m UNIVERSITY OF CALIFORNIA AT LOS ANGELES 'WE A TEXT-BOOK PLANE STJKVEYING BY WILLIAM G. RAYMOND, C.E. MEMBER AMERICAN SOCIETY OF CIVIL ENGINEERS; PROFESSOR OK GEODESY, ROAD ENGINEERING, AND TOPOGRAPHICAL DRAWING, IN THE REXSSELAER POLYTECHNIC INSTITUTE NEW YORK :. CINCINNATI : CHICAGO AMERICAN BOOK COMPANY COPYRIGHT, 1896, BY AMERICAN BOOK COMPANY. RAYMOND'S PL. SURV. \v. P. i TA Kut PREFACE. THIS book has been prepared to meet the needs of those beginning the study of surveying. The subject treated is a simple one, and an effort has been made to make its presen- tation clear. The book is a text-book, not a treatise, and it is hoped that the teachers who use it will find it possible to devote their lecture work to amplification, rather than to explanation, of the matter it embraces. So far as seemed necessary the plan of giving first the general method and then the details has been adopted, at the risk of some repetition, because I believe this to be the clearest method of presentation. A special effort has been made to render clear and comprehensible those points which an experience of fourteen years of practice and teaching has indicated to be the ones presenting the greatest difficulties to students. Simpler matters have been left to the student to work out from suggestions. The book can be read under- standingly by any one who has completed Trigonometry, two formulas only being given whose derivation requires anything beyond. These may be derived by the teacher for such students as are sufficiently advanced. Particular attention is called to the systematic arrangement of computations in Chapter VI. ; to the article on the slide rule ; to the discussion of practical surveying methods in Book II. ; to the full treatment of coordinates ; to the large number of examples ; and to the use of the terms " latitude difference " and " longitude difference " for the old terms "latitude" and "departure." 228321 4 PREFACE. The whole general scheme of terms is thought to be much more logical than that heretofore in use ; and in this I have the support of Professors Merriman and Brooks, who have adopted practically the same nomenclature in their " Hand- book for Surveyors," recently issued. The logarithmic tables are from Professor C. W. Crockett's "Trigonometry," and are particularly suitable for surveyors' use. I am indebted to many persons and books for valuable assistance. Especial acknowledgment is due to Professor H. I. Randall of the University of California, who drew Plate IV.; to Mr. J. J. Ormsbee, Mining Engineer, who drew Plate V. ; to Mr. John H. Myers, Jr., A.B., C.E., for the problems on coordinates and for many suggestions; and to Professors R. S. Woodward of Columbia, and Frank O. Marvin of the Uni- versity of Kansas, for valuable suggestions. Mr. E. R. Gary, C.E., Instructor in Geodesy, Rensselaer Polytechnic Institute, has given much help in the preparation of examples. I also acknowledge my indebtedness to the following instrument makers for the use of cuts : Messrs. Buff & Berger, Boston, Mass.; W. & L. E. Guiiey, Troy, N. Y.; Keuffel & Esser Company, New York ; G. N. Saegmuller, Washington, D. C. ; L. Beckman, Toledo, O.; Mahn & Co., St. Louis, Mo.; F. E. Brandis, Sons & Co., Brooklyn, N. Y. The principal instrument cuts, furnished by the Messrs. Gurley, Keuffel & Esser, and Mahn & Co., will be known by the firm name on the cut. Those of Buff & Berger are Figs. 19, 20, 48, 107, 148, 151, and 153. G. N. Saegmuller furnished Fig. 54. All of the cuts used are covered by copyright. The book is submitted to my fellow teachers and students of surveying in the hope that it may prove useful to them in their work. TROY, N. Y., Auerwr- 1896. WILLIAM G. RAYMOND. CONTENTS. BOOK I. PRINCIPAL INSTRUMENTS AND ELEMENTARY CHAPTER OPERATIONS. INTRODUCTION . . . . . *. -. ... ft I. MEASUREMENT OF LEVEL AND HORIZONTAL LINES . . 13 Instruments used . . . . , . . . ,- . 13 Methods . . . ... . ... .18 Errors Involved . , . . . . . . 22 II. VERNIER AND LEVEL BUBBLE . . . .. . . . . 31 Vernier 31 Level Bubble 35 III. MEASURING DIFFERENCES OF ALTITUDE, OR LEVELING . 40 Instruments 40 Use of the Level . 50 Adjustments of the Level .63 Minor Instruments 72 Leveling with the Barometer ...... 74 IV. DETERMINATION OF DIRECTION AND MEASUREMENT OF ANGLES .77 The Compass 77 Compass Adjustments 79 Use of the Compass .83 Magnetic Declination .86 The Transit . . . "* 95 Use of the Transit 100 Adjustment of the Transit 108 The Solar Transit . . . . . . . . .116 Adjustments of the Solar Transit 122 Saegmuller Solar Attachment 123 Meridian and Time by Transit and JSun .... 125 V. STADIA MEASUREMENTS . 127 VI. LAND SURVEY COMPUTATIONS Ill Balancing the Survey 144 Supplying Omissions 1 19 Areas 152 Coordinates 156 Dividing Land . . . . . . . . .163 Model Examples .... ... 165 The Planimeter 172 The Slide Rule . 179 6 CONTENTS. BOOK II. GENERAL SURVEYING METHODS. CHAPTER PAGK VII. LAND SURVEYS 201 Surveying with the Chain alone . . . f .' ' . v . 204 Farm Surveys . . . . . . x . . . 208 United States Public Land Surveys 219 City Surveying , 230 VIII. CURVES 238 IX. TOPOGRAPHICAL SURVEYING 244 Topography 244 Simple Triangulation 253 . Mapping 261 The Plane Table 268 X. EARTHWORK COMPUTATIONS 275 Ordinary Methods 275 Estimating Volumes from a Map ...... 281 XI. HYDROGRAPHIC SURVEYING 287 Soundings 289 The Sextant 294 Measuring Velocity and Discharge . . . . . 298 Direction of Current 304 XII. MINE SURVEYING 305 Surface Surveys 305 Underground Surveys 308 Connecting Surface and Underground Work . . . 316 Mapping the Survey 320 APPENDIX. I. PROBLEMS AND EXAMPLES 322 Chapter I 322 Chapter III 324 Chapter IV . . .324 Chapter V . . . . . .' . . . .326 Chapter VI . . . 326 Coordinates 328 Chapter VIII 335 Chapter IX . . .336 Chapter X 338 Chapter XI . . . , 339 Chapter XII . ... . . ' 340 II. THE JUDICIAL FUNCTIONS OF SURVEYORS . . . .341 III. THE OWNERSHIP OK SURVEYS, AND WHAT CONSTITUTES A SURVEY AND MAP . 351 IV. GEOGRAPHICAL POSITIONS OF BASE LINES AND MERIDIANS IN PUBLIC SURVEYS 357 V. TABLES ; .... 361 INDEX . . 471 BOOK I. PRINCIPAL INSTRUMENTS AND ELEMENTARY OPERATIONS. INTRODUCTION. 1. Preliminary conceptions. An ellipse of axes AB and CD (Fig. 1), being revolved around its shorter axis C'Z>, will gener- ate the surface of an oblate spheroid of revolution. If we imagine the sea to extend underneath the sur- face of the earth so that the visible solid portions of the earth will be, as it were, floating on a ball of water, the shape of that ball will be, approximately, that of an oblate spheroid of revolution. The surface of this ball is called the mean surface of the earth. The shorter axis is that connecting the poles ; the longer is the diameter of the circle called the equator. In the case of the earth these two axes do not differ much in length, and hence the earth is usually spoken of as a "sphere slightly flattened at the poles." It may seem strange to the student that a difference of twenty-seven miles should be spoken of as a slight difference. But when it is said that this difference is about one third of one per cent of the length of the longer axis, the meaning is clearer. The lengths of the two axes according to the latest deter- minations 1 are: Shorter or polar axis . . Longer or equatorial axis 41,709,790 feet. 41,852,404 feet. 1 Clarke's spheroid of 1880. The values as found for Clarke's spheroid of 1866 are those generally used by geodesists. They are: shorter, 41,710,242 feet; longer. 41,852,124 feet. IQ INTRODUCTION. If a plane is passed through an oblate spheroid of revolu- tion, parallel to its shorter axis, it will cut from the spheroid an ellipse. If passed parallel to the longer axis, it will cut a circle. So with the earth : a plane passed parallel to the polar axis cuts from the mean surface of the earth an ellipse, while one passed parallel to the equator cuts a circle. Hence me- ridians of longitude are ellipses, and parallels of latitude are circles. The surface of the sea, or that surface extended as before mentioned, forms what is called a level surface, and a line lying in this surface is a level line. A line perpendicular to this surface at any point is a verti- cal line for that point. (A plumb line at any point is a vertical line for that point.) A line perpendicular to a vertical line is a horizontal line. A tangent to the earth's mean surface at any point is per- pendicular to the vertical line at that point, and hence is a hori- zontal line for that point. An inclined line is a straight line that is neither vertical nor horizontal. A vertical plane at any point is a plane including the verti- cal line at that point. A horizontal plane at any point is a plane perpendicular to the vertical line at that point. A vertical angle is an angle formed by lines in a vertical plane. A horizontal angle is an angle formed by lines in a horizon- tal plane. At any point on the earth's surface there can be but one vertical line, but there may be an indefinite number of horizon- tal lines ; there can be but one horizontal plane, but there may be an indefinite number of vertical planes. If water collects upon the earth's surface in some depression above the mean surface, as in a lake or pond, or even as in a small glass, and if the water is still, its surface will be nearly parallel to that portion of the mean surface of the earth that is vertically below it ; hence it will be a level surface, and a line drawn on it will be a level line. Such a line will be longer than the corresponding line drawn on the mean surface of the SURVEYING DEFINED. H earth between the verticals through the extremities of the upper line. The visible solid parts of the earth above the mean sur- face and the invisible solid parts below, make up a very irregular body. It is customary to speak of the visible parts of the earth's surface, both fluid and solid, as the " surface of the earth." In the definition in Art. 2, however, this term must be understood to mean not only the visible parts of the earth's crust, but also those parts that must be reached in connection with the operations of mining, bridge build- ing, or other engineering works that extend below the visi- ble surface. 2. Surveying defined. Surveying is the art of finding the contour, dimensions, position, etc., of any part of the earth's surface, and of representing on paper the information found. The operations involved are the measurement of distances, level, horizontal, vertical, and inclined, and of angles, horizontal, vertical, and inclined ; and the necessary drawing and computing to represent properly on paper the information obtained by the field work. The drawn representation is called a map. It may be a map showing by conventional signs the shape of that part of the earth's surface that has been measured ; or it may be simply an outline showing the linear dimensions of the bounding lines, together with the angles that they make with the meridian, or with each other, and sometimes the position within the tract of structures, roads, or streams. A map of the former kind is called, a topographical map, and the operations necessary to its production constitute a topographical survey. A map of the latter kind is a land map, and the operations necessary to produce it constitute a land survey. Either one of these surveys is a geodetic survey, if the tract is so large that the curvature of the earth's surface must be taken into account. This limit is supposed' to be reached when the tract is greater than one hundred square miles, but many surveys of tracts of much greater area than this are made without considering the mean surface to be other than plane. 12 INTRODUCTION. Such surveys are of course inaccurate, but may be sufficiently correct for the purpose they are to serve. A plane survey is one made on the assumption that the mean surface of the earth is a plane, above which is the irreg- ular visible surface broken by hills and valleys. Almost all land surveys are plane surveys. Only plane surveys will be considered in this book. In plane surveying all measurements are referred to a plane. In geodetic surveying all measurements are referred to a sphere, or spheroid, according to the area covered and the accuracy desired. It must be borne in mind that no physical measurements are exact. The art of surveying makes it possible to deter- mine that a field of land contains a certain area, more or less, that a mountain is so many feet high, more or less, that a mine is so many feet deep, more or less, etc. That is to say, it is physically impossible to measure exactly either distance or angles. The precision attainable or desirable in any survey- ing operations will be discussed elsewhere in this book. CHAPTER I. MEASUREMENT OF LEVEL AND HORIZONTAL LINES. 3. The line to be measured. The distance between two points on the surface of the earth is the length of the level line joining the verticals through the points. If one of these points is much higher than the other (further from the mean surface), there may arise con- fusion as to which of several lines is meant by the above definition. In geodetic sur- veying it is customary to re- duce the distance, when meas- ured, to the length of the level line lying in the mean surface, and contained between the ver- ticals through the points. The distance as measured will always be approximately the length of the level line lying midway as to altitude between the two points ; and this length is that used in plane surveying. The length of this line is obtained by measuring a series of short horizontal lines ; the sum of these lines approximates to the length of the required level line, just as the regular polygon of an infinite number of sides approximates to the circle. Fig. 2 will serve to make the above statements clearer. FIG. 2. INSTRUMENTS USED. 4. Chains. The instruments used are chains, tapes, and wooden or metallic rods. Chains are of two kinds Gunter's chain and the engineer's chain. These chains are alike in form, 18 14 MEASUREMENT OF LEVEL AND HORIZONTAL LINES. but vary in the length of the links and the length of the entire chain. In Gunter's chain the links are 7.92 inches long, and in the engineer's chain they are 12.00 inches, or one foot, long. With this exception, one description will apply to both. A chain consists of one hundred " links " made of iron or steel wire. Number 12 steel wire is best. Fig. 3 shows the form of the links. A link includes one of the long pieces and two or three rings, according as there are two or three rings used to connect the long pieces. The rings are in- troduced to enable one to handle the chain more readily. Brass tags with the proper number of points mark the ten-link divisions from each end to- ward the center, and a round tag marks the center or fifty-link division. The handles are of brass, and are usually made adjustable, so that slight changes in the length of the chain may be cor- rected. A special form of handle is sometimes used, having a knife edge on which is filed a notch indicating the zero of the chain for the, day, the chain being compared daily with a standard kept for the purpose. The Gunter's chain, having 100 links of 7.92 inches each, is 66.00 feet long, and the engineer's chain is 100.00 feet long. The former was devised by Mr. Edward Gunter, for the purpose of facilitating the computations of areas that have been measured. Its length was so taken that 10 square chains make one acre. It is the chain referred to in the table of surveyor's square measure, which should be carefully memorized. This table may be found in almost any arith- metic. In all surveys of the public lands of the United States the Gunter's chain is used, and all descriptions of land, found in deeds or elsewhere, in which the word " chain " is used, are based on this chain. It is not convenient for use in con- nection with engineering works, such as railroad construction, FIG. 3. INSTRUMENTS USED. 15 canal building, bridge building, etc., where the unit of meas- ure is the foot, and hence in such work the engineer's chain is used. 5. Tapes. Steel tapes are better than any sort of chain for most engineering work and for all fine surveying. These tapes are made in various forms, from thin ribbons half an inch wide to flat wires about one eighth of an inch wide and one fiftieth of an inch thick. The ribbon tapes are graduated on the front to feet, tenths, and hundredths of a foot, or to feet, inches, and eighths, and on the back to links of 7.92 inches. They usually come in box reels and are from twenty-five feet to one hundred feet long. They are suitable for very nice work of limited extent, and particularly for measurements for struc- tures, such as bridges and buildings, both in the shop and in the field. They are not suitable for ordinary field operations of surveying, because they are easily broken. For such work the narrower, thicker tapes are preferable. These may be ob- tained in any lengths up to a thousand feet or more ; but the lengths usually kept in stock are fifty feet and one hundred feet. They are graduated, usually to ten feet and sometimes to fifty* feet only, but may be graduated to suit the purchaser. For surveying work the following graduation is recommended : Graduate to feet, numbering every tenth foot from one end of tape to the other, and not from each end to the middle, as in the chain. Have the tape one foot longer next the zero end than its nominal length and divide the extra foot into tenths. If it is required to make measurements closer than to tenths of a foot, carry a pocket steel tape from three to five feet long and graduated in feet, tenths, and hundredths. Hun- dredths can usually be estimated with sufficient precision. 6. Reels. For the narrow tapes there are a number of patterns of reels, most of them aiming to furnish an open reel, of form convenient to go in the pocket when not in use, and so constructed as to enable the surveyor readily to reel the tape. There is but one reel that has come to the author's attention that combines all three of these requisites. This is shown FIQ. 8. FIG. U. INSTRUMENTS USED. 17 in Fig. 6, and a modified form in Fig. 4. Other forms of reels are shown in Figs. 5, 7, 8, and 9. Fig. 5 is a reel for a tape from 300 feet to 1000 feet in length. Fig. 9 shows a tape fitted with a spring balance for measuring the pull on the tape when in use, a level to show when the tape is held horizontal, and a thermometer to give the temperature. The necessity for these attachments will appear hereafter. Such a tape is used for land surveys in the city of New York. Tapes should always be kept dry, and if wet by use, should be wiped dry and rubbed with a cloth or leather that has the smallest possible quantity of mineral oil on it. 7. Linen tapes. In addition to the steel tapes, linen and " metallic " tapes are used for rough work. The ordinary linen tape is well known to everyone. The metallic tape is a linen tape with a few strands of fine brass wire woven through it. The linen tape is subject to great change in length with changes of moisture in the atmosphere, is soon stretched, and is easily worn out. The metallic tape is not so subject to change in length with change of atmospheric con- ditions; it is soon stretched, but is not nearly so soon worn out as is the linen tape. Both these tapes, being easily stretched, soon become quite inaccurate for any but the commonest kinds of work, where the measurements are short and need not be closer than to the nearest tenth of a foot. They are gradu- ated in feet, tenths, and half-tenths, and on the reverse side in links of 7.92 inches. Sometimes they are graduated in inches. They are sold in paper or leather box reels. 8. Rods. While some rough measurements are made with the ordinary ten-foot pole or a similar arrangement, no other surveying measurements are now made with wooden or metallic rods, except measurements of base lines in connection with im- portant geodetic surveys ; in these the rods, usually metallic, are arranged with other devices into a very elaborate piece of apparatus. It is believed that the narrow steel and brass tapes will entirely supersede the elaborate base apparatus now in use. H'M'D SURV. 2 18 MEASUREMENT OF LEVEL AND HORIZONTAL LINES. 9. Pins. These are used with the chain for the purpose of marking chain lengths. They are about fourteen inches long, made of steel or iron wire somewhat less than a quarter of an inch thick (No. 4 to No. 6 wire gauge) with a ring at one end into which is fastened a strip of cloth to insure ready finding of the pin when stuck in tall grass or brush. The other end is pointed. Eleven of these pins constitute a set. They are usually carried on a ring like a large key ring, or loose in the hand. 10. Range poles. Poles are used to range out the line to be measured. They are usually of wood, round or hexagonal, six to eight feet long, tapering from the bottom to the top, shod with a pointed iron shoe, and painted red and white in alternate strips one foot long. Gas pipe is sometimes used, but is not recommended, because, while it does not break, and while from its weight such a pole is easily balanced on its point, it is also very easily bent, and very difficult to straighten, and is too heavy to be handled with ease. A good pole for nice work in cities or on railway surveys is made of hexagonal steel about three eighths to five eighths of an inch thick, painted like the wooden poles, and pointed at one end. METHODS. 11. Preliminary statement. The accurate measurement of a line on a comparatively level piece of ground is a task diffi- cult for a beginner and not simple for an expert chaimnan, however easy it may seem. The method of doing this work on ordinary farm surveys, where the smallest unit of measure is the link (7.92 inches), and where an error of one in three hundred to one in five hundred may be tolerated, will be described ; and the errors incident to this method with the necessary corrections, as well as the more precise methods applied to city work, will then be discussed. 12. Chaining. It will be noticed when the chain is received from the maker that it is so folded together as to be compact in the center of the bundle and somewhat bulky at the ends, in METHODS. 19 shape not unlike an hour glass. This results from doing up the chain as follows : Take the two links at the center of the chain in the left hand, with the fifty-link tag on the left. Take the right hand ends of the pair of links next but one to those in the left hand, in the right hand, and lay the right hand pair and the intermediate pair in the left hand diagonally across the pair already there. In like manner proceed to the ends of the chain, being careful always to place the new links diagonally across the links already in the left hand and always diagonally the same way. It is better, however, to do up the chain from one end instead of from the middle. The method is the same except that the two end links are first taken in the left hand, the handle end to the right. A little more time is required, but the chain is more readily loosened for service. It will be assumed that the ground on which the line is to be measured is comparatively level, and that the ends of the line are visible, one from the other. If there is no visible object to mark the further end of the line, a range pole is placed there, toward which the measurement is to be made. If the rear end of the line is also marked, the head chainman will be able, without difficulty, to put himself in approximate line, thus saving time. The strap with which the chain is fastened is removed, and, if the chain has been done up from the middle, the two handles are taken in the left hand of the forward chainman and the chain bundle in the right hand, allowing a few links next the handles to fall off. The chain bundle is then thrown out in a direction opposite to that in which the measurement is to be made, the chainman retaining the handles in his left hand. The chain should be thrown with sufficient force to straighten it out. The forward man, usually called the "head chainman," then takes the forward end of the chain and the pins, and starts toward the further end of the line, while the rear chainman allows the chain to slip through his hands to see that it is not kinked or bent. If he finds any bends he straightens them. If the chain has been done up from one end, it should be laid down near the starting point with one handle uppermost, the latter to be taken by the head chainman, who moves off toward the further end of the line. The rear chainman allows the chain to slip through his 20 MEASUREMENT OF LEVEL AND HORIZONTAL LINES. FIG. 10. hands as before. The chain gets enough rough service that can not be avoided, without subjecting it to the additional un- necessary wear arising from throwing it out forcibly, to be kinked or caught in the brush or other obstruction. One pin is left with the rear chainman. As the head chain- man walks out, he places one pin in the hand that carries the chain, the remaining pins being in the other hand. When the chain is almost out, the rear chainman calls "Chain." The head chainman then stops, turns, and straightens the chain while being put into approximate line by the rear chainman. The chain being taut and approxi- mately "lined," the head chainman assumes the posi- tion shown in Fig. 10, and the rear chainman, by mo- tions or the words " right " and " left," accurately aligns the pin held by the head chain- man and cries "Stick." The head chainman then forces the pin into the ground, taking care that it marks exactly the end of the chain, and cries "Stuck." The rear chainman then, and not till then, draws his pin, keeping hold of the chain, and follows the head chainman, who moves on toward the forward end of the line, and the whole operation is repeated. After one pin has been placed, the head chainman, on being stopped by the call of the rear chainman, can quickly put himself in approximate line by sighting back over the pin last set to the flag left at the starting point. The work thus proceeds till the further end of the line is reached, when the head chainman walks right on past the point till the chain is all drawn out. He then returns to the point and notes the fraction of a chain between the last pin and the point. This added to the number of chains gives the distance required. If the distance is more than ten chains, the head chainman, when he sticks his last or tenth pin, calls " Stuck out." He then waits by the pin till the rear chainman METHODS. 21 comes up with the pins he has collected, which should, with the pin he started with, be ten. He counts them, as does the head chainman, as a check, and they note one "tally." At any time the number of tallies plus the number of pins in the rear chainman's hands gives, in tens of chains and chains, the distance that has been measured. 13. Hints. The following hints may be of service to be- ginners : The rear chainman should not use the pin to brace himself. He should hold the outside edge of the handle flush with the rear side of the pin, without moving the pin. He should not stop the head chainman with a jerk. He should not sit down on the ground while holding the pin. Motions and words should be sharp and distinct. Motions and instructions should be proportionate to the dis- tance that the pin is to be moved ; for example, the arms should not be swung wildly when the pin is to be moved an inch. The head chainman should see that the rear chainman is looking when he tries to straighten the chain. The chain should not be jerked in straightening it ; it should be straightened by an undulatory motion. In straining the chain, the head chainman should pull steadily. Attention to these matters will greatly facilitate the work. 14. Chaining on slopes. In chaining up or down hill, one end of the chain is raised till both ends are as nearly as possible in a horizontal line. If the slope is so steep that one end of a full chain cannot be raised enough to bring both ends in a horizontal line, the chain is " broken," that is, the distance is measured by using a part of the chain at each measurement. To do this, the chain should be carried out as if a full chain were to be used, the head chain- man returning to such a point on the chain (preferably a ten- link point) that the portion of chain between himself and the rear chainman may be properly leveled. A measurement is made with this portion, then with the next succeeding portion, and so on till the whole chain has been used. Care must be 22 MEASUREMENT OF LEVEL AND HORIZONTAL LINES. taken not to get the pin numbering confused. The rear chain- man should have but one pin for the whole chain. The high end of the chain is transferred to the ground in one of several ways, according to the precision desired. If the work is to be done with care, a plumb line is used. If an error of a tenth of a link in each chain is not important, a pin may be dropped from the high end, and stuck in the ground where it is seen to fall. The pin should be dropped ring down. A small pebble will serve the purpose for rough work. In careful work the plumb bob should not be dropped and the pin placed in the hole made ; but it should be noticed where the bob will drop, and the ground should be made smooth with the foot, and the bob swung down till it is still and just clearing the ground ; then it should be carefully lowered till it touches. The chain- man should then lower his grasp on the string, hand over hand, keeping the bob steadily in its place, and place a pin in the ground at the point of the bob. The pin should be put in the ground in an inclined position across the line, so that the point where it enters the ground is that covered by the bob. The position should then be checked. In place of a pin a small wire brad may be used and left in the ground. In chaining up hill, the rear chainman must hold the bob directly over the pin which has been set in an inclined position, and must at the same time align the head chainman and see that he sticks at a moment when the bob is directly over the point. It will be at once inferred that it is easier to measure down hill correctly than up hill. Therefore, where close work is required on inclined ground, the measurements should always, if possible, be made down hill. ERRORS INVOLVED. 15. Classes. The errors involved in the method of chaining just described, whether the work is done with a chain or a tape, are of two classes : (a) constant or cumulative errors, and (5) accidental or compensating errors. (a) Cumulative errors are such as occur each time in the same direction. They are not necessarily equal, but may be so. Thus a line so long as to require that a chain one inch too short shall be applied to it ten times, will be recorded ten inches too ERRORS INVOLVED. 23 long, the error of the chain being added each time the chain is applied. In this case the errors are equal. (J) Compensating errors are such as tend to balance ; that is, they are as likely to be in one direction as in another. Thus the error that may be made in setting the pin, if it is attempted to set it just right, will be a compensating error, for it will be set ahead of the true point about as often as it will be set behind it. Error in plumbing is of the same character. 16. Causes. Cumulative errors arise from five causes : (a) erroneous length of chain, (6) errors in judgment in mak- ing the chain horizontal in chaining up or down hill, (c) erro- neous alignment of the chain, (cf) failure to straighten the chain for each measurement, (e) sag of the chain when not supported throughout its length. Compensating errors arise from accidental inaccuracies in setting the pin, and from irregularities in the pull exerted on the chain or tape. They are remedied by care, and, in fine work, by measuring the pull on the tape by a spring balance. Erroneous length of chain may arise from any one of six causes. (1) One or more links may be bent, making the chain too short. The remedy is to see that the links are straight or to use a tape. (2) Mud or grass may get in the links and rings with the same effect. The remedy is obvious. (3) A bent link that has been straightened has been per- manently lengthened, thus making the chain too long. The remedy is to compare the chain or tape frequently with a standard tape kept for this purpose. If the chain is found to be slightly too long, it may be adjusted by the nuts at the handle, or if such a handle as is described in Art. 4 is used, the stand- ard length of the chain for the day may be marked on the handle. (4) The links and rings wear, thus making the chain too long. While the wear is slight, it may be adjusted at the handle. When it becomes excessive, it must be known and allowed for as hereafter described. (5) The chain may be lengthened by too hard pulling, but this does not often occur. The remedy is the same as in (3). ^4 MEASUREMENT OF LEVEL AND HORIZONTAL LINES. (6) The chain may be too long or too short according as the temperature is higher or lower than that for which the chain is standard. The remedy is to know the temperature at which the chain is standard and that at w r hich the work is done and make the necessary correction to the recorded measurements. In general it may be said that erroneous length of chain may be corrected by adjusting the handles, or by comparing the tape or chain with a standard and correcting the records taken according to the errors found. It should be carefully noted that, in measuring the distance between two points, a long chain gives the distance too short and a short chain gives the distance too long, while in laying out a line of given length the errors are just reversed. Failure to appre- ciate this difference often causes confusion and error, and hence the student should thoroughly fix it in mind. Since similar figures are in area as the squares of their homologous sides, the erroneous area of a field determined from measure- ments with an erroneous chain, will be to the true area as the square of the nominal length of the chain is to the square of its true length. 17. Temperature. The coefficient of expansion of steel is about 0.0000065. (Tapes and chains being alike subject to this error, this discussion will do for both.) A tape or chain will expand or contract sixty-five ten-millionths of its length for each Fahrenheit degree change of temperature. Thus a line about ten chains long, if measured in the summer with the chain at a temperature of, say, 80 F., the chain being stand- ard at a temperature of 62 F., will be recorded 0.117 links too short ; while the same line measured with the same chain in midwinter with the chain at a temperature of F., will be recorded 0.403 links too long, making a total difference of 0.52 links between the two measurements. This is an error of one in two thousand for the extreme difference in temperature of 80 F. It is thus seen that for all ordinary work the tempera- ture correction may be neglected ; but in city work where an inch in frontage may be worth several thousand dollars, it is ERRORS INVOLVED. 25 very necessary that the temperature be determined and the standard temperature of the tape known. The tape shown in Fig. 9 is adjustable for the effect of temperature. A scale numbered to correspond to the thermometer readings indicates the proper setting of the adjusting screw. The spring balance insures a constant pull. 18. Sag. The effect of sag in shortening a tape that is un- supported except at the ends is given by the following formula in which I is the unsupported length of the tape, w the weight of a unit of length, and P the pull in pounds. _ I " This formula the student will have to accept until he has studied the elements of Mechanics and Calculus. It assumes that the tape is supported only at the ends, and that it is stand- ard for no pull when supported its entire length. If the tape is standard for a pull of P pounds, substitute in the formula for P the difference P - P Q , If the tape is applied n times in measuring a line and each time is supported only at the ends, and the pull is always the same, the correction for the whole line is n times the above expression. The formula gives the difference between the length of the curve of the unsupported tape and its chord. The real dis- tance measured is the chord, while the distance read is the length of the curve or of the whole tape. It is evident, there- fore, that the distance is read too long, and hence the formula is a negative correction. 19. Pull. If the chain were of constant cross section as is a tape, the amount that the chain would stretch for a pull of P pounds would be given by the following formula in which I is the length of the chain in inches, S is the area of its cross section in square inches, and E is the modulus of elasticity of the metal of which it is made : PI y= SE' 1 This formula has been developed by Prof. J. B. Johnson. 26 MEASUREMENT OF LEVEL AND HORIZONTAL LINES. E for steel is variable, but may be taken at 28,000,000. There is no such thing as a perfectly elastic material. If there were, the amount that a given length of the material would be stretched by varying pulls would be proportional to the pulls, and supposing the piece to be of unit cross section, as one square inch, the pull that would stretch it by its own length is known as E, the modulus of elasticity. For any other than a unit cross section the stretch for a given pull will be inversely propor- tional to the cross-sectional area. Hence the formula. The lengthening effect of a given pull on a tape would be as in the formula. In the case of a chain, the effect would be some- what greater, owing to the elongation of the rings. 20. Elimination of sag and pull. To find the pull that will just balance the effect of sag, equate the values of x and y and solve for P. Since the units are inches in the y formula, they must be inches in the x formula, and I must be the length of the tape in inches, and w the weight of an inch of the tape. The solution gives whence A good practical way to determine this value is as follows : Mark on a smooth level floor a standard tape or chain length, with the tape supported its entire length, and with only enough pull to straighten it. Raise the tape, and supporting it only at the ends, measure with a spring balance the pull necessary to bring the ends over the marks on the floor. It will be best to have one end fastened in a firm hook in the wall for the test, and afterward to have both ends held by the chainmen, that they may see just the difficulties involved. The test should be made for the whole chain, the half chain, and the quarter chain. The only way in which this work can be done with extreme nicety is by employing mechanical means to pull the chain steadily, and a telescopic line of sight to transfer the floor marks upward to the tape ends. As in all work, except the measurement of base lines for geodetic surveys, or elaborate ERRORS INVOLVED. 27 triangulation surveys of cities, the chain or tape is to be held in the hands of the chainmen, it will be unnecessary, except for comparisons, to resort to the nicer methods. Experiments of the kind noted above will demonstrate that the whole chain should never be used unsupported, and that the tape is by far the most satisfactory measuring instrument. In rough work, where a precision of one in five hundred, or even one in five thousand, as a maximum limit, is sufficient, the chain may be used. In close work requiring a precision of one in five thousand and upward, the tape should invariably be used. 21. Alignment. Errors due to inaccuracy of alignment of the chain are usually not great. In ordinary work no great pains need be taken to align the chain within an inch or two, except where stakes are to be driven on -the line. In close work, of course, the chain should be correctly aligned. 1 22. Slope. In chaining on slopes, errors of judgment in making the chain horizontal are eliminated by the use of a level tube fastened to one end of the chain, which tube, if properly adjusted, will indicate when the chain is horizontal. This is rarely used with a chain, but frequently with a tape. Much can be done without such a level by having a third man stand on one side of the chain and compare the parallelism of the chain and the horizon, or the horizontal lines of some building. If there is no horizontal line visible, he can still judge better from the side as to the horizon tality of the tape, than can the chainmen at the ends. It is almost always true that the lower end of the chain is not raised high enough, because a horizontal line on a hillside extending in the direction of the slope, always appears to dip into the hill. Hand levels (see Art. 52) carried by the chainmen are of great service in hilly country. The effect of neglecting the slope entirely, which is also the correction to be applied if the line has been measured on the slope instead of in horizontal lines, is given in Appendix, Table I., page 361. 1 Let the student compute the error arising in a ten-chain line from placing the end of the chain first six inches on one side of the line and then six inches on the other side, throughout the measurement. 28 MEASUREMENT OF LEVEL AND HORIZONTAL LINES- It will be seen that the error caused by neglecting a slope of five in one hundred is about one in one thousand, while a slope of ten in one hundred, which is not unusual in hilly country, causes an error of one in two hundred. Fifty feet in one hundred is about the steepest slope met with in nature, aside from rock cliffs, and the error here is more than one in ten. On a slope where close work is required, it is considered best to measure along the slope, keeping the tape or chain sup- ported throughout its entire length, and making the necessary reductions when the line has been measured. The reduction can be made exactly by the use of a table of versed sines if the angle of slope is known. It may be approximately obtained from Table I., page 361, by interpolating for the small angles, or it may also be approximately obtained by the use of the fol- lowing formula when the rise in a tape length or in the entire line, if it is of uniform slope, is known : The square of the rise divided by twice the known side, be it base or hypotenuse, gives the difference between the base and hypotenuse. Demonstration : Let B be the base, H the hypotenuse, and R the rise ; O being the difference between B and H. Then B H G and H = B + G. Assuming H known, there is written H*- (H- (7) 2 = #2. whence =' Neglecting <7 2 as a very small quantity, there results R* = 2H' Similarly if B is known, there may be written and as before, neglecting <7 2 , there results Hence the rule already given ERRORS INVOLVED. 29 23. Precision to be obtained. In measuring lines the degree of precision obtainable should be known by the surveyor. The author suggests the degrees of precision mentioned below as those that should be attained ordinarily before the surveyor can say he is doing good work. The figures given do not refer to the absolute lengths of the lines, involving a knowl- edge of the absolute length of the chain or tape, but merely to the probable error of the mean of two measurements of the same line made with the same tape under different conditions. Not all conditions of work are covered ; but only such as usually exist. The surveyor will be able to judge as to how closely the conditions under which he is working at any time correspond to those given. In good, fairly level ground, good work will be represented by differences between two measurements of one in twenty-five hundred, and excellent work by differences of one in five thou- sand, assuming the work to be done with a chain. These differences give the probable error of the mean value as one in seventy-five hundred and one in fifteen thousand, and the prob- able error of a single determination rather better than --$$-$ and TT5W On hilly ground, rough and covered with brush, one in one thousand might be considered good and one in five hundred passable, where the land is not of great value. These differences give the probable errors of mean and single meas- urement as goVo to Tinnf and 2uW to ToW respectively. It should be remembered that the value of the land measured, or the object of the survey, is a better basis for judgment as to passable work than the conditions under which the work is done.. In work in large cities the author thinks that a precision of one in fifty thousand should be obtained. That is, it is thought that the probable error of the mean of two measurements should not be greater than one in fifty thousand. This will require that the same line measured under totally different conditions as to weather should be recorded, after the necessary correc- tions for pull, grade, and temperature have been made, both times alike, within about one in seventeen thousand, or, in round numbers, three tenths of a foot in a mile. When but two observations of a quantity have been taken, 30 MEASUREMENT OF LEVEL AND HORIZONTAL LINES. the probable error of the mean is ^ D, where D is the differ- ence of values determined. The probable error of either of the observations is 0.47 D or, roughly, \ D. (See any treatise on Least Squares.) This supposes that all cumulative errors and mistakes have been eliminated by correction and that only accidental errors remain. The following are the requirements for securing a precision of one in five thousand and one in fifty thousand. For inter- mediate standards, the requirements will lie between those mentioned : For a precision of one in five thousand, using a tape, no cor- rections for sag, grade, pull, or small changes of temperature need be made. The tape may be stretched by hand, the pull and horizontality being estimated by the tapemen. The plumb line will be used on uneven ground as in close chaining. The temperature of the air may be compared with that for which the tape is standard, and a corresponding correction deduced. For a precision of one in fifty thousand, the temperature of the tape should be known within a degree or two Fahrenheit ; the slope should be determined by measuring over stakes whose elevations have been determined by a level, or by measuring on ground whose slope is known. The pull should be known to the nearest pound, and hence should be measured with spring balances. If the tape is held on stakes, the sag cor- rection must be considered. The work may be done in any ordinary weather, but is best done" on cloudy days, so that the temperature of the tape may be more constant. In sunny weather the mercurial thermometers attached to the tape may indicate a very different temperature from that of the tape. If the absolute length of the tape is not known, of course the absolute length of the line is not determined. CHAPTER II. VERNIER AND LEVEL BUBBLE. 24. Before proceeding with a description of surveying in- struments, it is necessary to describe two important attach- ments common to almost all such instruments. These are the vernier and the level bubble. VERNIER. 25. Vernier. This is a device for reading degree of precision than is possible with the finest convenient division of the scale. Thus a scale graduated to read tenths of an inch, may be read to hundredths of an inch by the aid of a vernier. This is done by making an auxiliary scale called a vernier, with divisions one one-hun- dredth of an inch smaller or larger than those of the main scale. If the divisions are larger than the main scale, the vernier is called a ret- rograde vernier ; and if the divisions are smaller, it is called a direct vernier. The reason for this distinction will appear hereafter. In Fig. 11 !S is a scale divided into inches and tenths. F"is the vernier made by dividing a space equal to nine of the small divisions of the main scale into ten equal parts, thus making each division on the vernier one one-hundredth of an inch shorter than a division of the main scale. The first division line of the vernier falls one one- hundredth of an inch toward zero from the first division line of the main scale. If then the first division line of the vernier is made to coincide with the first line of the main scale 31 to a greater . 5 - -v B 10 1 ^ \ o o . _ 5 2 - v *-i 10 FIG. 11. 32 VERNIER AND LEVEL BUBBLE. the vernier will have been moved one one-hundredth of an inch. Similarly the second division of the vernier is two one-himdredths of an inch toward zero from the second line of the main scale, and hence if the vernier is moved along till the second line of the vernier coincides with the second division of the scale, the movement has been two one-hun- dred ths of an inch, and so on. If the vernier is moved till the zero is opposite some other division of the scale than the zero division, the first line of the vernier will be one one-hundredth short of the line of the main scale next ahead of the zero of the vernier ; the second line of the vernier will be two one- hundredths short of the second line of the main scale, and so on. If the vernier is moved along a little further till, say, the fourth line of the vernier has been brought into coincidence with the fourth line of the main scale ahead, the vernier has been moved a further distance of four one-hundredths of an inch. Hence to tell how far the zero of the vernier has moved from the zero of the main scale, note the inches and tenths on the scale from zero to the zero of the vernier, and get the frac- tional tenth expressed in hundredths by looking along the ver- nier and finding the division that coincides with a division of the main scale. This vernier is called direct, because in read- ing it one looks forward along the vernier in the direction in which the vernier has moved. Let it be required to read the length of the bar B. Place one end of it opposite the zero of the main scale and vernier. It will be noticed that the other end is opposite a point on the main scale between one and three tenths inches, and one and four tenths inches. Move the vernier till the zero is opposite this end of the bar. To read the length of the bar, read on the main scale one and three tenths inches and look along the vernier and find that the sixth division coincides with a division of the scale and that therefore the length of the bar is one and thirty-six one-hundredths inches. It will be observed that the divisions of the vernier are one tenth of one tenth of an inch smaller than the divisions of the scale. That is, the value of the smallest division on the main scale divided by the number of divisions of the vernier gives the smallest reading that may be had with the vernier. This is called the least count. VERNIER. 33 Iii the retrograde vernier a space equal to a given number of divisions of the main scale is divided into a number one less on the vernier. Thus for a vernier reading to hundredths of an inch with a scale graduated to tenths, eleven divisions of the scale will be divided on the vernier into ten spaces, making each division one tenth of one tenth of an inch longer than those of the scale. The vernier is therefore placed as shown in Fig. 12, back of the zero of the scale instead of ahead of it, as in the direct vernier. "Back" and " ahead " are used with reference to the direction in which measurements are to be made. From the portion of the scale extended above the zero in the figure, it will be seen that the first line of the vernier is back of the first line of the scale by one one-hundredth of an inch, the second line by two one-hundredths, and so on. The principle of operation is the same as in the "direct vernier, except that one must look backward along the vernier to find the coinciding line. A vernier to read angles is generally used when the angles are to be read to the nearest minute or less. The principle of construction is the same as for linear verniers. A vernier to read minutes will usually occur with a circle graduated to read half degrees. If a space equal to twenty -nine of such divisions is divided on a vernier into thirty equal parts, each division of the vernier will be one thirtieth of thirty minutes, or one minute, less than a division of the main scale, and the instrument is said to read to minutes. If a circle is to be read to the nearest twenty seconds, it is usually divided into twenty minute spaces, and a vernier must then have sixty divisions, since n 3 ' n = 60 divisions. That is, fifty-nine parts of the scale must be divided on the vernier into sixty parts, making each part of the vernier one R'M'D SURV. 3 34 VERNIER AND LEVEL BUBBLE. sixtieth of twenty minutes, or one third of a minute, less than a division of the main scale. Figures 13, 14, and 15 show three double verni-> ers. They are called double, because there are really two verni- ers in each figure, one FlQ - 13 ' on each side of the vernier zero. They are thus arranged so that angles may be read in either direction, the circle graduations being numbered both ways for the same purpose. The student should deter- mine whether the first two are direct or retrograde, the least count of each, and their readings. The third is a peculiar FIG. 14. pattern found ordinarily only on compasses. It is a double vernier, direct as to division (though it is sometimes made ret- rograde), and the lower left-hand and upper right-hand por- tions form one vernier. It is used where there is lack of space to make the ordinary form. To read an angle measured to the right, read on the scale to the last division be- fore reaching the zero of the vernier, follow to the right along the *" FIG. 15. , . vernier, noting the lower line of figures for a coinciding line, and if none is found, pass to the extreme left end of the vernier and look along toward the right, noting the upper line of figures till a coin- ciding line is found. Thus the reading of the vernier in the figure is 355 20', or 4 40'. LEVEL BUBBLE. 35 LEVEL BUBBLE. 26. Description. The spirit level consists of a glass tube almost filled with ether, the remaining space being filled with the vapor of ether. The bubble of vapor will always seek the highest point in the tube. If the tube were perfectly cylindri- cal, the bubble would occupy the entire length of the tube when the tube is horizontal, and the same thing would be true if the tube were but slightly inclined to the horizon, thus making it impossible to tell when the tube is in a truly horizontal posi- tion. The tube is, therefore, ground on the inside so that a longitudinal section will show a circular arc. A line tangent to this circle at its middle point, or a line parallel to this tan- gent, is called the axis of the bubble tube. This axis will be horizontal when the bubble is in the center of its tube. Should the axis be slightly inclined to the horizon, the bubble will move toward the higher end of the tube, and if the tube is ground to the arc of a circle, the movement of the bubble will be propor- tional to the angle made by the axis with the horizon. There- fore, if the tube is graduated into divisions, being a portion of the circumference of a very large circle (so large in fact that the arc of a few seconds is quite an appreciable length), it will be possible to determine, within the limits of the tube, the angle that the axis may make at any time with the hori- zon, provided the angular value of one of the divisions of the tube is known. This is done by simply noting how many divisions the center of the bubble has moved from the center of the tube. It will be evident that divisions of uniform length will cover arcs of less angular value as the radius of the tube increases, and also that the bubble with a given bubble space will become more elongated as the radius is increased. Therefore the bub- ble is said to be sensitive in proportion to the radius of curva- ture of the tube, and this is also indicated by the length of the bubble. The length of the bubble, however, will change with change of temperature, becoming longer in cold weather and shorter in warm weather. In the best class of tubes there is a partition near one end, with a small hole in it at the bottom, so that the amount of liquid in the main tube may be regulated, 36 VERNIER AND LEVEL BUBBLE. thus regulating the size of the bubble. This is necessary, because independently of the effect of long radius a longer bubble is more sensitive than a shorter one. A bubble should settle quickly, but should also move quickly and easily. 27. Determining the angular value of one division. There are several methods of determining the angular value of one division of the bubble tube, all essentially the same in principle. The axis is moved through a small angle, and the move- ment of the bubble is recorded in divisions ; then the angular value of one division is at once found by dividing the angle by the number of divisions through which the bubble has moved. FIG. 16. It is not easy to measure the small angle exactly but it is not very difficult if closely approximate results are sufficient. In Fig. 16 is shown a level vial, as it is sometimes called, resting on a level trier. The construction of the level trier is perhaps sufficiently clear from the cut. It consists simply of a board resting on a knife edge at one end, and capable of being raised or lowered at the other by means of a screw so divided as to tell the angle of inclination of the board. The screw is called a micrometer screw, because it will measure a very small movement. Suppose the pitch of the micrometer screw is one sixtieth of an inch. Then the divisions on the vertical scale LEVEL BUBBLE. 37 attached to the movable board will be one sixtieth of an inch ; so that a single revolution of the screw will move the scale past the edge of the screw head by one division. If the circum- ference of the disk head of the screw is divided into one hun- dred parts, and the screw is turned only so much as will cause one division on the disk to pass the scale, the board has been moved vertically through one one-hundredth of one sixtieth of an inch. If now the length of the bar from pivot to screw is known, the angular movement of the bar may be computed. Thus, if the length of the bar is eighteen inches, and the bar is raised so that one division of the scale passes the micrometer head, and so that in addition ten divisions of the micrometer head pass the scale, the linear elevation of the end of the bar is li x gV inch = 0.018+ inch. Since there are 206,265 seconds in an arc equal in length to radius, there results the proportion, in which x is the angle in seconds, x = 0.018+ 206265 ' 18 Whence x = 206.265 seconds. If now a bubble tube were resting on the bar, and the run of the bubble were observed, for the above movement, to be ten divisions, the value of one division would be 20.6 seconds. Example. In the above example it is found that the run of the bubble is one inch. Find the radius of curvature of the bubble tube. Other methods of finding the angular value of a division of the tube will be suggested in the problems on Chapters III. and IV. Many of the level vials found on compasses, and on the lower plates of many other instruments, are not graduated, are ground to short radii, and not uniformly, and hence are not fit for accurately leveling the instruments to which they are attached ; but such bubbles are cheaper than others, and when placed on a compass or other instrument not intended for high- class work, they are sufficiently precise for the purpose for which they are used. 38 VERNIER AND LEVEL BUBBLE. 28. Principles. The proper adjustment of a bubble on an instrument so that one can determine when the instrument is level, depends on the following principles : I. If a frame carrying a bubble tube, and resting on two sup- ports that lie in a level line, is reversed end for end on the sup- ports, the bubble will occupy the same position in the tube for both positions of the frame. In Fig. 17 it is seen that the axis of the tube makes the same angle with the horizon in both positions, and the same end is higher. Conversely, if a frame to which a bubble tube is rigidly attached is reversed on two supports, and the bubble occupies the same position in its tube for both positions of the frame, the supports lie in a level line, or, as is usually said, are level. It should be noted in the above that the bubble is not necessarily Fl - 17- in the center of the tube, but merely retains the same position in the tube for the direct and reversed positions of the frame. If the axis of the tube in the foregoing cases is parallel to the line joining the supports, the bubble will lie in the center of the tube, and if not parallel, the deviation of the bubble from the center of the tube will be that due to the angle between the line of support and the axis of the tube. If in the latter case the bubble is brought to the center of the tube, the line of supports will make an angle with the horizon (be out of level) equal to that between the axis of the tube and the line of supports. If now the frame carrying the level is reversed, the movement of the bubble will be twice that due to the angle between the axis and the line of supports. 1 If the tube is now raised at one end, or lowered at the other, till the bubble has moved halfway back to its former position, the axis of the tube is made parallel to the line of support. The line of support may now be made level by raising the lower, or low- 1 Let the student make a diagram showing this. LEVEL BUBBLE. 39 ering the higher, support till the bubble stands in the center of its tube. If two levels are attached to a plate at right angles to each other and parallel to the plate, the plate will be level when both bubbles are centered. If the tubes are not parallel to the plate, it will be difficult to determine when the plate is level, as the position of each bubble for level plate must be determined by trial. If the tubes are so fastened to the plate as to permit of being adjusted, their parallelism may be tested and, if neces- sary, corrected by the method of this article. II. If a frame carrying a level tube is revolved about a verti- cal axis, the bubble will maintain a constant position in its tube. For the axis of the tube maintains a constant angle with the horizon. Conversely, if a frame carrying a bubble tube is revolved about an axis, and the bubble remains in one position in the tube, the axis of revolution is vertical. If the constant posi- tion occupied by the bubble is the center of its tube, the tube is horizontal, and consequently perpendicular to the axis of revo- lution. CHAPTER III. MEASURING DIFFERENCES OF ALTITUDE, OR LEVELING. 29. General principle. It will be evident that if by any means a line of sight may be made to revolve about a vertical axis to which it is perpendicular, it will describe a horizontal plane. Omitting consideration of the curvature of the earth, a rod graduated from the bottom up, and held at any point on the ground, will be cut by this horizontal plane at a distance above the ground equal to the height of the line of sight above the ground at the point where the rod is held. The dis- tance above the bottom of the rod, as indicated by the gradua- tions on the rod, is called the reading of the rod. If the eleva- tion of the line of sight above some assumed base or datum, as sea level, is known, the rod reading subtracted from that eleva- tion will give the elevation of the point where the rod is held, referred to the same base. Conversely, if the elevation of the point where the rod is held is known, and it is required to find the elevation of the line of sight, it is done by adding the rod reading to the elevation of the point. While there are many details to be considered, such as the curvature of the earth, the adjustment of instruments, atmospheric conditions, etc., the above contains the essential principle of leveling. INSTRUMENTS. 30. General description of level. Any instrument used for the purpose of securing a horizontal line of sight may be called a leveling instrument; or, as is more usual, simply a level. There are three comparatively common forms of levels, shown in Figs. 18-20. 40 Fie;. 111. INSTRUMENTS. 43 Fig. 20 is a precise level, or level of precision. Fig. 18 is known as a Y level. Fig. 19 is a dumpy level. The most common of these is the Y level, so called because the telescope rests in Y-shaped supports. The instrument consists essentially of a telescopic line of sight and an attached bubble tube, whose axis may, by adjustment, be made parallel to the line of sight, so that when the bubble is in the center of its tube it will be known that the line of sight is horizontal. These are com- bined with a leveling head which contains the vertical axis, and screws on to a tripod. A sectional view of a Y level is shown in Fig. 21. The dumpy level is so called because of its short telescope with large aperture. The precise level is simply a modification of the Y level, so improved as to make it capable of doing work to a greater degree of precision than can be obtained by the use of either the Y or the dumpy level. The dumpy level is sufficiently precise for all work that does not require the precise level, and it is considerably cheaper than a Y level of the same make. From the standpoint of the optician, the Y level is the more perfect instrument, because of its many easy adjustments ; but this very feature is to some extent an undesirable one .from the standpoint of the engineer, who wants, for all ordinary work, an instrument with few parts to get out of adjustment. The dumpy level can not be so easily and exactly adjusted for col- limation as the Y level, but, as has been before stated, it is sufficiently precise for all work not requiring a precise level. It is used almost altogether by English engineers, having been invented by an Englishman named Gravatt, whence the level is frequently called the Gravatt level. 31. Telescope. 1 The telescope of the level consists of a bar- rel in which slide two tubes. One of these tubes is the eye- piece tube carrying the eyepiece lenses LLLL, Fig. 21, and the other is the objective tube carrying the objective, or object glass 0. The objective tube is moved in and out by means of a pinion, which works in a rack attached to the sliding tube. The tube is made to move in the axis of the barrel by having 1 For a discussion of the principles of telescopes see any good book on Physics. 44 LEVELING. it pass through the ring at 6 Y , which is accurately centered in the barrel. The eyepiece tube moves in and out of the barrel at the other end in a similar way, and is centered by the ring shown at AA. Instead of a rack and pinion movement, the eyepiece should be moved by turning it around a small screw extending through the barrel and into a helical slot in the eyepiece tube. This is a better plan than the rack and pinion arrange- ment, because the eyepiece is but seldom changed, and when once set should not be easily disturbed. Some instruments, however, have the rack and ~"Y-. pinion movement, and it i] is preferred by some sur- veyors. In addition to these two tubes there is at R a ring (shown sep- arately in perspective in Fig. 22) which carries two fine wires, one verti- cal and one horizontal. These wires are either spider lines or fine plati- num wires. The spider lines are more common. This ring is centered in the barrel by means of the screws at BB. There are four of these screws, called capstan-headed screws, arranged as seen at Win Fig. 18. If the ring is to be moved to the right, the screw on the left is first loosened, and then the screw on the right is tightened, thus drawing the ring over to the right. Similarly for the vertical movement of the ring if the ring is to be moved up, the lower screw is first loosened and the upper screw is then tightened, and vice versa. The Fia. 21. FIG. 22. INSTRUMENTS. 45 difference between this telescope and the ordinary or Galilean telescope, is in the introduction of these wires and a difference in eyepiece necessitated by them. In the ordinary telescope, sucli as is found in field glasses, there are no wires and it is impossi- ble to say to what particular point in an object looked at the axis of the telescope is directed. Moreover, it is useless to put such wires in a field glass, since no image of the object looked at is formed in the tube of the telescope. With an angle-measuring instrument, or any instrument that must be pointed to a definite point, it is indispensable that the exact point to which the axis of the telescope is directed be known. It is perhaps inaccurate to say that the direction of the axis must be known, for any other fixed line in the telescope would do as well, provided the adjustments hereafter to be described could be made with that fixed line. It is more convenient to have the fixed line at least very close to the axis of the tele- scope, for reasons that will appear. The imaginary line joining the optical center of the object glass and the intersection of the cross wires is known as the line of collimation, and this is the line that is directed to the precise spot toward which it is desired to point the telescope ; or rather, in the level, it is the line that indicates the point towards which the telescope is directed. The area seen at one time through the telescope, or rather the angle between the rays of light from the extreme edges of this area, measured at the instrument, is known as the field of view. This field of view is larger as the magnifying power of the telescope is smaller, and varies from about one and one half degrees to about fifty minutes. The former is for the commoner kinds of surveyor's transits, and the latter corresponds to a magnifying power of about thirty-five diam- eters, or about what is found in the better leveling instruments. An image of an object within the field of view is formed at a point back of the object glass, and the glass is moved in or out till this image falls in the plane of the wires. This is called focusing the objective. The point in the image covered by the intersection of the wires is that toward which the telescope is directed. It would be practically impossible to tell when the focusing 46 LEVELING. had been done, were it not for the eyepiece, which is nothing more nor less than a microscope with which to obtain a magni- fied view of the wires and the image formed by the objective. This is done by first focusing the microscope eyepiece till the wires are plainly seen. Then, any other objects in the same plane with the wires will be seen at the same time, and objects not in that plane can not be distinctly seen, and hence it may be determined when the image formed by the objective is in the plane of the wires, and also what point in the field of view is covered by the intersection of the wires. The objective will need to be focused anew for each differ- ent object looked at, unless, as will rarely occur, all the ob- jects viewed are at the same distance. The eyepiece, on the other hand, since it has to be focused for only one distance, need be changed only for different individuals ; and hence, if only one person is to use the instrument for a long time, the eyepiece may be focused once, and not again disturbed during the time it is used by this person. Telescopes should be corrected for spherical aberration and should be achromatic. 1 The eyepiece shown in Fig. 21 has four "lenses, and is com- monly known as an erecting eyepiece. The image formed by the objective is inverted, and the eyepiece inverts the image so that the object appears right side up or erect. This eyepiece has been generally used in American surveying instru- ments because of a supposed difficulty in the use of one that shows the object inverted. Such an eyepiece, shown in Fig. 23, has two lenses less than the erecting eyepiece, and consequently absorbs less light and secures better definition of the object viewed. It is to be preferred for all surveying instruments, and is well-nigh indis- pensable for some kinds of work ; for instance, for stadia measurements, to be hereafter described. The inconvenience 1 Students unfamiliar with these terms can find their meaning in any good dic- tionary or encyclopaedia, in any text-book on Physics, or in Baker's "Engineer's Surveying Instruments." FIG. 23. INSTRUMENTS. 47 attending the use of a telescope that shows objects inverted is largely imaginary. A few hours with an inverting glass makes its use as natural as the use of an erecting glass. Dumpy levels and precise levels are almost -always inverting instruments ; while the Y level, the most commonly used in this coun- try, is almost always erect- ing. As has already been stated, there now seems to be little reason for the existence of the Y level. 32. Leveling Rods. There are three common patterns of leveling rods and an innum- erable number of uncommon ones. The three common patterns are shown in Figs. 24, 25, and 26. Each of these is made in two pieces about seven or less feet long. The New York rod (Fig. 24) and the Philadelphia rod (Fig. 25) differ in that the Philadelphia rod is so gradu- ated as to be easily read at ordinary distances by the lev- eler, while with the New York rod the target must be set and the reading taken and called out by the rodman. The tar- get of the New York rod is provided with a direct vernier, usually placed below the cen- ter of the target. This causes some confusion to the begin- ner, who has been taught to read the scale at the zero of Fm - ** FlG - w - FlG " 2G ' the vernier and the fractional reading by looking along the 48 LEVELING. vernier in the direction of motion for the coinciding line. There need be no confusion if the rodman remembers that it is the center of (he target that is set by the leveler, and not the zero of the ver- nier. A little study of the vernier will show him that the main scale is read oppo- site the ten of the vernier and the fractional reading by noting the number of mjm "( | m the vernier line coinciding ^* M.\ with a division of the scale. ^^| This rod is graduated to j^BH ""^ Wf hundredths of a foot, and ' ' the vernier permits read- 8 ings to thousandths. When M^\ il reading greater than 6.5 feet is required, the target is clamped at 6.5 and the rod extended. There will be found on the side a sec- ond vernier, which, when the rod is closed, reads 6.5 feet and which gives the readings when the rod is extended. This vernier is of the usual construction. The Philadelphia rod target has no vernier, but a tenth of a foot is divided into half-hundredths, per- mitting a direct reading of 0.005 foot and by estima- tion to 0.001 foot. The target is set at seven feet for greater readings than seven feet, and, if the rod is to be read by the leveler, it is extended to its full length, the A FIG. 27. 140 INSTRUMENTS. 49 graduation then being continuous from bottom to top. If the tar- get is to be set, the rod is extended until the target is in the line of sight; it is then clamped, and the rod is read by the rodman by means of a scale on the back. The target of the Boston rod (Fig. 26) is fixed to the rod. Read- ings are all obtained by extending the rod. It is held with the target end down, for readings less than 5.5 feet, and is inverted for read- ings greater than this. It is read altogether by vernier, the scales and vernier being on the sides. It is read to 0.001 foot. It is the lightest, neatest rod of the three, and the least used. The Phila- delphia rod, which is the heaviest of the three, is the most used be- cause of the fact that it may be quickly read by the leveler. In the great majority of sights, read- ings are taken to 0.1 foot only; and, on turning points (hereafter de- scribed), it is usual to read to 0.01 foot only. Readings to 0.001 foot are required in but a very small percentage of work done with the level, even on turning points or bench marks. For this reason many engineers prefer a " self-reading " rod without target, and made in one piece. Such a rod, the standard of the Lake Shore and Michigan Southern Railroad, is shown in Fig. 27. It will be noticed that the figures are so made as to mark the divisions into hundredths. With a target rod, much better work may be done if the target is painted with diamond-shaped figures instead of with quadrants, as is customary. The target may be set more pre- cisely if the wire has a sharp angle to bisect or sharp point to K'M'D SURV. 4 FJQ. 28. . T. 50 LEVELING. cover, than if it is to be made coincident with the edge of a dark surface. Professor Baker finds that at 300 feet the error of setting a quadrant target may be about 0.002 of a foot or more while that of setting a diamond-shaped target may be a little over 0.001 of a foot. Such a target is shown in Fig. 28. USE OF THE LEVEL. 33. Adjustment. The level has been said to be an instru- ment for securing a horizontal line of sight. It will be evident from the construction of the ordinary leveling instruments that this may be accomplished with those instruments if the line of collimation and bubble axis are made parallel ; because, if this condition exists, and the bubble is brought to the center of the tube, the line of sight will be horizontal. This introduces the idea of adjustment, and the adjustment of the level consists essentially in making the line of collima- tion and the bubble axis parallel. There are other adjustments for convenience, but this is the only necessary one. The gen- eral discussion of the adjustments will be deferred till the use of the adjusted level in doing simple leveling has been explained. 