 8357 :> 479 1197 7627 7874 -545 2177 568 079l| 3984 615 2041 24 458 1877 4774-5011266 523 1578 5951 4639 7910 6052 23 5397 8352 4906 5284 9727 8488 592 1839 6160064 22 8918 '480 1932) 8547 8990 5463503 569 23391 5768 4077 21 459 2439 5512 -502 2189'-524 2698 7281 619l| 9699 8092 20 5962 9093 5832 6407 547 1000 570 0045 593 3632 617 2108 19 9486 481 2675 9476 525 0117 4840 3899 7565 6126 18 460 3011 6258 5033121 3829 8621 7755 5941501 618 0145 17 6537 9842 6768 7541 548 2404 571 1612 5437 4166 16 461 0063 '482 3427 504 0415 526 1255 6188 471] 9375 8188 15 3591 7014 4063 4969 9973 9331 595 3314 619 2211 14 7119 4830601 7713 8685 549 3759 57231921 7255 6236 13 462 0649 4189 505 1363 -527 2402 7547 7054I-5961196-6200263 12 4179 7778 5015 6120 5501335 5730918 5140 4291 11 7710 '484 1368 8668 9839 5125 4783 9084 8320 10 463 1243 4959 506 2322 528 3560 8916 8649| -597 3030| -621 2351 1 9 4776 8552 5977 7281 551 2708 5742516 6978 6383 8 8310 485 2145 9633 529 1004 6502 6385 5980926-6220417 7 464 1845 5739 -507 3290 4727 552 0297 57502551 4877! 4452 6 5382 9334 6948 8452 4093 4126 8828 8488 5 8919 486 2931 -508 0607 530 2178 7890 7999 5992781-6232527 4 4652457! 6528 5996 -487 0126 4207 7929 5906 9634 553 1688 -576 1873 6735] 6566 3 5488 5748|-600069i;-6240607! 2 9536 466 3077 3726 7326 509 1591 5254 531 3364 7094 9288 9025 554 3091 -577 3503 4648 4650 1 8606 8694 o 65 64 63 62 61 GO 59 58 ' KAT. COTAN. NATURAL TANGENTS. 32 33 1 34 35 36 37 38 39 | ' 624 8694 6494076-6745085 "00 207;' 726 5425 53 5541 -781 2856 8097840 GO 625 2739 8212 9318 6411 9871 54 0102J 7542 810 2658 59 6786 650 2350 -675 3553 "01 0749 727 431* 4ur,r, -782 2229 7478 58 6260834 (5490 7790 5089 8767 9232J 6919 811 2300 57 4884 651 0631 -676 2028 9430 728 321f. 755 3799] -783 1611 7134 56 8935 4774 6268 "02 3773 7C71 8369 C305 812 1951 55 627 2988 891S.-677 0509 811!- 729 2125 "562941 "84100-2 6780 54 7042 6523064 4752 703 2404 65S2 7514 5700 813 1611 53 628 1098 7211i 8997 881E 730 1041 "57 2090 785 0400 6444 52 5155 6531360J-6783243 704 11CC 5501 6668 510:; 814 1280 51 9214 5511 7492 5515 9963 758 1248 9808 6118 50 629 3274 9663-6791741 9809 731 4428 5829 7864515 8150958 49 733C 6543817 5993 705 4224 8894 759 0413 9224 5801 48 630 1399 7972 680 0246 8581 732 3362 4999 787 39G5 816 0646 47 54G4 655.2129 4501 7062940 7832 9587 8649 5493 46 9530 6287 ' 8758 7301 733 230.3 7C04177 788 3364 817 0343 45 631 3598 6560447 681 3016 707 1664 6777 . 8769 8082 5195 44 7667 4609 7270 6028 7341253 761 3363 789 2802 SIS 0049 43 632 1738 8772 -682 1537 708 0395 5730 7959 7524 4905 42 5810 657 2937 5801 47 C^ 735 0210 762 2557 790 2248 9764 41 9883 7103 683 006b 913G 4G91 7157 6975 8194625 40 633 3959 65S-1271 4333 709 3504 9174 763 1759 791 1703 9488 39 so3f 5441 8G01 787,' 7363CCO 6363 6434 820 4354 38 634 2113 9012 6842871 710 225C 8147 7640969 7921167 9222 37 619P 659 3785 714: 6630 737 2C3C 5577 5902 821 4093 36 .635 0274 7960 6851416 711 1009 71'2" 7650188 793 0640 8965 35 435" 660 2136 5691 539: 738 1G20 4800 379 822 384() 34 8441 6313 9969 917-: 6115 9114 794 0121 8718 33 636 252" 661 0492 686424" 7124157 7390611 766 4031 - 48C5 .823 3597 32 6614 4673 8528 8643 51 11 8649 9611 8479 31 637 0703 8856 687 2S10 713 2931 9C11 767 3270 795 4359 824 3364 30 4793 8885 6623040 7093 7225-688 1379 7320 714171'. 740411. 801, 7893 768 2-517 9110 796 3862 8251 8253140 29 28 638 2978 6631413 566 61 Oi 741 312 7144 8617 8031 27 7073 5601 995o 715 0501 763 769 1773 797 3374 826292: 26 639 1169 9792 689 424 489S 742214 6404 8134 7821 25 526" 664 3984 853 9297 6C5 770 1037 798 289; 827 2719 24 9366 8178 690 283 716 369? 7431170 5672 7851 7fi2< 23 640 346" 665 2373 712 8100 568 771 0309 799 242;' 828 252b '22 7569 6570 691 142 717 250 744020 4948 7193 7429 21 641 167. 6660769 572 6911 472 9589 800 1903 829 2337 20 577 4969 692 002 718 1319 924 772 4233 673C 7247 19 988 9171 432 5729 745 377 8878 801 1511 830 2160 IS 642 399 667 3374 863. 7190141 829 773 3526 62SS 7075 17 810 7580 .693 293 4554 746 2S2 8176 802 106- 831 1992 16 6-13 221 668 17861 724 897< !'.'>:, 774 282" 584S 6912 15 632 5995-694155 720 3387 747 188 7481 803 063', 832 1834 14 .644044 669 0205] 586 780f 642 775 213" 5418 6759 13 456 4417 -695 018 721 2227 748 095 679J 804 020f 833 1686 12 867 8630 449 6650 549 776 14fw 499- 6615 11 645 279 670 2845 881 7221 07 fi 749 003 611? 979C 8341547 10 691 70611. 696 3131 550'. 457 777 078^ 805 458-J 6481 9 646 104 671 1280. 7451 993f 911 5448 9381 835 1418 8 516 5500-697 177C 723 4361 T50366 7780117 8064181 6357 7 929 9721 609' 879- r 821 4788 89Sr 836 129 6 647 341 672 3944 -698 042: 724 322- 751276 9461 807 378' 624 5 754 8169J 474< 766T 731 779413S 859: 837 118 4 648 167 673 2396 907* 7252101 752186 8812 808 340 613 3 580 6624! -6993404 654f 642 780 349-2 821 838 108 2 994 674 0854J 7741 726 098'^ 753098 817S 809 302, 604 1 649 407 5085 -700 207 542c 554 781 2856 1841 839 099 57 56 55 54 53 52 51 50 / NAT. COTAN. NATURAL TANGENTS, 261 40 41 42 43 44 45" 46 47 839 0996 869 2867 900 4040 932 5151 965 6888 00 00000 03 55303 07 23687 6 595o 797G 9309 933 0591 9662511 05819 1333 299435 8400915 870 3087 901 4580 6034 8137 11642 67367 36203 fi 5878 8200 9854 934 1479 967 3767 17469 73404 42467 5 841 0844 871 331C 902 5131 6928 1 9399 23298 79445 48734 5 5812 8435 903 0411 935 2380 9C8 5035 29131 85489 560065 842 0782 872 3556 5(9:, 7834 9690C74 34968 91538 61282,5 5755 8680 904 097! 936 3292 631G 40807 97589 6756115 8430730 8733806 6267 8753 97019C2 46051 0403645 73845:5 5708 8935 905 1557 937 4210 7610 52497 09704 80132 8440C8S 8744067 6S51 9oS3 971 3262 58348 15767 86423 5C70 9201 906 2147 938 5153 8917 C4201 21833 9271 8 .4 8450!;55 875 433S 744< 139 OC25 9724575 70058 27904 99018 '4 5G43 947 S 907 2748 6101 9730230; 75918 33977 1-08 05321 A 846 OC33 8764G20 8053 940 1579 5901 81782 40055 11628 N! 5G25 9705 908 33GO 7061 97415C9 87C49 40136 17939 4 847 OG20 877 4912 8071 941 2545 7240 93520 52221 24254 4 5017 878 0062 909 3984 8033 975 2914 99394 58310 30573.;-J 848 0017 5215 9300 9423523 8591 1-01 05272 C4402 36896 4 5G19 879 0370 910 4619 9017 976 4272 11153 70498 43223 4 849 0624 5528 9940 9434513 9956 17038 76598 49554 4 5631 880 OC8F 911 5265 944 0013 977 5643 22925 82702 55889, i 8500640 5S52 912 0592 5516 9781333 28817 88809 62228 !l 5053 881 1017 5922 945 1021 7027 34712 94920 68571 ,1 851 0067 61 SO 913 1255 6530 979 2724 40610 1-05 01034 74918 C 5684 882 1357 6591 946 2042 8424 4C512 07153 812f>9 .' 8520704 6531 914 1929 7556 980 4127 52418 13275 87024 i 5726 883 1707 7270 947 3074 9833 58320 19401 853075!; 6880 9152015 8595 981 5543 64239 25531 1-09 00347 : 5777 884 2068 7962 9484119 9821256 70155 31664 067 14J 8540807 7253 9163312 9646 6973 76074 37801 13085 : 5831 885 2440 8C65 949 517 C 983 2092 81997 43942 19460;: 855 0873 7030 917 4020 950 0709 8415 87923 50087 25840 i' 5910 8862S22 9379 6245 9844141 93853 5623- r - 32223 8560'J50 8017 918 4740 951 1784 9871 99780 623^,8 38610 5992 887 3215 919 0104 7326 985 5C03 1-02 05723 68544 45002 1 857 1037 8415 5471 9522871 986 1339 11664 74704 51397 6084 8883619 9200841 842( 7079 17608 803^7 57797 858 1133 8825 6214 953 3971 987 2821 23555 87035 64201 6185 889 4033 921 1590 9520 8567 29506 93206 70609 8591240 9244 6969 954 5083 9884316 35461 99381 77020 6297 890445? 922 2350 955 0644 989 0069 41419 1-0605560 83436 860 1357 9675 7734 6208 5825 47381 11742 89857 6419 891 4S94 9233122 9561774 990 1584 53346 17929 96281 861 14S4 8920116 8512 7344 734( 59315 24119 L-10 02709 6551 5341 924 390,' 957 2917 9913112 65287 30313 09141 8621621 893 0509 9301 8494 8881 71263 36511| 15578 6694 5799 9254700 958 4073 9924054 7724." 42713 22019 8631768 8941032 9260102 9655 993 0429 83220 48918 28463 6S4f 6268 5500 959 5241 6208 89212 55128 34912 864 192f 895 1500 927 0914 9600829 9941991 95203 61341 41365 7009 6747 6324 6421 7777 1-0301190 67558 47823 ! -865 2094 8961991 9281 7 3F 961 201 ( 995 3560 07194 73779 54284 > .7181 7231 7154 7C14 935S 13195 80004 60750 t -866227' 897 2487 929 257: 962 3215 9965154 19199 8G233 67219 > 736b 7739 7990 8819 997 0953 25208 9246f 73C93 ) -867 246Y 898 2994 930 3421 963 44 2~ 67 50 31220 98702 80171 7558 8251 8849 904 003" 998 2562 3723. 1-07 04943 86053 i -868 2659 899 351 _ 931 4280 5C51 8371 43254 11187 93140 ) 7762 8775 9714 965 120 ,999 4184 4927" 1743;' 99630 ) -869 2867 900 4040 932 5151 6SSS 1-000 OOOC 55303 23687 1-11 06125 49 48 47 46 45 44 43 42 NAT. COTAN, 262 I NATURAL TANGENTS. / 48-j 49 50 51 52 53 54 1-11 06125 1-15 03684 1-19 17536 1-2348972 1-27 99416 1-327044S 1-37 63819 1 1624 10445 24579 56319 1-28 07094 78483 72242 2 19127 17210 3162(i 63672 14776 0524 80672 3 25635 23979 38679 71030 22465 94571 89108 4 32146 30754 45736 78393 30160 1-3302624 97551 5 38662 37532 52799 85762 37860 10(584 1-38 06001 f 45182 44316 53866 93136 45566 18750 14458 7 51700 51104 66938 1-24 00515 53277 26822 22922 8 58235 57896 74015 07900 60995 34900 31392 9 64708 64693 81097 15290 68718 42984 39869 10 71305 71495 88184 22685 76447 51075 48353 11 77846 78301 95276 30080 84182 59172 5(W44 12 84391 85112 1-20 02373 37492 91922 67276 65342 13 90941 91927 09475 44903 99669 75386 73847 14 97495 98747 16581 52320 1-2907421 83502 82358 15 1-12 04053 1-1605571 23093 59742 15179 91624 90876 10 10616 12400 30810 67169 22943 99753 99401 17 17183 19384 37932 74602 30713 1-34 07888 1-3907934 18 23754 26073 45058 82040 38488 16029 16473 19 80329 32916 52190 89484 46270 24177 25019 20 36909 39763 59327 96933 54057 32331 33571 21 43193 46615 66468 1-25 04388 61850 40492 42131 22 50081 53472 73615 11848 69649 48658 50698 23 60674 60334 80767 19313 77454 50832 59272 24 63271 67200 87924 26784 85265 65011 67852 25 65872 74071 95085 34260 93081 73198 70440 20 7(478 809471-2102252 41742 1-3000904 81390 85034 27 83088 87827 09424 49229 08733 895S9 93636 28 69702 94712 16601 56721 16567 97794 1-4002245 29 96321 1-17 01601 23783 64219 24407 1-350(5006 10860 30 1-1302944 08496 30970 71723 32254 14224 19483 31 09571 15395 38162 79232 40106 22449 28113 32 16203 22298 45359 86747 47964 30680 36749 33 22839 29207 52562 94267 55828 38918 45393 34 29479 36120 59769 1-2601792 63699 47162 54044 35 36124 43038 66982 09323 71575 55413 62702 36 42773 49960 74199 16860 79457 63070 71367 37 49427 56888 81422 24402 87345 71934 80039 38 56085 63820 88650 31950 95239 80204 88718 39 62747 70756 95883 39503 1-31 03140 88481 97405 40 69414 77698 1-22 03121 47062 11046 96764 1-4106098 41 76086 84644 10364 54626 18958 1-3605054 14799 42 82761 91595 17613 62196 26876 13350 23506 43 89441 98551 1 24866 69772 34801 21653 32221 44 96126 1-18 05512 32125 77353 42731 29963 40943 45 1-1402815 12477 39389 84940 50668 38279 49673 46 09508 19447 46658 92532 58610 46602 58403 47 16206 26422 53932 1-27 00130 66559 54931 67153 48 22908 33402 61211 07733 74513 63267 75904 49 29615 40387 68496 15342 82474 71610 84662 50 ' 36320 47376; 75786 22957 90441 79959 93427 51 43041 54370 83081 30578 98414 88315 1-42 02200 52 49762 61369 90381 38204 1-32 06393 96678 10079 53 56486 68373 97687 45835 14379 1-37 05047 1976G 54 63215 753821-2304997 53473 22370 13423 28561 55 69949 82395 12313 61116 30368 21 800 37362 H 76687 89414 19634 68765 38371 30195 40171 57 83429 96437 26961 76419 40381 38591 549S8 58 90176 1-19 03465 34292 84079 54:>>97 40994 63811 59 96928 10498 41629 91745 62420 55403 72642 60 11 -la 03684 17536 48972 99416 70448 03S19 81480 ' 41 40 39 38 37 36 35 HAT. COTAN. NATURAL TANGENTS. 