34. Setting up. To "set up" the level is to place it in position for leveling, including making the line of sight hori- zontal. The level is an instrument that is rarely set "online," except in making certain adjustments. It is placed in that position that will command the greatest possible number of points whose elevations are to be determined. To set up, plant the legs firmly in the ground with the leveling plates approximately horizontal. Focus the eyepiece on the wires. Bring the telescope and attached level over one set of diagon- ally opposite leveling screws and, by the screws, bring the bubble to the center of its tube. Perform the same operation over the other set of screws. This will to some extent disturb the former work. Therefore turn the telescope again over the first set of screws and relevel ; again over the second set, etc.. till the instrument is level over both sets. If the instrument is in adjustment, the line of sight will now be horizontal in whatever direction it is pointed. USE OF THE LEVEL. 51 35. Differential leveling. To determine the difference in elevation between two points, both of which are visible from a possible position of the level, set up the level in a position such that a rod held on either point will be visible. Turn the telescope toward one point and read a rod held there by a rod- man. The rodman will then carry the rod to the other point, and the telescope will be directed toward that point and the rod read. The difference in readings will evidently be the difference of elevation required. Care must be taken to see that the bubble is in the center of its tube when each reading is taken. If the elevation above some base surface of one of the points is known, the difference of elevation applied to the known elevation gives the elevation of the second point. This operation is capable of further analysis, thus : The rod reading on the point of known elevation, added to the known eleva- tion, will give the elevation of the line of sight, and is there- fore called a plus sight. The r.od reading on the second point subtracted from the elevation of the line of sight will give the elevation of the second point. This reading is, there- fore, called a minus sight. From these considerations the fol- lowing definitions are formulated : A plus sight, or reading, is any reading taken on a point of known or assumed elevation for the purpose of determining the elevation of the line of sight. A minus sight, or reading, is any reading taken for the purpose of determining the elevation of the point on which the rod is read. A very bad custom of calling plus sights, "backsights," and minus sights, "foresights," has prevailed in the past. It has been a source of confusion to the beginner and is illogical. It probably arose from the fact that the work in leveling is considered to proceed from the point of known elevation toward the point of unknown elevation, and that, therefore, plus sights are taken in a backward direction, and minus sights in a forward direction. This is not always true, as will appear later, and hence the nomenclature, " backsight " and " foresight " is ill-chosen. It is, moreover, true that when a minus sight is taken in what may be considered a backward 52 LEVELING. direction, the beginner becomes confused and applies the wrong sign. Hence the terms should be abandoned. If consideration of curvature is neglected, the instrument should be set midway between the two points, in order to do correct work. It will be apparent that if this is done, the amount of the curvature of the earth, for half the distance between the two points will be added in the plus sight, and subtracted in the minus sight ; that is, each reading will be too great by the curvature. The amount of this curvature is about 8 inches in one mile and varies with the square of the distance. The student may determine the effect on a rod reading when the rod is held 528 feet from the instrument and when held 279 feet away. Three hundred feet is about as great a distance as will permit a definite reading of the rod with the average level ; though in work requiring no great exactness, much longer sights may be taken. If the points whose difference in altitude is required are so located that rod readings can not be had on both from one posi- tion of the instrument, an intermediate point is chosen that may be used with the first one, and the readings are taken on the first and on the intermediate point. The difference in altitude between the first and intermediate point is thus obtained. The level is then moved to a position between the intermediate and final point, and their difference of altitude is determined. The two differences added or subtracted, as the case may be, will give the required difference. It may be neces- sary to introduce several intermediate points. The work is sim- ply a succession of operations like those of the first case. It is unnecessary to determine the differences of altitude of each set of intermediate points, as the difference between the sum of all the plus sights and the sum of all the minus sights will give the difference of altitude of the first and final point. The student should show this. The intermediate points that are chosen should, if the work is to be well done, be firm, definite points, as the projecting part of a firm rock, the top of a peg firmly driven in the ground, etc. When extensive differential leveling operations are to be carried on, requiring close work (as the careful determination USE OF THE LEVEL. 53 of the altitude of an observatory or other point involving the carrying of levels over many miles and the introduction of many intermediate " turning points "), it is well for the rod- man to carry a " point " with him. A very convenient form is shown in Fig. 29. This is made of a triangular piece of boiler plate about three sixteenths of an inch to one quarter of an inch thick and about five or six inches on a side. The three corners are bent down to form, as it were, a three- legged stool, and a round-headed rivet is set in the center. A small hole is drilled in one side, in which to fasten a string or chain. When used, the points are firmly pressed into the ground with the foot, and the rod is held on the rivet. In some work, notably railroad leveling, this kind of point is not suitable, because it is well to leave every turning point so that it can be again found. In leveling down or up a steep hill, the distance from the instrument to the rod, called the length of the sight, may be greater or less for minus sights than for plus sights. This may be avoided by zig-zagging. Distances on each side of the instrument are made nearly enough equal by pacing. The form of notes that is kept in differential leveling is as simple as the work. There are three vertical columns, one for the name or number of the point observed, one for the plus sights, and one for the minus sights. The readings, both plus and minus, taken on any point, should appear in their proper columns opposite the number of that point. 36. Profile leveling. A profile of a line laid out on the ground is the bounding line of a vertical section that includes the line whose profile is desired. It shows the elevations and depressions along the line. Profile leveling differs from differ- ential leveling in that the elevations of a number of points along the line whose profile is required are obtained from a single setting of the instrument. The principle is the same as that of differential leveling, but the method of keeping the notes and of doing the work is a little different. 54 LEVELING. The line whose profile is required FIG. 30. is first marked out on the ground by stakes or other marks placed at such intervals as may be necessary. These in- tervals are usually reg- ular, and in railroad surveys are generally one hundred feet. In city streets the interval may be fifty feet. In other surveys the inter- val may be less or more, according to the nature of the survey. The object is usually to re- produce to scale on paper, the profile de- sired. For this purpose profile paper is gener- ally used, on which the notes are plotted, as will be described later. Fig. 30 shows a map of a line whose profile is desired, which may be assumed to be the center line of a road. It also shows in exaggerated form, and to no scale, the position of the level along the road, both in plan and elevation. The process of lev- eling is as follows : Some convenient point is chosen as a " bench mark," either USE OF THE LEVEL. 55 because its elevation above some accepted datum is known, or because it is a convenient permanent point whose elevation may be arbitrarily assumed. A bench mark in leveling is a perma- nent point of known or assumed elevation from which leveling operations may proceed. In the example given, the B. M. is the corner of the water table of a building, and its elevation is assumed to be 1000.000 feet above some arbitrary datum sur- face. The elevation of the B. M. should be so assumed as to avoid any minus elevations ; that is, the datum surface should LEVELS ALONG GRAFTON ROAD, QCACKENKILL TO CROPSETVILLE. LEVELEK TOUCEDA. ROD HIGGINS. March 24, 1895. Sta. +3. II. I. -S. Elev. B.M.No.l. 0.462 1000.4C2 1000.000 On the top of the water table at the 0.6 999.8 S.E. cor. of the brick store of P. Gosling. 1 2.2 98.3 Elev. assumed. 2 4.3 96.2 + 50 5.5 95.0 3 5.2 represent the latitude, there results at once from the principles of trigonometry the relation sine pole distance sine azimuth at elongation = : ; cosine The declination of Polaris is constantly increasing, and will continue to increase until the star is within about thirty min- utes of the pole, when it will begin to decrease. Table IV. Appendix, page 364, gives the polar distance of Polaris for a number of years. The latitude of the place may be taken from a good map, an error of a degree of latitude involving in the territory below 50 north latitude an error of 2' in latitude 50, 1-*-' in latitude 40, 1' in latitude 30, and J' in latitude 20. The calculation of the time of elongation involves a knowl- edge of astronomical terms that it is not thought wise to include here, and hence the times of elongation are given in Table VI. Appendix, page 365, for the year 1897, with rules for determining the time for other years and other latitudes than those for which the table was computed. The method of determining the meridian just given is suffi- ciently precise for compass work. A more precise method will be given in the discussion of the transit. As has been said, almost all of the original surveys made in the eastern part MAGNETIC DECLINATION. 93 of this country were made with the magnetic compass. Many of them were made and recorded with reference to the mag- netic meridian, without recording the magnetic declination at the place at the time of the survey ; and hence, if it were re- quired to retrace one of the old surveys, all but one of whose corners had been lost, it would be found to be a difficult thing to do. The method of procedure will be discussed further on. 79. Local attraction. The needle may be drawn from its mean position for a given locality by the attraction of large or small masses of iron or other magnetic substances. These may be either minerals hidden in the ground or manufactured arti- cles, as agricultural implements in a barn near by, railroad rails, nails in an adjacent house, etc. The chain used in the survey may influence the needle if brought too near, while keys in the pocket of the observer, or steel wire in the rim of a stiff hat, and steel button molds on a coat have been known to give trouble. Local attraction may be very troublesome, and must be carefully looked out for. It may be discovered and allowed for by simply occupying every corner of a survey, and reading the bearing of each line back as well as forward. It will be evident that at any one station where the compass is set, the local attraction, if any exists, will affect by the same amount all readings taken from that point ; and hence, if the bearings of two lines meeting at a point are both determined at that point, the angle of intersection is correctly deduced, even though there is local attraction and neither bearing is correctly deter- mined. If the bearing of a line is determined from each end, and it is found that the two readings are not numerically the same, and it is certain that the bearings are correctly read, there is probably local attraction at one end or the other, or possibly at both ends. If now another line is set off from one end of the first line, and the bearing of the new line is read at both ends, and the readings are found to agree, the local attraction affected the reading of the first line at the other end. If trouble is en- countered on the second line, a third line may be set off from the other end of the first line, the same observations being 94 DIRECTION AND ANGLES. made as on the second line. If the auxiliary lines are properly chosen, one will usually be sufficient. If it is feared that local attraction exists, the best method is to read the bearings of all lines at each elid, determining the angles be- tween adjacent lines from the bearings taken at their points of intersection. One line may then be assumed to be correctly read, or the true bearing may be determined, and the bearings of the others may be computed from the series of angles. 80. Special forms of compasses. Some compasses are made with an additional full circle of 360 and a vernier reading to minutes, with suitable clamps and tangent screws ; but it FIG. 45. may be said that the precision obtained in pointing with the open sights is hardly sufficient to warrant this. If angles are required to be read to single minutes, it is better to use a transit. The prismatic compass is a very convenient instrument to use on exploratory work. It has folding sights, and, by means of the prism, may be read while being pointed, which is a convenience when the instrument is held in the hand in- stead of being mounted on a Jacob staff or tripod. It is usually held in the hand, though it may be used either way. The in- strument is shown in Fkf. 45. THE TRANSIT. 95 THE TRANSIT. 81. Description. The instrument most used by surveyors and engineers for measuring horizontal angles, and, with cer- tain attachments, for measuring vertical angles and distance, and for leveling, is called a transit. The transit consists of a telescope attached to a pointer which may be moved around a graduated circle. There are suit- able attachments for controlling the motion of the telescope and the pointer, and for enabling the graduated circle to be made horizontal. The pointer may be clamped to the gradu- ated circle, and they may be revolved together. If this is done, and the telescope is pointed to a given object, and the graduated circle is clamped so that it will not revolve, and if the pofhter is then undamped from the circle, the pointer and telescope may be turned together till the telescope points to a second object. The number of degrees of the circle over which the pointer has passed will be the angle subtended, at the point where the transit is placed, by the two objects seen through the telescope. The pointer is the zero mark of a vernier. There are usu- ally two double verniers placed 180 apart. In the transit shown in Fig. 46, T is the telescope which may, by means of the horizontal axis A, be revolved in a vertical plane. The horizontal axis rests in bearings at the top of the stand- ards X, which are rigidly attached to the circular plate that carries the vernier V. On this plate are set two level tubes which, when adjusted to be parallel to the plate, will show whether the plate is level. This plate is made by the maker perpendicular to the axis on which it revolves, and hence, when the plate is level, the axis, of revolution is vertical. This axis is conical, and fits inside a conical socket which is the inside of the conical axis of the plate that carries the graduated circle. This latter conical axis re- volves in a socket that connects the top and bottom plates of the leveling head. The upper plate is leveled by the leveling screws L. The lower plate of this leveling head screws on to the tripod. The clamp C fastens together the vernier plate 96 DIRECTION AND ANGLES. (sometimes called the alidade^) and the plate carrying the graduated circle. The latter plate is called the limb. When the two plates are clamped together, the vernier plate may still FIQ. 46. be moved a small amount, relatively to the limb, by means of the tangent screw S. This is for the purpose of setting the vernier at a given reading, or pointing the telescope at a point, THE TRANSIT. 97 more precisely than would be ordinarily possible by simply turning the alidade by the hand. The collar K surrounds the spindle of the limb, and by the clamp screw C' may be fastened to that spindle. The lug M, which is attached to the collar, being held by the spring in the barrel P and the opposing screw /S", in turn fastened to the upper leveling plate, prevents the revolution of the limb, when the collar is clamped to its axis. The limb may, however, be moved a small amount by the tangent screw S' working against the pressure of the spring in the barrel P. Some instruments when put away in the box in which they are kept when not in use, are separated into two parts. The upper part of the instrument is separated from the leveling head, and the two parts are placed separately in the box. Others are so made as to be put away in one piece, being merely unscrewed from the tripod and screwed to a board that slides into the box. There are various forms of transits and various patterns for the different parts, according to the ideas of the different makers. A careful examination of an instrument should be made before putting it to use, that the user may become per- fectly familiar with its construction. The circles are usually so graduated that angles may be read to minutes. Many instruments are graduated to read 30 sec- onds, some to read 20 seconds, and a few to read 10 seconds. For ordinary land surveying, single minutes are sufficiently precise, while for fine city surveying 20 seconds, or even 10 seconds, is not too fine. The telescope is essentially the same as that used in the level, but is shorter and usually of lower magnifying power. It should be inverting, but is usually an erecting glass. The use of the inverting glass is growing. There is a proper rela- tion between the magnifying power of the telescope and the least count of the vernier used ; and it is not necessarily the best instrument that has the smallest count vernier, for the rea- son that the magnifying power of the telescope may be such that a movement of several of the smallest units that can be read by the vernier may give no perceptible lateral motion to the line of collimation. The power of the telescope and the II'M'D suuv. 7 98 DIRECTION AND ANGLES. least count of the vernier should be so adjusted to each other that a barely perceptible movement of the vernier will cause a barely perceptible movement of the line of collimation. The same is true of the magnifying power and the level under the telescope. Methods of testing these relations will suggest themselves. The verniers are usually read with small magnifying glasses known as reading glasses. In order that the transit may be set precisely over a point on the ground, there is fastened to the center of the lower part of the leveling head a ring or hook, from which may be sus- pended a plumb line. A sectional view of the lower part of the transit in Fig. 46 is shown in Fig. 47. It will be seen that the FIG. 47. upper leveling plate U in some forms four arms instead of a plate is attached to the socket in which the vertical axis re- volves, and that to this upper plate are attached the nuts in which the leveling screws work. These leveling screws rest in little cups, that in turn rest on the lower leveling plate L. The ball and socket jpint that permits the leveling of the instru- ment is separate from the lower leveling plate, and is not quite so large as the hole in the center of that plate. There is an extension plate to this ball and socket joint that extends under the lower leveling plate. When the leveling screws are tight- ened, this extension plate binds against the leveling plate, and THE TRANSIT. 99 the instrument can not be shifted on the lower plate. If the leveling screws are loosened, the upper part of the instrument may be shifted laterally on the lower plate by an amount de- pending on the relative diameters of the ball joint and the hole in the center of the lower plate. The device is called a shifting center and is found in all modern transits. 82. The tachymeter. Such a transit as has been described, while still in very general use, is being rapidly displaced by the complete engineer's transit, or tachymeter. This instru- ment consists of such a transit as has been described, with the following attachments : (1) A level under the telescope, making the transit a good leveling instrument. (2) A vertical circle or arc, by which vertical angles may be read. (3) A gradienter attachment by which very small vertical angles may be set off, as so many feet rise or fall in one hundred. (4) A pair of horizontal wires in the telescope, known as stadia wires. There are various other attachments for special purposes that will be found described in instrument makers' catalogues, but which it is not necessary to mention here. There are also special forms of transits for use underground. One form of the complete instrument is shown in Fig. 48. The level under the telescope and the vertical circle are readily recognized. The gradienter and slow-motion screw, for vertical motion of telescope, is shown at G. The telescope is clamped in vertical motion by the clamp C, It may then be moved slowly a small amount by the slow-motion screw. The head of this screw is graduated with reference to the pitch of the screw, so that a single revolution of the head corresponds to an angular motion of the telescope of one foot in one hundred, or half a foot in one hundred. In the figure, the head is graduated in the latter way, as is indicated by the double row of figures. The use of the gradienter is chiefly in setting out grades. Many transits are fitted with the slow-motion screw without the gradienter head. All makers make both the plain and com- plete transits. 100 DIRECTION AND ANIJI.KS. USE OF THE TRANSIT. 83. Carrying the transit. While in use, the transit is not removed from the tripod when it is carried from one point to another; but is carried with the tripod on the shoulder of the FIG. 48. " transitman." The plumb bob is carried in the pocket without removing the string from the instrument. Before the transit USE OF THE TRANSIT. ;(). can be properly set up and the angles measured, the various parts must be in proper adjustment. The discussion of the adjustments will, however, be deferred till the use of the ad- justed transit has been described. 84. "Setting up" the transit. To "set up" the transit over a point, let the plummet swing free, grasp in one hand the uppermost leg of the tripod, and plant it firmly in the ground. Next grasp the other two legs, one in' each hand, and place them symmetrically about the point so that the plumb bob covers the point as nearly as may be. The plates should be at the same time observed arid made approximately level. It should be noticed that a sidewise motion of a leg changes the level of the plates without much disturbing the plumb bob, and that a radial motion changes the level of the plates to a less degree, but does disturb the plumb bob. It is necessary to become expert in setting the transit, as much time may be lost by awkward work in setting up. After the transit is set approximately over the point, press the legs firmly into the ground so as to bring the bob more nearly over the point and to make the instrument firmer. Finish centering by the adjustable or shifting center, by loos- ening all the leveling screws and moving the whole upper part of the instrument till it is centered. If there is no ad- justable center, the instrument must be centered by manipu- lating the legs. When the instrument is finally centered, level the limb by the leveling screws. To do this, place one plate bubble parallel to one set of opposite leveling screws; then the other bubble will be parallel to the other set. Level one bubble at a time and note that leveling one will, to a slight extent, disturb the other, which must then be leveled again. If the lower plate is not fairly level, u leveling up " will dis- turb the centering of the plumb bob, which must then be cor- rected. Continue till both bubbles are in the centers of the tubes. Set the vernier that is to be used at the proper reading (this will often be zero), and the instrument is set up. 85. To produce a straight line with the transit. Set the instrument over one extremity of the line, and with the limb free to turn, turn the telescope on the other extremity or on 102 DIRECTION AND ANGLES. a point in the line. Clamp all motions except the vertical motion of the telescope, and, with either of the horizontal tan- gent screws, but preferably with the lower screw, bring the line of collimation to bisect exactly the distant point. Transit the telescope and set a point beyond it at any desired place, in the line of collimation. This point will, if the instrument is in adjustment, be in the straight line produced. If it is feared that the transit is slightly out of adjustment, unclamp the limb, turn the instrument till the line of collimation again cuts the first point, clamp and set precisely, and then transit and set a new point beside the first one set ; the point midway between the two set will be a point in the continuation of the straight line. This is called " double centering," and is the same operation as the test of the adjustment of the vertical wire, to be hereafter explained. The method of ranging out a number of points in a straight line in the same direction will be apparent without explanation. 86. To measure an angle with the transit. Set the instru- ment over the vertex of the angle, with the vernier at zero. With the lower motion bring the line of collimation to bear approximately on one of the distant points, clamp, and with the lower tangent screw make an exact bisection. Loosen the alidade and bring the line of collimation approximately to the second point, clamp, and complete the bisection with the upper tangent screw. Read the vernier. The reading will be the angle sought. This assumes the numbering to extend both ways from to 360. It is, of course, unneces- sary to set the vernier at zero. When the instrument has been set on the first point, a reading may be taken and noted and subtracted from the reading found on pointing the telescope to the second point. In this case the vernier may pass the 360 point, and it will then be necessary to add 360 to the final reading before subtracting the first. 87. Azimuth. The azimuth of a line is the horizontal angle the line makes with a reference line, as a meridian. It differs from bearing, in that it is measured continuously from to 360. If the azimuth of a point is mentioned, there is implied another point and a meridian through it from which the azi- muth is measured. USE OF THE TRANSIT. 103 In making surveys with the transit, except possibly city surveys and others where angles are to be repeated several times, it is better to use azimuths than bearings. It is neces- sary, however, in writing descriptions of property, to reduce the azimuths to bearings. It is customary to reckon azimuths from the south point around by way of the west, north, east, south, 360. This is the practice of astronomers and geodesists. It is believed to be more convenient for the surveyor to begin with zero at the north point and read to the right 360. The reason for this will appear in Chapter VI. 88. Traversing. The method of traversing with the transit is as follows : In Fig. 49, let it be required to determine the FIG. 49. lengths and azimuths of the courses of the crooked road A, B, (7, Z>, etc. It will be assumed that the magnetic meridian SN is the meridian of the survey. Set the transit over a tack in a stake driven at A, and, with the vernier set at zero, turn the telescope by the lower motion in the direction AN as denned by the needle, and clamp the lower motion. For the purpose of again using this line, if necessary, set a stake in the line AN, some distance toward N, and put a tack in it in the exact line. This is necessary because the needle would not give the direction twice alike to the nearest minute, which is desired in transit work. Unclamp the alidade and turn the telescope toward a point in the first turn of the road, as B. Clamp the alidade and set a stake at B. In most transit work all stakes marking stations to be occupied by the transit are " centered " by setting a tack in the precise line. Read and record the vernier and the needle. They should agree within the limit of precision of the needle reading. The angle read is that which has been turned to the right, and in the figure is greater than 270. (The mistake of reading the wrong circle must be 104 DIRECTION AND ANGLES. avoided. The needle reading may be recorded as bearing or may be mentally reduced to azimuth and so recorded. If the compass box is graduated continuously 360, the needle will give azimuths at once.) Measure AB, and, while this is being done, take the transit to B and set up over the tack that has been put in the stake. With the same vernier used at A (let it be called vernier A) bring the instrument to read the azi- muth AB plus 180, which is the "back azimuth" of AB. Point the telescope to A by using the lower motion, clamp, and make the exact pointing with the lower tangent screw. By so doing the instrument is oriented; that is, the zeros of the limb are made parallel to their positions at A. (The telescope has not been transited, and, if the alidade is now undamped and the telescope turned in azimuth just 180, the line of sight will be in AB prolonged, the vernier will read the azimuth of AB, and the zeros of the limb will be seen to lie in the meridian of the survey. This intermediate step is not usually taken separately in practice, but is intro- duced here for the clearer understanding of the method.) With the alidade undamped, turn the telescope in the direction of the next point, as (7. Clamp the alidade, set a stake and tack at (7, and read and record the vernier and needle. The reading of the vernier will be the azimuth of BC referred to AN, and the needle should read the same or the correspond- ing bearing. Measure BC, take the transit to (7, set the vernier at the back azimuth of BO, and proceed as at B. Thus con- tinue the work till the traverse is complete. 89. Transit vs. compass. The form for the field notes of the survey just outlined is the same as for a compass traverse, except that azimuths instead of bearings are recorded and another column is used in which to record the needle readings as a future evidence of the correctness of the work. Any land survey may be made with the transit, the object being, of course, the same as if made with the compass. The essential difference is that the work is done with a greater degree of precision with the transit and azimuths are used instead of bearings. One difference that should also be noted is this : USE OF THE TRANSIT. 105 the use of the compass makes every line independent of every other line, so far as direction is concerned, since the direction of each line is determined independently by the needle. With the transit this is not so. An error in one line is carried through the remainder of the survey ; and, being an angular error, the error of position of the final point of the survey is much greater than it would be if there were an error in the direction of bat one line of the survey. It may be said, however, that with the needle check always applied, there is very little probability of an error of this kind, and the use of the transit is advised in all work of importance. Where speed and roughly approximate results are the chief requisites, the compass is the better instru- ment to use, unless the measurements may be made with the stadia, in which case the transit still remains the better tool. The two may be combined. The directions may be determined with the compass attached to the transit, and the distances read with the stadia, thus securing a maximum economy of time. In this case the stakes would not be centered with tacks, one or two inches making little error in such work, since the compass may not be read to less than five minutes, and this by estimation. Five minutes of arc means 0.15 of a foot in one hundred feet. More- over, the distances, if read by the stadia, may not be determined closer than the nearest half foot or possibly the nearest foot. 90. Determining the meridian. In many surveys it is not necessary or common to determine the meridian before the survey is made. One line of the survey, usually the first run, is assumed as a meridian, and the direction in which the survey proceeds along this line is assumed as zero azimuth. The azimuths of all lines of the survey are then determined with reference to this one line. If, for any reason, it is desired to know the true bearing or azimuth of the lines of the field or survey, an observation for the meridian is made, and, when the meridian is determined, the angle that it makes with one side of the survey is found with the transit, and all the azimuths are corrected by this angle to make them read from the true meridian. The method of determining the meridian with the transit is the same as with the compass, except for the differ- ence in pointing. 106 DIRECTION AND ANGLES. The time of the elongation of Polaris is determined, and a few minutes before that time the transit is set up over a stake in an open space where the star may be observed. As this observation is usually made at night, it is necessary to make some provision for illuminating the wires in the transit so that they may be seen. This is usually done by throwing a faint light into the object end of the telescope by holding a piece of paper a little to one side and before it, and a lamp a bull's eye is best behind it so as to get a reflection of light from the paper in the telescope. It is troublesome to do this, and it takes two or three observers. A better way is to have a re- flector like one of those in Fig. 50, which fits on to the object end of the telescope and is illuminated by a lamp held back of the observer. A still better way, when the instrument is made for it, is to have the horizontal axis FlG - 50 - of the transit hollow, with a small mirror near the center of the telescope. A small bull's- eye lamp is placed on a stand that is fastened to the standards and throws a beam of light upon the mirror, by which it is reflected to the wires. The light must not be too strong or the star will be indistinct. This is particularly true when, as may be the case, another and less bright star than Polaris is used. The transit being set up and provision being made for illu- minating the wires, the telescope is turned on the star and both plates are clamped. The vertical wire is then made to cover the star as nearly as possible and is made by the tangent screw of the alidade to follow the star as it seems to move to the right or left, that is, in azimuth, until it seems to be stationary in azimuth and to be moving only vertically. The telescope is then plunged and a point set in the ground some distance away and left till morning. The setting of the point requires some patience. Perhaps the best way to accomplish this is to provide a box open on two sides. Cover one side with thin tissue paper, and place a candle in the box. This improvised lantern may be approxi- mately set from the transit in the proper line, and the point to USE OF THE TRANSIT. 107 drive the stake may be determined by holding a plumb line in front of the box, the papered side being turned toward the transit. This line will be put in position from the instrument and the stake driven. The stake will then be centered by the use of the line, and a tack driven. On the next day at about 10 o'clock the transit is set over the point occupied the previous evening; and the azimuth of Polaris at elongation, which has been previously computed as explained on page 364, is turned off from the line of stakes to the right, if western elongation has been observed, and to the left, if eastern elongation has been observed. Another stake is now set in the line thus determined and the line denned by the point occupied by the transit, and the last stake will be the true meridian. With the instrument set on this line the reading of the needle will give the magnetic declination in amount, but with the opposite sign. That is to say, the needle will read the mag- netic bearing of the true meridian ; and if the magnetic meridian lies west of the true meridian, thus making the declination west, the needle will read the bearing of the true meridian as east. 91. Needle checks on azimuths. In making surveys, if the compass box is not provided with a declination plate or ver- nier, on which the declination may be set off so that the needle will read north when pointing in the true meridian, it is often convenient to make the meridian of the survey, the magnetic meridian of the place, to facilitate checking by the needle the angles measured with the transit. It is inconvenient to do this even then, because the compass is graduated from two points 90 each way, while for azimuths, the transit is graduated con- tinuously through 360. This difficulty is obviated by making a similar graduation on the compass box. If one has a transit that is not graduated in this way, he may graduate a paper ring and paste it on the glass cover of the box. The gradua- tions need be only to degrees. It will be found that such a check on angle measurements is very valuable. If the ring has been properly set, and if the magnetic meridian has been chosen as the meridian of the survey, the reading of the compass should always agree practically with that of the transit. If, instead of the magnetic meridian some arbitrary meridian has 108 DIRECTION AND ANGLES. been assumed as the meridian of the survey, the paper circle should be placed so that the needle will read zero when the telescope is pointed in the direction of the assumed zero azi- muth, the vernier of the transit reading zero. The needle check is good to the nearest five minutes except where there is local attraction. Some transits have the graduations of the limb numbered as are those of the compass box. This is an old method. It is better to provide continuous numbering on the compass box. ADJUSTMENT OF THE TRANSIT. 92. Requirements. The transit is used for measuring hori- zontal angles, that is, angles subtended at a given point by the vertical planes through the two other observed points. If these two other points are not in the same horizontal plane, the vertical motion of the telescope must be used to bring the line of sight down from the higher point to the horizontal plane of the instrument and the line of sight of the lower point up to the same plane. Now, the line of collimation is the line in the instrument that is directed to the distant point, and the line that is revolved down or up, as the case may be, to the horizontal plane of the instrument, which is the horizontal plane through the center of the horizontal axis. It is evident that this line must revolve in a vertical plane, to properly pro- ject the distant points into the horizontal plane of the instru- ment. It will also be evident that the vertical axis of the transit must be truly vertical in order that the line of sight when projected into the horizontal plane of the instrument and then turned in azimuth, may move in a horizontal plane. If the axis of revolution is not vertical, the lateral motion of the instrument will be in an inclined plane. In order that these necessary conditions may obtain, certain adjustments of the instruments must be properly made. Every adjustment con- sists of two parts : the test to determine the error, and the rectification of the error found. 93. The plate bubbles. If the plate bubbles are perpen- dicular to the axis of revolution, that axis will be vertical when both bubbles are in the centers of their respective tubes. The ADJUSTMENT OF THE TRAXSIT. 109 test is made and the adjustment is performed as described for the compass. If one bubble gets broken, the other may be used for both, by first leveling with the bubble in one position and then in another, 90 from the first, exactly as in setting up the level, and by repeating the operation till the axis is vertical. But two operations would be necessary were it not for the fact that the second leveling will, to some extent, disturb the first. The adjustment being made, all points of the instrument will revolve in horizontal planes if the bubbles are both in the centers of their tubes. This adjustment should be tested every day, although it will probably be found correct for a considerable number of days in succession. If the transit is out of adjustment all around, it is better to make each adjust- ment approximately in order and then carefully repeat them all. 94. To make the line of collimation revolve in a vertical plane when the telescope is turned on its horizontal axis. This adjustment consists of several parts. One of these is to make the axis of revolution of the telescope horizontal. If the in- strument is provided with a striding level, this part may be performed first. If not, as is usually the case, it must be done last, and possibly at the expense of repeating some of those adjustments that have preceded it. It will be assumed that no striding level is used. If the line of collimation is not perpendicular to the axis of revolution of the telescope, it will describe, in revolving, the surface of a cone, whose axis is the horizontal axis of revo- lution of the telescope ; and if the line of collimation is per- pendicular to the axis and the axis is not horizontal, the line of collimation will describe, on being revolved, an inclined instead of a vertical plane. The student should make a mental picture of these conditions. The intersection of the wires should be in the line of motion of the optical center of the objective ; for, if not, then the line of collimation will not be fixed in its position in the telescope tube, and, if made perpendicular to the axis of revolution for one distance, it would not be so for some other distance that 110 DIRECTION AND ANGLES. would require a shifting of the object slide for new focusing. The object slide should move parallel to, if not absolutely coin- cident with, the axis of the telescope tube, in order that the line of collimation may be made perpendicular to the horizontal axis of revolution. Theoretically, its line of motion should pass through the horizontal axis. In the instrument shown in Fig. 48 the objective is permanently adjusted in motion by the maker. In some other instruments, however, this is not the case. The two wires are adjusted separately, and it is not unusual to omit the adjustment of the horizontal wire. This should not be omitted if the instrument is to be used for leveling or for measuring vertical angles. The first adjustment of the vertical wire is to make it vertical. This can be done only approximately, if the horizontal axis has not been previously adjusted. I. To make the vertical wire vertical. Carefully focus the eyepiece, level the instrument, turn the telescope on a sus- pended plumb line, and observe whether the vertical wire coincides or is parallel with it. The corner of a plumb build- ing will sometimes answer the purpose if the wind interferes too much with the swinging plumb line. If the wire is found to be out of plumb, it should be made plumb by loosening all four of the capstan-headed screws that hold the ring carry- ing the wire, and by moving the ring around by the screws till the wire is vertical. The holes by which the adjusting screws pass through the telescope tube are slotted for this purpose. The remaining part of the adjustments will be described as it is considered best to make them in the complete transit or tachymeter. (See Art. 114, page 127.) The adjustments that are to be made when only the plain transit is to be adjusted are those of Art. 93 and numbers I., III., and IV., of this article. II. To make the line of collimation parallel to or coincident with the geometric axis of the telescope. Construct of wood a pair of Y standards fastened to a firm block of wood and with such a distance between them that the telescope tube may, after being removed from its standards, rest in the Y's near its ADJUSTMENT OF THE TRANSIT. Ill ends. Having made the vertical wire vertical as described in the last adjustment, remove the telescope with its horizontal axis from the standards, and rest it .in the Y supports. Now adjust the wires (and the object slide if it is adjustable and needs it) in precisely the same manner as described in the first adjustment of the level. If the geometric axis of the tele- scope has been made exactly perpendicu- lar to the horizontal axis, and the two are coincident where they cross, the line of collimation is now perpendicular to the horizontal axis. The foregoing adjust- ment is, in all good instruments, sufficient for the horizontal wire. After replacing the telescope in its standards, the vertical wire is tested and, if necessary, corrected as explained in III. III. If the line of collimation as de- fined by the optical center of the objective and the vertical wire (which is the wire most used in the transit) is not perpen- dicular to the horizontal axis of revolu- tion, and the instrument is set over the middle one of three points that are in a straight line, the conditions will be as shown in Fig. 51. AB is the axis of rev- olution, Cl the line of collimation, show- ing only the object end to avoid confu- sion ; the latter makes with the axis the angle . P v C, and P 2 , are the three points in a straight line, the instrument being centered over C. XX 1 is a line drawn perpendicular to the axis of revo- lution. The error of perpendicularity is the angle /3 = 90 . The instrument is shown with the line of collimation directed toward the point P r If now the telescope is revolved, the axis of revolution remaining in the same straight line, the line of collimation will take the position CP 8 , making with the line CP V an angle equal to 2 (3. Since the lens can not be moved, the wire must X FIG. 51. 112 DIRECTION AND ANGLES. be moved so that the line of collimation will fall in the line CX found by taking a point halfway between P 3 and P 2 . Since the wire is nearer the observer than the lens, the wire must be moved in a direction opposite to that in which the forward end of the line of collimation is to be moved. With erecting instruments, since points that are really on the left or right appear so, this rule would be followed. With inverting instru- ments, in which an object on the right of the line of collimation ap- pears to be on the left, and vice versa, if the line should apparently be moved to the left, it should really be moved to the right ; and hence the wire should be moved in the direction in which it ap- pears that the correction should be made. If the position of P 2 , and hence also that of X, is not known, the explanation is as follows : If, while the line of collimation is pointing to P 3 , a point is established there, and the instrument is then turned on its vertical axis in the direction indicated by the arrow till the line of collimation again points to P x , the line of collimation and its axis of revolution have been turned in azimuth through an angle equal to 180 -2/8, and would be in the position shown at A S v in Fig. 52. The angle between the new u.iJ. former positions of the axis of revolution will be 2/8. If now the line of collimation is revolved on its axis, it will fall in the line (7P 4 as much to the right of (7P 2 as it was before to the left ; and the angle between its two positions pointing to P 3 and P 4 , will be 4/3; and hence, if P 2 had not been established, but only P x , (7, P 3 , and P 4 , the instrument would ADJUSTMENT OF THE TRANSIT. H3 be corrected by shifting the wire till the line of collimation should fall on JT 2 one fourth the distance from P 4 to P y It should be noted that unless the axis of revolution has been made horizontal, the points P v P 3 , and P 4 should all be in about the same horizontal plane ; that is, comparatively level ground should be used for this adjustment. From the foregoing explanation there results the fol- lowing rule for making this adjustment of the vertical wire, or, as it is usually called, the adjustment of the line of collimation : Set the instrument over a point in a comparatively level stretch of ground. After leveling, establish a point about two hundred feet in one direction and turn the line of collimation on this point, clamping all motions except the vertical motion of the telescope. Transit the telescope and set a point in the opposite direction, directly in the line of collimation. Loosen one clamp for motion in azimuth and turn the instrument in azimuth till the line of collimation cuts the first point, and clamp all motions except the vertical motion of the telescope. Transit the telescope, and set a point by the side of the sec- ond point. If the adjustment is perfect, the second and third points will coincide. If they do not, move the vertical wire to one side, loosening first one screw and then tightening the opposite, till the line of collimation cuts a point one fourth the distance from the third point toward the second. Repeat the test for a check. It will usually be found necessary to perform the opera- tion more than once. All bisections should be made with the greatest possible precision, using the clamps and slow- motion screws of all motions except the vertical motion of the telescope. The horizontal wire will probably not be appreciably dis- turbed by this adjustment, but may be tested as before. If it is found necessary to correct it, the vertical wire must again be tested as above. If the instrument is to be used for leveling or for reading vertical angles, it is just as neces- sary as in the level that the horizontal wire be properly adjusted so that the line of collimation shall be correct for all distances. R'M'D SCRV. 8 114 DIRECTION AND ANGLES. IV. The adjustment of the axis of revolution of the tele- scope is made in any one of several ways. Two will be given. (1) Hang a long plumb line from some tall and firm sup- port, as a second-story window sill. Having set up the transit a short distance, say twenty feet, from the line, turn the line of collimation, as defined by the intersection of the wires, on a point in the line near its upper end, and clamp azimuth motion. Swing the telescope vertically, noting whether the intersection of the wires remains on the plumb line. If not, raise or lower one end of the axis of revolution of the telescope till the inter- section of the wires will follow the plumb line. (2) Set up the instrument near some tall building or other high object, setting the intersection of the wires on some well- defined point near the top, and clamp the azimuth motion. Plunge the telescope downward and set a point on the ground near the building or object. Reverse in azimuth, transit the telescope, and set again on the point near the top and clamp in azimuth. Drop the telescope, and see whether the intersection of the wires falls on the same point as before near the bottom. If it does, the axis is horizontal and needs no adjustment. If not, set a point midway between the two points at the bottom and adjust the axis by raising or lowering one end till the wire will cut that point when plunged from the upper point. The student will be able to make a diagram showing the correctness of these methods and will be able to tell whether the axis is moved in the second adjustment, so as to make the line of collimation cut a point one fourth the distance from the second to the first lower point or one half that distance. Unless this adjustment is very badly out, the vertical wire will not be again disturbed. If thought necessary, the screws may again be loosened and the vertical wire may be made truly vertical, after which the adjustments for collimation must again be tested. The instrument as a transit is now adjusted. 95. Level under the telescope. To use the transit as a level, the level under the telescope must be adjusted by the peg method as described for the leveling instrument in Chap- ter III., adjusting the bubble, not the wire. ADJUSTMENT OF THE TRANSIT. 115 96. Vertical circle. If the transit has a vertical circle, that should be adjusted so as to read when the bubble under the telescope is in the center of its tube, after the last-mentioned adjustment for leveling has been made. For this purpose the vernier of the vertical circle is adjustable. To make the adjustment, loosen the small screws that fasten the vernier to the standard and slide the vernier along as is indicated by the position of the zero of the circle till the zeros of scale arid vernier coincide, the bubble being in the center of its tube. 97. Eyepiece. After the wires have been adjusted they may not appear in the center of the field of view. This is because the eyepiece is not properly centered. There need be no inaccuracy in work done with the transit if this is not corrected, but better seeing will result if it is corrected. This may be done by moving the ring through which the eyepiece slides, just as the wire ring was moved, there being a set of screws for this purpose next back of the wire screws. In the instrument shown in Fig. 48, this adjustment is per- manently made. In such an instrument, the horizontal wire may usually be adjusted with sufficient precision for ordi- nary work by merely bringing it by eye to the center of the field of view. The vertical wire is then adjusted as de- scribed in Art. 94, III., and it is unnecessary to remove the telescope from the standards for the adjustment of the line of collimation. In inverting instruments, in which the field of view is limited by the eyepiece itself, this may not be true unless the eyepiece is in adjustment. It usually is. 98. Eccentricity. There may exist errors of graduation ; but such as are likely to occur in modern machine-graduated instruments cannot be detected by ordinary means. The cen- ter of the graduated circle may not lie in the axis of rota- tion, and the line joining the zeros of the verniers may not pass through the center of the graduated circle. If the latter condition exists, the verniers will not read 180 apart, except possibly at some one point in case the first condition also obtains. The second condition simply means that the verniers are not 180 apart, and no error will result from this cause if 116 DIRECTION AND ANGLES. the same vernier is read for both pointings for the measure- ment of an angle. If the second condition exists and the first does not, the angular distance between verniers will be the same for all parts of the circle ; while, if the first condition exists, the angular distance for different parts of the circle will not be constant. These two facts furnish methods of testing for eccentricity that will be evident to the student who is familiar with the discussion of eccentricity in the compass. THE SOLAR TRANSIT. 99. What it is. The solar transit consists of a transit with an attachment for determining the true meridian by an obser- vation on the sun. The solar transit and the solar compass, essentially the same as the solar part of the solar transit, have been extensively used in laying out the public lands of the United States. The solar compass was invented by Wil- liam A. Burt of Michigan, and has become known as Burt's solar compass. The United States Land Office has specified that the work of subdividing the public lands must be done with solar instruments or transits. The solar compass is not now much used. When a solar instrument is used, it is usually the solar transit. 100. Fundamental conception. Before describing the solar transit, it will be necessary to explain the conceptions on which its action is based. For this purpose let the student imagine a celestial sphere, concentric with the earth and of infinite radius. This is not quite true to fact, but will assist the un- derstanding of the following statements : Let it be imagined that the equatorial plane and the axis of the earth are extended till one cuts from the celestial sphere a circle, called the celestial equator, and the other cuts the celestial sphere in two points that may be known as the north and south poles. Imagine further the meridian plane of the place of the reader extended to the celestial sphere. It will cut from that sphere a meridian circle. Let the earth be con- ceived to be very small as compared with the celestial sphere, so that points on its surface are practically at the center of the THE SOLAR TRANSIT. 117 sphere. If the reader imagines himself to be at the equator, the zenith will be the intersection of the celestial meridian and equator. If he now imagines that he moves north, his zenith point will move north by an angular amount equal to the lati- tude he covers. Moreover, his horizon, which at the equator included the poles, will be depressed below the north pole by an equal angular amount. Hence the altitude of the north pole will at any place indicate the latitude of the place, the angular distance from the pole to the zenith will be the co- latitude ; the angular distance zenith-equator will be the lati- tude, and the angular distance equator-south horizon will be the colatitude. It is well known that, because of the inclination of the ecliptic to the earth's axis, the sun is below the celestial equator for six months of the year and above it for six months. The amount that it is above or below it is con- stantly changing, and the angular distance of the sun from the celestial equator at any moment is known as the declination of the sun for that moment. It is the same as terrestrial latitude. Let it be forgotten for a time that the sun is fixed and that it is the revolution of the earth on its axis that causes the sun to appear to rise in the east and set in the west, and let it be imagined that the sun does the moving just as it appears to do. Then, if the sun were to maintain a constant declination for a whole day, its path in the heavens would correspond to a parallel of latitude above or below the equator by the amount of the sun's declination for the day. On the 21st of June it would correspond to the extension of the Tropic of Cancer, and on the 21st of December to the extension of the Tropic of Capricorn. If a pointer of any kind should be directed from the center of the celestial sphere toward the sun at any time during the day under consideration, it would make with the equatorial plane an angle equal to the declination, and with the polar axis an angle equal to the codeclination. If now this pointer is revolved about the polar axis, keeping the angle between them constant, the pointer will at all times point to some point in the path of the sun for the day ; and if it is revolved just as fast as the sun moves, it will all day point to the sun. 118 DIRECTION AND ANGLES. 101. Description. The solar attachment consists essentially of an axis that is made to be parallel with the earth's axis, and a line of sight (pointer) that is set at an angle to the instru- mental polar axis equal to the codeclination of the sun for the time of observation. FIG. 53. In Fig. 53 the polar axis is marked. It is made at right angles to the telescope tube, and hence if the telescope tube is THE SOLAR TRANSIT. 119 brought into the plane of the equator as shown, and is also in the meridian plane, the polar axis must be parallel to the terrestrial or celestial polar axis, or, as is commonly said, it must be pointing to the pole. If now the arm marked AB, which carries a line of sight, is brought to a zero reading on the declination arc, it will be perpendicular to the polar axis and practically coincident with the equatorial plane. If the sun is for the day on the equator, the line of sight, on being revolved about the polar axis, will cut from the celestial sphere the path of the sun, or it can be at any time turned on the sun. If the sun is a few degrees above the equator and its declination is set off on the declination arc, with the arc in the position shown in the figure, and the line of sight is then revolved about the polar axis, it will cut from the celestial sphere a circle parallel to the equator which will be for the day the path of the sun. So, as before, the line of sight may be at any time turned on the sun by simply revolving it about the polar axis. If now the whole transit is revolved in azi- muth so that the polar axis no longer points to the pole, the line of sight will not, on being revolved, cut from the heavens, the path of the sun and can not be set on the sun at any time by merely revolving it about the polar axis. There may be one instant at which it can be thus set. 102. Method of use. This, then, furnishes the key to the method of use of the instrument, which is as follows : The direction of the meridian plane being unknown and desired, set off on the vertical circle of the instrument the colatitude of the place, so that the polar axis, when in the meridian, may point to the pole. Set off on the declination arc the declination for the time of the observation. Now with the plates level, so that revolution about the vertical axis may be only in azimuth, revolve the instrument in azimuth and the line of sight about the polar axis till the sun is found to be in the line of sight. When this occurs, the polar axis and telescope lie in the merid- ian, and the instrument may be clamped and the line ranged out. There is a small circle which will then give the time of day, which was of course known beforehand, in order to com- pute the declination. 120 DIRECTION AND ANGLES. The declination is found in the " Nautical Almanac " J for the longitude of Greenwich and for noon of each day in the year, and with the hourly change. The longitude of the place being known, the declination for the time and place may be readily computed. It must be remembered that in most places standard time, which probably is not the same as local time, is used and, if very different from local time, allowance must be made. A difference of fifteen minutes will not ordinarily make any appreciable error in the resulting work. The latitude and longitude may be taken from any good map, or the latitude may be observed by measuring on a previous night the altitude of Polaris at culmination, and adding or subtracting from the result the pole distance of the star. See Table IV., page 364. It may also be observed with the solar transit, as will be explained later. The line of sight AS consists of a lens at A and a small silver disk at B. The line of sight is directed toward the sun by bringing the image of the sun formed by the lens into the center of a square ruled on the opposite silver plate. In order that the line of sight may be used when the sun is below the equator, there is a lens in each end, and a silver plate opposite each lens. If the declination were south, the declination arc would be reversed from the position shown in the cut; and the lens in B and the plate in A would be used. 103. Limitations. Since there is a horizontal, that is, azi- muth, and a vertical component to the sun's motion, except just at noon, when the motion appears to be all in azimuth, the image of the sun will appear to move off the square on the disk, if it is left stationary for a time, in a diagonal direction, and can be kept in the square only by revolving the line of sight about the polar axis and shifting the arm on the declina- tion arc, the polar axis being in the meridian. Just at noon, however, the motion of the sun is apparently all horizontal, and since at noon the line of sight will be in the same vertical plane as the polar axis and telescope, the sun's image will move out of the square horizontally or between two of the lines, so that J The "Nautical Almanac" is published by the Navy Department at Washing- ton. The portion relating to the sun's declination is published separately by W. & L. E. Gurley and by G. N. Saegmuller, and perhaps by other instrument makers. THE SOLAR TRANSIT. 121 it could be kept in the square for a little time by moving the polar axis a little in azimuth. At this time, therefore, the meridian cannot be correctly determined. It can not be well done within one or two hours either side of noon. 104. Latitude. For the same reason, since at noon the exact meridian is not needed to get the sun in the line of sight, this is the time to observe for latitude, as follows : Set up the instrument a little before noon. Set off the co- latitude (as nearly as known) on the vertical circle, and the declination for noon on the declination arc. Now bring the sun's image upon the silver disk between the horizontal lines by using the azimuth motion of the transit and the vertical circle tangent screw. The image will appear to get lower as the sun goes higher. Keep the image in the square on the disk till it appears to begin to move upward. The sun is then at its highest point ; and if the declination has been properly set off and the plates carefully leveled, the vertical circle will read the colatitude of the place. The solar transit may then be used at noon for finding lati- tude, and between 8 o'clock and 10.30 o'clock A.M. and 1.30 o ? clock and 5 o'clock P.M. for determining a meridian. It could be used earlier and later but for refraction, which is of unknown and very irregular amount near the horizon. 105. Refraction. Thus far nothing has been said of refrac- tion. It must always be considered in setting off the declina- tion. The effect of refraction has been discussed in the chapter on leveling. It there appears that the object is always seen higher than it really is. Hence, ii the sun's declination is north, and the exact amount is set off on the declination arc, and the polar axis is brought into the meridian, and the line of sight pointed toward the sun, the sun's image would not be formed exactly in the little square, because the sun's rays would seem to come from a higher point than its true place. It is true that for small differences, like that of refraction or small errors in setting the latitude, the sun's image may be brought on the square by slightly turning the instrument in azimuth, thereby destroying the correctness of the determination of meridian. Hence it is necessary to know and correctly set off the latitude 122 DIRECTION AND ANGLES. and the declination corrected for refraction. The correction for refraction is greatest near the horizon, and is nothing at the zenith. Since it always makes a luminous body appear higher than it is, the correction must be added to north declinations and subtracted from south declinations so as to result in set- ting the line of sight to point higher than if the correction were not applied. The corrections to be used are as given in Appendix, Table VII., page 366. ADJUSTMENTS OF THE SOLAR TRANSIT. 106. Named. The adjustments of the solar apparatus are simple. They consist in making the lines of collimation paral- lel to each other and at right angles to the polar axis when the declination arc reads zero ; and in making the polar axis perpen- dicular to the telescope. The transit is supposed to be adjusted. 107. Lines of Collimation. These are made parallel by mak- ing each line parallel to the edges of the blocks containing them. To do this, remove the bar carrying the line of collima- tion, and replace it with a bar called an adjuster, which is sim- ply a table upon which to rest the lines of collimation while adjusting. Rest the bar containing the lines of collimation on the adjuster, and, by any means, bring the sun into one line of collimation. Quickly turn the bar over (not end for end). If the image still falls in the square, the line of collimation is par- allel to the two edges of the blocks. If not, move the silver disk through one half the apparent error of position of the sun's image and try again till complete. Turn the bar end for end, and adjust the other line of collimation. The two now being parallel to the blocks are parallel to each other. Remove the adjuster, and replace the bar on the instrument. 108. Declination vernier. Bring the declination vernier to read zero, and, by any means, bring the sun into one line of collimation. Quickly and carefully revolve the bar on the polar axis, and note whether the sun is in the other line of col- limation. If so, the vernier is in adjustment. If not, move the arm till the image is centered, and note the reading of the vernier. Adjust the vernier by loosening it and moving it one half the apparent error. Test again. ADJUSTMENTS OF THE SOLAR ATTACHMENT. 123 109. Polar axis. To make the polar axis perpendicular to the telescope axis, first carefully level the plates and the tele- scope, and then level the solar apparatus by the capstan-headed screws shown underneath the attachment. This is precisely the same as leveling up an ordinary instrument, and is per- formed by the aid of an auxiliary level that rests on the blocks of the collimation bar. This bar is set so that the declination is zero, and is brought into the plane of the main telescope and leveled, then at right angles to this position and leveled again. This is an important adjustment, and should not be omitted. SAEGMULLER SOLAR ATTACHMENT. 110. Description and use. Fig. 54 shows another form of solar attachment, which consists simply of a small theodolite (so called because the telescope will not transit) attached to the top of the telescope of an engineer's transit. This is the invention of Mr. George N. Saegmuller, of Washington, D.C., and is made by him and furnished by other makers as well. In operation it is similar to the last-described attachment. The difference is that a telescopic line of sight is substituted for the lens and disk, and the small level is used in conjunction with the vertical circle for a declination arc. Suppose the transit to be turned with the object end of the telescope to the south, and the telescope level. Assume north declination. Turn the object end of the telescope down an amount equal to the corrected declination and bring the small bubble to the center of its tube. The angle between the main and small telescopes .then equals the declination. If the object end is now pointed to the equator by setting off the colatitude upward from zero (not from the declina- tion reading), the instrument is set ready for use. Turn the instrument in azimuth and the small instrument about its polar axis till the sun's image is seen in the small telescope ; then the large telescope and polar axis lie in the meridian. There is a diagonal eyepiece to the smaller tele- scope to facilitate observations. The instrument is approxi- mately pointed by the small disk sights above the level tube. The objective is turned up in setting off south decimation. MERIDIAN AND TIME BY TRANSIT AND SUN. 125 111. Adjustments. The adjustments of this instrument are two : the adjustment of the polar axis, and that of the line of collimation and small bubble. The first is performed, after the transit has been carefully adjusted, by making the main tele- scope level, then leveling the small telescope over one set of capstan-headed screws at the base of the attachment and adjust- ing, and then over the other precisely as in adjusting the plate bubbles of the transit, correcting first by screws and next by tipping the small telescope. The second adjustment is per- formed indirectly by making the two lines of collimation par- allel. Measure the distance between the centers of the two horizontal axes and draw on a piece of paper two parallel lines the same distance apart ; tack this up at some distance from the instrument and about on the same level, with the lines horizontal. Make the two bubbles parallel by making both level, with the telescopes pointing toward the paper. Set the line of collimation in the large telescope on the lower line on the paper and adjust the wires of the' small telescope until the line of collimation of the small telescope cuts the upper line. MERIDIAN AND TIME BY TRANSIT AND SUN. 112. Meridian. If the sun's altitude at any moment is measured with the transit, its azimuth at the same moment may be computed if its declination and the lati- tude of the place of observation are known. 1 For in the spherical triangle pole-zenith-sun, the three sides will be known and the angle Z (the azimuth) may be computed. By Trigonometry if s sin I Z = \T- sin b sin c from which, if 8 = declination, = latitude, and 7t = altitude, and if s' = ^(pole dist. (=90S)+< + cos cos h 1 The same is true of any known star. (1) 126 DIRECTION AND ANGLES. Set up the transit at a convenient hour, over a tack in a stake and, with vernier at zero, set a stake some distance away, approximately north, and set the line of collimation on this with the lower motion. Unclamp the alidade and turn the line of collimation toward the sun and measure its altitude, at the same moment clamping the alidade. A small piece of red glass placed inside the cap of the eyepiece will enable the observer to look at the sun; or, with the eyepiece drawn fully out, the image of sun and wires may be received on a card held just back of the eyepiece. To observe most accurately, bring the horizontal and vertical wires tangent to the sun's image, cor- rect the altitude by the sun's semi-diameter, =0 16', and the observed azimuth by 16' x sec h. The observed altitude must be corrected for refraction. The corrections given in Table IV., page 364, for Polaris, will answer if the column of latitude is taken as altitude. Substitute the corrected altitude, the latitude, and the declination, in Equation 1, and solve for Z. The dif- ference between the observed azimuth of the sun and Z is the azimuth of the line of stakes, from which the meridian may be laid off. 1 113. Time. The angle at P in Fig. 55 is called t, the hour angle. It is given by sin Z cos h Reduced to time, it is the true sun time, before or after noon, of the observation. This must be corrected by the equation of time (difference between true sun time and mean time) found in the " Nautical Almanac " to give mean local time. This must again be corrected by the difference between mean local and standard time. The result compared with the observed time of observation will give the error of the watch used. 1 Let the student show the error in azimuth resulting from an error of 01' in either latitude or altitude by computing Z for values of differing by 01' and the same for values of h. This should be done for latitudes near 30 and 50 and altitudes of 10 and 60 to show the effect of variations in these quanti- ties. Practically the same errors arise with the solar transit for like errors of latitude and declination. CHAPTER V. STADIA MEASUREMENTS. 114. Defined. The stadia is a device for reading distances by means of a graduated rod and auxiliary wires in a telescope. In many farm surveys where the ground is very rough and not very valuable, the measurements may be made with the stadia. When so made, they will probably be somewhat more accurate than if made with a chain or tape with the care that would ordinarily be taken in the kind of work mentioned. In fact, with the use of care and judgment, the stadia will serve for almost all farm surveys. The stadia is the best distance measurer for extensive topo- graphical surveys. The discussion of the stadia that follows will perhaps indicate its limitations. The term "stadia" has been very loosely used in this country. The word is the plural of the Latin word "stadium," which means a standard of meas- ure. " Stadia " was the word adopted by the Italian engineer Porro (who invented the method to be described) to indicate the rod used by him in the application of his method. The English have retained this use of the word and apply the word " tacheometer " to the transit equipped with the additional wires used in the method. " Tacheometer " means quick measurer, and is applied by at least one American manufac- turing firm to their high-class instruments that are equipped with vertical circle, level to telescope, and stadia wires. The word as used by them is "tachymeter." As generally used in this country the word " stadia " means the combination of in- strument and rod, but it is thought that it should be applied to the rod only, and the word "tacheometer" or "tachymeter" is a very suitable word to apply to the complete transit in- strument equipped for stadia work. Such a transit should 127 128 STADIA MEASUREMENTS. have an inverting telescope of comparatively high power for the best work. The term "stadia wires" is properly used to designate the wires used with the stadia. 