263 / 55 56 57 58 59 GO 61 / 1-42 83.480 48 25610 539CC60 1-60 03345 6642705 73 20508 1-80 40478 60 1 9032t 34916 5408160 13709 537 CO 32149 528CO 59 '2 99171 44,231 18280 24082 G4748 43803 65250 58 8 1-43 0803C 53554 28108 34405 75741 554GS 77664 57 4 16901 62884 3794G 44S5S 86744 7144 90086 56 5 257 SI 722 *: 47792 65260 97755 7SS33 1-81 02521 55 31G64 8157C 57047 65G72 67 08782 90533 14969 54 7 43554 9032;; C7510 76094 19818 7402245 27430 53 8 52451 4900238 77383 86525 30864 13969 39904 52 9 61350 09659 87264 96966 41921 25705 52391 51 10 70268 -19039 97155 1-61 07417 529S8 37453 64892 11 79187 2842 5507054 17878 640C7 49213 77405 49 12 88114 37822 1C9G3 28349 *5150 C03S4 89932 43 13 97049 47225 26880 38829 86256 727C8 1-8202473 47 14 1-4405991 56037 36806 49320 97307 84564 1502G 40 13 14941 C605J 4G741 59S20 680S4S9 9C371 27593 45 16 23897 754SC 5G685 70330 19621 7508191 40173 44 17 32862 84923 60639 80850 307C5 20023 527C7 43 18 4183 -i 94367 76601 91380 41919 31S6G 66374 42 19 50814 5003821 86572 1-6201920 53085 43722 77994 41 20 59801 13282 96552 12469 64261 55590 90628 40 21 6879G 22751 5606542 23029 75449 67470 1-8303275 39 '1-1 77798 32229 16540 3599 86647 79362 15930 38 23 86808 417K 26548 4417? 97S56 91267 28610 37 24 05825 51210 36564 54768 69 09077 7603183 41297 36 25 1-45 04850 60713 46590 65368 20308 15112 53999 35 26 13883 70224 5G625 75977 31550 27053 66713 34 27 22923 79743 66669 86597 42804 39007 79442 ?3 28 31971 89271 7 67 22 97227 54009 50972 92184 32 29 41027 98807 86784 1-63078G7 65344 62950 1-8404940 31 30 50090 1-51 08352 96856 18517 76631 74940 17709 30 31 59161 17905 1-57 06036 29177 87923 86943 30492 29 32 68240 2746f 17026 39847 99233 98953 43289 28 83 77326 3703C 271 20 50528 1-70 10559 77 10985 56093 27 34 86420 461)14 37234 61218 21890 23024 68923 20 35 95522 56201 47352 71919 33233 35076 81761 25 30 1-4601632 65796 57479 82630 44587 47141 94613 24 37 13749 75400 67615 93351 55953 59218 1-8507479 23 38 22S74 85012 77760 1-6404082 67329 71307 20353 22 39 32007 94632 87915 14824 78717 83409 33252 21 40 41147 1-5204261 98079 2557C 90116 95524 46159 20 41 50296 13899 1-58 08253 36338 1-71 01527 1-7807651 59080 19 42 59452 23545 18436 47111 12949 19790 72015 18 43 6861C 33200 28628 57893 24382 31943 84965 17 44 77738 4286? 38830 68687 35827 44107 97928 16 45 86967 52535 49011 79490 47283 56285 1-8610905 15 46 96155 62215 59261 90304 58751 68475 23896 14 47 1-4705350 71904 69491 1-65 01 128 70230 80678 36902 13 48 11553 81COL 79731 11963 81720 92893 49921 12 49 23764 91308 89979 22808 93222 1-79 05121 62955 11 50 329SP 1-5301023 1-5900238 33663 1-7204736 17362 76003 10 51 42210 1074 1050- 44529 16261 29616 89065 9 52 51445 20479 207S3 55405 27797 41SS3 1-8702141 8 53 60638 30219 31070 66292 39346 54102 15231 7 54 C9938 39969 41366 77189 50905 60454 28336 6 55 7919~ 4972" 51072 8809- 62477 78759 41455 5 56 88463 5949 61987 99016 74060 91077 54588 4 57 97738 6927 72512 1-660994, 85654 1-80 03408 67736 3 58 1-4307021 7905 82H47 2088 97260 15751 80898 2 59 16311 88S4 92991 31S3-J 1-7308878 2^103 94074 1 60 25610 9865 1-60 0334 4279 20508 40473 1-8307265 ' j 34 33 32 31 30 29 28 / NAT. COTAN NATURAL TANGENTS. 62 63 64 65 66 67 I 68 / 1-88 07265 96 26105 05 03038 2-1445069 24 60368 35 58524 2-47 50869 60 20470 40227 18185 61366 77962 77590 71612 59 33690 54364 33349 77683 95580 96683 92386 58 40924 68518 48531 94021 2513221 36 15801 48 13190 57 60172 82688 63732 2-1510378 30885 34946 b4023 56 73436 96874 78950 26757 48572 54118 54887' 55 86713 9711077 94187 43156 662*3 73316 75781 54 1-89 00006 25296 0609442 59575 81016 92540 96706 53 13313 39531 24716 76015 2601773 37 11791 49 17660 | 52 26635 53782 40008 92476 19554 310C8 38645 51 39971 68050 55318 21608958 37357 60372 59061 50 53322 82334 70646 254CO 55184 69703 80707 49 66688 96635 85994 41983 73035 89060 5001784 48 80068 98 10952 07 01359 58527 90909 38 08444 .22891 47 93464 25286 16743 75091 27 08807 27855 44029 46 1-90 OC874 39636 32146 91677 26729 47293 65198 45 20299 54003 47567 2-17 08283 44674 66758 86398 44 83738 68387 63007 24911 62643 86250 51 07629 1 43 47193 82787 78465 41559 80C36 39 057<:9 28890 1 42 60663 97204 93942 58229 98653 25316 50183 41 74147 99 11637 0809438 74920 28 16093 44889 71507 40 87647 26087 24953 91631 34758 64490 92863 39 1-91 01162 40554 40487 2-18 08364 52846 84118 52 14249 38 14691 55038 56039 25119 70959 4003774 35667 37 28236 69539 71610 41894 89096 23457 57117 36 41795 8405C 87200 58C91 2907257 43168 78598 35 55370 98590 09 02809 75510 25442 62906 2-5300111 34 68960 0013142 18437 92349 43651 82672 21655 ' 33 82565 27710 34085 2-19 09210 61885 41 02465 43231 i 32 96186 42295 49751 26093 80143 22286 64839 | 31 1-9209821 56897 65436 42997 98425 42136 86479! 30 23472 71516 81140 59923 2-30 1(5732 62013 2-5408151 29 37138 86153 96864 76871 35064 81918 29855 28 50819 01 00806 10 12607 93840 53420 2-4201851 51591 27 64516 15477 28369 2-20 10831 71801 21812 73359 26 78228 30164 44150 27843 90206 41801 95160| 25 91956 44869 59951 44878 2-31 08637 61819 2-551C992I 24 1-93 05699 59592 75771 61934 27092 81864 38S58 23 19457 74331 91611 79012 45571 2-4301938 607561 22 33231 89088 2-11 07470 96112 64076 22041 82686 21 47020 2-02 OS862 23348 2-21 13234 82606 42172 2-56 04649 20 60825 18654 39246 30379 2-3201160 62331 26645 19 74645 33462 55164 47545 19740 82519 48674 i W 88481 48289 71101 64733 38345 2-44 02736 7073&1 17 1-9402333 63133 87057 81944 56975 22982 92830 i 16 16200 77994 2-1203034 99177 75630 43256 2-5714957; 15 30083 92873 190302-2216432 94311 63559 37118 14 43981 2-03 07769 35046i 33709 2-3313017 83891 59312 13 57896 22683 51082i 51009 31748 2-45 04252 81539 12 71826 37615 67137 68331 50505 24642 2-58 03SOO 11 85772 52565 83213 85676 69287 45061 26094 10 99733 67532 . 99308 2-23 03043 88095 65510 48421 9 1-9513711 82517 2-1315423 20433 2-34 06928 85987 70782 8 2770 97519 31559, 37845 257 87 2.46 06494 9P177 i 7 4171 2-0412540 477 14' 55280 44672 27030 2-5915606 6 5573 2757 63890 72738 63582 4759f 38068 5 6978 42634 80085 90218 82519 68191 60564 4 8383 5770 963012-2407721 2-35 01481 88816 83095 3 9791 7280 2-14125371 25247 20469 2-47 09470 2-60 05659 i 2 1-961200C 8791 28793 42796 39483 30155 28258 1 2610 2-05 0303S 45069 60368 58524 50869 50891 27 26 25 24 23 22 21 ' NAT. COTAN. NATURAL TANGENTS. 265 69 70 71 72 73 74 75 2-60 50891 2-74 74774 2-90 42109 3-07 76835 3-27 08526 3-48 74144 3-73 20508 73558 99661 6957C 3-08 07325 42588 3-49 12470 63980 96259 2-75 24588 97089 37869 76715 50874 3-74 07546 2-61 18995 49554 2-91 24649 68468 3-28 10907 89356 51207 41766 74561 5225G 92122 451G4 3-5027916 949C3 64571 99608 79909 3-09 29831 79487 66555 3-7538815 87411 2-76 24695 2-9207610 60596 3-29 1J876 3-51 05273 82763 2-62 10286 49822 35358 91416 48330 44070 3-7626807 33196 74990 63152 3-10 22291 82851 82946 70947 56141 2-77 00199 90995 53223 3-3017438 3-5221902 3-77 15185 79121 25448 2-9318885 84210 52091 60938 59519 2-6302136 50738 46822 3-11 15254 86811 3-5300054 3-78 03951 25186 76069 74S07 46353 3-31 21598 39251 48481 48271 2-78 01440 2-94 02840 77509 56452 78528 93109 71392 26853 30921 3-1208722 91373 3-54 17886 3-7937835 94549 52307 59050 3U991 3-32 26362 57325 82661 2-64 17741 77802 87227 71.317 61419 96846 3-80 27585 40969 2-7903339 2-95 15453 3-13 02701 96543 3-55 36449 72609 64232 28917 43727 34141 3-3331736 7G133 3-81 17733 87531 54537 72050 65639 66997 3-5615900 62957 2-65 10867 80198 2-96 00422 97194 3-3402326 55749 3-8208281 34238 2-80 05901 28842 3-14 28807 37724 95C81 53707 57645 31646 57312 60478 73191 3-57 35690 99233 81089 57433 85831 92207 3-3508728 75794 3-8344861 2-6604569 83263 2-97 14399 3-15 23994 44333 3-58 15975 90591 28085 2-8109134 43016 55840 80008 56241 3-843C424 51638 35048 71683 877M 3-36 15753 90590 82358 75227 610(34 2-98 00400 316 19706 51568 3-5937024 3-85 28396 98853 87003 29167 51728 87453 77543 74537 2-67 22516 2-8213045 57983 83808 3-37 23408 3-6018146 3-8620782 46215 39129 86850 3-17 15948 59434 58835 67131 69951 65256 2-9915766 48147 95531 99609 3-87 13584 93725 91426 44734 80406 3-38 31G99 3-61 404C9 60142 2-68 17535 2-83 17639 73751 3-18 12724 67938 81415 3-8806S05 413S3 43896 3-0002820 451023-3904249 3-6222447 53574 65267 70196 SI 939 77540) 40631 63560 3-8900448 89190 2-6913149 96539 2-8422926 61109 90330 3-19 10039 77085 42598 3-40 13612 3-6304771 46064 47429 94516 37147 49356 3-61 19603 75217 50210 8744 1 3-9041710 61181 75831 48926 3-2007897 86882 3-64 28911 89011 85254 2-85 02349 78301 40638 3-41 23626 704C7 3-91 36420 2-70 09364 28911 3-0207728 73440 60443 3-65 12111 83937 33513 55517 37207 3-21 06304 97333 53844 3-9231563 57699 82168 66737 392283-4234297 95665 79297 81923 2-8608SG3 96320 72215| 71334 3-6637575 3-9327141 2-71 06186 35602 3-03 25954 3-22052633-4308446 79575 75094 30487 62386 55641 38373 45631 3-67 21665 3-9423157 54826 89215 85381 71546 82891 63845 71331 79204 2-87 160S8 3-0415173 3-23 04780 3-44 20226 3-6806115 3-95 19615 2-72 03620 43007 45018 38078 57635 48475 68011 28076 69970 74915 71438 95120 90927 3-96 16518 52569 96979 3-05 04S66 3-24 01860 3-45 32679 0-6933469 65137 77102 2-8824033 34870 38346) 70315 76104 3-97 13868 2-73 01674 51132 6492S 7 1 895 3-46 08026 3-70 1 8830 62712 262S4 78277 95038 3-25 05508 45813 1 61648 3-9811669 50934 2-8905467 3-06 25203 39184 83676 3-7104558 60739 75023 32704 55421 729243-4721616 47561 3-9909924 2-74 00352 599SO 85694 3-26 06728 j 59632 90658 59223 25120 87314 3-07 If 020 40596 1 977263-7233847 4-0008636 49927 2-90 14688 4(3400 745293-4835896! 77131 58165 74774 42109 76835 3-27 08526 74144 3-7320.508 4-01 07809 20 19 18 17 16 15 14 NAT. COTAN. 