115. Method explained. Stadia measurements depend simply on the proportionality of the corresponding sides of similar tri- angles. The telescope is fitted with two wires in addition to those already described, one above and one below the horizontal wire, and both parallel to it. When a rod is held at some distance from the instrument, and the telescope is properly focused on the rod, an image of the rod will be formed in the plane of the wires, and a certain definite portion of this image will be seen between the two stadia wires. The rod is usually held vertical. If the telescope is horizontal, the con- ditions will be as shown in Fig. 56. The two wires are shown at U and L. By drawing lines from U and L through the optical center of the objective 0, to the rod R, the space on the rod whose image is included between the wires is seen to be lu. FIG. 56. From the similar triangles OL U and Olu 7 = T (1) A law of optics is that the sum of the reciprocals of the con- jugate focal distances of a convex lens is equal to the reciprocal of the focal length of the lens. From this law, if / is the focal length of the objective, 1+1=1. A A f SPACING OF THE WIRES. 12i, Whence Ji g this value of results Equating this value of with that obtained from (1), there (3) This is the distance from the objective to the rod. It is usual to require the distance from the point over which the transit is set, to the rod. If the distance from the objective to the center of the horizontal axis, which is vertically over the plumb line. is represented by c, the total distance required is V=s + C/+0- (4) If, then, a graduated rod is held at an unknown distance from the instrument, the distance will become known by multi- plying the space intercepted on the rod by the ratio of / to i and adding to the product the constant quantity (/+ c). 116. Spacing of the wires. The value of ~ must be known. Sometimes the wires are made so that the space between them is adjustable. This is not considered by the author as good practice. The wires should be fixed. They are attached to the same ring that carries the cross wires, and should be spaced at equal distances from the horizontal wire. This is not neces- sary, but is very convenient in reading, because it will occa- sionally happen that but one stadia wire and the horizontal wire can be read. This is likely to occur in brushy country, but should not be permitted when it is possible, at an expense commensurate with the importance of the work, to avoid it. In case it should occur, it is convenient to have the wires equally spaced, because the reading of one wire multiplied by two will give a sufficiently accurate determination of the dis- tance. In case the wires are not equally spaced, separate values of the intervals may be determined for the wires. It is thought best to have the ratio 4 100 for convenience. If one is introducing stadia wires in his transit, he will find U'M'D SURV. 9 130 STADIA MEASUREMENTS. that it is practically impossible to accomplish this result accu- rately himself. It can rarely be done outside of the maker's shop. However, it is not absolutely necessary to make this ratio 100, as the rod may be graduated to suit the instrument. 117. To approximate the value of /. Focus the telescope on a distant point (theoretically the point should be infinitely distant, as a star) and measure the distance from the center of the objective to the plane of the wires. This is the value of /. 118. The value of c. This is not quite constant in most in- struments, since the objective is moved in and out in focusing for different distances. For almost all work done by the stadia the error from this cause will not exceed one tenth of an inch, which is inappreciable compared with the smallest unit readings taken by this method. Single distances may not be read in careful work much closer than the nearest half foot ; and, in a very large percentage of work done with the stadia, the dis- tances are read only to the nearest yard or meter. The value of c may be determined by focusing on an object at, say, two hundred feet distance, and measuring the distance from the objective to the center of the horizontal axis. In some inverting instruments the focusing of the image on the wires is accomplished by moving the eyepiece and wires together, instead of the objective. In such an instrument c is a constant. 119. The value of and (Y+c). If the value of -. is not known, it may be determined as follows : Select a strip of level ground, drive a stake and center it with a tack. Set the tran- sit over this point. From this point measure two distances, say, one hundred feet and two hundred feet. Hold a rod at each of the two distant points, and note the space intercepted on the rod at each point. (5) (6) In each of these equations, D and >S are known, and hence l j and (/ + '"A This will be the correctly closed field, and to each side will have been given such a portion of the entire error of closure, as that side is of the whole perim- eter. From the similarity of the triangles AEF, AE'F', BALANCING THE SURVEY. 147 AE'' F", etc., it will be seen that the same proportion of the error of latitude and the error of longitude has been given to each side, whence the following rule : RULE : Correct the latitude difference and longitude differ- ence of each course by an amount determined by the propor- tion ; the required correction in latitude (or longitude} is to the total error in latitude (or longitude) as the length of the course in question is to the entire perimeter of the field. 1 It will be observed that the courses nearly parallel to HA have been corrected mostly in length, while those nearly normal to EA have been corrected mostly in bearing. From the corrected latitude and longitude differences new bearings and lengths are computed. 132. The practice. This method is not strictly followed in practice. It is, as has been seen, based on certain assumptions as to the probable mode of occurrence of the errors. In a given case it may be that the surveyor knows, or is reasonably certain, that a greater portion of the error is due to the diffi- culty encountered in measuring one side, and in such a case he would give the greater portion of the error to that side. In determining that this is true, he must first ascertain the direction of the closing line to see whether the line supposed to be difficult to measure is parallel or nearly parallel to the closing line, so that an error in measurement would be respon- sible for the resulting closing line. It may be that it is believed that the error is not confined to one course but is distributed over all the courses, though not in the proportion of their lengths, as some may have offered greater difficulties to meas- urement than others. If this is so, the error is distributed in the following manner : RULE : Determine by judgment which line has offered the least difficulties to measurement and number this line 1. Give to each of the other sides a number that shall represent its rela- tive difficulty of measurement, as 2, 3, 2^, etc. Multiply each length by its number and find the sum of the multiplied lengths. Distribute the total errors of latitude and longitude by the follow- 1 Least square demonstration of this rule will be found in Wright's "Adjust- ment of Observations." 148 LAND SURVEY COMPUTATIONS. ing proportion : The correction to the latitude (or longitude) difference of any course is to the total error in latitude (or longitude} as the multiplied length of the course is to the sum of the multiplied lengths. This is termed weighting the courses in proportion to their probable error, and results in distributing the whole error in pro- portion to the difficulties encountered and the lengths of the lines. When the lines are all of equal difficulty of measurement, or when one who knows nothing of the field work balances the survey, the first method should be used. In case the person making the survey also balances it (as should always be the case, when possible), the second method should be followed. The original notes of the survey should not be destroyed. The new latitude and longitude differences and lengths and bearings of courses may be written in the notebook in red ink over the originals. The originals may be useful in the future in determining how the corrections were made. It is not com- mon to correct the observed bearings and lengths to correspond to the corrected latitude and longitude differences ; but it is well to do this. 133. When a transit is used. In case there is no reason to believe there is any error in bearings, as when a transit is used for determining angles, and therefore the entire error of closure is due to lack of uniformity in measurement, the survey should be balanced in such a manner that only the lengths shall be corrected. No satisfactory rule has yet been devised to accom- plish this, and it must usually be done by trial. If the field is rectangular and one side may be taken parallel to the meridian of the survey, or assumed so for the purpose, the following rule accomplishes the desired result : RULE : The correction in latitude (or longitude} difference for any course is to the total error in latitude (or longitude} as the latitude (or longitude) difference of the course is to the arith- metical sum of the latitude (or longitude) differences. It is believed that the student can demonstrate the correct- ness of this rule for the assumed conditions, if he remembers that the sides must be corrected in length only, their directions remaining unchanged. SUPPLYING OMISSIONS. 149 SUPPLYING OMISSIONS. 134. Necessity for. At times it is impossible to measure the length of a line ; at other times it is impracticable to deter- mine directly its bearing ; and sometimes it is impossible to do either. Moreover, it may happen that the length of one line and the bearing of another have not been determined. If any one of these conditions prevails, it becomes necessary to supply the omission in the notes. To do this leaves the work without a check, as all the errors are thrown into the quantity that is supplied ; hence no omission that can in any way be supplied in the field, should be permitted. The cases that may occur are the following : I. The length of one side may be wanting. II. The bearing of one side may be wanting. III. The bearing and length of one side may be wanting. IV. The bearing of one side and the length of another may be wanting. V. The lengths of two sides may be wanting. VI. The bearings of two sides may be wanting. 135. General discussion. It should be clear from what has preceded, that the algebraic sums of the latitude and longitude differences of the known sides are respectively equal to the latitude and longitude differences necessary to close the field (but are of opposite signs), or to the latitude and longitude differences of the wanting sides plus the errors in latitude and longitude of the known sides. Since there is no means of determining these errors, the sums are taken as the latitude and longitude differences of the defective sides. It is therefore possible to write two independent equations as follows : I. The algebraic sum of the products of the lengths of the defective courses by the cosines of their respective bearings equals the algebraic sum of the latitude differences of the known sides, but has the opposite sign. II. The algebraic sum of the products of the lengths of the defective courses by the sines of their respective bearings equals the algebraic sum of the longitude differences of the known sides, but has the opposite sign. 150 LAND SURVEY COMPUTATIONS. Having but two equations, there can be but two unknown quantities, and these may be as in Art. 134. If Zj and l z and 6 1 and 2 are the lengths and bearings of the defective sides, and if 2 is taken as a sign indicating the algebraic sum of any series of quantities, as in this case the latitude and longitude differences, and L and M represent lati- tude and longitude differences respectively, the above two equations may be written as follows : J 1 cos0 1 + Z a cos0 2 =-2.L, (1) /! sin l + l z sin 2 = - 'S.M. (2) These are, of course, equivalent to the following general equations : 136. Cases I., II., and III. These are most readily solved by applying equations (1) and (2). If but one quantity is want- ing, as the length or bearing of a side, either equation (1) or equation (2) will solve, one of the left-hand terms becoming part of the right member. If both the length and bearing of one side are wanting, one of the left-hand terms in e^ch equation becomes part of the right member, and equation (2) may be divided by equation (1), giving tan 6= ~ ,. - Having 0j, the length is found by substituting in either equation. The signs of the sums of the known differences will indi- cate the direction of the defective side. 137. Cases IV., V., and VI. The author thinks the remain- ing cases are better solved by special methods as follows : Case IV. 1. The two imperfect sides are adjacent. With the algebraic sums of the known latitude and longitude dif- ferences, compute the length and bearing of a closing side. This side will form with the two defective sides a triangle in which two sides and one angle are known. 1 The triangle may then be solved, giving the required quantities. 1 The student should draw a diagram showing this. SUPPLYING OMISSIONS. 151 FIG. 65. 2. The defective sides are not adjacent. Imagine some of the sides shifted till the defective sides are adjacent and pro- ceed as before. Thus, in Fig. 65, the defective sides are DE and AB. If the side AE is imagined shifted parallel to itself to the po- sition BF, a closing line DF may be computed, and this will form with ED and EF (= AB) a tri- angle. Case V. Treat in the same way as Case IV. In the triangle that results there will be known the three sides. This case is inde- terminate if the defective sides are parallel, unless the area is known. Case VI. is solved in the same way, and in this case there is known in the triangle one side and all the angles. Cases IV. and VI. are not determinate unless enough is known of the field to show which of two angles that correspond to a given sine or cosine is the correct one to use. This will be evident from the following algebraic discussion of Cases IV., V., VI. 138. Algebraic solution. Let the sum of the known latitude differences be denoted by L and the sum of the longitude dif- ferences by M. Then for these cases may be written, ^ cos 0j + ? 2 cos #2 = ~ LI (3) Zj sin l + l z sin 2 = - M. (4) In Case IV. let there be wanting Z x and # 2 . Solving (3) for cosine 2 and (4) for sine 2 , squaring and adding and then solving for l v there results -- - (L cos l It is evident that there are here two values for ? r If the proper value may be determined by a knowledge of the field, # 2 may be found by substitution in either (3) or (4). For Case V. there will be wanting in (3) and (4) l^ and 1 Y 152 LAND SURVEY COMPUTATIONS. Solve each for l v equate the resulting expressions, solve for l z , and derive M cos 1 L sin 0j _ M cos 1 L sin O l 2 "" sin 1 cos 2 cos 6 l sin 2 ~ sin (0 X - 2 ) from Avhich it is seen that the value is indeterminate when B l = 8^ or the defective sides are parallel. For Case VI. there will be wanting 6 1 and # 2 . Solve equa- tion (3) for cos^ and equation (2) for sin0 r Square both resulting equations and add, getting 7 2 _ 7J2 _ 71/2 7 2 T /I *>- /I ^1 - -I-/ - -if./ - In L cos 2 + M sm #2 = 3 ^ 2 -. Z< 2 Let the right-hand member be represented by P. Then 1TV1 - cos 2 2 = P - i cos 6> 2 ; square, and solve for cos 2 , deriving from which it is evident that there are two values, the correct one to be determined by a knowledge of the survey. In the algebraic solutions, careful attention must be paid to the signs of the trigonometric functions. Since no bearing is greater than 90, the signs, according to usual trigonometric concep- tions, would all be positive ; but in equations (3) and (4) those signs must be used that will .produce the proper sign for the latitude and longitude differences ; thus, for a S.E. bearing, the cosine would be negative and the sine positive. 139. Double longitudes. When the survey has been bal- anced, the area may be computed. This is usually done by the use of a formula involving the products of the latitude differ- ence and double longitude of each side of the survey. The double longitude of a line is simply twice its longitude, or the sum of the longitudes of its ends. The latitude differences are taken from the table of corrected latitude differences that has AREAS. 153 been prepared in balancing the survey. There must be found a convenient method for determining the double longitudes. In Fig. 66 let N8 be the reference meridian, the longitude differences of the various sides being as shown, with the signs prefixed to each. It is evident that the double longitudes of the first and last courses will be numerically equal to their respective longitude differences, but the sign of the D.L. of the final course will be opposite to that of its longitude difference. The D.L. of any other course, as BC, is the sum of the longi- tudes of its extremities. D.L. of BC=bB + cC = d l + d l + (- d 2 ). In words, the D.L. of BC is the D.L. of AB plus the longi- tude difference of AB, plus the longitude difference of BO. Again, the D.L. of CD is or the D.L. of CD equals the D.L. of BC plus the longitude difference of BC plus the longitude difference of CD. From the result of this investigation may be formulated the RULE : The double longitude of any course is equal to the double longitude of the preceding course, plus the longitude differ- ence of that course, plus the longitude difference of the course itself. In applying this rule, due attention must be paid to the signs of the quantities. This rule applies to the first course as well as to the others, if the preceding course is considered to be zero. To determine the D.L. of the various courses, select one course as the first. Its longitude difference is its D.L. To this add its longitude difference and the longitude difference of the second course, and obtain the D.L. of the second course ; to this add the longitude difference of the second course and the longitude difference of the third course, and obtain the I). L. of the third course, etc. The correctness of the work is proved if the D.L. of the final course is found numerically equal to its longitude difference. 154 LAND SURVEY COMPUTATIONS. With some it is customary to select the most westerly station of the survey as the point through which to pass the reference meridian. This practice makes all of the D.L.'s FIG. 66. positive, and consequently obviates the necessity of considering their signs in the subsequent work of finding areas, and to this extent simplifies the work. 140. To find the area of the field. The area in Fig. 66 evidently equals, 1. The triangles ABb, Cch, and EeA, plus 2. The trapezoids bBCc and EeDd, minus 3. The triangle Ddh. Beginning with the triangle ABb and taking the triangles and trapezoids in order around the figure to the right, there results the following equation : Area ABODE = \ I IJ>B + 1 2 (bB + cC} + 1 3 (c- 3d) + Z 4 (Ddf + Ee~)+l b Ee\: AREAS. 155 The third term is twice the difference of the triangles cCh and Ddh. 1 This equation may be written, paying attention to the signs of the latitude and longitude differences and remem- bering that distances to the right of the meridian are positive and to the left negative, Area ABCDE = } j - ^ - 1 2 (d l + d l - rf a ) - J 3 (2^- d 2 -d, - df 3 ) + / 4 (2 (rfj-rf,) - d, - d 3 - rf 4 ) - / 6 rf 5 f . It is seen that the coefficient of each of the latitude differences is the double longitude of the corresponding course, whence the RULE : To find the area of a closed survey : Multiply the latitude difference of each course by the double longitude of the course, and note the signs of the products. Divide the algebraic sum of these products by 2. The sign of the result is of no consequence and depends on the position assumed for the reference meridian and the direc- tion of survey around the field, whether clockwise or counter clockwise. As stated, the work is simpler if the reference meridian is chosen through the most westerly corner. 141. Irregular areas by offsets. It frequently occurs that one side of a field that is to be surveyed is bounded by a FIG. 67. stream or the shore of a lake, and that the bounding line is quite irregular and not easily run out. In Fig. 67 the irregular shore line is a boundary. To !The student should prove this. 156 LAND SURVEY COMPUTATIONS. determine the area, two auxiliary courses, AB and BC, are run and used with the remaining sides to compute the area on their left. Left and right refer to the direction in which the lines are run. The additional area between these lines and the shore is obtained by measuring offsets normal to these lines. These offsets are measured to the points of change of direction of the shore, and their lengths and distances from A, B, or (7, are noted. The area is then computed by considering the portions between the shore and base line and two adjacent offsets to be trapezoids or triangles. When the curve of the shore is such that the offsets may be taken at regular intervals along a base line, the area is found by applying the following rule : RULE : To the half sum of the initial and final offsets add the sum of all the intermediate offsets, and multiply the result by the common distance between offsets. 1 If the offsets are taken at irregular distances, the area may be found as described in Art. 146. COORDINATES. 142. Definitions. It is becoming common to call the lati- tude and longitude of a point the " coordinates " of the point. The base parallel and meridian are known as the " coordi- nate axes," and their intersection as the " origin of coordinates." The coordinates of the origin are evidently both zero. As with latitudes and longitudes, ordinates measured east, or to the right from the reference meridian, are considered positive; those measured west, or left, are negative; those measured north, or up from the base parallel, are positive; and those south, or down, are negative. It is not uncommon to speak of the latitude ordinates as " ordinates," and the longitude ordinates as " abscissas." In the city of New York, and in some other cities, many corners to public property, and many or all private property corners, are located by coordinates, the well-established line of some important thoroughfare being taken as a reference me- ridian, the origin and meridian being marked by monuments. 1 The student will be able to show the truth of this rule. COORDINATES. 157 The true meridian through a well-defined point would perhaps do as well, except that, if the subdivision of the city is on the rectangular plan, and not exactly " with " the cardinal points, a great deal of com- putation would be avoided by making the axes parallel to the street lines. 143. Elementary problems. A large amount of surveying work is much facili- x tated by the use of coordinates, particu- larly by the use of the methods of the two following prob- lems : I. Given the co- ordinates of two points, to find the bearing and length of the line joining them. 1 In Fig. 68 the coordinates of a and b are given. The angle cab is the bearing of ab, but the angle cba will be first found because the smaller angles can be found with greater precision. tan 1 (1) This equation expressed in words as a rule is : 160 LAND SURVEY COMPUTATIONS. RULE : To determine the area of a closed field when the coordinates of its corners are known : Number the corners consec- utively around the field. Multiply each ordinate by the following abscissa and sum the products. Multiply each ordinate by the preceding abscissa and sum the products. One half the difference of the two sums, subtracting the second from the first, is the area of the field. Equation (1) may be written, 8 - \ This equation expressed in words as a rule is : RULE : To determine the area of a closed field when the coordinates of its corners are known : Number the corners consec- utively around the field. Multiply each ordinate by the difference betiveen the following and preceding abscissas, always subtracting the preceding from the following. One half the sum of the product is the area required. The second rule ordinarily involves less work than the first, but there are certain cases in which the first is used to better advantage. 145. To make the coordinates all positive. The method of determining the coordinates will perhaps suggest itself to the student. To lessen the danger of making errors in signs it will be better to arrange the axes so that the field shall lie wholly within the northeast quadrant. All the coordinates will then be positive. This may be done as follows : Deter- mine in the usual way the latitudes and longitudes of the cor- ners of the field with reference to the meridian and parallel through one corner, preferably the most westerly corner. If the most westerly corner is chosen, the longitudes will all be positive without further arrangement. To make the latitude ordinates all positive, add to each latitude a quantity equal to the greatest southern latitude. This will move the reference parallel to the most southern point. If considered more con- venient, any round number, as 100, greater than the most southern latitude, may be added. An inspection will ordina- rily be sufficient to enable the computer to assume beforehand proper coordinates for the first corner. COORDINATES. 101 146. Elongated areas by offsets. A special case, in which the application of this method of coordinates is advantageously used, is the determination of elongated, irregular areas, the measurements for which consist of offsets at unequal intervals along a straight line. In this case, the line from which the offsets are measured is assumed as the reference parallel. Thus, in Fig. 71, it is required to determine the area between the line AH, the irregular line abcdefgh, and the two end off- FIG. 71. sets. The corners of the closed field are AabcdefghH and their coordinates are as shown in the figure. x 1 and y l and x 2 and y 10 are zero. The equation that would be written, follow- ing the second rule of Art. 144, would be as follows : + #8 o 9 - ^7) + y* too - * 8 ) + #10 to - %) I y l and y w being zero, the first and last terms will disappear ; x 1 and x 2 being zero, the second term becomes y^c y and the third term 2/ 3 4 . What is, perhaps, a less confusing system of writing these quantities is to write each ordinate and its cor- responding abscissa in the form of a fraction, connecting each ordinate with the abscissas whose difference is to be taken as a multiplier. Thus, calling the z's ordinates and the y's abscissas, The downward lines to the right show the following or positive abscissas, and the downward lines to the left show the R'M'D SURV. 11 162 LAND SURVEY COMPUTATIONS. preceding or negative abscissas. The lines may be omitted as soon as the student becomes thoroughly familiar with the work. This arrangement applies to any closed field and not alone to the elongated strip last described, though it is particu- larly applicable to that method since the quantities may be arranged for computation as they are taken in the field. Ap- plications of the coordinate method of surveying will be found in the problems, pages 328-335. 147. Zero azimuth. It is^ thought that the reason for sug- gesting that zero azimuth shall be the north point, will now be clear. Since north latitude and east longitude are considered positive, and south lati- tude and west longitude, negative, a system of azimuths should be so arranged that the signs of the trigonometric functions of any azi- muth shall agree with the signs of the corre- sponding latitude and longitude differences. Fig. 72 shows the ar- rangement of signs of coordinates, and the correspondence of the signs of the trigonometric functions. There is thus no necessity actually to convert azimuth into bearing in order to determine the signs of the latitude and longitude differences, nor to carry in mind any other than the ordinary scheme of signs given in any work on Trigonometry. The signs of the coordinates could, of course, be changed to suit a south zero azimuth, but the custom among all people to look to the north as the orienting point, and the long use of the signs given for coordinates, seems to make it better for the surveyor to use the north as zero azimuth. Were it merely a matter of changing technical terms used only by the surveyor, such as changing " departure " to "longitude difference," the case would be different. Cos.- - 72. DIVIDING LAND. 163 DIVIDING LAND. 148. Occurrence of the problem. It sometimes becomes nec- essary to divide a field into two or more parts of equal or known areas. This occurs when one man, as John Jones, sells to an- other, as Paul Smith, x acres to be laid off in the northeast corner of Jones's field. It also occurs in the division of inherited lands among the heirs, and in the determination of lands sold for taxes. When the taxes are not paid on a given piece of land, the land is sold to the lowest bidder. This means that the land is put up at auction for the taxes and expenses of sale, and that the person who agrees to take the least part of the whole piece, and pay therefor the taxes and expense of sale, is given a title to that portion of the land that he agrees to take. This title is redeemable by the original owner within a certain time, specified by law, after the expiration of which time, if the title has not been redeemed, it becomes vested in the purchaser forever. In any case that may arise the original tract will be fully known, either by previous surveys or by surveys made at the time and for the purpose of the subdi- vision. It will also be known in what way the land is to be divided, and the problem then becomes simply one of Geometry or Trigonom- etry. 149. Solution of the problem. The method of solving two common problems will be given, and others may be readily devised, on which the student may test his ingenuity. I. It is required to lay off from a given field A acres by a line beginning at a given point in a given side. Plot the field to scale. Let Fig. 73 represent the field so plotted. Let m be the given point. Imagine the line mD to have been run to the FIG. 73. 164 LAND SURVEY COMPUTATIONS. corner nearest the probable ending point of the required line mg. The point g on the line DE will first be determined as follows : Consider mBGD as a closed field with length and bearing, or azimuth, of one side, mD, wanting. Find bearing and length of mD, and area of mBCD, which call a. Then Area mDg = A a. In the triangle mDg, the angle at D is known, and the side mD. The area of the triangle is A a = | pk sin D. Whence k = A a ^ p sin D The triangle may then be further solved, giving the bearing and length of mg. II. It is required to lay off from a given field A acres by a line extending in a given direction. Let Fig. 74 represent the given plotted field. Select a corner of the field, as A, such that from this corner a line may be run jv in the given direction, cutting off, as nearly as may be, the required area A acres. Let mg be the required line, the area AFEgm being equal to A. If Am were known, the line mg could be run. Since the length I should be computed for a check, and since it is some- what simpler to deter- mine I first rather than Am, I will be first de- termined. Imagine the FIG. 74. . "J . line Ah run in the given direction. Consider AFEh a closed field with two wanting lengths viz., A h and Eh. Determine these and the area AFEh, which call a. The area Amgh = A a. Then MODEL EXAMPLES. 165 (1) J=p-&(tana + tan/3). (2) and /3 are known from the bearings. Whence k = -- -~ . (3) tan a + tan /3 Substituting in (1), tan + tan yS Whence I = V;? 2 - 2 ( J. - a) (tan a + tan /3) . (4) Z being known, k is determined from (1), or (3), and Am, or hg, from the small right triangles. In the field, find the point w, and run I on the given bearing to its intersection with DE at g. See that the length agrees with the computed length, and that gE as measured agrees with gE as computed. MODEL EXAMPLES. 150. Logarithms. When formerly, in land surveying, bearings were read only to quarter degrees, there were published for convenience what were known as traverse tables. These were nothing more nor less than tables of natural sines and cosines of the angles from to 90 for every quarter-degree multiplied by 1, 2, 3, 4, etc., to 10, and in some tables to 100. The use of such a table made it unnecessary to multiply the sines and cosines of the bearings to get the latitude and longitude differences, since for each digit in the number expressing the length of the course the differ- ences could be read from the table and brought to the right amount by moving the decimal point. The several quantities were then added. With modern methods of work the compass is read to the nearest five minutes, and when a transit is used the angles are determined to minutes. Such a table as has been described then becomes useless. The proper table to use is one of logarithmic sines and cosines. A single computation involv- ing not more than two or three figures can perhaps be more quickly per- formed without the use of logarithms, but any series of computations or a single computation involving five or more figures can be more quickly performed by logarithms. This does not mean that one unaccustomed to the use of logarithms can work with them so fast as without, but a very little practice with them will in any case substantiate the above claim. The student should familiarize himself with a good set of logarithmic tables. 1 The question will arise whether four-place, five-place, six-place, or seven- place tables should be used. The decision must be based on the size of the : The tables of Bremicker or Vega are recommended for work requiring great precision. Gauss or Crockett are recommended for five-place tables. 166 LAND SURVEY COMPUTATIONS. quantities involved in the computation and the precision required in the result. A four-place table will give results correct to three significant figures and almost correct to four significant figures, probably within one or two units in the extreme end of the table. A five-place table will give results correct to four significant figures and within one or two units in the end of the table, to five significant figures, and so on. For very many sur- veying computations four-place tables are good enough, but for the general use of surveyors five-place tables are considered better, and for use in connection with very accurate city surveys, six-place tables will not be too extensive, though almost all cases may be solved properly by the use of five- place tables. For general field use five-place tables are ample. Logarithmic tables should have auxiliary tables of proportional parts for quickly getting the logarithm of a number greater than any given in the table and for getting the number corresponding to a logarithm not in the table. Such tables, with proportional parts in the trigonometric functions for tenths of minutes instead of for seconds, will be found on pages 420-464, taken from Crockett's Trigonometry. 151. Example stated. The notes of the courses of a survey are as follows : N. 69 E 437.0 ft. S. 19 E 236.0 ft. S. 27 W 244.0ft. N. 71 W 324.0 ft. N. 19 W 183.5 ft. 1424.5 It is required to balance the survey and determine the area of the field. This example will be worked out in detail as a model for the student. He is advised to note carefully the systematic arrangement of the work, as by such system much time is saved. It is a case from practice. 152. Balancing. Letting L represent latitude differences and M longi- tude differences, the computation is arranged as shown on page 167. EXPLANATION. We first write the logarithm of the length of the course, and above it the logarithmic cosine of the bearing, and below it the loga- rithmic sine. The logarithm of L is then obtained by adding up, and the logarithm of M by adding down. The L's and M's are then taken out and placed in their respective columns with their signs, and each column added algebraically, giving the result 4.9 in L and +7.9 in M. The error of clo- sure is then found. In the example given it is entirely too large, and the field should be rerun. The errors in L and M are now distributed among the courses in proportion to the length of the sides. It is not necessary to be exact about this, and it is done by inspection. Thus is a little less than one third, hence the first corrections are 1.5 in L and 2.5 in M. The other fractions are treated in the same way, the second being somewhat less than one sixth, etc. If the corrections thus determined do not sum up exactly to MODEL EXAMPLES. 167 COMPUTATIONS FOR THE HARRINGTON SURVEY. QUANTITY. Loos. L M dZ's AND dJf'a. QUANTITY. Loos. L 2.19481 + 156.6 + 158.1 + 405.5 + 408.0 \ 43> x 4 9 15 435.2 2.63872 cos 69 9.55433 437.0 2.64048 sin 69 9.97015 ' 1424.5 X x 7.9 = 2.5 sin 68 42' 9.96927 405.5 2.60799 158.1 2.19893 M 2.61063 tan 68 42' 0.40906 L 2.34858 -223.1 - 222.3 + 76.5 + 76.8 1 "^ x 4 Q OR cos 19 9.97567 236.0 2.37291 sin 19 9.61264 ' 1424.5 X 7.9 = 1.3 M 1.88555 L 2.33727 - 217.4 - 216.6 - 112.1 - 110.8 1 244 . Q ~ Q cos 27 9.94988 244.0 2.38739 sin 27 9.65705 ' 1424.5 x 7.9 = 1.3 M 2.04444 L 2.02319 + 105.5 + 106.6 - 308.2 -306.4 1 324 *49 11 cos 71 9.51264 324.0 2.51055 sin 71 9.97567 ' 1424.5 X x 7.9 = 1.8 M 2.48622 L 2.23931 + 173.5 + 174.2 - 60.7 - 69.7 I 183 ' 6 - 4 9 07 cos 19 9.97567 183.5 2.26364 sin 19 9.51264 ' 1424.5 " x 7.9 = 1.0 M 1.77628 -440.5 +484.8 + 435.6 -476.9 -4.9 +7.9 Error of closure = V4.9 2 + 7.9 2 = 9.4 = 9.4 _ 1 1424.5 151 the respective total corrections, some one or more of them is slightly altered to make the sum correct ; thus, in the above example the corrections first written for the fourth L and the fifth M were 1.2 and 1.1 respectively, and these were changed to 1.1 and 1.0 to make the sums equal 4.9 and 7.9 respectively. The balanced L's and M's are now written in the column of L's and M's under the old L's and over the old M' s. The lengths and bear- ings are now inconsistent with the balanced L's and M's, and should be corrected to be consistent. This is done in this example for the first course 168 LAND SURVEY COMPUTATIONS. only. The tangent of the bearing is , hence write the log M and subtract L the log L and get log tan. Above log M write log sin and, subtracting up, get log length. This completes the balancing. 153. Areas by latitude differences and double longitudes. From the balanced L's and M's the double longitudes are computed, and from the L's and D's, as we may call the double longitudes, the double areas are com- puted. The work is systematized as in the following table : LOGS. DOUBLE AREAS. M of the First course is its D = 405.5 2.60799 + M = + 405.5 + L 2.19893 + M = + 75.5 4.80692 +64110 Second course D = + 886.5 2.94768 + M = + 75.5 - L 2.34694 + M = - 112.1 5.29462 - 197069 Third course Fourth course Fifth course D = + 849.9 2.92937 + M = - 112.1 - L 2.33566 + M = - 308.2 5.26503 - 184089 D = + 429.6 2.63306 + M = - 308.2 + L 2.02776 + M = - 60.7 4.66082 D = + 60.7 1.78319 + L 2.24105 4.02424 45795 10574 - 381158 + 120479 2)260679 43560)130339.5 2.992+ acres. It is believed that the work is self-explanatory. The results of the foregoing computations are usually tabulated in the following form : MODEL EXAMPLES. 169 jj CO CO g 3 -< 05 CO I H a US -# C5 1 + 1 DOUBLE T.nwfiT- TUDES. O O C2 O t 10 CO 05 05 O CO * CM CD ^ co co -^i g 10 i-H 01 t- 8 O >O CM CO O O 1-- i i O CO H fe 1 + + 1 1 1 i 1 CO CO CO Cl o M CO CM CO CO -t< O CM r- 1 O 1^ I 1 CM CM r-1 rl + 1 1 + + CO Tt< t^ O5 h 1 CO 05 CD Q ^ co ** fc CD CO co 5 + 5 *- co . 1 5 2 us d CM CM ^ Q H co >c o CO ^ + co 10 co >0 I- 1 STANCE, FEET. q q o o io 1-^ CO -^ -rf' CO co co -ti 01 co "* CM Cl CO i i 1 cs 1-1 j w w ^ ^ ^ c | 05 ^ 05 C , n o n t j 41 1 1 T3 '3 - i! a; u-i ^ o -111 2^3 n-i -|J .. . ^5 4) O II 170 LAND SURVEY COMPUTATIONS. 154. Areas by coordinates. We shall next work out by coordinates the area of the field just determined. The work is the same up to and including the determination of the balanced L's and J/'s. An inspection of these demonstrates that the first corner is the most westerly corner, and that the fourth corner is the most southerly, and that it is 283.9 feet south of the first. Therefore if it is desired to make all coordinates positive, the refer- ence meridian will be passed through the first corner, and the origin of coor- dinates will be taken on this meridian 300 feet south of the first corner. The coordinates of the corners and the area of the field are then found as in the table, in which the y's are the latitude ordinates and the x's are the longitude ordinates. CORNER T No. LOGS. DIFF. X's 300.0 2.47712 2.53757 -344.8=2-5 2 3 4 5 + 158.2 458.1 -222.3 235.8 - 216.6 19.2 + 106.6 125.8 + 174.2 300.0 5.01469 2.66096 2.68214 +481.0 = 3-1 405.5 405.5 + 75.5 481.0 -112.1 368.9 308.2 60.7 -60.7 0.0 5.34310 2.37254 1.56348 - 36.6 = 4-2 3.93602 1.28330 2.62356 -420.3 = 5-3 3.90686 2.09968 2.56691 -368.9 = 1-4 4.66659 X DOUBLE AREAS. SQ. FEET. + 103440 + 220341 - 8070 - 46408 + 323781 - 63108 2)260673 43560)130336.5 2.992+ acres. Rather more work is required by this method than by the method of double longitudes. This is not the case when the corners have been de- termined by random lines rather than by a continuous traverse around the field. When the coordinates of the corners are alone known, the method just given is by far the quickest, since it would be necessary to compute the L's and M's for each side before the D's could be obtained or the areas, just as it was here necessary to compute the coordinates from the L's and M 's. MODEL EXAMPLES. 171 155. Supplying an omission. I. In the example already used, let the bearing and length of the second course be wanting. Adding the origi- nal 's and M's, we get COURSE. L M 1 + 156.6 + 408.0 2 3 - 217.4 - 110.8 4 + 105.5 - 306.4 5 + 173.5 - 59.7 + 435.0 - 476.9 - 217.4 + 408.0 + 218.2 - 67.9 There is then to be added south latitude and east longitude in order to close the field, and hence the line to be supplied runs southeast. The tangent of its bearing is LOGS. tan0 = -6 1-83187 + 218.2 2.33885 t Log tan CU9302 = 17 17 ' + Log sin 9.47300 | Log length 2.35887 length = 228.5. The whole error of closure being thrown into this course, its bearing and length have been materially altered. II. Let the lengths of sides one and two be wanting. Adding the original L's and M' s, we get COURSE. L M 1 + 156.6 + 408.0 2 3 - 217.4 - 110.8 4 5 + 173.5 - 59.7 + 330.1 + 408.0 -217.4 - 170.5 + 112.7 + 237.5 It is seen that southwest bearing must be that of the closing line. Its bearing and length are obtained as in the last example. LOGS. ' 3756 112.7 2.05192 tan 8 = 2 ' 37566 Log tan 6 0.32374 6 = 64 37' S.W. Log sin 9.95591 Log length 2.41975 length = 262.9. 172 LAND SURVEY COMPUTATIONS. We now have a triangle composed of this closing side and the two wanting sides. This triangle 1 is formed by shifting one of the wanting sides par- allel to itself. In this triangle there are known the bearing and length of the closing side just found and the bearings of the two other sides, and hence there are known all the angles and one side. If the apexes are lettered A, B, C, and the sides opposite the apexes a, b, and c. respectively, and if a is the closing side just found, b side 2, and c side 4, we have from the known bearings Whence a sin B 262.9 sin 44 24' Angle A = 52 00' " B = 44 23' " C = 83 37' 180 00' Solve for course 4. The other problems are similarly solved. Log 262 Log sin Log sin Log b sin A .9 44 23' 52 00' sin 52 00' 2.41975 9.84476 2.26451 9.89653 2.36798 b = 233.3. THE PLANIMETER. 156. Description. The most elegant and rapid method to obtain the area of an irregular figure is to draw the figure to scale and measure the area with a planimeter. There are three kinds of planimeters, shown in Figs. 75, 76, and 77. FIG. 75. Fig. 75 shows the polar planimeter, the most commonly used. Fig. 76 is a suspended planimeter, which is a polar planimeter so arranged that the wheel is constantly changing, while with the circumferential motion the value of and the length of the line ed remains fixed. In Fig. 78 the component motions sd and sd f , and their resultant motion dd', are shown greatly enlarged. It will be evident, that, in cir- cumscribing a closed figure, each minute movement of d toward e will have its corresponding movement from e with the same value of (f>. Each element of right-hand circumferential motion will have its corresponding element of left-hand circumferential motion ; but, since d will be farther from e for one than for the other, these corresponding elements will not be made with equal values of . When the plane of the wheel that brings about the above result, is known as the zero circumference. Its radius, ed, may be easily shown to be (1) (2) It will be evident that if d could be moved outward till c should fall in the line ek, and then rotated clockwise, the 176 LAND SURVEY COMPUTATIONS. motion of c would be all rolling motion, and would be, looking from c to J, clockwise. The wheel is graduated so as to record positively for this kind of motion. Between these two posi- tions the motion of the wheel will be partly slip and partly roll, the amount of each depending on the value of <; and the roll will all be clockwise. It will be further evident that, if d were moved in till c should fall in the line ke produced, and then rotated clockwise, the motion of c would be all roll and would be counter-clockwise or left-handed, when looking as before from c to d. For d between the zero circumference and the last-named position, clockwise motion will produce a motion of c partly roll and partly slip. The amount of each will de- pend on the value of <, and the roll will be counter-clockwise. Hence, clockwise motion of c will be caused by positive motion of d outside the zero circumference, and by negative motion of d inside the zero circumference, and vice versa; arid, since the amount of roll for a given motion of the tracer depends on the value of <, any two equal infinitesimal motions in opposite directions with the same value of < will produce no resultant roll of the wheel, while, if made with unequal values of , there will be a resultant roll of the wheel that can be read. For these reasons, the radial components in cir- cumscribing a closed figure cause no resultant rolling of the wheel and may be neglected, while the circumferential components do cause a resultant roll and must be considered. It will be shown that the roll of the wheel for a given circumferential motion of d is proportional to the area included be- tween the path of d, the radial lines from e to the initial and final points of rf's path, and the arc of the zero circumference included between those lines. Let dd', Fig. 80, be a minute circumferential component of remaining constant. The wheel will move through the arc cc', and will partly roll and partly THE PLANIMETER. 177 slip. The rolling component of its motion will, of course, be normal to its axis, and may be represented by the line p. Therefore cs = A (j cos jp), (3) which is the roll of the wheel for the motion of d through the arc dd' '. To show that this is proportional to the area dd'o'o, there must be deduced an expression for that area. From Trigonometry ed = V/ 2 + h* + 2jh cos <, dd'=ed-A. The area of a sector of a circle is the product of one half its arc by its radius, whence Area edd' = J A (/> + A 2 + 2jh cos 0). (4) Using the value of the radius of the zero circumference given in equation (1), there results for the value of the area eoo', Area eoo' = 1 A (^ + A 2 + 2 pK). (5) Subtracting (5) from (4), there results Area dd'o'o = Ah (j cos - p). (6) R'M'D SURV. 12 178 LAND SURVEY COMPUTATIONS. This area is equation (3), the roll of the wheel, multiplied by the length of the adjustable arm, arid hence is proportional to the roll of the wheel. Q. E. D. It is now to be shown that, in tracing a closed area, the record of the wheel is correctly summed. In Fig. 81 let the tracing point move about the area dd^d^d clockwise. Motion from d to d^ will cause a clockwise roll of the wheel propor- tional to the area dd^o'o. The motion from d 1 to d 2 will be neu- tralized by motion from d s to d. Motion from d 2 to d 3 will cause counter-clockwise roll of the wheel proportional to the area c? 2 c? 3 0o', and the resulting roll will therefore be proportional to the area dd-^d^dy The student may reason similarly x for the other areas. Since the roll of the wheel is proportional to the area lying between t?'s path and the zero circumference and is positive, or clockwise, when d is outside the zero circumference and moves to the right, and negative when d is inside and FlG - 81g moves to the right, it follows that if an area is traced with e inside that area, so that d must com- plete a revolution about e, there must be added to the area ob- tained by multiplying the roll of the wheel by A, the area of the zero circumference. This area is 159. To find the zero circumference. This area is usually furnished with the instrument when it comes from the maker, but may be found thus : Measure a known area with the point e within it and com- pare the result by the instrument with what is known to be the correct area. The difference is Z. This should be done a number of times, and a mean value of the several determina- tions used. 160. To find the circumference of the wheel. If n is the number of revolutions, and c is the circumference, Roll of wheel = no. THE SLIDE RULE. 179 In a given circumscribed area with e outside, A = hnc. If c is not known, measure a known area with any con- venient length of arm and note the reading of the wheel, which is n. From the known quantities compute c. This should likewise be done a number of times. 161. Length of arm. It is very convenient to make h such a length as will reduce the work of multiplying hnc to a mini- mum. Most instruments are so made that the length of the arm may be such that To find what this length is for a given instrument in which c is known, let it be assumed that one revolution of the wheel shall correspond to ten square inches ; then 10 = Ac, 10 and 7, A = -. If the arm A is not graduated, it may be set by trial so that A = 10 n and the value c will not be required. Some cheap forms of the instrument are made with the arm h fixed in length. When so made they are usually proportioned so that A = 10 n. The drawing on which the instrument is to be used should be perfectly smooth THE SLIDE RULE.i 162. Described. The slide rule is an instrument for mechanically performing multiplication, division, involution, and evolution. It is merely a series of scales, which are the logarithms of numbers laid off to scale, so arranged that by sliding one scale on the other the logarithms may be mechanically added or sub- tracted. The divisions are numbered with the numbers to which the plotted logarithms correspond. The rule is constructed in many forms, but the principles involved are FIG. 82. 1 Written by C. W. Crockett, C.E., A.M., Professor of Mathematics in the Rensselaer Polytechnic Institute. 180 LAND SURVEY COMPUTATIONS. the same in all. The ordinary rule, about ten inches long, consists of a framework called the rule and a movable part called the slide, arranged as shown in Fig. 82. On their surfaces, which should be in the same plane, are scales at I and IV on the rule, and at II and III on the slide. The initial points of these scales are in a line perpendicular to the upper edge of the rule. A runner, acting on the principle of a T-square, assists in finding points on the scales that are at a common distance from the initial points of the scales. The slide may be inverted turned end for end so that II is adjacent to IV, and III to I, or reversed so that the other side of the slide becomes visible. One form of the slide rule is shown in Fig. 83. FIG. 83. 163. Historical. In 1624 Gunter proposed the use of the logarithmic scale, as shown in Art. 165. In 1630 Oughtred suggested that two scales, sliding by each other, could be used. In 1685 Partridge fastened two scales together by bits of brass, another scale sliding between them. In 1851 Mannheim intro- duced the runner. 164. Construction of the scales. A logarithmic scale is one on which the distance from the initial point to any division is proportional to the mantissa of the logarithm of the number corresponding to that division. The slide rule usually bears two of these scales, constructed as follows : B I T ? f" - ? } ? 9 f~ "? 1 7 i I ? i ? 9 T i ? i 'a 6 c d e f g h i 3 FIG. 84. On scale B, Fig. 84, let a distance of 5 inches represent unity in the logarithm, so that, if a'j' = 5", we have THE SLIDE RULE. 181 log 1 = 0.00; and the beginning of the scale is marked 1. log 2 = 0.30; and at b', so that a'V = 1.5", we mark 2. log 3 = 0.48, " '< c', " " a'c' = 2.4", " 3. log 4 = 0.60; " " d', " " a' Opp. x. ( A. C. Read y. i Inverted. 190 LAND SURVEY COMPUTATIONS. 10. y=*^. x .'. log y (log ft 2 colog a) + colog x. B. Read y. Iny ( A. Seta. Opp. x. \ A . ~C. Opp. h. Ex. a = 15, b = 2 ; x = 3, y = 20.0 ; x = 5, y = 12.0 ; z = 8, y= 7.5; .'. logy = I {(log fc 2 - colog a) + colog a;}. B. j ( A . Set a. Opp- x - I A. C. Opp. b. Read y. In the other two cases the result must be read on the slide, so that the distances- from the left index (1) will be the mantissas of the cologarithms of the corresponding numbers, while the distances on the rule will correspond to the mantissas of the logarithms. 12 , = *. a: 2 .'. colog y = colog a + colog b colog z = (colog a log b) + log x 2 . B. Opp. b. Inv (/l. Seta. Ready. ~CT~ Opp. x. Ex. a = 5, 6 = 12; x=2, y = 15.00; x = 4, #= 3.75; a; = 5, y= 2.4. 13. y = x 2 .'. colog y = colog a + colog 6 2 color x 2 = (colog a - log 6 2 ) + log x 2 . THE SLIDE RULE. I A. C. Opp. b. Opp. x. Ex. a = 15, 6 = 2 ; x = 3, y = 6.67 ; x = 5,' y = 2.4. ' NOTE. Expressions similar to (1), (8), and (10), Art. 173, may be solved iu such a way that the result may be read on the slide A. 176. Use of the runner in complicated expressions. If we have found the value of an expression, say , and marked its place on the rule by placing the runner there, then to multiply it by a frac- tion - we bring the denominator e on the slide to the runner and read the result on the rule opposite d on the slide. For this subtracts log e from log and then adds log d to the remainder. _ abcde _ ab c d e ghk g h k 1 Denoting the runner by R, we have B. Opp. a. Read y. A. Set g. R to b. h to R. R to c. k to R. R to d. 1 to R. Opp. e. EXAMPLES. 4 5 6 . 7_4 - 5 6 7 =g5 2-3-4 "~ 2 ''1 ' 5-6.7-8 5-6781 2-4 5- 6 2 4 8-9 8-91 = 0.5. 2-3-4-6 2 34 An expression of the form _a(m 2 + n 2 + r + s/> + 9 3 ) y ~ b may be written 6 b b b b and the values of the several terms found separately. Notice that one setting will answer for the first three terms, if the results are read on the slide. 192 LAND SURVEY COMPUTATIONS. 177. Gage points. When a formula is often used, as that for computing the horse power of an engine, it will be found convenient to combine the constant factors, sometimes using the resulting constant, and sometimes its reciprocal. Such a constant is called a gage point. Thus the formula for the horse power of an engine is UP _ P x 0.7854 d 2 x 2s x r 33000 where p = mean effective steam pressure in the cylinder in-lb. per sq. in., d = diameter of piston in inches ; s = stroke in feet, r = number of revolutions per minute. 0-7854 x 2 _ 1 d>r 33000 " 21008 ' ~ 21008' where 21008 is the gage point for the formula B. Read H.P. A. Set 21008. R to p. 1 to R. R to s. 1 to R. Opp. r. A. C. Opp. d. 178. Extraction of cube roots. y = j/b. Since ^8. 000 = 2, ^80.000 = 4 +, ^800.000 = 9 +, the cube root of any sequence of figures will depend upon the position of the decimal point, so that the first figure of the root should be known approximately in order that the wrong number may not be taken. This first figure may be found by dividing the given number into sections containing three digits each, commencing at the decimal point, and extracting the cube root of the left section. Then, with the slide inverted, opposite b of B, set one of the extreme indices (1) of A, and find the number on C that is opposite the same number on A. This will be the cube root required. Sometimes the right index of A must be used, at others the left index. Opp. ' SetL PP-y- A. or Opp. y. C. Read y THE SLIDE RULE. 193 179. Slide reversed. The reverse of the slide sometimes has along one edge a scale of logarithmic sines and along the other a scale of logarithmic tangents, the scales being so ar- ranged that the numbers (degrees) increase from left to right when, the slide being reversed, the corresponding edge of the slide is adjacent to B. 180. Scale of logarithmic sines, S. If we reverse the slide and place S adjacent to B so that their initial points coincide, we find that the beginning of B corresponds to 34' + on S, the middle index of B to 5 44' +, and the right index of B to 90; for log sin 34' + = 8.00 ( = - 2), log sin 5 44' + = 9.00 (=-1), log sin 90 = 0.00, unity in the characteristic being represented by the same distance on S and on B. In this position the natural sine of an angle may be found by reading the number on B corresponding to the angle on S. The distance from the beginning of 5 to any division, in the direction of increasing numbers, represents the mantissa, or the mantissa + 1, of the logarithmic sine of the angle corresponding to that division ; and the dis- tance from any division to the other end (90) of S represents the mantissa, or the mantissa + 1, of the cologarithm of the sine. In expressions containing cos x, since cos x = sin (90 x), we may sub- tract the angle x from 90 and then use the sine of the remainder, finding it on S. 1. a = c sin x. .-. logo = logc + log sin x = logc - colog sins. Opposite c of B set the beginning of S, and opposite x of S read a on B, for this adds log sin x to log c. If x should fall outside of the rule, we would set the right end of S opposite c of B, and then opposite x of S read a on B, for this subtracts colog sin x from log c. 2. 6 = ccosx. . . log b = log c + log sin (90 - a:) = log c - colog sin (90 - x), and the methods given for the multiplication of a sine by a number (a = c sin x) are applicable, using 90 x instead of x. 3. c = a -4- sin x. . . log c = log a log sin x = log a + colog sin x. Opposite a of B set x of S, and opposite the end of S read c on B ; for this subtracts log sin x from log a when we read opposite the left end, and adds colog sin x to log a when we read opposite the right end. R'M'D SURV. 13 194 LAND SURVEY COMPUTATIONS. 4. c = b -4- cos x. :. log c = log b - log sin (90 - a;) = log b + colog sin (90 - x), and the methods for c = a *- sin a; are applicable, using 90 z instead of x. 5. a c sin z -4- sin z. .'. log a = log c log sin z + log sin x. Opposite c of B set 2 of 5, and opposite x of 5 read a on .6 ; for this first subtracts log sin z from log c, and then adds log sin x to the remainder. If x falls outside the rule, set the runner over the end of S that is in the rule, shift the slide until the other end of S comes to the runner, and then opposite x of S read a on B. 1 The expression for a becomes c sin x when z = 90, and c -f- sin z when x = 90. Compare the setting just given with those for c sin x and a -f- sin x. 6. sin x = a sin z -T- c. .'. log sin x = log sin z log c + log a. Opposite c of .B set z of 5, and opposite a oi B read a; on S ; for this subtracts log c from log sin z, and adds log a to the remainder. This becomes sin x = a -t- c when z = 90. 7. cos a: = 6 -i- c. .'. log sin (90 - z) = log b - log c = log sin 90 - log c + log b. Opposite c of B set 90 of S, and opposite b of read 90 - x on 5. 181. Scale of logarithmic tangents, T. If we reverse the slide and place T adjacent to B so that their initial points coincide, we find that the beginning of -B corresponds to 34' + on T, the middle index of B to 5 42' + , and the right index to 45 ; for log tan 34' + = 8.00 (=-2), log tan 5 42'+ = 9.00 (=-1), log tan 45 = 0.00. Unity in the characteristic, therefore, corresponds to the same distance on T, A, and S. In this position the natural tangent of an angle less than 45 may be found by reading the number on B opposite the given angle on T. To find the tangent of an angle greater than 45, invert the slide so that T is adja- cent to C, their ends coinciding, and read the number on B corresponding to the complement of the angle on T. The distance from the beginning of T to any division, in the direction of increasing numbers, represents the mantissa, or the mantissa + 1, of the logarithmic tangent of the angle corresponding to that division ; while the 1 In all cases when the result cannot be read, the slide is shifted over the length of its scales, since this merely changes the characteristic. THE SLIDE RULE. 195 distance from that division to the end (45) of the scale represents the man- tissa, or the mantissa + 1, of the cologarithm of the tangent Since tan x = 1 -f- cot z. we have log tan x colog cot x and colog tan x = log cot x. Hence the distance that represents log tan x also represents colog cot x, and that representing colog tan x also represents log cot x. When x is greater than 45, subtract x from 90 and use the cotangent of the remainder instead of tan x, and the tangent of the remainder instead of cot a;. 1. a = b tan x. x < 45. .'. log a = log b + log tan x = log b colog tan x. Opposite b of B set the end of T, and opposite x of T read a on B ; for this adds log tan x to log b when the left end is used, and subtracts colog tan x from log b when the right end is used. x > 45. .'. log a = log b + log cot (90 - x) = log b - log tan (90 - x) = log b + colog tan (90 - z). Opposite b on B set 90 - x of T, and opposite the end of T read a on B; for this subtracts log tan (90 z) from log b when the left end is used, and adds colog tan (90 - x) to log b when the right end is used. 2. b = a -f- tan x. x < 45. .'. log b = log a log tan x = log a + colog tan x. Opposite a of B set x of T and opposite the end of T read b on B ; for this subtracts log tan x from, or adds colog tan x to, log a when we read opposite the left or the right end respectively. x > 45. /. log b = log a - log cot (90 - x) = log a + log tan (90 - x) = log a - colog tan (90 - x). Opposite a of B set the end of T and opposite 90 -x of T read b on J5; for this adds log tan (90 - x) to, or subtracts colog tan (90 - x) from, log a when we use the left or the right end respectively. 3. b = a cot x. /. 6 = a -=- tan x, and the settings given under (2) are used. 4. b = a +- cot x. :.b = a tan x, and the settings given under (1) are used. 196 LAND SURVEY COMPUTATIONS. 182. The Thacher rule. This rule (Fig. 92) consists of a woovlen base bearing two upright metallic standards with a large circular opening in each, the line joining the centers of the openings being perpendicular to the planes of the standards. Attached to the standards are two circular plates of metal, FIG. 92. each with a large circular opening concentric with those in the standards, so arranged that they can revolve around the line joining the centers of the openings as an axis. These plates are united by twenty bars a little more than 18 inches long, triangular in section, and perpendicular to the plates. These bars are arranged at equal dis- tances around the circular openings, with their vertices outward so that their bases form a cylindrical envelope, the distances between the bars being ap- proximately equal to the width of their bases. A cylindrical slide fits in this cylindrical envelope, moving with either a rotary or a longitudinal motion. The system of bars and plates may be rotated about the axis of the cylindrical envelope without disturbing the relative position of the bars and the slide. The system of bars is the rule, and the cylinder is the slide. The latter bears only one scale A, while the former contains the scales B and C. The Thacher rule is equivalent to a carpenter's rule in which the slide bears only one of the two A scales. The arrangement of the scales is practically the same as that in a straight rule of the form shown in Fig. 93. The scale A is laid down on the cylinder as follows : THE SLIDE RULE. 197 Let M and N be two logarithmic scales 360 inches long, and let each be divided into forty equal parts, a, b, , a', b', . Draw forty equidistant elements on a cylinder and lay off on them the segments of the scales as shown in Fig. 95, which represents the development of the cylinder. Then the scales will read continuously from left to right. The twenty bars are graduated on each side and bear two different FIG. 94. scales. The one nearer the cylindrical slide B is constructed in the same way as that on the slide. The outer one C is formed by dividing a loga- rithmic scale 720 inches long into eighty equal parts and placing them in order above a, 6, -, k, 1; a', b', ..., k', V. The Thacher rule gives results that should never be in error by more than one unit in the fourth significant figure, while the fifth figure can often be found with only a small error. The error should not exceed one part in ten thousand, so that this rule is intermediate in accuracy between a four- and a five-place logarithmic table. 183. Settings for the Thacher rule. The settings given for the carpenter's rule will answer for the Thacher rule. 1 All the cases that can be solved with a single setting of the instrument have been men- tioned. Problems that can not be solved with one setting may sometimes be easily computed by the use of the runner, and sometimes by computing the different * 1Q * ^ parts of the expression, reading the results, and then combining them with the aid of the rule. 184. Fuller's slide rule. This rule is shown in Fig. 96. The hollow sleeve (7, which bears the graduations, is capable of sliding along and revolving about the continuous cylinder HH, the latter being held by the handle attached to it. F is 1 Mr. Thacher has devised two simple statements that give the settings for all the expressions in Art. 173 with the exception of the third. 198 LAND SURVEY COMPUTATIONS. a fixed index fastened to H, and by moving the sleeve, any division of the scale may be made to coincide with F. The cylinder H is hollow, forming a guide for the motion of a third cylinder that is attached to the flange 6r, its axis being coinci- dent with the common axis of and H. A and B are two indices fixed to and moving with 6r, the distance AB being FIG. 96. equal to the axial length of the scale. This scale, 500 inches in length, is wrapped on the sleeve O in the form of a helix, its beginning being at the end towards 6r. Stops are provided, so that the indices may be readily made to coincide with the beginning of the scale. To illustrate the use of the indices, 1 suppose that we wish to find the value of a x ft. Move the sleeve C until a is opposite F, and then by mov- ing G, place A at the beginning of the scale ; the distance from A to F, along the helix, will be log a. Move the sleeve C, being careful not to change the relative positions of G and H, until b is opposite A ; then the distance along the helix from the beginning of the scale to A is log b. Hence the distance from the beginning of the scale to F, along the helix, is log b + log a = log ab, so that the product will be found on the scale op- posite jP. 185. The Mannheim rule. With the Mannheim rule (Art. 166) we can find with one setting the value of any expression in a fractional form with two factors in the numerator and one in the denominator, one of the numbers being variable and one or all of the numbers being squared, and also the value of the square root of such an expression, the result being always read on the rule. The carpenter's and the Thacher rules do not possess this power, since the slide does not bear a scale similar to 0. 1 This is intended only to illustrate the principle of the instrument. BOOK II. GENERAL SURVEYING METHODS. CHAPTER VII. LAND SURVEYS. 186. Obstacles and problems. In the field work of the sur- veyor, various obstacles arise which must be overcome. A few of the more common difficulties, with the methods of surmount- ing them, will be given. I. In Fig. 97 it is required to produce the line XA in distance and direction. (1) At A erect the perpendicular AC, to which erect the perpendicular CD, which make long enough to pass the obstacle. At the point D erect the perpen- r- dicular DB equal to AC, and at B erect the perpen- dicular BY, which is the line pro- duced. The dis- tance A B equals CD. (2) At A set off an angle of 60 and measure A C of such length that a line making an angle of 60 with AC will pass the obstacle. At C set off from A C an angle of 60 and measure CB equal to A C. At B set off the angle ABC 60. The instrument will then be in the line XA produced, pointing toward A, and the distance AB equals A C or CB. (3) At A set off any convenient angle BA C and measure any distance AC. At C set off any angle ACB, making it FIG. 97. 202 LAND SURVEYS. 90 if convenient, and measure CB, a distance determined by solving the triangle ABC, and at B set off an angle ABO de- termined in the same way. FIG. 98. II. In Fig. 98 it is required to determine the distance AB, B being an inaccessible point, or the stream too wide to be spanned by a single chain or tape length. (1) At A set off the angle BA C equal to 90 and measure A O a convenient distance, not so short as to make the angle FIG. 99. ABC less than 20, unless this is unavoidable, and at C measure the angle BOA. Solve the right-angled triangle. (2) At A set off any convenient angle BA C and measure any convenient distance AC, observing the previous caution TWO COMMON PROBLEMS. 203 with regard to the angle at B. Measure the angle at C and solve the triangle ACB. III. In Fig. 99, it is required to determine the length and bearing of the inaccessible line AB. (1) Measure the line CD, from each end of which both A and B may be seen, and determine its bearing. With the instrument first at O and then at D measure three angles at each point. Solve the triangle A CD for AC, BCD for CB, and ABC for AB, and an angle at A and B. Another com- bination of triangles could be used. IV. In Fig. 100, it is required to determine the length and bearing of the line AB. FIG. 100. (1) Run the random line AbcdB, noting bearings and distances. If the survey is treated as a closed field, AB will be the error of closure, or the closing line, or a wanting side which may be fully determined in the manner previously given for finding the bearing and length of a wanting side. 187. Two common problems. I. In Fig. 101, it is required to determine the lengths and bearings of the sides of the field ABCDEF, the lines being through woods and no two corners visible, the one from the other. The positions of the corners are, however, known. Run the random line AabBcdCefDgEhikFIA, and com- pute the latitudes and longitudes, or the coordinates, of the points ABCDEF, referred to the true or any convenient meridian, preferably the true meridian through the most westerly point of the survey. From these coordinates deter- 204 LAND SURVEYS. mine the latitude and longitude differences and the bearing and length of each course. II. If a corporation, as a mining company, proposes to purchase a considerable tract of ground containing many small parcels owned by different individuals, and a description of each parcel with its area is wanted, there are two ways of obtaining the information. (1) Each parcel may be separately surveyed, determining the bearings and lengths of the courses and the areas. (2) By far the quicker and better way is to run a random field, touching on the various corners of the various parcels, to compute the coordinates of these corners, and from these the bearings, or azimuths, and the length of the sides and the areas. If a separate description of each piece is not required, but only the area, the computation of bearing and length of course may be omitted and the areas may be obtained at once from the coordinates of the corners. SURVEYING WITH THE CHAIN ALONE. 188. In most surveys the chain or tape is used in connec- tion with some instrument for measuring angles, since, when the sides and angles of a polygon (a field) are known, the polygon may be drawn and its area computed. But it is frequently convenient in approximating to make an entire sur- vey, usually of a very small tract, with the chain alone. This entire survey may be a part of a larger survey, but is, never- SURVEYING WITH THE CHAIN, 205 theless, being a closed survey, complete in itself. A method of making such a survey will therefore be described briefly, intro- ducing some methods that are employed as well when the com- pass or transit is used with the chain. 189. Preliminary examination. Let Fig. 102 represent the map of a farm, a survey of which is desired, and let it be sup- posed that there is no instrument available except a chain or tape. It will of course be impossible to determine bearings. It is assumed that it is the area that is desired. The first thing to do in any land survey is to make a rough sketch of the tract to be surveyed, drawing it as nearly as pos- sible in correct proportion, from an inspection made by walking over the field, or from a description of the field taken from the deed, if one can be obtained. From the deed will be obtained only the description of the boundary ; and the other features that may be desired must be sketched in the field. On inspec- tion it is found that Mr. Miller owns a farm bounded on one side by the center of the road, on another by the Green River, and on three remaining sides by fences, broken on one side by a pond, which is owned partly by Mr. Miller and partly by his neighbor. But one convenient way to get the area of the field is known to the surveyor, and that is to divide the field into triangles, measure the sides of the triangles or two sides and an included angle of each, whereupon the area of each may be computed and the whole summed. It is evident in the case of the Mil- ler farm that what might be considered as two sides, those formed by the river, are not straight, and therefore can not be taken as sides of a triangle. Two auxiliary sides, DE and EF, are chosen, lying as nearly as may be parallel to the two sides. It is found impossible in the field to choose the point E so that A may be seen from it, hence it is so chosen that it shall lie on a perpendicular to the line A F drawn through F. This makes the triangle AFE a right-angled triangle, and hence AE need not be measured. If the house were so located that the line BF could be conveniently laid out, it might be measured and the two triangles ABF and BFE might be used instead of AFE and AEB. 206 LAND SURVEYS. 190. Survey. To begin the survey, select a point, F, in the center of the road, and, placing a flag in the fence at B, or merely sighting along the fence, if it is straight, locate a point in the center of the road at A, in Ba prolonged. Measure FA. Measure AB. If the fence aB is not straight, or is a, MAP OF FARM OF E.MILLER. PUYALLUP CO., WASHINGTON, Kote: Surveyed by A.B.Wooct with chain nnltf Oct.i, I8t. Scale; 2 chains per inch . FIG. 102. rough rail fence, so that it is not convenient to measure on the line AB, measure along a parallel line offsetting a short dis- tance from the fence, making no note of the offset, but record- ing merely the length AB and the distance from A to a. Meas- ure along the line BO. or on a small offset parallel to it toward (7, until the pond is reached. Note the distance, and erect a SURVEYING WITH THE CHAIN. 207 perpendicular offset long enough to permit a line parallel to BO to pass the pond. At the extremity of this offset erect a perpendicular which will be parallel to BC, and measure along the perpendicular far enough to clear the pond, and then, by a process like that just used, get back on the line CB. When measuring along the parallel line, take offsets to the pond as often as may be necessary, noting the distance to each offset arid the length of the offset. Having reached C, measure along the line CD to D, then along DE to E, noting the dis- tance to and the length of each offset that it appears necessary to take to correctly locate the river; and having reached E, measure EF, noting similarly the offsets and also the distance to the fence at the roadside. Perhaps the ordinary method of procedure now would be to measure the diagonals DB and BE ; then the field would be divided into four triangles, ABE, BCD, BED, and AEF, the areas of which could be computed, and to their sum, the area between the lines DE and EF and the river (computed as described in Chapter VI.) could be added, and the area of the entire field would thus be obtained. The line DB would be a difficult one to run, because both D and B being lower than the ground between them, the line must be ranged out, and, more- over, the intervening woods make it even more difficult to determine. If the woods were thick, it could be obtained only by running a trial line and correcting that to the true line. Moreover, the method above described gives no check on the work ; and the constant thought of the surveyor should be, " Where can I find a check on my work ? " Not more than one check on one piece of work is, however, necessary ; though, if more may be obtained without waste of time, they may some- times prove advantageous. Having measured the boundary of the field, it is probably best now to measure the line AC, finding by trial the points at which perpendiculars to this line will pass through F, B, E, and D. Measure these perpendiculars, and now four sides of the boundary are the hypotenuses of right-angled triangles, of which the other two sides are known. The other two sides are sides of trapezoids, or may be considered the hypotenuses of right triangles. The computed value of each of the sides of 208 LAND SURVEYS. the boundary should agree with its measured length. If it does not, there is an error in the measurement of either the line AC, one or more of the perpendiculars, or one or more of the sides of the boundary. If all is right except one side, the error is in that side. The work should then be remeasured in so far as it may be found wrong. If this latter method is not adopted, a rough check may be had on the former method by measuring the angles made by the sides. These should meas- ure on a drawing the same as found in the field. The work necessary to obtain the area of the farm is now completed. If it is desired to locate the drives, buildings, and other objects within the inclosure, it may be done by running auxiliary lines at known angles and from known points on the various sides. For instance, a line could be run from E, at right angles to EF, and at stated points on this auxiliary line offsets could be measured to the corners of the objects it is desired to locate ; or if it were the driveway, the offsets could be taken at frequent noted intervals to the sides of the drive. As the point D can not be seen from the line A 0, a point may be chosen at random, as near as possible to the proper position, and a perpendicular run out to a point opposite .Z), whereupon the length of the perpendicular and the distance that the point on AC must be shifted to be in its proper place become known. 191. Notes. The measurements should be written on the sketch, which must be made large enough to permit this to be done without confusion. It is believed that no form of notes for such work is so good as a sketch on which all information is written. For the auxiliary lines locating drives, etc., the sketch would consist of a straight line, with distances to offsets marked along it, and offsets and objects offsetted to sketched in with dimensions. These need not be to any scale. FARM SURVEYS. 192. Classes. All land surveys may be divided into three classes, original surveys, resurveys, and location surveys. These surveys are made with a compass or transit. The sur- veys of former years, in the older settled portions of the United FARM SURVEYS. 209 States, were all made with the compass, and almost always with reference to the magnetic meridian. 193. Original surveys. Original surveys are those made for the purpose of mapping a" field whose boundaries are marked in some way, for determining its area, and for making a descrip- tion from which it could be again laid out if the boundaries should be destroyed. Thus, Mr. Brown may own a farm, a portion of which is timber land, which Mr. Black wishes to buy. The boundaries are sufficiently marked by the edges of the growth of timber. After the timber is gone, however, there will be nothing to mark the boundaries, unless the tract is fenced. There is no way in which Mr. Black may know how many acres he is to buy, unless the tract is surveyed. Neither can Mr. Brown give Mr. Black a deed to the property that would contain a definite description from which the tract could be laid out on the ground. A surveyor is called in and shown the tract and asked to make a survey of it, to compute the acreage, and to write a description of the plot. 194. Making an original survey. At the corners that are shown him he sets monuments, preferably of stone. Too many surveyors set merely small stakes, which soon rot or are pulled out. These monuments are " witnessed " by trees, or other natural objects near by, whose positions relative to the corner are observed and noted. The witness points are marked as indicated in Art. 196. The surveyor then determines the bearing and length of each side, usually beginning at one corner and working round the field in one direction till he closes on the corner from which he started. He keeps notes of the work in a notebook, preferably in the form shown on page 229, or on a sketch, which is afterwards "written up" in the form just mentioned, and from the notes he " tables " the sur- vey and computes the area as explained in Chapter VI. 195. Making the map. He may, and usually does, make a map of the survey. This map should conform to the require- ments stated in the Appendix, page 355. The easiest and most rapid method of plotting the map is by " latitudes and longi- tudes." Two reference lines at right angles are drawn, and K'M'B simv. 14 210 LAND SURVEYS. one is assumed as the meridian and the other as a line of zero latitude. The total latitudes and longitudes of the corners are determined from the tabling work, arid are laid off from these base lines. The latitudes are measured from the zero parallel, and the longitudes from the meridian. It will be convenient to make the reference meridian pass through the most westerly point of the survey. This is not necessary. To avoid negative signs for latitude, compute all latitudes as if the reference par- allel were through the same point as the meridian ; then add to all latitudes a sum equal to the greatest negative latitude; or assume the latitude of the most westerly point to be so large that there shall be no negative latitudes. When the corners are all plotted they are connected by right lines, and the outline is complete. It remains to number the corners, to write the bearings and lengths along the sides, and to put on the necessary descriptive matter. The work should all be done with the utmost neatness, the lettering being preferably the simple Roman, the most effective and most diffi- cult letter that is made. 196. Description. Having made the map, the surveyor writes a description, somewhat in the following fornr: Beginning at a post marked B.I., at the S.E. corner of the land of Joseph Brown, from which post a hard maple tree, 8 inches in diameter, bears S. 10 W. 10 links, and a white ash, 12 inches in diameter, bears N. 70 W. 50 links, both of which trees are blazed and marked B.l.B.T. (Black 1 Bearing Tree), and running thence N. 10 30' W. along the easterly line of said Joseph Brown, six and forty-two one-hundredths ( 6-j 4 ^- ) chains to a post marked B.2., from which post a hickory tree, 10 inches in diameter and marked B.2.B.T. bears S. 68 W. 30 links ; thence N. 84 W. seven and fifty one-hundredths (7i 5 ooO chains to a stone about 12 inches long and 6 inches square, set flush with the ground and marked B.3., from which stone a beech tree, 15 inches in diameter and marked B.3.B.T. bears, etc., thence, etc., to the point of beginning, containing acres, more or less. The surveyor usually does his tabling and writes his descrip- FARM SURVEYS. 211 tions in a book which he keeps for the purpose, for future reference. Copies are made, and, with a tracing of the map, are furnished the person for whom the survey is made. This is the simplest kind of land surveying. 197. Resurveys. These are far more difficult than original surveys. A resurvey consists in tracing on the ground an original survey, from a description similar to that just given. The difficulties arise from the destruction of monuments and from errors in original work, from change in declination of the needle (if the surveys were run by the magnetic compass and referred to the magnetic meridian), from insufficient data in the original description, such as failure to state whether the survey is referred to the true or magnetic meridian, failure to state the declination on which the survey was made, lack of bearing trees or other reference points, etc., and from conflict- ing testimony of interested owners as to where the corners were. It is impossible to point out all the difficulties that will be encountered by a surveyor in his attempt to reestab- lish the monuments of an old survey. The more of this work that he does, the more firmly will it be impressed upon him that the only kind of corners to establish are those that will be as nearly as possible permanent, and that minuteness of detail and accuracy in all descriptions are well worth the time they take. 198. Reasons for a resurvey. A resurvey becomes necessary for various reasons. Among others might be the following, referring to the original survey previously described: After Mr. Black has owned the wood lot for a number of years, and perhaps has cleared it, and it has descended to his son and his son's son, and the corners are mostly obliterated, it is sold to Mr. Johnson who, desiring to fence it off, and, moreover, to see whether the land he is paying for is all there, employs a surveyor to " run it out," giving him the description written by the original surveyor as he finds it in his deed, with possibly some errors in copying. 199. Procedure. If the surveyor can recover a single line of the original survey, and the notes are correct, his work will be 212 LAND SURVEYS comparatively easy. (He simply has to establish a series of lines of given bearing and length.) He therefore endeavors to do this. If this can not be done, the next best thing is to recover any two corners and determine in the field the bearing and length of the line joining them. From the original notes he will then compute the bearing and length of this line, and the angle made with it by one of the sides of the field joining it, and he then can lay off this angle from his field-determined line and locate that side. As there will be four sides joining the line between the two corners, it will be seen that the sur- veyor has a good beginning for his work. 200. Change of declination. It might seem that if he could find but one corner, he could run out the field, knowing the bearings. This would be true if he knew also the declination on which the original survey was made and the declination at the time of his resurvey. The following consideration will explain this : Let it be assumed that the original survey was made with reference to the magnetic meridian and that the declination at that time was east 10. Then, any line that is recorded due north will be 10 east of north. Any line recorded due south will be 10 west of south, etc. That is, all the points of the compass are turned 10 to the right. Let it be assumed that the declina- 'tion at the time of the resurvey is 8 east. A line recorded as due north will be 8 east of north, and a line recorded as due south will be 8 west of south. The points of the compass have been turned back two degrees to the left. And hence, if a line were run out with the original bearing, it would lie two degrees to the left of the true place. It would be necessary to run the line with a bearing two degrees to the right of its originally recorded bearing. Thus, if its original bearing were north, it must be rerun with the needle reading N. 2 E. Hence, the following rule is serviceable : RULE : To rerun lines recorded by their magnetic bearings, change the bearings by an amount equal to the change in declina- tion and in a direction opposite to that change that is, left or right. If the compass has a declination vernier, that vernier may FARM SURVEYS. 213 be set so that the compass, when the line of sight is pointing in a known bearing, as magnetic north, shall read a bearing as many degrees to the right or left of the known magnetic bear- ing as the declination has changed to the right or left. It must be remembered that, in using the declination vernier, the compass box moves with reference to the line of sight. In the above case it should read N. 2 W., when the sights are point- ing to magnetic north. Hence, to be able to retrace the survey with a compass, using the original bearings, the following rule is serviceable : RULE : Set the declination vernier so that the compass shall read magnetic bearings erroneously by an amount equal to and in the direction (left or right) of the change in declination. To find the change in declination, if it is not known, deter- mine the bearing of a line of the original survey and compare this with the bearing recorded. The difference is the change in declination in amount and in opposite direction (left or right). If one line can not be found, but two corners may be, con- nect the two corners by a random line as described in Art. 186. Find the bearing of the closing line and compare it with the bearing of the same line computed from the original notes. 201. Transit or compass. When a transit is used, or in- deed a compass, the angles at the corners of the field may be determined from the recorded bearings and, when one side has been recovered, the angles and distances may be measured, to recover the other sides. This does away with all consideration of, or change in, declination. It is by far the better method to pursue with lands of any considerable value. The angles de- termined from compass readings can not be depended on to minutes, and may be found to vary five or ten minutes. It would be well if, when a transit is first used on a resurvey, the corrected bearings and distances could be introduced in a cor- rection deed which could be filed with the proper authority. If angles are to be used, and one side can not be directly re- covered, but a closing line joining two corners is determined, proceed as follows: Over one known corner, set the line of 214 LAND SURVEYS. sight in the closing line mentioned, and turn off an angle deter- mined, by computation from the description, to be the angle between this line and a side adjacent to the corner occupied. This should give the direction of that side. Measure the recorded length of that side, and look about over a consider- able area for evidences of the corner. Look for recorded wit- ness trees, or their stumps. Carefully shovel off the top of the soil do not dig it up by spadefuls. A careful shoveling will frequently reveal the hole formerly occupied by a stake, now rotted entirely away, with evidences of the decayed wood. Continue the work till all corners are found or satisfactorily relocated. The only thing that can absolutely insure the correct- ness of the resurvey is the finding of the old corners. 202. Report. Upon the completion of the resurvey, the surveyor should report to his employer just what he finds. It is not his business to decide controversies. He may advise, just as an attorney would do ; but he has no authority to cor- rect errors or to establish corrected corners as the corners. 1 His business is to make an examination, to reset lost corners in their original positions, if he can find them, and to report his method of procedure and the reasons for his action to his em- ployer, who may then take such action as he chooses. A neat arid explicit map should accompany the report. The surveyor may possibly be assisted in his work by an understanding of the principles of Art. 204, deduced from many court decisions. 2 203. Application of coordinates. In all of this work the coordinate system is very helpful. Usually the true corners of a tract of land can not be occupied by the instrument, nor can the lines be seen throughout their length. If it is known where the corners are, and it is desired to make a survey for a map or a description, a random survey is made with corners as near as practicable to the true corners. . This survey is bal- anced and the coordinates of its corners are determined. From such of the corners as are near true corners, angles or azimuths, 1 See "Judicial Functions of Surveyors," by Judge Cooley, Appendix, pages 341-350. 2 These and many other decisions may be found in a valuable " Manual of Land Surveying," by Hodgman and Bellows. FARM SURVEYS. 215 and distances to the true corners are noted, and from these the coordinates of the true corners are determined. From these coordinates, lengths and bearings of true lines may be found. If the survey is for the purpose of relocating lost corners, and the bearings of the lines and at least one point in each are known, the corners may be located even though the lines can not be occupied by the instrument, provided the corners and known points are accessible. This would be done by running a random survey as before with pointings to the known points (and such corners as are known) then finding the coordinates of the known points, and by Problem II., Art. 143, the coor- dinates of the wanting corners. Knowing now the coordinates of the points in the random survey and those of the true corners, by Problem I., Art. 143, find the bearing and distance from a corner of the random to the nearest wanting true corner, and locate the corner. 1 204. Principles for guidance in resurveys. I. Construing de- scriptions. The following principles have been applied to the construing of descriptions that are inconsistent, obscure in meaning, or imperfect. (1) The description is to be construed favorably to the purchaser, unless the intent of both parties can be certainly ascertained. If that intent can be ascertained, the description will be construed accordingly. (2) The deed must be construed according to the condi- tions existing, and in the light of the facts known and in the minds of the parties, at the time the instrument was drawn. (3) Every requirement of a description must be met, if pos- sible. Nothing is to be rejected if all requirements are mutually consistent. (4) If some parts are evidently impossible and, by rejecting such parts, the remainder forms a perfect description, such impossible parts may be rejected. (5) A deed is to be construed so as to make it effectual rather than void. 1 For model example, applied to city property, see Appendix, page 328. The stu- dent may tell how to proceed if the bearings and lengths are all given, with but two corners known, and lines so grown over or occupied by structures that they can not be run out directly without great labor, 216 LAND SURVEYS. (6) If land is described as that owned and occupied by an individual, the actual line of occupation is a requirement or call to be met in the location. (7) A line described as running a definite distance to a definite known line or object, will be construed as running to that object, whatever distance is required. If the known object is uncertain as to position, the written distance may be used. (8) The terms " northerly," " southerly," " easterly," and " westerly," are to be construed, in the absence of other infor- mation, as meaning due north, south, east, and west. (9) When a definite quantity of land is sold and nothing appears to indicate its form as, for instance, ten acres in the northeast corner of B's land, the land will be laid out as a square, unless this is manifestly impossible. (10) A description by " metes and bounds " will convey all the land within the boundaries, be it more or less than the area mentioned in the deed. (11) Property described as bounded by a highway extends to the center of the highway, unless specifically noted other- wise. (12) A description by metes and bounds, followed by a statement that the land described is a particular well-known parcel, will be construed to convey the well-known parcel, though the metes and bounds do not fulfill the necessary conditions. II. Water boundaries. (1) Local laws of different states give different constructions to the word " navigable " and the surveyor must examine the laws of the state in which he works. The United States statutes provide as follows for the streams within the area known as the public lands : " All navigable rivers, within the territory occupied by the public lands, shall remain and be deemed public highways ; and, in all cases where the opposite banks of any streams not navigable belong to different persons, the stream and the bed thereof shall become common to both." (2) Grants of land bordering on navigable streams carry only to high-water mark, while on non-navigable streams they carry to the center, or "filum aquee." FARM SURVEYS. 217 (3) The common law holds those streams only to be navi- gable in which the tide ebbs and flows. The civil law con- siders a stream navigable that is capable of being used as a commercial highway. The courts of Pennsylvania, North Carolina, South Carolina, and Alabama follow the civil law, while those of Maine, New Hampshire, Massachusetts, Connecticut, New York, Maryland, Virginia, Ohio, Illinois, Indiana, and Michigan follow the common law. (4) The bank is the continuous margin where vegetation ceases. The shore is the sandy space between it and low- water mark. (5) A description reading "to the bank," or "along the bank," is construed to mean "the bank," and to include no portion of the stream. (6) Islands in rivers fall under the same rule as the land under water, and belong to one adjoining proprietor or the other unless previously lawfully appropriated according as they are on one side of the center or the other. The filum aqute is midway between lines of ordinary low-water mark, without regard to the position of the main channel. (7) Riparian rights, unless expressly limited, extend to the middle of the navigable channel. (8) In some states the tide lands are held to belong in- alienably to the people of the state and may not be sold to individuals. In others a different policy has been pursued. (9) A boundary by the shore of a millpond carries to low- water mark. (10) Boundary lines of lots fronting on a river extend into the river at right angles to the thread of the stream, without regard to the form of the bank. (11) Land made by the drying up of a lake or the deposit of alluvium along a river accrues to the adjacent owners and should be so distributed among them that each will receive such a portion of the made area as his former frontage on the water was of the entire former frontage. If, however, the water front is the valuable item, as it would be along a navi- gable river or lake in a city, the new frontage is to be dis- tributed according to the old frontages. 218 LAND SURVEYS. III. Special field rules. (1) Monuments control courses and distances. That is, if the location of an original monu- ment can be certainly ascertained and the recorded distance does not reach that monument, the line must, nevertheless, be run to the monument. In the absence of sufficient evidence to determine the monument, the description will govern. (The surveyor should use every effort to find evidence as to the location of the lost corner.) (2) Adverse possession of land for a definite period of time (varying in different states), even without color of title, constitutes title in fee ; but the possession must be adverse ; that is, the true line must be known to the parties or the line of occupation must be acquiesced in by them. If the true boundary is unknown, and each claims to own only to the true line, no adverse possession can arise. (3) Boundaries and monuments may be proved by any evidence that is admissible in establishing any other facts. (4) A resurvey after original monuments have been lost is for the purpose of finding where they were, and not where they should have been. (5) Purchasers of town lots have a right to locate them according to the stakes which they find planted and recognized, and no subsequent survey can be allowed to unsettle them. The question afterwards is not where they should have been, but where they were planted with authority, and where lots were purchased and taken possession of in reliance on them. (6) Of two surveys that disagree, made many years apart the monuments being lost the original survey will be pre- ferred, particularly if the line of the first survey has remained unquestioned for many years. (7) When streets have been opened and long acquiesced in, in supposed conformity to a plot, they should be accepted as fixed monuments in locating lots or blocks contiguous thereto. (8) A beginning corner is of no greater dignity or impor- tance than any other corner. (9) A call for a lot by a name or number that it bears on a mentioned plot will prevail over courses and distances, and sometimes over monuments. UNITED STATES LAND SURVEYS. 219 205. Location surveys. These consist in laying out on the ground lines previously determined by computation or draw- ing. Such surveys are not infrequently connected with either original surveys or resurveys. Surveys for the partition and division of land are of this class. If Joseph Brown sells to John Black five acres of land in the shape of a square, one corner and the direction of one side being fixed, the location of the sides and corners would be a location survey. Such surveys are comparatively simple, involving sometimes the running of one or more trial lines for data to compute the location. Many complex problems, however, arise. UNITED STATES PUBLIC LAND SURVEYS. 206. Value and character of work. The original surveys for the subdivision of the public lands of the United States are location surveys. The method adopted, imperfectly as the work has been done, has been of incalculable value in definitely describing each separate tract or parcel of land sold to individ- uals, and in providing that lines once established by the United States deputy surveyors shall remain as the lines they purport to be, even though found to be improperly placed. The latter provision has been of particular value in making the land lines permanent. The work of subdividing the public lands is almost completed, and hence the work of the surveyor of the future will be largely resurveying, that is, relocating corners that have been or are supposed to be lost, and dividing into smaller par- cels the areas already located. If the original surveys had been properly executed, the work would not be difficult. In some instances the work was most wretchedly done, either willfully or through ignorance ; and the corners, when established, were placed far from their proper positions. The notes have generally been returned in a form indicating correct work ; and hence has arisen more or less difficulty in relocating the corners. 207. General scheme of subdivision. Certain points have been selected in different parts of the country through which true north and south lines, called " principal meridians," have been run and marked out on the ground. Intersecting these prin- 220 LAND SURVEYS. cipal meridians at the initial point are run parallels of lati- tude, known as "base lines." On either side of these meridians the land is laid out in approximately square parcels, six miles on a side, called "townships." A tier of these townships run- ning north and south is called a "range." The townships are described as being " Township No. south or north of a named base line and range No. east or west of a named principal meridian." This definitely locates every township. The lines that divide the ranges are called "range lines," and those that divide a southern from a northern township are called "township lines." Each township is further divided into thirty-six " sections," each approximately one mile square. These sections are numbered from one to thirty-six. Each section, then, is definitely located by its number, township, and range. The sections have been further divided into quarter sections as, the northeast quarter, the southwest quarter, etc. Sometimes the sections have been divided into halves, described as the north half or the east half, etc. The Government does not divide the land into smaller divisions than quarter sections, but it sells less areas than this, and, in such cases, and when original purchasers sell a portion of their purchase, it is usual to sell a quarter of a quarter, or half of a quarter, or even a quarter of a quarter of a quarter section ; and the method of describing these fractional portions is the same as that used to describe the quarter sections. The description would be writ- ten as follows for the piece described : The N. E. ^ of the S. W. of Sec. 26, Tp. 8 N., R. 4 E., Mt. Diablo Meridian. The positions of the meridians being chosen more or less at random and at different times, as the necessity for surveys in different localities developed, it is to be expected that the surveys extending east from one meridian will not close with regular full sections or townships on the surveys extended west from the next easterly meridian. The same is true of tiers of townships extended north and south from adjacent base lines. The result is fractional townships and sections. Some- times these are larger than the standard division, sometimes smaller. If very much larger, the surplus in a township is divided into lots which are numbered. These lots are made to contain as nearly as possible 160 acres. There are other UNITED STATES LAND SURVEYS. 221 circumstances that will appear, that cause a departure from the ordinary method of subdividing and describing. The work in a given state is under the direction of a United States surveyor general. The work has all been done by contract, a surveyor taking a contract to perform a definite portion of work for a specified sum per mile. This has been the principal cause of much bad work. The lands classified as public lands and subdivided accord- ing to the method outlined include all land north of the Ohio River and west of the Mississippi River, except Texas, and including Mississippi, Alabama, and Florida, except in all the above-named territory such lands as belonged to individuals at the time the territory became a part of the United States. The public lands of Texas are, by a provision of the laws admitting Texas into the Union, the property of the state. No general scheme for w their subdivision -has been developed. They have been sold in parcels as nearly square as may be. The Spanish vara is the unit that has been adopted w for measurement. The vara is, in Texas, 33^ inches. There are 5645 square varas in an acre. The following is very much condensed from the s " Manual of Surveying FlG - 103 - Instructions " issued by the General Land Office in Washington. 208. Historical note. The first surveying of the public lands was done in Ohio under an act passed by Congress in 1785. The territory included in this early survey is now known as "The Seven Ranges." The townships were divided into thirty-six sections one mile square, numbered consecu- tively, as in Fig. 103. In 1796 the method of numbering sections was changed to that shown in Fig. 104 , and this method is still in use. 36" 30 24" 18 12 6 35 29 23 17 11 5 34 28 22 16 10 4 33 27 21 15 9 8 32 26 20' 14 8 2 31 25 19 13 7 1 222 LAND SURVEYS. 6 5 4 3 2 1 7 8 9 10 11 12 18 17 16 15 14 13 19 20 21 22 23 24 30 29 28 27 26 25 31 32 33 34 35 36 s FIG. 104. An act of 1805 directs the subdivision of public lands into quarter sections, and provides that all the corners marked in the public surveys shall, be established as the proper corners of sections which they were intended to designate, and that cor- ners of half and quarter sections not marked shall be placed, as nearly as possible, " equi- distant from those two corners which stand on the same line." This act further provides that "the boundary lines actually run and marked . . . shall be established as the proper boundary lines of the sec- tions or subdivisions for which they were intended; and the. length of such lines as returned by . . . the surveyors shall be held and considered as the true length thereof. ..." An act of 1824 provides " that whenever, in the opinion of the President of the United States, a departure from the ordi- nary mode of surveying land on any river, lake, bayou, or water course would promote the public interest, he may direct the surveyor general in whose district such land is situated ... to cause the lands thus situated to be surveyed in tracts of two acres in width, fronting on any river, bayou, lake, or water course, and running back the depth of forty acres." l An act of 1820 provided for the sale of public lands in half- quarter sections, the quarters to be divided by lines running north and south ; and an act of 1832 provided for the sale of the public lands in quarter-quarter sections, and that the half sec- tions should be divided by lines running east and west. The latter act also provided that the secretary of the treasury should establish rules for the subdivision of fractional sections. 209. Legal requirements inconsistent. Existing law requires that the public lands be laid out in townships six miles square by 1 Let the student determine the width and depth in chains of such a strip. This provision is carried out where the water front rather than area is the valuable item. UNITED STATES LAND SURVEYS. 223 lines running due north and south, and others east and west, also that the township shall be divided into thirty-six sections, by two sets of parallel lines, one governed by true meridians and the other by parallels of latitude, the latter intersecting the former at right angles, at intervals of one mile ; and each of these sections must contain, as nearly as possible, six hundred and forty acres. These requirements are manifestly impossible because of the convergency of the meridians, and the discrep- ancies will be the greater as the land divided is farther north. The law also provides that the work of subdivision shall be so performed as to throw all shortages or surplus into the northern and western tiers of sections in each township. To harmonize these various requirements as much as possible, the following methods have been adopted by the general land office. 210. Principal reference lines. Initial points are established astronomically under special instructions. From the initial point a " principal meridian" is run north and south. Through the initial point a "base line" is run as a parallel of latitude east and west. 1 On the principal meridian and base line the section and quarter corners, and meander corners at the inter- section of the line with all streams, lakes, or bayous, prescribed to be meandered, will be established. These lines may be run by solar instruments, but methods involving the use of the transit with observations on Polaris at elongation are now preferred. 2 211. Standard parallels. Such parallels, called also cor- rection lines, are run east and west from the principal merid- ian as parallels of latitude at intervals of twenty-four miles north and south of the base line. " Where standard parallels have been placed at intervals of thirty or thirty-six miles, re- gardless of existing instructions, and where gross irregularities require additional standard lines, from which to initiate new, or upon which to close old, surveys, an intermediate correction line should be established to which a local name may be given, e.g., 'Cedar Creek Correction Line'; and the same will be run, in all respects, like the regular standard parallels." 1 A list of base lines and principal meridians will be found in the Appendix, pages 357-360. 2 See the " Manual of Surveying Instructions" for detail of methods. 224 LAND SURVEYS. 212. Guide meridians. Guide meridians are extended north from the base line, or standard parallels, at intervals of twenty- four miles east and west from the principal meridian. When existing conditions require the guide meridians to be run south from a standard parallel or a correction line, they are initiated at properly established closing corners on the given parallel. This means that they are begun from the point on the parallel at which they would have met it if they had been run north from the next southern parallel. The point is obtained by computation, and is less than twenty-four miles from the next eastern or western meridian by the convergence of the meridians in twenty-four miles. In case guide meridians have been improperly located too far apart, auxiliary meridians may be run from standard corners, and these may be designated by a local name, e.g., " Grass Val- ley Guide Meridian." 213. Angular convergence of two meridians. This is given by the equation 6 = m sin L, (1) where m is the angular difference in longitude of the meridians, and L is the mean latitude of the north and south length under consideration. The linear convergence in a given length I is c = I sin 6. (2) The derivation of equation (1) is as follows, assuming the earth to be a sphere, which will introduce no error of consequence in this work. In the figure, R is the mean radius, r is the radius of a parallel, S and S' are tangents to the two meridians. The angle 6 between these is the angular convergence of the meridians in the latitude L. m is the difference in longitude. UNITED STATES LAND SURVEYS. 225 r = R cos L. (3) S = R cot L. (4) S and S' may be considered as radii of the arc AB, which has also the radius r. Since a given length of arc subtends angles inversely proportional to the radii with which it may be drawn, 6 = r _ R cos L ,_. m~S RcotL' whence 6 = m sin L, which was to be found. Since the distance between meridians is usually given in miles, this must be reduced to degrees. This is done by first finding the linear value of one degree for the mean latitude, using the value of r given in equation (3). To make (3) and (4) strictly correct, the normal at A should be used instead of R ; but the mean radius will give results sufficiently close for land surveying. See Tables IX. and X., pages 371, 372, for values of 6 and length of 1' of longitude in various latitudes. 214. Township exteriors. Each twenty-four mile "square" block is, when practicable, subdivided into townships at one time, the work being done as follows : Beginning with the southwestern township, the meridional boundaries or range lines are first run, and on these are estab- lished the section and quarter corners. These are run as true meridians from south to north. Next the east and west lines, or township lines, are run from east to west between corre- sponding corners of the range lines. On each such township line the section and quarter corners are established at full distances from the eastern range line of each range, the short- age being thrown into the most westerly half mile in each range. A random line is run from east to west, temporary corners being set at correct distances. The distance north or south by which the random line fails to reach the proper corner is observed, and from this and the known length and bearing of the random line a correct bearing is determined and the correct line run eastward, on which are placed the permanent corners. If a random line fails to meet the required corner by more than three chains, it must be rerun. Deviations from the foregoing methods are sometimes made necessary by the topo- graphical features of the territory. R'rD SURV. 15 226 LAND SURVEYS. 215. Subdivision of townships. Each township is next sub- divided into sections as follows : Beginning on the south line of the township, at the corner common to sections 35 and 36, run northerly parallel to the eastern range line of the range in question, one mile, setting a quarter corner at the half mile. Establish the corner common Toivnship No. 5 North, Range No. 9 West, of a Principal Meridian Eat 7 -80K 80(00 40JJO,40_1_4 | >.^ -8o|ou T o _SeA %l_ -^ Sea^S u ir- ilmUt 80|00 Seci It tjSecj|4_ " 6ft ; ! TFe su.oo fSecl 2_ ^ 6$ 7 TPe.s -80,00- 80l002 . l^j 10 40 i 40? r 80(00 i " T ?JBU_1 g ~ 640 g 80OO r "* 80|00 ^ 80.OO gaeLl^f \ West 8000 80JOO _ ! Wes 80(00 ;_Sec|25__g g Seel 36 g "~ ~1 7T Sec. 6 J\'rst Sec. 5 Standard Sec. U Parallel Sec. 3 North Sec. 2 T^ above plot represents a theoretical township with perfect subdivisions, contiguous to the north side of a Standard Parallel ; in assumed Latitude hi 15' N., and Longitude 10000' W. of Gr. Area 2S02U.16 A. FIG. 106. to sections 25, 26, 35, and 36, and run east on a random line to the corner common to sections 25 and 36 on the range line, set- ting at the half mile a temporary quarter corner. From the observed failure to meet the corner on the range line, compute the bearing of a true line, and run this from the corner west, UNITED STATES LAND SURVEYS. 227 setting the permanent quarter corner at the middle point of the line. Proceed thus with each succeeding section to the north till section 1 is reached. From the corner common to sections 1, 2, 11, and 12, the meridional line is run to the section corner on the next township line, by a random line corrected back ; but the quarter corner is set at forty chains from the south end so as to throw whatever discrepancy there may be into the north half mile. In case the township is the most northerly of a twenty-four mile block, the line between sections 1 and 2 is not run to the corner on the correction line, but is run parallel to the range line, to an intersection with the correction line where a closing corner is established and its distance from the section corner noted. This process is repeated for each tier of sections until the fifth tier is completed. The east and west lines of the sixth tier are run from east to west as random lines and corrected back, but the quarter corner is set just forty chains from the eastern boundary so as to throw the discrepancies into the most westerly half miles. Fig. 106 represents a township with perfect subdivisions. The directions written along the various lines indicate the directions in which they are run. Probably no townships that have been surveyed are like that in the figure, all of them being more or less distorted by inaccuracies, and some of them being very much distorted. When very bad work is discovered in time to correct it, before such alteration will interfere with acquired rights of individuals, the correc- tions may be made under rules prescribed by the General Land Office. When individuals have bought the land, the corners as actually set must remain the corners, however erroneously placed. 216. Meandering a stream. This consists in running lines along its bank to determine its direction and length. The left and right banks of a stream refer to the banks as they would be to one passing down stream. The " Instructions " provide for the meandering of naviga- ble streams and those whose width is three chains and upward. 228 LAND SURVEYS. They are to be meandered on both banks. Lakes, deep ponds, bayous, etc., of twenty-five acres or more are also meandered. Corners called " meander corners " are established wherever a meander crosses standard base lines, township lines, or section lines. A meander of tidal waters follows the high water line. 217. Corners. Stones, posts, trees, and earth mounds are used to mark the corners. The kind of corner post depends on the material afforded by the country. The " Instructions " designate the kinds of corners to be established and the method of marking them. When it is impossible to establish a corner in its proper place, auxiliary corners are established within twenty chains of the true place, on each of the lines approaching the corner. These corners are called " witness corners." The center quarter corner of a section is located by connecting the opposite north and south quarter corners by a straight line, and placing the center corner at the middle of this line. Exceptions that will suggest themselves are necessary in the northern and western tiers of sections. Any person having to do with surveys of lands that have been subdivided as part of the public lands, should procure the " Instructions " and study them carefully. These instruc- tions have varied from time to time and it is well to consult those in force at the time the original surveys in hand were made. 1 218. Notes. The following sample page of the field notes prescribed in the instructions is given as an excellent form of notes for any land survey. For field purposes the author pre- fers a sketch on which all notes are made, arid from which each night the field book may be written up in the form given. Variations in items noted and in nomenclature will suggest themselves to the maker of other kinds of surveys. 1 He will also be much helped by a book entitled " Manual of Land Surveying" by Hodgman and Bellows. This work contains the gist of many court decisions and instructions, resulting from many years' experience in this work. Dorr's "Surveyor's Guide" will also be found to contain many practical suggestions. UNITED STATES LAND SURVEYS. 229 GTH GUIDE MERIDIAN EAST, THROUGH TPS. 13 N., BETWEEN Rs. 24 AND 25 E. Survey commenced August 29, 1890, and executed with a W. & L. E. Gur- ley light mountain transit, No. , the horizontal limb being provided with two opposite verniers reading to SO" of arc. I begin at the Standard Corner of Township 13 North, Ranges 24 and 25 East, which I established August 29, 1890. Latitude 45 34'.5 N., longi- tude 107 24' W. At this corner, at 8 h 54 m P.M., by my watch, which is 3" 49 fast of local mean time, I observe Polaris at eastern elongation in accordance with instructions in the manual, and mark the point in the line thus deter- mined by a tack driven in a wooden plug set in the ground, 5.00 chs. north of my station. August 29, 1890. August 30: At f> h 30 A.M., I lay off the azimuth of Polaris, l49'.5to the west, and mark the TRUE MERIDIAN thus determined by a cross on a stone firmly set in the ground, west of the point established last night. The magnetic bearing of the true meridian is N. 18 05' W., which re- duced by the table on page 100 of the Manual gives the mean mag. decl. 18 02' E. From the standard cor. I run North, bet. Sees. 31 and 36. Descend over ground sloping N. W. Creek 10 Iks. wide in ravine, 45 ft. below the Tp. cor., course N. 32 W. To edge of table land, bears N. E. and S. W. ; thence over level land. Bluff bank, bears N. 58 W. and S. 58 E. ; descend abruptly 40 ft. Bottom of ravine, course S. 58 E. ; ascend 50 feet to Edge of table land, bears S. 58" E. and N. 58 W. ; thence over level land. Difference between measurements of 40.00 chs., by two sets of chainmen, is 18 Iks. ; position of middle point By 1st set, 40.09 chs. By 2d set, 39.91 chs. ; the mean of which is Set a limestone, 16 X 7 X 5 ins., 11 ins. in the ground, for J sec. cor., marked 4 on W. face, and raise a mound of stone, 2 ft. base, 1J ft. high, W. of cor. Stream, 6 Iks. wide, in ravine 15 ft. deep, course N. 60 W. Enter heavy oak timber, bears E. and W. An oak, 30 ins. diam., on line, I mark with 2 notches on E. and W. sides. Creek, 20 Iks. wide, 1 ft. deep, course N. 83 W. Right bank of creek, begin very steep rocky ascent. Top of ridge, 250 ft. above creek, bears N. 80 W. and S. 80 E. Begin descent. Difference bet. measurements of 80.00 chs., by two chainmen, is 22 Iks. ; position of middle point By 1st set, 79.89 chs. By 2d set, 80.11 chs. ; the mean of which is The point for sec. cor., 150 ft. below top of ridge, falls on a flat rock in place, 10 ft. E. and W. by 6 ft. N. and S., on which I Cut a cross (X) at the exact cor. point, for cor. of sees. 25, 30, 31, and 36, marked with 5 grooves on N. and 1 groove on S. sides ; from which An oak, 10 ins. diam., bears N. 22 E., 54 Iks. dist., marked T. 13 N., R. 25 E., S. 30, B. T. A dogwood, 5 ins. diam., bears S. 64J E., 40 Iks. dist., marked T. 13 N., R. 25E..S. 31, B. T. An ash, 13 ins. diam., bears S. 51 W., 37 Iks. dist., marked T. 13 N., R. 34 E., S. 36, B. T. An oak, 9 ins. in diam., bears N. 34" \V., 42 Iks. dist., marked T. 13 N., R. 24 E., S. 25, B. T. Land, level and mountainous. Soil, gravel and rock ; 4th rate. Timber, oak. Mountainous or heavily timbered land, 33.00 chs. 230 LAND SURVEYS. CITY SURVEYING. 219. Precision required. Surveying to determine land lines in a town or city does not differ in principle from surveying for similar purposes in the country. The difference lies in the degree of precision required. In the country a square foot of land is worth from one cent to, perhaps, in extreme cases, five or ten cents. In the heart of a great city it may be worth many dollars, and a single inch frontage of a lot one hundred feet deep may be worth several hundred dollars. The brick wall of a modern office building of ten or mote stories, built an inch over the line of the owner's land, would be a costly wall to move, and an error so locating it would virtually place the owner at the mercy of the possessor of the adjoining property. It is therefore readily seen that an error in closure of one in three hundred, or one in five hundred, which may be tolerated in farm surveying, would be altogether out of the question in surveys in the heart of a great city. In such situations a precision of one in fifty thousand should be secured. In villages and small towns a precision of one in five thousand will frequently be sufficiently close, though, if the place has "prospects," it may be well to make closer surveys. The methods of making measure- ments to the various degrees of precision required are noted in Chapter I. To lay off an angle so that the position of a line may not depart .from its true position by more than -5-5-5-5-^ of its length, requires that the angle have no greater error than about four seconds. A transit reading to thirty seconds will nearly accom- plish this if the angle is measured three times before reading, the reading being divided by three. An instrument reading to twenty seconds will ordinarily do a little better than the requirement, and an instrument read- ing to ten seconds will usually accomplish the required result with a single measurement of the angle, though it should always be measured more than once, if for no other reason than to secure a check on the work. Of course the magnify- ing power of the telescope must correspond to the fineness of the graduations. CITY SURVEYING. 231 It has been stated that angles are read. Bearings are not used, nor are regular traverses run out in city surveys. Bear- ings are usually worked up from the angles for all the lines of a survey,. in order to compute the latitudes and longitudes for determining the error of the survey. Azimuth would do as well. 220. Extent of survey. City land surveys are usually, ex- cept where " additions " are being laid out, of small extent, covering perhaps a single lot of 25 feet by 100 feet, more or less. In a well-monumented city the entire survey may be confined to the block in which the lot lies. In more cases, it will be necessary to go several blocks away to obtain monu- ments for determining the necessary lines. In cities that have been laid out with irregular lines, it may often be necessary to carry the lines and measurements over the roofs of the solid blocks of buildings in order to determine the angle points in the side lines of the lots. Each case involves new difficulties that the surveyor must meet by exercising his ingenuity. Many difficulties are solved by the proper use of the coordinate system, following the general methods outlined in Art. 203. 221. Instruments. A surveyor who undertakes to do city work should be supplied with the best instruments for meas- uring lines and angles. The tape mentioned in Chapter I., as used in New York City, is a very good tool. It should not be used, however, until it has been tested by the surveyor himself to verify the " pull scale " for various temperatures. The city surveyor should have, in his office, a standard length marked out on the floor, or some other place not subject to change, by which standard he may test his tapes. The United States Coast and Geodetic Survey Department in Washington, D.C., will test tapes sent there for that purpose, and will report the tem- perature, pull, etc., for which they are standard, and their con- stants. For this service a very small fee is required. The city surveyor's transit should be of high magnifying power, and should read to thirty seconds and preferably to twenty seconds or even ten seconds. It should be well made, and of a pattern to insure stiffness and permanency of adjust- ment. A good form is shown in Fig. 107. FIG. 107. CITY SURVEYING. 233 The compass being of little use in city surveying, the space usually occupied by the compass box is used to form the base of the horseshoe-shaped standards. The transit is made with either three or four leveling screws. The advantage of three screws is wider leveling base, and therefore, with given pitch of thread, a finer adjustment of the horizontal plate for a given turn of the screw. The author prefers four screws, perhaps because he has never become accustomed to the use of three. 222. Description of a city lot. The description of a city lot as found in a deed is either by "lot and block" or by "metes and bounds." By the former method it will be described as "Lot No of Block No of the original survey of the city of " Or, "Lot No of Block No of. 's addition to the city of. as the same is shown and delineated on a certain map entitled (Then follows the title of the map) filed in the office of the recorder of county, --^"Tiy-- ( ^y)__. _._(y s Jl_." By the method of metes and bounds it will be described as : "Beginning at a point in the (N., S., E., or W.) line of. street distant thereon (Direction) (Distance) feet from the cor- ner formed by the intersection of the said line of street, and running thence along the said line of. street feet, thence at an an- gle of degrees (Direction written easterly, northerly, etc.) feet ; thence at an angle of degrees (Direction) feet ; thence at an angle of. degrees (Direc- tion) feet to the point of beginning," To this will sometimes be added: "being Lot No of block No ," etc., as was written above. 223. Finding a city lot. If the description is by lot and block, it will be necessary to refer to the map of the survey mentioned for the data as to the widths of streets, angles in the lines, and positions of lots, before the survey can be begun. The map should also show the character and position of mon- uments located at the time of the original subdivision of the 234 LAND SURVEYS. tract. This it will rarely do if the survey is an. old one, and it may not do so even if the survey is a late one. 1 It will often happen that even though the map shows the positions of the monuments that originally marked the lines, those monuments are not now to be found. In such cases, the best that can be done is to find any monuments having any bearing on the survey in question, and further to note existing lines of permanent improvements that have been long in place, and to endeavor to reconcile these with the figures shown on the map. This work requires the best judgment of the sur- veyor, for it will almost always be found that the discrepancies are considerable. This is not so true in the centers of very large cities, for here surveys have been gone over and over by most careful workmen, and the lines have become pretty defi- nitely fixed ; but in smaller and newer cities there is an endless amount of difficulty. 224. Marking corners. When the boundaries of the re- quired lot are finally fixed upon, each corner is marked. If the ground is open, a stake with a small tack may mark each corner, if the marking is for temporary use. The marking may be a mark made on a building or in the stone flagging of the sidewalk, or otherwise. A sketch of the lot with a note of the marks made and a certificate of survey should be furnished the person for whom the survey is under- taken. 225. Discrepancies. If any discrepancies are discovered, they should not be "fudged in" and hidden, but mention should be made of just what is found. If discrepancies are found, the surveyor's work is doubled, for he must be sure the apparent errors are not in his own work. If the discrepancies found are small, no greater than the surveyor would expect to find in repeating his own work that is, if they are within the limit of precision required no note need be made of them. When, for instance, a block that is recorded 500 feet long is found to be 500.04 feet long, and it is required to lay out a lot in any portion of the block, the measurement from one end of the block to locate the lot should be to the recorded i Consult Appendix, page 354. CITY SURVEYING. 235 measurement as 500.04 is to 500. Such discrepancies may be thus distributed. If the street lines on both sides of the block are well de- nned, and the block is found to overrun or fall short of the recorded length, and if the description of the lots sold has been only by lot and block, the error should ordinarily be di- vided proportionally among the various lots. If the descrip- tion is by metes and bounds, it is more difficult to say what to do, and the best the surveyor can do is to report what he finds and ask his employer for further instructions. He must, in no case, locate the lot in what he thinks should be its proper place and attempt to defend such an action as the only proper course. Such procedure has proved a source of much litigation for the owner and loss of reputation to the surveyor. If the description is by metes and bounds only, the surveyor must determine the starting point by finding the street line on which it lies and the other street line which, with the former, forms the intersection from which the beginning point is located. This is frequently difficult to do because of lack of monuments or marks. When it is accomplished, the surveyor's work is clear : he follows the description. If he finds discrep- ancies, he reports what he finds. If the description is both by metes and bounds and by block and lot, it is more difficult properly to locate discrepancies, and such as are found must be reported. The interpretation of the deed would perhaps usually be the intent of the buyer and seller, if that could be discovered. The surveyor must remember that he has no right to interpret the deed. That is the business of a court. Descriptions in deeds are often inconsistent in themselves and indicate impossible plots. 226. Planning additions. Whenever possible, additions should be so laid out that the streets may be continuations of those in the adjacent subdivided portion of the city. Not enough consideration has been given in the past to this impor- tant feature. Unless the ground is very irregular and the addition is a remote suburb, the subdivision should be rectan- gular. If the ground is irregular and on the extreme outskirts of the city, in a part that will probably never be thickly built 236 LAND SURVEYS. up, it may be laid out in curved lines to conform more nearly to the surface of the ground, involving less work in grading streets and lots, and making the tract more beautiful. In such cases the tract will not be cut up into very small lots. No lot should have less than one hundred feet front, and it maybe said that such subdivision should not ordinarily be undertaken if the lots are to be less than one acre in area. In planning a curved subdivision, the streets should usu- ally be located so as to give good drainage to the lots. This will generally mean that the streets will occupy the lower ground. To do this work most satisfactorily, a contour map of the tract is desirable. This is made as described in Chap- ter IX. 227. Making the survey and map for an addition. The first step in subdividing an addition is to make a complete survey of the entire tract, using the same care as for city work. Many discrepancies of greater or less amounts will be found. The descriptions of the tract are probably from old farm surveys. If no greater discrepancies are found than might be expected from such work, and the corners are well established, no trouble need arise. The new distances and angles are the ones that will be recorded on the map that is to be made. The surveyor will carefully note the intersection of the bound- ary with the street lines of the adjoining tract, and the direc- tions of those lines. The survey will be very carefully balanced and mapped. Probably the best method of making the original map is to divide the sheet into squares of, say, one hundred to one thousand units on a side the number will depend on the size and scale of the map. One of the lines of division will be assumed as the reference meridian, and another, at right angles, as the base parallel. Their intersection may be taken as the origin or may be given any arbitrary number of hundreds as its coordinates. Each of the corners of the squares will then have for its ordinates quantities expressed in whole hundreds. The coordinates of any point in the survey being determined, it is at once known what square the point falls in, and the point may be measured in from a corner of that square, and be cer- CITY SURVEYING. 237 tainly within reach of a single rule length. If the map is small, this method has no advantage over a single meridian on which latitude ordinates are measured and perpendicular to which longitude ordinates are measured. A system of subdivision will then be planned and drawn on the map, 1 and then located on the ground. The data for placing the street lines on the ground will not be scaled from the map, but will always be computed. The coordinate system is recommended for this work. The location on the ground consists in placing monuments at all intersections of street lines. These are sometimes placed at the intersection of center lines and sometimes at the block corners. Neither practice is good. Probably the best place to locate them is in the sidewalk area, at a given offset, say five feet, from the property line. They are less likely to be dis- turbed here than elsewhere. If the addition is to be laid out on curved lines, the monuments should mark the intersections of street lines and the points on the arcs where the radii change. For methods of locating curves, see Chapter VIII. The monuments should be of stone, durable in quality and about two feet long, with the top dressed to about six inches square and the bottom left rough. The top face should be smooth, with a hole drilled in its center, in which is set with lead a copper bolt with a cross marked in its head. The stone should be set below the disturbing action of frost. A piece of iron pipe may reach to the surface of the ground and terminate with a cast cover. Where stone is costly, concrete may be used, formed of sand and cement, inside of a piece of stove pipe. The copper bolt should be set as before. Other forms may suggest themselves to the surveyor. A carefully constructed map finishes the work. This map should show the position of each monument, preferably by giv- ing its coordinates referred to some origin, the relation of the monuments to the property lines, and such further information as is indicated in the Appendix, page 355. If curves are used, the coordinates of the centers and beginning and ending points of the curves should be noted. 1 See an excellent paper by J. Stiibben, of Cologne, Germany, in Vol. XXIX., " Transactions American Society Civil Engineers." CHAPTER VIII. CURVES. 228. Use of curves. When a road, railroad, or canal is built, the center line is laid out on the ground. The center line is, in the case of a railroad, always a series of straight lines connected by arcs of circles to which the straight lines are tangent. In the case of a canal or wagon road, this is not always true, as the line is sometimes irregular, following more closely the con- formation of the ground than does a railroad. It would be better if canals and wagon roads were always laid out as railroads are. In the case of wagon roads, the radii of connecting curves may, of course, be short. Many park roads and suburban streets are located as arcs of circles and straight line tangents. Park drive curves are often irregular and sometimes are successive arcs of varying radii, known as compound curves. Railroad curves are frequently compound curves. 229. Principles. Only the fundamental principles of the laying out of these curves will be here given. The straight PRINCIPLES OF CURVES. 239 lines are called technically "tangents." The curves are known by the angle subtended at the center by a chord of one hundred feet. Thus, if such a chord subtends 4, the curve is known as a 4 curve. Any two tangents not parallel intersect at some point, and the angle at the point, between one tangent produced and the other tangent, measured in the direction in which the line is to bend, is known as the intersection angle. Curves are measured in chords of one hundred feet. This dis- tance is a " station," whether used on curve or tangent. A curve four hundred feet long means a curve of four stations. It does not mean that the arc is just four hun- dred feet long, but that there are four chords, each of one hundred feet. In Fig. 108, two tangents, A V and VB, are connected with a curve of radius R. AB is known as the long chord, O. DE is the middle ordinate, M. AV=VB is the tangent distance, T. VE is the external distance, E. From the figure it is evident that (1) Jf=72versiA, (3) (2) ^=J2exsecJA. (4) External secant is a trigonometrical function not commonly mentioned in Trigonometry, but very useful in handling curves. It is the "secant minus one." In the above equations it will be seen that the radius R is used, while it has been said that curves are known by their "degrees." If AB is one hundred feet, A becomes D, the degree of the curve, and from (2) 7? 50 _ jB = sh^p' < 5) For any other curve of degree D 1 ', For small angles it may be assumed that the sines vary as the angles ; whence R D' sin 1 D' ~ ~ ' R' The radius of a 1 curve is 5729.65 feet. Except in close land 240 CURVES. surveys or surveys of city streets, the radius may be taken as 5730.00 feet. In using curves in land surveys, the term "de- gree" should be abandoned and all circles designated by their radii. Since the values of all the functions given in equations (1), (2), (3), (4), vary directly as R, A remaining constant, these functions will all, assuming the correctness of equation (7), vary inversely as D. They would vary exactly in this way if the one hundred feet that measures D were measured on the arc instead of as a chord. This will be clear from the principles of Geometry. The length of a curve for a given value of A and D is in stations, ' ff = | (8) and in feet measured in chords of one hundred feet, =100-- (9) It is frequently required in practice, to know the values T and E for given or assumed values of A and D. If a table is made of values of these quantities for a great number of val- ues of A and for D = 1, the value of T or E for any other value of D would be at once obtained by dividing the value found in the table opposite the given A, by the given value of T>. In practical railroad work this method may be used for curves of less than 8 or 10 without serious error. For sharper curves than these the quantities should be computed by equations (1), (2), (3), and (4). If, however, the sharper curves are measured in fifty-foot chords, up to say 14 to 16 curves, and in twenty-five-foot chords thereafter, the approxi- mate method may be used up to at least 20 curves or 24 curves without serious error. This means that a 14 curve will be defined as one in which a chord of fifty feet subtends at the center an angle of 7. A 20 curve is one in which a twenty-five-foot chord subtends an angle of 5 ; etc. 230. Laying out the curve. The point at which the curve is to begin is known when the point V and the tangent distance T are known. For the latter, A and D must be known. Let the point of beginning, called the P. C. (point of curve), be A. A transit is set at A, Fig. 109, an angle = | D turned from the LAYING OUT CURVES. 241 tangent A V to C. The chain or tape is stretched from A and the farther end made to coincide with the line of sight at C. 1 From the line CA an angle equal to | D is laid off, and the chain stretched from (7 and the far- ther end put in the line of sight at D. Thus the points one hundred feet apart are located. The final deflection angle VAB is g A. If the curve is not of a length ex- pressed by a whole number of stations, there must be turned off for the "subchord " a de- flection usually made proportional to the length of the sub- chord ; thus in the figure, if EB is fifty feet, the deflection EAB is one half of l D, etc. This again assumes that the curve is measured on the arc, and the method is practically correct for curves of less than 10. For sharper curves than this a computation should be made. Thus, since A is known, there remains to turn off from AE an angle equal to \ A VAE. This being half the angle subtended by the chord EB, that chord may be computed. If the curves of higher degree are laid out with short chords as before described, the proportionality of chord to deflection angle may still be assumed. If the whole of the curve can not be seen from the beginning, the transit may be moved to a point on the curve already located and the work continued as follows: Let it be assumed in Fig. 109 that there is an obstruction in the line AE so that E can not be seen from A. Move the transit to D, and with the verniers set at zero, with the lower motion, turn the line of collimation on A. Now lay off to the right an angle equal to D (the deflection from A to 7)) and the line of collimation will 1 Let the student show that this procedure locates the point C on the curve. R'M'D SURV. 16 242 CURVES. lie in the tangent to the curve at D. Transit the telescope and deflect to the right for the station E, etc., as usual. Not much more than 90 of an arc should be located from one point. 1 Points occupied by the transit are usually solid " hubs " driven flush with the ground with a small tack to mark the exact point. The other points are, in railroad and common road surveys, merely marked with stakes. In close land sur- veys and city work, each point should be marked with hub and tack. 231. Location by chain alone. A method for locating arcs approximately by the use of the chain or tape is as follows, and is known as the method by " chord off- sets " : In Fig. 110, A V is a tangent to which a curve of degree D is to be joined at A. If AS is one station, 5 = 100 sin ID. Imagine the curve continued back one sta- tion to E. eE = b, and /, in EA produced, is distant from b by an equal amount, therefore Bb is known as the tangent offset t, and fB is the chord offset = 2 t. To locate the curve, swing the chain from A and place the farther end of it a distance t from the tangent. Stretch the chain from B in line with AB and make a mark at Maps of land sur- veys, or others made to show to non-technical persons, should be made at so many feet, or chains, or miles, to the inch ; and the top of the map should be north. Maps of large territories, mainly for the use of those who are conversant with surveying methods, may be made with advantage to the natural scale. The original map may be copied by tracing on cloth or on paper, and transferring to other drawing paper. THE PLANE TABLE. 254. Description. Detail topographical surveys are some- times made with the plane table , indeed, this is the standard instrument of the United States Coast Survey for such work, and is also largely used by the United States Geological Survey. The plane table consists essentially of a drawing board with a suitable leveling device, mounted on a tripod, and a ruler for drawing. The ruler is attached to a line of sight, usually telescopic, but sometimes merely open sights, like those of the compass. The combined ruler and telescope is called the alidade. Fig. 124 is a complete plane table. Clips for holding the paper are shown, as well as the alidade, compass box, and levels, and a device for setting a point on the paper over a point on the ground. The plane table is, in its ordinary THE PLANE TABLE. 269 form, a very awkward instrument, and is used very little out- side the two surveys mentioned. A much simpler form of leveling head than that generally used is one invented by Mr. FIG. 124. W. D. Johnson, and this device has been approved by the topographers of the United States Geological Survey. It is shown in Fig. 125. By loosening the wing nut d, the table may be leveled, and when leveled, the nut d is tightened. When the nut g is loosened, the table may be turned in azi- muth without disturbing its hori- zontality. The whole arrange- ment is very light. It does not permit of as close leveling as does the ordinary form with leveling screws, and should not FIG. 125. be used where contours are to be carefully determined by vertical angles. In making a map of a park showing the location of each tree, bush, etc., for planning new work, and when work may be, without loss, performed on pleasant days and omitted on wet or damp days, 270 TOPOGRAPHICAL SURVEYING. the plane table may be used with advantage, and, aside from the United States Coast and Geological Surveys, it is for this class of work that it is most used. The author prefers for contour work to use the transit and stadia. If the advantage of mapping at once in the field is desired, it may be obtained by having an assistant and a light drawing board. It is be- lieved that a lack of notes is rather a disadvantage than, as is generally assumed, an advantage. 255. Use. The plane table is used for the immediate map- ping of a survey made with it, no notes of angles being taken, but the lines being plotted at once on the paper. The simplest case is the location of a number of places from one point by azimuth and distance. The table is set up so that some con- venient point on the paper is over a selected spot on the ground, and clamped in azimuth. The ruler is then brought to the point on the paper and swung about it till the line of sight which is parallel in azimuth to the ruler 1 is directed toward a point that is to be located. A scale for the drawing is deter- mined, and a line is drawn along the ruler and made to scale, equal to the distance to the desired point, which distance is found by measurement or by the stadia. The point is then located. A similar procedure locates other points. This is called the method of radiation. If a. plane table is set up in the interior of a field all of whose corners are visible from the position of the table, the corners may be thus located and con- nected and a map of the field is at once obtained. There is evidently nothing by which to determine the area except to scale the map for additional data or to use a planimeter. Traversing may be performed with the plane table as fol- lows : If it is a field that is to be " run out," set the table over one corner, choosing a point on the paper to represent that corner in such position that the drawing of the field to the scale selected will come on the board. In the four-sided field shown in Fig. 126, it would be necessary to occupy but two corners to map the field, though it would be better for a check to occupy the third. The figure shows the field and, to a very exaggerated scale, the plane table. 1 It is only necessary that the two shall have a fixed angle in azimuth. THE PLANE TABLE. 271 The table is first set over B and a point b marked on the paper directly over B. The instrument being clamped, the alidade is brought to bear on A, and a line is drawn to scale equal to BA, which is found by measurement or by stadia. The alidade is then directed to C, and the line be is drawn to scale. The table is then removed to C, and so set up that the point c shall be over the point C and the line cb in the direction CB. This is hard to do with the plane table. If s a. FIG. 126. the scale is large, it must be done ; but if the scale is very small, the following is a sufficiently close approximation : Set the table level, with c over C and with cb approximately in line with CB. Loosen the azimuth motion and, with the alidade on the line <$., bring the line of sight on B. Clamp in azimuth, and the table is set. The point c is not exactly over (7, but the error will be inappreciable. The table being set over C and oriented on CB, turn the alidade toward D and draw cd. da may now be connected, giving the entire field. It will be better to occupy D and see whether the line DA will pass through a on the paper, and whether the length da equals to scale the length DA. 272 TOPOGRAPHICAL SURVEYING. Another method very much used with the plane table is known as the method of intersections. Let it be required to locate the points A, B, (?, Fig. 127, from the points D and E. Measure DE, and lay off to scale on the paper a line equal to DE, in proper position so that the points desired will fall on the paper. Set the table over .D and orient on E. Swinging the alidade about d, draw lines toward (7, B, and A. Set the table over E and orient on D. Swinging the alidade about e, draw lines toward 0, B, and A, and note the intersections of these lines with those drawn from d to corresponding points. These intersections locate the points. 256. The three-point and two-point problems. In filling in topography that is hung on a system of triangles, it is common to complete the triangulation and map it. There will usu- ally be drawn on a single sheet on the plane table only the B topography adja- / \ cent to one or two A^ / \ triangulation sta- i x \ / \ tions, the work \ \ % / \ being carried on 1 \ N / \ precisely as in the \ A' X \ ^'-f use of the transit and stadia, except that it is mapped at once. The first sheet will ordina- rily have plotted on it at least one triangulation sta- tion and a line toward another. These, indeed, might be drawn at random, so that they be located conveniently on the sheet for the work to be done. Usually there will be two or more triangulation stations, or previously occupied plane table stations, mapped on a new sheet. The table may be set over one and oriented on another. It not unfrequently happens that the table can not conveniently be set over any one of the stations mapped on the sheet. Then, if the table is set over any point at random and FIG. 127. THE PLANE TABLE. 