266 NA TURAL TANGENTS. / 76 77 78 79 80 81 i 82 ' 4-01 07809 4-33 14759 4-70 46301 5-1 445540 5'6 712818 63 137515'7'1 153697 60 1 57570 72316 4-71 136S6 525557 809446 256601 304190 59 i 4-020744U i-34 30018 81256 605813 906394 376126 455308 58 8 57440 878C6 4-72 49012 686311 5-7 003663 496092 607056 57 4 4-03 07550 4-35 45S61 4-73 16J54 167051 101256 616502 759437 56 6 57779 4-36 04003 850f,3 848035 199173 737369 912456 55 6 4-0408125 62293 4-74 53401 9292G4 297416 858665 -2 066116 54 7 585,0 4-37 20731 4-75 21907 5-2 010738 395988 980422 . 220422 53 8 4-05 09174 79317 90603 092459 494889 6-4 102633 375378 52 9 59877 4-38 38054 4-76 53490 174428 594122 225301 530987; 51 10 4-06 10700 93940 4-77 28568 256647 883688 348428 687255 50 11 61 043 4-39 559^7 97837 339116 793588 472017 844184 49 VI 4-07 12707 4'40 15164 4-78 C7300 421836 893825 596070 7-3-001780 48 13 G3892 74504 4-79 30957 504809 994400 720591 160047 47 14 4-08 15199 4-41 33996 4-80 06808 588035 5-8095315 845581 318989 46 15 66627 93641 70854 671517 196572 971043 478010 45 ie 4-09 18178 4-42 53439 4-81 47096 755255 298172 6-5 096981 638916| 44 17 69852 4-43 13392 4-82 17536 839251 400117 223396 799909, 43 18 4-10 21649 73500 88174 923505 502410 350293 961595! 42 19 73569 4-4433762 4-83 59010 5-3 008018 605051 4776727-4123978! 41 20 4-1125614 94181 4-8430045 092793 708042 605538 287064 40 21 77784 4-45 54V50 4-85 01282 177830 811386 733892 450S55J 39 '2.1 4-12 30079 4-4615489 72719 263131 915084 862739 615357 38 23 82499 76379 4-86 44359 348696 5-9019138 992080 780576 37 24 4-13 35046 4 : 47 37428 4-87 16201 434527 123550 6-6 121919 946514 36 25 87719 98636 88248 520626 228322 252258 7-5 11 3178 35 2(1 4-1440519 4-48 60004 4-88 60499 606993 333455 383100 280571 34 27 93446 4-49 21532 4-89 3295*6 693030 438952 514449 448699 33 2* 4-15 46501 83221 4-90 05620 780538 544815 646307 617567 32 29 99885 4-50 45072 78491 867718 651045 778677 787179 31 30 4-16 52998 4-51 07085 4-91 51570 955172 757644 911562 957541 30 3i 4-17 06440 69261 4-92 24859 5-4 042901 864614 l 6-7 044966 7-6128657 29 32 60011* 4-5231601 9S358 130300 971957 178891 300533 28 3:; 4-18 13713 94105 4-93 72068 219188 6-0079076 313341 473174 27 34 67546 4-53 56773 4-9445990 307750 187772 448318 646584 26 SO 4-19 21510 4-54 19608 4-95 20125 396592 296247 583826 820769 25 86 75600 82608 94474 485715 405103 ' 719867 995735 24 37 4-20 29835 4-5545776 4-90 69037 575121 514343 856446 7-7 171486 23 3S 84193 4-5609111 4-97 43817 664812 623967 993565 348028 22 39 4-21 38690 72615 4-9818813 754788 733979 6-8 131227 525366 21 40 93318 4-57 36287 94027 845052 844381 269437 703506 20 41 4-22 48080 4-58 00129 4-90 69459 935604 955174 408196 882453 19 42 4-23 02977 64141 5-0045111 5-5 026446 6-1 0663HO 547508 7-8062212 18 4:) 58009 4-59 28325 5-01 20984 117579 177943 687378 242790 17 44 4-24 13177 92680 97078 209005 289923 827807 424191 16 45 68482 4-60 57207 5-02 73395 300724 402303 968799 606423 15 46 4-25 23923 4-61 21908 50349935 392740 515085 6-9 110359 789489 14 47 79501 86783 5-04 26700 485052 628272 252489 973396 13 48 4-26 35218 4-62 51832 5-05 03090 577663 741865 395192 7-9 158151 12 49 91072 4-63 17056 80907 670574 855867 538473 343758 It 50 4-27 47066 82457 5-06 58352 763786 970279 682335 530224 10 51 4-28 03199 4-64 48034 5-07 36025 857302 6-2085106 826781 717555 9 52 59472 4-6513788 5-08 13928 951121 200347 971806 905756 8 53 4-29 15885 79721 92061 5-6045247 316007 7-0 117441 8-0 094835 7 51 72440 4-66 45832 5-09 70426 139680 432086! 2n: 1 ,r,i;2 284796 6 5.") 4-30 29136 4-67 12124 5-1049024 23-4421 548588 410482 475647 5 66 85974 78595 5-11 27855 329474 665515 557905 667394 4 57 4-31 42955 4-68 45248 5-12 OG921 424838 782868 705934 860042 3 5S 4-32 00079 4-69 12083 86224 520516 9006511 854573 8-1 053599 2 59 57347 79100 5-13 65763 616509 6-30188687*1003826 248071 1 60 4-3314759 4-70 46301 5-14 45540 712818 137515 153697 443464 / 13 12 11 10 9 8 7 / NAT. COTAN. NATURAL TANGENTS. 267 83 84 85 86 87 88 89 / 8-1443464 9-5143645 11.430052 14-300066 19-081137 28-636253 57-289962 60 639786 410613 468474 360696 187930 877089 58-261174 59 837041 679068 507154 421230 295922 29-122006 59-265872 58 8-2 03523U 949022 546093 482273 405133 371100 60-305820 57 234384 9-6 220486 685294 543833 515584 124499 61-382905 56 434485 493475 624761 6U5916 627296 * 8225 19 02-499154 55 635547 708000 664496 668529 740291 30-144119 63-656741 54 837579 9-7 044075 704500 731679 854591 4115SO 64-858008 53 8-3 040586 321713 7-14779 795372 970219 6&3307 66-105473 52 244577 600927 78533Q 859616 20-OS719U 959928 67-401854 51 449558 881732 8261C7 924417 205553 31-241577 68-750087 50 655536 9-8 164140 867282 989784 325308 52S392 70-153346 49 862519 448166 908682 15-055723 440486 820516 71-615070 48 8-4 070515 733823 950370 122242 509115 32-118099 73-138991 47 279531 9-9 021125 992349 1S9349 693220 421295 74-729165 46 4S9573 310088 12.034022 257052 818828 730265 76?390009 45 700651 600724 077192 325358 945966 33-045173 78-126342 44 912772 893050 120002 394276 21-074064 360194 79-943430 43 8-5 125943 10-018708 163236 463814 204949 693509 81-847041 42 340172 048283 206716 533981 336851 34-027303 83-843507 41 555468 078031 250505 604784 470401 367771 85-939791 40 771838 107954 294609 076233 605630 715115 88-143572 39 989290 138054 339028 748337 742569 35-069546 90-463336 38 8-6207833 108332 383768 821105 881251 431282 92-908487 37 427475 198789 428831 804545 22-021710 800553 95-489475 36 648223 229428 474221 968667 163980 36-177596 98-217943 35 870088 260249 519942 16-043482 308097 562659 101-10690 34 8-7 093077 291255 565997 118998 454096 950001 104-17094 33 317198 322447 612390 195225 602015 37-357892 107-42648 32 542461 353827 659125 272174 751892 76S613 110-89205 31 768874 385397 706205 349855 903766 38-188459 114-58865 30 996440 417158 753634 428279 23-057077 617738 118-54018 29 8-8 2251S6 449112 801417 507456 213066 39-056771 122-77396 28 455103 481261 849557 587396 371777 505895 127-32134 27 686206 513607 898058 668112 532052 905400 132-21851 26 918505 546151 946924 749614 694537 40-435837 137-50745 25 8-9 152009 578895 .9961 CO 831915 859277 917412 143-23712 24 386726 611841 13-045769 915025 24-026320 41-410588 149-46502 23 622608 644992 095757 998957 195714 915790 156-25908 22 859843 678348 146127 17-083724 367509 42-433464 163-70019 21 9-0 098261 711913 196883 109337 541758 964077 171-88540 20 337933 745/587 248031 255809 718512 43-508122 180-93220 19 578867 779673 299574 343155 897826 44-066113 190-98419 18 821074 813872 351518 431385 25-079757 638596 202-21875 17 9-1 064564 848288 403867 520516 264361 45-226141 214-85762 16 309318 882921 456625 610559 451700 8293.J1 229-18166 15 555436 917775 509799 701529 641832 40-448Sr>2 245-55198 14 802838 952850 563391 793442 834823 47-085343 264-44080 - 13 9-2051564 988150 617409 886310 26-0307^6 739501 286-47773 12 301627 11-023676 671856 980150 229638 48-412084 312-52137 11 553035 059431 726738 18-074977 431000 49-10P881 343-77371 10 805802 095416 783060 170807 636090 315726 381-97099 I 9 9-3 059936 131635 837827 267054 844984 50-548500 429-71757 8 315450 168089 894045 305537 27-056557 51-303157 491-10600 7 572355 204780 950719 464471 271486 52-080673 572-95721 6 830663 241712 14-007856 564473 489853 882109 68^54887 5 9-4 090384 278885 065459 605562 711740 53-708587 859-43630 4 351531 316304 123686 767754 937233 54-561300 1145-9153 3 614116 353970 182092 871068 28-166422 >5-441517 1718-8732 2 878149 391885 241134 975523 399397 56-350590 3437-7467 1 9.5 143645 430052 300666 19-081137 636253 57-289962 Infinite. 6 5 4 3 2 1 / NAT. COTAN. TABLE XVI. CHORDS, VERSED SINES, EXTERNAL SECANTS, AND TANGENTS OF A ONE-DEGREE CURVE. The angles of the table are the intersection angles, I, equal to the total central angle included between the tangent points. 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. 270 CHORDS, VERSED SIXES, EX TERN A 7. SECANTS. 55 O N -<*-VO CO O N ^-vO 00 O N ^O OO O N TJ-\O OO O N ^VD 00 O IN --. ro Tho 00 O M ro irivo OO O CN ^t-vo o M ro moo O ro inoo O ro\o oo M ? K . A 0660006600066600660600066606066 1 ui 6666666066666666666666666666666 Q K *"OvQ O rOvO O rovO O f ")VO O rovO O rovD O rovO O rovO O rovo O ro-O CO ro\O O rovO O OTO O ro^O O roo O ro\Q O ro\Q O nTO O ro\O O ro^O O OOMM^oiMNrororoTTTj-TMom u^vo vo vo t-, i^ t-co CO CO O^ ON Os & 55 W ^"O CO O N ^vO 00 O N ThvO 00 M -^"O OO O N ^-vO CO O (N Tj-O CO O 5 ^VOOO ..VOCO .VOOO N .VOOO ...00 . * x - O^O roOvO roO"O roQ^O roO*O roO ^O- roOO rooO roO'O rOO^O roO .< O M co io\o oo O M ro iOvo co O >-< ro tovo CO O - ro ir>\o oo O w ro iovO oo I OOOQOO-'i-iMNvNro-t-*J- 1O-O t^ t^co O O *-i o -f IAO r^. O O >-i ^OOOOOOOOOOOOOOOOOOOOwtHHMMMMWtNOJ . 3 0066666666666666066666666666666 o c/5 OgOOo" V 00 N 2^2 N ctr?r^ ^-5- SvS & Sco ^ 5" ^ ^TH ^ g"^ 1 ooooooooooooooooooooooooooooooo M 8 rovo O rovo* 8 ro^O 8 rovO O rovO* 8 roO O ro^ 8 rOvO* 8 roO 8 rovO* 8 O X CJ O rovO O rovO O rovo O rovo O rovo O rovO O rOvO O rovO O rovo O rovO O n M M c* CN M'rororOT}--^-rj-u->u-> i/^vo \O vo C^. Os r^.OO OO CO Q> O^ O^ O f N -^VO CO N "4-vO CO O N -4-vO CO N -^vO OO O CJ -^-vO OO O M ^vO OO O CHORDS, VERSED SINES, EXTERNAL SECANTS. 271 g s STR^ilTt^ftRfft^ ^jtl5^9^8^a .ONNOrOO..O.. M CO,. N ON H O M ro n^o co O w ro m-o co O w ro IA ^0 cS 0?C?oS C SN SN 3 S^^ 8 u M % 8 o cT M *B S"8 R 5- 5: & vo NO ?l^o3ooaloNO C ?M8^?or^?? X W MCMNOPICMCMCMCMCMCMINCMCMCMCM CMCMCMCSCMCMrororororororocoro w SB 14 MCMCMCMCSCMCMCMCMffCMCMCMCMCMCM CM CM CM N CM CM rorororororororOfO . v ' K "ON C^NO c5 C?vS ON'M VO Ox^ NO ON'CM VO O^ O s c^rorororocororororororororororo roo O roo O ro\O ON rOvO O rovO O Z MMM.lMCMCMCMCMCMrO W *-v> 00 O d ^-VO CO O W ^*"O 00 O rororoco^-^-^-^-^inmioio mvO Z O W 'f^O OO O CM -^-vO OO O CM Tj-vO 00 O CM^OOO.