273 properly oriented (assuming that the latter could be done by the needle or otherwise) and the alidade is then revolved sepa- rately about each plotted point and directed toward the place in the field thereby represented, a line being drawn on the paper in this direction, the lines so drawn from the various points should all intersect at one point, which o would be the properly mapped point over which the instrument is set. In Fig. 128, if points A, B, (7, have been properly mapped \ \ in a, 5, and , q, r will be in embankment. The points tf, J, n, p, and r are "at grade," being neither in cut nor fill. Connecting corresponding grade points, there result the dotted grade lines 84, /, b, 81 ; , w, w ; w, p, v, which are bounding lines of bodies of cut and fill. Thus, the whole central portion of the figure is in excava- tion, while the upper left corner is in embankment. The horizontal area in excavation or embankment at any level is the area included between original and finished surface contours at that level. 1 The student should show how to do this, using Fig. 135. 284 EARTHWORK COMPUTATIONS. To get the volume of any single body of cut or fill, measure successive areas with the planimeter, and assume these to be end areas of figures that are as nearly prismoids as anything else, with altitudes equal to the contour interval. If the pris- moidal formula is to be used for computing the volume, the 80 81 82 83 84 S5U t 86 87 FIG. 136. altitude of a single prismoid is taken as twice the contour interval, that is, two layers are taken to make one prismoid. The volumes may be computed by the formulas of the last article. Assuming average end areas, the small body of cut on the right is found to have areas : At elevation 81, 00. At elevation 82, acb. At elevation 83, def. At elevation 84. 00. Volume = + def + = ESTIMATING VOLUMES FROM A MAP. 285 The portion of fill between this cut and the large central body of cut has areas: At 88-foot level, 00. At 87-foot level, ghi. At 86-foot level, klm. etc., etc. 268. Application to structures. One example of the appli- cation of this method of estimating volumes to regular struc- tures built on irregular ground will be given. The form of the FIG. 137. structure is not presented as an example of good engineering, but merely to show the method of estimating volumes. A small reservoir is to be built on a hillside, and will be partly in excavation and partly in embankment. Fig. 137 shows such a case. The contours, for the sake of simplicity, are spaced five feet apart. The top of the reservoir (shown by 286 EARTHWORK COMPUTATIONS. the heavy lines making the square) is 10 feet wide, and at an elevation of 660 feet. The reservoir is 20 feet deep, with side slopes both inside and outside of two to one, making the bottom elevation 640 feet, and 20 feet square, the top being 100 feet square on the inside. The dotted lines are contours that would be invisible if both original surface and completed reser- voir were supposed to exist at the same time. The areas of fill all fall within the broken line marked abcdefghik, and the cut areas all fall within the broken line marked abcdefyo. These broken lines are grade lines. The areas of fill and cut are readily traced by following the closed figures formed by con- tours of equal elevation: At 640-foot level area in fill is pst. At 650-foot level area in fill is Imnuvxl. At 650-foot level area in cut is 1 2 3 ux 1, The other areas are easily traced. In the figures given, the lines have all been drawn in black for printing. In practice they should be drawn in different colors to avoid confusion. CHAPTER XL HYDROGRAPHIC SURVEYING. 269. Definition. A hydrographic survey is a survey having to do with any body of water. A topographic survey may be, and frequently is, partly a hydrographic survey. 270. Objects. A hydrographic survey may be undertaken for any one of the following purposes : (1) To determine the topography of a portion of the bed of the sea, a bay, or harbor, or river, in order that it may be mapped for the information of seamen. In this case it is necessary merely to locate the channels, dangerous rocks, and shoals. (2) To determine definitely the configuration of a small portion of the bed of the sea, a bay, harbor, or river, for the purpose of planning works to rest on or in the bed, such as lighthouses, sea walls or docks, bridge piers, etc., or to esti- mate the quantities of materials to be moved by dredging or blasting, for the improvement of the channels or harbor. (3) To determine the flow of a given river or estuary and the direction of the currents, for the purpose of studying the physics of the stream or for planning waterworks or drainage systems. (4) To secure the necessary information for planning the improvement of marshy shores of a lake or stream by lowering the water level or otherwise. 271. Work of the surveyor. The survey ends with the determination of the required information. The planning of the works and their execution is hydraulic engineering. The laying out of the plan on the ground requires the methods of the surveyor. The work of the surveyor, then, is to make a 287 288 HYDROGRAPHIC SURVEYING. topographical map of the area to be covered, to compute mate- rial moved, to determine cross sections of streams, their ve- locities, their discharge, the direction of their currents, and the character of their beds, and to lay out projected improvements. 272. General statement of methods. The configuration or topography of the bed of a body of water is determined by sounding, that is, measuring the depth of water. If many points are observed, a contour map of the bottom may be drawn, the water surface being the plane of reference. The cross section of a stream is determined by taking soundings along a fixed transverse line. The velocity of flow of a stream is determined by noting how long a float requires to drift a known distance, or by the use of a current meter. Since the velocity is not the same at all parts of a given cross section, many determinations for velocity must be made before an average for the whole cross section can be obtained. The discharge of a large stream like the Mississippi, Mis- souri, Ohio, or Hudson River is determined by measuring a cross section and finding the velocity of flow past it. The discharge of a small stream, like any one of the thousands of creeks, is obtained by weir measurements. A weir is a notch cut in a dam. A dam is constructed across the stream, and a notch is built in it, through which the water flows. From the known size of the notch and the depth of water flowing over it, the flow may be calculated by methods that are explained in full in any work on hydraulics. Insignificant streams may be measured by any simple means that may occur to the surveyor. The direction of surface currents may be determined by means of floats. Subcurrents are best determined by means of the direction meter. The character of the bed of a body of water may be ascer- tained by securing samples in a cup attached to the sounding lead, or by putting tallow in a cavity in the bottom of the lead. Some of the bottom will adhere to the tallow. If the character of the bed for some distance below the bottom is required, as when bridge piers are to be founded on firm bottom which is SOUNDINGS. 289 at an unknown depth, various methods are resorted to, accord- ing to the depth of water, the soil below, and the difficulties to be overcome. Simple cases may be handled with gaspipe rods driven through the bottom by hammers to a hard layer of material. In other cases drills driven by pile drivers are used, or diamond drills if rock is encountered, and it is neces- sary to determine its character and depth. The method of making weir measurements, being properly a part of the study of hydraulics, and requiring considerable space to treat adequately, will not be given here ; but such methods as require the use of surveying instruments and meters will be described in sufficient detail to enable the student to undertake such work. SOUNDINGS. 273. Making soundings. Deep-sea soundings are made with special elaborate apparatus which will not be described. Soundings in moderately deep water are made with a weight, known as a lead, attached to a suitable line. For depths of less than fifteen or twenty feet a pole is best. The lead may be any heavy weight. It is preferably of lead, molded about an iron rod. The form is shown in Fig. 138. The cup at the bottom is for collecting samples of the bottom. There is a leather washer that slides up and down the rod between the bottom of the lead and the top of the cup. This keeps the soil that is collected, in the cup, while the lead is being raised. Very often the cup is omitted, and if samples of the bottom are required, tallow is used as already described. The line for very deep work may be of wire, but must be used with a reel. For ordinary work, a hemp line or a chain is best. A line must be stretched, but not too much, as it may shrink. It must be frequently tested. For soundings for navigation charts, the depths are taken in feet to four fathoms, and thereafter in fathoms. The party required to make soundings consists of a leadsman, a recorder, one or more oarsmen, and, when soundings are located from a R'M'D SURV. 19 FIG. 290 HYDROGRAPHIC SURVEYING. boat, one or two observers to read the angles. If the boat is located by angles read on shore, there must be one or more observers on shore. The leadsman casts the lead and announces the depths. For this purpose the line is graduated by suitable tags, so that the depths can be readily determined. In still, deep water the boat may be stopped while the sounding is being takes and recorded, but more often it is not stopped. The leadsman stands in the bow of the boat and throws his lead ahead, so that, as he judges, it will be on the bottom and the line will be vertical when the boat is over it. He will become expert at this after a time. The recorder notes the number of the sounding and the depth that the leadsman gives him. The method of using rods will be obvious. The reference plane for depths is the surface of the water, and as this is continually changing, except in still ponds or lakes, it is necessary to know the stage of the water at the time the soundings are taken. This is done by establishing a gage in the vicinity of the work. The soundings on all navigation charts are referred to mean low tide. The zero of the gage should be set, if possible, at this elevation. The gage may be a graduated board nailed to a pile in a pier in the vicinity of the work and read at hourly intervals during the progress of the sounding. Gages for rivers are frequently inclined, being laid along the shore at right angles to the stream, and the points of equal differences of altitude are determined by level- ing. A tide gage that is to remain permanently in place should be self -registering. Such a gage may consist essentially of a float protected by a surrounding house or tube, and attached by suitable mechanism to a pencil that has a motion propor- tional to the rise and fall of the float. The pencil bears against a piece of graduated paper fastened to a drum that is revolved by clockwork. There will thus be drawn on the paper a pro- file in which the horizontal, units are time, and the vertical units are feet, rise, and fall. The stage of the tide for any instant can be read from the profile. 274. Locating the soundings. (1) The soundings may be located by two angles read simultaneously from the opposite SOUNDINGS. 291 ends of a line on shore, the recorder in the boat signaling the observers on shore when a sounding is about to be taken, and again when it is taken. (2) The soundings may be located by establishing two flags on shore in a line nearly normal to the shore line and taking the Fto. 139. soundings in line with the range thus formed, the position of the boat being determined by an angle read on shore, or by time in- tervals. A series of such ranges is laid out, and soundings are taken along each range. The point selected for . occupation by the angle- ^ s> measuring instrument on shore should be so chosen that the angles of inter- section with the range lines will be large enough to give good locations. If the shore is narrow and precipitous, the range may be established by placing a flag on shore and a buoy in the water. The position of the buoy is determined as was described in the first method for locating soundings. The distances be- tween the shore points will be measured for base lines, and the angles will be measured from these base lines. Fig. 139 will explain the method. The buoys may be any visible float, an- chored in place by any heavy weight attached to the buoy by a rope. The rope must be sufficiently long to permit the buoy to FIG. 140. 292 HYDROGRAPHIC SURVEYING. be seen at high tide. At low tide the position of the buoy will not be certain, for obvious reasons. A good kind of buoy is a tapering spar of wood. If it is not readily visible, a flag may be set in the end. (3) If many soundings are to be taken at different times on one cross section of a river, they may be located by taking them at the intersection of a series of ranges so laid out as to have their intersections on the required cross section. Fig. 140 will sufficiently explain. (4) Again, the soundings may be located by stretching a rope or wire across the channel and taking them at marked FIG. 141. intervals along the line. This method is adapted to narrow channels, and is used somewhat in connection with measure- ment of materials dredged in such channels. (5) Again, they may be located by measuring in the boat, at the time the soundings are taken, two angles to three known points on shore. The angles are measured with a sextant. There are several methods of obtaining from the data thus secured the location of the point where the sounding is taken. The problem to be solved is known as the "three-point prob- lem." The most convenient way to plot the point on a map on which have been already plotted the known shore points, is by the use of a three-arm protractor, shown in Fig. 141. The two SOUNDINGS. measured angles are set on the protractor and the arms aro made to coincide with the plotted points. The center then gives the required position of the sounding. If no three-arm protractor is available, draw three lines on a piece of tracing paper so that the two measured angles are included between the lines. Shift the paper over the map till the three lines pass through the three points. The point from which they radiate is now over the position of the sound- ing, and may be pricked through. The position of the point may be found al- gebraically as follows : In Fig. 142, the known points are A, B, C, de- termined by the angle A, and the sides a and b. The measured angles are a and /3, and the required point is P. and GPB there is obtained FIG. 142. From the triangles APB PB = a sin I b sin m Also, whence, and sin a sin /? I + m = 360 - (A + m = S - I, sin m sin S cos I cos S sin I. (1) (2) (3) From (1) and (3) determine Z, which being found, the triangle APB may be solved and P located. To solve for I, substitute in (1) the value of sin m found in (3), reduce to common denominator, divide by sin Z, transpose, and get cot I = I sin sin S = cot s b sin cos S (4) 275. Occurrence of methods. The fifth method is used almost exclusively in locating soundings in bays, harbors, and 294 HYDROGRAPHIC SURVEYING. off the seacoast. The others are used in connection with river and small lake surveys. 276. Survey of a harbor, river, etc. In making a complete map of a harbor, coast, or river, a system of triangles of greater or less extent is laid out. The triangulation stations thus locate the prominent points. These points are then used as a basis for locating the soundings. If too far apart, other points are located along shore by running a traverse between triangu- lation stations. Such a traverse is usually run in any event to determine the detail configuration of the shore line and the topography adjacent thereto. This work is best done with the transit and stadia or with the plane table. The map is finished like any topographical map. The contour lines of the bottom are frequently drawn as dotted lines to a depth of four fathoms, and beyond that depth the soundings are given in figures expressing fathoms and quarters. Charts of the various harbors and many portions of the coast of this country are published by the United States Coast and Geodetic Survey, and may be purchased for a small price. On these charts are located buoys and beacons. These are estab- lished to denote shoals and rocks, and are so arranged that in entering a harbor a red buoy with even number is to be passed on the right; a black buoy with odd number is to be passed on the left, and buoys with red and black stripes may be passed on either the right or the left. Buoys in channels are painted with black and white vertical stripes. Beacons and buoys are different things. A beacon is a per- manent fixed signal, usually on a shoal or dangerous rock ; while a buoy is a float of some kind, anchored by a chain. It is used to denote either danger or channel. THE SEXTANT. 277. Description. It has been said that when the angles are read in the boat, they are measured with a sextant. The sextant is shown in Fig. 143. It consists of an arc of sixty degrees, but, from the principle of the instrument, reading angles up to one hundred and twenty degrees. I is a mirror attached to the movable arm that carries the vernier V. The THE SEXTANT. 295 arm is centered under the mirror. This mirror is called the index glass. The vernier reads to ten seconds. H is another glass, called the horizon glass. Its lower portion is a mirror, and its upper portion is unsilvered. CrCr are colored glasses to protect the eye when making observations on the sun. They may be turned back out of the way when not needed. 278. Use. To use the sextant to measure an angle between two terrestrial objects, hold the plane of the arc in the plane of the observer's eye and the two points. The telescope should be directed toward the fainter object. It may be necessary to hold the sextant upside down to do this. Swing the vernier arm till the image of the second point reflected from I to IT to the telescope is seen superimposed on the fainter object seen directly. Clamp the vernier, bring the images into exact coincidence, using the tangent screws, and read the vernier. The reading is the angle sought. It will be observed that the angle read is riot horizontal, unless the distant points are in the same level with the observer. It requires some practice 296 HYDROGRAPHIC SURVEYING. to become expert in the use of the sextant. The eyepieces shown in the figure are of different kinds. There is usually one astronomical eyepiece, one for terrestrial work, and one without magnifying power. The instrument is that used by seamen for observing for latitude and longitude. 279. Theory. The principle on which the sextant is con- structed is as follows : If a ray of light is reflected successively FIG. 144. from two plane mirrors, the angle between the incident and finally reflected ray is twice the angle of the mirrors. Referring to Fig. 144, since the angles of incidence and reflection are equal, i = r and i' = r\ and from Geometry and V (i + r}- (i 1 + /) = 2 (r - r'), (90 - i') - (90 - r) - (r - r'). Therefore E = 2 I 77 , which was to be shown. To show that the angle read by the vernier on the arc or limb is the angle F 7 , suppose the two mirrors to be parallel and the telescope to be directed to an object infinitely distant, so THE SEXTANT. 297 that the rays from it are parallel. An image of the object will be seen in the same line as the object, or will appear superim- posed on the object. This will be evident from the equality of the angles a. The vernier will then be in the position V and this point of the arc is graduated zero. If now the vernier is moved to another point V for the purpose of getting the reflection of a second point to cover the direct image of the first point, the angle VIV will equal that at F 7 , the angle of the mirrors. It is, however, the angle at E that is required, and this is twice the angle through which the vernier has moved. Hence the arc FT 7 , instead of being numbered to give the angle V, is numbered to give at once the angle E. That is, each degree is marked two degrees, etc. 280. Adjustments. There are four adjustments of the sextant: (1) To make the index glass perpendicular to the plane of the limb. (2) To make the horizon glass perpendicular to the limb. (3) To make the line of collimation of the telescope parallel to the plane of the limb. (4) To make the vernier read zero when the mirrors are parallel. (1) Bring the vernier to about the middle of the arc. Hold the eye at about the upper Gr of Fig. 143, and observe the arc near the zero point directly, and the reflected image of the other end of the arc in the index glass. If the glass is perpen- dicular to the plane of the limb, the reflected and direct por- tions will seem to form one continuous arc. Adjust the glass, if necessary, by means of the screws at its base. It may be necessary to place slips of thin paper under the base. (2) Direct the telescope toward a star or other very distant object and note whether the direct and reflected images seem, when the vernier is moved, to separate or overlap laterally. If they do not, the horizon glass is in adjustment. The adjust- ment is made, when necessary, by the screws of the glass. (3) Place the sextant on a plane surface and direct the tel- escope to a point not far away. Place the two peep sights shown in Fig. 143 on the extreme ends of the limb and note whether the line of sight through them coincides in elevation 298 HYDROGRAPHIC SURVEYING. with the line of sight of the telescope. If not, adjust the tele- scope by the screws in its collar. Any other objects of equal altitude will serve as well as the peep sights. (4) Bring the di- rect and reflected images of a very dis- tant point to coincide, and read the vernier. It should read zero. If it does not, note the error as an index cor- rection to be applied to all angle readings. The error may be cor- rected by adjusting the horizon glass, but this is not usually done. Fro. 145. 281. Other forms. By a double sextant, Fig. 145, two angles can be measured quickly by one observer, one sextant measuring one angle and the other sextant, the other. MEASURING VELOCITY AND DISCHARGE. 282. Position of maximum velocity. The velocity of a stream varies in different portions of a cross section and in different cross sections. The maximum in a cross section is, when there is no wind, at about one third the depth in the middle of the channel. The surface velocity may be greater if the wind is favorable. To determine the mean velocity in a given cross section, the velocity in many parts of the section must be found. This is best done with a current meter. 283. Current meters. These are of various patterns. Those shown in Figs. 146-148 are considered good forms. Fig. 146 is a meter devised by W. G. Price, United States assistant engineer. There is an electric connection with a register that 300 HYDROGRAPHIC SURVEYING. indicates the number of revolutions of the wheel. The number of revolutions per second is a function of the velocity of the current through the meter. Fig. 147 is a modification of the Price meter, called an audible, or acoustic, meter, and for streams is probably as con- venient as any form yet devised. It is so constructed that at each tenth or twentieth revolution a small gong is struck, and the sound is carried through a rubber tube to the ear of the observer above in a boat. This meter requires no registering apparatus at all. It is not recommended for deep-sea work, in which a strong, heavy instrument is needed, since it is very light. Fig. 148 is a different form of meter with electric register- ing attachment. 284. Use of the meter. In use the meter is simply held in different positions in a cross section whose velocity is required, and in each position the number of revolutions per second is noted. The position of the meter is located by any one of many possible methods that will suggest themselves to the surveyor. The number of revolutions per second bears some relation to the velocity of the current going through the propeller-like wheel of the meter, and the determination of this relation is called "rating the meter." Every meter must be rated before the velocity of current corresponding to an observed velocity of wheel can be told. 285. Rating the meter. The meter is rated by moving it through still water at a known rate and noting the revolutions per second. By moving it at different velocities it will become apparent that the velocity is not strictly proportional to the speed of the wheel but bears a relation that must be expressed by an equation. From a number of observations at different velocities, a diagram may be drawn that will give the velocity corresponding to any observed speed of the wheel. This is done as follows: The meter is moved over a known distance and a stop watch is used to determine the time occupied. The distance divided by the time in seconds gives the velocity in feet per second. The number of revolutions made during the time is recorded, and this divided by the time gives the number MEASURING VELOCITY AND DISCHARGE. 301 of revolutions per second. The following observations were made by Mr. Price, all on a distance of two hundred feet. In the table, R is the whole number of revolutions in the two hundred feet, T the whole time, r the revolutions per second, and v the velocity per second. No. T r v 1 100 53 1.886 3.774 2 101 44 2.295 4.544 3 101 41 2.464 4.878 4 96 124 0.774 1.613 5 94 152 0.618 1.316 6 90 193 0.466 1.036 7 91 181 0.503 1.105 8 103 28 3.678 7.142 9 100 53 1.886 3.774 10 98 73 1.342 2.740 11 193 29 3.552 11)19.464 38.818 1.769 The average speed of the wheel is seen to be 1.769 revolu- tions per second, and the average velocity of flow, or current, is 3.529 feet per second. Two axes at right angles are now drawn. On one of these axes are laid off the several values of r, and parallel to the other are laid off the corresponding values of v. In Fig. 149, the horizontal axis is that of r, and the vertical axis is the velocity axis. In the first observation, r is 1.886 and v is 3.774. Assuming suitable scales the observations are plotted in the same way as a point by latitudes and longitudes. Thus, the latitude of the first observation is 3.774 and its longitude is 1.886. If cross-section paper is used the work will be facili- tated. Each observation being plotted, it is observed that they all fall nearly on a straight line. It may be assumed that they should all fall on the straight line and that they do not because of small errors in the observations. If the mean values of r and v are plotted it may be assumed that the line should pass through the point thus found and that it must then be swung to average the other points as nearly as may be. A piece of fine thread is stretched through the point of average r and u, and swung 302 HYDROGRAPHIC SURVEYING. around till it seems to the eye to average the other points. This line is that on which it is most probable that the observa- Axis of Revolutions per Second. FIG. 149. tions should all fall, and hence if it is drawn and the meter is held in a running stream and the revolutions per second are MEASURING VELOCITY AND DISCHARGE. 303 noted, the corresponding velocity may be determined by laying off the observed r on its axis and measuring the corresponding v up to the line. If a horizontal line is drawn on the diagram through &, where the line cuts the axis of v, it will be seen that for any value of r the corresponding value of v is r tan a + bo. If b represents 60, we may write v = r tan a + b. From the values of the average r and v in the example given v = 1.904 r + 0.16. In its general form this equation is v = ar + 5, and this is known in Analytical Geometry as an equation of a straight line. This equation is the general equation of the rela- tion between v and r. The rating of the meter then consists in finding values for a and b from observations that give v and r. Since there are two unknown quantities, two independent observations should be sufficient ; but since no observation can be perfect, many are taken, and mean values are determined for a and b. This may be done analytically by the method of Least Squares. 286. Rod floats. A very good method of obtaining veloci- ties, when a current meter is not available, is to observe the velocities of rod floats in various parts of the stream. For this purpose two ranges are established on shore, one above and one below the section in which it is desired to measure the velocity. The ranges are laid out normal to the stream. A transit is placed on each range, and the rod is started just above the upper range. The transit man on the upper range signals the lower observer when the rod is about to cross the upper range, and just as it crosses ; and the lower observer reads an angle to the rod at the instant it crosses the upper range. The opera- tion is repeated when the rod crosses the lower range, the lower observer making the signals, and the upper observer reading the angle to the rod. Each notes the time of crossing the range on which he is stationed. In this way the path of the rod and the time it takes to travel that path are known. The rod may be of wood, or it may be a long cylindrical tin can. In either case it should be weighted so as to remain 304 HYDROGRAPHIC SURVEYING. vertical and just clear the bottom. The velocity of the float will then be approximately that of the vertical filament of the stream in which it floated. The immersion of the rod should be about nine tenths of the depth of the stream. If this rule is observed, the mean velocity in the vertical filament will be given by V m = V [1 - 0.116 (VI) - O.I)] 1 in which V is the observed velocity of the rod, and D is the ratio of the depth of the water below the bottom of the rod to the total depth of the water. In small streams, wires or ropes graduated with tags may be stretched across the stream at the upper and lower ends of the stretch of river to be observed, and the time of passing of the float under each wire or rope may be observed. By observing the velocities in many vertical filaments, the mean velocity of the section may be determined. It would not be a mean of the several observed velocities unless all "fila- ments " had equal areas. The mean velocity is that which multi- plied by the area will give the discharge, hence it is - -r - 287. The discharge. To get the discharge, simply multiply the mean velocity of each filament by its cross-sectional area, and add the products. If the velocity is determined by meter, the work need be done in but one cross section ; if by rod floats, it will usually be best to measure two or more sections, preferably at the quarter points of the stretch, from which to get a mean section. DIRECTION OP CURRENT. 288. How determined. It is sometimes necessary to know the direction of both surface and subsurface currents. The necessity may arise in the determination of the proper place to discharge sewage, or in surveys for the improvement of har- bors and the approaches thereto. The surface currents may be determined by watching floats. The subsurface currents are best found by a direction meter. This is a form of meter that not only gives velocities, but also shows the direction of the current in which it is placed. It was devised by Messrs. Ritchie and Haskell. It is said to be one of the best forms of meters for measuring currents alone. 1 Francis's " Lowell Hydraulics." CHAPTER XII. MINE SURVEYING. SURFACE SURVEYS. 289. Coal mines. The surface surveys in connection with coal mining operations consist in making land surveys of 'the property that may be owned by the mining company, and locat- ing all shafts, tunnel openings, buildings, railways, roads, and other structures. In addition to this it is well, if the area owned is extensive, to make a complete topographical survey of the property. Frequently a mining company mines coal under the lands of many small holders and pays a royalty for the coal so mined. It is therefore necessary to know the posi- tions of the land lines of all owners under whose land it is probable that operations may extend. The location of these lines is best performed by careful stadia surveys reading the stadia to the smallest possible unit. (There may be cases of sufficient importance to warrant most careful transit and tape work.) The topographical survey may be carried on at the same time. Indeed, the location of the land lines becomes merely an incident in the topographical survey. The positions of the corners are determined by computing their latitudes and longitudes. Among other points that will be located in the topograph- ical survey or land survey will be the positions of exploration drill holes that have been put down in advance of mining oper- ations for the purpose of locating veins, or beds, as they are frequently called. The entire survey may or may not be hung on a system of triangles. If it is of any considerable extent, it is best to reference it thus for the sake of the checks it gives on the field work, even though not more than two or B'M'U SCRV. W 305 306 MINE SURVEYING. three triangles are used. The whole work, except the side shots, is best plotted by latitudes and longitudes, these being computed with sufficient accuracy for this purpose by the aid of a trigonometer. The side shots, except those to land corners, may be plotted with a protractor. The surface map should show all outcroppings of mineral. 290. Metal mines. Surface surveys for metal mines may be of the same character as those for coal mines, but in the United States there are certain laws governing the size and form of mining claims that are located on public land, and the method of surveying them. 291. Form of mining claim. By the provisions of these laws, a mining claim may not exceed fifteen hundred feet in length along the vein nor six hundred feet in width (three hundred feet each side of the vein). It' may be of any length or width less than these limits, according to local custom ; but no rule may be made that will compel the claim to be less than twenty-five feet on each side of the vein. The side lines need not be straight or parallel, but the end lines must be both. The miner has the right to follow the vein he has discovered into the ground to any depth, even should it depart so much from the vertical as to pass beyond the'vertical planes through the side lines ; but he may not mine beyond the vertical planes through the end lines. When claims intersect, the subsequently located claim is said to conflict with the previous one and carries with it only the area that is not in conflict. The foregoing rules apply to what are called lode claims. A " lode " is the name sometimes given to a vein of ore. A " placer " claim is a claim taken for the purpose of ob- taining gold or silver or other metal from the surface materials, as in washing gold from a river bed or from gravel in the mines worked by hydraulic process in California. Such a claim may not exceed twenty acres, and, if located on surveyed lands, its lines must conform to the legal subdivisions. A placer claim carries no right to mine beyond the vertical planes through its bound- ing lines. It carries the right to mine a vein discovered within SURFACE SURVEYS. 307 its boundaries after the claim is located, but not if the vein is known to exist when the claim is located, unless specially mentioned. 292. Surveying the claim. When a miner has found a vein of ore, he has a claim staked out on the ground, conforming to the legal requirements, using a width and length that he thinks best. He also posts the notices required by law. This survey may be made by any surveyor. After the miner has expended five hundred dollars in working his claim he is entitled to have -it deeded to him by the government. This deed is called a "patent," and the claim is said to be "patented." For the purpose of this patent, a survey is made, from which to write a description. This final survey, on which the deed to the miner is based, must be made by a United States offi- cer, known as a United States deputy mineral surveyor. Such officers are under bonds of ten thousand dollars, and are re- quired to pass an examination before receiving their licenses. They make the final survey" and tie it to a corner of the public surveys or to a special mineral survey monument, and return to the surveyor general of the state the notes, with an affidavit that the required five hundred dollars have been expended on the mine. The form of field notes and the instructions issued to United States deputy mineral surveyors by the surveyor generals of the various mining states may be had on application to the surveyor general's office. 293. Surface monuments. The surface monuments of a mine that is to be worked for any length of time should be of a permanent character. At least two of them should be located near the entrance to the mine, and these two should be particularly permanent, for use in connecting the under- ground and surface surveys. These may be auxiliary monu- ments located for the express purpose of connecting the two surveys. They may be built of stone or brick, with a properly centered copper bolt to mark the exact spot. The bolt may be removable for the purpose of substituting a signal. These monuments must be constructed with a view to being proof 308 MINE SURVEYING. against disturbance by frost, or travel, or by the operations about the mine. A true north and south line permanently monumented is perhaps the best base for all surveys. UNDERGROUND SURVEYS. 294. General statement. Underground surveys consist of traversing, leveling, and sometimes measuring volumes. The necessary traverses are run to determine the location of the various portions of the mine, so that a map or a plan of the mine can be made. Leveling is done to determine the relative elevations of the different parts of the mine, the grade of the tunnels or drifts, the direction and amount of dip of the vein, etc. The quantity of ore mined is sometimes determined by measuring the volume excavated in the mine, and the meas- urement of the quantity of ore " in sight " is usually by the volumetric method. The ore mined is usually measured by weight after it is brought out of the mine. The only differences between underground and surface surveys are due to the difficulties encountered in working in dark, cramped passages. These necessitate certain modifica- tions in surface methods to make them applicable to under- ground work. 295. Definitions. The following are a few definitions of technical terms used in mining. 1 There are many others that the mining surveyor must know. Shaft. A pit sunk from the surface. Adit. A nearly horizontal passage from the surface, by which a mine is entered and through which the water that collects in it is removed. In the United States an adit is usually called a tunnel, though the latter, strictly speaking, passes entirely through a hill, and is open at both ends. Drift. A horizontal passage underground. A drift follows the vein, as distinguished from a cross-cut, which intersects it, or a level or gallery, which may do either. Level. A horizontal passage or drift into or in a mine. It 1 Taken from " A Glossary of Mining and Metallurgical Terms " by R. W. Ray- mond, mining engineer. UNDERGROUND SURVEYS. 309 is customary to work mines by levels at regular intervals in depth, numbered in their order below the adit or drainage level. Cross-cut. A level driven across the course of a vein. Winze. An interior shaft, usually connecting two levels. Strike. The direction * of a horizontal line, drawn in the middle plane of a vein or stratum, not horizontal. Dip. The inclination of a vein or stratum below the hori- zontal. The dip is necessarily at right angles 2 with the strike or course, and its inclination is steeper than that of any other line drawn in the plane of the vein or stratum. Pitch. The inclination of a vein, or of the longer axis of an ore body. Incline or Slope. A shaft not vertical ; usually on the dip of a vein. Stope. To excavate ore in a vein by driving horizontally upon it a series of workings, one immediately over the other, or vice versa. Each horizontal working is called a stope (prob- ably a corruption of step), because when a number of them are in progress, each working face being a little in advance of the next above or below, the whole face under attack as- sumes the shape of a flight of steps. The process is called overhand stoping, when the first stope is begun at a lower corner of the body of ore to be moved, and, after it has ad- vanced a convenient distance, the next is commenced above it, and so on. When the first stope begins at an upper corner, and the succeeding ones are below it, it is underhand stoping. The term " stoping " is loosely applied to any subterranean ex- traction of ore except that which is incidentally performed in sinking shafts, driving levels, etc., for the opening of the mine. 296. Location and form of station marks. In underground work, stations can not be stakes driven in the floor of the mine. The reasons are obvious. Where the roof is solid rock, holes may be drilled in it and wooden plugs inserted, in which may be driven a nail or tack. A good form is a horseshoe nail with flattened head, in which is cut a triangular hole with an apex of the triangle toward the head end of the nail. Such a nail 1 Azimuth. 2 That is, the direction of the vertical plane in which the amount of the dip is measured is at right angles. 310 MINE SURVEYING. driven in the roof insures that a plumb line suspended from it will always hang from the same point. A small round opening about one eighth inch in diameter is preferred by some. When the roof is soft and the drift timbered, temporary stations may consist of nails driven in the timbers. It is probably better in such cases to reference the station by two points on the side of the drift. If the two points are properly located, and the dis- tance from the vertical through the station to each of them is recorded, the station may be recovered at any time. When about to occupy a station, transfer it to the floor with a plumb line and set there a temporary mark consisting of a piece of slate with a cross scratched on it, or a small metallic cone carried for the purpose. Stations underground should be plainly marked. White paint is good if kept bright. The stations in a mine should be consecutively numbered, and the same number should not be used twice. They should be numbered at the station points as well as in the notebook. Probably as good a way to do this as any is to drive round- headed nails or tacks in the timbering near the station, locating them all in the same position relative to the station. The tacks may be driven in a small board or plug fastened to the side of the drift near the station. The nails are arranged to give the number of the station, thus :: for 324 and 0: for 302, a washer being used for the 0. Practically this method was followed in the New Almaden mine in California. The nail heads can be felt and read, even though not clearly visible. As the mine is extended and new stations are added, they should receive their proper numbers and be permanently lo- cated. There is no branch of surveying where method and system count for so much as in mine surveys. 297. Instruments used. For all angle work in mine surveys, except the location of short tortuous passages, the transit is the only suitable instrument. The work must -be done with great precision, llesults need not be so precise, perhaps, as in high class city work, but a precision of one in five thousand should certainly be secured. It is best to use azimuths and perform the traverses as already described under traversing with the transit. UNDERGROUND SURVEYS. 311 The transit should be a complete transit or tachymeter, so that leveling may be done either directly or by vertical angles. The vertical angle method is the better ordinarily, as it saves much time in the mine. The tripod should be either short or adjustable in length. A diagonal eyepiece is desirable for pointings at high inclinations. The wires are made visible by means of a reflector described in Art. 90, or by a hollow axis and reflector within the telescope. Often enough light is secured by having an assistant flash a candle near the objective. It is sometimes necessary to point upward or downward at a considerable inclination, as when an incline is to be traversed. Pointing up is accomplished by means of an auxiliary diagonal eyepiece, but pointing down can not be done when the angle is so great that the line of sight cuts the plates of the instrument. Various devices have been invented to overcome this diffi- culty. Sometimes an aux- iliary telescope is mounted above, and parallel with, the main telescope of the tran- sit, as in Fig. 150. In some transits, the auxiliary telescope is mounted on the end of the horizontal axis, as in Fig. 151. Some tran- sits are made with the standards leaning forward far enough to permit the line of sight to clear the FlG - 150 ' plates when turned vertical. Others are made with double standards, one as usual, and one bracketed to the ordinary standards. The telescope is made so as to be readily changed from one bearing to the other. If the side telescope is used, the transit should be set over a point to one side of the station occupied, or the line of collimation of the auxiliary telescope should be directed to a point as much to one side of the true station as the auxiliary telescope is to one side of the main telescope. This is conveniently done by having the object sighted to, as a candle, fastened to a flag at the proper distance from it. The flag is held on the true station. UNDERGROUND SURVEYS. 313 For tortuous passages in which a transit can not be set up, so crooked that one end can not be seen from the other, resort is had to what is sometimes called a German dial, and a hanging clinometer. One of the best forms of the dial is shown in Fig. 152. 1 The compass is hung on a wire or line stretched from one station to the next, and as it levels itself like a mariner's com- pass, it may be properly read. The bear- ings are not depended on, but angles are read. Passages so tortuous as to require the use of such methods are not usually of great length, and the resulting errors are small. They are not likely to occur in coal mines, but occur frequently in quicksilver mines, and sometimes in gold and silver mines. Coal mines usually FlG - 152 - involve fewer irregularities of 'alignment than any other class of mines. The linear measurements are made with a steel tape. In some cases of long, straight drifts, a long tape, two hundred feet, or three hundred feet, may be used to advantage ; but usually a tape either fifty feet or one hundred feet long is best. The tape should be as long as possible, and no more stations should be used than seem necessary. It is very necessary to expedite work in a mine, because mining work in a drift must be largely suspended while the survey is in progress. The flat wire tapes are best for mine work, since they are not so easily broken as the ribbon tapes. A pocket tape, graduated to hundredths of feet, may be carried for the necessary plus meas- urements at stations. If a level rod is used, it must be a short rod. Bench marks are best placed on the side of the drifts where they will not be disturbed. 298. Devices for making stations visible. The object ob- served at a station is usually a plumb line made visible by holding a piece of oiled paper or milk glass behind it and a candle behind the transparency. The plumb line may be illu- i From Brough's " Treatise on Mine Surveying." 814 MINE SURVEYING. minated by holding a light in front of it, shielded from the observer. Another method is to sight the flame of a candle or lamp, properly centered over the station. Another way is to use a plummet lamp, which is a lamp made in the form of a plummet. Perhaps the neatest method, at least in coal mines, is to use a second and third tripod on which can be mounted a target lamp, such as is shown in Fig. 153. The lamp is be- hind the target from the ob- server, and the target may be accurately centered over the station point. The target and transit are interchangeable on the tripods, the particular form shown being used with a special transit that is interchangeable above the leveling screws. The method of use is some- what as follows : The back station is occupied by a target, as is also the forward station, and that at which the obser- vation is to be made is occu- pied by the transit. When the observation is complete, the tripod at the back station is carried to the next forward station, the others remaining in place. The back lamp is placed on the tripod that was occupied by the transit while the latter is being carried to the tripod occupied by the previous forward lamp, which in turn is carried to the next forward station. The lamps should neither of them be moved till the observations at the station occupied by the FIG. 153. UNDERGROUND SURVEYS. 315 transit are complete. This is a very expeditious method of work. 1 The entire angular work may be done first, and the linear measurements made afterward, or vice versa, or tho two may be carried on at the same time. The latter method will probably consume more of the miner's time, since the linear work would retard the angle work. The linear work may be carried on without seriously interfering with the work of the mine. The extent or character of the work Avill guide the surveyor in choosing his methods. When lateral drifts depart at frequent intervals from one long tunnel or drift, the stations at the angle points of the main drift should be first located and the substations to be used in running the lateral drifts should be set in the line of the main drift. This may be done at the time the angles of the main drift are measured. In this case, the angle work should be done first. When an extensive survey of a mine is to be made, the sur- veyor should go through the mine and lay out his programme of work as closely as may be. It will almost always be possible to mark out the main stations before taking the transit into the mine, or at least before setting it up. 299. Notes. In an underground traverse in which notes are to be taken at intervals along each course to objects on the sides, a good form is as follows : Prepare the page of the notebook by ruling a column down the center, in which to write distances and azimuths, and leave the sides to note the objects measured to, and such other information as need be recorded. The form would be similar to that shown on page 229, except that the column for distances would also contain the azimuths and would be in the middle of the page instead of at one side. Notes taken on the right of a line that is run out should be noted on the right of the column, and those taken on the left should be recorded on the left. 1 An important point in favor of the three-tripod system, when long tapes are used for measuring, is that all ordinary distances may be measured from center of the telescope to center of back-sight or fore-sight flame, the vertical angle being read and noted. Having then the length of the line of sight and its angle, the true horizontal distance between stations is easily calculated. Oftentimes this is a much safer course than to rely on presumably horizontal measurements along the floor of the gangways. J. J. ORMSBEE, Mining Engineer. 316 MINE SURVEYING. A line should be drawn across the page between each two courses. For the purpose of computation of latitudes and longitudes, the azimuths and distances may be copied in more compact form in the office notebook. If elevations are carried by vertical angles, a good form is that given in Art. 127 for traversing with the transit and stadia, with sufficient vertical space given each course to write the details of the line along the side or right-hand page under remarks. The center line up the middle of the right-hand page may be used to represent the transit line, and the objects may be sketched to scale on this page. There should always be two sets of notes in existence, kept in different places, and when not in use they should be stored in fire-proof vaults. CONNECTING SURFACE AND UNDERGROUND WORK. 300. When the mine is entered by a tunnel. In order that the underground workings may be plotted with reference to the surface lines, or even so that they may be properly oriented, it is necessary to connect the underground surveys with those on the surface. The simplest case occurs when the mine is entered by an adit or drift. From a point in the surface sur- vey, located conveniently for the purpose, run a traverse into the drift, 301. When there are two shafts. If the mine is entered by shafts, and there are two or more of these some distance apart, the surface and underground surveys are connected by running a traverse on the surface and another underground between the shafts. The azimuth of the line joining the shafts is computed from the surface survey ; and that of the same line underground, from the underground survey, a zero azimuth being assumed for this purpose. The two computed azimuths will differ by the angle between the true zero and the assumed zero, and the assumed underground azimuths may be corrected accordingly. The pre- cise points at the shafts are made the same by using plumb lines. In deep shafts the lines are best made of piano wire, and the plummet must w.eigh from ten to twenty-five pounds. CONNECTING SURFACE AND UNDERGROUND WORK. 317 The line is first lowered with a light bob, and the heavy one is attached at the bottom. The line may be lowered by a reel. Signals may be given for lowering or raising by tapping on the guides or pipes in the shaft. Wherever possible, this is an excellent method, since small errors of position of the bobs are not multiplied. Another method, that may be applied in shallow mines, is to use the transit and auxiliary telescope, either setting the points in the bottom in line with a given line on top, or setting the transit in the bottom by trial, so that the line of collimation may be in line with a surface line. Perhaps a better way than the latter is to set the transit on the bottom and, with the horizontal motion clamped, set two points in a line on opposite sides of the shaft at the surface, which line is produced into the mine as the first course of the traverse. The points located on the surface are afterward connected with the surface survey. This method of using the transit and auxiliary telescope is adapted only to shallow shafts that are tolerably free from smoke. The better method is that using the plummets. Work may be carried from one level to another through winzes and air shafts by the same method. As high vertical angles occur in this work, double centering is an abso- lute essential to correct work. Levels are carried down the shafts by measuring with a steel tape, or, what is the same thing, by lowering a weighted wire and then measuring the wire. If the tape is used, and is shorter than the shaft, it is first hung on a nail near the surface, the elevation of the nail being determined by leveling. The reel and observers are then slowly lowered in the cage until the tape is all paid out, when a second nail is driven at a measured distance from the first. A piece of white cloth is fastened to this second nail, to make it visible ; and the cage is raised, the tape unhooked, the cage again lowered to the second nail, and the operation repeated from there down. A nail is left for a bench mark opposite each level. 302. When the mine is entered by one shaft. To connect a surface survey with an underground survey through a single 318 MINE SURVEYING. deep shaft is the most difficult task the mining surveyor will encounter. If the mine is of small extent and the shaft shal- low, it may be done with the transit and auxiliary telescope, as already described under the preceding article. But if the shaft is deep, it can not be done this way. The method is to use two plumb lines. The utmost care is necessary in their use, for from a base line from six to twelve feet long must be produced a survey one or two or more miles long. It may be necessary to drive a second shaft a mile or two distant, and it must be driven to meet an underground drift. An error of one tenth of an inch in a base line ten feet long will cause a shaft two miles distant to be driven nearly nine feet out of place. Two methods of using two plumb lines have given satisfactory results. (1) In Fig. 154, M and M' are surface monuments, and AB the two plumb lines. The dis- tance AB is care- fully measured, and the distance MA and MB and the angles at M. The azimuth of AB is then deter- mined. T is the transit set in the mine below. The angles at T, and the distances AT and BT are meas- ured. The triangle TAB is then solved, from which the azi- muth of TD becomes known. (2) This method is probably not so good as the second, which consists in suspending the plumb lines so that the transit may be set up in line with them below. The diffi- culty in all this work is that the plumb lines are never still, but continually oscillate. The mean position of the line is best determined by placing a fine scale behind it and noting the amplitude of the vibrations. The transit is pointed to the FIG. 154. CONNECTING SURFACE AND UNDERGROUND WORK. 319 mean position. The oscillations are not in a straight line, but are more or less elliptical; hence the scale must be ^placed a little behind the wire, but should be as close as possible to it, in order to avoid parallax. The transit is set up in the mine as close to the nearest line as possible (this will be from ten to fifteen feet distant), and each line is separately observed and the transit gradually brought by trial into line with the mean positions of both wires. The two plummets should hang in water, or, better, in some more viscous liquid, as oil, or even molasses. The oscillations will be slow, a line three hundred and twenty-five feet long requiring about ten seconds for one complete vibration in air. Of course the cages can not be in use during the operation. The points from which the bobs are suspended must be as firm as possible and as well defined, so that the line join- ing them may be connected with the surface surveys. Two transits are really required to insure good work. One is left on the surface with an observer to see that there is no movement of the points of suspension during the operation, and to note what -change, if any, takes place. It must be known that the lines do not hang against any projections on the sides of the shaft. This may be told, in some instances, by looking up the shaft ; in others, by passing a candle or other light slowly around the wire at the bottom, and observing from the top that it is visible throughout the motion. In case neither method is practicable, a system of movements of the wire may be arranged. When the observer below is ready, he will signal by rapping on the pipes or guides, and the observer above will, at agreed intervals of time, move the wire out one inch, two inches, etc. The observer below notes whether cor- responding movements take place there. On account of the stoppage of machinery this connection of surface and underground surveys is an expensive operation. It should nevertheless, in important cases, be performed several times, and a mean of the results taken for the first course of the underground traverse. When the bottom of the shaft can be seen from the top, a method used at the Severn tunnel will give good results. A wire is stretched from one side of the shaft, about one hundred or more feet into the drift, and accu- 320 MINE SURVEYING. rately aligned for the few feet visible by a transit on the sur- face. In the case of the Severn tunnel the wire was stretched over horizontal screws, the wire lying in the groove between the threads. The wire was then moved laterally for aligning by turning the screws. MAPPING THE SURVEY. 303. Metal mines. Three maps are necessary for a com- plete representation of a mine with many levels running in different directions (1) the plan ; (2) a longitudinal sec- tion ; (3) a transverse section. Such a set of maps is shown in Plate IV. at the back of the book. The notes taken by the surveyor should be such as to enable him to make these three maps. These maps may be in addition to one of the surface survey, and, if so, there would be four maps necessary for a complete representation of a mining property. It is usual to tint the portions shown on the plans as worked, and the differ- ent levels are sometimes tinted with different colors to distin- guish them. If several lodes are shown on one map, it will be better to color each lode, working with a single color to dis- tinguish it from the other lodes. 304. Coal mines. The representation of a coal mine is usually a simpler matter than the mapping of a metal mine. The coal ordinarily lies in beds nearly horizontal, and the work- ings may often be represented by a plan alone, the different levels being distinguished by colors, and the elevations being written in. It is customary to show on the map the direction and amount of dip of the seams. There are a number of sym- bols used to represent various objects ; but the practice is not uniform. Arrows may be used to show the direction of air currents, blue for inlets, and red for outlets, etc. The plan should show all the details of the mine, all stop walls, all doors, all sumps or reservoirs, all pumps or machinery, the location of faults, etc. It is extremely necessary that complete, correct maps should be kept up, so that if any connections are to be made, they may be correctly directed. Very serious accidents have occurred from incorrect maps of mines. MAPPING THE SURVEY. 321 305. Scale. The state of Pennsylvania provides that there shall be kept at each mine a plan on a scale of one to twelve hundred, or one hundred feet per inch, for the use of the state inspector. The English law requires the plan to be on a scale of twenty-five inches per mile, or about one half that required in this country. Other countries require different scales. The maps of the mining claims made by the deputy mineral surveyors are to a scale of two hundred feet to the inch. This would seem to be large enough for working drawings, except for details of structures. As in the case of notes, two maps should be in existence and kept in different places. A good example of a coal mine map is shown in Plate V. at the back of the book. 306. Problems. The problems that arise in mining survey- ing call for some ingenuity in handling Trigonometry and Descriptive Geometry, although they may all be solved without a knowledge of the latter subject by name. Some problems, such as seeking a lost vein, require more the principles of Geology than those of Mathematics. The commonest problems are the following : (1) The determination of the course and distance from a point in one compartment of a mine to a point in another com- partment. By course is meant azimuth and vertical angle. This includes such problems as the determination of the dis- tance and direction to run a tunnel to tap a shaft. (2) Knowing the strike and dip of a vein, to determine the length of a shaft or drift started at a given point and run in a given direction to meet the vein, or to determine the direction to run a tunnel from a given point so as to cut the vein in the shortest possible distance, and to find the distance required. It is believed that the student can solve such problems as are given in the Appendix, page 340, if he remembers that the pitch or dip is always measured in a vertical plane normal to a horizontal element of the vein, and that the strike is the direc- tion of such horizontal element. R'M'D SURV. 21 APPENDIX. I. PROBLEMS AND EXAMPLES. CHAPTER T. THE problems in the use of the chain may be solved on the blackboard with string and chalk. 1. To range out a line between two points on opposite sides of a hill : Mark the points so that they will be visible from the top of the hill. Two men with range poles then place themselves near the top of the hill, facing each other, so that each can see one end of the line. They then alternately align each other with the visible ends of the line until both are in the line. The same principle is used in "fudging" one's self into line between two distant visible points. 2. To range a line across a deep ravine or valley : The inability to do this by eye alone arises from the fact that the eye can not carry a vertical line on a hillside, nor can it be sure of transferring a point vertically down- ward, even in level country. The observer stands on one side of the valley, swinging a plummet over a point in the line to be ranged. He places his eye in line with the plumb line and a distant point in the line to be ranged, and, casting his eye down the plumb line, directs the placing of a flag held by an assistant in the valley. 3. To erect a perpendicular to a line : A triangle whose sides are in the ratio 4, 5, 6, is a right triangle. Hence, fasten one end and the thirty-link division at one point in the line and the ten-link division at another point. Carry the eighteen-link division out till the three portions of the chain used form a triangle. If the line is to be erected at a given point, the method will suggest itself to the student. 4. Using the chain and pins in the way a string and pencil or a pair of dividers would be used on the drawing board, perform the above problem in a different way. 5. To drop a perpendicular on a line from a given point : Run an inclined line from the point to the given line, erect a perpendicular to the latter, and produce it till it intersects the inclined line. The required PROBLEMS AND EXAMPLES. 323 perpendicular is parallel to the perpendicular erected, and the consideration of the similar figures will enable the student to solve. 6. To run a line parallel to a given line : Erect two equal perpendiculars to the given line. If the parallel is to pass through a given point, drop a perpendicular from the point to the line and erect a perpendicular to this at the point. Otherwise : The diagonals of a parallelogram bisect each other. 7. To measure an angle with a chain : In an isosceles triangle of legs , included angle A, and base b, 2 a sin \ A = b. 8. To lay out a given angle : Reverse the process just suggested. 9. A line is measured with a chain that is afterward found to be one link too long, and is found to be 10.36 chains long. What is its true length ? 10. A line is measured with a 100-foot tape and found to be 723.36 feet long. The tape is afterward found to be 0.02 foot short. What is the true length of the line? 11. A triangular field is measured with a chain that is afterward found to be one link too long. The sides as measured are chains, 4 chains, and 3 chains, respectively. What is the resulting area and what is the true area? 12. An irregular field is measured with a chain three links short. The area is found to be 36.472 acres. What is the true area? 13. A 100-foot tape is standard length for a pull of 10 pounds, when supported its entire length, and at a temperature of 62 F. A line is measured and found to be 1000.00 feet long. The tape is of steel, has a cross section of 0.002 square inch, and weighs 0.00052 Ib. per inch. In the measurement it is at a temperature of 80 F., and is unsupported except at the ends. The pull is 25 pounds. "What is the true length of the line ? 14. A steel tape weighs 0.0061 pounds per lineal foot. It is 100 feet long and is standard for no pull when entirely supported, at 62 F. It has a cross section of 0.002 square inch. What pull must be exerted to keep the tape standard length if it is supported only at its ends, the temperature remaining unchanged? 15. A line is measured along the surface on a hillside. The first 300 feet has a vertical angle of 3, the next 250 feet has a vertical angle of 4 30', and the last 700 feet has a vertical angle of 1. What is the true length of the line ? 16. A similar line, of equal surface lengths, is measured, and the rise per hundred feet of each stretch is found to be respectively 5 feet, 8 feet, and If feet. What is the true length of the line? 17. Measure a line of about 1000 feet or more in length, at least twice with a chain and twice with a tape, and determine the difference of measure- ment. Do this, if possible, over two lines of about equal length, but offering very unequal difficulties to measurement, and note the differences. 324 APPENDIX. CHAPTER III. 1. Determine the angular value of one division of each bubble avail- able in the school collection of levels. The method is in brief as follows : Hold a rod at a known distance ; read the rod and bubble with the bubble near one end of its tube, and then read with the bubble near the other end. By the differences of readings and the known distances, determine the value of one division. If there are any striding levels, use these as described for the level with metal base. 2. Other exercises with the level will suggest themselves to the student or teacher. A considerable amount of differential and profile leveling should be done, and profiles drawn. The leveling should always be checked by rerunning in the opposite direction. CHAPTER IV. 1 . Radiating from a point A , are eight lines of bearings as follows : AB, N. 53 45' W. AF, S. 86 45' E. A C, N. 36 42' W. A G, S. 24 36' E. AD, N. 18 34' E. A H, S. 20 30' W. AE, N. 34 28' E. A I, S. 36 20' W. Required the angles between the following bracketed lines always considering the smaller angle : \AB ^(AB (AB (AB (AB (AB (AB \AC \AD \AE (AF (AG (AH (AI (AC (AC ^AC (AC (AC (AC (AD \AE (AF (AG I AH ( AI (AD (AD (AD (AD (AD '(AE (AF (AG (AH ( AI (AE (AE (AE (AE (AF (AG (AH (AI (AF (AF (AF (AG (AH (AI (AG (AG \AH (AI (AH (AI 2. A traverse is run with the following bearings : N. 42 E., N. 36 E., S. 1 W., N. 50 W. Determine the interior angles. If the traverse is of a closed field, what should be the sum of the interior angles? PROBLEMS AND EXAMPLES. 325 3. A traverse is run with the following azimuths, zero azimuth being north : 42, 144, 181, 230. What are the bearings? 4. Another is run with the following azimuths : 306 15', 18 34', 93 15', 200 30'. What are the bearings? 5. What are the azimuths of the following lines of Example 1 : AC, AE,AG,AI1 6. Determine the exterior or deflection angles in Example 4. 7. Let the eight lines of Example 1 be the first eight consecutive courses of a traverse. Determine the deflection angles. 8. The magnetic declination at a given place is found to be 10 30' E. What will be the bearing of the true north ? the true south ? the true east ? the true west? 9. The magnetic declination is 7 10' W. What will be the bearings of the four true cardinal points ? 10. A line of an old survey is recorded as N. 18 E. mag. bear. It now reads N. 16 30' E. What has been the change in declination in direction and amount ? 11. A line of an old survey is recorded as N. 36 15' E. mag. bear. It now reads N. 34 30' E., and the magnetic declination is now 10 30' W. What was the declination at the time of the original survey? 12. A line of an old survey is recorded as N. 36 15' E. mag. bear., and the declination is recorded as having been 10 30' W. at the time of the sur- vey. The declination is now 12 00' W. What magnetic bearing should the line now show ? 13. A line of an old survey is recorded as S. 26 00' E. mag. bear., dec- lination 10 30' W. The declination is now 9 W. What magnetic bear- ing should be used to retrace the line? 14. The magnetic declination is 10 30' E. If the declination vernier is attached to the south side of the compass box, in what direction and by what amount should it be moved so that true bearings may be read by the needle ? With sights pointed to the magnetic north, what would the needle read after moving the vernier ? 15. A line of an old survey is recorded as N. 30 15' E. mag. bear. The declination is now 4 W., and the same line reads N. 30 E. It is desired to set the declination vernier so that the remainder of the survey may be retraced by the recorded bearings. The vernier being attached to the south side of the compass box, what is its movement in amount and direction? 16. Suppose, in the above example, the former bearing had been S. 26 W., and the present bearing S. 26 30' W. What would be the movement of the vernier? 17. Is it necessary in the foregoing examples to know the present decli- nation ? 18. Determine the angular value of the plate bubbles and the telescope bubble of each transit in the school collection. Do this work as suggested 826 APPENDIX. for the level, and also for the telescope bubble, by the use of the vertical circle. 19. Determine the meridian by an observation on Polaris and by the solar transit, and compare the results. 20. Measure all the angles of a polygon that has been laid out on the ground and note whether they sum up properly. 21. Set over each station in succession and run the polygon as a traverse, using azimuths, and, reoccupying the first station, redetermine the azimuth of the first course from that of the last, and note whether the azimuth found agrees with that first used. In fairly good transit work, the error in a perimeter of a mile or more should not exceed one minute. 22. The adjustments of the transit should be made by the student. CHAPTER V. 1. Determine for the transits in the school collection the value of 4 and 2. Reduce the horizontal distances and differences of elevation in the set of notes shown on pages 337-338. CHAPTER VI. Find the error of closure, balance the survey, plot, and compute the areas of the following fields : STA. BEARING. CHAINS. A S. 51 10' E. 5.05 B S. 58 10' W. 4.63 C X. 29 15' W. 4.16 D' X. 45 30' E. 2.87 Area, 1.676 + A. A X. 84 00' W. 9.04 B S. 21 15' W. 12.34 C X. 72 15' E. 12.92 I) X. 9 30' E. 6.68 S. 7 25' W. S. 62 05' W X. 2 35' E. X. 34 25' E. X. 83 00' E. Area, 9.264 4.35 6.94 4.01 3.64 4.51 Area, 3.124 + A. PROBLEMS AND EXAMPLES. 327 4. STA. BEARING. CHAINS. 1 S. 62 15' E. 5.12 2 S. 71 15' E. 4.66 3 S. 5 30' W. 12.00 4 S. 80 15' W. 12.46 5 N. 2 15' E. 9.46 6 N. 25 45' E. 9.40 Area, 17.683 + A. STA. AZIMUTH. CHAINS . 1 107 15' 16.40 2 196 15' 24.10 3 246 05' 19.60 4 13 55' 37.00 Area, 48.36 + A. C). The latitudes and longitudes of a series of points are as follows, measured in chains: ( Latitude + 13.63 1 Longitude + 10.20 5 Latitude + 1 .26 Longitude + ( Latitude - 1 Longitude + < Latitude 1 Longitude 5.10 3.30 7.20 3.30 ( Latitude + 7.10 I Longitude - 10.20 Find the area included by the broken line joining the points. (Shift the coordinate axes so that the coordinates are all positive.) 7. The following offsets were taken on the sides of a line indicated, at points 100 feet apart. Required the area between extreme boundaries. L. 137.4 93.5 49.3 78.2 102.0 100.5 96.6 144.2 111.9 71.4 DIST. R. 55.1 100 76.5 200 83.3 300 79.9 400 09.7 500 59.5 600 83.3 700 70.5 800 54.4 900 101.2 Area, 153,085 square feet. 328 APPENDIX. 8. The following offsets were taken on one side of a base line, to deter- mine the area between it and an irregular boundary line. The distances are all from zero : Distance* Offset. Feet. Feet. 20.7 202 31.5 358 42.6 825 53.2 984 3G.1 1223 40.7 Area, 49,698.25 sq. ft. 9. In Example 1, let it be supposed that the second course is wanting. Supply it. 10. In Example 2, let it be supposed that the bearing of the second course and the length of the third course are wanting. Supply them. 11. In the third example, let it be supposed that the bearings of the sec- ond and fourth courses are wanting. Supply them. 12. In the fifth example, let it be assumed that the lengths of the second and fourth courses are wanting. Supply them. 13. The wheel of a planimeter has a circumference of 2.26 inches. What should be the length of tracer that the number of revolutions multi- plied by 10 shall give in square inches the area circumscribed ? 14. An area is circumscribed by a planimeter with the fixed point inside, and the resulting reading is found to be 11.26 square inches. The area is known to be 120 square inches. What is the area of the zero cir- cumference ? COORDINATES.^ MODEL EXAMPLE. 1. To determine the coordinates of the corners Abcdef (Fig. 155) and plot the survey, the corner e and bearing fe being unknown, but point e' in prolongation of fe and bearing of de, as well as other corners, known : Run the random traverse ABCDEF. Make a rough sketch of the traverse and place on it lengths of lines and angles measured. (Note that angles and not azimuths or bearings are read. This is careful work where angles are repeated several times, and hence it is impracticable to read azimuths.) Balance the angles so that their sum will be " twice as many right angles less four as the figure has sides," distributing any error equally among the various angles. In the example the angles "close." 1 The problems here given were furnished the author by Mr. John H. Myers, Jr., A.B., C.E., Assistant Engineer Department of Public Works, Brooklyn, N.Y. The examples are from practice in New York City and were treated as here indicated. FIG. 155. 330 APPENDIX. Following a previous survey in this district, assume bearing of DE to be S. 9 31' E., and compute bearings of remaining lines and place them on the sketch. Assume coordinates of point A 200 N. and 100 E. to bring all coordi- nates positive. Compute the coordinates of the other corners of the traverse as shown on page 331. The student should note the systematic layout of the work. The error of closure, 0.05 E. and 0.14 N., is distributed as in balancing a survey, and the balanced coordinates are used in all subsequent work. Plot the traverse as shown in Fig. 155. Check the plotting by scaling the lengths of the lines. To locate the property corners ; Corner A is a corner of the random. Course .4/was actually run on the ground. Calculate from the recorded angles the bearings of the lines joining ran- dom corners and true corners, and compute the latitude and longitude differences of these lines, and from these the coordinates of the corners. They will be found to be : b, 597.42 E., 282.05 N. c, 585.10 E., 353.95 N. d, 881.69 E., 403.66 N. /, 118.59 E., 70.33 N. The computations are systematically arranged as follows, the computa- tions for c, d, and /, being shown : Dd N. 34 32' 30" W. 29.81 c . . S. 89 51' 30" W. 11.49 // . . 9.753587 1.474362 602.00 E. 16.90 W. 9.915776 1.474362 1.390138 7.417968 1.060320 329.40 N. 24.55 N. 1.227949 9.999990 1.060320 585.10 E. 893.18 E. 11.49 W. 353.95 N. 403.69 N. 0.03 S. 1.060319 881.69 E. 403.66 X. S. 8 09' 30" E. 131.00 / 9.152010 2.117271 100.00 E. 18.59 E. 118.59 E. 9.995582 2.117271 200.00 N. 129.67 S. By Prob. 1, Art. 143, find the length and bearing of the courses the coordinates of whose ends are known. They will be found to be as indi- cated on the figure. The arrangement of the work for a single course is shown on page 332. PROBLEMS AND EXAMPLES. 331 fc tes fesste M i-H 00 ' fc Ci TO TO . s TO TO :n So os cs .0 J ryi M fV, rV] 8 oS: oS: + + + KrU W C^ O O, r-l < T-H *>l. ' fri 1-1" TO ei it s g S 8t sssi- 05 1> OS CN Ci !> O iC 2 5 OS 1-H 05 05 OS 125' 720 - 247' 35.0 65.0 5. 1540' 635 - 242' 30.0 70-0 6. 2650' 985 - 158' 33.8 66.2 7. 3000' 1175 - 148' 37.0 63.0 8. 4200' 1370 - 142' 40.5 59.5 9. 4830' 1120 - 145' 34.2 65.7 10. 3520' 730 - 215' 28.7 71.3 11. 5250' 825 - 208' 30.7 69.3 12. In ravine running) northerly. i 17 23230' 710 - 301' 37.5 62.5 45. 22325' 640 - 300' 33.7 66.3 46. . 20335' 835 - 225' 35.2 64.8 47. In ravine. 22935' 945 - 231' 41.5 58.5 48. 22415' 1025 -221' 42.0 58.0 49. In ravine. 229 15' 1185 - 213' 46.0 54.0 50. 34300' 960 - 212' 37.0 63.0 51. 33100' 1180 - 159' 40.5 59.5 52. 32005' 1350 - 152' 44.0 56.0 53. 31530' 1115 - 201' 38.9 61.1 54. 32630' 780 - 222' 32.2 67.8 55. 34320' 440 - 316' 25.2 74.8 56. 30005' 795 - 220' 32.5 67.5 57. 28715' 930 - 158' 31.8 68.2 58. 230' 315 - 401' 22.0 78.0 59. 30030' 300 - 355' 20.4 79.6 60. 27445' 770 - 227' 33.0 67.0 61. 27655' 470 - 259' 24.5 75.5 62. 34045' 140 - 543' 13.8 86.2 63. 28530' 175 - 437' 14.0 86.0 64. 26255' 275 - 351' 18.5 81.5 65. Head of ravine 24020' 225 - 510' 20.0 80.0 66. 25345' 570 - 249' 28.0 72.0 67. 24300' 715 - 242' 33.8 66.2 68. 25555' 900 - 225' 38.0 62.0 CHAPTER X. 1. Assume in Example 1, Chapter IX., that the area shown is to be graded to a plane surface of uniform elevation of 160 feet. Determine the depth of cut or fill at each corner of the small squares and from these the volumes. 2. Determine the volumes by the approximate graphical method, using the planimeter for the areas. PROBLEMS AND EXAMPLES. 339 CHAPTER XI. The following observations 1 for rating a Price acoustic current meter were made by Mr. W. G. Price, June 21, 1895, by the method described by him in "Engineering News," Jan. 10, 1895. The student will use the first, fourth, fifth, and sixth columns. The second and third columns will be understood by a reference to the article in " Engineering News." Get the equation for rating the meter. RATING OF PRICE ACOUSTIC CURRENT METER No. 4, JUNE 21, 1895. No. IST WIRE. Feet. 2D WIRE. Feet. DISTANCE. Feet. REV. TIME. Second: 1 2 31.09 24.84 79.72 74.28 48.63 48.44 20 20 24.4 22.4 3 37.50 86.32 48.82 20 35.0 4 32.76 82.50 49.74 20 56.0 5 26.21 76.54 50.33 20 56.0 6 13.61 64.21 50.60 20 99.0 7 27.55 77.78 58.23 20 94.6 8 13.45 62.31 48.86 20 25.0 9 23.28 71.27 48.09 20 15.8 10 32.22 80.35 48.13 20 10.2 11 27.84 76.01 48.17 20 7.4 12 32.08 103.95 71.87 30 20.4 13 39.99 87.98 47.99 20 6.0 14 35.08 82.68 47.60 20 5.8 15 24.20 . 72.84 48.64? 20 6.0 16 38.86 86.39 47.53 20 5.8 17 24.67 72.47 47.80 20 5.2 18 39.13 86.85 47.72 20 5.2 19 35.74 83.51 47.77 + 20 11.2 20 20.15 67.97 47.82 + 20 11.2 21 30.92 79.17 48.25 20 13.4 22 21.86 70.30 48.44 20 11.6 23 23.67 72.09 48.42 + 20 19.4 24 14.47 62.71 48.24 + 20 20.0 25 39.27 88.65 49.38 20 32.4 26 46.18 85.82 49.64 20 31.4 27 40.82 90.57 49.75 20 37.4 28 26.14 101.15 75.01 30 64.4 The center of the meter wheel was 2.1 feet below the water surface. In Nos. 19, 20, 23, and 24 I felt sure the time interval used to start the watch and stop the first wire from passing out, after I had heard the click of the i Kindly furnished the author by Mr. Price. 340 APPENDIX. meter, was longer than the time consumed for the same purpose at the end of the run. I therefore put a + mark to indicate that the distance was too small. No. 15 was doubtful, as the skiff rocked during the trip. The six columns are an exact copy of the field notes. CHAPTER XII. 1. From the following data of the survey of a tunnel and shaft for a connection, determine the azimuth, length, and grade of the connecting drift. First draw a map of the survey and indicate the connecting drift. From a monument at the mouth of the tunnel run in the tunnel, azimuth 36 30', 436 feet, vert. Z + 1 00' ; thence azimuth 52 10', 200 feet, vert. Z + 1 10', to point near breast of tunnel. From the monument at the mouth of the tunnel run on the surface, azimuth 86 30', 232 feet, vert. Z - 3 30' ; thence azimuth 40 20', 636 feet, vert. + 13 50', to center of shaft 110 feet deep. 2. The strike of a vein of ore observed on the point of an outcrop is N. 36 W. or S. 36 E. The dip of the vein is found to be 13 from the vertical and to the northeast (right angle to the strike). From a point which bears from the before-mentioned point of outcrop N. 36 E. 300 feet, vert. Z 15 a cross-cut tunnel is to be driven level and in a direction S. 55 W. How long must it be to reach the vein ? What will be the difference if the tunnel is run on a one-per-cent grade ? 3. If, in the above example, the tunnel is to run S. 60 W., how long must it be ? 4. How long must it be, if the dip is to the southwest, the tunnel being run S. 54 W. ? 5. How long must it be, if the dip is to the southwest and the tunnel is run S. 50 W. ? 6. In Example 4 what would be the azimuth, grade, and length of the shortest tunnel that could be run to reach the vein ? THE JUDICIAL FUNCTIONS OF SURVEYORS. 341 II. THE JUDICIAL FUNCTIONS OF SURVEYORS. 1 BY JUSTICE COOLEY OF THE MICHIGAN SUPREME COURT. WHEN a man has had a training in one of the exact sciences, where every problem within its purview is supposed to be susceptible of accurate solution, he is likely to be not a little impatient when he is told that, under some circumstances, he must recognize inaccuracies, and govern his action by facts which lead him away from the results which theoretically he ought to reach. Observation warrants us in saying that this remark may frequently be made of surveyors. In the state of Michigan all our lands are supposed to have been sur- veyed once or more, and permanent monuments fixed to determine the boundaries of those who should become proprietors. The United States, as original owner, caused them all to be surveyed once by sworn officers, and as the plan of subdivision was simple, and was uniform over a large extent of territory, there should have been, with due care, few or no mistakes ; and long rows of monuments should have been perfect guides to the place of any one that chanced to be missing. The truth unfortunately is that the lines were very carelessly run, the monuments inaccurately placed ; and, as the recorded witnesses to these were many times wanting in permanency, it is often the case that when the monument was not correctly placed it is im- possible to determine by the record, with the aid of anything on the ground, where it was located. The incorrect record of course becomes worse than useless when the witnesses it refers to have disappeared. It is, perhaps, generally supposed that our town plots were more accu- rately surveyed, as indeed they should have been, for in general there can have been no difficulty in making them sufficiently perfect for all practical purposes. Many of them, however, were laid out in the woods ; some of them by proprietors themselves, without either chain or compass, and some by imperfectly trained surveyors, who, when land was cheap, did not appre- ciate the importance of having correct lines to determine boundaries when land should become dear. The fact probably is that town surveys are quite as inaccurate as those made under authority of the general government. It is now upwards of fifty years since a major part of the public surveys in what is now the state of Michigan were made under authority of the United States. Of the lands south of Lansing, it is now forty years since the major part were sold and the work of improvement begun. A genera- tion has passed away since they were converted into cultivated farms, and few, if any, of the original corner and quarter stakes now remain. The corner and quarter stakes were often nothing but green sticks driven into the ground. Stones might be put around or over these if they were handy, but often they were not, and the witness trees must be relied upon after the stake was gone. Too often the first settlers were careless in fixing 1 A paper prepared for the Michigan Society of Surveyors and Engineers. 342 APPENDIX. their lines with accuracy while monuments remained, and an irregular brush fence, or something equally untrustworthy, may have been relied upon to keep in mind where the blazed line once was. A fire running through this might sweep it away, and if nothing were substituted in its place, the adjoining proprietors might in a few years be found disputing over their lines, and perhaps rushing into litigation, as soon as they had occasion to cultivate the land along the boundary. If now the disputing parties call in a surveyor, it is not likely that any one summoned would doubt or question that his duty was to find, if pos- sible, the place of the original stakes which determined the boundary line between the proprietors. However erroneous may have been the original survey, the monuments that were set must nevertheless govern, even though the effect be to make one half-quarter section ninety acres and the one adjoining but seventy; for parties buy or are supposed to buy in reference to those monuments, and are entitled to what is within their lines, and no more, be it more or less. Mclver v. Walker, 4 Wheaton's Reports, 444 ; Land Co. v. Sounders, 103 U. S. Reports, 316; Cottingham v. Parr, 93 111. Reports, 233 ; Bunton v. Cardwell, 53 Texas Reports, 408 ; Watson v. Jones, 85 Penn. Reports, 117. While the witness trees remain there can generally be no difficulty in determining the locality of the stakes. When the witness trees are gone, so that there is no longer record evidence of the monuments, it is remark- able how many there are who mistake altogether the duty that now devolves upon the surveyor. It is by no means uncommon that we find men whose theoretical education is supposed to make them experts, who think that when the monuments are gone the only thing to be done is to place new monuments where the old ones should have been, and where they would have been if placed correctly. This is a serious mistake. The problem is now the same that it was before, to ascertain, by the best lights of which the case admits, where the original lines were. The mistake above alluded to is supposed to have found expression in our legislation ; though it is possible that the real intent of the act to which we shall refer is not what is com- monly supposed. An act passed in 1869, Compiled Laws, 593, amending the laws respect- ing the duties and powers of county surveyors, after providing for the case of corners which can be identified by the original field notes or other un- questionable testimony, directs as follows : " Second. Extinct interior section-corners must be reestablished at the intersection of two right lines joining the nearest known points on the original section lines east and west and north and south of it. " Third. Any extinct quarter-section corner, except on fractional lines, must be reestablished equidistant and in a right line between the section corners ; in all other cases at its proportionate distance between the nearest original corners on the same line." The corners thus determined, the surveyors are required to perpetuate by noting bearing trees when timber is near. THE JUDICIAL FUNCTIONS OF SURVEYORS. 343 To estimate properly this legislation, we must start with the admitted and unquestionable fact that each purchaser from government bought such land as was within the original boundaries, and unquestionably owned it up to the time when the monuments became extinct. If the monument was set for an interior-section corner, but did not happen to be " at the intersec- tion of two right lines joining the nearest known points on the original section lines east and west and north and south of it," it nevertheless deter- mined the extent of his possessions, and he gained or lost according as the mistake did or did not favor him. It will probably be admitted that no man loses title to his land or any part thereof merely because the evidences become lost or uncertain. It may become more difficult for him to establish it as against an adverse claimant, but theoretically the right remains ; and it remains as a potential fact so long as he can present better evidence than any other person. And it may often happen that, notwithstanding the loss of all trace of a section corner or quarter stake, there will still be evidence from which any surveyor will be able to determine with almost absolute certainty where the original'boundary was between the government subdivisions. There are two senses in which the word " extinct " may be used in this connection : one the sense of physical disappearance ; the other the sense of loss of all reliable evidence. If the statute speaks of extinct corners in the former sense, it is plain that a serious mistake was made in supposing that surveyors could be clothed with authority to establish new corners by an arbitrary rule in such cases. As well might the statute declare that if a man lose his deed he shall lose his land altogether. But if by extinct corner is meant one in respect to the actual location of which all reliable evidence is lost, then the following remarks are pertinent : (1) There would undoubtedly be a presumption in such a case that the corner was correctly fixed by the government surveyor where the field notes indicated it to be. (2) But this is only a presumption, and may be overcome by any satis- factory evidence showing that in fact it was placed elsewhere. (3) No statute can confer upon a county surveyor the power to "estab- lish " corners, and thereby bind the parties concerned. Nor is this a ques- tion merely of conflict between state and federal law ; it is a question of property right. The original surveys must govern, and the laws under which they were made must govern, because the land was bought in refer- ence to them ; and any legislation, whether state or federal, that should have the effect to change these, would be inoperative, because disturbing vested rights. (4) In any case of disputed lines, unless the parties concerned settle the controversy by agreement, the determination of it is necessarily a judicial act, and it must proceed upon evidence, and give full opportunity for a hear- ing. No arbitrary rules of survey or of evidence can be laid down whereby it can be adjudged, 344 APPENDIX. The general duty of a surveyor in such a case is plain enough. He is not to assume that a monument is lost until after he has thoroughly sifted the evidence and found himself unable to trace it. Even then he should hesitate long before doing anything to the disturbance of settled possessions. Occupation, especially if long continued, often affords very satisfactory evi- dence of the original boundary when no other is attainable ; and the sur- veyor should inquire when it originated, how, and why the lines were then located as they were, and whether a claim of title has always accompanied the possession, and give all the facts due force as evidence. Unfortunately, it is known that surveyors sometimes, in supposed obedience to the state statute, disregard all evidences of occupation and claim of title, and plunge whole neighborhoods into quarrels and litigation by assuming to " establish " corners at points with which the previous occupation can not harmonize. It is often the case that where one or more corners are found to be extinct, all parties concerned have acquiesced in lines which were traced by the guidance of some other corner or landmark, which may or may not have been trustworthy ; but to bring these lines into discredit when the people concerned do not question them, not only breeds trouble in the neighborhood, but it must often subject the surveyor himself to annoyance and perhaps discredit, since in a legal controversy the law as well as com- mon sense must declare that a supposed boundary line long acquiesced in is better evidence of where the real line should be than any survey made after the original monuments have disappeared. Stewart v. Carleton, 31 Mich. Reports, 270 ; Diehl v. Zanger, 39 Mich. Reports, 601 ; Dupont v. Starring, 42 Mich. Reports, 492. And county surveyors, no more than any others, can conclude parties by their surveys. The mischiefs of overlooking the facts of possession must often appear in cities and villages. In towns the block and lot stakes soon disappear ; there are no witness trees and no monuments to govern except such as have been put in their places, or where their places were supposed to be. The streets are likely to be soon marked off by fences, and the lots in a block will be measured off from these, without looking farther. Now it may per- haps be known in a particular case that a certain monument still remaining was the starting point in the original survey of the town plot ; or a surveyor settling in the town may take some central point as the point of departure in his surveys, and assuming the original plot to be accurate, he will then undertake to find all streets and all lots by course and distance according to the plot, measuring and estimating from his point of departure. This procedure might unsettle every line and every monument existing by acqui- escence in the town ; it would be very likely to change the lines of streets, and raise controversies everywhere. Yet this is what is sometimes done ; the surveyor himself being the first person to raise the disturbing questions. Suppose, for example, a particular village street has been located by acquiescence and use for many years, and the proprietors in a certain block have laid off their lots in reference to this practical location. Two lot owners quarrel, and one of them calls in a surveyor that he may be sure THE JUDICIAL FUNCTIONS OF SURVEYORS. 345 that his neighbor shall not get an inch of land from him. This surveyor undertakes to make his survey accurate, whether the original was, or not, and the first result is, he notifies the lot owners that there is error in the street line, and that all fences should be moved, say, one foot to the east. Perhaps he goes on to drive stakes through the block according to this con- clusion. Of course, if he is right in doing this, all lines in the village will be unsettled ; but we will limit our attention to the single block. It is not likely that the lot owners generally will allow the new survey to unsettle their possessions, but there is always a probability of finding some one dis- posed to do so. We shall then have a lawsuit; and with what result? It is a common error that lines do not become fixed by acquiescence in a less time than twenty years. In fact, by statute, road lines may be- come conclusively fixed in ten years ; and there is no particular time that shall be required to conclude private owners, where it appears that they have accepted a particular line as their boundary, and all concerned have cultivated and claimed up to it. McNamara v. Seaton, 82 111. Reports, 498; Bunce v. Bidwell, 43 Mich. Reports, 542. Public policy requires that such lines be not lightly disturbed, or disturbed at all after the lapse of any con- siderable time. The litigant, therefore, who in such a case pins his faith on the surveyor, is likely to suffer for his reliance, and the surveyor himself to be mortified by a result that seems to impeach his judgment. Of course nothing in what has been said can require a surveyor to conceal his own judgment, or to report the facts one way when he believes them to be another. He has no right to mislead, and he may rightfully express his opinion that an original monument was at one place, when at the same time he is satisfied that acquiescence has fixed the rights of parties as if it were at another. But he would do mischief if he were to attempt to "establish" monuments which he knew would tend to disturb settled rights ; the farthest he has a right to go, as an officer of the law, is to ex- press his opinion where the monument should be, at the same time that he imparts the information to those who employ him, and who might otherwise be misled, that the same authority that makes him an officer and entrusts him to make surveys, also allows parties to settle their own boundary lines, and considers acquiescence in a particular line or monument, for any con- siderable period, as strong, if not conclusive, evidence of such settlement. The peace of the community absolutely requires this rule. Joyce v. Williams, 26 Mich. Reports, 332. It is not long since that, in one of the leading cities of the state, an attempt was made to move houses two or three rods into a street, on the ground that a survey under which the street had been lo- cated for many years had been found on more recent survey to be erroneous. From the foregoing it will appear that the duty of the surveyor where boundaries are in dispute must be varied by the circumstances. (1) He is to search for original monuments, or for the places where they were originally located, and allow these to control if he finds them, unless he has reason to believe that agreements of the parties, express or implied, have rendered them unimportant. By monuments in the case of government sur- 346 APPENDIX. veys, we mean, of course, the corner and quarter stakes : blazed lines or marked trees on the lines are not monuments; they are merely guides or finger posts, if we may use the expression, to inform us with more or less accuracy where the monuments may be found. (2) If the original mon- uments are no longer discoverable, the question of location becomes one of evidence merely. It is merely idle for any state statute to direct a surveyor to locate or "establish" a corner, as the place of the original monument, according to some inflexible rule. The surveyor, on the other hand, must inquire into all the facts; giving due prominence to the acts of parties con- cerned, and always keeping in mind,jirst, that neither his opinion nor his survey can be conclusive upon parties concerned ; second, that courts and juries may be required to follow after the surveyor over the same ground, and that it is exceedingly desirable that he govern his action by the same lights and rules that will govern theirs. On town plots if a surplus or defi- ciency appears in a block, when the actual boundaries are compared with the original figures, and there is no evidence to fix the exact location of the stakes which marked the division into lots, the rule of common sense and of law is that the surplus or deficiency is to be apportioned between the lots, on an assumption that the error extended alike to all parts of the block. O'Brien v. McGrane, 29 Wis. Reports, 446 ; Quinnin v. Reixers, 46 Mich. Reports, 605. It is always possible when corners are extinct that the surveyor may usefully act as a mediator between parties, and assist in preventing legal controversies by settling doubtful lines. Unless he is made for this purpose an arbitrator by legal submission, the parties, of course, even if they consent to follow his judgment, can not, on the basis of mere consent, be compelled to do so; but if he brings about an agreement, and they carry it into effect by actually conforming their occupation to his lines, the action will conclude them. Of course it is desirable that all such agreements be reduced to writing; but this is not absolutely indispensable if they are carried into effect without. Meander Lines. The subject to which allusion will now be made is taken up with some reluctance, because it is believed the general rules are familiar. Nevertheless it is often found that surveyors misapprehend them, or err in their application; and as other interesting topics are somewhat connected with this, a little time devoted to it will probably not be altogether lost. The subject is that of meander lines. These are lines traced along the shores of lakes, ponds, and considerable rivers as the measures of quantity when sections are made fractional by such waters. These have determined the price to be paid when government lands were bought, and perhaps the impression still lingers in some minds that the meander lines are boundary lines, and all in front of them remains unsold. Of course this is erroneous. There was never any doubt that, except on the large navi- gable rivers, the boundary of the owners of the banks is the middle line of the river; and while some courts have held that this was the rule on all fresh-water streams, large and small, others have held to the doctrine that THE JUDICIAL FUNCTIONS OF SURVEYORS. 347 the title to the bed of the stream below low-water mark is in the state, while conceding to the owners of the banks all riparian rights. The prac- tical difference is not very important. In this state the rule that the center line is the boundary line is applied to all our great rivers, including the Detroit, varied somewhat by the circumstance of there being a distinct channel for navigation in some cases with the stream in the main shallow, and also sometimes by the existence of islands. The troublesome questions for surveyors present themselves when the boundary line between two contiguous estates is to be continued from the meander line to the center line of the river. Of course the original survey supposes that each purchaser of land on the stream has a water front of the length shown by the field notes ; and it is presumable that he bought this particular land because of that fact. In many cases it now happens that the meander line is left some distance from the shore by the gradual change of course of the stream or diminution of the flow of water. Now the divid- ing line between two government subdivisions might strike the meander line at right angles, or obliquely ; and in some cases, if it were continued in the same direction to the center line of the river, might cut off from the water one of the subdivisions entirely, or at least cut it off from any privi- lege of navigation, or other valuable use of the water, while the other might have a water front much greater than the length of a line crossing it at right angles to its side lines. The effect might be that, of two government subdivisions of equal size and cost, one would be of very great value as water front property, and the other comparatively valueless. A rule which would produce this result would not be just, and it has not been recognized in the law. Nevertheless it is not easy to determine what ought to be the correct rule for every case. If the river has a straight course, or one nearly so, every man's equities will be preserved by this rule : Extend the line of divi- sion between the two parcels from the meander line to the center line of the river, as nearly as possible at right angles to the general course of the river at that point. This will preserve to each man the water front which the field notes indicated, except as changes in the water may have affected it, and the only inconvenience will be that the division line between differ- ent subdivisions is likely to be more or less deflected where it strikes the meander line. This is the legal rule, and it is not limited to government surveys, but applies as well to water lots which appear as such on town plots. Bay City Gas Light Co. v. The Industrial Works, 28 Mich. Reports, 182. It often happens, therefore, that the lines of city lots bounded on navigable streams are deflected as they strike the bank, or the line where the bank was when the town was first laid out. When the stream is very crooked, and especially if there are short bends, so that the foregoing rule is incapable of strict application, it is sometimes very difficult to determine what shall be done; and in many cases the surveyor may be under the necessity of working out a rule for 348 APPENDIX. himself. Of course his action can not be conclusive; but if he adopts one that follows, as nearly as the circumstances will admit, the general rule above indicated, so as to divide, as near as may be, the bed of the stream among the adjoining owners in proportion to their lines upon the shore, his division, being that of an expert, made upon the ground and with all avail- able lights, is likely to be adopted as law for the case. Judicial decisions, into which the surveyor would find it prudent to look under such circum- stances, will throw light upon his duties and may constitute a sufficient guide when peculiar cases arise. Each riparian lot owner ought to have a line on the legal boundary, namely, the center line of the stream, proportioned to the length of his line on the shore ; and the problem in each case is, how this is to be given him. Alluvion, when a river imperceptibly changes its course, will be apportioned by the same rules. The existence of islands in a stream, when the middle line constitutes a boundary, will not affect the apportionment unless the islands were sur- veyed out as government subdivisions in the original admeasurement. Wherever that was the case, the purchaser of the island divides the bed of the stream on each side with the owner of the bank, and his rights also extend above and below the solid ground, and are limited by the peculiari- ties of the bed and the channel. If an island was not surveyed as a govern- ment subdivision previous to the sale of the bank, it is of course impossible to do this for the purposes of governmental sale afterwards, for the reason that the rights of the bank owners are fixed by their purchase : when mak- ing that, they have a right to understand that all land between the meander lines, not separately surveyed and sold, will pass with the shore in the gov- ernment sale ; and having this right, anything which their purchase would include under it can not afterward be taken from them. It is believed, how- ever, that the federal courts would not recognize the applicability of this rule to large navigable rivers, such as those uniting the Great Lakes. On all the little lakes of the state which are mere expansions near their mouths of the rivers passing through them such as the Muskegon, Pere Marquette, and Manistee the same rule of bed ownership has been judi- cially applied that is applied to the rivers themselves ; and the division lines are extended under the water in the same way. Rice v. Ruddiman, 10 Mich. 125. If such a lake were circular, the lines would converge to the center ; if oblong or irregular, there might be a line in the middle on which they would terminate, whose course would bear some relation to that of the shore. But it can seldom be important to follow the division line very far under the water, since all private rights are subject to the public rights of navigation and other use, and any private use of the lands inconsistent with these would be a nuisance, and punishable as such. It is sometimes im- portant, however, to run the lines out for some considerable distance, in order to determine where one may lawfully moor vessels or rafts, for the winter, or cut ice. The ice crop that forms over a man's land of course belongs to him. Lorman v. Benson, 8 Mich. 18 ; People's Ice Co. v. Steamer Excelsior, recently decided. THE JUDICIAL FUNCTIONS OF SURVEYORS. 349 What is said above will show how unfounded is the notion, which is sometimes advanced, that a riparian proprietor on a meandered river may lawfully raise the water in the stream without liability to the proprietors above, provided he does not raise it so that it overflows the meander line. The real fact is that the meander line has nothing to do with such a case, and an action will lie whenever he sets back the water upon the proprietor above, whether the overflow be below the meander lines or above them. As regards the lakes and ponds of the state, one may easily raise ques- tions that it would be impossible for him to settle. Let us suggest a few questions, some of which are easily answered, and some not: (1) To whom belongs the land under these bodies of water, where they are not mere expansions of a stream flowing through them ? (2) What public rights exist in them? (3) If there are islands in them which were not surveyed out and sold by the United States, can this be -done now ? Others will be suggested by the answers given to these. It seems obvious that the rules of private ownership which are applied to rivers can not be applied to the Great Lakes. Perhaps it should be held that the boundary is at low-water mark, but improvements beyond this would only become unlawful when they became nuisances. Islands in the Great Lakes would belong to the United States until sold, and might be surveyed and measured for sale at any time. The right to take fish in the lakes, or to cut ice, is public like the right of navigation, but is to be exercised in such manner as not to interfere with the rights of shore owners. But so far as these public rights can be the subject of ownership, they belong to the state, not to the United States ; and, so it is believed, does the bed of a lake also. Pollard v. Hagan, 3 Howard's U. S. Reports. But such rights are not gener- ally considered proper subjects of sale, but, like the right to make use of the public highways, they are held by the state in trust for all the people. What is said of the large lakes may perhaps be said also of many of the interior lakes of the state ; such, for example, as Houghton, Higgins, Che- boygan, Burt's, Mullet, Whitmore, and many others. But there are many little lakes or ponds which are gradually disappearing, and the shore pro- prietorship advances part passu as the waters recede. If these are of any considerable size say, even a mile across there may be questions of con- flicting rights which no adjudication hitherto made could settle. Let any surveyor, for example, take the case of a pond of irregular form, occupying a mile square or more of territory, and undertake to determine the rights of the shore proprietors to its bed when it shall totally disappear, and he will find he is in the midst of problems such as probably he has never grappled with, or reflected upon before. But the general rules for the exten- sion of shore lines, which have already been laid down, should govern such cases, or at least should serve as guides in their settlement. 1 1 Since this address was delivered, some of these questions have received the attention of the Supreme Court of Michigan in the cases of Richardson v. Prentiss, 48 Mich. Reports, 88, and Backus v. Detroit, Albany Law Journal, vol. 26, p. 428. 350 APPENDIX. Where a pond is so small as to be included within the lines of a private purchase from the government, it is not believed the public have any rights in it whatever. Where it is not so included, it is believed they have rights of fishery, rights to take ice and water, and rights of navigation for business or pleasure. This is the common belief, and probably the just one. Shore rights must not be so exercised as to disturb these, and the states may pass all proper laws for their protection . It would be easy with suitable legislation to preserve these little bodies of water as permanent places of resort for the pleasure and recreation of the people, and there ought to be such legislation. If the state should be recognized as owner of the beds of these small lakes and ponds, it would not be owner for the purpose of selling. It would be owner only as a trustee for the public use ; and a sale would be incon- sistent with the right of the bank owners to make use of the water in its natural condition in connection w r ith their estates. Some of them might be made salable lands by draining ; but the state could not drain, even for this purpose, against the will of the shore owners, unless their rights were appro- priated and paid for. Upon many questions that might arise between the state as owner of the bed of a little lake and the shore owners, it would be presumptuous to express an opinion now, and fortunately the occasion does not require it. I have thus indicated a few of the questions with which surveyors may now and then have occasion to deal, and to which they should bring good sense and sound judgment. Surveyors are not and can not be judicial offi- cers, but in a great many cases they act in a quasi judicial capacity with the acquiescence of parties concerned ; and it is important for them to know by what rules they are to be guided in the discharge of their judicial functions. What I have said can not contribute much to their enlightenment, but I trust will not be wholly without value. THE OWNERSHIP OF SURVEYS, ETC. 351 III. THE OWNERSHIP OF SURVEYS, AND WHAT CONSTITUTES A SURVEY AND MAP. 1 THERE seems to be a difference of opinion among surveyors as to how much of the information obtained, and how much of the work done in mak- ing a survey shall be furnished to the individual for whom the survey is made. Many surveyors keep what are called "private notes." All men doing business as surveyors keep notes of all surveys in a convenient form for ready reference. The extent to which these notes are private has not been rightly comprehended by all surveyors, and hence has resulted the difference of opinion mentioned. This article is an attempt to present a side of this question that has not heretofore been fully considered. An endeavor has also been made to point out to the young surveyor a line of action expedient for him to follow, which will at the same time be found advantageous to the community in which he works. In this discussion the question arises at once, "What constitutes a sur- vey?" and the answer obviously depends on the object of the survey. This discussion will be confined to land surveys ; that is, to surveys made for the purpose (1) of subdividing a large tract of land into smaller parcels to be sold ; (2) of determining the boundary of a tract the description of which is known ; (3) of determining the description when the boundaries are known. The principle to be enunciated applies to any other survey as well, be it railroad, canal, bridge, or topographical survey. Indeed, it is well under- stood in all such surveys, but seems to be ignored by many engineers having to do with land surveys. A survey is the operation of finding the contour, dimensions, position, or other particulars of any part of the earth's surface, . . . tract of land, etc., and representing the same on paper. In making a survey it is necessary to set certain points, called monu- ments or corners, and to determine a description of these points. These items therefore become a part of the survey. Then a map must be drawn. This map, to be a faithful representation of the ground and the work done, should, together with the notes, show all of the items mentioned. The object of establishing monuments or corners and describing them is twofold : (1) to mark on the ground the boundaries of the tract, and (2) to secure definite information as to its location with reference to other points or tracts, so that from this information the land may at a future time be found. For a complete survey, therefore, the corners must be fixed, information that will preserve their location must be obtained, and the facts must be delineated on a map with accompanying notes. To whom belongs this survey? It would appear to be evident that it belongs to the individual who pays to have it made. It is not readily seen in 1 A paper prepared by the author for "The Polytechnic," the student journal of the Rensselaer Polytechnic Institute, from which it is taken. 352 APPENDIX. what way the survey, or any part of it, becomes the sole property of the sur- veyor. He may keep a copy of his notes to facilitate his future work ; but he has not the shadow of a claim to a single note, the time for taking which has been paid for by his employer. If his charge for his work is on a time basis, there can be no question as to the correctness of the above statements. If he contracts to do the work for a definite sum for the entire job, he may take as much time as he likes, and may keep as many private notes as he desires ; but he is bound in honor to return to his employer the complete survey; and, if he does so, the private notes would thereafter be of no great assistance to him in securing further employment, particularly when it is remembered that men of repute do not bid against each other for professional work. His reputation for accuracy and honesty will be a far more potent factor in securing employment than any set of private notes fairly obtained. It is true that a great many surveyors hold a different opinion, and purposely return their maps and notes in such condition, that, while they may serve the purpose for which they are primarily made, they do not tell the whole story, nor enough to make it easy for another surveyor to relocate the tract surveyed. When this is done, the person ordering the survey does not receive what he pays for. Something is withheld. It seems to need no argument to show that this is radically wrong. But there is another reason for condemning this practice. The correct and permanent location of all public land lines, as streets, alleys, etc., as well as the permanent location of party lines between private owners, is a matter of the gravest importance, and no information that will at all serve to fix such lines in their correct positions for all time, should be withheld from the owner who pays for the survey, be it private citizen, municipality, county, or state. The records of monuments and street lines made by a city engineer are no more his private property than are the records in the offices of the clerk, auditor, or treasurer, the property of the individuals who held office at the time the records were made. The correctness of the position assumed has been indicated by court decisions. A great deal of laxity is shown in the conduct of offices of city engineers and county surveyors. The methods of regulating the pay of these officers has doubtless had much to do with this. It is frequently the case that the surveyor receives no salary, but is allowed to collect certain specified fees for work performed, and this gives color to his claim that his work is private and belongs to him. That this is not true concerning the public work he does is evident from what has preceded. That the records of work done for private citizens are not the property of the public needs no demon- stration ; but such work belongs to those citizens for whom it was done. A different policy should be pursued with regard to these offices. In every case such an office should be a salaried one, with such salaried assistants as may be necessary. Certain fees should be prescribed for performing the various kinds of work that the surveyor may be called upon to do within the THE OWNERSHIP OF SURVEYS, ETC. 353 limits of the territory of the political division whose servant he is. These fees should cover all work connected with public construction and public or private land lines, and should be returned to the public treasury. Their amount may be regulated, from time to time, so that they shall aggregate a sum sufficient to pay the expenses of the office. They should, of course, not cover work of a private character not having to do with land lines. But the entire public is interested in the permanency of land lines, and all records concerning them made by a public official should become public property. The permanency of land lines is too important a matter to be subject to avaricious and jealous rivalry ; and all the surveyors in a given district should cooperate to preserve, in their correct places, all lines within the district. To this end, the returns of every surveyor made to the owner should be thoroughly complete. Maps made for filing as public records should be so finished as to enable any surveyor to relocate the land without the least uncertainty as to the correctness of his work. That this is done in very few instances is well known to every surveyor who has had occasion to examine public records for data for surveys which he has been called upon to make. Because of the fact that in most cases neither owners nor attorneys have been fully posted as to what constitutes a complete description, suffi- cient for relocation, and because surveyors have been willing to let matters stand as they were, great carelessness has arisen in the practice of making and filing maps for record. While in some states good laws exist prescribing what shall appear on a map before it will be received as a public record, in more states there is nothing whatever to guide either owner, surveyor, attorney, or recorder in the matter. In the county records in such states, anything that is made up of lines and figures and labeled " this is a map," is considered a sufficient basis for a correct description and location of the property it purports to represent no matter whether it is drawn by hand, photo-lithographed, or simply printed with " rule " and type. The records are full of auctioneers' circulars, manufactured in a printing office from information coming from nobody knows where, filed at the request of the auctioneer's clerk, with no name of owner or other interested party attached, except as the name of the auctioneer appears in the accompanying advertisement. Further than this, these maps are frequently purposely distorted to create a favorable impres- sion of the property to be sold. Wide streets are shown where only narrow ones exist, streets appear opened for the full width where they have been opened for but half their width, subdivisions are indicated as rectangles that really may not be even parallelograms, etc. Such maps as these frequently form the only basis for the description and location of the prop- erty they are supposed to represent. Such misrepresentations are bad, very bad for those who buy; but is the information given by these circulars much worse than that furnished by many of the maps made by surveyoi'S and filed at the request of the owners? K'M'D SURV. 23 354 APPENDIX. On these plots, if of " additions," we find lines indicating the boundaries of blocks and lots, all of which blocks and lots are numbered; the names of streets appear in neat letters; a few dimensions, possibly all linear dimen- sions, will be given; the streets or blocks may be delicately tinted, and the whole set off with a fine border and title. As an exhibition of the draughtsman's skill, these maps are perhaps valuable. As a source of in- formation as to the location of the lines they purport to show, they are worth little more than the auctioneer's circular. Perhaps they have a few more figures, and the presumption may be a little stronger that the figures are correct. Examine one of these maps closely. There will be found no evidence that a monument has been set in the field; not an angle is recorded, though the lines may cross at all sorts of angles ; and dimensions are given that do not agree among themselves, so that the angles can not be cal- culated. There will be found no name signed except, possibly, that of the surveyor, who thus advertises what we shall charitably call his stupidity. Frequently no monuments are set except small stakes at the corners of the blocks; but even the fact that such stakes have been set is not recorded on the plot. One who is acquainted with the practice of surveyors in a given district knows at what points to look for such stakes, and if they have been set and not pulled out to make room for a fence post or building, he may succeed in finding them. Some surveyors are accustomed to set stakes a certain distance away from the point the stake is supposed to mark, but no mention of this fact appears on the map. In fact, the map is so drawn that no one but the surveyor who made it can write a description of any one of the parcels of land shown, or correctly locate it on the ground. Furthermore, the surveyor himself finds it impossible, after the lapse of a few years and the destruction of his "private marks," to rerun any one of the lines exactly as originally laid out. It is easy to see to what this leads impossible descriptions of property, giving opportunity for differences in judgment as to interpretation of what was intended; disputes as to position of party lines; costly litigation and expensive movement of structures begun or completed ; and the actual shifting of lines back and forth by different surveyors, or even by the same surveyor, honestly trying to locate the lines properly. The writer has seen enough trouble of this sort to indicate to him that a radical change is needed in the field work and mapping of cities, towns, and additions, not to mention farms and other tracts of land that it may be necessary to lay out and describe. So long as fallible man is responsible for the accuracy of surveys, maps, and descriptions of properties, so long will there be errors ; but that it is possible greatly to reduce their number by proper regulation the writer is fully persuaded. What we have been de- scribing are not maps at all, or at most they are very imperfect maps, and "what constitutes a map?" thus seems to be a very pertinent question. A map of a city, town, or addition, or other tract of land, serving as a basis for the description of property, should furnish all the information THE OWNERSHIP OF SURVEYS, ETC. 355 necessary for the proper description and location of the various parcels shown, and also of the whole piece. It should further show the exact loca- tion of the whole tract relatively to the lands immediately adjoining ; particularly should this be done when an offset or angle in a street line occurs. To accomplish these things, there should appear on the map the following items : (1) The lengths of all lines shown. (2) The exact angle made by all intersecting lines. (3) The exact position and character of all monuments set, with notes of reference points. (4) The number of each block and lot. (5) The names of all streets, streams or bodies of water, and recognized land marks. (6) The scale. (7) The direction of the meridian and a note as to whether the true or magnetic meridian is shown. (It should be the true meridian.) (8) The angles of intersection made by the lines of adjoining property with the boundaries of the tract mapped. (9) The exact amount of offset in lines that may extend from the out- side through the tract mapped. (10) A simple, complete, and explicit title, including the date and the name of the surveyor. All this is necessary to make the map valuable for description and loca- tion of the property it represents. Of course monuments will not be shown if none have been set, and very frequently none are set, either from carelessness on the part of the surveyor, or an unwillingness on the part of the owner to pay their cost. Monuments of a permanent character should be set at each corner of a tract surveyed, and at least two, visible the one from the other, should be on the line of each street. If these monuments are not placed on the center lines of the streets, they should be at uniform distances from the center or property lines. If placed with reference to the center line they should all be on the same side of the center. In streets extending east and west the monuments should all be on the north of the center, or they should all be on the south, and at uniform distances. In streets extending north and south the monuments should all be on the east of the center or all on the west. Uniformity in such practice saves a vast amount of time. Monuments may be set at uniform distances from the block lines, in the sidewalk area, and this is an excellent practice. The stakes or monuments set at the corners of the blocks in additions, or town sites, should never be the only stakes or monuments set in the tract. That the map may be reliable there should appear on it the following : (1) The certificate of the surveyor that he has carefully surveyed the 356 APPENDIX. land, that the map is a correct representation of the tract, and that he has set monuments (to be described) at the points indicated on the map. (2) The acknowledged signature of all persons possessing title to any of the land shown in the tract, and, if possible, signatures of adjoining owners. (3) If of an addition, the acknowledged dedication to public use forever of all areas shown as streets or roads. (4) If a street of full width, whose center line is a boundary of the tract, is shown, the acknowledged signature of the owner of the adjoining property, unless his half of the street has been previously dedicated. It has been already stated that, in some states, a map may be filed at the request of any person, and without signature. This practice frequently leads to trouble. The writer knows of cases in which owners of large tracts of land have had those tracts subdivided and have taken land of adjoining non-resident owners for street purposes without the consent or knowledge of those owners. When, at a later day, the owners of the land so taken have objected and attempted to close half of the street, trouble of a serious character has arisen. The same trouble has occurred where streets have been run through narrow gores of land and have subsequently been completely closed, leaving houses built on the mapped property with- out outlet. Time and again have cases of this sort come to the knowledge of the writer. Having pointed out certain evils, it remains to suggest a remedy. It lies in the enactment of laws governing these matters. There should be in- cluded in the statutes of every state a law explicitly defining what shall appear on every map filed for reference, and making it a misdemeanor to file a map that does not strictly conform to the requirements. In the absence of such laws it is believed that the young surveyor can assist greatly in a much-needed reform, by following the principles suggested in this paper as the correct ones, and avoiding the errors here indicated. It is hoped that those graduates of our engineering schools who drift into this class of work will be guided by a higher principle than that which actuates the surveyor who covers up his tracks, at the expense of his em- ployer, in order to secure a monopoly of the business of the locality. The young surveyor can spend his energies to greater advantage in devising new and better methods of work, than in inventing ways for hiding information that belongs to his employer. Certainly a thorough education should so broaden the young man's views as to make it impossible for him to be controlled by those meaner instincts which, if indulged, lead ever to narrow the vision and prevent one from perceiving the greater problems that con- tinually present themselves for solution. GEOGRAPHICAL POSITIONS OF BASE LINES, ETC. 357 IV. GEOGRAPHICAL POSITIONS OF BASE LINES AND PRINCI- PAL MERIDIANS GOVERNING THE PUBLIC SURVEYS. THE system of rectangular surveying, authorized by law May 20, 1785, was first employed in the survey of United States public lands in the state of Ohio. The boundary line between the states of Pennsylvania and Ohio, known as " Ellicott's line," in longitude 80 32' 20" west from Greenwich, is the meridian to which the first surveys are referred. The townships east of the Scioto River, in the state of Ohio, are numbered from south to north, com- mencing with No. 1 on the Ohio River, while the ranges are numbered from east to west, beginning with No. 1 on the east boundary of the state, except in the tract designated " U. S. military land," in which the townships and ranges are numbered, respectively, from the south and east boundaries of said tract. During the period of one hundred and nine years since the organization of the system of rectangular surveying, numbered and locally named prin- cipal meridians and base lines have been established, as follows: The first principal meridian begins at the junction of the Ohio and Big Miami rivers, extends north on the boundary line between the states of Ohio and Indiana, and roughly approximates to the meridian of longitude 84 48' 50" west from Greenwich. The ranges of the public surveys in the state of Ohio, west of the Scioto River, are, in part, numbered from this meridian. For further information in regard to numbering of townships and ranges of the early surveys in Ohio, the reader is referred to the state map prepared in the General Land Office. The second principal meridian coincides with 86 28' of longitude west from Greenwich, starts from a point two and one half miles west of the confluence of the Little Blue and Ohio rivers, runs north to the northern boundary of Indiana, and, with the base line in latitude 38 28' 20", governs the surveys in Indiana and part of those in Illinois. The third principal meridian begins at the mouth of the Ohio River and extends north to the northern boundary of the state of Illinois, and with the base line in latitude 38 28' 20", governs the surveys in the state east of the third principal meridian, with the exception of those projected from the second principal meridian, and the surveys on the west, to the Illinois River. This meridian is nearly coincident with 89 10' 15" of west longi- tude from Greenwich. The fourth principal meridian begins at a point on the right bank of the Illinois River, in latitude 40 00' 30" north, and longitude 90 28' 45" west from Greenwich, and with the base line running west from the initial point, governs the surveys in Illinois west of the Illinois River and west of that part of the third principal meridian which lies north of the river. The fourth principal meridian also extends north through Wisconsin and northeastern Minnesota, and, with the south boundary of Wisconsin as 358 APPENDIX. its base line, governs all the surveys in the former and those in the latter state lying east of the Mississippi River, and the third guide meridian west (of the fifth principal meridian system), north of the river. The fifth principal meridian starts from the old mouth of the Arkansas River, and with the base line running west from the old mouth of the St. Francis River, governs the surveys in Arkansas, Missouri, Iowa, North Dakota ; those in Minnesota, west of the Mississippi River and west of the third guide meridian north of the river ; and in South Dakota all east of the Missouri River, and the surveys on the west side of the river to a limit- ing line following the third guide meridian (of the sixth principal meridian system), White River, and the west and north boundaries of the Lower Brule Indian Reservation. This meridian is nearly coincident with 91 03' 42" longitude west from Greenwich. The sixth principal meridian, which is approximately the meridian of 97 23' west longitude from Greenwich, extends from the base line coinci- dent with the north boundary of Kansas in latitude 40 north, south through the state to its south boundary, in latitude 37 north, and north through Nebraska to the Missouri River ; and governs the surveys in Kansas and Nebraska; the surveys in Wyoming, except those referred to the Wind River meridian and base line, which intersect in latitude 43 01' 20" north, and longitude 108 48' 40" west from Greenwich ; the surveys in Colorado, except those projected from the New Mexico and Ute meridians, the latter intersecting its base line in latitude 39 06' 40" north and longitude 108 33' 20" west from Greenwich ; and the surveys in South Dakota extended, or to be extended, over the tract embracing the Pine Ridge and Rosebud Indian reservations. In addition to the above-mentioned numbered principal meridians, other principal meridians with local names have been established as follows : The Michigan meridian, in longitude 84 22' 24" west from Greenwich, with a base line in latitude 42 26' 30" north (eight miles north of Detroit), governs the surveys in Michigan. The Tallahassee meridian, in longitude 84 16' 42" west from Greenwich, runs north and south from the initial point on the base line at Tallahassee, in latitude 30 28' north, and governs the surveys in Florida. The Saint Stephen's meridian, in longitude 88 02' west from Greenwich, begins at the initial point (Ellicott's corner), on the base line, in latitude 31 north, extends south to Mobile Bay and north to latitude 33 06' 20", and governs the surveys in the southern district of Alabama, and in Pearl River district lying east of the river and south of the Choctaw base line, in latitude 31 52' 40" north, in the state of Mississippi. The Huntsville meridian begins on the northern boundary of Alabama, in latitude 34 59' north, longitude 86 34' 45" west from Greenwich, extends south to latitude 33 06' 20" north, and governs the surveys in the northern district of Alabama. The Choctaw meridian begins on the Choctaw base line, latitude 31 GEOGRAPHICAL POSITIONS OF BASE LINES, ETC. 359 54' 40" north, longitude 90 14' 45" west from Greenwich, runs north to the south boundary of the Chickasaw cession, in latitude 34 19' 40" north, and governs the surveys east and west of the meridian, and north of the base line. The Chickasaw meridian begins on the north boundary of Mississippi in latitude 34 59' north, longitude 89 15' west from Greenwich, extends south to latitude 33 48' 45" north, and governs the surveys in north Mississippi. The Washington meridian begins on the base line in latitude 31 north, longitude 91 9' 15" west from Greenwich, extends north to the Mississippi River, and governs the surveys in the southwestern angle of the state of Mississippi. The Saint Helena meridian begins at the initial point of the Washington meridian, in latitude 31 north, and longitude 91 09' 15" west of Greenwich, extends south to the Mississippi River, and governs the surveys in the Greensburg and southeastern districts of Louisiana, east of the Mississippi River. The Louisiana meridian, in longitude 92 24' 15" west of Greenwich, extends from the Gulf of Mexico to the north boundary of Louisiana, and with the base line through the initial point, conforming to the parallel of 31 north latitude, governs all the surveys in the state west of the Missis- sippi River. The New Mexico meridian, in longitude 106 53' 40" west from Green- wich, extends through the territory, and with the base line, in latitude 34 15' 25" north, governs the surveys in New Mexico, except those in the north- west corner of the territory, referred to Navajo meridian and base line, which have their initial point in latitude 35 45' north, longitude 108 32' 45" west from Greenwich. The Salt Lake meridian, in longitude 111 54' 00" west from Greenwich, has its initial point at the corner of Temple Block, in Salt Lake City, Utah, extends north and south through the territory, and, with the base line, through the initial, and coincident with the parallel of 40 46' 04" north latitude, governs the surveys in the territory, except those referred to the Uintah meridian and base line projected from an initial point in latitude 40 26' 20" north, longitude 109 57' 30" west from Greenwich. The Boise meridian, longitude 116 24' 15" west from Greenwich, passes through the initial point established south 29 30' west, nineteen miles distant from Boise City, extends north and south through the state, and, with the base line in latitude 43 46' north, governs the surveys in the state of Idaho. The Mount Diablo meridian, California, coincides with the meridian of 121 54' 48" west from Greenwich, intersects the base line on the summit of the mountain from which it takes its name, in latitude 37 51' 30" north, and governs the surveys in the state of Nevada, and the surveys of all central and northern California, except those belonging to the Humboldt meridian system. 360 APPENDIX. The Humboldt meridian, longitude 124 08' west from Greenwich, inter- sects the base line on the summit of Mount Pierce, in latitude 40 25' 12" north, and governs the surveys in the northwestern corner of California, lying west of the Coast range of mountains, and north of township 5 south, of the Humboldt meridian system. The San Bernardino meridian, California, longitude 116 56' 15" west from Greenwich, intersects the base line on Mount San Bernardino, latitude 34 07' 10" north, and governs the surveys in southern California, lying east of the meridian, and that part of the surveys situated west of it which is south of the eighth standard parallel south, of the Mount Diablo meridian system. The Willamette meridian, which is coincident with the meridian of 122 44' 20" west from Greenwich, extends south from the base line, in latitude 45 31' north, to the north boundary of California, and north to the inter- national boundary, and governs all the public surveys in the states of Oregon and Washington. The Black Hills meridian, longitude 104 03' west from Greenwich, with the base line in latitude 44 north, governs the surveys in the state of South Dakota, north and west of White River, and west of the Missouri River (between latitudes 45 55' 20" and 44 17' 30"), the north and west bounda- ries of the Lower Bruld Indian Reservation, and the west boundary of range 79 west, of the fifth principal meridian system. The Montana meridian extends north and south from the initial monu- ment on the summit of a limestone hill, eight hundred feet high, longitude 111 38' 50" west from Greenwich, and with the base line on the parallel of 45 46' 48" north latitude, governs the surveys in the state of Montana. The Gila and Salt River meridian intersects the base line on the south side of Gila River, opposite the mouth of Salt River, in latitude 33 22' 40" north, longitude 112 17' 25" west from Greenwich, and governs the surveys in the territory of Arizona. The Indian meridian, in longitude 97 14' 30" west from Greenwich, extends from Red River to the south boundary of Kansas, and with the base line in latitude 34 30' north, governs the surveys in the Indian Terri- tory, and in Oklahoma Territory all surveys east of 100 west longitude from Greenwich. The Cimarron meridian, in longitude 103 west from Greenwich, extends from latitude 36 30' to 37 north, and with the base line in latitude 36 30' north, governs the surveys in Oklahoma Territory west of 100 west longitude from Greenwich. TABLES. 361 V. TABLES. TABLE I. CORRECTION TO ONE HUNDRED UNITS MEASURED ALONG THE SLOPES GIVEN. UNITS RISE IN 100. CORRESPONDING VERTICAL ANGLE. CORRECTION. 1.02 o 35' 0.005 2.01 i 09' O.O2O 3-3 i 44' 0.046 4-02 2 1 8' 0.081 5-01 2 5 2' 0.125 6.00 3 26' 0.179 7.00 4 oo' 0.244 8.02 . 4 35' 0.320 9.01 5 9' 0.404 10.01 5 43' 0.497 2O.OI n 19' 1.617 30.00 1 6 42' . 4.218 4O.OO 21 48 7.151 50.00 26 34' 10.559 TABLE II. 1 CORRECTION COEFFICIENT FOR TEMPERATURE AND HYGROMETRIC CONDITIONS. This correction is used when no hygrometric observations have been made. To the difference in altitude found in Table III. for the given barometer readings is added algebraically the product of that difference and the correction below given, according to the formula, Diff. Alt. = (Diff. by Table III.) (1 + c). SUM 0. T.2 CORR. COEFF.3 SUM O. T. CORR. COEFF. SUM O. T. CORR. COEFF. O.IO24 70 0.0273 I 4 0.0471 10 0.0915 80 0.0166 150 0.0575 20 0.0806 90 0.0058 160 0.0677 30 0.0698 100 0.0049 170 0.0779 40 O.O592 110 0.0156 1 80 0.0879 5 0.0486 1 20 0.0262 60 O.OjSo 130 0.0368 1 Computed from Tables I. and IV., Appendix 10, " U. 8. Coast Survey Report " for 1881. * Sum of Observed Temperatures. Correction Coefficient. 362 APPENDIX. TABLE III.' BAROMETRIC ELEVATIONS. Giving altitudes above arbitrary sea level (barometer reading 30 inches) for various barometer readings B. To determine difference of elevation of two points having barometer readings B and B v take from the table the altitudes corresponding to B and .Bj, and correct their difference by Table II. The corrected difference is the quantity required. B. A. DlFF. FOR .01. B. A. DJFF. FOR .01. B. A. DlFF. FOR .01. Inches. Feet. Feet. Inches. Feet. Feet. Inches. Feet. Feet. II. 27,336 24.6 14.0 20,765 -19-5 17.0 15,476 16.0 II. I 27,090 24.4 14.1 20,570 "9-3 17.1 5>3 l6 '5-9 II. 2 "3 ii. 4 26,846 26,604 26,364 24.2 24.0 21.8 14.2 14-3 14.4 20,377 20,186 19,997 19.1 18.9 18.8 17.2 J 7-3 17.4 15,157 14,999 14,842 15.8 '5-7 15.6 11.5 26,126 ^ J " 23.6 14-5 19,809 18.6 17-5 14,686 I c. c ii. 6 25,890 23-4 I 4 .6 19,623 18.6 17.6 H,53i J J 11.7 ii. 8 25,656 23.2 23.0 14.7 I 4 .8 19,437 19,252 18.5 18.4 17.7 17.8 "4,377 14,223 15.4 11.9 25,194 22.8 14.9 19,068 1 8. 2 17.9 14,070 I c 2 I2.O 24,966 22.6 15.0 18,886 18.1 18.0 13,918 1 j i * 12. 1 12.2 12.3 24,740 24, 5 l6 24,294 22.4 22.2 22. 1 I5-I 15.2 '5-3 18,705 18,525 18,346 18.0 17.9 17.8 18.1 18.2 18.3 13,767 13,617 13,468 15.0 14.9 14.9 12.4 24,073 21.9 15-4 18,168 17.6 18.4 13,319 14.7 12-5 12.6 23,854 23,637 21.7 21.6 15-5 15.6 17,992 17,817 17-5 17.4 18.5 18.6 13,172 13,025 14.7 14.6 I2. 7 12.8 23,421 23,207 21.4 21.2 15-7 15-8 17,643 17,470 17-3 17.2 18.7 18.8 12,879 12,733 14.6 14.4 12-9 22,995 2I.O 15-9 17,298 17. i 18.9 12,589 14.4 13.0 I3-I I 3 .2 22,785 22,576 22,368 2O-9 20.8 20. 6 16.0 16.1 16.2 17,127 16,958 16,789 16.9 16.9 16.8 19.0 19.1 19.2 12,445 12,302 12,160 14-3 14.2 14.2 13-3 13-4 22,162 21,958 20.4 20. i 16-3 16.4 16,621 i6,454 16.7 16.6 19-3 16.4 12,018 11,877 14.1 14.0 13-5 I 3 .6 13-7 13-8 2I ,757 2I ,557 21,358 2 1 , 1 6O 20. o 19.9 19.8 19.8 16.5 16.6 16.7 16.8 16,288 16,124 15,961 15,798 16.4 16.3 16.3 16.2 19-5 19.6 19.7 19.8 ",737 ",598 ",459 11,321 13-9 13-9 ,3.8 13.7 13-9 14.0 20,962 20,765 -19.7 16.9 17.0 15,636 15,476 16.0 19.9 20. 11,184 11,047 -'3-7 i Taken from Appendix 10, " U. S. Coast and Geodetic Survey Report " for 1881 . TABLES. 363 TABLE III. (continued). B. A. DlFF. FOR .01. B. A. DlFF. FOR .01. B. A. DlFF. FOR .01. Incites. Feet. Feet. Inches. Feet. Feet. Inches. Feet. feet. 20.0 11,047 -13.6 23-7 6,423 -II-5 27.4 2,470 9-9 20.1 10,911 13.5 23.8 6,308 11.4 27-5 2,371 9-9 20.2 20.3 10,776 10,642 13-4 13.4 23-9 24.0 6,194 6,080 11.4 1 1-3 27.6 27.7 2,272 2,173 9-9 9.8 20.4 10,508 13-3 24.1 5,967 "3 2 7 .8 2,075 9.8 20-5 20.6 20.7 I0 >375 10,242 10,110 13-3 13.2 24.2 24-3 24.4 5,854 5,741 5,629 "3 II. 2 I I.I 27.9 28.0 28.1 1,977 i, 880 1,783 9-7 9-7 9 7 20.8 9,979 * 24-5 5,518 1 1. 1 28.2 1,686 y / 9.7 20.9 21.0 9,848 9,718 13.0 12.9 24.6 24.7 5,296 II. I I I.O 28.3 28.4 1,589 i,493 y* / 9.6 9.6 21. I 21.2 9,589 9,460 12.9 1.2.8 24.8 24.9 5,186 5>77 10-9 28.5 28.6 ',397 1,302 9-5 Q.C 21.3 9,332 12.8 25.0 4,968 IO-9 28. 7 1,207 s J 9-5 21-4 9,204 12.7 25.1 4,859 10.8 28.8 1,112 9-4 2I -5 9,077 12.6 25.2 4,75' 10.8 28.9 1,018 9-4 21.6 8,951 12.6 25-3 4,643 10.8 29.0 924 9-4 21.7 21.8 8,825 8,700 12.5 12.5 25-4 25-5 4,535 4,428 10.7 10.7 29.1 29.2 830 736 9-4 9-3 21.9 8,575 12.4 2 5 .6 4,32i 10.6 29-3 643 9-3 22.O 8,45 * 12.4 25-7 4,215 10.6 294 55 9.2 22.1 22.2 8,327 8,204 12.3 12.2 25.8 25-9 4,109 4,004 10.5 10.5 29-5 29.6 458 366 9-2 9.2 22.3 22.4 8,082 7,960 12.2 12.2 26.O 26.1 3,899 3,794 10.5 10.4 29.7 29.8 274 182 9.2 9.1 22.5 7,838 12. 1 26.2 3,690 10.4 29.9 9 1 9.1 22.6 7,717 12.0 26. 3 3,586 10.3 30.0 oo 9.1 22.7 22.8 7,597 7,477 I2.O 1 1.9 26.4 26.5 3,483 3,38o 10.3 10.3 3 O.I 30.2 -91 181 9.0 9.0 22. 9 7,358 II.9 26.6 3,277 IO.2 30.3 271 9.0 23.0 7,239 ii. 8 26.7 3,175 IO.2 30.4 361 9.0 23.1 7,121 11.7 26.8 3,073 10. 1 30-5 45 * 8.9 23.2 7,004 1 1.7 26.9 2,972 IO. I 30.6 540 8.9 23-3 6,887 1 / II-7 27.0 2,871 IO. I 30-7 629 8.8 23-4 6,770 ii. 6 27.1 2,770 IO.O 30.8 717 8.8 23-5 6,654 n.6 27.2 2,670 IO.O 30.9 805 -8.8 23-6 6,538 11.5 27-3 2,570 IO.O 3 I.O -893 23-7 6,423 27.4 2,470 364 APPENDIX. TABLE IV. POLAR DISTANCE OF POLARIS. For January 1 of years named. 1894 1897 1900 1903 1906 1909 1912 1915 1918 1921 10 I S43' i 14-5' i 13-55' i 12.62' i 11.68' i 10.75' i 09.82' i 08.88' i 07.97' i 07.03' sin polar distance Sm of azimuth at elongation = cosine latitude Latitude = altitude of Polaris at culmination polar distance refraction correc- tion given below. LATITUDE. CORRECTION, MINUTES. LATITUDE. CORRECTION, MINUTES. 20 3 40 2.60 I.6 S I-I3 50 60 0.80 o-SS TABLE V. 1 AMOUNT AND VARIATION OF THE MAGNETIC NEEDLE FROM ITS MEAN DAILY POSITION. The letters E and W indicate which side of the mean position the needle points. LOCAL MEAN TIME; MORNING HOURS. 6* 7 h 8 9 h ioh It h I2 h December, January, February : Latitude 37 to 49 Latitude 25 to 37 March, April, May : Latitude 37 to 49 . . i 0.7 E o.iW 2 6E 1 i.iE o.iE 3 8 E I 1.9 E i.oE 4dE I 2.2 E 2.0 E , e E 1 i-5 E 2.2 E I 2 E o.i W i.i E i 6 E 1.8 W 0.5 W 3 8 W Latitude 25 to 37 June, July, August : Latitude 37 to 49 Latitude 25 to 37 September, October, November: Latitude 37 to 49 Latitude 25 to 37 1.6 E 4.0 E 24 E 1.8 E 0.9 E 2.8 E 5-6 E 4.0 E 2.6 E 21 E 3-3 E 5-7 E 4.2 E 3-i E 26E 2.6 E 4-5 E 2.9 E 2.5 E 21 E i.iE 1.7 E 0.5 E i.oE 06 E 0.6 W 1.6 E 1.6 W i-S E OQ \V 1.9 W 4.1 W 2.8 W 3-3 W 21 W SEASON AND POSITION IN LATITUDE. LOCAL MEAN TIME; AFTERNOOM HOURS. oh I* 2 h 3 h 4 h 5 h 6* December, January, February : Latitude 37 to 49 Latitude 25 to 37 March, April .May: / 1.8 W 0.5 W 3.8 W 1.9 W 4.1 W 2.8 W 3-3 W 2.1 W / 2.9 W 1.5 W 4.8 W 2.6 W 5.6 W 3.2 W 4.0 W 2.3 W i 2.8 W 1.8 W 4.6 W 2.8 W 5-6 W 3.1 w 3-4 W 1.9 W / 2.1 W 1.6 W 3-8 W 2.4 W 4.6 W 2.4 W 2.3 W 1.2 W 1 13 W i.oW 2-5 W 1.6 W 3.0 W i.S W 1.2 W 0.7 W / 0.7 W 0.4 W 14 W 0.9 W 14 W 0.8 W 0.6 W 0.4 W 1 0.2 W o.i W 0.7 W 0.5 W 0.6 W 0.4 W o.i W 0.2 W Latitude 25 to 37 June, July, August : Latitude 37 to 49 . Latitude 25 to 37 September, October, November : Latitude 37 to 49 Latitude 25 to 37 i From " Manual of Instructions " issued by the U. S. Land Office to Surveyors General. TABLES. TABLE VI 365 APPROXIMATE LOCAL MEAN TIMES (COUNTING FROM NOON 24 HOURS) OF THE ELONGATIONS AND CULMINATIONS OF POLARIS IN THE YEAR 1897 FOR LATITUDE 40 N. ; LONGITUDE 6 h W. FROM GREENWICH. DATE. EAST ELONGATION. WEST UPPER ELONGATION. CULMINATION. LOWER CULMINATION. h. m. h. in. Jan. i 38.2 12 2 7 .8 6 33-6 18 31-6 v '5 23 39-o II 32.5 5 38.6 17 36.3 Feb. i 22 3'-8 IO 25-4 4 31.2 16 29.2 '5 21 36.6 9 30.2 3 35-9 15 33-9 Mar. i 20 41.4 8 34-9 2 40.7 H 38.7 15 19 46-3 7 39-8 I 45-7 13 43-7 Apr. i 18 39-3 6 32.8 O 38.6 12 36-7 15 May i \l 44-3 41-5 5 4 37-8 35-o 23 22 39-7 36.8 II 10 41.7 38.8 15 15 46.6 3 40.1 21 41.9 9 43-9 June i 14 39-9 2 33-4 2O 35-3 8 37-3 *5 13 45-o i 38.5 19 40.4. 7 42.4 July i 12 42.4 35-9 18 37-8 6 39-8 *5 II 47-5 23 37-i 17 42.9 5 44-9 Aug. i IO 41.0 22 30.6 16 36.4 4 38.4 s 9 46.1 21 35-7 15 4i-5 3 43-5 Sept. i 8 39-5 2O 29.1 14 34-9 2 36.9 15 7 44-6 19 34-2 3 40.0 I 42.0 Oct. i 6 41.8 18 31-4 12 37-2 39-2 is 5 46.8 17 36-4 II 42.2 23 40.3 Nov. i 4 40.0 16 29.6 IO 35-4 22 33-4 15 3 44-8 15 34-4 9 40.2 21 38-2 Dec. i 2 41.8 H 3i-4 8 37-2 2O 35-2 15 I 46-5 13 36.1 7 41.9 19 39-9 To refer to any calendar day other than the first and fifteenth of each month, subtract 3.94 m for every day between it and the preceding tabular day, or add 3.94 m for every day between it and the succeeding tabular day. To refer the tabular times to any year subsequent to the year 1897, add 0.25 m (nearly) for every additional year (after 1900, 0.2 m ). Also, For the second year after a leap year, add, 0.9 m . For the third year after a leap year, add, 1.7 m . For leap year before March 1, add, 2.6 m . For leap year on and after March 1, subtract, 1.2 m . For the first year after a leap year the table is correct, except for the regu- lar annual change. To refer the tabular times to other longitudes than six hours, add when east, and subtract when west of six hours, 0.16 m for each hour. To refer to any other than the tabular latitude between the limits of 25 and 50 north, add to the time of west elongation 0.13 m for every degree south of latitude 40, and subtract from the time of west elongation 0.18 m for every degree north of 40. Reverse these signs for corrections to the times of east elongation. For latitudes as high as 60, diminish the times of icest elongation and increase the times of east elongation by 0.23 m for every degree north of latitude 40. 