^OCOOCM^OOOO s z" ONO MONO roO^O ro O NO ro O t^ ro o" t^-^MOO iriNCO IAN OvO ro O t^ gHfll .H.H.H u III CO ro W N N ro ^-so oo H ^co M \O w co ^> t^. O rovo O N u*>6o w moo w m o w \O COONO^O^ONOOOHHtHNWn ro eo O\t^orow CN w M w o* -^-^ oo M $- o ^t- "!j- m m mvo \o t>. t*-. r^co oo ON o> . w z' w . tN, rovo O\ (N irico N moo N m O w *O COO^t^O^O O O MM MO (N w roro SSS&SftSVS- < 8^ id 0000 d K OO M Tftx.M -^-hxO ^^0 ro f^. O rO'O C?NS S C?NS ^ C?N2 ^ C?NS ^ 8\S ^ X u i?tr?i?lif?li?if reH*fieff z' OfM^-NOOOOfM^-vOOOOCM^NOMO N^ J^^CO^ 0_ N_ ^^^ 0^ CM^ ^^OO^g s 272 CHORDS, VERSED SINES, EXTERNAL SECANTS. CN| "J-xO CO O CM -*NO OO O CM t^ -* M oo in EM o NO -3- O r-. "* H o MOO in M 00 in CM OO IT> CM CO in CM C O 1-1 ro ir>vo OO O w ro in\O OO O ir> in in m in in^o vovQvovoso t^i m o fx * M ONO m O t- ir, CM o CM ON in CM ON in CM ONNO N ONNO CO O H ro UINO 00 O w m in -t^M moro )CN O t^-^-N - iri^o vO t^oo , O rovO ON CM moo w -* t^ o m t^ o rovo ON CM moo M * t^ o m\o O n mco H moo M moo H moo w moo w M-OO M -^-oo M -^-co H - ON O <*-CO mOO * O tx CMOO IOMOO tOM rj- t^ O fO\O ON M VO O CM triCO M -*OO M * t^ O m\O ON rONO ON CM 1000 w -*OO ONCMNO 0>CM IOOCN1 ITlONCM U1ONCN! IDC3SCM ITlONCM >AOO CM LOCO CM IT1OO CM ITICO ON. fONO ON mNO ON mvO ON rONO ON mvO ON rONO ON rONO ON m\O ON rONO ON rONO ON O*O O O H M M M CM M romro-^-^j-^-mio IONO vo NO i^ i^ r^co oo oo ON ON ON rO'OOMMMNNrorO'<$- mvo t^ r>.GO ON O ** N ro ^t" *o\o r>.oo O O w \O ^OJ^OOt-tNfOTj- irjO t^OO O O M CM CO -J-^O t^OO O> O W ro ^- in 1^00 O\ CHORD. j CTSaJWS SS ?Mfr ff M SS 5ft?*SS-* z O N -*vO OO O 01 -*NO OO O W *-vO OO O C) ^vO OO O N -^vO OO O N 'frvO OO O < i o a e . . g . o ^oo o ^NOOO o , * X co o^O ro O^O ro O*O roO'O roOO fOO'O roO t>.roO *> ^" O t^**"O t-^^ usmusimKttnuBis&iuigsft y CO t^ t^oo oooooooooooooocooooo OOXONOSONO^ONOO^O^O o o o o o o o 2 - os w > OOOOOOOOOOO-*ON rooo 01 r-i-<\O O in **<*> OOONOi-oiroro-* fnvo t-^o? S O* N ro 5- m>O NO" So?^ ON O ST o? S ? S>vO* t- t^.00 03000000000000000000 ONONONONONONONONONONONO O O O O.O O CHORD. SIH^l^llS^^ll^Hlt^lSslH 55 O W ^-\O OO O W ''t-vO OO O W ^f-vO OO O C* ^vO OO O PI ^^O OO O W ^vO OO O 274 CHORDS, VERSED SINES, EXTERNAL SECANTS. 2 S 55 10 ro M oo MD ^N O t^ 10 ro M oo vQ ro O OO v> ** ro M o 1-^.vD * N 3 OO vO *O N H o^^^o^o u C/2 t^*oo O M ovo CTNVO ON\O o -^oo w vo O "^oo N vo o -<* o^ \O r>.c>O - ro-<*-io t^oo o M w TJ- iovo oo O -* 0) ro u^sO oo o O E??^8 M t^. t^ t^OO OCJOOCQOOOOOOOO C^ONONO>OO'<>O 6 6 6 6 6 6 M g O w ^- r*^ Qs H ^t-\o co M ro irjoo O N m t^. o c* ^vo O M ro^O oo O ro m t- O moo M -foo O a o ON tOON IOO I O O N -*vO oo O w -*vO oo O ro moo O ro moo O ro moo M <> t-s ON M S ro^*- m tioo o< O M ro * mvo oo o O M ro * m\o oo o O rri^J-^-^^-Tj-^^-^mmmioi^iom mo vOvOvOvO^OO^O t^ o N moo M X W 2 C/5 O O H ro ^ mvO t^ O> O w 01 ro mvO t^oo O w N ro mvo c^oo O * tv. o rovO M N -< mvD ro-4-'4-4--4"4-4'-4--4-mmmmmmm m\o \OMDvOvO\OvDvO r> ( r^ t^ r^ i^- > Q O N * (N. O M M-vOOO M romoo O f> v> t~* O N '^O O> M rovo oo i*- 1--. rovo rovo Ox rovO O> N vo Os N moo M moo moo M ^ O ro m f~ O u OO^O O^M^S 2^N * * 5n Jn ro 5- * >* in S mvo vo vo f-. r~ r^co t-*OO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO SSa?*?*? 55 ^>1 . ^vooo ^oo N ^vooo ^co N it-\D 00 O a CHORDS, VERSED SINES, EXTERNAL SECANTS. 275 O PI ^-vO OO O N -<*-NO OO O PI **"NO OO O PI 'f-NO OO Mt-iHi-tHp*PiP4PiPirorocororo O PI -*"O CO O PI -^-VO 00 O -* P) O co '.o 1- CM OoovO TJ-N oco r>. in ro P) O O< fc*w a e W *3j H M m m\o oo O M ro mvo oo O " ro >n\D oo O Pi ro In in in in in^^^^^S)^ in in In in in En m^ m S'Si.Siisis^ij I O \O ro ONVO PICO mwoo Tj-o t^-^MCO -4-Hoo m i-f^>&m M . w v s v v S"8NNprNpTcr t s < s ( s c s c s < SprprprM ^prrororororororororo *H in NO oo O M ro m ^co P) ro K M iiM^iiiM*t 8>?ftftftAAfta^ a S S U i rw"*t O N M-^O OO O N ^^O OO O i ^O 0* -^"O OO O N ^\O OO O N ^'O CO O N -^-vO OO umufeM . fcW *j RM **S**88fc BSf^gg^^S H WttiW4444iiStt*aa CA! c??^ S'WS.8'8 ff^KS^SJ^a^^ MVONCOM-OVONCO^O ON P) ro m i^oo w ro in 1 X w aasaasaas?5s?j?j?s?ffjfjfjrffjr pt p? p? p? c? p? p?^ ^S ^S ^ H 2 en O <* ON * ON -*CO rOOO COCO OOOO MOO <*- ON * ON * 00 ON O P) rO u-ivO 00 ON w P) -^- in t^OO O M ro -^"O oo C?M Pi -00 t^vO ^ r^ r^ X) u~) o* O^O coo t^roo t^ ^t* M oo "*> H JNNNNfSSSSSS***** Htitt&si&tii y C/3 Sr*OT!,S88W!,88 ssszszw&ss^ss t->. t^ r^ t^ c-xoo oooooooo oo>ac>a>O OOOOOMi-iMf4NDDPlfO e . hi * 55 CD ONM rmot^ONM romr^OM ro^t^ H ^0 OOMM H -M O 1 Sro\O rO\O O r*ivO O rO\O rOvO i irj in\O \o ^O t^ t^. t*NOO OO OO O O> O S5 ^ fc O N --vO 10 * N H O ONOO vO >O in ^ ro H ooo r^ m ^ ^ w M ONOO ^ CN ON'O ro O^O COO l>* ^** M OO -^HOO H Waa^WSB^WW S SSSJNfSittttSSX u Id C/3 10 t>. o O N ^'O oo OM r^iorxOO w m -^- w M o^ t^^o m ** ro w o 000 t^. d rororororococo^rororocorororoco xxmasassrftmMfi H 2 M 10 t^ o O N -?^O 00 O M ro to vO OO ^j- N o oo r-*M3 ^ ro M o o> t-x\c ^ fo M H aaaaRananftaaaaas assssaaaas***** Q K = " S2 " 22 = 5S "" IfriifhHiilfi 55' S "^^^ ^**2!!!S! CHORDS, VERSED SINES, EXTERNAL SECANTS. 277 Z W>co c. ^vooo o cj j}-o N^^CO ^.^vooo o^c^^voco Z assess ^ AS fc*?K32tf ag^ag^^asr -S> 5; So vg v^S so K. R S'g.'R wco S o o? SN S S'Ss" 8 3 o 1 o , s ^i g ' ffW . 8 ,-8 M siaift4 M> . ! rh^r8.a X w ss^as.aa^^^^s.s.Kssssss^sisiss^ H VER. SIN. I u 2" M O N ^-vO OO O N ^-\D OO O CJ -^-vO OO O N ^*vO CO O W ^-vO CO N ^^ OO O w S MM M ci N w w w corofOfnco^-Ti-^-^-^uSint?5ir> *VC 55 O N ^"O CO O N "^"O CO W ^t"\O CO N **"VO OO O N ''t'SO OO 0* ^^0 OO O s RS?S,gAS^iSa$8^S^^S9SBa8^8rSS8S H ?o lv 8 c 2EJ^^^ < 2g?!Ncr t S?>^^^P;M5-^?^^?SN^^ CO ro ro J- invo t^oo ON-0 M w fo Tt-yD t^oo O w N * >n^O oo O f> -*^o oo w * O N ^NO OO O W ^t^.ONi-t COlOt^ONw ^t-vO CO W ^>O ONM fOlOC^O W ^*" X ^*^^*^^^^^ 2 2 c/3 M w M mroH-^ui invo t^-oo o\ O M 01 f^i ^t* i^ivo t^oo o w PO -^"O t-*. o\ o N Id K O s O\ o\ O O O M *-i 1-1 N N w mrofO^^j-^j-iotri i/^o \O vO t^. r^. t-^oo oo oo o ON 1 ^^^2^S 278 CHORDS, VERSED SIXES, EXTERNAL SECANTS. 2 O N -*-vO CO -*^O CO O N i*-vO CO O N -*vC CO N M-vO OO O N -*\O CO O z S re^^rag g, 0,0,0 & M g ro.^. .co .0 o M H co oo 00*00 co oo oo oo oo oo'oo^oo^oo'oo'oo oo oo oo oo oo oo" oo* 00^00" co^co^ ON ^ o\ o^ o> u Cd ^0 , X W m^^m^$^u&iiin H JB CO S, 55 g. wg>&wa , w$ , ws -f B88,*g? S u a^^i^^^^*!#^*S5S88^fca8. ri K O a OOO O O"-i M >- ci CN w roroco-*''J-'i-Tt-ioio invo ^O O t^ rx, r^oo oo oo ON U as O 0* ^\O CO O N **-vO CO O N ^fvOOO O N **"O 00 O M -<*-vO 00 O n ^vO 00 C z - S 1 1 2- vO ro ^ -I- M co u^ n ONVO ro 1^ 1- * M -^-OO M ^O "^OO M ^O O ^ O^ rOOO M *^O H *A O ^ 1 t>* o\ 04 Tf h*. cr M ^-\o oo M ro moo o roiot^o 01 iot^.ON -^-t>.ooi ^t^.o^ j M ^^^^^^^ 1 aU IS ^ sg 1 sg II"!* ^l^^H ! ! J^ SS&S U 5 -^^^^^ CHORDS, VERSED SINES, EXTERNAL SECANTS. 279 55 s N -4-vooo e. jvo oo jj-vgo, S^oco o 2^00 o voco o z ?o8 S.SITS'fc^O J^^^S^^O it'S-'S 8N.S Sooo MC. ovo * <, ^v vS vo'vo'S'vo' R. S. ^"R 1^ ^C ( 00 M 28 <" 0> S O^'Sv'ov 8 O ? Q 1 ^ 'o 2 oosCT>o>a\o>aNOONONOooNOM>ooNa\o>oCT>(^oo>o c \ u 0> -O 5 O\ S in t^O rO'O 1 ON N mc 8^ rovo (^^S moo HI *t- t~- O rovO ON W "loo O m . x W S ?^( C(OOOOOOOOOOOOCXDOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOD H ^ do 5.S ?5-S v &crs,g:S t SvS& < ^ v R.5S?vS ^ ^KS-^vS SN?^^? K Id > ^^ K S g Keg eg < cS c5 eg ^^^^^c^o^^c^^^S S g ^ K O M -4-00 " -4-t^M -^-t-xM -4-i^.d -^-t~-d -4-r-~6 pt^-6 rot^d ro^ d ^^ o OOOOOO'ONONONONONONO^ONONONONONONONON^ONO^ONO^ONONONONONONONON 2 f) -*VO 00 O N -*VO OO O N -*^O OO O N -*-vO OO O M *\O OO O N -<*-yD CO O s 55 <3 O N "*^O 00 O W -*VO OO O N -4-vO OO O d -*-vO OO O N -*vO OO O N -*>O OO O H M M M H N N (S N N ft' C* fO *J f ****** *A OO IOVO S5. N rr> -OVO t^OO O M * mvO OO ON O H * XTlvO in (N ON^O m O t^ * N ON^D ro O t^-^woovo roO t^-4-MOO iriroo t^-^-i-oo < r^ ON O N -^-vO t^* ON M CJ TJ--O OO ON M CO Tt-vO OO O w CO invO OO O W CO in t^.cO ONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONON 8 . W3 CJ ON^O POO t^-^-MOO mCJ ONf^-^M ONt^^C* ONt-N-Tf-CSl OOOvO COM ONt^lO j -j-vo OMN 10 r- m TIOO M MVO ON M ^- r- O ro >r.oo M -1- t^ ON N inoo O ro^o jj 2 c/5 ^OVO rOONin(NOO -^-M t^rOO t~s-rJ-M (^-J-MOO ION ONt^-^-H ONO fO M OO moo O ro moo -t ro^ ONM -^-t^.ONW xnr-^O (^ moo H rovo ON N -^- t^ O ro in H > ^^^^^r^^^^^^^^r^^ K O s wvo ONW mONCJ nONCM moo ct moo c* moo c* moo H moo w moo M -**-oo M ON ON ONOOOwMCN]piNr OO O ! H t-t M H HI N N N (N N rororororo^--^-r)-Tj--*-vO 00 O N - O.O>ONO o 8 o o o o o o o^^o'o'o'o'o'o'o 1 o^^o ^ ^o o" ! o * - M c/5 mo^^^o^gNOOg^OO^^ o^^^^ S: & %%% SlSS^SSSooooooooo'o'oooo Q O 5 iffi!inni?ii s s iiiiffnfiifiifi 55 O N ^"O CO O N ^t-vO OO O N *^-vO OO O N ^-'O OO O N ^*O OO O N ***-*> OO O s 5 O N ^O OO O N -^-VO OO O N Tf-\O OO O C ^\O OO O N -*-\O OO O W ^t"O OO O * 5oooooSSooooo 5 o?o > ?o 1 oS'S"3'S'oooo 1 o^ v 8 v S U CO MvO N moo M -0- r^ o 1- t^ rovo ON d moo M -o- r-^ O -* t^ O rove. ON . roo ON N moo M ^t- r-, O rovo O> PI moo t^ o rovo" S OS U oooooocooo^oo^oocoooo...^.......^........ Q O rovo oo M * r^ o w moo M * t^ o N moo M * t^ ON N moo M -*-\o ON w m O inio iii f si fttjitiifftti SB .^M4,, M M Mi W iM CHORDS, VERSED SINES, EXTERNAL SECANTS. 281 z' S ~^^^~^S^ Z a ^- t^ M x*-OO M xnoo CM in ON CM NO ON CONO O m t~. M -*OO H moo CM NO O fONO O m* x M z c/5 ro^o ^^0 ^^M ^^H ^-oo M ^-oo M moo M xnoo CM moo CM m ON CM OS Id sssallaaslsssG^sSJHsiSsSsssJJJS Q 1 ^oo H ^t^.w -*t-t-^O ^t^o mr^O mt^-O fiNO O fONO ON rONO ON CM NO ON CM CJ Z 3 N ^NO OO O CM ^NOCO'O CM -*VO 00 O N -*NO OO O CM -*NO OO O CM -*NO OO O i z ! ^5 Z -J??E-?8g3-S'S'8^S> S%% S5 5- ? 5^-5- & 5 m^ N v^NO 1 y C/3 CMvoosCMNOONCMvOONCMNOONCMNOONWNOON^NOaNcnNOOM^Orr.^O^ ... X W j N z' M Id 2 2 8 IIS 2 2 2 2 2 2 2 2 2 2\ M S S SS SS SS SS SS 3S OS O S 8, Si 85^ 8"^ S^ci ff$S,&&i < is>K < v5v!?S ^K^cSocT CJ CNdMC^WtNWWNWWWNWNNCiWWWNNWONWWNCiNW z S O W -^-vO 00 N ^^ OO O N ^'O 00 N ^"O 00 O N "*-vO OO O N -*t"\O OO O 282 CHORDS, VERSED SINES, EXTERNAL SECANTS. 55 ^ H ^^iSIIlllI^llllllIllIOIlHI u C/3 .. H N 2 ti xovo vo vo h- rxoo c ojc>a;o^OH N MM 4-4-4- .o.n^o . j K O X CJ 6 ro\O O roo o I^JVD c>cio ON mow >oco N iooD M inoo M ^*-co 4- r^ '. . s R1 ^**^' W ****^*** | W 55 j " fc 00 O^ M ro -*-MD 00 O M i-O 10 t^OCJ O f< -4- in t^. O M W -j-vp CO O M ro in^O CO 6 ' H w N S S PI N rJ c?r?p?c?r?p?crN"N"M"M'cr(N 1 c? M**M 8 jj JJ <" "" tj C/) cococo^^^O^p^MNNNrn^ro^^.o-n jjjvo vo vo p. t^oo co co Oj . o . W ! IN 2 0! i*?^lf ^l^f^^l^lllllllliljll^ll i ^ Q O X u 55 o w ^\o oo o w ^^o oo o w -s|iii W W ^ K i^l^f ljj&^j^jjJRRSSSRi^^^ > Q s O 55 s ,.wiirmo>S>oxSSo^o^o^o! > a o a U i 5 i ^^0 Of N ^^O 00 N -^-\O 09 N -"t-O 00 N -*^O OO O f) ^O OO O s 5 O N >*-vO 00 O N -*^O 00 O N -*^O 00 O N -*vO OO O M -*vO OO M -<*-^O OO O * ^^^^^^^^.^^^^^K^^^^-S^^^^^^^^^ogcS C/3 R tC'R^ ^ [^^ SScicgoooowoooo o? o? oo"oo oo w 1 S S cS-^oo w . w Ct c 00'*OONvOO'*-^ f .^O.^nO*OOM00*OONvOO^ ft! H ---Jrjr^^^ir^ir^^^^^^^^^^^^^ c s c i2 < s c s < s i 2 ( s 3 o s H ro in t^-co ." 4-0 t^. <> M ro invo coOC<4-in^.<>Hro H % $ Si S^ Si % Jn\o ovOvSovooooovSovooovooooooo o"o" u C/2 O o t^ t^co coc^oOO.'-iM.foro-^-'^-in mo O t^ c^co co co o\ &> O O . W .... H W 2 as u ..N.N...N..NN. ................. K O M C^ N ww H moo' M' 4- .- M 4- t^ 6 M r^ o ^^ c^ r^O <^ M O <^ M moo X CJ &&&&cornrorororoc^ro<^roco<^e^c^c^rorofOro<^ 2 O N ^"VO CO N ^->O CO O W '^'O CO O N M-^O OO O W ^vO OO O N ^*O OO O 2 i N "*vO OO O N ^t-vO 00 O N ^^O OO O W "*"O OO O W ^-O CO O N ^-vO OO 2 8Hs51;s$KS S HH^'S S; S ^H^'SS'S y C/5 M0momo*o^oom^. ^MvOHioOtnOTt-ON^av^ooMooro 8S888.88888g8888^^ S!:;:55 ^ S553 w 2 as m lOvO ^O t> t* h-OO CO OOOO O * M N Ol rOrOro-^''^"iO r>^ vO i^- ^ t^OO OOOO^OC>C?*ONOOC>OOOOOOOOOOOOOOOOOOOO Q o O . 2 * 280 CHORDS, VERSED SINES, EXTERNAL SECANTS. 55' S < ?jlI^ils.g'gssis.zil6l8iIiliHill y to n W SB C/5 vO vO vO t^. r^CO OOOOOOMMNWrOrO't-'^-iO ir>\O -O t^. r^.CO CXD OO O O O U Q pj O (J .ifiiilittliiiiiiiiiiiliiliiiili z s -0 5 O N ^-VO 00 O N ^-VO 00 O N ^-vO OO O N ^VO OO O N ^\O CO O N -^t-VO CO O * - z, H ^H-vll^^ls 1 si 1 J-HH ITS 1 s S 1 HJ^I^ y 0! lH9?tM^lt^iilll}m % 2 w aj ?!S?! f ?P??rcT^c? v ^?r^ c s c s t s^^^^^^^^s^?;^^^^,; 1 Q 1 i u ^iis^fiiiiiiiiraiiisfliii^ X O 04 -+\C 00 O N ^^C 00 O N ^t-vO CO O N ^VO CO O W ^-vO CO O N Tt-vO CO O H M M H M o* w OJ c* o* rororofOro^-^-'^-^f^-iommio tovo CHORDS, VERSED SINES, EXTERNAL SECANTS. 287 g ******;*** 55 VQ.OO ro in^> oo* O N *-vO r"~ O ro 10 t^-co O w ^^O oo o M H oococooooooooooooooococooooooooocooocooooooooooooococococooooo !Q CO X oojx>^ o o^o ^o ^ ^ ^ o?o?oo"co'( 1 oo > oo co"oo"oo oo cooo ooooooSoo CO 55 a , ....... .... ....... a* o ^- ? Tn ^^vS vo S P. * < oo"cS- & S^ 8 ?" o 2 "2 ? ?! K 1 *S ?J> ^?o 5. (J rocorocOfOrororoMrorocococofOfOrOfn^romMfororocofOfOfOroro 2 O N *^O OO O N -*VO OO O N "t-vO 00 O M 'J'^O OO O M -*vO OO O N -^-vO OO O i s O N ^VO 00 O N "-vO 00 O N -*\O 00 O N M-O OO O N -*\O OO O IN -*0.nHxM<> 1 nMlxMOirW><*-OvOW>*OvO a 2 2 !? ^ 2" ^ J?^ ^ ^S, c 2?2 N Mp o K O ^ ^oo ^cx^^ *^ M jj- ^ ^g g^vo g g g, c. u.00 M g- g. U ftm^w% rt m n %. nM ^wK. M K^^ftft^ 2 O C( "*"vO 00 O C* ^**vO OO O N '^'O OO O N ^vO OO O N -^vO CO O N -^-\O OO O H M M M M e* w o w w focorofOco'^-'^'^-'^^iooioin 100 5 ^00 ..00 . S oo oo oo oo oo oo'o^'oo'oo'cS'co oo oo oo oo oo ao ao moo ao v&&o>o>ovovc*oc> o X W >t- mvo vo ^ t^oo oooOQ*- (l -'WWcOTt-^t-in nvo (^ t^oo oo O o^ O n ^ w 03 PS U U Tt- TJ- -^- 1^1 to mvo vo vO *O r^. rs, ix,oo ooooocTtONOOOt-'i-MC O 0) ro -^vO t^OO O w N ^ r> O K u oo oo c!?oo > o?oo"oo h oo h ( 1 a?aD ) oS o o? oo oo"oo"o^"oo oo oo oo o^oo^ o o o c> o^ o> o> 2 i O N -*-O OO O N Tj-^O OO O -*^O 00 N -*vO OO O N -*v> OO O N -*^O OO O M H tH M M CM O CM CJ CM fOCOfOCOfO 1 ^- 1 ^' 1 ^- < M A^t^^M M 4-v^oo o 4-vo t^c>M M^r^(> H o\oc>ooo>ooooSlo t^OO O H f> - H S55555555555S3S3555555SSS55555S a . W3 roo ooo>O O *-" N P* ro ^i- ^ o\o o i^co cooOO'-'NMro'^'^ mvo ^ i^ * 5 os M Q ^ VO ON ^*-U3 r-00 ON -* v> t^CO O PI -J- A I-.OO ON O a U M M M pi PI Pi co ro ro -<4- ~-t- -r to in invo vO vo ^O t^ r-~ i^co coco ONO^OO O O OOOOOOOOOOOOOOOOOOOOOOOOOOOOM z O CN! -OP,OOVO OO* CO* 00* 00* < 3 niiu&&m^^tittMt4ifi^ 1 '2 irivo ^O t^. t^.OO OO OO O t- 1 >H N rorO-**--^-r) mvO t^- t^OO OOOOO'-i-ww M 1 ri OS o s ||ll|illi|l|s|l|||llil|iii| 8 s s ; u x O OJ ^st-'sO 00 O W -^vO OO O N -^-vO OO N ^*-vO OO W ^^O OO N -*i-O CO O CHORDS, VERSED SINES, EXTERNAL SECANTS. 291 55* s z SS?lNclc!wflc!ciNN?l?lflpJpiOCO>O M (M N ro-*m mvo r^ t^oo O O * o r^vO O n irj o N moo M <* t^ O r<*i t** Q u z _ 2S? ,, 8 ^,,^^^,,^ 292 CHORDS, VERSED SINES, EXTERNAL SECANTS. s N -*vO co o PI ^*-^o at o N -"J-^o co o PI -<*-vo co o PI -*vo oo o PI i-vo co g i M H M M pi pi pi pi PI rOr^rornrO^--<-Tj-T}--4-u-)LOU-)U-> iO\O ! i . ggggs^^r^ONi-' r^orxOM ^> J C/3 &|||||||^^5iIJlJ|v|^^|||^||||| W z K r^* t^.co o^O>O'- < wwro i ^-'^' io^o *O t^ t^.oo oo>O M M ^ mrn^-OCO ^VOCO VO CO ^ CHORDS, VERSED SINES, EXTERNAL SECANTS. 293 s z O N -*vO 00 O P) -^-O OO O PI -*VO 00 O N -*VO OO O M -*-^O OO O P) "*-\O 00 O z H iliiiiiiiiiiiliilllfilllliliiii y C/3 zzzizzzzzzzzzzzxzxzxzzxzzszxzzx Z c/5 M w OIOWGO mwoo IOM t^roO r^roO t>.mO I^^-MOO u")N ON^O ro t>> ** CHORD. O w moo M -^oo "- -^- r^. O roo O N u~ioo >-> -^- t^. O fOO O N u^ O N ioco M z O W ^-vO CO O N "^"O OO O N ^^O OO O N -^-vO CO N ^vO OO O N -^vO OO - z O N ^vO 00 O N ^-^O CO O (N 4~\O CO O N ^^O OO O W -*-\O CO W ^-"O CO O z H & 3; % o 5. 5^5- ? S, S S E? RvS vo'^^vg s ^ R R ^ S<5 Z NNNNN P ) C, N P *** * ^ ^ '"^ "^""""^ " RARa& *^ z' C/3 U iA\O -O t^OO OOOO O WWW CO-^TT \r>\o \O l^ t^oo OOO - - W <-oro^-u-> CHORD. Hfllfllllllilllllllliltiilllll z* W **"O 00 N "*t\O 00 N ^^O OO N ^"O QO O N -^-vO OO O N ^vO OO O 294 CHORDS, VERSED SINES, EXTERNAL SECANTS. z <5 M m ^ m . + + + ^. + ^ ^ vOvO^OvO H 1 1 II f II 1 1 1 ^ ^ll'l^l^ff ^ ^^ 1^111" J CO M iMIIilllllililMllllIliM^lil z' M ^s^sllalllllllslgissllilsiil 3 o X CJ z* N -*vO 00 O f) -*^O 00 N -<*-vO 00 O N -*^O 00 O N -*-^O OO f) -*^O OO S z a ^co o . ^ M o . ^ o . o . ^, o . ^co^ z in 10 in 10 \TI\O 'OvOvOvO O. t^. t>. l^. r-^co OOOOOOOO O^OOOOO O O O O M w CO X N CO CO ^ mvO t^.OO OOOOt-'NtNro^t- irvO t^ I->.CO O -* N ro^?o 5- " *^* 2> E?> ^^^vS 2* IjSyi u u tn X W in ro PI M - 1 M oo oo <> O S?S>$3- IOGO M -^ r^ 6 m\o c> w mco M ^- r^ 6 m\o <> g gico M ^ IN. O en X ??$? U I O PI -4-*O OO O d ^vO OO O W ^vQ OO O W ^^O OO O W ^-vO OO O W *^-vO OO O w ^ M N N ON rofocOfOco^-^^--i-^-toioion o^ 1 2 O M -4-vO OO O S -^"O OO O W *-4-vO OO O W "J-vO OO O N -4-vO OO O C* "^-"O OO O w M w w N w cococomco^ 4--j-^^inioiou^ invo 3 K.R^'R oi^oTSS 8>^3;^^8 s rs< 2 s j^s^ s ct^ ^?>s 1 C, N N . C, C, . C, N j c/> r X m to m u- 8 * > ro ro% o rIIl|irj|liilliillisllsls? ! Si O s ^^^J ^oTrl Js l-SHs si o^d,? ,Sl:3,s!&|$|||i CJ 2 O M *<3 OO O M ->-vO OO O P ^->O OO O P - -4-\O OO O A 296 CHORDS, VERSED SINES, EXTERNAL SECANTS. 2 O N -.oo o\ O M W ro ^*- ovD r^oo O\ O O OOOOOOOOOOOOOOOOOO OOONOOvO^ONOOO \OvOvOvO\OvOvOvOvOvOvOvOvOvOvOvOvOvOvO t~t>. C c/5 000 u.roooo ^roOOO > IIHH2111 ^^^^^^^ Q H O ^ 2^ ?! S"S i 54 9 R S^vS J\ g. R JC^cS oo'dS- i Si g; 8 U io>oioioiomio>oioio 10101010101010OIOIO001010101010^1IOIOIO 1 , o S yo. g 3 r8-~-:ia*rt g s rg's RRjjr^^^^aasir&'s^ i^*fft- u . o> X M to^O ^ t^.00 O O M W ro \OvOvOvOvOvOvOvOvOvO IO IO 1O , j 10 55 OS Id r^co oo oo oo oo oo oo^oo^oo Q 1 K a ff^^S.S^^^iJS, IO 1O O\O VO VO IN. IN. r%.CO OOOOOO OOOO O O M W U 55 O *^O 00 O N "*VO OO SW H-O 00 O W ^*-VO 00 O N ^-\O OO O N ^VO 00 O w W W w rocorororo^t-^^ir^f^ioioioio io\o CHORDS, VERSED SINES, EXTERNAL SECANTS. 297 55 O M *-VO OO O O O 'O^OOvO'O^OvO^O tx. vOvOvOvO^OvOvO^O^O Q ^VOVO^-*-^- N 00 1 g^g% 2 O N "^"O OO O W -^-VO OO O N 'i-vO CO O N ^f-vO OO O W MMMiHMC40iNCrx,(>M m U-100 8S . w 2 M ^J- u->\o \O l^CO OOO - W roro^ iO\O vO "r^-OO O ON O M N N rr> ^- ID icvO t^ Q 9 X o S o nc M ^S ff| 5 s ffS 5 ??? J^ffff?^^ 1 OO O ro\O O O i/^OO M o r>^ t^. r> i-^oo oo oo o 55 O C ^VO 00 N ^-0 00 N ^-VO 00 0^ N ^-vO CO O W ^^ ^ 10 lO lO^^vS 298 CHORDS, VERSED SINES, EXTERNAL SECANTS. z * 8 < g&as*'9.* < assaa IIIiai!!ll-*slit!lllllltflillll u C/3 W 10 * i 8 S S.S ^^S | g. S ^$& ^ ( ?c c $ y y^ 8 o o ^ " M c.coco^cncococ.c.coc.coc.cococ.coc.^coco.ocococo^c.^coc.^ u w 0000 CD o vS vo ^"vo % *o vg* 8. ^. R. R S R^ K'S. So? cS cooS'oo'S c-g o? & 55' as H O <*O V O CO M -^- t^ O fO'O O Sf, ^^vS z' H u M fO -*t- i/lvQ ^ O O w N fO -^ tn\D t--CQ O^ "- 1 N f) ** ^>vO x/^ m m in in mvo ^OvOvO^O^ovOvOvO'O t^t^t^r^.t^t^. oooococooooooooooooooooooooooooooooocooooooo 00 00*00 CO OO 00 00 00 CO z t III 5. 5 ? ^^. <3. ? & R R jf, 5 5; -^i Js% gjj 3 vSWS !8 CHORD. 513H! 5 ! si JHI 1 ! i 1 ! f i^S* 8.StSS.s!R8.SS z' -*-O CO O N - ocroTc?ci? <*-^O t^oo o O w M ro M-il-M-Tf^-iniriinio z 1 tglisagjj^| < i j l||||||ll ipj^siiiii CHORD. 000 fxvO >0 * m N N M 000 t^vo >0 ^- r^ N H O 000 00 1-VO IA TJ- M N w u-> in\o vo o t^. t^ t^ t^oo cooooNOOoooO'-i'-ii-) O n Tioo H * r^ O ro ********* z -:^!m^wtt.MM *VO 00 O N -*^O OO O * * -^- u-> 10 in in irj^O 300 CHORDS, VERSED SINES, EXTERNAL SECANTS. g O C* -^"O 00 O N -J-vO OO O N -^"O OO O N ^^% 5. ^. ?^.5- g) ^ ^"g,^^ * .< iO t** O i- 1 ^'O OO O f> u" r*s. O . t^.oo oooooo OOOOOO O O O M ^^oodpI^t^OM^^o^dN rorororocororocororomoocororororo i M^iOMOOOOHf.ro.o.COOOHro^vO r-OHO^-.o^ooONro.oOOO X* &&&&&&&&&8888538S zzzzzzzzzssszz o z - ro -^- tovo t>.oo oooQi-twrofO^t- iovO H' ' ' ' 6 M N m 4- .0 J^M3 r^oi U] S-S S ^ K ^ S ^ Soo oo oo oo oo oo oo cS oooooooooooooooooooo o ac U M M cs M (N ro ro rn r*^ <* -^- -^t- m m in irvO oooooooooooooooooooooooooooooooooo U^iOiniOiOlOiOiOiAiOU^lOlDlOiOiOiO 0000000000000000000000 00*00* O mu-)ioininir>iDu-)u-iiomiOioio z O W "^"O CO O N -^t-vO OO O M ^t-'O OO O N ^^^ 5- $. 5^.^. g> Jo S^ 1 ^^ S z OC.*0 PJ^OOOPJ^-OOOO N M^'m 5- SJ. ^"^-^ Si S> ^i^'SjvS z H fornroroc^coro^c^focorororororofo S5S33SSS^SSS u U) x W ScoSc^^^So^^^&^Soooo" oooooooooooooooooooooooooo aooch O z r^OO O O "- 1 M M <"*"> "^ "^^O vD t^.CO O O i- 1 oooooooooooooooooooo oooo H ( ^- r^ N 000 t-^0 -* CO N H OS O CJ Illllilllslllflil t~-00 OOOOOOONOOOOi-"-- * O N ^-O CO O N M-VO OO O N ***-vO OO O N ***>* CHORDS, VERSED SINES, EXTERNAL SECANTS. 301 s ss O N TT\O CO O PJ -*\O OO O N -*-\O 00 O PJ 'J-VO 00 O N -*^O OO O W S'&'S.vB 3 rO\O O P) toco H *t- i> O rovo O PJ 1000 w ^- o* fOvO MSO W moo H Th M ro moo o PI 10 c^ o P* -^-vo oo M m 1000 O cj ii Hill i romoo t^ t^. rxco jji Cfl M dMNr^mvO^ooOMN^^ioAcoodcj' ro -^ iovc OO O O >- $t$i oooolooCTlooS 8 8 8 8 8 8 8 5 S 00000000 c/5 w * rr> N M ooo rwo * ro N M ooo txvO <* ro PJ M 000 fv\0 ro P M OOO !>. OOCOCOOOOOOOOOOOOOOOOOOOCQOOOOOOOOOOCO Ov>*OvOvO^O^OvO 3OOOOOOOCOOOOOOO OO OO CO OO CHORD. oo>oinooo o O o^o oooooooooooooooo > o > o > o > m 10 10 m irjvC vOvOvO'O ( OvOUDvOvO^O v OUD v O OvOvSvOvS^OvOvO **- t^ O PJ g O N -*\O 00 O CJ -4-^0 00 O N -*'O OO O PJ *vO x> o PJ ^-vo oo o PJ **> OO O w 2 O N ^"^O 00 O CJ ^vO 00 O PJ 't'VO OO O CJ -^-\O 30 O P) -*-^O 00 O CJ fOTj-^-rt-'^-'^-lolO V *: CJ 10 t^ o CJ -4-^0 oo M roint^O PJ -^-t^Oi-i ro ooo O roiot^OP* o'g ?S in 10 in 10 rOMrorororororofororocororofornrornro """""" C/2 M ^^O t>-OO O O M ro <} irj'O I^-OO O M N ro ^vO c^oo o O w n -<* 10 ^o t^* o o oo oo oo o^ 1 U OOOO "-" W f*"> ^f- irivo tx t>*cO O O -* N f^ ro oooooooooooooooooooooooooooooooocooooo <- 10^0 r^oo ooo oc?oo oo'c 1 oo > oToT CHORD. O O O >- M M M (N N w rorororr>-!t--!*-- i * t^ t^ u^kX)iou^"^u^u^u^inir>ioiOu^u^u^u^iou^tou^iOiO s O N ^-O 00 O N M-^O CO O W "*"vO OO O N ^-^O OO O N ^VO CO O O +&00 1010 IOVO 302 CHORDS, VERSED SINES, EXTERNAL SECANTS. 2" O C4 -^-vO OO O N ^fvO CO O M ^'O OO O N ""t-vO OO O N -^VO OO O N ^-vO OO O 55 ^-OOH^OOO.^ONNvOON01vOONrONOO*rxM^OOO.^ONrONOOrO^H O 0) IT) t^* O 0) -vt-vO O\ "-" flvO OO O ro lOOO M lO t*^. O^ OO>OOO O O Qt-< t-t M iHC4 u' X f^^?l&|f^.gM^4^?fl'S b ff?H'Sis W 2' * AoOONOO-N^^^vo't^O^ONONOMNn^^^^oo'oo'oNOHN^^- u cooooo^^a,^o,o,o.