1 Computed from information contained in the " Manual of Instructions " issued by the General Land Office. The information was furnished by the U. S. Coast and Geodetic Survey. 366 APPENDIX. TABLE VII. 1 REFRACTION CORRECTIONS TO DECLINATION OF THE SUN. The hour angle is the time either side of noon. LATI- TUDE. HOUR ANGLE. DECLINATIONS. + 20 +*> + 10 +5 -5 10 -15 -~ b. 25 oo O o 05 10 o 15 O 21 o 27 o 33 o 40 o 48 o 57 2 o 08 o 14 o 19 o 25 o 31 o 38 o 46 o 54 i 05 3 O 12 o 18 o 24 o 30 o 37 o 44 o 53 i 04 i 18 4 o 23 o 29 o 35 o 45 o 53 i 03 i if) i 31 1 52 5 o 49 o 59 I IO i 24 1 52 2 07 2 44 346 5 43 27 30 o 2 o 08 II o 13 O 10 o 18 O 22 S3 o 30 o 34 o 36 o 41 o 44 o 49 o 52 I 00 i 02 I 10 3 4 o 28. O 22 o 35 28 o 42 o 35 o 50 o 42 I OO o 50 I II I 00 i 26 I II i 43 i 26 2 9 5 o 54 I 05 i 18 i 34 1 54 2 24 3 " 438 8 15 30 oo 10 o '5 21 o 27 o 33 o 40 o 48 o 57 i 08 2 o 14 o 19 o 25 o 31 o 38 o 46 o 54 i 18 3 o 20 o 26 o 32 o 39 o 47 o 55 i 06 i 19 i 36 4 o 32 o 39 o 46 o 52 i 06 i 19 i 35 * 57 2 29 5 I OO I IO I 24 i 52 2 07 2 44 3 46 5 43 13 06 32 30 o 13 o 18 o 24 o 30 o 36 o 44 o 52 i 02 I 14 2 o 17 O 22 o 28 o 35 o 42 o 50 I OO i ii I 26 3 o 23 o 29 o 35 o 43 5 1 I 01 I 13 i 28 i 47 4 o 35 o 43 o 51 I OI i 13 I 27 1 46 2 13 2 54 5 i 03 i 15 i 3' 1 53 2 20 3 05 4 25 736 35 o 15 21 o 27 o 33 o 40 048 57 I 08 I 21 2 2O o 25 o 32 o 38 46 55 1 05 i 18 i 35 3 o 26 o 33 o 39 o 47 o 56 i 07 I 21 i 38 2 00 4 o 39 o 47 o 56 i 07 I 20 i 36 1 59 2 32 3 25 5 i 07 I 20 i 38 2 OO 2 34 3 29 5 *4 10 16 1 Computed from formula 57"cot (& + N). in which 8 is the declination, plus when north, and minus when south ; and N an auxiliary angle found by tan N = cot cos t, in which is the latitude of the place, and t the angle between the meridian of the place and the meridian through the sun at the given time, called the " hour angle." The formulae are from Chauvenet's " Spherical and Practical Astronomy," vol. I., p. 171. The table was computed by Mr. Edward W. Arms, C.E., for Messrs. W. & L. E. Gurley, of Troy, N.Y., and is here used by their permission. TABLES. 367 TABLE VII. (continued). LATI- TUDE. HOUR ANGLE. DECLINATIONS. + 20 +15 -HO- +5 -5 -10 -S 2O 37 30 o o 18 o 24 o 30 o 36 o 44 o 52 I 02 I H I 2 9 2 22 o 28 o 35 o 42 o 50 I OO I 12 I 26 i 45 3 o 29 o 36 o 43 o 52 I 02 I 14 I 29 i 49 2 16 4 o 43 o 51 I 01 I 13 I 2 7 i 49 2 14 2 54 4 05 5 I II I 26 i 44 2 IO 2 49 3 55 6 15 H 58 40 oo 21 o 27 o 33 o 40 048 o 57 i 08 I 21 i 03 2 o 25 o 32 o 39 o 46 o 52 i 06 i 19 i 35 i 57 3 o 33 o 40 o 48 57 I 08 I 21 i 38 2 O2 2 36 4 o 47 55 i 06 i 19 1 36 I 58 2 30 3 21 4 59 5 i 15 i 3i i 5i 2 2O 3 05 4 25 7 34 25 18 42 30 o o 24 o 30 o 36 o 44 o 52 i 02 i 14 i 29 i 49 2 o 28 o 35 o 39 o 50 I 00 I 12 i 26 i 45 2 II 3 o 36 o 43 o 52 I 02 i 13 I 2 9 i 49 2 17 2 59 4 o 50 I OO i ii I 26 I 44 2 10 2 49 3 55 6 16 5 i 16 1 36 i 58 2 30 3 22 5 oo 9 24 45 o o 27 o 33 o 40 o 48 o 57 I 08 I 21 i 39 2 O2 2 o 32 o 39 o 46 o 52 I O6 I 19 i 35 i 57 2 29 3 o 40 o 47 o 56 I 7 I 21 1 38 2 OO 2 34 3 29 4 o 54 i 04 i 16 i 33 I 54 2 24 3 ii 438 8 15 5 i 23 i 4i 2 05 2 41 3 40 5 40 12 O2 47 30 o o 30 o 36 o 44 o 52 i 02 I 14 I 29 i 49 2 18 2 o 35 o 42 o 50 I OO I 12 I 26 i 45 2 OI 2 51 3 o 43 o 51 I 01 I 13 I 28 i 47 2 15 2 S 6 4 08 4 o 56 I 09 I 23 I 40 2 5 2 40 3 39 5 37 ii 18 5 i 27 1 46 2 12 2 52 4 oi 6 30 16 19 50 oo o o 33 o 40 4 8 o 57 I 08 I 21 i 39 2 02 236 2 o 38 o 46 55 i 06 I 18 i 35 i 57 2 28 3 '9 3 o 47 o 56 i 06 i 19 i 36 2 2 9 2 31 3 23 5 02 4 I 02 I 14 i 29 i 48 2 16 258 4 18 6 59 19 47 5 I 3 I 51 2 I 9 3 04 4 22 7 28 24 10 52 30 o o 36 o 44 o 52 I 02 I M i 29 i 49 2 18 3 05 2 o 43 o 50 o 59 I II I 26 i 42 2 23 2 49 3 55 3 o 50 I OO i ii I 26 i 45 2 II 2 51 258 6 22 4 i 05 i 18 i 35 2 IO 2 28 3 19 4 53 842 5 i 34 i 56 2 27 3 16 4 47 852 55 o o o 40 o 48 57 i 08 I 21 i 39 2 O2 236 3 33 2 o 46 o 55 i 18 I 34 i 56 2 30 3 15 4 47 3 o 55 i 06 i 19 i 35 i 58 2 30 3 21 4 58 9 19 4 I 10 i 23 i 42 2 06 2 43 3 44 5 49 12 4 I 5 1 37 2 OI 2 34 3 28 5 '5 10 18 368 APPENDIX. TABLE VIII. 1 MAGNETIC DECLINATION. Formulas giving approximately the magnetic declination at the places named and for any time within the limits of the period of observation. The places are di- vided into three groups, as follows : GROUP I. Magnetic stations on the eastern coast of the United States and in- clusive of the region of the Appalachian range, with some additional stations in Newfoundland and other foreign localities. GROUP II. Magnetic stations mainly in the central part of the United States between the Appalachian and Rocky Mountain ranges, with additions in British North America, Canada, the West Indies, and Central America. GROUP III. Magnetic stations on the Pacific coast and Rocky Mountain re- gion ; also in Mexico and Alaska and in some foreign countries. D stands for declination, + indicating west, and east declination ; m stands for t 1850.0 or for the difference in time, expressed in years and fraction of a year, for any time t and the middle of the century ; a * indicates uncertainty. NAME OF STATION AND STATE. LATI- TUDE. WEST LONGI- TUDE. THE MAGNETIC DECLINATION EXPRESSED AS A FUNCTION OF TlME. GROUP I. ' ' 00 Saint Johns, New- 47 34-4 5241.9 D= + 21.94+ 8.89 sin ( i. 05 771 + 63.4)* foundland. Quebec, Canada. 46 48.4 71 14.5 D= + 14.66+ 3X>3sin(i.4 771 + 4.6) + 0.61 sin (4.0 771+ 0.3) Charlottetown.P.E.I. 4614 6327 D= + 15.95+ 7.785111(1.2 771 + 49.8) Montreal, Canada. 45 30-5 73 34-6 D= + n.88+ 4.i7sin(i.5 771 18.5) Eastport, Me. 4454-4 6659.2 D= + i5.i8+ 3.79 sin(i. 25 771+31.1)* Bangor, Me. 44 48.2 68 46.9 D= + 13.86+ 3.55 sin(i.3om+ 8.6) Halifax, Nova Scotia. 4439.6 63 35-3 D= + i6.i8 + 4.53sin(i.o 771+46.1)* Burlington, Vt. 4428.5 73 12.0 D= + 10.81 + 3.65 sin(i. 30 m 20.5) + o.i8sin(7.o 771 + 132) Hanover, N. H. 4342-3 7217.1 D=+ 9.80+ 4.O2Sin(i.4 771 14.1)* Portland, Me. 43 38.8 70 16.6 D= + 11.40+ 3.28sin(i.3om+ 2.7) Rutland, Vt. 43 36-5 7 2 55-5 P= + 10.03+ 3.82sin(i.5 771-24.3) Portsmouth, N. H. 43 04-3 70 42.5 D= + 10.71+ 3.36 sin (i. 44 m- 7.4) Chesterfield, N. H. 42 53-5 7224 D=+ 9.60+ 3.843^(1.35 m 16.1)* Newburyport, Mass. 4248.9 7 49-2 D= + 10.07+ 3-Q2sin(i.35 m i.o) Williamstown, Mass. 4242.8 73 13-4 D = + 8.84+ 3.i3sin(i.4 77114.0)* Albany, N. Y. 42 39-2 7345-8 D=+ 8.17+ 3.02 sin (i. 44m- 8.3) Salem, Mass. 4231-9 7 52-5 D=+ 9.98+ 3.85sin(i.4 m 5.1)* Oxford, N. Y. 42 26.5 75 40-5 D=+ 6.19+ 3.24 sin(i.35 771-18.9) Cambridge, Mass. 42 22.9 71 07.7 D = + 9-54+ 2.69sin(i.3om+ 7.0) + o.i8sin(3.2 771+44) Boston, Mass. 4221.5 7 1 3-9 D=+ 9.48+ 2.94 sin (i. 3 m+ 3.7) Provincetown, Mass. 4203.1 7011.3 D = + 9.67+ 3.04 sin (i. 3 7)i+ii.o)* Providence, R. I. 41 50.2 71 23.8 D=+ 9.10+ 2.99 sin (i. 45 m- 3.4) + 0.26 sin ( 7 771 + 84) Hartford, Conn. 4i 45-9 72 40.4 D=+ 8.06+ 2.90 sin(i. 25 m 26.4) New Haven, Conn. 41 18.5 D=+ 7-78+ 3.11 sin(i.4O77i 22.1) Nantucket, Mass. 41 17.0 70 06.0 D= + 8.61+ 2.83 sin(i.35 m + 19.7) Cold Spring Harbor, Long Island, N. Y. New York City, N.Y. 4052 4042.7 7328 7400.4 D=+ 7.19+ 2.52 sin(i. 35 m 11.4) D=+ 7.04+ 2.77 sin(i. 3077118.1) + 0.14 sin (6.3 wi + 64) From Appendix 7, " U. 8. Coast and Geodetic Survey Report " for 1888. TABLES. 369 NAME OF STATION LATI- WEST THE MAGNETIC DECLINATION AND STATE. TUDE. LONGI- TUDE. EXPRESSED AS A FUNCTION OF TlME. 00 Bethlehem, Pa. 40 36.4 75 22.9 D=+ 5.40+ 3.13 sin(i. 55 7/1-38.3) Huntingdon, Pa. 4031 7802 = + 3-76+ 2.93sin(i. 4 8m-35.2) New Brunswick, N.J. 40 29.9 74 26.8 D=+ 5.11+ 2.94 sin (1.307/1+ 4.2) Jamesburg, N. J. 40 21 7427 D=+ 6.03+ 2.94 sin (i. 40 7/i -22.4) Harrisburg, Pa. 40 15.9 76 52.9 D=+ 2.93+ 2.98 sin (i. 50 m+ 0.2) Hatboro, Pa. 4012 75 7 D=+ 5.17+ 3. 1 6 sin (i. 54 7/1-16.7) + 0.22 sin (4. i 771 + 157) Philadelphia, Pa. 39 56-9 7509.0 D = + 5.36+ 3.i7sin(i.5om-26.i) + o.i9sin(4.o 7/1+146) Chambersburg, Pa. 3955 7740 D=+ 2.79+ 3. 10 sin (i. 55 7/1-30.6) + o.2osin(4.6 7/1+124) Baltimore, Md. Washington, D. C. 39 17-8 38 53-3 76 37-o 7700.6 D = + 3.20+ 2.57 sin(i.45 m 21.2) D=+ 2.73+ 2.5 7 sin (i. 45 m -2 1. 6) + o.i4sin(i2 7/4 + 27) Cape Henlopen, Del. Williamsburg, Va. 3846.7 37 1 6.2 75 05- 76 42.4 D=+ 4.01+ 3.22 sin (1.35 7/1-25.2) D=+ 2.33+ 2.56301(1.5 7/1 38.1) Cape Henry, Va. 7600.4 |D=+ 2.42+ 2.25 sin( 1.47 7/1 30.6) Newbern, N. C. 35 6 7702 D = + 0.63+ 2.56 sin (1.45 7/i 18.2)* Milledgeville, Ga. 33 04-2 8312 D= 3.10+ 2.53 sin(i.4om 61.9)* Charleston, S. C. 32 46.6 7955.8 jD= 1.82+ 2.75 sin(i.4om 12.1)* Savannah, Ga. 32 04.9 81 05.5 D= 2.13+ 2.55 sin(i.40 7/1-40.5)* Paris, France. 48 50.2 2 20.2E D= + 6.479 + i6.oo2 sin (0.765771 + 1 1 8 46'.5 + [0.85 - 0.35 sin (0.69 )] sin [(4.04 + 0.0054 n + .000035 ' 2 )n] St. George's, Ber- muda 3223 6442 D=+ 6.95 +0.0145 771 + 0.00056 ? 2 * Kiode Janeiro, Brazil -2254.8 4309.5 D=+ 2.19+9.91 sin(o.8o m 10.4)* GROUP II. York Factory, Brit- ish North America 5 6 59-9 92 26 D = + 7.34+16.03 sin (1.107/1 97.9) Fort Albany, British North America. 5222 8238 D= + 15.78+ 6.95 sin (i.207/ -99.6)* f Duluth, Minn., and \ Superior City, Wis. 46 45-5 46 39-9 92 04.5 1 92 04.2 / D= 7.70+ 2.41 sin (1.4 7/1 120.0)* Sault Ste. Marie, Mich. 46 29.9 842O.I D=+ 1.54+ 2.70 sin(i.45 7/1-58.5) Pierrepont Manor, N. Y. 4344-5 76 03.0 D=+ 5.95+ 3.788^(1.4 7/1-22.2) Toronto, Canada. 43 39-4 79 23.5 D=+ 3.60+ 2.82sin(i.4 m 44.7) + 0.09 sin (9.3 7/1+136) + 0.08 sin (19 7/1 + 247) Grand Haven, Mich. 435-2 86 12.6 D= 4.95+ 0.0380 7>i +0.00 1 20 m' 2 Milwaukee, Wis. 43 02.5 8754-2 D= 4.12+ 3.60 sin ( i. 45 m 64.5)* Buffalo, N. Y. 4252.8 D=+ 3.66+ 3.478^(1.4 7/127.8) Detroit, Mich. 42 20.0 83 03.0 D=- 0.97+ 2.2isin(i.5 7/1-15.3) Ypsilanti, Mich. 42 14 8338 D= 1.20+ 3.40 sin (i. 40 m 4.1) Erie, Pa. 4207.8 8005.4 D = + 2.17+ 2.69 dn( 1.5 7/127.3) Chicago, 111. 41 50.0 87 36.8 D=- 3.77+ 2. 4 8sin(i.45?-62.5) Michigan City, Ind. 4i 43-4 86 544 D=- 3.23+ 2.42 sin (i. 4 7/1-48.0) Cleveland, 0. 41 30.4 8141-5 D=+ 0.47+ 2.393111(1.30771-14.8) Omaha, Neb. 4i 15-7 95 56.5 D=- 9.30+ 3.34 sin (i. 30 m- 54.7) Beaver, Pa. 4044 8020 D=+ 1.41+ 2.72 sin (i. 40 m- 39.6) K'AI'D SURV. 24 370 APPENDIX. NAME OF STATION LATI- WEST LONGI- THE MAGNETIC DECLINATION AND STATE. TUDE. TUDE. EXPRESSED AS A FUNCTION OF TlME. Pittsburg, Pa. 40 27.6 8000.8 00 D = + 1.85+ 2.45 sin (1.45 77128.4) Denver, Col. 3945-3 104 59-5 D= 15.30+ o.o ii 771+0.0005 ?7i' 2 Marietta, O. 3925 8128 D = + 0.02+ 2.89sin(i.4 771 40.5) Athens, O. 8202 D= 1.51+ 2.63sin(i.4 771 24.7) Cincinnati, O. 39 08.4 84 25.3 D=- 2.59+ 2.43 sin (1.42 m -37. 9) Saint Louis, Mo. Nashville, Tenn. 38 38.0 36 08.9 9O 12.2 8648.2 D=- 5.91+ 3.oosin(i.4om-5i.i)* D=- 3-57+ 3-33 sin(i.35 771-68.5)* (Florence, Ala. 3447-2 874L5 D=- 4.25+ 2.33sin(i.3 771-52.8) Mobile, Ala. 3041.4 88 02.5 D=- 4.38+ 2.69 sin (i. 45 731-76.4) Pensacola, Fla. 30 20. 8 87 18.3 D=- 4.40+ 3.i6sin(i.4 771-59.4) New Orleans, La. 2957-2 9003.9 D= 5.20+ 2.98 sin (1.4077? 69.8) San Antonio, Tex. 29 25.4 98 29.3 D=- 7.40+ 2.88sin(i.35?7i-8i.8)* Key West, Fla. 24 33-5 8 1 48.5 D= 4.31+ 2.86 sin ( i. 30 m 23.9) Havana, Cuba. 23 09.3 8221.5 D=- 4.25-+ 2.74 8dn(i. 25 m- 23.3)* Kingston, Port Royal, Jamaica. '755-9 76 50-6 D= 3.81+ 2.39 sin ( 1. 10 ?7i 10.6) Barbados, Caribbee Islands. 1 3 05-7 59 37-3 D= 1.38+ 2.84 sin( 1. 10771+09.4) Panama, Colombia. 79 32-2 D= 5.66+ 2.22 sin(i.iom 27.8) GROUP III. Acapulco, Mexico. 1 6 50.5 99 52-3 D=- 4.48+ 4.4isin(i.o 771-85.7)* Vera Cruz, Mexico. 1911.9 9608.8 D=- 5.09+ 4.22sin(i.2 771-63.4)* City of Mexico, Mex. 19 26.0 9911.6 D =- 5-34+ 3-28sin(i.o 7/1-87.9)* San Bias, Mex. 21 32.5 105 18.4 D=- 5.21+ 4.26 sin(i. 15 771-96.5) ' San Lucas, Lower Cal. 22 53-3 109 54.7 D= 5.94+ 3.68sin(i.2om- 116.8)* Mazdalena Bay, Lower Cal. 24 38-4 1 1 2 08.9 D= 6.33+ 4.17 sin(i. 15 m 119.2)* Cerros Island, Lower Cal. 2804 115 12 D= 7.40+ 4.61 sin (1.05 731107.0) El Paso, Tex. 3i 45-5 1 06 27.0 D = 9.08+ 3.40 sin( 1.3 731108.4) San Diego, Cal. 3242.1 "7 4-3 D= 10.32+ 3.00 sin( i. lorn 126.5) Santa Barbara, Cal. 34 24-2 11943.0 D= 11.52+ 3.32 sin(i. 10 m 123.1) Monterey, Cal. 3636-1 121 53.6 D= 13.25+ 2.83 sin (1.10771144.0) San Francisco, Cal. Cape Mendocino, Cal. 37 47-5 40 26.3 12227.3 12424.3 D=- 13.94+ 2.655^(1.05771-135.5) D= 15.25 + 2.45 sin ( 1. 10771 128.0)* Salt Lake City, Utah. 4046.1 III 53.8 D= 12.40+ 4.255^(1.4 771 121.6)* Vancouver, Wash. 45 37-5 12239.7 D = -i7.93+ 3.12 sin(i.35m- 134.1)* Walla Walla, Wash. 4604 II822 D= 17.80+ 3.3osin(i.3 731129.0)* Cape Disappointment, Wash. 46 16.7 12402.8 D=- 19-39 + 2.54 sin(i.25 m- 158.7) Seattle, Wash. 47 35-9 12220.0 D= 19.19+ 3.143111(1.4 731136.1)* Port Townsend, Wash. 48 07.0 12244.9 D=- 18.84+ 3.00 sin (i. 45 m -i 22.1) Neah Bay, Wash. 48 21.8 12438.0 D= 19.83+ 2.91 sin(i.4O 771 141.6) Nootka, Vancouver Isl. 49 35-5 12637.5 D= -21.25+ 2.74 sin(i.30 m- 152.0)* Captain's and Iliuliuk Harbors. 53 52-6 16631.5 D=- 18.01 + i.82sin(i.3 771-69.6)* Sitka, Alaska. St. Paul, Kadiak Island. 572-9 5748.0 135 '9-7 15221.3 D= 25.79+ 3.30 sin (1.30 TO 104.2) [)= -22.21+ 5.l8sin(l.3573l-72.5) Port Mulgrave, Alaska. 59 33-7 13945-9 [)= 24.03+ 7. 77 sin( i. 30 77185.8) Port Etches, Alaska. Port Clarence, Alaska. 60 20.7 65 16 146 37.6 1 66 50 D=- 23.71 + 7.89 sin (i. 35731-80.9) D = 18.98+ 7.99801(1.3 771 68.4)* Dhamisso Isl., Alaska. 6613 161 49 [)= 23.62+ 7.64 sin (1.3 771 64.0)* Petropaulovsk, Siberia. 5301 201 17 D = - 3.35+ 2.97'sin(i.3 771+12.2) TABLES. 371 TABLE IX. ANGULAR CONVERGENCES AND DISTANCES BETWEEN MERIDIANS. 1. Angular convergence of meridians per mile of easting or westing. 2. Distance between meridians converging by one minute. LATI- TUDE. ANGULAR CON- VERGENCE PER MILE. MINUTES. DISTANCE FOR CONVERGENCE OFl'. FEET. LATI- TUDE. ANGULAR CON- VERGENCE PER MILE. MINUTES. DISTANCE FOE CONVERGENCE OF 1'. FEET. o I 0.015 34S733 31 0.521 10140 2 .030 I743H 32 542 975 3 045 116150 33 .563 9382 4 .061 87052 34 585 9034 5 .076 69578 35 .607 8703 6 .091 57917 36 .630 8387 7 .107 49578 37 653 8087 8 .122 43337 38 .677 7801 9 137 38436 39 .702 7527 10 '53 34525 40 .727 7265 ii .169 31320 4i 753 7013 12 .184 28642 42 .780 6770 13 .200 26371 43 .808 6 537 14 .216 24419 44 .836 6313 15 .232 22723 45 .866 6097 16 .249 21234 46 .897 5888 1 7 .265 19916 47 .929 5686 18 .282 18740 48 .962 549 '9 .299 17685 49 , 99 8 53i 20 .316 16731 5 .032 5"8 21 333 15864 5 1 .069 4940 22 35 I573 52 .108 4766 23 .368 14348 53 .149 4597 24 .386 13680 54 .191 4433 25 .404 13062 55 .236 4271 26 423 12488 56 .283 4"5 27 .442 "955 57 333 3962 28 .461 "457 58 385 38i3 29 .480 10990 59 1.440 3666 3 .500 '552 60 1:499 35 2 3 372 APPENDIX. TABLE X. 1 LENGTH OF ONE MINUTE OF LATITUDE AND ONE MINUTE OF LON- GITUDE TO THE NEAREST WHOLE FOOT. LATI- TUDE. 1' LATITUDE. FEET. 1' LONGITUDE. FEET. LATI- TUDE. 1' LATITUDE. FEET. 1' LONGITUDE. FEET. I 6046 6086 3i 6062 5222 2 6046 6083 32 6063 5167 3 6046 6079 33 6064 5110 4 6046 6072 34 6065 552 5 6046 6064 35 6066 4992 6 6047 6054 36 6067 4930 7 6047 6042 37 6068 4867 8 6047 6028 38 6069 4803 9 6047 6013 39 6070 4737 10 6048 5995 40 6071 4670 ii 6048 5976 4i 6072 4601 12 6049 5955 42 6074 453i 13 6049 5932 43 6075 4459 H 6050 5908 44 6076 4386 15 6050 5881 45 6077 43" 16 6051 5853 46 6078 4236 17 6051 5823 47 6079 4159 18 6052 579i 48 6080 4081 19 6052 5758 49 6081 4001 20 6053 5722 50 6082 3921 21 6054 5685 5 1 6083 3839 22 6055 5 6 47 5 2 6084 375 6 23 6055 5606 53 6085 3671 24 6056 5564 54 6086 3586 25 6057 5520 55 6087 3499 26 6058 5475 56 6088 3412 27 6059 5427 57 6089 3323 28 6059 5379 58 6090 3234 29 6060 5328 59 6091 3"43 30 6061 5276 60 6092 3051 1 Abbreviated from the Smithsonian Geographical Tables. TABLES. 373 TABLE XL TRIGONOMETRIC FUNCTIONS AND FORMULAS. SOLUTION OF TRIANGLES. By definition, if R = 1, p O c ED = sine a. OD - cosine a. DA = versed sine a. HF = coversed sine a. BA = tangent a. jPC = cotangent a. OB = secant a. OC = cosecant . If R is other than 1, it follows from the above definitions and the propor- tionality of similar figures, that 5. BA = R tan a. 6. FC = Rcota. 7. OB = R sec a. 8. OC = R cosec a. 2. OD = Rcosa. 3. DA = R versin a. 4. HF = R co versin a. from which also in any right triangle of angles a and ft, if o be the side opposite the angle a, a the side adjacent thereto, and h the hypotenuse, 9. sin a = - = cos ft. h 10. cos a = - = sin ft. h, 11. tana = - = cot . 12. cot a = - = tan fl. 13. sec a = - = cosec ft. 14. cosec a = - = sec ft. 15. vers a = = covers /3. 16. covers a = ^ = vers /?. Hence, So = A sin a = h cos 5. ft= o = o sin a cos ft !a = h cos a = h sin /?. f*rs ft. si n ^ 19. sin/3 o = a tan a = a cot /J. ~~ tan ~ cot y8' 20. 2. a = o cot a = o tan ft. o = a = a cot a tan /?' h = a sec a = a cosec ft. __ __. sec a cosec /3 h = o cosec a = o sec /J * = h cosec a sec /?' 374 APPENDIX. 23. o = VA 2 _ a 2 _ V(A + a)(A - a). 24. a = VA 2 - o 2 = V(A + o) (A - 0). 25. A = Vo 2 + a 2 . 26. Oblique triangles may be solved by some one of the following formulas GIVEN. I SOUGHT. FORMULAS. 27. A, B, a, | C, b, c, C = 180- (A + J5), = ." sin B, sin 4 c a sin ( 4 1 5") 28. A, a, b, 29. C, a, 6, 30. 31. 32. 5, C, c, i (4 + B), A,B, c, sin B = ?HLd&, C = 180 - (A + B), a c = - sin C. sin A $(A + B) = 90 - J C. tan J (^4 .B) = a ~ tan $(A + B). a + b B = %(A + B) - $ (A - B). j\ cos K^4 + -B) /s'n S Kl+?) 33. 34. a, 0, c, Area, Area = \ ab sin C. If s = i (a 4 + c), sin i 4 * /&2-Z ^)( s ~~ c ^ V C ' tan! 4 ./C* -)(*-) V .(.-a) 2 Vs(s a)(s ft)(s c) be ver3 4 2(s-o)(s-c) 35. 36. A, B, C, a, Area, Area, Area = Vs(s - a)(s - o)(.v - c). ,. a 2 sin B sin C 2 sin 4 TABLES. 375 From the definitions of the trigonometric functions, the geometrical properties of right triangles and in some cases algebraic transformations, it may be shown that if A is any angle and B any other angle, 37. sin. 9 A +cos 2 -4 = 1. 38. sin A = - - = Vl cos' 2 A = tan A cos A vers A cot | A sec A sin 2 A sin 2 A = cos 2 ^ -sin 2 .4 = 40. tan A = sin A _ 1 cos A cot A 1 + cos 2 A = cot .4 (sec ,4 -1). 41. cot A = 2Ld. = _L_ = Vcosec 2 A - 1 sin A tan A sin 2 A _ sin 2 A _ 1 + cos 2 J. _ tan ^ A ~ 1 - cos 2 ^4 ~ vers 2 A ~ sin 2 ^4 ~ sec A - 1 42. vers A = 1 -cos ^4 = sin A tan J A = 2 sin 2 J ^4 = cos ^4 (sec ,4 - 1). 43. sin (A ) = sin ^4 cos J5 sin B cos .4 . 44. cos (A B) = cos A cos 5 =F sin /I sin 5. 48. cos 2 4 = 2 cos 2 ,4 - 1 = cos 2 A - sin 2 A = 1 - 2sin 2 ^. tan.4 1 + sec A 49. tanM=-^^- = cosec^-cot^ = 1 - C09 ^ 50. 1 - tan 2 4 51. sin A cosec A cot ^4 376 APPENDIX. 52. cot 2 A = cot 2 .4-1 53. vers A A = 1 - cos ^4 2+V2(l vers A 1 + VI - vers J. 54. vers 2 A =2 sin 2 .4. 55. sin A + sin B = 2 sin J (/I + B) cos (4. -B). 56. sin ^4 sin B = 2 cos J (J. + -B) sin (^4 B). 58. cos B cos A =2 sin (4. + B) sin ( J. B) . 59. sin 2 A - sin 2 B = cos 2 B - cos 2 .4 = sin (A + B) sin (A 60. cos 2 ,4 - sin 2 B = cos(J. + B)cos(4 - B). 61. tan^+tanB = sin (f + B \ cos A cos B 62. tan A tan B = cos ^4 cos TABLE XII. LENGTHS OF CIRCULAR ARCS OF RADIUS 1, AND VARIOUS CIRCULAR MEASURES. No. DEGREES. MINUTES. SECONDS. No. DEGREES. MINUTES. SECONDS. i 0174533 .0002909 .0000048 6 .1047198 .0017453 .0000291 2 .0349066 .0005818 .0000097 7 .1221730 .0020362 0000339 3 0523599 .0008727 .0000145 8 .1396263 .0023271 .0000388 4 .0698132 .0011636 .0000194 9 .1570796 .0026180 .0000436 5 .0872665 .0014544 .0000242 10 1745329 .0029089 .0000485 Degrees in arc of length equal to radius, 57. 295 780. Degrees in arc of length equal to IT, 180. 000 000. Circumference = 2 irr = 360. 000 000. Area = Trr 2 . If I = length of circular arc d number of degrees in same r = radius of same c = chord of same m = middle ordinate 180 7T ' "180" Area of sector = \ Ir. Area of sector = - Trr 2 . o60 Approximate area of segment = | cm. TABLES. 377 TABLE XIII. LINEAR TRANSFORMATIONS. 1. Gunter's Chains to Feet. CHAINS. 0.0 0.01 .02 03 .04 05 .06 .07 .08 .09 0.0 .66 1.32 I. 9 8 2.64 3-30 3-96 4.62 5 .28 5-94 .1 6.60 7.26 7.92 8.58 9.24 9.90 10.56 11.22 u.88 12.54 .2 13.20 13.86 14.52 I5.I8 15.84 16.50 17.16 17.82 18.48 19.14 3 19.80 20.46 21.12 21.78 22.44 23.10 23.76 24.42 25.08 25-74 4 26.40 27.06 27.72 28.38 29.04 29.70 30-36 31.02 31.68 32.34 5 33-oo 33-66 34-32 34.98 35.64 36-30 36-96 37.62 38.28 38.94 .6 39.60 40.26 40.92 41.58 42.24 42.90 43-56 44.22 44.88 45-54 7 46.20 46.86 47-5 2 48.18 48.84 49.50 50.16 50.82 51.48 52.14 .8 52.80 53-46 54.12 54.78 55-44 56.10 56.76 57-42 58.08 58.74 9 59-40 60.06 60.72 61.38 62.04 62.70 63-36 64.02 64.68 65.34 0.0 I.O 2.O 3-0 4.0 5-o 6.0 7.0 8.0 9-0 o.o 66 132 198 264 330 396 462 528 594 10.0 660 726 792 858 924 990 1056 1 1 22 1188 1254 20.0 1320 1386 1452 1518 1584 1650 1716 1782 1848 1914 30.0 1980 2046 2112 2178 2244 2310 2376 2442 2508 2574 4O.O 2640 2706 2772 2838 2904 297 3036 3102 3168 3234 5O.O 3300 3366 3432 3498 3564 3630 3696 3762 3828 3894 60.0 3960 4026 4092 4158 4224 4290 4356 4422 4488 4554 70.0 4620 4686 4752 4818 4884 495 5016 5082 5H8 5214 80.0 5280 5346 5412 5478 5544 5610 5676 5742 5808 5874 378 APPENDIX. ro u-> f^ ON fO O NO ro Q NO f> O NO ro 3 ftS&S-gi&S ? vONONOvONOvONONO t^t^ - ro if> r^. ON ro 10 l^. ON >vOOO OOO ~ c5 ro 4 1 i>i vD 00 00 00 00 00 OO OO 00 M co ""> t>. ON n CO I Ox^f^^-^r^ONOW i ^OO\O f^iOsvO CS S?*^ M coi^r^d\w roint^cf\ o i ij 5 $< 5:^ o, O\O N"^M NVO fO OO O . O-ur)*otoo gqqqoqqoq - . o N f*i"->vO r-.oo O*O eoooooooooooooooooo ON c5 4 0\o" V ?.| R|RRRR|R HNNft ** S ' teg 'S.co'co 8. S: o C o 6 6 6 6 I M 8,^5 ARRR s, e m 5w 1? r wo aaaaa 8.SI8? sra's? s-?ss;s; as ftftftjj ft JK t>. r-*oo oo || ^^.SS^ sa 5,3.S,i aaaa a aaaaa aaaa > ft K R^tf y?? ,S,SvS- 5TS-S-& acS-o?cS .&&& * 4=> So'SITg "-t* i 8>8,^8> 8- .S.S. & a ^5 ^ R iS! SSSSJg JTJefJf ss jefjfe sss g'R.R sssa S'gS'S 9: S-StS"? K.'Rfifi- " 12 "S 1 c8^S SlSff'g gggg Sm H R 8 S 1 382 TABLES XV, XVI. COMMON LOGARITHMS OF NUMBERS. LOGARITHMS OF TRIGONOMETRIC FUNCTIONS, EDITED BY C. W. CROCKETT. NOTE. THE well-known tables of Gauss, Hotiel, Becker, and Albrecht have been taken as the standards, and the figures compared with the more extensive tables, the doubtful cases being recomputed. 384 EXPLANATION OF THE TABLES INTRODUCTORY. 1. When we have a number with six or more decimal places, and \ve wish to use only five : (a) If the sixth and following figures of the decimal are less than 5 in the sixth place, they are dropped; thus, 0.46437 4999 is called 0.46437. (6) If the sixth and following figures of the decimal are greater than 5 in the sixth place, the fifth place is increased by unity and the sixth and following places are dropped ; thus, 0.46437 5001 is called 0.46438. (c) If the sixth figure of the decimal is 5, and if it is followed only by zeros, make the fifth figure the nearest even figure ; thus, 0.46437 500 is called 0.46438, while 0.46438 500 is also called 0.46438. The number is thus increased when the fifth figure is odd, and decreased when it is even, the two operations tending to neutralize each other in a series of computations, and hence to diminish the resultant error. 2. Hence any number obtained according to Art. i may be in error by half a unit in the fifth decimal place. 3. When the last figure of a number in these tables is 5, the number printed is too large, the 5 having been obtained according to Art. i (If) ; if the 5 is without the minus sign, the number printed is too small, the figures following the 5 having been dropped according to Art. i (#). 4. The marginal tables contain the products of the numbers at the top of the columns by i, 2, 3, 9 tenths, and may be used in multiply- ing and dividing in interpolation. (a) To multiply 38 by .746 : 38 38 x. 7= =26.6 3 38 x .4= 15-2; .'. 3 8 X .04 = 1.52 38 x .6 = 22.8 ; .-. 38 x .006 = .228 6 38 X .746 = 28.348 3-8 7.6 11.4 '5-2 19.0 22.8 26.6 3-4 9 ! 34-2 In multiplying by the second figure (hundredths), the decimal point in the table is moved one place to the left ; in multiplying by the third (thousandths), two to the left ; and so on. (385) 386 EXPLANATION OF THE TABLES. To divide 28 by 38 : Dividend, Next less, Remainder, Next less, Remainder, Nearest, .'. Quotient, 28 26.6 i 4 i 1.4 26 corresponding to .7 corresponding to .03 2 6.6 corresponding to .007 .737 38 3.8 7-6 11.4 2 2 !8 26.6 30.4 9 34-2 to the nearest third decimal place. The decimal point is moved one place to the right in each remainder, since the next figure in the quotient will be one place farther to the right. To divide 23 by 38 : Dividend, 23 22. corresponding to .6 o.o corresponding to .00 2 o. Nearest, i 9.0 corresponding to .005 .-. Quotient, .605 The computer should use the marginal tables mentally. LOGARITHMS. 5. The logarithm of a number is the exponent of the power to which a given number called the base must be raised to produce the first number. If A = e a , a is called the logarithm of the number A to the base log 37300 = 4.5 71 71, and and an <3 log 3.73 =0.57171 logo.373 =1.57171 ^0.0373 = 2.57171 Hence the position of the decimal point affects the characteristic alone, the mantissa being always the same for the same sequence of figures. For this reason the common system of logarithms is used in practice. 10. The characteristic is found as follows : When the number is greater than i, the characteristic is positive, ami is one less than the num- ber of digits to the left of the decimal point ; when the number is less tlian i, the characteristic is negative, and is one more t/ian t/ie number of zeros between the decimal point and the first significant figure. 11. To avoid the use of negative characteristics we add io to the characteristic and write io after the mantissa, i.e. adding and subtract- ing the same quantity, io. Thus log 0.2 ='1.30103 would be written 388 EXPLANATION OF THE TABLES. 9.30103 10. The 10 is often omitted for brevity when there is no danger of confusion, but its existence must not be forgotten. Such logarithms are called augmented logarithms. In this case the characteristic of the logarithm of a pure decimal is 9 diminished by the -number of ciphers to the left of the first significant figure. Thus the characteristic of log 0.004 > s 9 2 , or 7> an d that of log 0.94 is 9 o, or 9. 12. The arithmetical complement of the logarithm (written colog} of a number is the logarithm of its reciprocal, and is found by subtract- ing each figure of the logarithm from 9, commencing at the left, except the last significant figure on the right, which is subtracted from 10. For log- = log.r = 10 log.r 10; thus, if log x= 2.46403, colog x= 7-53597 I0 ; if log .# = 8.43000 10, colog x = 1.57000. TABLE XV. 13. Page 397 contains the logarithms of numbers from i to 100, to five decimal places. Pages 398-415 contain the mantissas of the logarithms of numbers from 1000 to 10009, to five decimal places. Pages 416,417, contain the mantissas of the logarithms of numbers from 10000 to 11009, to seven decimal places. NOTE. The mantissas of the logarithms of numbers, except those of the integral powers of 10, are incommensurable, the mantissas in the tables being found as shown in Art. i. To find the Logarithm of a Number. 14. The characteristic is found by the rules in Arts. 10 and n, and the mantissa from the tables, as shown in Arts. 15, 16, 17, 18. 15. When the number has four figures. Find on pages 398-415 the first three figures in the column marked N, and the fourth at the top of one of the other columns. The last three figures of the mantissa are found in this column on the horizontal line through the first three figures of the given number in column N. The first two figures of the mantissa are those under L in the same line with the number, or else those nearest above it, unless the last three figures of the mantissa as given in the tables are preceded by a *, when the first two figures are found under L in the first line below the number. Thus (page 398), log 1 136 = 3.05538; log 1137 = 3.05576; log 1 138 = 3.05614; log 1370 = 3.13672 ; log 1371 = 3.13704 ; log 1372 = 3.13735 ; log 1380 = 3.13988 ; log 1381 = 3.14019 ; log 1382 = 3.14051. EXPLANATION OF THE TABLES. 389 16. When the number has less than four figures, annex ciphers on the right and proceed as in Art. 15. Thus, log 1.13 = 0.05308 ; log 12.8 = 1.10721 ; log 130 = 2.11394 ; log 15 =1.17609; log 1 6 =1.20412; log 17 =1.2304^. 17. When the number has more than four figures, as 11.4672. Since the mantissa is independent of the position of the decimal point, point off the first four figures and find the mantissa of log 1146. 72. This will be between the mantissas of log 1146 and log 1147. Hence find from the tables the mantissas corresponding to 1146 and 1147; multiply the difference between them (called the tabular difference) b .72, and add the product (called the correction) to log 11.46; the result will be the logarithm required. Mantissa of log 1146 = 05918 log 11.46 = 1.05918 Mantissa of log 1147 =05956 correction = 38 x .72 = 2 7-36 Tabular difference = 38 .-. log 11.4672 = 1.05945 36 or = 1.05945 XOTK. Since any mantissa in the tables may be in erroj by half a unit in the fifth decimal place (Art. 2), no advantage is gained by using the sixth place in the interpolated logarithm. Thus, according to Art. I, we drop the .36, and call log 1 1. 4672= 1.05945. NOTE. The marginal tables should be used in multiplying the tabular difference to find the correction (Art. 4). NOTE. It is assumed that the change in the mantissa is proportional to that in the number, as the latter increases from 1146 to 1147. An increase of I in the num- ber causes an increase of 38 in the mantissa; hence an increase of .72 in the number will cause an increase of 38 X .72 in the mantissa. NOTE. We could also find the mantissa of log 11.4672 by subtracting the prod- uct of the tabular difference by .28 (or i.oo .72) from the mantissa corresponding to 1147; that is, the required mantissa is 05956 (38x .28)^05956 10.64 05945 as before. 18. The general rule is : Find the mantissa corresponding to the first four figures of the number; multiply the tabular difference by the fifth and following figures treated as a decimal; and add the product to the mantissa just found. The tabular difference is the difference between the mantissas corre- sponding to the two numbers in the tables, between which the given number lies. log 1.62163 = 0.20995 ; log 0.38024 = 1.58006 ; log 0.085 2 763 = 2.93083 ; log 189. 524 = 2. 27767 ; Iogo.386o2=l. 58661 ; ^0.0085938 = 3.93419 ; log 19983.4 = 4.30067 ; Iog3-98743 = o. 60070 ; logo. 090046 =2.95446. 390 EXPLANATION OF THE TABLES. NOTE. Page 397 is used when the number contains less than three figures, the number being found in the column N, and the logarithm in the column headed Log. The characteristic is given for whole numbers, and must be changed for decimals. NOTE. When a number is composed of three figures, find on pages 398-415 the number in the column N, and the mantissa corresponding in the column L. o. To find the Number corresponding to a Given Logarithm. 19. From the tables we find the sequence of figures corresponding to the given mantissa, as shown in Arts. 20, 21, and 22, the position of the decimal point being determined by the characteristic (Arts. 10, n). 20. When the given mantissa can be foimd in the tables. Find on pages 398-415 the first two figures of the mantissa under L in the column headed L. o. The last three figures of the mantissa are then sought for in the columns headed o, i, 2, 9, in the same line with the first two figures, or in one of the lines just below, or in the line next above (where they would be preceded by a *). The first three figures of the required number will be found in the column headed N, in the same horizontal line with the last three figures of the mantissa, and the fourth figure of the number at the top of the column in which the last three figures of the mantissa are found. Thus (page 398), 0.06221 = log 1.154 ; 0.06558 log 1.163 i 0.06893 = log 1.172 ; 0.07004 = log 1.175 > 0.07188 = log 1.180 ; 0.08063 = log 1.204. 21. When the given mantissa can not be found in the tables. If we wish to find the number whose logarithm is 2.16531, we enter the tables with 16531, and find that it lies between 16524 and 16554, so that the given mantissa corresponds to a number between 1463 and 1464. Also 16531 exceeds 16524 by 7, and this difference, divided by the tabular difference 30, gives .23" as the amount by which the required number exceeds 1463. Hence 2.16531 = log I46.323---, which we call 146.32, according to Art. i, the incompleteness of the tables making the sixth figure uncertain. NOTE. The marginal tables should be used in dividing the difference between the given mantissa and the one next less in the tables by the tabular difference. 22. The general rule is : Find the number corresponding to the mantissa in the tables next less than the given mantissa ; divide the excess of the given mantissa over the one found in the tables by the tabular difference; and annex the quotient to the first four figures already found. The tabular difference is the difference between the two mantissas in the tables, between which the given mantissa lies. T.i66oo = log 0.14656 ; 0.18002 = ^1.5136; 2.18200 = log 152.06 ; 1.19000 = log 15.488 ; 4.19680 = log 15773 ; 1.20020 = log 15.856. 23. For the use of the numbers S', T', S", T", see Arts. 35-38. EXPLANATION OF THE TABLES. 391 TABLE XVI. 24. This table (pages 420-464) contains the logarithms, to five deci- mal places, of the trigonometric sines, cosines, tangents, and cotangents of angles from o to 90, for each minute. The logarithms in the columns headed L. Sin, L. Tan, and L. Cos, are augmented, and should be diminished by 10 (Art. n), while those in the columns headed L. Cot are correctly given. 25. Since secx = , and cosec.# = - , the logarithms of the cos x sin* secant and cosecant of an angle are the arithmetical complements of those of the cosine and sine respectively (Art. 12). To find the Logarithmic Functions of an Angle Less than 90. 26. When the angle is less than 45, the degrees are found at the top of the page, and the minutes on the left. The numbers in the same horizontal line with the minutes of the angle are the logarithmic functions indicated by the notation at the top of the columns. Thus (page 428), log sin 8 4' = 9.14714 10, log tan 8 4' = 9.15145 10, log cot 8 4' = 0.84855, log cos 8 4' = 9.99568 10. 27. When the angle is greater than 45, the degrees are found at the bottom of the page, and the minutes on the right. The numbers in the same horizontal line with the minutes of the angle are the logarithmic functions indicated by the notation at the bottom of the columns. Thus (page 428), log sin 81 25' = 9.99511 10, log tan 81 25' = 0.82120, log cot 81 25' = 9.17880 10, log cos 81 25' = 9.1 7391 10. 28. When the angle is given to decimals of a minute. In finding log sin 30 8 '.48, for example, we see that it will lie between the logarithmic sines of 30 8' and 30 9', that is, between 9.70072 and 9.70093, their difference 21 being the change in the logarithmic sine caused by a change of i' in the angle. Hence, to find the correction to log sin 30 8' that would give log sin 30 8'-48 we multiply 21 by .48 (Art. 4). The product 10.08 added to log sin 30 8', since log sin 30 9' is greater than log sin 30 8', gives log sin 30 8'.48 = 9.70082 (Art. i). Similarly, log tan 30 8'.48 = 9.76391, log cot 30 8'-48 = 0.23609, log cos 30 8'.48 = 9.93691, the correction being subtracted in the last two cases, since the cotangent and the cosine decrease as the angle increases. 392 EXPLANATION OF THE TABLES. 29. The general rule is : Find the function corresponding to the given degrees and minutes; multiply the tabular difference by the decimals of a minute ; add the product to the function corresponding to the given degrees and minutes when finding the logarithmic sine or tangent, and subtract it when finding the logarithmic cosine or cotangent. The tabular differences are given in the columns headed d. and c. d., the latter containing the common difference for the L. Tan and L. Cot columns. The difference to be used is that between the functions cor- responding to the two angles between which the given angle lies. For 30 39'.38 : log sin = 9.70747 ; log cos = 9.93462 ; log tan = 9.77285 ; log cot 0.22715. For 59 43'-46 : log sin = 9.93632 ; log cos = 9.7025 7 ; log tan = 0.233 75 ; log cot = 9.76625. 30. When the angle is given to seconds, the correction may be found by multiplying the tabular difference by the number of seconds, and dividing the product by 60. To find the Acute Angle corresponding to a Given Logarithmic Function. 31. The column headed L. Sin is marked L. Cos at the bottom, while that headed Z. Cos is marked L. Sin at the bottom ; hence, if a logarith- mic sine or cosine were given, we should expect to find it in one of these two columns. Similarly, we should expect to find a given logarithmic tangent or cotangent in one of the two columns headed Z. Tan and Z. Cot. 32. When the function can be found in the tables. If a logarithmic sine is given, find it in one of the two columns marked Z. Sin and L. Cos ; if found in the column headed Z. Sin, the degrees are taken from the top, and the minutes from the left of the page ; if in the column headed Z. Cos but marked Z. Sin at the bottom, the degrees are taken from the bottom, and the minutes from the right of the page. Thus, 9. 701 15 = log sin 30 10'; 9.9345 7 = log sin 59 20'; 9.93 724 = log cos 30 4'; 9.70590 = log cos 59 28'; 9. 76406 = log tan 30 9'; 0.23130 = log tan 59 35'; 0.23420 = log cot 30 15'; 9.76870 = log cot 59 35'. 33. When the function can not be found in the tables. If we wish to find the angle whose logarithmic sine is 9.70170, we see on page 450 that the given logarithmic sine lies between 9.70159 and 9.70180, and EXPLANATION OF THE TABLES. 393 hence the angle is between 30 12' and 30 13'. The given logarithmic sine differs from log sin 30 12' by 11, and this difference, divided by the tabular difference 21, gives .52+ as the decimal of a minute by which the angle exceeds 30 12'. Hence 9.70170 = log sin 30 i2'-52, which we call 30 12'.$, since the incompleteness of the tables (Art. i) makes the hundredths of a minute uncertain. 34. The rule is : For a logarithmic sine or tangent find the degrees and minutes corresponding to the function in the tables next less than the given function ; divide the difference between the given function and the one next less by the tabular difference ; and the quotient will be the decimal of a minute to be added to the degrees and minutes already found. For a logarithmic cosine or cotangent find the degrees and min- utes corresponding to the function next greater than the given function, since tJte cosine and cotangent decrease as the angle increases, and divide the difference between the given function and the one next greater by the tabular difference, to find the decimal of a minute. The tabular difference is the difference between the two functions in the tables, between which the given function lies. 9.70000 = log sin 30 4 '.7; 9. 93500 = log sin 59 25'.;; 9.93400 = log cos 30 4y'.6 ; 9.70500 = log cos 59 ^'.2 9.77000 log tan 30 29 '.5 ; 0.23200 = log tan 59 3 7 '.4 ; 0.23300 = log cot 30 19'.! ; 9.76400 = log cot 59 51 '.2. Angles Near o or 90. 35. The assumption that the variations in the functions are propor- tional to the variations in the angles if the latter are less than i ' fails when the angle is small, shown by the rapid changes in the tabular differences on pages 420, 421, and 422. 36. The quantities S' and T' which are used in this case are defined by the equations , , sin a' 7"=log tan /'-, a where ' is the number of minutes in the angle. Their values from o to i 40' (=100') are given at the bottom of pages 397-415 ; from i4o' to 3 20' at the left margin of pages 398 and 399, the first three figures being found at the top ; and from 3 to 5 on page 418. Thus, for i'= i' (page 399), S' = 6.46 373, T' = 6. 46 373; for 15'= 15' (page 399), ' = 6.46372, T' = 6.46 373; for 2 40'= T 60' (page 399), S' =6.46 35 7, T' = 6.46 404; for 4 20' = 260' (page 418), ' = 6.46331, 7" = 6.46 456. Each of these numbers should have 10 written after it (Art. n). 394 EXPLANATION OF THE TABLES. NOTE. The logarithmic cosine of a small angle is found by the ordinary method. The cotangent of an angle is the reciprocal of the tangent, and hence the logarithmic cotangent is the arithmetical complement of the logarithmic tangent. The formulas for finding the logarithmic cosine, tangent, and cotangent of angles near 90 are given on page 419. 37. To find the logarithmic sine or tangent of a small angle. From Art. 36, we have log sin a = S' + log a', log tan = T' + log '. Hence, to find the logarithmic sine or tangent of an angle less than 5, find the value of the S' or T 1 corresponding to the angle, interpolat- ing if necessary, and add it to the logarithm of the number of minutes in the angle. Find log sin o42'.6. Since the angle is nearer 43' than 42', we take S' = 6.46 371 log 42. 6 = 1.62 941 .. log sin o 42 '.6 = 8.09 3 1 2 Find log tan i53'.2. Since the angle is nearer i53 r (= 113') than 114', we take T' = 6.46 388 log 113.2 = 2.05 385 .-. log tan i53'.2 = 8.51 773 NOTE. When the angle is given in seconds, either reduce the seconds to deci- mals of a minute, or use the values of S" and T" given at the bottom of pages 397-417 and on page 418. They are defined by the equations c(/ , sin o , -,,, , tan a S " = log , and T" log , a a where a" is the number of seconds in the angle. Hence log sin o = S" + log a", and log tan a = T" + log a". 38. To find the small angle corresponding to a given logarithmic sine or tangent. From Art. 36, log ' = log sin a S', log a' = log tan T', or log ' = log sin a + cpl .S ', log ' = log tan a -\- cpl T'. When the angle is less than 3, find on pages 420-422 the value of cpl S' (or cpl 7") corresponding to the function, interpolating if neces- sary, and add it to log sin (or log tan ) ; the sum will be the loga- rithm of the number of minutes in the angle. In finding the angle whose logarithmic sine is 8.09006, we see from EXPLANATION OF THE TABLES. 395 the L. Sin column (page 420) that the angle is between o 42' and o 43', and that the value of cpl S' must be either 3.53628 or 3.53629. The given logarithmic sine is nearer that of 42' than that of 43'; hence we take cpl 5' =3.53628 log sin a = 8.09006 log a' = 1.62634 .'. ' 42'. 300. When the angle is between 3 and 5, we may find S' and T' from page 418 after finding the angle approximately from pages 423 and 424. Thus in finding the angle whose logarithmic tangent is 8.77237 we find from page 423 that the angle is between 3 23' and 3 24', being nearer 3 23'. Then on page 418 we have * T'= 6.46423 log tan a = 8.77237 .-. log tan a T' = log '== 2.30814 .-. '= 2O3'-3O = 3 23'. 30. Angles Greater than 90". 39. To find the logarithmic sine, cosine, tangent, or cotangent of an angle greater than 90, subtract from the given angle the largest multi- ple of 90 contained therein. If this multiple is even, find from the tables the logarithmic sine, cosine, tangent, or cotangent of the remain- ing acute angle. If the multiple is odd, the logarithmic cosine, sine, cotangent, or tangent, respectively, of the remaining acute angle will be the function required ; thus, sin 120 = sin (90 + 30) = cos 30. 1. QUADRANT. 11. QUADRANT. III. QUADRANT. IV. QUADRANT. jr = a 90 + 180 + a 270 + a sin x = + sin a + cos a sin a cos a cos x = -f cos a sin a cos a + sin a tan x = + tan a cot a + tan a cot a cot x = + cot a tan a + cot a tan a Or we could find the difference between the angle and 180 or 360, and find from the tables the same function of the remaining acute angle ; thus, cos 300 == cos (360 60) = cos 60, etc. I. QUADRANT. II. QUADRANT. III. QUADRANT. IV. QUADRANT. X = <* 180 -a i&J'+a 360 -a or - a sin x = + sin a + sin a sin a sin a COS J" = + cos a cos a cos a -f cos a tan x = + tan a tan a + tan a tan a cot x = + cot a - cot a + cot a cot a To indicate that the trigonometric function is negative, n is written after its logarithm. 396 EXPLANATION OF THE TABLES. 40. To find the angle corresponding to a given function, find the acute angle a corresponding thereto, and the required angle will be a, i8o, or 360 a, according to the quadrant in which the angle should be placed. 41. There are always two angles less than 360 corresponding to any given function. Hence there will be ambiguity in the result unless some condition is known that will fix the angle ; thus, if the sine is positive, the angle may be in either of the first two quadrants, but if we also know that the cosine is negative, the angle must be in the second quadrant. Given One Function of an Angle, to find Another without finding the Angle. 42. Suppose log tan = 9.79361, and log cos is sought. On page 451 the tabular difference for log tan a is 28, and that for log cos a is 8, the given logarithmic tangent exceeding 9.79354 by 7. Hence 28:7 = 8:.*; /. x = 2 = correction to 9.92905, giving log cos = 9.92903. In the margin are tables to facilitate the process. In the column headed ^ 8 g, the numerator is the tabular difference for the logarithmic cosines, and the denominator that for the logarithmic tangents 1 . The correction for the logarithmic cosine will be o when the given logarith- mic tangent exceeds the next smaller logarithmic tangent, found in the tables, by less than 1.8, i for an excess between 1.8 and 5.2, 5 for an excess between 15.8 and 19.2, etc. In the example above, the excess was 7, which is between 5.2 and 8.8, so that the correction is 2. For example, if we have given the logarithms of the sides of a right-angled triangle, log # = 2. 98227 and log b = 2.90255, to find the hypotenuse, we use the formulas tan = 2, and; = -- = --. b sin cos a The value of log tan ft. being found in log a = 2.98227 (i) the column marked L. Tan at the bot- .-. log sin =9.88571 (4) torn, the right column will contain the log = 2.90255 (2) logarithmic sine of the corresponding .-. log tan = 0.07972 (3) angle. Also, the correction to 9.88563 .-. log<: = 3.09656 (5) is 20 x \\, which we find to be 8 from the table headed $. 1 For angles < 45. TABLE XV. COMMON LOGARITHMS OF NUMBERS FROM I TO IIOOO. Log. Log. N. Log, N, Log. N, Log. 17 18 '9 2O 0.30 103 0.47 712 0.60 206 0.69 897 0.77815 0.84510 0.90 309 0.95 424 1.04139 1.07 918 1.11394 .14613 .17609 .20412 23 045 2 5 S 2 ? .27 875 1.30 103 2O 21 22 23 24 25 26 27 28 29 3O 32 33 34 35 36 37 38 39 4O 3 103 1.32222 1.34242 I-36 173 1.38021 1-39794 1.41 497 1-43 136 1.44716 i .46 240 .49 136 1.51851 1.53 148 1.54407 I-55630 1.56820 I-57978 1.59 106 i. 60 206 4O 49 50 5' 52 53 54 i .60 206 6O 1.61 278 1.62325 I-63347 -64 345 .65 321 .66 276 .67 210 .68 124 .69 020 .69 897 70757 .71 600 73 239 74036 .74819 75 587 76 343 -77085 1-77815 6O 61 62 63 64 79 8O [.77815 I-78533 1.79239 J-79934 1.80618 1.81 291 1.81 954 1.82 607 1.83251 1.83885 8O 81 82 83 84 87 88 89 90 1.90309 IOO 1.84510 1.85 126 1-85 733 1.86332 1.86923 1.87506 1. 88 08 1 1.88649 1.89209 1.89 763 90309 1.90849 1.91 381 1.91 908 1.92428 1.92942 1.93450 1-93952 1.94448 1-94939 1.95424 1.95904 1.96379 1.96848 I.973I3 1.97772 1.98 227 1.98677 1.99123 1.99564 S'. 6.46 373 373 T'. 373 373 o o' = o" o i = 60 O 2 = 120 S". T". 4-68 557 557 557 557 557 557 (397) 398 S'. T'. N. L. 1 2 3 4 5 6 7 8 9 P, P. 366 385 100 ooooo 043 087 130 '73 217 260 303 346 389 AA AQ AO 366 1 385 101 432 475 518 561 604 647 689 732 775 817 366 366 3^5 3 86 102 103 860 oi 284 945 368 988 410 "030 452 *0 7 2 494 *SsI *i57 578 *i 99 620 *242 662 2 8.8 8.6 8.4 u '366 366 386 386 386 I0 4 I0 5 1 06 703 02 119 745 1 60 572 787 202 612 828 243 653 870 284 694 912 735 953 366 776 991 407 816 *o 3 6 449 857 *o 7 8 49P 898 4 17.6 17.2 16.8 5 22.O 21.5 2I.O 6 26.425.825.2 366 387 107 938 979 *oi9 *o6o *IOO *i4i *i8i *222 *262 *302 730.830.1 29.4 365 387 1 08 03342 383 42.3 463 503 S43 583 623 66^ 703 835.234.433.6 ,365 387 109 743 782 822 862 902 941 981 *O2I *o6o *IOO 9!39-6 38.7 37-8 365 387 no 04139 179 218 258 297 336 3/0 415 454 493 365 ! 388 in 532 57i 610 650 689 727 766 8o S 844 883 365 388 112 922 961 999 *07S *077 *ii5 "192 *27I *2faq 1 4.1 4.0 3.9 365 388 "3 05308 346 385 423 461 538 576 614 6 S 2 2 8.2 8.0 7.8 365 365 364 389 339 389 114 "5 116 690 06070 446 483 767 145 521 805 183 558 843 221 595 881 258 633 918 296 670 956 333 77 994 37 1 744 408 7 8l 3 11.3 i^.u 11.7 4 16.4 16.0 15.6 5120.5 2O.O 19.5 6 24.6 24.0 23.4 364 389 117 819 856 893 930 967 *oo4 *04i *o;8 *ii5 *I5I 7 28.7 28.0 27.3 364 390 118 07188 225- 262 298 33 S 372 408 44 S 482 51* 8(32.832.031.2 364 390 119 55J. 591 628 664 700 737 773 809 846 882 9:36.936.035.1 364 39 I2O 918 954 990 '171 '243 364 391 121 08279 3H 3 So 386 422 4S8 493 S29 565 600 38 37 36 ! 363 39 1 122 636 672 707 743 778 814 849 884 920 955 i 3- 3-7 3-o 363 39 1 123 991 *026 *o6i *o 9 6 *i6 7 *202 *237 *2 7 2 307 2 7-6 7-4 7-2 363 363 363 39 1 39 2 392 124 125 126 09342 691 10037 377 726 072 412 760 106 447 795 140 482 830 209 55 2 243 587 934 278 621 968 312 656 346 3 4 i 11.4 1 1. 1 10.8 15.2 14.8 14.4 19.0 18.5 18.0 363 363 362 392 393 393 I2 7 128 I2 9 380 721 11059 415 755 093 449 789 126 483 823 160 857 193 551 890 227 585 924 26l 619 958 294 653 992 327 687 *36? I 9 26.6 25.9 25.2 30.4 29.6 28.8 74-2 77.7 72.4 362 393 ISO 394 428 461 494 528 561 594 628 66 1 694 362 S94 131 727 760 793 826 860 893 926 9S9 992 *O24 35 34 33 362 362 394 394 132 133 12057 385 123 45 156 483 189 516 222 548 $ 287 613 646 352 678 I 3-5 3-4 3-3 7.0 6.8 6.6 362 '361 395 395 134 710 13033 III 775 098 808 170 840 162 8 7 2 194 226 937 2 S 8 969 290 *OOI 322 3 4 10.5 10.2 9.9 14.013.613.2 395 136 354 386 418 450 481 513 545 577 609 640 5 17.5 17.0 16.5 '361 396 361 396 361 396 137 138 139 672 988 14301 * 7 4 333 * 735 767 *082 395 799 *u 4 426 * 83 - 457 862 *i 7 6 489 520 925 *2 39 551 956 *2 7 582 I 9 24.5 23.8 23.1 28.0 27.2 26.4 71. C 7Q.6 2Q.7 361 360 397 140 141 613 644 if ^06 *OI 4 737 768 "076 799 *io6 829 860 8 9 I 32 31 30 397 922 953 *45 *i37 *i68 *I98 360 | 397 142 15229 259 290 320 35' 381 412 442 473 S03 i 3.2 3-1 3.0 360 3y8 534 564 594 625 685 715 746 776 806 2 6.4 6.2 6.0 36o )98 144 836 866 897 927 9S7 987 *oi7 *047 *io7 3 9.6 9.3 9.0 360 398 16137 167 197 227 256 286 716 346 376 406 4 12.8 12.4 12.0 360 399 146 435 465 495 524 554 584 613 643 673 702 5 1 6.0 15.5 15.0 359 1359 J359 399 399 .100 4s 149 ISO 732 17026 761 056 348 791 085 377 820 114 406 696 850 143 435 879 '73 464 909 202 493 938 231 -iff 967 260 55 1 997 289 580 6 19.2 15.0 is.o 7 22.4 21.7 21.0 8:25.624.824.0 9 28.8 27.9 27.0 | 359 4o 609 638 667 725 754 782 840 869 N. L. 1 | 2 3 4 5678 9 P. P. S.' T.' S." T." S." T." i' 6.46373 373 o i'= 60" 4-68557 557 o 19'= 1 140" 4-68557 558 2 373 373 o 2 = 120 557 557 o 20=1200 557 558. 10 373 373 o 3 = 180 557 557 o 21 =1260 557 558 13 373 373 o 16 = 960 557 558 o 22-1320 557 558 '4 372 373 o 17=1020 557 558 o 23=1380 557 558 15 372 373 o 18=1080 557 558 o 24=1440 557 558 o 19=1140 557 558 o 25 =1500 557 558 399 ! s/ 6. 359 359 358 358 358 358 358 358 357 357 357 357 357 356 356 356 356 356 355 355 T'. 4 6 401 4 o 401 409 402 403 4<>3 404 N. L. 1 23 4 5 6 7 8 9 P.P. ISO '5 1 '53 154 |55 157 158 159 I6O 161 162 163 164 169 170 171 172 173 174 '75 176 177 178 179 180 181 182 183 184 185 1 86 187 1 88 189 190 191 192 193 194 199 200 17609 898 18 184 469 752 19033 312 590 866 20 140 412" 638 667 696 725 754 *0 4 I 327 611 893 173 45 ! * 728 276 782 81 840 869 i 2 3 4 7 s 9 i 2 3 4 5 6 8 9 i 2 3 4 1 9 i 2 3 4 1 7 8 9 2S 2 II 14 17 20 23 26 27 2 IO 13 16 18 21 24 2 3 4 5 6 i 24 2 4 7 9 12 14 1 6 19 21 22 2 4 6 8 ii 13 '5 17 19 28 9 2.8 8 5.6 7 8.4 6 II. 2 5 HO 4 16.8 3 19.6 2 22.4 I 25.2 26 7 2.6 4 5-2 i 7.8 8 10.4 5 13- 2 15.6 9 1 8.2 6 20.8 3 23.4 25 2-5 5-o 7-5 IO.O 12.5 15.0 17-5 20.0 22-5 23 4 2.3 8 4.6 2 6.9 6 9.2 o 11.5 4 13-8 8 16.1 2 18.4 6 20.7 21 2 2.1 4 4-2 6 6.3 8 8.4 o 10.5 2 12.6 4 H-7 6 1 6.8 8 18.9 926 213 498 780 06 1 340 618 893 167 439 955 241 526 808 089 368 645 921 194 984 270 554 837 117 396 673 948 222 "013 298 583 865 H5 424 700 976 249 "070 355 639 921 20 1 479 303 384 667 949 229 57 *o 8 8 330 602 -127 412 696 977 257 535 811 *o85 358 "156 441 724 *oo 5 I 5 562 838 *II2 385 404 466 493 520 548 575 629 656 *I92 458 722 246 505 J 6 3 404 405 405 4 u6 406 406 407 407 408 408 683 952 21 219 484 748 22 01 1 272 531 789 710 978 245 5 11 775 037 298 si 737 *00 5 272 537 80 1 063 324 583 840 763 *0 3 2 299 564 827 089 350 608 866 *o 9 325 590 854 "5 376 634 891 *o8 7 35 2 617 880 141 401 660 917 844 *II2 378 643 906 167 427 686 943 871 *i 39 405 669 932 194 453 712 968 898 *i6 5 696 958 220 479 737 994 355 23045 070 096 350 603 855 105 353 601 846 091 334 575 121 M7 172 198 223 249 274 355 354 354 354 354 354 353 353 353 408 409 4<>y 410 4 1 4 1 300 553 805 24055 304 797 25042 285 325 578 830 080 329 576 822 066 310 376 629 880 I 3 378 625 8 7 I 115 3S8 401 654 905 155 403 650 895 139 382 624 426 679 930 1 80 428 674 920 164 406 452 704 955 204 452 699 944 1 88 477 729 980 229 477 724 969 212 455 502 754 *oo3 254 502 748 993 237 479 528 779 "030 279 527 773 *oi8 261 J03 744 353 4 1 527- 768 26007 245 482 717 95' 27184 416 646 600 648 672 696 720 353 352 352 352 S !35 35 350 350 350 349 349 349 349 348 4 1 41 41 4i 4 1 4 1 792 031 269 505 975 207 439 669 816 055 293 529 764 998 231 462 692 840 079 ii *02I 254 485 715 864 102 340 576 811 277 508 738 888 126 364 600 834 *o68 300 761 912 150 387 623 858 *0 9 I 323 554 784 935 411 647 88 1 *U4 346 577 Jo7 959 198 435 670 ^905 370 600 830 983 221 458 6 94 928 *i6i 393 623 852 *o8i 4 1 418 418 419 419 420 420 421 421 875 28 103 330 SS 6 780 29003 226 447 667 885 898 921 944 623 847 070 292 513 732 967 989 *OI2 126 803 026 248 469 688 907 149 375 601 825 048 270 491 710 929 194 421 646 870 092 3H 535 754 973 217 443 668 892 "5 336 557 776 994 2 4 4 66 6 9 I 914 '37 358 579 798 *oi6 262 488 7'3 937 601 820 "038 285 5" 735 959 181 403 623 *o6o 307 533 758 981 203 425 645 863 *o8i 348 439 30103 '23 146 1 68 190 211 233 255 276 298 N. L. 1 2 i 3 4 5678 9 | P.P. S.' T.' i' 6.46 373 373 2 373 373 o 2' = o 3 = S." T." 120" 4-68557 557 '80 557 557 240 557 SS 8 S." T." o 28'= 1680" 4-68557 558 o 29=1740 557 559 o 30=1800 557 559 15 372 373 20 372 373 .} o 25 =1500 557 558 o 26=1560 557 558 o 27=1620 557 558 o 28=1680 557 558 o 31 -1860 557 559 o 32 =1920 557 559 o 33 =1980 . 557 559 o 34=2040 557 550 400 N. L. I 2 3 4 5 6 7 8 9 P. P. 200 20 1 202 203 204 205 206 208 20 9 210 211 212 213 214 215 216 2I 7 218 219 220 221 222 22 3 224 225 226 22 7 228 229 230 231 232 233 234 235 236 237 238 239 24O 241 242 243 244 245 246 247 248 249 250 30103 125 146 1 68 190 211 233 253 8| 899 *II2 323 534 744 952 160 276 298 i 2 3 4 5 i 6 i 7 i 8 i 9 i i 2 3 4 5 6 I 9 i 2 3 4 i 1 9 i 2 3 4 I 7 8 9 i 2 3 4 I 9 22 21 2.2 2.1 4-4 4-2 6.6 6.3 8.8 8.4 i.o 10.5 3.2 12.6 5-4 14-7 7.6 16.8 9.8 18.9 20 2.0 4.0 6.0 8.0 1 0.0 12.0 14.0 1 6.0 1 8.0 19 i-9 3-8 \l 9-5 11.4 13-3 J 5- 2 17.1 18 1.8 3-6 5-4 7-2 9-o 10.8 12.6 14.4 16.2 17 i-7 3-4 8 8. 5 10.2 II.9 I 3 .6 15-3 320 535 75 963 3i 175 387 597 806 32015 341 557 771 984 197 408 618 827 035 363 57 792 *oo6 218 429 639 848 056 34 600 814 *02 7 239 45 660 86 9 077 406 621 835 *o 4 8 260 47' 681 890 098 428 643 856 *o69 281 492 702 911 118 449 664 878 *09i 302 513 723 93i 139 492 707 920 *i33 343 555 763 973 181 54 728 942 *i54 366 576 785 994 201 222 ^28" 634 8 3 8 33041 244 445 646 846 34044 243 263 284 305 325 346 366 387 408 449 654 858 062 264 465 666 866 064 469 675 879 082 284 486 686 885 084 490 695 899 I O2 304 5 06 7 06 905 104 5 10 715 919 122 325 526 726 925 I2 4 53 I 736 940 H3 34 I 546 746 945 143 552 756 960 163 365 566 766 965 163 572 777 980 183 385 586 786 983 183 593 797 *OOI 203 405 606 806 "005 203 613 818 *02I 224 425 626 826 *02? 223 242 439 635 830 35 2 3 218 411 603 793 984 36i73 262 282 3 OI 321 34i 36i 380 400 420 459 655 850 044 238 43 622 813 *oo3 479 674 869 064 257 449 641 832 *02I 498 6 94 889 08 3 276 468 660 5' "040 518 7'3 908 102 295 4 88 679 870 *059 537 733 928 122 315 507 698 88 9 *o 7 8 557 753 947 141 334 526 717 908 *097 577 772 967 160 353 545 736 927 *n6 596 792 986 180 37 2 564 755 946 *i35 616 811 -005 199 774 965 *'54 192 211 229 248 267 286 303 324 342 361 549 736 922 37 I0 7 291 475 658 840 3802T 380 568 754 940 125 310 493 676 858 399 586 773 959 % % 876 418 605 791 977 162 346 S3o 712 894 436 624 810 996 181 365 548 73i 912 455 642 829 "-014 199 383 566 749 93i 474 66 1 847 *o 3 3 218 401 583 767 949 493 680 866 *0 5 I 236 420 603 785 967 5 11 698 884 "070 254 438 621 803 985 1 66 530 717 903 *o88 273 457 639 822 *oo3 039 57 75 093 112 130 148 184 202 382 56i 739 917 39094 270 445 620 794 220 399 578 757 934 in 287 463 637 238 417 596 775 952 129 305 480 655 256 435 614 792 970 146 322 498 672 274 453 632 810 987 164 340 5'5 690 2 9 2 471 650 828 *oo5 182 358 533 707 310 489 668 846 *023 199 375 550 724 328 57 686 863 *0 4 I 217 393 568 742 346 523 703 88 1 "058 233 410 585 759 364 543 721 899 "076 252 428 602 777 811 829 846 863 881 898 915 933 95 N. L. 1 234 5 6 7 8 9 P.P. S.' T.' 2' 6.46373 373 3 373 373 S." T." 3'= 180" 4-68557 557 4= 240 557 558 5 = 3oo 557 558 . o . o S." T." 56' = 2160" 4.68 557 559 57 = 2220 557 559 58 = 2280 557 559 59 = 2340 557 559 jo = 2400 557 559 \i = 2460 556 560 12 = 2520 556 560 20 372 373 25 372 373 o o 33 = 1980 557 559 o 34 = 2040 557 559 o 35 = 2100 557 559 o 36 = 2160 557 559 o o < o < o 401 N. L. 1 234 5 6 7 8 9 P.P. 25O 251 252 253 254 2 55 256 257 258 259 26O 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 2 77 278 279 280 281 282 283 284 285 286 287 288 289 29O 291 292 293 294 295 296 297 298 299 300 39794 811 829 846 863 88 1 898 9'5 933 95 i 2 3 4 I 9 2 3 4 5 6 I 9 i 2 3 4 I 7 8 9 i 2 3 4 5 6 1 9 2 3 4 9 18 1.8 3-6 5-4 7-2 9.0 10 8 12.6 14.4 16.2 17 i-7 3-4 8 8-5 10.2 1 1. 9 13-6 S-3 16 1.6 it 6. 4 8.0 9.6 II. 2 12.8 14.4 15 i-5 3-o tl 7-5 9.0 10.5 12.0 '3-5 14 1.4 2.8 4.2 5-6 7.0 8.4 9.8 II. 2 12.6 967 40 140 312 483 654 824 993 41 162 330_ 497 664 830 996 42 160 325 488 651 813 975 985 157 329 500 671 841 *OIO 179 347 *002 175 346 5^8 688 858 *027 196 363 *oi9 192 3 6 4 535 705 875 *o 44 212 380 *037 209 38i 552 722 892 *o6i 229 397 *54 226 398 569 739 909 *078 246 414 *0 7 I 243 415 586 756 926 *? 263 430 *o88 261 432 603 773 943 *ui 280 447 *io6 278 449 620 790 960 *I28 296 464 "123 295 466 637 807 976 *I45 3i3 481 647 5'4 531 547 504 58i 597 614 631 68 1 847 *OI2 177 341 504 66 7 830 991 697 863 *029 193 357 521 684 846 *oo8 714 880 *45 210 374 537 700 862 *024. & *062 226 390 553 716 878 *04O 747 913 *078 243 406 57 732 894 ""056 764 929 "095 259 423 586 749 911 "072 780 946 *ui 275 439 602 765 927 *o88 797 963 *I27 292 455 619 781 943 "104 814 979 *i 44 308 472 63? 797 959 *I2O 43 136 152 169 329 489 648 807 965 122 2 7 9 436 592 ~34T 505 664 823 981 138 295 45 l 607 201 217 233 249 265 425 584 743 902 "059 217 373 529 685 28l 297 457 616 775 933 44091 248 404 560 3U 473 632 791 949 107 264 420 576 361 521 680 838 996 154 3" 467 623 377 ig 854 *OI2 170 326 483 638 393 553 712 870 *028 185 342 498 654 409 569 727 886 *o 44 20 1 358 5'4 669 441 600 759 917 *75 232 389 545 700 716 73i _747_ 902 056 209 362 818 969 1 20 762 778 793 809 963 117 271 423 576 728 879 *030 1 80 824 840 855 871 45025 179 332 484 637 788 939 46090 886 040 194 347 500 652 803 954 105 917 071 22? 378 53 -4682 834 984 135 932 086 240 393 545 697 849 *000 150 948 102 255 4 08 561 7 I2 864 "015 165 979 *33 286 439 59i 743 894 *45 '95 994 148 301 454 606 758 909 *o6o 210 *OIO 163 317 469 621 773 924 *75 225 240 255 270 285 300 315 330 3.45 359 374 389 538 687 835 982 47 I2 9 276 422 j>67_ 712 404 553 702 830 997 144 290 436 582 419 568 716 864 *OI2 159 305 451 596 434 583 731 879 *026 173 319 465 611 449 598 746 894 *0 4 I 1 88 334 480 625 464 613 761 909 "056 202 349 494 640 479 627 776 923 *O70 217 363 59 654 494 642 790 938 "085 232 378 524 669 509 657 805 953 *IOO 246 392 538 683 523 672 820 967 *M 4 261 407 13 727 74 I 756 77 784 799 813 828 842 N, L. 1234 5 6 7 8 9 P. P. S.' T.' 2' 6.46373 373 3 373 373 S." T." o 4'= 240" 4.68 557 558 o 5 = 3o 557 558 o S." T." tS'=-27oo 4-68556 560 $6 = 2760 556 560 17 = 2820 556 560 j8 = 2880 556 560 W = 2940 556 560 jo = 3000 556 561 25 372 373 26 372 373 27 372 374 30 372 374 o 41 = 2460 556 560 o 42 = 2520 556 560 o 43 = 2580 556 560 o 44 = 2640 556 560 o 45 = 2700 556 560 o R'M'D SURV. 26 402 N. L. 1 234 6 7 9 P.P. 300 301 302 303 304 305 306 307 308 309 310 3" 312 313 3H 315 316 317 3i8 319 32O 321 322 323 324 9 % 329 330 33i 332 333 334 335 336 337 338 339 340 34i 342 343 344 345 346 347 348 349 350 47712 727 741 756 770 914 058 202 344 487 629 770 911 *052 784 799 8i3 828 842 i 2 3 4 I I 9 i 2 3 4 9 2 3 4 7 8 9 i 2 3 4 5 6 9 15 !-5 3<> 4-5 6.0 7-5 9.0 10.5 12.0 '3-5 14 1-4 2.8 4.2 5-6 7.0 8.4 9.8 II. 2 12.6 13 i-3 2.6 3-9 6: 5 2 7 .8 9-i 10.4 11.7 12 1.2 2.4 3-6 4 .8 6.0 7.2 8.4 9.6 10.8 857 48001 144 287 430 572 7 J 4 8 55 996 871 015 59 302 444 586 728 869 *OIO 885 029 173 316 4 5 8 601 742 883 *024 900 044 187 330 473 6i5 756 897 *o 3 8 929 073 216 359 501 643 783 920 *o66 943 087 230 373 5'5 657 799 940 *o8o 958 101 244 387 530 671 813 954 *094 234 972 116 259 401 6-81 827 968 *io8 986 130 273 416 558 700 841 982 *I22 49136 !5<> 164 178 192 206 220 248 262 276 415 554 693 831 969 50 106 243 379 290 429 568 707 845 982 120 2 5 6 393 304 443 582 721 8 5 9 996 133 270 406 3i8 457 596 734 872 *OIO 47 284 420 332 47 1 610 748 886 *O24 161 297 433 346 485 624 762 900 *37 174 3" 447 360 499 638 776 914 "051 188 323 461 374 5'3 651 790 927 -065 202 338 474 388 SI 803 941 *079 215 $ 402 679 817 * 955 *092 229 365 5I 5^5 529 542 556 569 583 596 610 623 637 651 786 920 5'53 1 88 322 455 587 720 85J_ 983 52114 244 375 54 634 763 892 53020 664 799 934 068 202 335 468 601 733 678 8i3 947 081 215 348 481 614 746 691 826 961 093 228 362 495 627 759 703 840 974 108 242 375 508 640 772 718 853 987 121 IS 521 654 786 732 866 *OOI 133 268 402 534 667 799 & "014 148 282 4i3 548 680 812 759 893 *028 162 295 428 56i 693 825 772 907 *04i 175 308 441 574 706 838 86? 878 891 904 917 93 943 957 970 996 127 257 388 517 647 776 905 033 *oo9 140 270 401 530 660 789 917 046 *022 '53 284 414 543 673 802 930 058 "035 1 66 297 427 556 686 8i3 943 071 *048 1 79 3 IO 440 569 6 99 827 95 6 084 *o6i 192 323 453 582 711 840 969 097 *073 205 336 466 595 724 853 982 no *o88 218 349 479 608 737 866 994 122 *IOI $ 492 621 75 879 *00 7 i35 148 161 173 186 199 212 224 237 230 263 275 403 529 656 782 908 54033 158 283 288 415 542 668 794 920 045 170 295 301 428 555 68 1 807 933 058 183 307 3H 441 567 694 820 945 070 *95 320 326 453 580 706 832 958 083 208 332 339 466 593 719 845 970 095 220 343 352 479 605 732 857 983 108 233 357 364 491 618 744 870 995 1 20 245 370 377 54 631 757 882 *oo8 133 258 382 390 5i7 643 769 895 *O20 '45 270 394 407 419 432 444 456 469 481 494 506 5i8 N. L. 1 2 3 4 5 6 7 8 9 P. P. S.' T.' 3' 646373 373 4 373 373 S." T." o 5'= 300" 4-68557 558 o 6 = 360 557 558 o o S." T." 54' =3240" 4-68556 561 55 = 3300 556 561 56 = 3360 556 561 57 = 3420 555 561 58 = 3480 555 562 59 = 3540 555 562 30 372 374 35 372 374 o 50 = 3000 556 561 051= 3060 556 561 o 52 = 3120 556 561 o 53 = 3180 556 561 o 54 = 3240 55 6 5 6 ' o o 403 N. | L. | 1 2 3 4 5 6 7 i 8 9 P.P. 35O 351 35 2 353 354 355 356 357 358 359 36O 36i 362 363 364 3 ^I 366 367 368 369 370 37i 372 373 374 375 376 377 378 379 38O 38i 382 383 384 385 386 387 388 389 390 39i 392 393 394 395 396 397 398 399 400 54407 419 432 444 456 469 481 494 506 5'8 i 2 3 4 1 I 9 i 2 3 4 5 6 I 9 i 2 3 4 9 2 3 4 7 8 9 13 1.3 2.6 3-9 7-8 9.1 10.4 u. 7 12 1.2 2.4 3-6 4.8 6.0 7-2 8.4 9.6 10.8 11 2.2 3-3 4-4 5-5 6.6 88 O.O 9-9 10 I.O 2.0 3-o 4.0 5-o 6.0 7-o 8.0 9.0 lli 777 900 55 2 3 H5 267 388 509 543 667 790 913 035 157 279 400 522 555 679 802 925 047 169 291 4'3 534 568 69 1 814 937 060 182 303 425 546 580 704 827 949 072 194 3*5 437 558 593 716 839 962 084 206 328 449 57 605 728 851 974 096 218 340 461 582 617 74i 864 986 1 08 230 352 473 594 6^0 753 8', 6 998 121 2 4 2 364 485 606 6^2 765 888 *OII 133 ^55 376 497 618 630 642 654 666 678 691 703 715 727 739 75i 871 991 56 no 229 348 n 703 763 883 *00 3 122 241 3 60 47 8 597 7H 775 95 "015 134 253 372 49 608 726 787 907 ="027 146 263 384 502 620 738 799 919 ""038 158 277 396 5H 632 750 811 93i ="050 170 289 407 526 644 76! 823 943 *O62 182 301 419 538 656 773 835 * 955 *074 194 312 43i 549 667 785 847 9 6 7 *o86 205 324 443 561 679 797 859 979 *098 217 336 455 IS 808 820 832 8 44 855 867 879 891 902 914 926 937 57054 171 287 403 5'9 634 749 864 978 949 066 183 299 415 530 646 761 875 961 078 194 310 426 542 657 772 887 972 089 206 322 438 553 669 784 898 984 101 217 334 449 565 680 795 910 996 "3 229 345 461 576 692 807 921 *oo8 124 241 357 473 588 73 818 933 *oi9 136 252 368 484 600 7'5 830 944 *O3i 148 264 3 8o 496 611 726 841 955 *043 *59 276 392 57 623 738 852 967 990 *OOI "013 *02 4 *35 *047 ="058 *O7O *o8i 58092 206 320 433 546 659 771 883 995 104 218 331 444 557 670 782 894 *oo6 "5 229 343 456 569 68 1 794 906 "017 127 240 354 467 580 692 805 917 *028 138 252 365 478 591 704 816 928 *040 149 263 377 490 602 7'5 827 939 *0$I 161 % 726 838 95 *062 172 286 399 5 12 62? 737 850 961 *0 7 3 184 297 410 524 636 749 861 973 *o84 195 309 422 535 647 760 872 984 *095 59106 118 129 140 I 5 l 162 273 384 494 6o ? 71? 824 934 *043 152 173 184 195 207 218 329 439 550 660 77 879 988 60097 229 340 450 561 671 780 890 999 1 08 240 $ 57 2 682 791 901 *OIO 119 % 472 583 693 802 912 *02I I 3 262 373 483 594 704 813 923 *0 3 2 I 4 I 284 395 506 616 726 835 945 *54 163 295 406 5*7 627 737 846 956 *o6J 173 306 417 528 638 748 857 966 "076 184 3i8 428 539 649 759 868 977 *o86 1 9S 206 217 228 239 249 260 271 282 293 304 N, L. 1 2 3 4 5 6 7 8 9 P. P. S.' T.' 3' 6.46 373 373 4 373 373 S." T." 5'= 300" 4.68 557 558 6= 360 557 558 7 = 420 557 558 S." T." i'=366o" 4-68555 562 2 = 3720 555 562 3 = 378o 555 562 4 = 3840 555 563 5 = 3900 555 563 6 = 3960 555 563 7 = 4020 555 563 35 372 374 39 372 374 40 372 375 o 58 = 3480 555 562 o 59 = 3540 555 5 62 i o = 3600 555 562 i i = 3660 555 562 404 N. L. 1 2 3 4 5 6 7 8 9 P. P. 400 401 402 403 404 405 407 408 409 410 411 412 413 414 415 416 417 418 419 42O 421 422 423 424 425 426 427 428 429 43O 43i 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 60 206 ^I4~ 423 531 638 746 853 959 61066 172 217 325 433 54i 649 756 863 970 077 183 228 239 249 260 271 379 487 595 73 810 917 *02 3 130 236 282 293 401 509 617 724 831 938 *o 4 5 'Si 257 304 i 2 3 4 5 6 9 i 2 3 4 I 9 i 2 3 4 1 9 11 i.i 2.2 3-3 4-4 9.9 10 I.O 2.0 3-o 4.0 5.0 6.0 7-o 8.0 9.0 9 0.9 1.8 2-7 3-6 4-5 5-4 6-3 7.2 8.1 336 444 55 2 660 767 874 981 087 194 347 455 563 670 778 8S 5 991 098 204 358 466 574 68 1 788 895 *002 I0 9 215 369 477 584 692 799 906 *oi3 119 225 390, 498 606 7 J 3 821 927 *o 34 140 247 412 520 627 735 842 949 *55 162 268 278 289 300 310 321 33i 342 352 363 .574 384 490 595 700 80? 909 62014 118 221 395 500 606 711 815 920 024 128 232 405 5" 616 721 826 93 034 138 242 416 521 627 $ 941 045 149 252 356 4 26 532 637 742 847 95 055 159 263 437 542 648 752 857 962 066 170 273 448 553 658 763 868 972 076 180 284 458 Ig m 982 086 i 9 o 294 469 574 679 784 888 993 097 20 1 304 479 584 690 794 899 *00 3 107 211 3'5 32JL 428 g 737 839 941 63043 144 246 335 439 542 644 747 849 95 053 155 256 346 366 377 387 397 408 418 449 55 2 653 757 859 961 063 165 266 459 562 665 767 870 972 073 '75 276 469 572 675 77 8 880 9 82 1 480 583 685 788 890 992 094 1 95 296 490 593 696 798 900 *OO2 I0 4 205 306 407 500 603 706 808 910 *OI2 114 215 3'7 I U 613 716 818 921 *022 I2 4 22 5 327 521 624 726 829 931 *033 '34 236 337 347 357 367 377 387 397 417 428 438 448 548 649 749 849 949 64048 47 246_ 345 458 558 659 759 859 959 058 '57 256 468 568 669 769 869 969 068 167 266 478 579 679 779 879 979 078 177 276 488 589 689 789 889 988 088 I8 7 286 498 599 699 799 899 998 098 197 296 5 08 60 9 70 9 80 9 909 *oo8 108 207 306 5 X 8 619 719 819 919 *oi8 118 217 3i6 528 629 729 82 9 929 *028 128 227 326 538 639 739 839 939 "038 137 237 335 355 365 375 385 395 404 414 424 434 444 542 640 738 836 933 65031 128 225_ 3 2I 454 552 650 748 846 943 040 137 234 464 562 660 758 856 953 050 H7 244 473 572 670 768 865 963 060 157 254 483 582 680 777 875 972 070 167 263 493 59 1 689 787 885 982 079 176 273 503 601 699 895 992 089 1 86 283 5'3 709 807 904 *002 99 196 292 523 621 719 816 914 *on 108 205 302 532 631 729 826 924 *02I 118 2i5 312 33i 34i 350 360 369 379 389 398 408 N. L. 1 2 3 4 5 6 7 8 9 P.P. S.' T.' 4' 6.46373 373 5 373 373 S." T." o 6'= 360" 4.68 557 558 o 7 = 420 557 558 o 8= 4 8o 557 558 9'= 41 10 = 42 ii = 42 S." T." 40" 4-68555 563 oo 554 563 60 554 564 20 554 564 80 554 564 40 554 564 oo 554 564 40 372 375 42 372 375 43 37i 375 44 37 1 375 45 37i 375 i 6 = 3960 555 563 i 7=4020 555 563 i 8 = 4080 555 563 i 9 = 4140 555 563 12-43 13 =43 14 =44 15=45 405 N. L. 1 2 3 4 5 6 7 8 9 P.P. 45O 45 * 45 2 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 47O 47i 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 49O 491 492 493 494 495 496 497 498 499 500 65321 33' 34i 35 360 369 379 389 398 408 i 2 1 I 7 8 9 i 2 3 4 7 8 9 2 3 4 5 6 I 9 10 I.O 2.0 3-o 4.0 5.0 6.0 7.0 8.0 9.0 9 0.9 1.8 2.7 3-6 4-5 % K 8 0.8 1.6 2.4 3-2 4.0 4.8 5-6 6.4 7-2 418 5H 610 706 80 1 896 992 66087 181 427 523 619 7'5 8u 906 *OOI 096 191 437 533 629 72? 820 916 *OII 1 06 200 447 543 639 734 830 925 *020 "5 210 45 6 & 744 839 935 *030 124 219 466 562 658 753 849 944 *39 134 229 475 57 1 667 763 858 954 *o 4 9 M3 238 485 f 677 III 963 "058 153 247 495 59i 686 782 877 973 *o68 162 257 504 600 696 792 887 982 *0 77 172 266 3t'i '276 370 464 558 652 745 839 932 67025 117 285 380 474 567 66 1 $ 941 034 127 295 304 3H 323 332 342 35i 389 483 577 671 764 857 95 043 J36 398 492 5 86 680 773 867 960 052 145 408 502 596 689 783 876 969 062 154 417 5" 605 699 792 885 978 071 164 427 S2i 614 708 80 1 894 987 080 173 436 530 624 717 811 904 997 089 182 445 539 633 727 820 913 *oo6 099 191 455 549 642 736 829 922 *oi5_ 1 08 2OI 210 219 228 237 247 256 265 274 284 293 302 394 486 578 669 761 852 943 68034 3" 403 495 587 679 77 86 1 952 Q43 133 321 4U 504 596 688 779 870 961 052 33 422 5'4 605 697 788 879 97 06 1 339 43 1 523 614 706 797 888 979 070 348 440 532 624 Q 8 97 9 88 079 357 449 54i 633 724 8i5 906 997 088 367 459 55 642 733 825 916 *oo6 097 376 468 560 651 742 834 925 01 5 1 06 385 477 569 660 752 843 934 *024 "5 124 215 35 395 485 574 664 753 842 93i 142 I5 1 1 60 i6 9 178 187 196 205 224 3H 404 494 583 673 762 851 940 233 323 4i3 502 592 681 771 860 949 242 332 422 ! 690 780 869 958 251 34i 43' 520 610 699 789 878 966 260 350 440 529 6i 9 7 o8 797 886 975 269 359 449 538 628 717 806 895 984 278 368 458 547 637 726 815 904 993 287 377 467 556 646 735 824 9U *002 296 386 476 l!f 744 833 922 *OII 69 020 028 037 04(1 055 0(14 073 082 090 099 1 08 197 285 373 461 548 636 723 810 897 117 205 294 3fl 469 557 644 732 819 126 214 302 390 478 566 653 740 827 135 223 3" 399 487 574 662 749 836 144 232 320 408 496 583 671 11 152 241 329 417 54 592 679 767 854 161 249 338 425 I' 3 60 1 688 775 862 170 258 346 434 522 609 697 784 871 179 26 7 355 443 Si 705 793 880 1 88 276 364 452 539 627 7H 801 888 906 914 923 932 940 j 949 958 966 975 N. L. 1 | 2 3 4 5 | 6 7 8 9 P. P. S.' T.' 4' 6.46 373 373 5 373 373 o S." T." 7'= 420" 4-68557 558 8 = 480 557 558 9 = 540 557 55 8 i i 8'=46 9=47 S." T." 80" 4-68554 5 6 5 40 554 565 00 554 565 60 553 566 20 553 566 80 553 5 6 6 40 553 566 45 37 1 375 48 37 1 375 49 37' 37 6 50 37' 376 o =48 i =48 2=49 3 =49 14=50 i 15 =4500 554 564 i 16 =4560 554 565 i 17 =4620 554 565 i 18=4680 554 565 2 406 N. L. I 2 3 | 4 5 j 6 7 09 P.P. 500 5i 502 53 504 507 508 59 510 5" 512 5'3 5H 5i5 5i6 5i7 5'8 5i9 52O 52i 522 523 524 525 526 III 529 530 53i 532 533 534 535 536 537 538 539 54O 54i 542 543 544 545 546 547 548 549 550 69897 906 914 923 932 940 949 958 966 *053 140 226 312 398 484 569 655 740 975 i 2 3 4 5 6 9 2 3 4 5 6 I 9 i 2 3 4 I I 9 9 0.9 1.8 3 4-5 5-4 6-3 C 8 0.8 1.6 2.4 3-2 4.0 4.8 5-6 6.4 7-2 7 o-7 1.4 2.1 2.8 3-5 4-2 4-9 5-6 6-3 984 70070 157 243 329 4i'5 5 01 I 86 672 992 079 165 252 338 424 509 595 680 *OOI 088 174 260 346 432 5i8 603 689 *OIO 096 183 269 355 441 526 612 697 *oi8 105 191 278 364 449 535 621 706 *027 114 200 286 372 458 544 629 74 "036 122 209 295 381 467 SS 2 638 723 *044 I3i 217 33 389 475 56i 646 73' *002 I 4 8 234 3 2I 400 492 578 66 3 749 757 766 774 783 791 800 808 817 825 834 842 927 71 012 096 181 265 349 433 M_ 600 851 935 020 105 189 273 357 441 _5^5_ 609 859 944 029 "3 198 282 366 450 533 868 952 037 122 206 290 374 458 542 876 961 046 130 214 299 383 466 55 885 969 054 139 223 307 39i 475 559 642 893 978 06 3 H7 2 3 I 315 399 483 567 902 986 071 155 240 324 408 492 575 910 995 079 164 248 332 416 5oo 584 919 *oo3 088 172 257 34i 4^5 508 592 617 625 634 650 659 667 675 684 767 850 933 72016 099 181 263 346 692 775 858 941 024 107 189 272 354 700 784 867 950 032 "5 198 280 362 709 792 875 958 041 123 206 288 370 717 800 883 966 049 132 214 296 378 Kl 892 975 057 140 222 304 387 734 817 900 983 066 148 230 313 395 742 825 908 991 074 156 239 321 403 750 834 917 999 082 165 247 329 411 759 842 925 *oo8 090 73 255 337 419 428 59 59i 673 754 916 997 73078 I59_ 239 436 444 452 460 469 477 485 493 50i 5i8 599 68 1 762 843 925 *oo6 086 167 526 607 689 770 852 933 *oi4 094 175 534 616 697 779 860 941 *022 102 183 542 624 75 949 "030 in 191 550 632 7'3 795 876 957 *o 3 8 119 199 558 640 722 803 884 965 *O46 127 207 567 648 730 811 892 973 *054 135 2I 5 III 738 819 900 981 *062 143 223 583 665 746 827 908 989 *o-o I5 1 231 312 247 255 26 3 272 280 288 296 304 320 400 480 560 640 719 799 878 957 328 408 488 568 648 727 807 886 965 044 336 416 496 576 656 735 8i5 894 973 344 424 504 584 664 743 823 902 981 352 432 5^2 592 672 75' 830 910 989 360 440 520 600 679 759 838 918 997 076 368 448 528 608 687 767 846 926 *oo5 "084" 376 456 536 616 695 775 8 5 4 933 *Qi3 092 384 464 544 624 703 783 862 941 *020 392 472 552 632 711 791 870 949 *028 107 74036 052 060 068 099 N. L. 1 2 3 | 4 5 6 7 8 9 P, P. S.' TV 5' 646 373 373 6 373 373 S." T." 8'= 480" 4-68557 558 9= 540 557 558 o= 600 557 558 26'= 5 27 = 5 S." T." 160" 4-68 553 567 220 553 567 280 553 567 340 553 5 6 7 400 553 567 460 552 568 520 552 568 5 37 1 37 6 55 37i 376 I 28-5 29 = 5 30 = 5 3i =5 32 = 5 i 23 = 4980 553 566 i 24 = 5040 553 566 i 25 = 5100 553 566 i 26 = 5160 553 567 407 1 N * L. 1 1 2 | 3 4 5 6 7 8 9 P.P. 55O 551 552 553 554 555 556 557 558 559 56O 56i 562 563 564 566 567 568 569 570 57i 572 573 574 575 576 577 578 579 5SO 583 584 585 586 587 588 589 590 59i 592 593 594 595 596 597 598 599 6OO 74036 044 052 060 '39 218 296 374 453 S3' 609 687 764 ooS 07(1 084 092 090 107 2 3 4 1 7 8 9 I 2 3 4 5 6 '.9 8 0.8 1.6 2-4 3-2 4.0 4-8 5-6 6.4 7-2 7 0.7 1.4 2.1 2.8 3-5 4.2 4-9 I' 6 6-3 "5 194 273 35i 429 57 586 663 74i 819 123 202 280 359 437 5S 593 671 749 13* 2IO 288 367 445 523 601 679 757 147 225 304 382 461 539 617 695 772 ^55 233 312 390 468 547 624 702 780 [fa 241 320 398 476 554 632 710 788 170 249 327 406 484 562 640 718 796 178 257 335 414 492 570 648 726 803 1 86 263 343 421 300 578 656 733 811 827 834 842 850 858 865 873 881 889 896 974 75051 128 205 282 358 435 ill &L 66 4 740 815 8 9 i 967 76042 118 193 268 904 981 059 136 213 289 366 442 519 912 989 066 H3 220 297 374 450 526 920 997 074 IS* 228 305 38i 458 534 927 *cos 082 159 236 312 389 465 542 935 *OI2 08 9 1 66 243 320 397 473 549 943 *O20 097 174 2 5 J 328 404 4 8l 557 95 *028 105 182 259 335 412 488 _J65 641 958 *035 "3 189 266 343 420 496 648 966 *043 1 20 197 274 35' 427 504 580 595 603 679 755 831 906 982 057 133 208 283 610 618 626 633 656 671 747 823 899 974 050 J 25 200 275 686 762 838 914 989 065 140 215 290 694 770 846 921 997 072 148 223 298 702 778 853 929 *oo3 080 J 55 230 305 709 785 86 1 937 *OI2 08 7 163 238 313 717 m 944 *020 095 I 7 245 320 724 800 876 952 *027 103 178 253 328 IS 884 959 *035 no '85 260 335 343 350 35 8 365 373 380 388 395 403 410 418 492 567 641 716 790 864 938 77012 085 425 ?00 574 649 723 797 871 945 019 433 507 582 656 730 805 879 953 026 440 5i5 589 664 738 812 886 960 034 448 522 597 67. 745 819 893 967 041 455 530 604 678 753 827 901 975 048 462 537 612 686 760 834 908 982 056 47 545 619 693 768 842 916 989 063 477 K 701 S3 923 997 070 483 559 634 708 782 856 930 "004 078 093 100 107 "5 122 129 137 144 IS' '59 232 305 379 45 2 52? 597 670 143_ 8i5 1 66 240 313 386 459 532 6o ? 677 75o 822 173 247 320 393 466 539 612 68 5 757 181 254 327 401 474 546 619 692 764 1 88 262 335 408 481 554 627 699 772 '95 269 342 415 488 56i 634 706 779 203 276 349 422 495 568 641 7H 786 2IO 283 357 430 503 576 648 7.21 793 217 291 364 437 5io 583 656 728 801 223 298 371 444 5*7 590 663 830 37 844 851 859 860 873 880 ! N. L. 1 2 3 4 56789 P.P. S.' T.' 6' 6.46 373 373 S." T." )'= 540" 4-68557 558 D = 600 557 558 3 2 5'= 57 S." T." oo" 4.68 552 569 60 552 569 20 552 569 80 552 569 4 551 569 oo 551 570 55 37 1 37 6 56 371 376 57 37i 377 58 371 377 59 37 377 60 37 377 O I 6-57 7 = 58 8 = 58 9 = 59 o = 60 31 = 5460 552 568 32 = 5520 552 568 33 = 558o SS 2 568 34 = 5640 552 568 35 = 57 55 2 569 3 3 3 4 408 N. L. 1 2 3 4 5 6 7 8 9 p.p. 600 601 602 603 604 605 606 607 608 609 610 6u 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 63. 