^<.ON.o,o,o,ONONO,ON ,a,c>o>ONONO,SNS,S, ri o t^, ro ^100 w 4- t^. ON oi inco M rovO ON o! 1000 rONO CJN oi 4- t^, 6 rove' oo M a $ 2 O N TfvQ CO O N ^-vO CO O N ^vO CO O W ^\O CO O N ^^D CO O N ^VO OO O w M M n M oi w W w M rocrjroforO'*-'^-'^-'*-' i o > o > ??o"o"o"o"o"o"o"o'(?o > o 1 c?o^o^ 1 c?o i o^o v o v o f * X 2* s O n N m ^ >n 100 r^-oo ON O M w rn m -^- 10^0 r^.oo o O -" ^ w ro -^- mvo IN E o N .noo' M 4- A ON M .nco g rovg ON ol ,noo* O r^vo ON ol 4- A o ^NO" ON M 4- t^ U NSNSNONONONO.NONONONONONONON^NONONONONONO....^.^ 2' 2 O 04 -l-vO OO O N ^f-vO CO O 01 -*hvO CO O 0) ^vO OO O OJ -*$-\O OO O OJ -^*-vO OO O MMHw> OO O N -^vO CO O N - H sssssftRSRSSSSSSftssssssssssRSRRS (J H in 8 S 1 ^ S"8 8 S 2 ?^ ^?8 5 S 1 ^^^ ^MM^M^P;%5.i & 1 z 1 4-mv0^r^oo'(>dNM4-mvo^oooods6NM^-m^^ooON6vc5M id 0,0,^^^^0,0,0,0,0,0,^0,0,0,0,0,^0,0,0,0,0,0,0,0,0,^0,0, E o \ 5 O N -^"O OO O OJ -^VO OO O W -^-^O CO O N -^-vO OO O N '^vO CO O W *^-vD OO O rorococorr-c^^-^Tj-rnuiinin io\o j 304 CHORDS, VERSED SINES, EXTERNAL SECANTS. & -1 O W ^*-vO OO O N <*-vO OO O N ^**vO OO O W ^\O CO O N M-vO CO O N '*-'O OO O S a u C/5 CM CN Ci 04 N M fO TO CO CO rO TO TO ^ Tj- ^ -^- ^ Tf Tj- IO 1O 1O 1O >O \T) lOMD VO VO *O d w . c * OT 8oooooooooooooooooooooooo 1 o'o > o ) v?o > o 1 Q* VO 1 O 55* ^vOOO ^vooo o ^co C, ^vooo . ^VOOO , ^-vooo S i O N -*VO 00 O N -*vD 00 O N -*VD 00 O N -*VO 00 O N **-vO OO O N -*vO OO O fc 4-t^O M 4-VD 00 M rovo OO M mv^ OO M ro l>10O 6 <^ <^ H 11 wfitfiat . .< n^ *Pla s sllllaSIa u Id M ro -*J- 10 t^.CO O M d ro tO\O CO O O N ro -^-vO tx. o\ O H ro ^f in t^OO O -< N 00 W g a- U Q o 00 H rnvo OMN -4- t^ 6 ro 1000 M 4-vo Ov N inoo" 6 roo ov M 4- t^ o IN .000 M X o W^4|M53;|3WJ3Si;^*l s O W "Tj-vO CO O N "^vO CO x > N ^O CO O W ^J-vO CO O N -^-vD CO W ^vO CO O CHORDS, VERSED SINES, EXTERNAL SECANTS. 3Q5 z O N ^^O 00 O (M -*\O 00 O N -*VO 00 O W ^vO OO O CJ *"& OO O N "^-VO OO O , H , ^.co ,,00 coco ^0,^0.^0 ^0^ M^C, r.cnoo ^o^-o H y CO l^HSSalliSHSilSSS'SSSiHti'iiSS vH W *~ ss CO ci ooooooooooooooooooooooooooooooo H . o s^vff^s tiKiScSo^^ii'^g s 1 ^ ?s js-?a ff's &si u ^^^^^^^^^^.^^^^^.^^^^^^^^^^^,0^^ 1 O W ThvO 00 O N -*vO 00 O N -*^O 00 O -*vO OO O (N -*\O CO O C -*VO 00 O ss- O N -*VO 00 O N -ihvOOO O N -J-'O 00 O N <>> OO O N -^VO OO O N -*MD OO O S iiflfiiiliiiiifiliiffitllillttt A to x t^lilli^frfiffiililllllliljHI b* W *" z CO ?^?lji?jf^H'fi^S > ???f ^^i'l'fi'i' > Q .0 AM 3 35 O N ^"O OO O N - *H *! t^. O w moo O ro\o OO M TJ-\O O> 01 ^ i-% O CJ moo O ro^D CO HI Tj-vo 56 o -^-vo co o N ^-vo co o oj ^MD co o Tt-\o co o w ^vo co o (N -d-vo co o 2 z . ..00 , ^00 . ^000 ^ H VO W t^ fOOO ^ O- "^ -* \O N t^ COOO * O ^> H t^% COCO Tt- O ^^ *-* ^ fOCO . O J ' oorooo-l-O^OinOvOH * a s^ g'&^^s ^ g. R R^-g as g ^; CN * t^ O -ON mt^O fo IACO M ro\O jl W -^-vO CO O N ^vO CO O N "'^'O OO O N -^-VO OO O 04 -^-VD 00 O N ^O OO O 308 CHORDS, VERSED SINES, EXTERNAL SECANTS. 2 2 ****' 10ini< "'" t^x O fOvO OO M -rj-vO ON vo tx. O roO CO H Tf t>, ON O lOOO O f*")\O O M ^- t-^ O ir^vo vO ^O O t^ (^ t^ i^oo oooo OOOO>O O O O M MMW g w rn ?i fo ^- s sis! J 1 vi'vS s8 1 J 1 sSvSvS^ J 1 li J'ii| i Ji > | ! JJ- 5 . w t- 2' tt If Hf&SI'S^^NWWW'Sl'fl'S'tSfFI' M Q OO PICO -<*-OO PIOO -^-OO MOO ^-OO NOO -^OvO WOO ^i-OO WOO ^O X ^O ONM -1-t^ONPJ ^J-t^O PI lOt^O ro lOOO O mo G/3 M COO Os M -^-O ON PI rororo- t^oo oooo OsONONOsO O O O IH w M M PI pi PI ron-)rom-*---^--4-ioioio u w c/) 5- * * * * rt- JO To 10 vo 10 10O OO OOO t^Kl^t^t^ Soo 0?Oo"oO 00 OS ON . W * 2' ^.,. ^cococo ONOSOSO o o H H H ... rrt ? ^ u o X 1000 mo oo M mo ON H -\o CHORDS, VERSED SINES, EXTERNAL SECANTS. 309 ON-fvOOOON^OOON^-vOWON vO < . ^ W O N * CO ^^^^^'^'^'^"'^'^"^"'^" T 1"'*t"'^"*"'^"'^"'^"' < *" T *"'^"'^" > * "* "* ^ ^ "* "* "" >O N r^ 1/1 tx, ON >-* N "^" ON O O Q * N^t-xON-H-tx.nninhxOro moo O ro moo H roj^ co w ^*-\c o M ^O O S JMM c! cTw^rnroS-^-'? ^^- lo S lOlSnvO vS \O S ?> ^ t-, KB OO CO OO r K ^^^^^^^^^^^^^^^t^^^^^^^^^^^^'^^tl^ O N ^"^O OO O W ^VO CO O OO O 1000 O ro 1000 O ro moo ro 'oco O ro moo U 3- * " V s Z OO O rovO ON W IOOO O ro\O ON N n t^. O rovO ON N UIOO O roo ON O moo M ro O -t t-t M M c^ N N rorororO't--^--^-iriu-)in ioO -O "O t^ t->. t^ t^oo CO CO ON ON Id cn W 00 x - tt> O H W TO -^-vO !>-CO ON O M &SUS'fc& < 3,&8vo2v2\J?* CHORD. ^smis^^s K^t^tvt-^fxt^t^t^t-* DO M rovo oo O co moo O ro moo O ro m X) O m irico 1/1VO vO VO vO O VO vD VO VO I W -4-vOCO O N -hvOOO 8d -^-\O OO O N -^-vO CO O W *<*"O OO O 04 04 04 04 COCOCOfOCO^-^^-^-^m 0*1 00 X O 04 ^vO OO O 04 ^J-vO OO O N M-vOOO O N -*VO CO O M H M M M 7. M rovo o\ N moo M' ^- ix. OOOOOOOO ONONOO O O OOOO^OOOO O O O fOvO O> 04 moo M rovo ON 04 moo M ** ,&&l,?,ISSaSSISSSS t^. O royD O cT'o o^o o 10 in m 10 >o $ m X w ^f.HwOOOCO^t^vOmm^^^C.MMOOOOOOOOOOO^^^vo ilSiUfffl 1 00 00 00 00 00 oo'oo'oo'oo'oo' ON 0\ 0\ & ON O^ M - Q O iJss&ssM'ji.'SE: JiMi^i.^si'ij w% - - SB O N ^-\O 00 O N -^-VO OO O M **> 00 O N -*\O OO O Pi -. t>. t^OO OOOOOO ONONO*O O O ^ w *-" N N \O t^CO Os O M ro ^ irjvQ t-^OO O O -* <"*") * m\O t^OO O O W r*i tf 10 5 Q mco N vo vo o JIli^l^^SSN s ^ s """ 8 8 ^^ """ . & 00 55* K I oo O r^i tnoo O romt^O N int^Oxw ^^ONM -rt-^o O^ -i r*i\o oo M rovo oo O a oo oo'oo'oo'oo' ^^^^^g^P^^ggg^gg^P^g; ^ ^ ^ ^^^^ ^ O 5 O Pi ^-vO OO O N ^vO OO O N -^O OO O N ^VO OO O N -^-vO OO O W -^-vO OO O M M M HI M o s ^S5SSSSSS- 1 S*S-SS3:*S**JS5S S C. >M CJ t^OO <>H ^ 0,0 , t ^0 ^O.H ?? d s 2 000 C/3 s|||illllii^is^||s|i|illf||f Q O S o oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo . 55 O *vO 00 O *VO OO N *vO OO O N -*-vO OO O N -*vO OO O N ^VO OO ^ 314 CHORDS, VERSED SINES, EXTERNAL. SECANTS. 2 O N ^vOOO W -4-vOOO N ^-^000 O W ^J-vO OO O N ^-vO OO O N **vO OO O M M M M M o* w w w w rofOroroco-^-^t-^-Tt-^-irjir>ioio iovO H u gj C/2 X M 00 g 5 a ^OO M u-)O\ovO O -^t^H moo W vO ON ro l^ w rfoo W in ON ro\O O ^J-OO M 10 o s O d "^"VO OO O W ^J-vO OO O W ^vO OO O N -^-vO OO O N -^-\O OO O N -^vO OO O 2 O W TfvO 00 O 0* -"t-vO OO O N -^"O OO O 0* ^-vO OO O W ^h%O OO O N ^hvO OO O * 2: CO\O ON fO\O ON M vO O N lOCO C< tOOO M moO M -* O rO\O 00 C/3 in t^. O N -^-vO O> M ro moo O W ^ t^ ON w rOMD CO O ro >r, (^ o\ c* -^"O O M <^> 00 55* *? i I s j ^ vs ss vs i vg ^ s& i -sio"! f ii^ii|sj im > Q O \OOOO t^t^f^.^ t^OO OOOOOOONONONONOOOO-' w '- t ^- t MNNfOfO SB O ^NO 00 O (M -^-vO OO O N -*VOOO O (M -<-vO OO O (M -*NO OO O -*O OO O SLOPES, FOR TOPOGRAPHY. 315 TABLE XV. SLOPES, FOR TOPOGRAPHY. Degrees. Vertical Rise in 100 Feet Horizontal. Horizontal Distance to a Rise of 10 Feet. Degrees. Vertical Rise in TOO Feet Horizontal. Horizontal Distance to a Rise of 10 Feet. i i-75 572-9 19 34-43 29.0 2 3-49 286.4 20 36.40 27-5 3 5-24 190.8 21 38.40 26.0 4 6.99 143.0 22 40.40 24.7 5 8-75 114.3 23 42.45 23-5 6 10.51 95-i 24 44-52 22.4 7 12.28 81.4 25 46-63 21.4 8 14.05 71.2 26 48.77 20.5 9 15-83 63.1 2 7 50.95 19.6 10 17-63 56-7 28 53-17 18.8 ii 19.44 5i-4 29 55-43 18.0 12 21.25 47.0 3 57-73 17-3 13 23.09 43-3 35 70.02 14.2 14 2 4-93 40.1 40 83.91 11.9 IS 26.79 37-3 45 100.00 10. 16 28.67 34-9 50 119.17 8.4 17 30-57 '32-7 55 142.81 7-o 18 32.49 30-7 60 173-20 5-7 TABLE XVII. RISE PER MILE OF VARIOUS GRADES. 318 RISE PER MILE OF VARIOUS GRADES. Grade per Station. Rise per Mile. Grade c P er Station. Rise per Mile. Grade c P er Station. Rise per Mile. Grade per Station. Rise per Mile. .01 .528 .61 32.208 I. 21 63.888 1.81 95-568 .02 i .056 .62 32-736 I .22 64.416 1.82 f / 96 . 096 03 1.584 .63 33-264 1.2 3 64.944 1.83 96.624 .04 2. 112 .64 33-792 1.24 65-472 1.84 97.152 05 2.640 65 34-320 I - 2 5 66.000 1.85 97.680 .06 3.168 .66 34-848 1.26 66.528 1.86 98.208 .07 3.696 .67 35.376 1.27 67.056 1.87 98.736 ,08 4.224 .68 35'94 1.28 67.584 1.88 99.264 .09 4-752 .69 36.43 2 1.29 68.112 1.89 99.792 .10 5.280 .70 36.960 1.30 68.640 i. go 100.320 .11 5.808 7i 37.488 *$ 69.168 1.91 100.848 .12 6.336 72 38 -Ol6 1.32 69.696 1.92 101.376 13 6.86 4 73 38.544 !-33 70.224 *-93 101.904 .14 7.392 74 39.072 1-34 70.752 1.94 102.432 !5 7.920 75 39.600 i35 .71.280 i-95 102.960 .16 8.448 .76 40. 128 1.36 71.808 1.96 103.488 I? 8.976 77 40.656 J -37 72-336 1.97 104.016 .18 9-504 .78 41.184 1.38 72.864 1.98 104.544 .19 IO.O32 79 41.712 i>39 73.392 1.99 105.072 .20 10.560 .80 42.240 i. 4 o 73.920 2.OO 105.600 .21 I I. 088 .81 42.768 1.41 74.448 2.10 110.880 .22 II .6l6 .82 43-296 1.42 74.976 2.20 116.160 23 12.144 .83 43.824 1.43 75-504 2.30 121.440 .24 12.672 .84 44-352 1.44 76.032 2.40 126.720 .25 13.200 8 5 44.880 1.45 76.560 2.50 132.000 .26 13.728 .86 45.408 1.46 77.088 2.6o 137.280 A I4.2 5 6 14.784 .87 .88 45.936 46.464 1.47 1.48 77.616 78.144 2. 7 2.8o 142.560 147.840 .29 15.31-2 .89 46.992 1.49 78.672 2.90 153.120 30 15.840 .90 47.5 2 o 1.50 79.200 3-00 158.400 31 16.368 .91 48.048 1-51 79.728 3-10 163.680 32 16.896 .92 48.576 1.52 80.256 3 .20 168.960 33 17.424 93 49.104 53 80.784 3-30 174.240 34 17.952 94 49.632 J -54 81.312 3-40 179.520 35 18.480 95 50 . i 60 i>55 81.840 3-50 184.800 36 19.008 .96 50.688 1.56 82.368 3-60 190.080 37 I9.536 97 51.216 -57 82.896 3-7 195.360 .38 20.064 .98 5!-744 1.58 83.424 3-80 200.640 39 20.592 99 52.272 1-59 83.952 3-9 205 . 920 .40 21.120 1. 00 52.800 i. 60 84.480 4.00 211.200 .41 21.648 .01 53-328 1.61 85.008 4.10 216.480 .42 22. 176 .02 53-856 1.62 85-536 4.20 221 .760 43 22.704 3 54.3 8 4 1.63 86.064 4-30 227.040 44 23.232 .04 54-912 1.64 86.592 4.40 232.320 45 23.760 05 55.440 1.65 87.120 4-5 237.600 .46 24.288 .06 55.968 1.66 87.648 4.60 242.880 47 24.816 .07 56.496 1.67 88.176 4.70 248.160 .48 2 5.344 .08 57-024 1.68 88.704 4.80 253.440 49 25.872 .09 57.552 1.69 89.232 4.90 258.720 5 26.400 .10 58.080 1.70 89.760 5.00 264.000 5 1 26.928 .11 58.608 1.71 90.288 5.10 269.280 52 27.456 . 12 59- l 36 1.72 90.816 5.20 274.560 53 27.984 13 59-664 i-73 9 J -344 5-30 279.840 54 28.512 .14 60. 192 1.74 91.872 5-40 285.120 55 29.040 !5 60.720 1 . j r 92.400 5-5 290.400 56 29.568 .16 61.248 1.76 92.928 5-6o 295.680 57 30.096 !? 61.776 1.77 93-456 5-7 3OO . 960 58 30.624 .18 62.304 1.78 93-984 5-80 306.240 59 S 1 '^ 2 I.I9 62.832 1.79 94-512 5-9 311.520 .60 31.680 I .20 63.360 i. 80 95.040 6.00 316.800 INDEX. PAGE Abbreviations explained ix Acres, roods, and perches in square feet, Table VI 152 Adjustment and use of instruments 23 Angles of frogs, to find 129 index, to find 69 intersection, to find 55 plane 12 to read on verniers 43 tangential and deflection . 50 of switch-rails 130 Apex distance of curves, to find 52 Arc, functions of, to find 13 Arithmetical complement 6 Axemen, duties of 84 Azimuths of North Star, Table III 150 Barometer, levelling by 29 Bench-marks, proper intervals for 83 Bubble, to adjust on level 25 to adjust on transit 40 Chain, to lay out curves with 63 Chainman, duties of 82 Chief engineer, duties of 79 Chords, to calculate 54,58 Table XVI 269 Circle, propositions concerning 49 Circular arcs to radius of 1, Table VII 152 Complement of an angle 12 arithmetical 6 Compound curves. See Curves. Contour maps, utility of 85 Correction for curvature and refraction in levelling 28 Cosines defined 12 Crossings, plain rules for laying off 139 Cross-hairs, to adjust 24, 26, 40 eccentricity of " 24 to put in new 44 319 320 INDEX. PAGE Cross-sectioning. See Slope stakes. Cubes and cube roots of numbers, Table XI 161 Curves, circular, on railroad defined 51 to find radius, length, degree, apex distance, chord, versed sine, and external secant 52, 56 form for field notes 70 Curves, how to lay out on the ground, with the chain only 63 with transit and chain 66 hints as to field-work 82 protractor for 84 slackening grade on 87 terminal 88 Curves, simple, location of, how to proceed when the P. C. is inaccessible 93 to shift the P. C. in order to strike a fixed tangent 96 to change radius from same P. C. in order to strike a fixed tan- gent 97 to triangulate on 94 to pass through a fixed point 127, 128 Curves, compound, how to proceed when the P. C. C. is inaccessible 95 to compound a curve in order to strike a fixed tangent .... 98 to shift a P. C. C. in order to strike a fixed tangent 99 summary of rules for 101 to compound into a tangent intersecting main curve on concave side 102 to compound into a tangent intersecting main curve on convex side . . . ' 103 Curves, reversed, parallel tangents, radii equal , 115 parallel tangents, radii unequal . . . . ! 117_ angles unequal, radii equal 119 angles unequal, tangent points fixed, radii equal 120 divergent tangents, radii equal, advancing towards intersection . 123 receding from intersection 124 to shift a P. R. C. in order to strike a fixed tangent 125 Curves, miscellaneous, elevation of outer rail . 141,142 degree of, to find by calculation 52, 55 to find on ground 145, 146 to connect curves of contrary flexure by short tangents ... 89 to locate a Y from a tangent 103 from a convex curve 104 from a concave curve 106 to locate a tangent to a curve from a fixed point 108 to two curves already located 109 to substitute a curve for a tangent connecting two curves . . . 109 terminal curves 88 INDEX. 321 PAGE Curves, miscellaneous continued. trackmen's table of curves and spring of rails . 143 vertical curves, to calculate 36 to project 39 Datura in levelling 27 Decimals of an acre per 100 feet for various widths, Table V 151 Deflection angles and distances explained 50 to find 57, 64, 68 short rule for sub-deflections 68 limit in field-practice 82 Degree of curve, to calculate 52, 55 to find on ground 145, 146 Deviations from project admissible on location 81 Distances, tangential and deflection, defined 50 table of 155 of frogs from toe of switch 130, 132 tables of 135, 136 Elevation of outer rail on curves 141 table of 142 Excavation and embankment, to stake out 30 External secants, to find 54 of a 1 curve, Table XVI 269 Extreme elongations of North Star, Table II 150 Feet in decimals of a mile, Table VIII 153 Field-work, suggestions concerning 79, 85 Field-book, form of, for level 27 for transit 70 for slope stakes 33, 34, 35 Frogs and switches . . 129 rules for angles and distances . . . ., 130 table of, switch-rails straight 135 switch-rails curved 136 plain rules for locating, switch-rails straight 132 switch-rails curved 133 on narrow gauges 134 patterns for 134 Functions, trigonometrical, defined 12 logarithmic, of arcs, to find 14 General propositions in trigonometry 15 as to circles 49 Grade, to slacken on curves 87 rise per mile, Table XVII. 317 Grade lines, how to project on map 86 how to trace in field ..... 81 Heights, to find by barometer and thermometer 322 INDEX. PAGE Inches in decimals of a foot, Table IX 153 Index angles, to determine 69 Instruments, adjustment and use of 23 Intersection angles of tangents, to find 55 desirable to fix on ground 66 Level, to adjust 24 Leveller, duties of ~ . 83 Levelling, art of 26 by barometer and thermometer 29 correction for curvature and refraction 28 form for field-book 27 rules for exact work 27 rules for survey and location 28 suggestions concerning 83 Location, problems in field 94 admissible errors on ground 81 form of record for 81 projects, hints concerning 84 of terminal curves 88 of a Y 103, 104, 106 Logarithms explained 3 multiplication by 5 division by 6 of numbers, to find 4 Table XII 179 roots and powers by 7 Logarithmic sines, tangents, &c., to find 13 table of, XIII 197 Maps, contour, utility of 85 notes for ..... ^ 82, 83 not sufficient for intelligent projects 79 Meridian, to establish 44 by equal shadows 45 by North Star 45 times of passage of North Star, Table I. 149 Multiplication by logarithms 5 Natural sines, tangents, &c., defined 12 Table XIV 243,256 Needle, magnetic, to adjust 41 to re-magnetize 44 hints as to management 44 bearings should always be noted 82 North Star, to establish meridian by 45 times of meridian passage, Table 1 149 extreme elongations of, Table II 150 azimuths and natural tangents, Table III 150 INDEX. 323 PAGE Obstacles in the field to vision 71 to measurement 73 Ordinates of circular curves, to find 58, 59 of parabola, to 4ind 36 of a r curve, Table X 155 Parabola, ordinates of 36, 59 Plane trigonometry 12 Powers and roots of numbers by logarithms 7 Propositions, general, in trigonometry 15 Protractor for curves described 84 how to make 85 Rails, table of spring for trackmen 143 Radius of a curve, how to find 52, 54, 56 of a turnout curve 129 plain rule for, on curves 133 for narrow gauges 134 Radii and their logarithms, Table X 155 Records, forms for 81 Refraction and curvature, correction for 28 Reversed curves. See Curves. Rise per mile of various grades, Table XVII 317 Rod, levelling 28 how to read 42 Rodman, duties of 83 Roods and perches in decimals of an acre, Table iV 151 Roots and powers of numbers by logarithms 7 Senior assistant, duties of 80 equipment for 81 Sines defined 12 Shadows, to fix true north by 44 Slopes for topography, Table XV 315 Slopeman, duties of 84 Slope stakes, to set for earth excavation 31 for embankment 33 for hillsides and rock 35 field record of work 34 Spring of rails, table for trackmen 143 Squares, cubes, and roots of numbers, Table XI. . 161 Supplement of an angle .' 12 Survey, form for record 81 to facilitate 82 Switch-rails, angles of _ . . . 130 tables of 135,136 Tangent, or apex distance of curve, to find 52, 54 324 INDEX. PAGE Tangent of a 1 curve, Table XVI 269 to curve from a fixed point, how to locate 108 to two curves on the ground, how to locate 109 Tangential angles and distances explained 50 how to find 57,58,64 Thermometer, levelling by 29 Track problems 115 Trackmen's plain rules for finding frog distances 132, 133 tables of turnouts . 135, 136 plain rules for laying off turnouts with tape-measure and pins . 137 crossings on straight lines and on curves 139 elevation of outer rail 142 instructions how to put in missing stakes on curves with tape- measure 144 table of curves and spring of rails 143 explanation of the trackmen's tables 144 how to find the degree of a curve 145, 146 Transit, adjustment of 40 cross-hairs 24 Transitman, duties of 82 Triangles, solution of, two angles and a side given 16 two sides and an angle given 17 three sides given 18 Triangles, right-angled, solution of 19 Trigonometry, plane 12 general propositions 15 Turnouts. See Trackmen. Vernier explained 42 on transit 43 Versed sines defined 12 to calculate 54, 58 of a 1' curve, Table XVI . 269 Vertical curves, to calculate 36 to project 39 TABLES : Ordinates of a 1 curve 60 For locating terminal curves 88 Tangents between curves of contrary flexure 89 Turnouts, switch-rails straight 135 switch-rails curved 136 Elevation of outer rail on curves 142 Curves and spring of rails 143 I. Time of meridian passage of North Star 149 II. Time of extreme elongations of North Star ....... 150 INDEX. 325 PAGE HE. Azimuths of North Star, and their natural tangents .... 150 IV. Roods and perches in decimal parts of an acre 151 V. Decimals of an acre in one chain length of 100 feet, and of various widths 151 VI. Acres, roods, and perches in square feet 152 VII. Circular arcs to radius of 1 152 VIII. Feet in decimals of a mile 153 IX. Inches reduced to decimal parts of a foot 153 X. Radii and their logarithms, middle ordinates, and deflection distances 155 XI. Squares, cubes, roots, and reciprocals of numbers, from 1 to 1,042 161 XII. Logarithms of numbers from 1 to 10,000 179 XIII. Logarithmic sines, cosines, tangents, and cotangents .... 197 XIV. Natural sines and cosines 243 Natural tangents and cotangents 256 XVI. Chords, versed sines, external secants, and tangents of a 1 curve 269 XV. Slopes for topography 315 XVII. Rise per mile of various grades 817 SCIENTIFIC BOOKS, PUBLISHED BY D. VAN NOSTRAND, 23 MURRAY STREET, AND 27 WARREN STREET, WIESBACH (JULIUS). A Manual of Theoretical Mechanics. Translated from the fourth augmented and improved German edition, with an Introduction to the Calculus, by ECKLEY B. COXE, A.M., Mining Engineer. 1,100 pages and 902 woodcut illustrations. 8vo, cloth .... $10 00 FRANCIS (J. B.). Lowell Hydraulic Experiments. Being a selection from Experi- ments on Hydraulic Motors, on the Flow of Water over Weirs, in Open Canals of Uniform Rectangular Section, and through submerged Orifices and diverging Tubes. Second edition, revised and enlarged, with twenty-three copperplate engravings. 4to, cloth 15 00 ROEBLING (J. A.). Long and Short Span Railway-Bridges. By JOHN A. ROEBLING, C.E. Illustrated.' Imperial folio, cloth 25 00 CLARKE (T. CJ. Description of the Iron Railway Bridge over the Mississippi River at Quincy, 111. Twenty-one lithographed plans. 4to, cloth . . 7 50 TUNNER (P.). A Treatise on Roll-Turning for the Manufacture of Iron. By PETER TUNNER. Translated and adapted by JOHN B. PEARSE, of the Pennsylvania Steel Works. With numerous engravings, woodcuts, and folio atlas of plates 10 00 GILLMORE (Gen. Q. A.). Practical Treatise on the Construction of Roads, Streets, and Pavements. With seventy illustrations. 12mo, cloth . . . 2 00 D. VAN NOSTRAN&S SCIENTIFIC BOOKS. THE KANSAS-CITY BRIDGE. With an account of the Regimen of the Missouri River, and a description of the methods used for founding in that river. By O. CHANUTE, Chief Engineer, and GEORGE MORRISON, Assistant Engineer. Illustrated with live lithographic views and twelve plates of plans. 4to, cloth $6 00 LEDOUX (M.). Ice-making Machines. The theory of the action of the various forms of cold-producing, or so-called Ice-Machines (Machines a Froid). 18mo, boards. Illustrated 50 WATTS (ALEX.). Electro-Metallurgy Practically Considered. New revised and en- larged edition. [This new edition of this standard work, which has been in the market many years, and is well known by its intrinsic merits, has been very much increased in size, and has been added to in many of its departments, bringing it up to the latest developments of the art.] 12mo, cloth 1 00 McCULLOCH (R. S.). Elementary Treatise on the Mechanical Theory of Heat, and its Application to Air and Steam Engines. 8vo, cloth . . . . 3 50 AXON (W. E. A.). The Mechanics' Friend; a Collection of Receipts and Practical Suggestions relating to Aquaria, Bronzing, Cements, Drawing, Dyes, Electricity, Gilding, Glass - Working, Glues, Horology, Lacquers, Locomotives, Magnetism, Metal-Working, Modelling, Photography, Pyrotechny, Railways, Solders, Steam-Engine, Teleg- raphy, Taxidermy, Varnishes, Waterproofing, and Miscellaneous Tools ; Instruments, Machines, and Processes connected with the Chemical and Mechanical Arts. With numerous diagrams and woodcuts. Fancy cloth 1 50 SHREVE (S. H.). A Treatise on the Strength of Bridges and Roofs. Comprising the determination of algebraic formulas for strains in horizontal, inclined or rafter, triangular, bowstring, lenticular, and other trusses, from fixed and moving loads. With practical applications and examples, for the use of students and engineers. Eighty- seven woodcut illustrations. Second edition. 8vo, cloth . . 5 00 MERRILL (W. E.). Iron Truss Bridges for Railroads. The method of calculating strains in trusses, with a careful comparison of the most promi- nent trusses, in reference to economy in combination, &c. Illus- trations. 4to, cloth 5 00 8 D. VAN NO STRAND'S SCIENTIFIC BOOKS. STUART (C. B.). The Civil and Military Engineers of America. With nine finely executed portraits of eminent engineers, and illustrated by en- gravings of some of the most important works constructed in America. 8vo, cloth . . . $5 00 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 1 50 A Manual of Rules, Tables, and Data for Mechanical Engineers. Large 8vo. 1,012 pages. Illustrated. Cloth 7 50 Half morocco 10 00 WILLIAMSON (R. S.). On the Use of the Barometer on Surveys and Reconnoissances. Part I. Meteorology in its Connection with Hypsometry. Part II. Barometric Hypsometry. With illustrative tables and engravings. 4to, cloth 15 00 Practical Tables in Meteorology and Hypsometry, in connection with the use of the Barometer. 4to, cloth 2 50 CLEVENGER. A Treatise on the Method of Government Surveying, as prescribed by the U. S. Congress and Commissioner of the General Land Office; with complete Mathematical, Astronomical, and Practical Instructions for the use of the United States Surveyors in the Field. By S. V. CLEVENGEE. Morocco 2 50 RICHARDS'S INDICATOR. A Treatise on the Richards Steam-Engine Indicator, with an Appendix by F. W. BACON, M. E. Third edition, revised and en- larged. 18mo, flexible, cloth 1 00 BOW (R. H.). A Treatise on Bracing, with its application to Bridges and other Structures of Wood or Iron. One hundred and fifty-six illustra- tions. 8vo, cloth . .'.".'.' 1 50 HAMILTON (W. G.). Useful Information for Railway Men. Sixth edition, revised and enlarged. 562 pages. Pocket form, morocco, gilt . . . . 2 00 STUART (B.). How to become a Successful Engineer. Being Hints to Youths intending to adopt the profession. Sixth edition. 12mo, boards . 50 D. VAN NOSTRAN&S SCIENTIFIC BOOKS. SCRIBNER (J. M.). Engineers' and Mechanics' Companion. Comprising United-States Weights and Measures, Mensuration of Superficies and Solids, Tables of Squares and Cubes, Square and Cube Roots, Circumfer- ence and Areas of Circles, The Mechanical Powers, Centres of Gravity, Gravitation of Bodies, Pendulums, Specific Gravity of Bodies, Strength, Weight, and Crush of Materials, Water Wheels, Hydrostatics, Hydraulics, Statics, Centres of Percussion and Gyra- tion, Friction Heat, Tables of the Weight of Metals, Scantling, &c., Steam and the Steam-Eugine. Eighteenth edition, revised. Morocco $1 50 Engineers', Contractors', and Surveyors' Pocket Table-Book. Comprising Logarithms of Numbers, Logarithmic Signs and Tan- gents, Natural Signs and Natural Tangents, the Traverse Table, and a full and complete set of Excavation and Embankment Ta- bles, together with numerous other valuable tables for Engineers, &c. Tenth edition, revised. Morocco 1 50 SCHUMANN (F.). A Manual of Heating and Ventilation in its Practical Application, for the use of Engineers and Architects. Embracing a series of tables and formulae for dimensions of heating, flow and return pipes, for steam and hot-water boilers, flues, &c. Illustrated. Morocco 1 50 SHIELDS (J. E.). Embracing Discussions of the Principles involved and Descrip- tions of the Material employed in Tunnelling, Bridging, Canal and Road Building, &c 1 50 MORRIS (ELWOOD). Easy Rules for the Measurement of Earthworks, by means of the Prismoidal Formula. Illustrated. 8vo, cloth 1 50 GILLMORE (Gen. Q. A.). Treatise on Limes, Hydraulic Cements, and Mortars. With numer- ous illustrations. 1 vol. 8vo, cloth 4 00 HARRISON (W. B.). The Mechanic's Tool Book, with Practical Rules and Suggestions for use of Machinists, Iron- Workers, and others. Illustrated. 12mo, cloth 1 50 HENRICI (OLAUS). Skeleton Structures, especially in their application to the Building of Steel and Iron Bridges. With folding^plates and diagrams. 8vo, cloth 1 50 HEWSON (WILLIAM). Principles and Practice of Embanking Lands from River Floods, as applied to the Levees of the Mississippi. 8vo, cloth . . . 2 00 4 D. VAN NOSTRANJyS SCIENTIFIC BOOKS. HOLLEY (A. LJ. Railway Practice. American and European Railway Practice, in the Economical Generation of Steam, including the Materials and Construction of Coal-burning Boilers, Combustion, the Variable Blast, Vaporization, Circulation, Superheating, Supplying and Heating Feed-water, &c., and the Adaptation of Wood and Coke burning Engines to Coal-burning; and in Permanent Ways includ- ing Road-bed, Sleepers, Rails, Joint-fastenings, Street Railways, &c. With seventy-seven lithographed plates. Folio, cloth $12 00 MINIFIE (WILLIAM). Mechanical Drawing. A Text-Book of Geometrical Drawing for the use of Mechanics. With illustrations for Drawing Plans, Sec- tions, and Elevations of Railways and Machinery; an Introduction to Isometrical Drawing, and an Essay on Linear Perspective and Shadows. Illustrated. Ninth edition. With an Appendix on the Theory and Application of Colors. 8vo, cloth 4 00 STILLMAN (PAUL). Steam-Engine Indicator, and the improved Manometer Steam and Vacuum Gauges, their Utility and Application. New edition. 12mo, flexible cloth 1 00 WALKER (W. H.). Screw Propulsion. Notes on Screw Propulsion : its Rise and His- tory. 8vo, cloth ...75 MAC CORD (C. W.). A Practical Treatise on the Slide-Valve by Eccentrics, examining by methods the action of the Eccentric upon the Slide- Valve, and explaining the practical processes of laying out the movements adapting the valve for its various duties in the steam-engine. For the use of engineers, draughtsmen, machinists, and students of valve-motions in general. Illustrated. 4to, cloth . . . . 3 00 KIRKWOOD (JAMES P.). Report on the Filtration of River Waters for the Supply of Cities, as practised in Europe. Made to the Board of Water Co&mi8- sioners of the city of St. Louis. Illustrated by thirty double-plate engravings. 4to, cloth 15 00 BURGH (N. P.). Modern Marine Engineering, applied to Paddle and Screw Propul- sion. Consisting of 36 colored plates, 259 practical woodcut illus- trations, and 403 pages of descriptive matter; the whole being an exposition of the present practice of James Watt & Co., J. & G. Rennie, R. Napier & Sons, and other celebrated firms. Thick 4to vol. Cloth 25 00 Half morocco 30 00 5 D. VAN NOSTRANIfS SCIENTIFIC BOOKS. SIMMS (F. WJ. A Treatise on the Principles and Practice of Levelling, showing its application to purposes of Railway -Engineering and the Con- struction of Roads, &c. Illustrated. 8vo, cloth . . . . $2 50 BURT (W. A.). Key to the Solar Compass, and Surveyor's Companion. Compris- ing all the rules necessary for use in the field. Second edition, pocket-book form, tuck . 2 50 THE PLANE-TABLE. Its Uses in Topographical Surveying. From the papers of the United-States Coast Survey. Illustrated. 8vo, cloth . . . 2 00 HOWARD (C. R.). Earthwork Mensuration on the Basis of the Prismoidal Formula. Containing simple and labor-saving method of obtaining pris- moidal contents directly from end areas. Illustrated by examples, and accompanied by plain rules for practical uses. Illustrated. 8vo, cloth 1 50 GRUNER (M. L.). The Manufacture of Steel. Translated from the French by LENOX SMITH. With an Appendix on the Bessemer process in the United States, by the translator. Illustrated by lithographed drawings and woodcuts. 8vo, cloth 3 50 BARBA (J.). The Use of Steel for Constructive Purposes ; Method of Working, Applying, and Testing Plates and Brass. With a Preface by A. L. HOLLEY, C.E. 12mo, cloth 1 50 DUBOIS (A. J.). The New Method of Graphical Statics. With sixty illustrations. 8vo, cloth . 1 50 EDDY (H. T.). Researches in Graphical Statics, embracing New Constructions in Graphical Statics, a new General Method in Graphical Statics, and the Theory of Internal Stress in Graphical Statics. 8vo, cloth, 1 50 WEALE'S RUDIMENTARY SCIENTIFIC SERIES. These highly popular and cheap series of books, now comprising over two hundred distinct works in almost every department of Science, Art, and Education, are recommended to the notice of Engineers, Architects, Builders, Artisans, and Students generally, as well as to those interested in Workmen's Libraries, Free Libra- ries, Literary and Scientific Institutions, Colleges, Schools, Science Classes, &c. Fully illustrated by wood-cuts where necessary. Full list, with prices, furnished upon application. 4~ General Catalogue of Scientific Books, with Index to Authors, 76 pages, 8vo, sent on receipt of 10 cents. 6 TF THE UNIVERSITY OF CALIFORNIA LIBRARY