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 778i5 822 830 837 844 851 859 866 873 880 I 2 3 4 I 7 8 9 2 3 4 1 9 3 4 i 9 8 0.8 1.6 2.4 3-2 4.0 4.8 5.6 6.4 7-2 7 1.4 2.1 2.8 3-5 4.2 4-9 5-6 6-3 6 0.6 1.2 1.8 2.4 3- 3-6 ti 5-4 887 960 78032 104 176 247 319 390 462 533 604 675 746 817 888 958 79029 099 169 895 967 039 in 183 254 326 469 902 974 046 118 190 262 333 405 476 909 981 053 I2 5 197 269 340 412 483 916 988 06 1 132 2-04 276 347 419 490 924 996 068 140 211 283 426 497 075 219 290 362 433 504 938 *OIO 082 154 220 297 369 440 945 *OI 7 089 161 233 305 376 447 5'9 952 *025 097 1 68 240 312 383 455 526 540 547 554 561 569 576 583 590 597 611 682 753 824 895 965 036 106 176 618 689 760 831 902 972 043 183 625 696 767 838 909 979 050 1 20 190 633 704 774 845 916 986 57 127 197 640 711 781 852 923 993 064 134 204 647 718 79 859 930 *ooo 071 141 211 654 7 2 5 796 866 937 *00 7 078 148 218 66 1 732 803 873 * 944 o8 5 155 225 668 739 810 880 951 *02I 092 162 232 239 246 253 260 267 274 28l 288 295 302 309 379 449 588 657 727 796 865 316 386 456 525 I? 5 664 734 803 872 323 393 463 532 602 671 741 810 879 330 400 470 539 609 678 748 817 SS6 337 407 477 546 616 685 754 824 893 344 414 484 553 623 692 761 83' 900 351 4 2I 491 560 630 699 7 68 837 9 06 358 428 498 567 637 706 775 844 9i3 365 435 505 574 644 7'3 782 920 372 442 5" 581 650 720 789 858 927 934 941 948 9 = 5 962 969 975 982 989 996 80003 072 140 209 414 482 55 618 686" 95 6 81 023 090 158 224 010 079 H7 216 284 353 421 489 557 017 085 223 291 359 428 496 564 024 092 161 229 366 434 502 570 030 099 1 68 236 305 373 441 509 577 037 1 06 243 448 584 044 "3 182 250 387 455 523 59i 5 l 120 i83. 257 325 393 462 530 598 058 127 J 95 264 332 400 468 536 604 i34 202 2 7 I 339 407 475 543 6n 625 638 645 652 659 665 672 679 693 760 828 895 963 030 097 164 231 699 767 835 902 969 037 104 171 238 706 774 841 909 976 043 in 178 245 78i 848 916 983 050 117 184 251 720 III 922 990 057 124 191 258 726 794 862 929 996 064 \gS -26? 733 801. 868 93 6 070 137 204 271 740 808 875 943 *OIO 077 144 211 2 7 8 345 747 814 882 949 *OI 7 084 218 285 291 298 305 3" 325 338 35 ! N, L. | 1 2 3 4 5 6 789 P. P. S.' T.' 6 6.46 373 373 7 373 373 S." T." o io'= 600" 4.68 557 558 o ii 660 557 558 44' = 62 45 = 6 3 S." T." 40" 4.68 551 571 oo 551 57i 60 551 571 20 550 572 80 550 572 40 550 572 60 37 377 63 37 377 64 370 378 65 370 378 40 =6000 551 570 41 =6060 551 570 42 =6120 551 570 43 =6180 551 570 44 =6240 551 571 46-63 47 = 6 4 48=64 49 =65 409 N. L. 1 2 3 4 5 6 7 8 9 P .P. 650 651 652 653 654 657 658 659 660 66 1 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 68 1 682 C3 4 63 5 686 687 688 689 690 691 692 693 694 695 696 697 698 699 ?oo 81 291 298 305 3" 3i8 323 331 338 343 35 1 i 2 3 4 5 6 I 9 2 3 4 7 8 9 7 0.7 i-4 2.1 2.8 3-5 4-2 4-9 6 0.6 1.2 1.8 2.4 3-6 4.2 4-8 5-4 425 491 558 624 690 757 823 889 363 43' 498 564 631 697 763 829 895 961 371 438 503 57 1 637 704 770 836 902 378 443 5" 578 644 710 776 842 908 385 584 651 717 783 849 913 4?8 523 657 723 790 856 921 398 463 53i 598 664 730 796 862 928 403 47 1 538 604 671 737 803 869 933 411 478 544 611 677 743 809 875 941 418 485 55 1 617 684 750 816 882 948 954_ 82020 086 217 282 347 $ 543 968 974 981 987 994 *000 *oc>7 *oi4 027 092 158 223 289 354 419 484 549 033 099 164 230 295 360 426 491 556 040 I0 5 171 236 302 367 432 497 562 046 112 I 7 8 243 308 373 439 54 569 053 119 184 249 313 38o 445 5 10 575 060 i*5 191 256 321 387 452 5'7 582 066 132 197 263 328 393 458 523 588 073 138 204 269 334 400 463 530 593 079 145 210 2 7 6 341 4 06 47* 536 60 1 607 672 737 802 866 930 993 83059 123 187 614 620 627 633 640 646 653 659 666 679 743 808 872 937 *OOI 065 129 '93 685 750 814 879 943 *oo8 072 136 200 692 756 821 885 * 95 078 142 206 698 763 827 892 956 *O20 083 149 213 703 769 834 898 ^963 091 155 219 711 776 840 903 969 "033 097 161 225 718 782 847 911 * 97 o 104 1 68 232 724 789 853 918 982 ="046 no 174 238 730 795 860 924 988 *0 5 2 181 243 3'3 378 442 506 569 632 696 759 822 257 26 4 270 2 7 6 283 289 296 302 308 321 383 448 5 12 575 639 702 765 828 327 391 455 582 645 708 771 835 334 398 461 65^ 715 778 841 340 404 467 S3 1 594 658 721 784 847 910 347 410 474 537 6bi 664 727 790 853 353 480 544 607 670 734 797 860 923 359 423 487 55 677 740 803 866 366 429 493 556 620 683 746 809 872 372 436 499 563 626 689 753 816 879 883 891 897 904 916 929 935 942 948 84011 073 J36 8 448 954 017 080 142 267 330 392 454 960 023 cS6 148 211 273 336 398 4 60 967 029 092 155 217 280 342 404 466 973 036 098 161 223 286 348 410 473 979 042 167 230 292 354 417 479 985 048 in 173 236 298 361 423 4*3 992 053 117 1 80 242 305 367 429 491 998 061 123 1 86 248 3" 373 435 497 067 130 192 253 317 379 442 504 5 10 5 I() 5 22 528 535 541 547 553 559 566 N, L. 1 2 3 | 4 5 6789 P, P. S.' T.' 6' 6.46 373 373 7 373 373 o S." T." io'= 600" 4.68 557 558 ii = 660 557 558 12= 720 557 558 5 i i S." T." i' =6660" 4-68530 573 2 = 6720: 530 573 3 = 6780 530 573 4 = 6840 530 573 5 = 6900 549 574 6 = 6960 549 574 7 = 7020 549 574 65 37 378 69 37 378 7 37 379 I 48 = 6480 550 572 i 49 = 6540 550 .572 i 50 = 6600 550 572 i 51 =6660 530 573 c 410 N. L. 1 2 3 4 5 6 7 8 9 P. P. 7OO 701 702 703 704 705 706 707 708 709 7IO 711 712 713 7H 75 716 717 7 I 719 720 721 722 723 724 725 726 727 728 729 730 73i 732 733 734 735 736 737 738 739 740 741 742 743 744 3 747 748 749 750 84510 56 522 528 533 54i 547 553 559 566 2 3 4 5 6 I 9 2 3 4 7 8 9 i 2 3 4 1 7 8 9 7 0.7 1.4 2.1 2.8 3-5 4-2 4-9 5-6 6-3 6 0.6 1.2 1.8 2.4 3-o 3-6 4-2 4.8 5-4 5 -5 I.O i-5 2.0 2-5 3-o 3-5 4.0 4-5 572 634 696 757 819 880 942 85 00} 065 578 640 702 763 825 887 948 009 o? * 132 584 646 708 770 831 893 954 016 077 590 652 7'4 776 837 899 960 022 083 597 658 720 782 844 905 967 028 089 603 665 726 788 850 911 973 034 095 156 609 671 733 794 856 917 979 040 101 6. 5 677 739 800 862 924 985 046 107 621 683 743 807 868 93 991 052 114 628 689 751 813 874 936 997 058 120 126 w 248 309 370 431 491 552 612 673 138 144 IS? 163 169 175 236 297 358 418 479 540 600 661 721 181 193 254 315 376 437 497 558 618 6 79 199 260 321 382 443 53 564 625 68? 3 327 388 449 509 57 631 691 211 2 7 2 333 394 455 5'6 576 637 697 217 278 339 400 461 522 582 643 73 224 285 345 406 467 528 588 649 709 230 291 352 412 473 534 594 655 715 242 303 364 425 48 S 546 606 667 727 ~W 733 739 745 75i 757 763 769 775 781 794 8 5 4 914 974 86034 094 153 213 273 800 860 920 980 040 100 '59 219 279 806 866 926 986 046 106 165 225 283 812 872 932 992 052 112 I 7 I 2 3 I 291 818 878 938 998 058 118 177 237 297 824 884 944 *OO4 064 124 183 243 303 830 890 95 *OIO 070 130 189 249 308 836 896 956 *oi6 076 136 195 253 3H 842 902 962 *022 082 I 4 I 201 26l 320 848 908 968 *02S 088 147 207 267 326 332 392 45 1 5 10 57 629 688 747 806 864 923 338 344 350 356 362 3 (>8 374 3 80 386 398 457 5'6 576 3 K 8 7 o 404 463 522 700 759 817 876 ~933~ 410 469 528 587 646 705 7 6 4 823 882 415 471 534 593 652 711 770 829 888 421 481 540 599 658 717 776 835 894 427 487 546 605 664 723 782 841 900 433 493 55 2 611 670 729 788 847 906 439 499 558 617 676 735 794 853 9" 445 504 564 623 682 74i 800 859 917 929 941 947 953 958 964 970 976 982 87040 099 '57 216 274 332 390 448 506 9 88 046 105 163 221 280 338 396 454 994 052 in 169 227 286 344 402 460 999 058 116 17? 233 291 349 408 466 *00 5 064 122 181 239 297 355 4U 47i *OII 070 128 1 86 245 303 361 419 477 "017 075 134 192 2 5 39 367 425 483 *023 08 1 140 198 256 315 373 43i 489 "029 087 146 204 262 320 379 437 493 *035 093 I 5 l 2IO 268 326 384 442 5 00 5 12 5i8 523 529 535 54i 547 552 558 IM, L. 1 2 3|4 5 6 7 8 I 9 P. P. | S.' T.' 7' 6.46373 373 8 373 373 I I S." T." i'= 660" 4.68 557 558 2 = 720 557 558 3= 780 557 558 i 59'= 7 2 = 7 S." T." 140" 4.68 549 575 200 549 575 260 549 575 320 548 576 380 548 576 440 548 576 500 548 577 7 37 379 7 1 37 379 72 369 379 74 369 379 75 369 38o I 2 i -7 2 = 7 3 = 7 4 = 7 5 = 7 i 56 = 6960 549 574 i 57 = 7020 549 574 i 58 = 7080 549 575 i 59 = 7140 549 573 2 2 2 411 N. L. 1 2 | 3 4 5 6 7 8 [ 9 P. P. 75O 75 1 75 2 753 754 755 75 6 757 758 759 760 761 762 763 764 7 ^ 766 767 768 769 77O 77 1 772 773 774 775 776 777 778 779 78O 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 80O 87506 5 12 5i8 523 529 535 541 547 552 558 6 i 0.6 2 1.2 3 1.8 4 2.4 ill ill 9 5-4 5 i 0.5 2 I.O 3 i-5 4 2.0 5 2.5 6 3.0 7 3-5 8 4.0 9 4-5 564 622 679 737 795 852 910 967 88024 osT 57 628 685 743 800 858 915 973 030 576 633 691 749 806 864 921 978 036 58i 639 697 754 812 869 927 984 041 587 645 73 760 818 875 933 990 047 593 651 708 766 823 88 1 938 996 053 599 656 7H 772 829 887 944 *OOI 058 604 662 720 777 835 892 95 *oo7 064 610 668 726 783 841 898 955 "013 070 127 616 674 73i 789 846 904 961 *oi8 076 ~^33 087 093 098 104 IIO 116 121 138 '95 252 309 366 423 480 536 593 649 144 201 2 5 8 315 372 429 485 542 598 15 207 264 321 377 434 491 547 604 156 213 270 326 383 440 497 553 610 101 218 2 75 332 389 446 502 559 615 167 224 281 338 395 45 i 508 564 621 73 230 287 343 400 457 5'.3 57 627 I 7 8 235 292 349 406 463 5 '2 576 632 184 241 298 355 412 468 52? 58i 638 [90 247 304 360 417 474 53 587 643 655 660 666 672 677 683 689 694 75 807 863 919 975 *0 3 I 087 H3 198 254 700 70S 762 818 874 930 986 89042 098 '54 209 7 II 767 82 4 880 936 992 048 104 159 717 773 829 885 941 997 053 109 165 722 779 835 891 947 *oo3 59 "5 170 728 784 840 897 953 *009 064 120 _rj6^ 232 734 790 846 902 958 *oi 4 070 126 182 237 739 795 852 908 964 *O2O 076 131 I8 7 745 80 1 857 9i3 969 *02 5 08 1 137 193 756 812 868 925 981 "037 092 148 204 215 221 ~^6 332 387 49 S 553 609 664 719 226 243 2 4 8 260 315 37' 426 481 537 592 647 702 757 265 321 376 432 487 542 597 653 708 2 7 I 326 382 437 492 548 658 7'3 282 337 393 448 504 559 614 669 724 779 259 "0480 *09Oi *i323 *I744 *2I&5 0132587 6797 014 1003 5205 9403 015 3598 7788 016 1974 6155 0170333 3008 7218 1424 5625 9823 4017 8206 2392 6573 3429 7 6 39 1844 6045 *0243 4436 8625 2810 6991 3850 2264 6465 "0662 4855 9044 3229 7409 4271 8480 2685 6885 *I082 5274 9462 3647 7827 4692 8901 3103 7305 * I 5 I 5 6 93 9881 4065 8245 5"3 9321 3525 7725 *I920 6112 ="0300 4483 8663 5534 9742 3945 8144 *2 34 6531 *07i8 4901 9080 5955 *0l62 4365 8564 *2759 6950 *"37 53i9 9498 6376 *o 5 8 3 4785 8984 *3'78 7369 *'555 5737 9916 0751 1168 1586 2003 2421 2838 3256 3673 4090 4507 8677 018 2843 7005 019 1163 5317 9467 0203613 1751 021 1893 4924 9094 3259 7421 1578 5732 9882 4027 8169 5342 95" 3676 7837 1994 6147 *O2g6 4442 8583 5759 9927 4092 8253 2410 6562 *o7ii 4856 8997 6176 *344 4508 8669 2825 6977 *II26 5270 94" 6593 ="0761 4923 9084 3240 7392 *i5 4 o 5684 9824 7010 *"77 5341 9500 3656 7807 *i953 6099 *0238 7427 *i594 5757 9916 4071 8222 *2 3 6 9 6513 ="0652 7844 *20IO 6173 *0 33 2 4486 8637 *2 7 8 4 6927 *io66 8260 *2427 6589 *747 4902 9052 * 3 i 9 8 734i *i 4 79 2307 2720 3 J 34 3547 3961 4374 4787 5201 5 6l 4 N. L. 1 2 3 4 5 6 789 S." T." S." T." 2 46' = 9960" 4.68541 591 2 51' = 10260" 4.68540 593 2 47 = 10020 540 592 2 52 = 10320 539 594 2 48 = 10080 540 592 2 53 = 10380 539 594 2 49 = 10140 540 592 2 54 = 10440 539 595 2 50 = 10200 540 593 2 55 = 10500 539 595 41T N. L. 1 2 | 3 | 4 5 6 | 7 | 8 9 IO5O 1051 1052 i53 1054 I0 55 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 IO7O 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 IO9O 1091 1092 i93 1094 i95 1096 1097 1098 1099 MOO 02 1 1893 2307 2720 6854 0983 5109 9230 3348 7462 1572 5 6 78 9780 334 _3547_ 7680 1808 5933 *oo54 4171 8284 2393 6498 *o6oo 396i 4374 4787 8919 3046 7170 "1289 5405 95'7 3625 7729 *i829 5201 i 5614 6027 0220157 4284 8406 023 2525 6639 024 0750 4857 8960 02 5 359 6440 0570 4696 8818 2936 7050 1161 5267 9370 7267 1396 5521 9642 3759 7873 1982 6088 *oi90 8093 2221 6345 *0466 4582 8695 2804 6909 *IOIO 8506 2634 6758 ="0878 4994 9106 3214 7319 *I 4 I9 9332 3459 7582 "1701 5817 9928 4036 8i39 ^2239 9745 3871 7994 *2U3 6228 *0339 4446 8549 *2<>4y ! 3468 3878 4288 4697 5 I0 7 55i6 5926 6335 6744 7^54 026 1245 5333 9416 027 3496 7572 028 1644 5713 9777 75 6 3 1654 574i 9824 3904 7979 2051 6119 "0183 7972 2063 6150 *0233 4312 8387 2458 6526 *059o 8382 8791 2472 1 2881 6558 6967 *o64i *iO49 4719 5127 8794 9201 2865 | 3272 6932 7339 "0996 *I402 9200 3289 7375 *'457 5535 9609 3679 33 9609 3698 7783 "1865- 5942 *ooi6 4086 8152 *22I 4 *ooi8 4107 8192 *2273 6350 *0 4 2 3 4492 8558 *262O "0427 4515 8600 *268o 6757 "0830 4899 8964 *3026 *o836 i 4924 9008 * 3 o88 7165 *I237 5306 9371 *3432 029 3838 4244 4649 I s55 ! 5461 5867 6272 6678 7084 7489 7*95 030 1948 5997 031 0043 4085 8123 0322157 6188 0330214 8300 2353 6402 0447 4489 8526 2560 6590 0617 8706 2758 6807 0851 4893 8930 2963 6993 1019 9111 3163 7211 1256 5296 9333 3367 7396 1422 95 l6 3568 7616 1660 5700 9737 3770 7799 1824 9922 3973 8020 2064 6104 *oi4O 4173 8201 2226 ="0327 4378 8425 2468 6508 *0544 4576 8604 2629 "0732 4783 8830 2872 6912 *0947 4979 9007 3031 -1138 5188 9234 3277 7315 *I 35 5382 9409 3433 * I 543 5592 9638 3681 7719 * I 754 5785 9812 3835 4238 4640 5042 5444 5846 6248 6650 7052 7453 7855 8257 034 2273 6285 035 0293 4297 8298 036 2295 6289 037 0279 8659 2674 6686 0693 4698 8698 2695 6688 0678 9060 3075 7087 1094 5098 9098 3094 7087 1076 9462 3477 7487 1495 5498 9498 3494 7486 1475 9864 3878 7888 1895 5898 9898 3893 7885 1874 *02&5 4279 8289 2296 6298 *0297 4293 8284 2272 "0667 4680 8690 2696 6698 '0697 4692 8683 2671 *io68 5081 9091 3096 7098 *i097 5091 3070 *J47 5482 9491 3497 7498 *I496 5491 9481 3468 *l87I 5884 9892 3897 7898 "1896 5890 9880 3867 4265 4663 5062 5460 5858 6257 6655 7053 745 i 7849 8248 038 2226 6202 0390173 4141 8106 040 2066 6023 9977 041 3927 8646 2624 6599 0570 4538 8502 2462 6419 "0372 9044 3022 6 99 6 0967 4934 8898 2858 6814 -0767 9442 3419 7393 1364 5331 9294 3254 7210 1162 9839 3817 7791 1761 5727 9690 3650 7605 *i557 *02 37 4214 8188 2158 6124 *oo86 4045 8001 *i95 2 "0635 4612 8585 2 554 6520 "0482 4441 8396 *2347 *IQ33 5009 8982 295 * 6917 *o8 7 8 4837 8791 *27 4 2 *H3i 5407 9379 3348 7313 *I274 5232 9187 *337 *i829 5804 9776 3745 7709 "1670 5628 9582 *3532 4322 4716 5'" 5506 5900 6295 6690 7084 | 7479 N. L. 1 2 3 4 5 6 7 8 9 S." T." S.' 1 T." 2 55' = 10500" 4.68539 595 3o'=io8oo" 4.68538 597 2 56 = 10560 539 595 31= 10860 537 598 2 57 = 10620 538 596 32= 10920 537 598 2 58 = 10680 538 596 33= 10980 537 599 2 59 =10740 538 597 3 4 =11040 537 599 K'.M'U SLKV. '2~i 418 i M, | S'. T', Sec. S". T". 2 3 4 1 1 9 IO ii 12 13 H 15 16 1 7 18 19 20 21 22 23 24 3 27 28 29 3O 3i 32 33 34 P 37 38 39 4O 4i 42 43 44 11 47 48 49 50 5i 52 53 54 55 5 6 57 58 59 60 1 80 181 182 183 184 185 186 187 1 88 189 190 191 192 193 194 '95 196 199 6. 353 *6 412 10800 538' 68 597 353 352 352 35 2 352 35 i 35 i 35i 35L 35 413 413 414 414 4i5 415 415 416 416 10860 10920 10980 11040 IIIOO 11160 1 1 220 II280 II340 537 537 537 537 537 536 536 53 ^ 536 ^35_ 535 535 535 534 534 534 534 533 J33. _533_ 533 532 532 532 532 53i 53i 53i 53i 598 598 599 599 599 600 600 601 601 602 602 603 603 604 604 605 605 606 606 417 II400 350 350 350 350 349 349 349 349 348 417 418 418 419 419 420 420 421 421 11460 II520 11580 11640 II7OO 11760 II820 II880 II940 200 348 348 348 347 347 347 347 346 346 346 422 12000 607 607 608 608 609 609 610 610 611 611 201 202 203 204 20 5 206 207 208 209 422 423 423 424 424 425 425 426 426 427 12060 I2I2O I2I80 12240 I23OO 12360 12420 12480 12540 210 346 12600 530 530 530 530 529 529 529 529 528 528 528 612 211 212 213 2I 4 215 216 217 218 219 345 345 345 345 344 344 344 344 343 343 427 428 428 429 429 430 430 43i 43i 12660 12720 12780 12840 I290O 12960 I3O20 13080 I3I40 612 613 613 614 614 615 615 616 616 ~677~ 2 2O 432 13200 221 222 223 224 225 226 227 228 229 343 342 342 342 342 34i 34i 34i 340 432 433 434 434 435 435 436 436 437 13260 13320 13380 13440 13500 13560 13620 13680 13740 528 527 527 527 526 526 526 526 ^i. 5 2 5 525 525 524 524 524 5 2 3 523 5 2 3 522 617 618 618 619 620 620 621 621 622 230 340 437 13800 622 231 232 233 234 235 236 237 2 3 8 239 340 340 339 339 339 338 338 338 338 438 439 439 440 440 441 441 442 443 13860 13920 13980' 14040 I4IOO 14160 I422O 14280 I 434 623 623 624 625 625 626 626 627 628 240 337 443 14400 522 628 > M. S'. T'. Sec. S". T". l i 2 3 4 5 6 I 9 IO ii 12 13 H 15 16 1? 19 20 21 22 23 24 11 29 30 3 1 32 33 34 P 37 38 39 40 41 42 43 44 45 46 47 48 49 5O 5 1 5 2 53 54 | 58 59 6O 240 6. 337 337 337 336 336 336 336 335 335 335 443 14400 4.68 522 628 241 242 243 244 245 246 247 248 249 444 444 445 446 446 447 447 448 449 14460 14520 14580 14640 14700 14760 14820 14880 14940 522 522 521 52i 521 520 520 520 _520_ 5!9 629 629 630 631 631 632 632 633 634 250 334 449 15000 634 635 635 636 637 637 638 638 639 640 640 25 1 252 253 254 255 256 257 258 259 334 334 333 333 333 332 332 332 332 45o 450 451 452 452 453 454 454 455 15060 15120 15180 15240 15300 15360 15420 15480 15540 519 59 5i8 5i8 5i8 5 J 7 5i7 5'7 _ni 516 260 33i 456 15600 261 262 263 264 265 266 267 268 269 33i 33i 330 330 330 329 329 329 328 456 457 457 458 459 459 460 461 461 15660 15720 15780 15840 15900 15960 16020 16080 16140 516 515 5*5 515 5H 5H 54 5i3 5"3 5U 641 642 642 643 644 644 645 646 646 "647 270 328 | 462 16200 271 272 273 274 275 276 2 77 278 279 328 327 327 327 326 326 326 325 3 2 5 "325" 463 463 464 465 465 466 467 467 468 16260 16320 16380 16440 16500 16560 16620 16680 16740 512 512 5'2 5" 5" 5" 5io 5io 510 509 648 648 649 650 650 651 652 652 653 654 280 469 16800 281 282 283 284 285 286 287 288 289 324 324 324 323 323 323 322 322 321 469 47 471 47 2 472 473 474 474 475 16860 16920 16980 17040 17100 17160 17220 17280 17340 509 59 508 508 508 57 57 57 506 654 ^ 5 I 656 656 657 658 659 659 660 290 321 476 17400 506 66 1 "66T 662 663 664 664 665 666 666 667 668 291 292 293 294 295 296 297 298 299 300 321 320 320 320 3i9 3i9 3 J 9 3i8 3i8 3i7 477 477 478 479 479 480 481 482 482 483 17460 17520 17580 17640 17700 17760 17820 17880 17940 18000 506 55 505 505 504 504 503 53 53 502 411) TABLE XVI. THE LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS FOR EACH MINUTE. Formulas for the Use of the Auxiliaries 5 and T, 1. When a is in the first five degrees of the quadrant : log sin a = log a' + S. 1 log tan a = log a' + T. 1 log cot a = cpl log tan a. log sin a = log a" + S." log tan a = log a" + T. 1 ' log cot a = cpl log tan a. log a' log a" log sin a + cpl S.' log tan a + cpl TV : cpl log cot a + cpl T.' log sin a + cpl S." log tan a + cpl 7'." : cpl log cot a +cpl T." 2. When a is in the last five degrees of the quadrant : log cos =log(90-o)' + 5.' log cot =log(90- a)'+ T.' log tan = cpl log cot a. log cos = log(90 - a)" + S." log cot = log(90 - a)" + T." log tan = cpl log cot a. a = 90 -(90 -a). log(9O a)' = log cos a + cpl S.' = log cot a + cpl T. ' = cpl log tan o + cpl T. 1 log(9O o)"= log cos a + cpl S." = log cot a + cpl T." cpl log tan a + cpl T." 420 J L. Sin. d. Cp\.S'. Cpl. T'. L. Tan. c. d. L. Cot. L. Cos. I 2 3 4 I s 9 IO 30103 17609 9691 7918 6694 5800 5"5 457 4139 3779 3476 3218 2997 2802 2633 2483 348 227 119 021 930 848 773 704 639 579 524 472 424 379 336 297 259 223 190 158 128 100 072 046 022 999 976 954 934 914 896 877 860 843 827 812 797 782 769 755 743 730 ! i| 30-3 17609 12494 969* 7918 6694 5800 5"5 457 4 J 39 3779 3476 'I 2996 2803 2633 s 2228 2119 2020 I 93 I I8 4 8 1773 1704 I6 39 1579 1524 M73 1424 !379 1336 1297 1259 1223 1190 "59 1128 1072 1047 1022 998 976 955 934 9i5 895 878 860 843 828 812 797 782 769 756 742 730 o.oo ooo 60 59 58 57 56 55 54 53 5 2 5i 50 49 48 47 46 45 44 43 42 4i 40 39 38 37 36 35 34 33 32 3i 30 27 26 2 5 24 23 22 21 20 19 18 17 16 15 H 13 12 II IO 7 6 5 4 3 2 I 6o 240 300 360 480 540 6-46 373 6.76476 6.94 085 7.06579 7.16 270 7.24 188 7.30 882 7.36 682 7.41 797 3-53 627 3-53 627 3-53627 3-53 627 3-53 627 3-53627 3-53 627 3-53627 3-53 627 3-53 627 3-53 627 3-53627 3-53 627 3-53 627 3-53627 3-53627 3-53627 3-53627 3-53627 6.46 373 6.76476 ! 6.94 085 i 7- 6579 7.16 270 7.24 188 ; 7.30 882 i 7-36 682 7-4i 797 7-46 373 i 7-50512 7.54291 7-57 767 7.60 986 ! 7.63 982 7.66 785 ! 7.69418 7.71 900 i 7.74 248 3-53627 3-23 524 3-05915 2.93421 2.83 730 2.75812 2.69 118 2.63318 2.58 203 0.00000 O.OO 000 0.00 OOO O.OO 000 0.00 OOO 0.00000 0.00000 O.OO 000 O.OO OOO 600 746 373 3-53 627 2.53627 2.49488 2-45 79 2.42 233 2.39014 2.36018 2.33215 2.30 582 2.28 loo 2.25 752 2.23 524 O.OO OOO 660 720 780 840 900 960 1020 I080 1140 ii 12 13 14 \l 17 18 19 7-5 5 12 7.54291 7-57 767 7.60985 7.63 982 7.66 784 7.69417 7.71 900 7.74 248 3-53 627 3-53627 3-53 627 3.53628 3-53 628 3-53 628 3-53 628 3-53628 3-53628 3-53627 3-53 627 3-53 627 3-53627 3-53627 3-53627 3-53627 3-53627 3-53627 O.OO OOO o.coooo O.OO 000 0.00000 O.OO 000 0.00 OOO 9-99 999 9-99 999 9-99 999 1200 20 21 22 23 24 3 27 28 29 7.76475 "7-78 594 7.80615 7-82 545 7-84 393 7.86 166 7.87 870 7.89 509 7.91 088 7.92612 3-53 628 3-53627 i 7-76476 7-78 595 7.80615 i 7-82 540 7-84 394 7.86 167 7.87871 i 7.89510 7.91 089 : 7-92613 9.99 999 1260 I3 8o 1440 1500 1560 1620 1680 1740 3-53 628 3.53628 3-53 628 3.53628 3-53 628 3-53 628 3-53 628 3-53628 3-53 628 3-53 627 3-53627 3-53627 3-53627 3-53 627 3-53 627 3.53626 3-53 626 3-53 626 2.21 405 2.19385 2.17454 2.15 606 2.13833 2.12 129 2.IO490 2.08 9 1 1 2.07 387 9.99 999 9-99 999 9-99 999 9-99 999 9-99 999 9-99 999 9 99 999 9-99 999 9-99 998 1800 1860 1920 1980 2040 2160 2220 2280 2340 30 3i 32 33 34 P s 39 7-94 084 3-53 628 3-53 626 7.94086 2.05 9'4 2.04 490 2.03 1 1 1 2.01 775 2.00478 1.99219 1.97996 1.96806 1.95 647 94 5 '9 9-99 998 7-95 58 7.96 887 7.98 223 7.99 520 8.00 779 8.02 O02 8.O3 192 8.04 350 8.05 47 8 3-53 628 3-53 628 3-53 628 3-53628 3-53628 3.53628 3-53 628 3-53 628 3.53 628 3-53 626 3-53626 3-53 626 3-53 626 3-53 626 3-53 626 3-53 626 3-53 626 3.53626 7-95 5 10 j 7-96889 7.98 225 i 7-99522 8.00781 8.02004 803194 ! 8.04 353 i 8.05481 9-99 998 9-99 998 9-99 998 9-99 998 9-99 998 9-99 998 9-99 997 9-99 997 9-99 997 2400 40 8.06 578 3-53 628 3.53625 j 8.06581 93419 9.99 997 2460 2 5 80 26 4 2700 2760 2820 2940 4i 42 43 44 3 :i 49 8.07 650 8.08 696 8.09718 8.10717 8.II693 8.12647 8.13581 8.14495 8.I539I 3-53 628 3-53 628 3-53 629 3-53 629 3-53 629 3-53 629 3-53 629 3-53 629 3-53 629 3-53 625 3-53 625 3-53625 3-53 625 3-53 625 3-53 625 3-53 625 3-53 625 3-53 624 8.07 653 8.08 700 8.09 722 8.10 720 8. 1 1 696 8.12651 8-13 585 | 8.14500 8.15 395 8.16273 8.17133 | 8.17976 8.18 804 8.19616 8.20413 8.21 195 8.21 964 8.22 720 8.23 462 8.24 192 92 347 .91 300 .90 278 .89 280 .88 304 87 349 .86415 .85500 .84 605 9.99 997 9-99 997 9-99 997 9-99 996 9-99 996 9-99 996 9-99 996 9.99 996 9-99 996 3000 50 8.16268 3-53 629 3-53 624 83 727 9-99 995 3 060 3180 3240 33 3360 3480 354 5i 52 53 54 ii 57 58 59 60 8.17 128 8.17971 8.18798 8.19 610 8.20407 8.21 189 8.21 958 8.22713 8.23456 3-53 629 3-53 629 3-53629 3-53 629 3-53 629 3-53 629 3-53629 3-53 629 3-53 630 3-53 630 3-53 624 3-53 624 3-53 624 3-53 624 3-53 624 3-53 624 3-53 623 3-53 623 3-53 623 3-53 623 .82 867 .82 024 .81 196 .80 384 79 587 .78805 .78036 .77280 76538 9-99 995 9-99 995 9-99 995 9-99 995 9-99 994 9-99 994 9-99 994 9-99 994 9-99 994 3600 8.24 1 86 .758o8 9-99 993 L. Cos. d. || L. Cot. |c. d. L. Tan. L. Sin. ' 421 I ; L. Sin. d. Cpl. S'. Cpl. T, L. Tan. c. d. L. Cot. L. Cos. 3600 I 2 3 4 1, 7 8 9 IO ii 12 J 3 15 16 18 19 8.24 1 86 717 706 695 684 673 663 653 644 634 624 616 608 599 590 583 575 568 560 553 547 539 533 526 520 5*4 508 502 496 491 485 480 474 470 464 459 455 450 445 441 436 433 427 424 419 416 411 408 404 400 396 393 39 386 382 379 376 373 369 367 363 3-53 630 3-53 623 8.24 192 718 706 696 684 673 663 654 643 634 625 617 607 599 p4 575 568 553 546 540 533 527 520 509 502 496 491 486 480 475 470 464 460 455 45 446 441 437 432 428 424 420 416 412 408 404 401 397 393 390 386 383 380 376 373 37 367 363 1.75808 9-99 993 6O 59 58 57 56 55 54 53 50 49 48 47 46 45 44 43 42 4 1 40 39 38 37 36 35 34 33 32 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 '5 H 13 12 II 10 9 8 7 6 5 4 3 2 I 3660 3720 3780 3840 3900 3960 4020 4080 4140 8.24 903 8.25 609 8.26 304 8.26 988 8.27661 8.28 324 8.28977 8.29621 8.30 255 3-53 630 3-53 630 3-53 630 3-53 630 3-53 630 3-53 630 3-53 630 3-53 630 3-53 630 3-53 623 3-53 623 3-53623 3-53622 3.53622 3.53622 3-53622 3-53622 3-53622 8.24910 8.25616 8.26312 8.26 996 8.27 669 8.28 332 8.28 986 8.29 629 8.30 263 1.75090 1.74384 1.73688 1.73004 1-72331 1.71 668 1.71 014 1.70371 1.69 737 9-99 993 9-99993 9-99 993 9-99 992 9.99 992 9-99 992 9-99 992 9-99 992 9.99991 8.30 879 3-53 630 3-53621 8.30 888 1 .69 1 1 2 9-99 99i 4260 4320 4380 4500 4560 4620 4680 4740 8.31 495 8.32 103 8.32 702 8-33 292 8.33875 8.34 450 8.35018 8-35 578 8.36 131 3-53 630 3-53631 3-53631 3-53631 3-53631 3-5363I 3-5363I 3-53 631 3-53631 3-53621 3-5362I 3-53621 3-53621 3-53 620 3-53 620 3-53 620 3-53 620 3-53 620 8-31 505 8.32 112 8.327II HiSl 8.34461 8.35 029 8.35 590 8.36 H3 1.68495 1.67 888 1.67 289 1.66698 1.66 114 !-65 539 1.64971 1.64410 1.63857 1.633" 9.99991 9.99 990 9-99 990 9-99 990 9-99 990 9.99 989 9.99 989 9-99 989 9-99 989 4800 2O 21 22 23 24 2 5 26 2 7 28 29 30 8.36 678 8.37217 8-37 75 8.38 276 8.38 796 8.39310 8.39818 8.40 320 8.40816 8.41 307 3.53631 3-53620 8.36 689 9-99 988 9.99 98"8 9-99 988 9-99 987 9-99 987 9.99 987 9-99 986 9-99 986 9.99 986 9-99 985 4860 4920 4980 5040 5280 5340 3-5363I 3-53632 3-53 632 3-53632 3-53632 3-53632 3-53 632 3-53632 3-53632 3-53619 3-536I9 3-536I9 3-536I9 3-53619 3-536i8 3-53618 3.53618 3-536I7 8.37 229 8-37 762 8.38 289 8.38 809 8-39 323 8.39832 8.40 334 8.40 830 8.41 321 1.62771 1.62238 1.61 711 1.61 191 1.60 677 1.60168 1.59 666 1.59 170 1.58679 5400 8.41 792 3-53 632 8.41 807 1.58193 9-99 985 5460 5520 5580 5640 5700 5760 5820 5880 5940 32 33 34 35 36 39 4O 41 42 43 44 45 46 47 48 49 5O 52 53 54 55 56 57 58 59 8.42 272 8.42 746 8.43 216 8.43 680 8-44 139 8.44 594 8.45 044 8.45 489 8-45 93 8.46 300 3-53632 3.53633 3-53633 3-53 633 353633 3-53 633 3-53 633 3-53 633 3-53 633 3-53617 3-536I7 3-53617 3-53617 3-536i6 3-536i6 3.53616 3-53615 8.42 287 8.42 762 8.43 232 8-43 696 8.44 156 8.44611 8.45 06 1 8-45 507 8.45 948 S. 4 6 385 8.46817 8.47 245 8.47 669 8.48 089 8.48 505 8.48917 8-49 325 8.49 729 8.50 130 I-577I3 1-57238 1.56768 1.56304 1.55844 i-55 389 !-54939 1-54493 JJ4052. 1-53615 9.99 985 9-99 984 9.99 984 9-99 984 9-99 983 9-99 983 9-99 983 9-99 982 9-99 982 9-99 982 6000 3^3634 3-53 615 3-S36I3 3.53615 3.536I4 3-536I4 ' 3-53614 3-536H 3-536I3 3.53613 .v536i;, 6060 6120 6180 6240 6300 6360 6420 6480 6540 8.46 799 8.47 226 8.47 650 8.48 069 8.48 485 8.48 896 8.49 304 8.49 708 8.50 108 8.50 504 '8.50 897 8.51 287 8-51 673 8-52055 8.52434 8.52810 8-53 183 8-53552 8.53919 3-53 634 3-53 634 3-53 634 3-53 634 3-53 634 3-53 634 3-53 635 3-53 635 i-53 183 I-52755 1.51911 i. 5 i 495 1.51083 1.50271 1.49 870 9-99 98i 9-99 981 9-9998I 9.99 980 9.99 980 9-99 979 9-99 979 9-99 979 9.99 978 6600 3jj|3 <>35 3.536I3 8.50 527 1-49473 1.49080 1.48690 1.48304 1.47921 I-4754I i-47 '65 1.46792 1.46422 1-46055 1.45 <)()_> )-W 97 s 6660 6720 6780 6840 6900 6960 7020 7080 7140 3-53 635 3-53 635 3-53635 3-53 635 3.53635 3-53 636 3-53 636 3-53 636 3-53636 3-53612 3-536I2 3-53612 3-536ii 3.536H 3-536ii 3.536H 3-536io 8.50 920 8.51 310 8.51 696 8.52079 8-52459 8-52835 8.53 208 8.53578 8-53 945 9-99 977 9-99977 9-99977 9-99 97 6 9-99 976 9-99975 9-99 975 9-99 974 9-99 974 7200 60 8.54 282 3-53 636 3.53610 8.54308 9-99 974 L. Cos. d. || ! L. Cot. c.d. | L.Tan. L. Sin. ' 88 422 2 // / L. Sin. d. Cpl. S'. Cpl. T', L. Tan. c. d. L. Cot. L. Cos. 7 200 o 8.54 282 360 3-53 636 3.53610 8.54 308 361 1.45 692 9-99 974 60 7260 7320 7380 7440 7500 7560 7620 7680 7740 I 2 3 4 I 9 8.54 642 8.54 999 8-55 354 8-55 75 8.56054 8.56 400 8.56 743 8.57084 8.57421 357 355 35i 349 346 343 34i 337 116 3-53 636 3-53 637 3-53 637 3-53 637 3-53 637 3-53 637 3-53 637 3-53 637 3-53 638 3-53 609 3-53 609 3-53 609 3-53 609 3-53 608 3-53 608 3-53 608 3-53 607 3-53 607 8-54 669 8.55 027 8-55 382 8-55 734 8.56 083 8.56 429 8.56 773 8.57114 8-57452 358 355 352 349 346 344 34i 338 116 i-45 33i 1-44973 1.44618 1.44 266 I-439I7 i-43 57 1 1.43 227 1.42886 1.42 548 9-99 973 9-99 973 9.99972 9-99 972 9.99971 9-99 97i 9-99 97 9-99 97 9-99 969 59 58 57 56 55 54 53 52 5i 7800 10 8-57757 3-53 638 3-53 607 8.57 788 1.42212 9-99 969 50 7860 7920 7980 8040 8100 8160 8220 8280 8340 ii 12 13 H 15 16 3 19 8.58 089 8.58419 8.58 747 8.59072 8-59 395 8-59 7 1 S 8.60 033 8.60 349 8.60 662 330 328 325 323 320 3i8 316 3U 3-53 638 3-53638 3-53 638 3-53 638 3-53 639 3-53 639 3-53 639 3-53 639 3-53 639 3-53 606 3.53 606 3.53 606 3-53 605 3-53 6 5 3-53 605 3-53 604 3-53 604 3-53 604 8.58 121 8.58451 8.58 779 8.59 105 8-59428 8-59 749 8.60 068 8.60 384 8.60 698 33 328 326 323 321 319 3i6 3H 1.41 879 1.41 549 I.4I 221 1.40893 -40 572 .40251 39 932 .39616 -39 32 9.99 968 9-99 968 9.99 967 9-99 967 9-99 967 9.99 966 9-99 966 9.99 965 9-99 964 49 48 47 46 45 44 43 42 41 8400 20 8.60 973 3-53 639 3-53 603 8.61 009 3899' 9-99 964 40 8460 8520 8580 8640 8700 8760 8820 8880 8940 21 22 23 24 2 5 26 11 29 8.61 282 8.6 1 589 8.61 894 8.62 196 8.62 497 8.62 795 8.63091 8.63 385 8.63 678 39 37 35 302 301 298 296 294 293 3-53 640 3-53 640 3-53 640 3-53 640 3-53 640 3' 5 3 640 3-53 641 3-53 641 3-5364I 3-53 603 3-53 603 3-53 602 3-53 602 3.53602 3-536oi 3-53 601 3-53 601 3-53 600 8.61 319 8.61 626 8.61 931 8.62 234 8-62 535 8.62 834 8.63 131 8.63 426 8.63 718 307 305 303 301 299 297 295 292 .38 681 38 374 .38069 -37 766 %% .36869 36 574 .36 282 9-99 963 9-99 963 9-99 962 9-99 962 9-99 961 9-99 96i 9-99 96o 9.99 960 9-99 959 39 38 37 36 35 34 33 32 3i 9000 30 8.63 968 290 ,00 3-53 641 3.53600 8.64 009 291 ?0>0 35 99 1 9-99 959 3O 9060 9120 9180 9240 9300 9360 9420 9480 9540 3i 32 33 34 P 37 38 39 8.64 256 8.64543 8.64 827 8.65 no 8.65 391 8.65 670 8.65 947 8.66 223 8.66497 287 284 283 281 279 277 276 274 3-53 641 3-53 642 3-53 642 3-53 642 3-53 642 3-53 642 3-53 642 3-53 643 3-53 643 3-53 599 3-53 599 3-53 599 3-53 598 3-53 598 3-53 598 3-53597 3-53597 3-53596 8.64 298 8.64 585 8.64 870 8.65 154 8.65 435 8.65715 8.65 993 8.66 269 8.66 543 287 285 284 281 280 278 276 274 35 7 2 35415 35 130 -34 846 34 565 34285 .34007 33 73i 33 457 9.99 958 9-99 95 8 9-99 957 9-99 956 9.99 956 9-99 955 9-99 955 9-99 954 9-99 954 29 28 27 26 2 5 24 23 22 21 9600 4O 8.66 769 272 3-53 643 3-53 596 8.66816 273 33 184 9-99 953 20 9660 9720 9780 9840 9900 9960 10020 10080 IOI40 4 1 42 43 44 !i 47 48 49 8.67 039 8.67 308 8-67 575 8.67 841 8.68 104 8.68 367 8.68 627 8.68 886 8.69 144 270 269 267 266 263 263 260 259 258 3-53 643 3-53 643 3-53 644 3-53 644 3-53 644 3-53 644 3-53 644 3-53 645 3-53 645 3-53 596 3-53595 3-53 595 3-53 594 3-53 594 3-53 594 3-53 593 3-53 593 3-53592 8.67 087 8.67 356 8.67 624 8.67 890 8.68 154 8.68417 8.68 678 8.68 938 8.69 196 269 268 266 264 263 261 260 258 32913 32644 32 376 .32 no .31 846 -3i 583 .31 322 .31 062 .30 804 9-99 952 9-99 95 2 9-99 95 J 9-99 95 9-99 95 9-99 949 9-99 949 9.99 948 9.99 948 19 18 17 16 15 H 13 12 II IO2OO 50 8.69 400 250 3-53 645 3-53 592 8.69 453 257 30 547 9-99 947 10 I02&0 10320 0380 0440 0500 0560 0620 0680 0740 5' 52 53 54 H H 59 8.69 654 8.69 907 8.70 159 8.70 409 8.70 658 8.70 903 8.71 151 8.71 395 8.71 638 254 253 252 250 249 247 246 244 243 3-53 645 3-53 646 3-53 646 3-53 646 3-53 646 3-53 646 3-53 647 3-53 647 3-53 647 3-53 592 3-53591 3-53591 3-53 590 3-53 590 3-53 589 3-53 589 3-53 589 3-53 588 8.69 708 8.69 962 8.70 214 8.70 46? 8.70714 8.70 962 8.71 208 8.71 453 8.71 697 255 254 252 25 1 249 2 4 8 2 4 6 245 244 .30 292 .30 038 .29 786 -29 535 .29 286 .29038 .28 792 28 547 .28 303 9-99 946 9-99 946 9-99 945 9-99 944 9-99 944 9-99 943 9-99 94 2 9-99 942 9-99 94i 1 7 6 5 4 3 2 10800 60 8.71 880 242 3-53 647 3-53 588 8.71 940 243 1.28060 9-99 940 L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. 1 87 423 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. P. P. 8.71 880 8.71 940 1.28060 9.99 940 60 I 8.72 120 8.72 181 1.27819 9-99 94 V> 2 3 8.72 359 8-72597 2 39 238 8.72420 8.72659 2 39 239 1.27580 I-2734I 9-99 939 9-99 938 5* S7 1 241 239 237 236 234 24.1 23.9 23.7 23.6 23.4 4 5 8.72 834 8.73069 237 235 8.72 896 8-73 132 2 37 236 1.27 104 1.26868 9-99 938 9-99937 56 55 3 t 48.2 47.8 47.4 47.2 46.8 72.3 71.7 71.1 70.8 70.2 96.4 95.6 94.8 94.4 93.6 6 8-73 303 2 34 8-73 366 2 34 1.26634 9-99 936 54 S f. 120.5 "9-5 "8.5 118.0 117.0 7 8-73 535 8.73 600 2 34 1.26400 9-99 936 51 7 168.7 167.3 i 6 5-9 165.2 163.8 8 9 8.73 767 8-73 997 230 8-73 832 8.74063 232 231 1.26168 J -25937 9-99 935 9-99 934 52 5i 9 192.8 191.2 189.6 188.8 187.2 | 216.9 2'5-i 213-3 212.4 210.6 IO 8.74 226 228 8.74 292 1.25708 9-99 934 50 232 231 229 227 226 ii 8-74454 226 8.74521 1.25479 9-99 933 49 i 23.2 23.1 22-9 22.7 22.6 12 13 8.74 680 8.74 906 226 8-74 748 8-74 974 227 226 1.25 252 1.25 026 9-99 932 9-99 932 48 47 3 4 46.4 46.2 45.8 45.4 45.2 69.6 69.3 68.7 68.1 67.8 92.8 92.4 91.6 90.8 90.4 14 'S 8.75 130 8-75353 223 8-75 J 99 8-75 423 225 224 222 1.24801 1-24577 9-99 93i 9-99 93 46 4S I 7 116.0 115.5 "4-5 "3-5 "3- 139.2 138.6 137.4 136.2 135.6 .162.4 161.7 160.3 I 5^-9 158.2 16 8-75575 8-75 645 1-24355 9-99 929 44 8 185.6 184.8 183.2 181.6 180.8 K 19 8-75 795 8.76015 8.76 234 2 2O 219 8.75 867 8.76087 8.76 306 22O 219 1-24133 1.23913 1.23694 9-99 929 9-99 928 9-99 927 43 42 4' 9 208.8 207.9 206.1 204.3 23-4 224 222 220 219 217 20 8.76451 216 8-76 525 1-23475 9.99 926 40 2 44.8 44.4 44.0 43.8 43.4 21 22 2.3 8.76667 8.76 883 8.77097 216 214 8.76 742 8.76958 8-77 173 216 2I 5 1.23258 1.23042 1.22827 9-99 926 9-99 925 9-99 924 39 38 37 3 4 5 6 67.2 66.6 66. o 65.7 65.1 89.6 88.8 88.0 87.6 86.8 112. III.O 110.0 109.5 108.5 134-4 133-2 I32-0 131.4 130.2 24 8.77310 213 8.77387 214 1.22 613 9-99 923 36 I 156.8 155.4 154.0 153.3 I5L9 179.2 177.6 176.0 175.2 173.6 2 5 8.77522 8.77 600 I.224OO 9-99 923 JS 9 201.6 199.8 198.0 197.1 195.3 26 8 -77 733 211 8.77811 1.22 189 9-99 922 34 2 7 8-77943 2O9 8.78022 2IO I.2I 97 8 9.99921 33 216 214 213 211 2O9 28 8.78152 2O8 8.78232 I.2I 768 9-99 920 V 29 8.78360 2O8 8.78441 209 i. 21 559 9.99 920 3i 3 64.8 64.2 63.9 63.3 62.7 30 8.78 568 206 8.78 649 206 i-2i 35 1 9.99919 3O 4 86.4 85.6 85.2 84.4 83.6 31 8.78 774 8.78 855 206 I.2I 145 9.99918 29 6 129.6 128.4 2 7-8 126.6 125 4 32 8.78979 205 8.79061 1.20939 9.99917 2S 7 151.2 149.8 49.1 147.7 M6.3 33 8.79 183 204 8.79 266 205 1.20734 9.99917 27 (1 194.4 192-6 91.7 189.9 1 88. i 34 8.79 386 203 8-79 47 1.20530 9.99916 26 35 8.79 588 8.79673 203 202 1.20327 9-999I5 2^ 2O8 206 2O3 2O1 199 3& 8-79 789 8-79 875 1. 20 125 9.99914 24 I 20.8 2 .6 20.3 20.t 19.9 37 38 8.79990 8.80 189 199 8.80076 8.80 277 201 1.19924 1.19723 9-999I3 9-99913 23 22 3 4 41.6 4 .2 40.6 40.2 39.8 62.4 6 .8 60.9 60.3 59.7 83.2 8 .4 81.2 80.4 79.6 39 8.80 388 199 8.80 476 199 108 !-i95 2 4 9.99912 21 , 5 104.0 10 .0 101.5 100.5 99-5 1 40 S.So s 67.5 67.0 665 66.0 3 1 8.98 288 8.98 490 i.oi 510 9-99 798 2Q 6 81.0 80.4 798 79.2 32 33 8.98419 8.98 549 130 8.98622 8.98 753 132 13* i.oi 378 i.oi 247 9-99 797 9.99 796 28 27 I 9 94-5 93-8 93-i 924 108.0 107.2 106.4 105.6 121.5 120.6 119.7 u8.8 34 8.98 679 130 8.98 884 I 3 I i.oi 116 9-99 795 2(> 35 8.98 808 129 8.99015 I 3 I 1.00985 9-99 793 2^ 131 130 129 128 36 8.98 937 129 8-99 J 45 130 1.00853 9-99 792 24 1 13.1 13.0 12.9 12.8 37 8.99066 129 8.99 275 130 i.oo 723 9 99 79 1 23 3 39-3 39-o 38-7 38-4 38 39 8-99 194 8.99 322 128 128 8.99 403 8-99 534 130 129 128 1.00595 1 .00 466 9-99 790 9-99 788 22 21 4 5 6 52.4 52.0 51.6 51.2 65.5 65.0 64.5 64.0 78.6 78.0 77.4 76.8 4O 8.99 430 8.99 662 i.oo 338 9.99 787 20 I 91.7 91.0 90.3 89.6 41 42 8-99 577 8.99 704 127 8.99 791 8.99919 128 1 .00 209 1.00081 9-99 786 9-99 785 19 18 9 117.9 "7- "o- 1 JI 5-2 43 8.99 830 8.99 956 126 9.00 046 9.00 174 127 128 0.99 954 099 826 9-99 783 9.99 782 17 1 6 127 126 123 124 45 9.00082 126 9.00301 127 0.99 699 9-99 78i 15 25.4 25.2 25.0 24.8 46 9.00 207 125 9.00427 0-99 573 9 99 780 14 4 50.8 50.4 500 496 47 9.00 332 125 9.00 553 126 0.99 447 9-99 778 i.S 63 5 63 o 62 5 62 o 48 9.00456 124 9.00 679 0.99 321 9 99 777 12 7 88.9 88.2 87.5 868 49 9.00 581 125 9.00 803 0.99 19 s 9-99 776 II S 101.6 100.8 loo o 99.2 50 9.00 704 9.00 930 I2 5 0.99 070 9-99 775 10 9 114.3 "3-4 1125 "1-6 51 9.00 828 9.01 055 I2 5 0.<>S 945 9-99 773 9 123 122 121 120 52 9.00951 9.01 179 124 0.98821 9-99 772 8 12 3 12.2 12. 1 12 53 9.01 074 '-3 9.01 303 0.98 697 9.99771 7 j 24.6 24.4 242 24.0 54 55 9.01 196 9.01 318 122 122 9.01 427 9.01 550 124 123 0.98 573 0.98 430 9-99 7 6 9 9-99 7 6 8 5 3 4 \ 36.9 36.6 36.3 36.0 49-2 48.8 484 48.0 61.5 61.0 60.5 60.0 56 57 9.01 440 9.01 561 121 9.01 673 9.01 796 123 0.98 327 0.98 204 9-99 7 6 7 9-99 7 6 5 4 I 73-8 73-2 7 2 -6 7 2 - 86.1 85-4 847 84.0 98.4 97-6 96.8 96.0 58 9.01 682 9.01 918 0.98 082 9-99 764 2 9 1,0.7 109-8 1089 108.0 59 9.01 803 9.02 040 0.97 960 I 6O 9.01 923 j 9.O2 102 0.97 838 9.99761 L. Cos. | d. L. Cot. |c. d. L. Tan. L. Sin. ' P. P. 84 426 i / L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. P. P. o 9.01 923 9.02 162 0.97 838 9.99 761 60 I 2 3 4 I 9 9.02 043 9.02 163 9.02 283 9.02 402 9.02 520 9.02 639 9.02 757 9.02 874 9.02 992 120 120 119 118 119 118 117 118 9.02 283 9.02 404 9-02 525 9.02 645 9.02 766 9.02 885 9.03 005 9.03 124 9.03 242 121 121 1 2O 121 119 120 119 118 0.97717 0.97 596 0-97 475 0-97 355 0-97 234 0.97115 0.96 995 0.96 876 0.96 758 9.99 760 9-99 759 9-99 757 9-99 75 6 9-99 755 9-99 753 9-99 752 9-99 75 9-99 749 59 5* 57 56 55 54 53 52 5 1 I a 3 4 I 121 12O 119 118 12. 1 12.0 11.9 ii. 8 24.2 24.0 23.8 23.6 36.3 36.0 35.7 35.4 48.4 48.0 47.6 47.2 60.5 60.0 59.5 59.0 72.6 72.0 71.4 70.8 IO 9.03 109 9.03361 1 18 0.96 639 9.99 748 50 I ii 12 13 14 9.03 226 9-03 342 9.03 458 9-03 574 9-03 690 9.03 803 9.03 920 117 116 116 116 116 "5 "5 9-03 479 9-03 597 9.03 714 9.03 832 9.03 948 9.04 065 9.04 181 118 117 118 116 117 116 0.96 521 0.96 403 0.96 286 0.96 1 68 0.96052 0-95 935 o 95 819 9-99 747 9-99 745 9-99 744 9.99 742 9-99 74i 9.99 740 9 99 7 "?8 49 48 47 46 45 44 9 108.9 108.0 107.1 106.2 117 116 115 114 11.7 ii. 6 11.5 11.4 19 9.04 034 9.04 149 114 "5 9.04 297 9.04413 116 116 0.95 703 0-95 587 9-99 737 9-99 736 42 3 4 5 35.1 34.8 34.5 34.2 46.8 46.4 46.0 45.6 58.5 58.0 57.5 57.0 20 9.04 262 9.04 528 0.95 472 9-99 734 40 6 70.2 69.6 69.0 68.4 21 22 23 24 3 27 28 29 9.04 376 9.04 490 9.04 603 9-4 7 1 5 9.04 828 9.04 940 9.05052 9.05 164 9-05 273 114 113 112 "3 112 112 112 III 9.04 643 9.04 758 9.04 873 9.04 987 9.05 101 9.05 214 9.05 328 9.05 441 9-05 553 "5 "5 114 114 "3 114 "3 112 0-95 357 0-95 242 0.95 127 0.95013 0.94 899 0.94 786 0.94 672 0-94 559 0.94 447 9-99 733 9-9973' 9-99 730 9-99 728 9.99 727 9-99 726 9.99 724 9-99 723 9.99721 39 38 37 36 35 34 33 32 8 9 3 93.6 92.8 92.0 91.2 105.3 104.4 103.5 102.6 113 112 111 110 II. 3 II. 2 II. I II. 22.6 22-4 22.2 22.0 33-9 33-6 33-3 33-O 30 9.05 386 9.05 666 o-94 334 9.99 720 3O 5 56.5 56.0 55.5 55.0 32 33 34 I 39 9-05 497 9.05 607 9.05 717 9.05 827 9-05 937 .9.06 046 9-o6 155 9.06 264 9.06 372 IIO IIO IIO IIO I0 9 109 109 1 08 9.05 778 9.05 890 9.06 002 9.06 113 9.06 224 9-o6 333 9.06 445 9.06 556 9.06 666 112 112 III III III IIO III IIO 0.94 222 0.94 110 0.93 998 o-93 887 0.93 776 0.93 665 0-93 555 0.93 444 0-93 334 9-99 7 J 8 9.99717 9-99 7 l6 9-99 7H 9-99 7'3 9.99711 9.99 710 9-99 78 9-99 77 29 28 2 7 26 25 24 23 22 21 6 I 9 2 67.8 67.2 66.6 66.0 79.1 78.4 77.7 77.0 90.4 89.6 88.8 88.0 101.7 100.8 99.9 99.0 109 108 107 106 10.9 10.8 10. 10.6 21.8 21.6 21. 21.2 40 9.06 48 1 1 08 9.06 775 0.93 225 9.99 705 2O 3 32.7 32-4 32- 31-8 41 42 43 44 45 46 47 48 49 9.06 589 9.06 696 9.06 804 9.06911 9.07018 9.07 124 9.07 231 9-07 337 9.07 442 107 1 08 107 I0 7 106 107 1 06 9.06 8&5 9.06 994 9.07 103 9.07 211 9.07 320 9.07 428 9.07 536 9.07 643 9.07751 IIO 109 109 108 109 108 108 107 108 o.93"5 0.93 006 0.92 897 0.92 789 0.92 680 0.92 572 0.92 464 0.92357 0.92 249 9-99 74 9-99 702 9.99 701 9-99 699 9.99 698 9-99 696 9.99 695 9-99 693 9.99 092 19 18 '7 16 15 13 ii 1 I 9 54-5 54-o 53- 53-O 65.4 64 8 64. 63.6 76.3 75.6 74.9 74 2 87.2 86 4 85.6 84.8 98.1 97.2 96.3 95.4 105 104 103 5O 9.07 54 S 9.07 858 107 1 06 0.92 142 9.99 690 IO 2 21 208 20.6 5 1 5 2 53 54 P H 59 9-0? 653 9.07 758 9.07 863 9.07 968 9.08 072 9.08 176 9.08 280 9.08 383 9.08 486 I0 5 105 I0 5 104 104 104 103 103 9.07 964 9.08071 9.08177 9.08 283 9.08 389 9.08 493 9.08 600 9.08 705 9.08810 107 106 106 106 106 I0 5 105 0.92 036 0.91 929 0.91 823 0.91 717 0.91 611 0.91 505 0.91 400 0.91 293 0.91 190 9.99 689 9.99 687 9.99 686 9-99 684 9-99 683 9.99 68 1 9.99 680 9.99 678 9-99 677 9 8 7 6 5 4 3 2 I 3 1 3 1 31-2 3-9 4 42 41.6 41.2 6 63 62.4 61.8 7 73-5 72.8 72.1 8 84.0 83.2 82.4 9 94-5 93-6 927 6O 9.08 589 9.08914 0.91 086 9-99 675 O L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. ' P. P. 83 s L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. p. p. I 2 3 4 I 7 8 9 IO ii 12 13 14 \l 17 18 19 20 21 22 23 24 2 5 26 2 7 28 29 30 3i 32 33 34 3 3 39 40 4i 42 43 44 s 47 48 49 50 5' 52 53 54 55 56 H 59 60 9.08 589 9.08 692 9.08 795 9.08 897 9.08 999 9.09 101 9.09 202 9.09 304 9.09 405 9.09 506 103 103 1 02 102 102 101 IO2 101 101 100 IOI 100 100 99 100 99 99 98 98 98 98 97 97 97 97 96 97 96 96 95 96 95 95 95 94 95 94 94 93 94 93 93 93 93 93 9 2 9 2 92 92 9i 92 9i 9i 90 9i 90 91 90 9.08914 I0 5 104 104 103 104 103 103 102 103 IO2 IO2 IOI IO2 IOI IOI JOI IOI IOO IOO IOO IOO 99 99 99 99 99 98 98 98 98 97 98 97 97 96 97 96 96 96 96 95 95 95 95 95 94 95 94 94 93 94 93 93 93 93 92 92 93 9i 92 0.91 086 9-99 675 60 59 58 57 56 55 54 53 52 5' 50 4<> 48 47 4" 45 44 43 4-2 4i 40 39 38 37 36 35 34 33 32 31 30 29 28 -1 26 25 24 23 22 21 20 19 iS '7 1 6 '5 4 13 12 n IO 9 8 7 6 5 4 3 2 I 2 3 7 2 3 4 i I 9 2 3 4 I I 9 I 2 3 4 7 8 9 i 2 3 4 I I 9 105 104 103 10.5 10.4 10.3 21.0 20. 8 2O.6 31.5 31.2 30.9 42.0 41.6 41.2 52.5 52.0 51.5 63.0 62.4 61.8 73.5 72.8 72.1 84.0 83.2 82.4 94-5 93-6 92-7 102 101 99 10.2 IO.I 9.9 20.4 20.2 19.8 30.6 30.3 29.7 40.8 40.4 39.6 51.0 50.5 49.5 6l.2 60.6 59.4 71.4 70.7 69.3 8l.6 80.8 79.2 91.8 90.9 89.1 98 97 96 9.8 9.7 9.6 19.6 19.4 19.2 29.4 29.1 28.8 39.2 38.8 38.4 49.0 48.5 48.0 58.8 58.2 57.6 68.6 67.9 67.2 78.4 77.6 76.8 88.2 87.3 86.4 95 94 93 9-5 9-4 9-3 19.0 18.8 18.6 28.5 28.2 27.9 38.0 37.6 37.2 47.5 47.0 46.5 57-o 5 6 -4 55- 8 66.5 65.8 65.1 76.0 75.2 74.4 85.5 84.6 83.7 92 91 90 9.2 9.1 9.0 18.4 18.2 18.0 27.6 27.3 27.0 36.8 36.4 36.0 46.0 45.5 45.0 55.2 54.6 54.0 64.4 63.7 63.0 73.6 72.8 72.0 82.8 81.9 81.0 9.09019 9.09 123 9.09 227 9.09 330 9-09 434 9-09 537 9.09 640 9.09 742 9.09 845 0.90981 0.90 877 0.90 773 0.90 670 0.90 566 0.90 463 0.90 360 0.90 258 0.90 155 9-99 (J 74 9-99 672 9-99 670 9-99 669 9-99 667 9.99 666 9-99 664 9-99 663 9.99661 9.09 6oO 9.09 947 0.90053 9-99 659 9.09 707 9.09 807 9.09 907 9.10006 9.10 106 9.10205 9.10304 9.10402 9.10501 9-10599 9.10697 9.10795 9.10893 9.10990 9.11 087 9.11 184 9.11 281 9-" 377 9.11474 9.10049 9.10 150 9.10252 9-10353 9.10454 9-10555 9.10656 9.10756 9.10856 0.89951 0.89 850 0.89 748 0.89 647 0.89 546 0.89 445 0.89 344 0.89 244 0.89 144 9-99 658 9-99 656 9-99 655 9-99 653 9.99651 9-99 650 9-99 648 9-99 647 9-99 645 9.10956 9.11 056 9-" 155 9.11354 9-" 353 9-II452 9-n 55 1 9.11649 9. ii 747 9.11845 0.89 044 9-99 643 0.88 944 0.88 845 0.88 740 0.88 647 0.88 548 0.88 449 0.88351 0.88 253 0.88 155 9-99 642 9.99 640 9-99 638 9-99 637 9-99 635 9-99 633 9-99 632 9-99 630 9-99 629 9.11 570 9.11 666 9.11 761 9.11857 9.11952 9.12047 9.12 142 9.12 236 9-12331 9.12425 9- "943 9. 1 2 O4O 9.I2I38 9.12235 9.12332 9.12428 9.12525 9-12 621 9.12717 9-12 813 9-I2909 0.88057 9-99 627 0.87 960 0.87 862 0.87 765 0.87 668 0.87572 0-87 475 0.87 379 0.87 283 0.87 187 9-99 625 9.99 624 9.99 622 9.99 620 9.99 618 9.99617 9.99615 9.99613 9.99612 9.12519 9.12 612 9.12 706 9.12799 9.12892 9.12985 9.13078 9-I3I7 1 9.13 263 9-13355 0.87091 9.1)9 bio 9.13004 9.13099 9-13 194 9.13 289 9-I3384 9-I3478 9-13573 9.13667 9.13761 0.86 996 0.86 901 0.86 806 0.86711 0.86616 0.86 522 0.86427 0.86333 Q-86 239 p.86 146 9.99 608 9-99 607 9-99 605 9-99 603 9.99 60 1 9.99600 9-99 598 9-99 596 9-99 59? 9-'3447 9-I3854 9-99 5'>3 9-13539 9.13630 9.13722 9-13813 9.13904 9-13994 9.14085 9-HI75 9.14266 9.13948 9.I404I 9.14 134 9.14227 9.14320 9.I44I2 9.14 504 9-H597 9.14688 0.86052 0-85 959 0.85 866 0-85 773 0.85 680 0.85 588 0.85 496 0.85 403 0.85 312 0.85 220 9-99 59' 9-99 589 9-99 588 9-99 586 9-99 584 9-99 582 9-99 581 9-99579 9^9577 9-99 575 9-H35 6 u.I.l -S<> L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. ' P. P. 82' 428 8 1 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. p. p. 9-I4356 80 9.14780 0.85 220 9-99 575 60 I 2 3 4 I 9 9.14445 9-14535 9.14624 9.14714 9.14803 9.14891 9.14980 9.15069 9-15 157 90 89 90 89 88 89 88 9.14872 9.14963 9-15 54 9-I5 145 9.15236 9.15327 9-15417 9.15 508 90 90 0.85 128 0.85 037 0.84 946 0.84 855 0.84 764 0.84673 0.84 583 0.84 492 0.84 402 9-99 574 9-99 572 9-99 570 9-99 5 6 8 9-99 566 9-99 5 6 5 9-99 563 9-99 561 9-99559 59 58 57 56 55 54 53 52 5 1 92 91 90 i 9.2 9.1 9.0 2 18.4 18.2 18.0 3 27.6 27.3 27.0 4 36.8 36.4 36.0 5 46.0 45.5 45.0 6 55.2 54.6 54.0 7 64.4 63.7 63.0 IO 9-15245 88 9.15688 0.84312 9-99 557 50 ii 12 13 15 16 19 9-15 333 9.15421 9.15 508 9-15 596 9.15 683 9.15 770 9-I5857 9.I5 944 9.16030 88 87 88 87 87 87 87 86 86 9-15777 9.15867 9.I5956 9.16046 9-16133 9.16224 9.16 312 9.16401 9.16489 *N O O\ O O\ ON 00 ONOO 0< 3 ONOO O 00 OO 00 00 00 0( 0.84 223 0.84 133 0.84 044 0-83 954 0.83 865 0.83 776 0.83 688 0.83 599 0.83511 9-99 SS 6 9-99 554 9-99552 9-99 55 9-99 548 9.99 546 9-99 543 9-99 543 9-99 54i 49 48 47 46 45 44 43 42 89 88 i 8.9 8.8 2 17.8 17.6 3 ; 26.7 26.4 4 35-6 35-2 5 i 44.5 44.0 2O 9.16 116 87 9.16577 88 0-83 423 9-99 539 40 6 ! 53-4 52-8 21 22 23 24 25 26 27 28 29 9.16 203 9.16 289 9-16374 9.16460 9-16545 9.16631 9.16 716 9.16801 9.16886 86 85 86 85 86 85 8 5 9.16665 9-16753 9.16 841 9.16928 9.17016 9.17 103 9.17 190 9.17277 9-I7363 88 88 87 88 87 87 8 7 86 0-83 335 0.83 247 0-83 159 0.83 072 0.82 984 0.82 897 0.82810 0.82 723 0.82 637 9-99 537 9-99 535 9-99 533 9-99 532 9-99 53 9-99 528 9-99 526 9-99 524 9.99 522 39 38 37 36 35 34 33 3i 8 71.2 70.4 9 80. i 79.2 87 86 85 i 8.7 8.6 8.5 2 17.4 17.2 17.0 3 26.1 25.8 25.5 30 9.16970 Or 9.17430 S 7 86 0.82 550 9-99 520 30 4 34-8 34-4 34- 32 33 34 35 36 38 39 9-I7055 9-17 J 39 9.17223 9.17307 9-I739I 9-17474 9-I7558 9.17641 9.17724 5 8 4 84 84 8 4 83 8 4 83 ^ 9-I7536 9.17 622 9.17708 9.17 794 9.17880 9.17965 9.18051 9.18 136 9.18 221 86 86 86 86 85 86 11 0.82 464 0.82 378 0.82 292 0.82 206 0.82 1 20 0.82035 0.81 949 0.8 1 864 0.81 779 9.99518 9-99 5 1 7 9.995*5 9-99 5 J 3 9-99 5 11 9-99 59 9-99 57 9-99 55 9-99 53 29 28 27 26 2 5 24 23 22 21 6 52.2 51.6 51.0 7 60.9 60.2 59.5 8 69.6 68.8 68.0 9 78-3 774 76-5 84 83 i 8.4 8.3 40 9.17807 8 3 Si 9.18 306 0, 0.81 694 9-99 5 01 2O 42 43 44 11 47 48 49 9.17890 9-17973 9.18055 9-18137 9.18220 9.18 302 9-18383 9.18465 9-18547 3 OOOOOOOOOOOOOOOOCX 9.18391 9-I8475 9.18560 9.18644 9.18728 9.I88I2 9.18896 9.18979 9.19063 84 85 84 84 84 84 83 84 Ci 0.81 609 0.8 1 525 0.81 440 0.81 356 0.81 272 0.8 1 1 88 0.81 104 0.81 021 0.80 937 9-99 499 9-99 497 9-99 495 9-99 494 9-99 492 9.99 490 9-99 488 9.99 486 9-99 484 19 18 17 1 6 15 13 12 II 4 33-6 33-2 5 42.0 41.5 6 | 50.4 49.8 7 58.8 58.1 8 67.2 66.4 9 75-6 74-7 82 81 80 50 9.18628 81 9.19 146 Si 0.80 854 9-99 482 10 i 82 8 i 80 5' 52 53 54 55 S 6 58 59 9.18709 9.18790 9.18871 9.18952 9-19033 9.19113 9.19193 9.19273 9-19353 81 81 81 81 80 80 80 80 80 9.19229 9.I93I2 9-19395 9.19478 9.19561 9.19 643 9.19 725 9.19807 9.19889 3 OOOOOOOOOOOOOOOOOI 0.80 771 0.80 688 0.80 605 0.80 522 0.80 439 0.80 357 0.80 275 0.80 193 0.80 1 1 1 9-99 480 9-99 478 9.99476 9-99 474 9.99472 9-99 47 9.99 468 9.99 466 9-99 464 9 8 7 6 5 4 3 2 2 16.4 1 6.2 1 6.0 3 24.6 24.3 24.0 4 32.8 32.4 32.0 5 41.0 40.5 40.0 6 49.2 48.6 48.0 7 57-4 56-7 56-0 8 65.6 64.8 64.0 9 ; 73.8 72.9 72.0 6O 9-19433 9.19971 0.80 029 9-99 462 O ' L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. ' P. P. j 81 C 429 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. P. P. 9-19433 9.19971 0.80 029 9.99 462 60 I 2 3 4 I 7 8 9 9-i95'3 9.19592 9.19672 9.19751 9.19830 9.19909 9.19988 9.20067 9.20 145 79 80 79 79 79 79 79 78 9.20053 9.20 134 9.20 216 9.20 297 9.20 378 9.20459 9-20 540 9.20621 9.20 701 81 82 81 81 81 81 81 80 XT 0.79 947 0.79 866 0.79 784 0.79 703 0.79 622 0.79 541 0.79 460 0-79 379 0.79 299 9.99 460 9-99 458 9-99 45 6 9-99 454 9-99452 9-99 450 9.99 448 9-99 44 6 9.99 444 !I 57 56 55 54 53 5 2 Si 2 7 82 81 80 8.2 8.1 8.0 16.4 16.2 16.0 24.6 24.3 24.0 32.8 32.4 32.0 41.0 40.5 40.0 49.2 48.6 48.0 57-4 56-7 56-0 IO 9.20 223 7 8 9.20 782 Sr> 0.79 218 9-99 442 50 8 65.6 64.8 64.0 ii 12 13 H !S 17 18 19 9.20 302 9.20 380 9.20458 9-20 535 9.20613 9.20691 9.20 768 9.20 845 9.20922 79 78 78 77 It 77 77 77 9.20 802 9.20 942 9.21 022 9.21 IO2 9-21 l82 9-21 26l 9.21 341 9 .2I 4 20 9.21 499 so so So 80 79 80 79 79 0.79 138 0.79 058 0.78978 0.78 898 0.78818 0.78 739 0.78 659 0.78 580 0.78 501 9-99 44 9-99 438 9-99 436 9-99 434 9-99 432 9.99 429 9.99427 9.99425 9-99 423 49 48 47 46 45 44 43 4- 4 1 9 I 2 3 4 I 73.8 72.9 72.0 79 78 77 7-9 7-8 7-7 15.8 15.6 15.4 23.7 23.4 23.1 31.6 31.2 30.8 39-5 39-o 38-5 47.4 46.8 46.2 20 9.20999 77 9-21 578 0.78422 9.99421 4O I 55-3 54-6 53-9 21 22 23 24 2 5 26 27 28 29 9.21 076 9.21 153 9.21 229 9.21 306 9.21 382 9.21 458 9-2i 534 9.21 610 9.21 685 77 M 77 76 76 76 76 75 9.21 657 9-21 736 9.21 814 9.21 893 9.21971 9.22 049 9.22 127 9.22 205 9.22 283 79 78 79 78 78 78 78 7 * 0.78 343 0.78 264 0.78 1 86 0.78 107 0.78029 0.77951 0.77 873 0.77 795 0.77717 9.99419 9.99417 9-994I5 9-994I3 9.99411 9.99 409 9-99 47 9.99 404 9.99 402 39 38 37 36 35 34 33 32 3i 9 i 2 3 4 I 71.1 70.2 69.3 76 75 74 7- 6 7-5 7-4 15.2 15.0 14.8 22.8 22.5 22.2 30.4 30.0 29.6 38-0 37-5 37-o SO 9.21 761 / 9.22 361 7 0.77 639 9.99 400 30 45.6 45.0 44.4 31 32 33 34 P 37 38 39 9.21 836 9.21 912 9.21 987 9.22062 9-22 137 9-22 211 9-22 286 9.22361 9-22435 75 76 75 75 75 74 75 75 74 9.22 438 9.22 516 9-22 593 9.22 670 9-22 747 9.22 824 9.22 901 9.22977 9.23 054 77 78 77 77 77 77 77 76 11 0.77 562 0.77 484 0.77 407 o-77 33 o-77 253 0.77 176 0.77099 0.77023 0.76 946 9-99 398 9-99 396 9-99 394 9-99 392 9-99 390 9-99 388 9-99 385 9-99 383 9-99 38i 29 28 27 26 2 5 24 23 22 21 i 9 i 2 3 4 S3- 2 5 2 -5 5 J -8 60.8 60.0 59.2 68.4 67.5 66.6 73 72 71 7-3 7-2 7-i 14.6 14.4 14.2 21.9 21.6 21.3 29.2 28.8 28.4 40 9-22 509 74 9.23 130 76 0.76870 9-99 379 20 f 36.5 36.0 35.5 4i 42 43 44 9.22583 9.22657 9.22731 9-22 805 9 22 878 74 74 74 74 73 9.23 206 9.23 283 9-23 359 9-23435 Q 2"? SIO 76 H 76 75 0.76 794 0.76717 0.76 641 0.76 565 9-99 377 9-99 375 9-99 372 9-99 370 9 99 3^8 19 18 17 16 9 51.1 50.4 49.7 58.4 57-6 56-8 65.7 64.8 63.9 47 48 49 9.22952 9.23025 9.23 098 9.23 171 74 73 73 73 9.23 586 9.23 661 9-23 737 9.23812 76 75 76 75 0.76414 0-76 339 0.76 263 0.76 188 9-99 366 9-99 364 9-99 3 6 2 9-99 359 I 4 13 12 II o 333 79 78 77 13 2 13.0 12.8 SO 9.23 244 73 9-23 887 75 0.76 113 9-99 357 10 i 39-5 39- 38-5 5i 52 53 54 !i 12 59 9.23317 9.23 390 9.23 462 9-23 535 9.23 607 9.23 679 9-23752 9.23823 9-23 895 73 73 72 73 72 72 73 7i 7 2 9.23 962 9.24037 9.24112 9.24 1 86 9.24 261 9-24 335 9.24410 9.24484 9-24558 75 75 75 74 75 74 75 74 74 0.76038 o-75 963 0.75 888 0.75 814 o-75 739 0.75 665 0.75 590 o-75 5' 6 0.75 442 9-99 355 9-99 353 9-99 35 * 9-99 348 9-99 346 9-99 344 9-99 342 9-99 340 9-99 337 7 6 5 4 3 2 I 3 o i 2 3 65.8 65.0 64.2 333 76 75 74 12.7 12.5 12.3 38-0 37-5 37-o 63.3 62.5 61.7 60 9.23967 9.24632 0.75 368 9-99 335 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. ' P. P. 80 430 10 / L. Sin. d. L. Tan. |c. d. L. Cot. L. Cos. d. p. p. 1 I 2 3 4 9 IO ii 12 13 H 16 19 20 21 22 23 24 3 2 29 30 3i 32 33 34 P 11 39 4O 4i 42 43 44 8 3 49 50 5i 52 53 54 P P 59 6O 9.23 967 9-24 039 9.24 no 9.24 181 9.24 253 9.24 324 9-24 395 9.24 466 9-24 536 9.24607 7 2 7i 7i 72 7i 7i 7i 70 7i 70 7i 70 70 70 70 70 70 69 70 69 69 69 69 69 69 69 68 69 68 68 68 68 68 68 68 67 68 67 ^ 67 67 9.24 632 74 73 74 73 74 73 73 73 73 73 7 2 73 72 73 72 7 2 72 72 72 7 1 72 7 1 7 2 7i 7i 7 1 7 1 70 7i 7i 70 7 7 1 70 70 70 70 69 7 69 7 g 69 69 69 69 69 69 68 69 68 68 68 68 67 68 68 67 0.75 368 9-99 335 2 2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2 3 2 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 3 2 2 3 2 3 2 2 3 2 3 2 3 2 2 3 2 60 59 58 57 56 55 54 53 52 5 1 50 49 48 47 46 45 44 43 42 41 40 37 36 35 34 33 32 3i 30 29 28 2 7 26 25 24 23 22 21 2O 19 18 17 16 '5 M 13 12 II 10 9 8 7 6 5 4 3 2 I o 2 3 4 6 7 8 9 i 2 3 4 1 I 9 2 3 4 1 7 8 9 74 73 72 7-4 7-3 7-2 14.8 14.6 14.4 22.2 21.9 21.6 29.6 29.2 28.8 37-o 36-5 36-0 44.4 43.8 43.2 51.8 51.1 50.4 59-2 58-4 57-6 66.6 65.7 64.8 71 70 69 7.1 7.0 6.9 14.2 14.0 13.8 21.3 21.0 20.7 28.4 28.O 27.6 35-5 35- 34-5 42.6 42.0 41.4 49.7 49.0 48.3 56.8 56.0 55.2 63.9 63.0 62.1 68 67 66 6.8 6.7 6.6 13.6 13.4 13.2 20.4 20. i 19.8 27.2 26.8 26.4 34-o 33.5 33.0 40.8 40.2 39.6 47.6 46.9 46.2 54.4 53.6 52.8 61.2 60.3 59.4 65 3 i 6.5 0.3 2 13.0 0.6 3 19.5 0.9 4 26.0 1.2 5 32.5 i-5 6 39.0 1.8 7 45-5 2.1 8 52.0 2.4 9 58.5 2.7 9.24 706 9.24 779 9.24 853 9.24926 9.25000 9-25 73 9.25 146 9.25219 9.25 292 0.75 294 0.75 221 o-75 H7 0.75 074 0.75 ooo 0.74927 0.74 854 0.74 781 0.74 708 9-99 333 9-99 33i 9.99 328 9-99 326 9-99 324 9-99 322 9-99 3'9 9-993I7 9-993I5 9.24 677 9-25 365 0.74 635 9-993I3 9.24 748 9.24818 9.24 888 9.24958 9.25 028 9.25 098 9.25 1 68 9.25 237 9-25 37 9-25 437 9-25 5 10 9.25 582 9-25 655 9.25 727 9-25 799 9-25 871 9-25 943 9.26015" 0.74 563 0.74 490 0.74418 0-74 345 0.74 273 0.74 2OI 0.74 129 0.74057 0-73 985 9.99310 9-99 38 9-99 306 9-99 304 9-99 30i 9.99 299 9.99 297 9-99 294 9.99 292 9-25 376 0.739H 9.99 290 9-25 445 9-25 5H 9-25 583 9.25 652 9.25 721 9.25 790 9.25 858 9.25 927 9-25 995 9.26 158 9.26 229 9.26 301 9.26372 9.26443 9.26514 9.26 585 9.26655 9.26 726 0-73 842 0.73 771 0.73 699 0.73.628 0-73557 0.73 486 0-734I5 o-73 345 0.73 274 9.99 288 9-99 285 9-99 283 9.99 281 9-99 278 9.99 276 9-99 274 9.99271 9-99 269 9.26 063 9.26 797 0.73 203 9-99 267 9.26 131 9.26 199 9.26 267 9-26 335 9.26 403 9.26 470 9.26 538 9.26 605 9.26672 9.26 867 9.26937 9.27 008 9.27 078 9.27 148 9.27218 9.27 288 9-27357 9.27427 0-73 J 33 o-73 063 0.72 992 0.72922 0.72 852 0.72 782 0.72712 0.72 643 0-72 573 9-99 264 9.99 262 9.99 260 9-99 257 9-99 255 9-99 252 9-99 250 9.99 248 9-99 245 9.26 739 9.27496 0.72 504 9-99 243 9.26 806 9.26 873 9.26 940 9.27 007 9.27 073 9.27 140 9.27 206 9.27 273 9-27 339 67 67 67 66 67 66 67 66 66 66 66 65 66 66 65 65 66 65 65 9.27 566 9.27 635 9.27 704 9-27 773 9.27 842 9.27911 9.27 980 9.28049 9.28 117 0.72 434 0.72 365 0.72 296 0.72227 0.72 158 0.72089 0.72020 0.71 951 0.71 883 9-99 241 9-99 238 9-99 236 9-99 233 9-99 231 9-99 229 9.99 226 9-99 224 9.99 221 o i 2 3 S 7 l" $ 333 74 73 72 12-3 12.2 12.0 37-o 36.5 36.o 61.7 60.8 6O.O 333 1 70 69 68 .8 11.7 11.5 11.3 5 35-o 34-5 34-o 258-357-55 6 -7 9.27 403 9.28 1 86 0.71 814 9-99 219 9.27471 9-27 537 9.27 602 9.27 668 9-2? 734 9.27 799 9.27 864 9.27 930 9-27 995 9.28 254 9.28 323 9.28 391 9.28 459 9.28527 9.28 595 9.28 662 9.28 730 9.28 798 0.71 746 0.71 677 0.71 609 0.71 541 0-71 473 0.71 405 0-71 338 0.71 270 0.71 202 9.99217 9-99 214 9.99212 9-99 209 9.99 207 9-99 204 9.99 202 9.99 200 9 99 197 9.28 060 9.28 865 0.71 US 9.99 195 L. Cos. | d. L. Cot. c. d. L. Tan, L. Sin. d. ' P.P. 79 11 431 1 L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. p. p. 9.28060 6c 9.28 865 68 0-71 135 9-99 195 j 60 I 9.28 125 9-28933 f.- 0.71 067 9-99 192 59 2 9.28 190 5 9.29000 6* 0.71 ooo 9-99 190 2 58 68 67 66 3 9.28 254 6c 9.29 067 7 67 0.70933 9-99 187 57 i 6.8 6.7 6.6 4 9.28319 5 fie 9-29 134 67 0.70 866 9.99 185 56 2 13.6 13.4 13.2 9.28 384 g 9.29 2OI 0.70 799 9.99 182 SS 3 20.4 20. i 19.8 6 9.28 448 04 9.29 268 7 0.70 732 9.99 1 80 54 4 27.2 26.8 26.4 ; I 9 10 9.28 512 9.28577 9.28 641 04 65 I 4 64 61 9-29 335 9.29 402 9.29 468 07 67 66 67 66 0.70 665 0.70 598 0.70 532 9-99 177 9-99 175 9-99 172 3 2 3 2 3 53 52 5i 50 I Q 34-0 33-5 33-o 40.8 40.2 39.6 47.6 46.9 46.2 54-4 53-6 52-8 9.28 705 9-29 535 0.70465 9-99 170 ii 9.28 769 {.. 9.29 601 0-70 399 9-99 167 49 12 13 9.28 833 9.28 896 63 9.29 668 9-29 734 66 0.70332 0.70 266 9.99 165 9.99 162 3 48 47 65 64 63 14 9.28 960 64 64 9.29 800 66 0.70 200 9.99 1 60 46 i 6.5 6.4 6.3 ; IS 9.29024 6? 9.29 866 66 0.70 134 9-99 157 2 4S 2 13.0 12.8 12.6 16 9.29 087 9.29 932 0.70068 9-99 155 44 3 19.5 19.2 18.9 ;; 19 9.29 150 9.29214 9.29 277 3 64 63 ()! 9.29 998 9.30 064 9.30 130 66 66 6tr 0.70 002 0.69 936 0.69 870 9-99 IS 2 9-99 15 9-99 M7 3 2 3 2 43 42 41 4 I 26.0 25.6 25.2 32.5 32.0 31.5 39.0 '38.4 37.8 20 9.29 340 6-. 9.30 195 66 0.69 805 9-99 145 } 40 I 52.0 51.2 50.4 21 22 9.29 403 9.29 466 63 9.30 261 9.30 326 6 A 0.69 739 0.69 674 9-99 142 9-99 HO 2 39 38 9 58-5 57-6 5 6 -7 23 9.29 529 3 62 9-3039I 5 66 0.69 609 9-99 137 3 37 62 61 60 24 3 29 3O 9.29 591 9.29 654 9.29716 9.29 779 9.29 841 9.29 903 63 62 63 62 62 63 f>-> 9-30457 9.30522 9-30 5 8 7 9.30 652 9.30717 9.30 782 65 65 65 65 ^ S 64 fir 0.69 543 0.69 478 0.69413 0.69 348 0.69 283 0.69 218 9-99 135 9.99 132 9-99 130 9.99127 9-99 124 9.99 122 3 2 3 3 2 3 2 3 ft 35 34 33 32 3i 30 I 2 3 4 5 6 7 6.2 6.1 6.0 ' 12.4 12.2 I2.O 1 8.6 18.3 1 8.0 ! 24.8 24.4 24.0 31.0 30.5 30.0 37.2 36.6 36.0 43.4 42.7 42.0 9.29 966 9.30 846 0.69 154 9.99119 3 9.30028 62 9.30911 6A 0.69 089 9.99117 29 8 49.6 48.8 48.0 , 32 9.30 090 fii 9-30 975 04 6e 0.69 025 9-99 I H 28 9 55-8 54-9 54-o 33 9.30 151 62 9.31 040 64. 0.68 960 9.99112 27 34 9.30213 62 9.31 104 f.. 0.68 896 9-99 109 26 59 3 35 9.30 275 61 9.31 1 68 fie 0.68 832 9.99 106 2S 30 9-30 336 62 9-3i 233 5 6d. 0.68 767 9-99 104 24 i 5-9 -3 P 9-30 398 9-3 459 61 fi-> 9.31 297 9-3i 36i 64 64 0.68 703 0.68 639 9.99 101 9.99 099 2 23 22 3 17-7 -9 4 ! 23.6 1.2 39 9-305 2 i 61 9-3i 425 64 0.68 575 9-99 096 21 5 ! 29.5 1.5 40 9.30 582 61 9.31 489 h-i 0.68 511 9-99 093 2 2O 6 35.4 1.8 4 1 9-30 643 61 9.31 552 i " 0.68 448 9.99091 IP 7 4i-3 2.1 42 9.30 704 61 9.31 616 1 2 0.68 384 9.99 088 3 1 8 8 47.2 2.4 43 9.30 765 61 9.31 679 W J 6/1 0.68321 9.99 086 i? 9 53-i 2.7 44 9.30 826 61 9-3i 743 fit 0.68 257 9-99 083 16 45 9.30887 60 9.31 806 jg 0.68 194 9.99 080 15 46 9-3 947 61 9-31 870 | 0.68 130 9.99078 14 333 47 9.31 008 60 9-31 933 6 \ 0.68 067 9-99 075 13 67 66 65 48 9.31 068 61 9-3i 996 A 0.68 004 9.99 072 12 o 49 9.31 129 60 9-32059 1 0.67 941 9-99 070 II i 1 1.2 1 1.0 10.8 | 50 Si 9.31 iS<> 6l 60 9-32 122 63 fi-j 0.67 878 9-99 067 3 IO Q 2 3 33-5 33-o 32-5 i 55-8 55-0 54-2 9.31 250 9.32 185 0.67815 9.99 064 S2 9-3I3IO 60 9.32 248 ^3 0.67 752 9.99 062 2 8 53 9-3i 37 60 9-323II j 62 0.67 689 9-99 059 7 3 3 3 54 9-3i 430 60 9.32373 61 0.67 627 9-99 056 6 64 63 62 II P 9.31 490 9-3i 549 9.31 609 9.31 669 59 60 60 9-32436 9.32 498 9.32561 9-32 623 62 63 62 62 0.67 564 0.67 502 0.67 439 0.67 377 9-99 054 9.99051 9-99 048 9.99 046 3 3 2 5 4 3 2 o 10.7 10.5 10.3 32.0 31.5 31.0 53-3 52-5 5' -7 59 9.31 728 59 60 9-32 6S 5 62 0.67 31? 9^9043 I I 60 9.31 788 9-32 747 0.67 253 9-99 040 L. Cos. d. L. Cot. c. d.j L. Tan. L. Sin. d. ' P.P. 78 432 12 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d. p.p. 9.31 788 CO 9-32 747 61 0.67 253 9.99 040 2 60 I 2 3 4 5 6 9 9-3i 847 9.31 907 9.31 966 9.32 025 9.32 084 9-32 H3 9.32 202 9.32 26l 9-323J9 60 59 59 59 59 59 59 58 9.32 810 9.32 872 932933 9-32 995 9-33057 9-33II9 9-33 1 80 9-33 242 9-33 303 62 61 62 62 62 61 62 61 62 0.67 190 0.67 128 0.67 067 0.67 005 0.66 943 0.66881 0.66 820 0.66 758 0.66 697 9-99 038 9-99 35 9-99 032 9-99 030 9.99027 9.99 024 9-99 022 9.99019 9.99016 3 3 2 3 3 2 3 3 59 58 57 56 55 54 53 52 Si I 2 3 4 s 6 7 63 62 61 6.3 6.2 6.1 12.6 12.4 12.2 18.9 18.6 18.3 25.2 24.8 24.4 31.5 31.0 30.5 37.8 37.2 36.6 44-1 43-4 42.7 10 9-32378 9-33 365 01 0.66 635 9.99013 2 50 50.4 49.6 48.8 ii 12 13 H 16 17 19 9-32437 9-32 495 9-32 553 9.32612 9.32 670 9.32728 9.32 786 9-32-844 9.32 902 $ 59 58 58 58 58 58 eg 9-33426 9-33487 9-33 548 9-33 609 9-33 670 9-33 73i 9-33 792 9-33853 9-339I3 61 61 61 61 61 61 61 60 61 0.66 574 0.66513 0.66452 0.66 391 0.66 330 0.66 269 0.66 208 0.66 147 0.66 087 9.99OII 9-99 OO8 9-99 005 9.99 002 9.99 ooo 9.98 997 9.98 994 9.98991 9.98 989 3 3 3 2 3 3 3 2 i 49 48 47 46 45 44 43 42 41 60 59 i 6.0 5.9 2 I2.O 1 1. 8 3 18.0 17.7 I 24.0 23.6 5 30-0 29.5 3 36.0 35.4 20 9.32960 r8 9-33 974 fio 0.66 026 9.98 986 } 40 1 4 o' 4 ' <3 21 22 23 24 25 26 11 29 9.33018 9-33075 9-33 133 9-33 190 9-33 248 9-33 305 9-33 362 9-33 420 9-33477 57 58 57 58 57 57 58 57 9-34 034 9-34095 9-34 155 9-342I5 9.34276 9-34 336 9-34 396 9-344S 6 9-34 5 l6 61 60 60 61 60 60 60 60 60 0.65 966 0.65 905 0.65 845 0.65 785 0.65 724 0.65 664 0.65 604 0.65 544 0.65 484 9.98 983 9.98 980 9.98 978 9-98 975 9.98972 9.98 969 9.98 967 9.98 964 9.98961 3 2 3 3 3 2 3 3 P 37 36 35 34 33 32 3i j 54.0 53.1 58 57 i 5-8 5-7 2 1 1.6 11.4 3 17-4 I7- 1 j. 23.2 22.8 ') 29.0 28.5 3 T.A. 8 "'42 3O 9-33 534 9-34576 0.65 424 9-98958 3O 1 40.6 39-9 3 1 32 33 34 P 37 38 39 9-33 59i 9-33 647 9-33 74 9-33 76i 9.338i8 9-33 874 9-33 93i 9-33 987 9-34 043 56 57 57 57 56 57 56 56 C-7 9-34 635 9-34 695 9-34 755 9.34814 9-34 874 9-34 933 9-34 992 9-3505 1 9-35 1 " 60 60 59 60 59 59 59 60 eg 0.65 365 0.65 305 0.65 245 0.65 1 86 0.65 126 0.65 067 0.65 008 0.64 949 0.64 889 9-98955 9-98 953 9.98950 9.98 947 9.98 944 9.98 941 9.98 938 9.98 936 9-98 933 2 3 3 3 3 3 2 3 -i 29 28 27 26 25 24 23 22 21 < i 2 3 4 5 46.4 45- 6 ? 52-2 51.3 56 55 3 5- 6 5-5 -3 1 1. 2 II.O 0.6 16.8 16.5 0.9 22-4 22.O 1.2 28.0 27.5 1.5 4O 9-34 ioo c6 9-35 '?o 0.64 830 9.98 930 2O g 33- 6 33- J -8 4i 42 43 44 9.34156 9.34212 9-34 268 9-34 324 56 56 56 1-5 9-35 229 9-35 288 9-35 347 9-35 405 59 59 58 0.64771 0.64 712 0.64 653 0.64 595 9.98927 9.98924 9.98921 9.98919 3 3 2 19 18 17 16 7 8 9 39.2 38.5 2.1 44.8 44.0 2.4 50.4 49.5 2.7 4 1 46 47 48 49 9-34 38o 9-34 436 9-34491 9-34 547 9.34 602 56 55 56 y 9-35 464 9-35 523 9-35 58i 9-35 640 9-35 698 59 58 59 58 0.64 536 0.64477 0.64419 0.64 360 0.64 302 9.98 916 9.98913 9.98910 9.98 907 9.98 904 3 3 3 3 15 H 13 12 II o 333 62 61 60 10.3 10.2 10.0 50 9-34 658 re 9-35 757 rS 0.64 243 9.98 901 -? IO 2 51 52 53 54 55 56 H 59 9-34 7!3 9-34 769 9-34 824 9-34 879 9-34 934 9-34 989 9-35 044 9-35 099 9-35 154 56 55 55 55 55 55 55 55 cc 9-358i? 9-35 873 9-35 93i 9-35 989 9.36047 9.36 105 9.36 163 9.36 221 9.36 279 riOnt-nLn Ln Cn Oi Ln ^ t. a oooooooocoocoooo 0.64 185 0.64 127 0.64 069 0.64011 0.63953 0.63 895 0.63 837 0:63 779 0.63 721 9.98898 9.98 896 9.98 893 9.98 890 9.98 887 9.98 884 9.98 88 1 9.98 878 9-98875 2 3 3 3 3 3 3 3 7 6 5 4 3 2 I 3 o I 2 3 333 59 58 57 9-8 9-7 9-5 29.5 29.0 28.5 49.2 48.3 47.5 60 9-35 209 9-36 336 0.63 664 9.98 872 L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. d. ' P.P. 77 13 433 ; L. Sin. | d. L. Tan. c. d. L. Cot. L. Cos. d. p. p. ! 1 o 9-35 209 9-36 336 c8 0.63664 9.98 872 3 60 2 3 9-35 263 9-35 373 55 55 9-36 394 9.36452 9-36 59 58 57 0.63 606 0.63 548 0.63 491 y.yS 8(,y 9.98 867 9.98 864 2 3 59 58 57 j 58 57 56 5-8 5-7 5-6 4 1 9 9.35427 9-35 48i 9-35 536 9-35 590 9-35 6 44 9-35 698 54 54 55 54 54 54 9.36 566 9.36 624 9.36681 9-36 738 9-36 795 9.36852 W 58 57 57 57 57 r-j 0-63 434 0.63 376 0.63 319 0.63 262 0.63 203 0.63 148 9.98 861 9.98 858 9-98 853 9.98 852 9.98 849 9.98 846 3 3 3 3 3 3 56 55 54 53 52 2 11. 1) II.4 II. 2 3 17.4 17.1 16.8 4 | 23.2 22.8 224 5 29.0 28.5 28.0 6 34.8 34.2 33.6 7 40.6 39.9 39.2 8 ! 46.4 4^-6 44.8 10 9-35 752 ! ^ 9.36 909 0.63091 9.98 843 5O 9 5 2.2 51.3 504 ii 9-35 806 ?. 9.36 966 0.63 034 9.98 840 49 12 9-35 860 34 9-37023 57 0.62977 9.98837 3 48 55 54 53 3 9-35 9H 54 9.37 080 0.62 920 9.98 834 47 14 9-35 968 54 9-37 137 5/ 1 rf. 0.62 863 9.98831 3 46 1 5-5 5-4 5o \l 9.36022 9-36 075 54 53 9-37 !93 9-37 2 5 5 57 rf. 0.62 807 0.62 750 9.98828 ? 9-98825 3 45 44 3 16.5 16.2 15.9 4 22.O 21.6 21.2 17 9.36 129 9-37 3 6 0.62 694 9.98822 43 5 27.5 27.0 26.5 19 9.36 182 9-36 236 54 9-37 363 9-374I9 ^6 0.62 637 0.62 581 9.98819 9.98816 3 3 42 4i 6 33.0 32.4 31.8 7 .5 s -5 37-8 37-1 2O 9-36289 ^ 9-37476 r,6 0.62 524 9-98 813 - 40 8 44.0 43.2 42.4 21 9.36342 " 9-37 532 0.62 468 9.98 810 39 9 49-5 48-6 47.7 22 9.36395 r^ 9-37 588 rG 0.62412 9.98 807 3 J8 23 9-36449 ; ; 9-37 644 5 0.62 356 9.98 804 3 37 52 51 24 9.36 502 ; 9-37 700 5 rf. 0.62 300 9.98 80 1 3 36 i 5-2 5-i 2 5 26 9-36555 9.36608 53 9-37 75 6 9.37812 5^ 56 s6 0.62 244 0.62 1 88 9.98 798 9-98 795 3 3 35 34 2 3 10.4 10.2 '5-6 15-3 27 9.36 660 ; 9.37 868 0.62 132 9.98 792 33 4 20.8 204 28 9-36713 H 9o7924 rG 0.62 076 9.98 789 3 V 5 26.0 25.5 29 9.36 766 53 9.37 980 5 cc 0.62 020 9.98 786 3 6 31.2 30.6 3O 9.36819 t2 9-38035 r6 0.6 1 963 9-98 783 ? 30 7 36.4 35-7 32 9.36871 9.36 924 53 9.38091 9-38 H7 56 0.61 909 0.61 853 9.98 780 9.98 777 3 29 28 9 46.8 45-9 33 9-36 976 5 2 9.38 202 55 0.61 798 9-98 774 3 27 34 9-37 028 5 2 9-38 257 55 0.6 1 743 9.98771 3 26 4 3 35 9-37 081 9.38313 56 0.6 1 687 9.98 768 * 25 I 0.4 0.3 36 9-37 133 9.38 368 55 0.61 632 9.98 765 \ 24 2 0.8 0.6 37 9-37 J 85 ;>< 9-38 423 55 c6 0.61 577 9.98 762 \ 23 3 1.2 O.g 38 9-37 237 5 2 9-38 479 5 0.61 521 9-98 759 ^ 22 4 1.6 1.2 39 9.37 289 5 2 C2 9-38 534 55 cc 0.61 466 9.98 756 -J 21 5 2.0 1.5 4O 9-37 34i C2 9.38 589 ce 0.61 411 9.98 753 " 20 2.4 1.8 7 8 ? I 42 43 9-37 393 9-37 445 9-37 497 52 5 2 9.38 644 9.38 699 9-38 754 -> irlui 0.6 1 356 0.61 301 0.61 246 9.98 730 9-9S 746 \ 9.98 743 \ 19 18 17 9 3-2 24 3-6 2.7 ^ 9 38 808 i4 s In/ , 45 9.37600 5 1 9-38 863 55 0.61 137 9.98 740 9-98 737 * IS 46 9-37 6 52 5 2 9.38918 55 0.61 082 9-98 734 j 4 433 47 9-37 703 i* 1 9.38972 54 0.61 028 9-98 73i 3 13 55 54 58 57 48 49 9-37 755 9-37 806 5 2 C2 9.39027 9-39 082 55 55 p.6o 973 0.60918 9-98 728 \ 9-98 723 I \ 12 II | 6.9 6.8 9.7 9.5 '120.620.229.028.5 SO 9-37 858 y.j<; 136 0.60 864 9-98 722 , IO 34433.848.347.5 5i 9-37 909 5 9-39 190 0.60810 9.98719 9 348.147.2 5 2 9.37960 5 1 9-39 245 55 0.60 755 9-98 7'5 8 53 9.38011 5 9-39 299 54 0.60 701 9.98712 3 7 333 54 9.38062 * 9-39 353 0.60 647 9.98 709 6 56 55 54 11 9.38 164 5i 9-39 407 9.39 461 54 54 0.60 593 0.60 539 9.98 706 9.98 703 3 5 4 9-3 9-2 9-0 H 9.38215 9.38 266 5 1 51 9-39 5 J 5 9-39 569 54 54 0.60 483 0.60431 9.98 700 9.98 697 3 3 3 2 28.0 27.5 27.0 46.7 45-8 45-o 59 9-383I7 51 9-39 623 54 54 0.60 377 9 98 694 I [60 9.38 368 9-39 677 0.60 323 9.98 690 L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. d. ' P.P. 76 434 14 1 L. Sin. | d. L. Tan. c. d. L. Cot. L. Cos. d. p.p. o I 2 3 4 5 6 1 9 10 ii 12 13 M \l 17 18 19 20 21 22 23 24 3 3 29 30 3i 32 33 34 35 36 % 39 9.38 368 5 5 1 5 5 1 So 5 5 1 5 5 5 50 50 50 5 50 49 50 5 49 5 49 49 50 49 49 49 49 49 49 49 49 49 48 49 48 49 48 49 48 9-39677 54 54 53 54 53 54 53 54 53 53 54 53 53 53 53 53 53 52 53 53 53 52 53 52 53 S 2 52 52 53 S 2 5 2 52 52 5 2 5 2 S 2 5i 5 2 5 2 5 1 52 5 1 52 5i 5i 52 5 1 5i 5 1 5i 5 1 5i 5i 5 1 5i Si So 5i 5i 5 0.60 323 9.98 690 3 3 3 3 3 4 3 3 3 3 3 4 3 3 3 3 4 3 3 3 4 3 3 3 4 3 3 3 4 3 3 3 4 3 3 4 3 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 4 60 59 58 57 56 55 54 53 52 5i 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 3i 30 29 28 27 26 2 5 24 23 22 21 2O 19 18 17 16 15 14 13 12 II 10 9 8 7 6 4 3 2 I 2 3 4 7 8 9 2 3 4 i 7 8 9 i 2 3 4 7 8 9 2 3 4 1 9 54 53 5-4 5-3 10.8 10.6 16.2 15.9 21.6 21.2 27.0 26.5 32.4 31.8 37-8 37-i 43-2 42-4 48.6 47.7 52 51 50 5.2 5.1 5.0 10.4 10.2 IO.O 15.6 15.3 I 5 .0 20.8 20.4 20.0 26.0 25.5 25.0 31.2 30.6 30.0 36-4 35-7 35- 41.6 40.8 40.0 46.8 45.9 45.0 49 48 47 4-9 4-8 4-7 9.8 9.6 9.4 14.7 14.4 14.1 | 19.6 19.2 18.8 : 24.5 24.0 23.5 29.4 28.8 28.2 34-3 33-6 32.9 39.2 38.4 37.6 44.1 43.2 42.3 4 3 0.4 0.3 0.8 0.6 1.2 0.9 1.6 1.2 ,2.0 1.5 2.4 1.8 2.8 2.1 3-2 2.4 3-6 2.7 9.38418 9.38 469 9-38 5 J 9 9-38 57 9.38 620 9.38 670 9-38 7 21 9-38 77 1 9.38821 9-3973' 9-39 785 9 39 838 9-39 892 9-39 945 9-39 999 9.40052 9.40 1 06 9.40 159 0.60 269 0.60215 0.60 162 0.60 1 08 0.60053 0.60001 0.59 948 0.59 894 0.59841 9.98 687 9.98 684 9.98 68 1 9.98 678 9.98 675 9.98671 9.98 668 9.98 665 9.98 662 9.98 659 9.38871 9.40 212 0.59 788 9.38921 9-38971 9.39021 9.39071 9.39121 9-39 17 9.39 220 9.39 270 9-393I9 9.40 266 9.40319 9.40 372 9.40423 9.40 478 9-40 531 9.40 584 9.40 636 9.40 689 -59 734 0.59681 0.59 628 -59 575 0.59522 0.59 469 0.59416 0-59 364 0.59311 9.98 656 9.98652 9.98 649 9.98 646 9.98 643 9.98 640 9.98 636 9-98 633 9.98 630 9-39 369 9.40 742 0.59 258 9.98 627 9.39418 9-39 467 9-39 5 X 7 9-39 5 66 9-39 61? 9-39 664 9-397I3 9-39 762 9-39 81 1 9-40 793 9.40 847 9.40900 9.40952 9 . 4I 003 9-41 057 9.41 109 9.41 161 9.41 214 0.59 205 0-59 153 0.59 loo 0.59 048 0-58995 0.58 943 0.58891 0.58 839 0.58 786 9.98 623 9.98 620 9.98617 9.98614 9.98610 9.98 607 9.98 604 9.98 601 9-98 597 9.39 860 9.41 266 0.58 734 9-98 594 9-98 59 1 9.98 588 9.98 584 9.98 581 9.98 578 9-98 574 9.98571 9.98 568 9.98 563 9.98561 9-39 909 9-39 95 s 9.40 006 9.40 053 9.40 103 9.40152 9.40 200 9.40 249 9.40 297 9.41 318 9.41 370 9.41 422 9.41 474 9.41 526 94i 578 9.41 629 9.41 681 9-41 733 0.58 682 0.58 630 0.58578 0.58 526 0.58 474 0.58422 0.58371 0.58319 0.58 267 40 9.40 346 AS 9.41 784 0.58216 4i 42 43 44 45 46 47 48 49 9.40 394 9.40 442 9.40 490 9-4 538 9.40 586 9.40 634 9.40 682 9.40 730 9.40 778 48 48 48 48 48 48 48 48 47 48 48 47 48 47 48 47 47 47 48 9.41 836 9.41 887 9-41 939 9.41 990 9.42 041 9.42 093 9.42 144 9-42 195 9.42 246 0.58 164 0.58 113 0.58061 0.58010 0-57959 0.57 907 0.57 856 0-57 803 0-57 754 9.98558 9-98553 9.98551 9.98 548 9-98 543 9-98 54i 9.98 538 9-98 535 9-98 53i 4444 54 53 52 51 6.8 6.6 6.5 6.4 ( 2O.2 19.9 19.5 ig.I ,33-8 33-i 32-5 3i-9 ^47.246.445.544.6 3 3 3 j[ 54 53 52 51 9.0 8.8 8.7 8.5 t 27.0 26.5 26.0 25.5 145.044.2 43.3 42.5 50 9.40 825 9.42 297 0.57 703 9.98 528 5' 52 53 54 P 57 58 59 60 9.40 873 9.40921 9.40 968 9.41 016 9.41 063 9.41 in 9.41 158 9.41 205 9.41 252 9.42 348 9-42 399 9.42 450 9.42501 9-42 55 2 9.42 603 9.42653 9.42 704 9-42 753 9.42 805 o-57 652 0.57 601 o-57 53 0-57499 0.57 448 o-57 397 o-57 347 0.57 296 -57 245 9-98 523 9.98521 9.98518 9.98513 9.98511 9-98 508 9-98 53 9.98 501 9.98 498 9.41 300 o-57 195 9.98 494 L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. | d. ' P.P. 75 C 15 C / L. Sin. | d. L. Tan. c. d.| L. Cot. L. Cos. | d. p.p. o I 2 3 4 5 6 7 8 9 IO ii 12 '3 M 1 20 21 22 23 24 3 3 29 3O 3i 32 33 34 3 9 39 4O 4 1 42 43 44 % % 49 50 5 1 52 53 54 55 56 P 59 6O 9.41 300 47 47 47 47 47 47 46 47 47 46 47 46 47 46 47 46 46 47 46 46 46 46 46 46 45 46 46 46 45 46 45 46 45 46 45 45 46 45 45 45 45 45 45 45 44 45 45 45 44 45 44 45 44 45 44 44 44 45 44 44 9.42 805 5 1 5 5 1 50 50 5i 50 5 5 50 50 5 50 50 5 49 5 50 49 5 49 5 49 5 49 49 49 5 49 49 49 49 49 49 49 48 49 49 48 49 49 48 49 48 48 49 48 48 48 49 48 48 48 48 48 48 47 48 48 48 -57 '93 9.98 494 3 3 4 3 4 3 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 A 3 4 3 4 4 3 4 3 4 3 4 3 4 3 4 4 3 4 3 4 3 4 4 3 4 3 4 4 3 4 60 P 57 56 55 54 53 52 5i 5O 49 48 47 46 45 44 43 4 2 41 4O 39 38 37 36 35 34 33 32 3i 30 29 28 27 26 25 24 23 22 21 2O 19 18 1 7 16 15 H 13 12 II 10 9 8 7 6 5 4 3 2 I 2 I 3 i 4 2 i\ \\ 9 A 2 3 i 4 i 3 \ 7 3 8 3 9 4 i 2 3 4 5 6 9 2 3 4 5 6 I 9 51 50 49 5.1 5.0 4.9 O.2 IO.O 9.8 5-3 15-0 '4-7 0.4 20.0 19.6 5.5 25.0 24.5 0.6 30.0 29.4 5-7 35-o 34-3 0.8 40.0 39.2 5-9 45-o 44-i 48 47 46 4.8 4.7 4.6 9.6 9.4 9.2 4.4 14.1 13.8 9.2 18.8 18.4 40 23.5 23.0 8.8 28.2 27.6 3.6 32.9 32.2 8.4 37.6 36.8 3.2 42.3 41.4 45 44 4-5 44 9.0 8.8 13-5 13-2 18.0 17.6 22-5 22.0 27.0 26.4 31.5 30.8 36.0 35.2 40.5 39.6 4 3 0.4 0.3 0.8 0.6 1.2 0.9 1.6 1.2 2.0 1.5 2.4 1.8 2.8 2.1 3-2 2.4 3-6 2.7 9-4i 347 9-4i 394 9.41 441 9.41 488 9-4i 535 9.41 582 9.41 628 9.41 675 9.41 722 9-42 856 9.42 906 y-42957 943 007 943057 943 108 943 15 s 943 208 943 258 0.57 144 0.57094 0.57043 0.56 993 0.56 943 0.56 892 0.56 842 0.56 792 0.56 742 9.98491 9.98 488 9.98 484 9.98481 998477 9.98 474 9.98471 9.98 467 9.98 464 9.41 768 9-43 308 0.56 692 9.98460 9.41 815 9.41 861 9.41 908 9-41 954 9.42001 9.42 047 9.42 093 9.42 140 9.42 186 942 232 9.42 278 9.42 324 9.42 370 9.42416 9.42 461 9-42 57 9-42 553 9.42 599 9-42 644 943 35 8 9-43 408 9-43 45 8 943 58 943 55 943 607 9-43 657 943 707 943 756 0.56 642 0.56 592 0.56 542 0.56492 0.56 442 0.56393 0-56 343 0.56 293 0.56 244 9.98457 9-98453 9.98 450 9.98 447 9-98 443 9.98 440 9.98436 9-98433 9-98 429 9.43 806 0.56 194 9.98 426 943 855 943 903 9-43 954 9.44004 9-44053 9.44 102 9.44I5I 9.44 201 9-44 250 0.56 143 0.56095 0.56 046 0-55 996 o-55 947 0.55 898 0.55 849 o-55 799 o-55 75 9.98 422 9.98419 9.98415 9.98412 9.98 409 9.98 405 9.98 402 9.98 398 9-98 393 944 299 o-55 7 01 9.98391 9-42 735 9.42 781 9.42 826 9.42 872 9.42917 9.42 962 9.43 008 9-43 053 9.43 098 9-44 34 9-44 397 9.44446 944 493 9-44 544 944 592 9.44641 944 690 9-44 738 0-55 652 o-55 603 0-55 554 o-55 55 0.55 456 0.55 408 0-55 359 0-55 3'o 0.55 262 9.98 388 9.98 384 9.98381 9-98 377 9-98 373 9.98 370 9.98 366 9-98 363 9-98 359 9-43 143 944 787 _ 944 836 9-44 884 9-44 933 9.44981 945 029 945 o?^ 945 I26 945 J 74 9-45 222 o-55 213 0.55 164 -55 n6 0.55 067 0-55 019 0.54971 0.54 922 0.54 874 0.54 826 0-54 778 9.98 356 9.98352 998349 9-98 345 9.98 342 9.98 338 9-98 334 9-9833I 9.98 327 9.98 324 9-43 1 88 9-43 233 9.43 278 9-43 323 9-43 367 9.43412 943457 9-43 502 9-43 546 4444 50 49 48 47 o' 6.2 6.1 6.0 5.9 i S. Si 8.4 1 8.0 1 7.6 ,31.230.630.029.4 j 43.8 42-9 42-0 41. i 3333 51 50 49 48 | 8.5 8.3 8.2 8.0 , 25.5 25.0 24.5 24.0 ; 42.5 4 1. 7 40.8 40.0 9-43591 945 271 0.54 729 9.98 320 9-43 635 9-43 680 9-43 724 9-43 7 6 9 9-438I3 9-43 857 9-43 901 9.43 946 9-43 990 945 3'9 945 367 945413 945 463 9455H 9-45 559 945 606 9-45 6 54 945 702 0.54681 0-54 633 0-54 5 8 5 0-54 537 0.54489 0.54441 0.54 394 0.54 346 0.54 298 9-98317 9-983I3 9.98 309 9.98 306 9.98 302 9-98 299 9.98 295 9.98 291 9.98 288 9-44 034 945 73 0.54 250 9.98 284 L. Cos. d. L. Cot. lc. d. L, Tan, L. Sin. | d. 1 P. P. 74' 436 16 ' L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. p. p. 2 3 4 I 9 IO n 12 13 14 17 18 19 2O 21 22 23 24 2 5 26 3 29 30 3i 32 33 34 i 39 40 42 43 44 45 46 47 48 49 50 53 54 P 59 60 9.44 034 44 44 44 44 43 44 44 44 43 44 44 43 43 44 43 44 43 43 43 43 43 44 43 42 43 43 43 43 43 42 43 42 43 42 43 42 42 42 43 42 42 42 42 4 2 42 42 4i 42 42 4 2 4 42 4i 42 4i 42 4i 42 9-45 73 47 48 47 48 3 47 48 47 47 47 47 47 47 47 47 47 47 46 47 47 47 46 47 47 46 47 46 46 47 46 46 46 46 46 46 46 46 46 46 45 46 46 46 45 46 45 S 45 8 45 o-54 250 9.98 284 3 4 4 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 4 3 4 4 3 4 4 4 3 4 3 4 4 4 3 4 4 4 4 3 4 4 4 4 4 3 4 4 4 4 4 4 3 60 59 58 57 56 55 54 53 52 5i 50 49 48 47 46 45 44 43 42 40 38 37 36 35 34 32 30 27 26 25 24 22 21 2O 19 18 16 15 14 13 12 IO 7 6 5 4 3 2 I o 2 3 i 4 i 5 2 6 2 11 9 A i 2 3 i 4 i 1 \ 8 2 9 4 i 2 3 4 I I 9 i i i i 9 48 47 46 4.8 4.7 4.6 9-6 9-4 9-2 4.4 14.1 13.8 9.2 18.8 18.4 4.0 23.5 23.0 8.8 28.2 27.6 3.6 32.9 32.2 8.4 37.6 36.8 3.2 42.3 41.4 45 44 43 4-5 4-4 4-3 9.0 8.8 8.6 3.5 13.2 12.9 8.0 17.6 17.2 2-5 22.0 21.5 7-0 26.4 25.8 1.5 30.8 30.1 6.0 35.2 34.4 o-5 39-6 38-7 42 41 4.2 4.1 8.4 8.2 12.6 12.3 16.8 16.4 21.0 20.5 25.2 24.6 29.4 28.7 33-6 32-8 37-8 36-9 4 3 0.4 0.3 0.8 0.6 1.2 0.9 1.6 1.2 2.0 1.5 2.4 1.8 2.8 2.1 3-2 2.4 3-6 2.7 9.44078 9.44 122 9.44 1 66 944 210 9-44 253 9-44 297 9-44 341 9-44 385 9-44 428 9-45 797 9-45 845 9-45 892 9.45 940 9-45 987 9.46 035 9.46 082 9.46 130 9-46 177 9.46 224 0.54 203 o-54 155 0.54 108 0.54060 0.54013 0-53 965 0.53918 0.53 870 0-53823 9.98 281 9.98 277 9-98 273 9.98 270 9.98 266 9.98 262 9.98 259 9-98 255 9.98251 9.44472 0.53 776 9.98 248 9.44516 9-44 559 9.44 602 9.44 646 9-44 689 9-44 733 9-44 776 9.44819 9.44 862 9.46 271 9-463I9 9.46 366 9.46413 9.46 460 9.46 507 946 554 9.46 60 1 9.46 648 0.53 729 0.53681 0-53 634 o-53 587 o-53 540 o-53 493 0.53 446 o-53 399 0.53352 9-98 244 9.98 240 9.98 237 9-98 233 9.98 229 9.98 226 9.98 222 9.98218 9.98215 9-44 905 9-46 694 0.53 306 9.98 211 9.44 948 9-44 992 9-45 35 9-45 077 9.45 120 9-45 163 9 45 206 9-45 249 9-45 292 9.46 741 9.46 788 9.46 835 9.46881 9.46928 9.46 975 9.47021 9.47 068 9-47 "4 0-53 259 0.53212 o-53 165 0.53119 0.53072 0-53025 0.52979 0.52932 0.52886 9.98 207 9.98 2O4 9.98 200 9.98 196 9.98 192 9.98 189 9-98 185 9.98 181 9.98177 9-45 334 9.47 1 60 0.52 840 9.98 174 9-45 377 9-45 4i9 9-45 462 9-45 54 9-45 547 9-45 589 9-45 632 9-45 674 9-45 7i6 9-45 758 9.45 801 9-45 843 9-45 885 9-45 927 9-45 969 9.46011 9.46053 9.46095 9.46 136 9-47 207 9-47 253 9-47 299 9-47 346 9-47 392 9-47 438 9-47 484 9-47 530 9-47 576 0-52 793 0-52 747 0.52 701 0.52654 0.52608 0.52 562 0.52 516 0.52 470 0.52424 9.98170 9.98 1 66 9.98 162 9.98 159 9.98 155 9.98151 9.98 147 9.98 144 9.98 140 9.47 622 0.52378 9.98 136 9.47 668 9.47 7'4 9-47 76o 9.47 806 947852 9-47 897 9-47 943 9-47 989 9.48 035 0.52 332 0.52 286 0.52 240 0.52 194 0.52 148 0.52 103 0.52057 0.52011 0.51965 9-98 132 9.98 129 9.98125 9.98 121 9.98117 9 . 9 8lI3 9.98 no 9.98 106 9.98 102 4444 48 47 46 45 6.0 5.9 5.8 5.6 1 8.0 17.6 17.2 16.9 2 30.0 29.4 28.8 28.1 3 42.0 41. i 40.2 39.4 3333 48 47 46 45 8.0 7.8 7.7 7.5 24.023.523.022.5 3 40.0 39.2 38.3 37.5 9.46178 9.48 080 0.51 920 9.98098 9.46 220 9.46 262 9-46 303 946 386 9.46 428 9.46469 9.46511 9-46552 9.48 126 9.48171 9.48217 9.48 262 9.48 307 948 353 9-48 398 9-48 443 9.48 489 0.51874 0.51 829 0.51 783 0-51 738 0.51 693 0.51 647 0.51 602 o-S 1 557 0.51511 9.98 094 9.98 090 9.98087 9.98 083 9.98 079 9.98 075 9.98071 9.98 067 9.98063 9-46 594 9-48 534 0.51 466 9.98 060 L. Cos. d. L. Cot. c. d.i L. Tan. L. Sin. I d. ' P.P. 73 17 437 1 L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. P. P. o 9.46 594 41 948 534 41: 0.51 466 9.98 060 4 60 I 2 3 4 I 7 8 9 946 635 9.46 676 9.46717 9.46 758 9.46 800 9.46841 9.46 882 9.46 923 9.46 964 4i 4i 4i 42 4i 4i 4i 4i 948 579 9.48 624 9.48 669 9.48 714 9-48 759 9.48 804 9.48849 9.48 894 9-48 939 45 45 45 45 45 45 45 45 0.51421 0.51 376 0-51 33i 0.51 286 0.51 241 0.51 196 0.51 151 0.51 106 0.51 06 1 9.98 056 9.98052 9.98 048 9.98 044 9.98 040 9.98 036 9.98 032 9.98 029 9.98 025 4 4 4 4 4 4 3 4 59 58 57 56 55 54 53 5 2 5 1 2 3 4 5 6 7 45 44 43 4-5 4-4 4-3 9.0 8.8 8.6 13.5 13.2 12.9 18.0 17.6 17.2 22-5 22.O 21.5 27.0 26.4 25.8 31.5 3O.8 30.1 IO 9-47 oo5 9.48 984 0.51 016 9.98021 5O 8 36.0 35.2 34.4 12 13 14 15 16 19 9-47 45 9.47 086 9.47 127 9.47 1 68 9-47 209 9.47 249 9-47 29 9-47 330 9-47 37i 4i 4i 4i 4i 40 4i 40 4i 40 9.49 029 9-49 073 9.49 118 9.49 163 9.49 207 9.49 252 9-49 296 9-49 341 9-49 385 44 45 45 44 45 44 45 44 41: 0.50971 0.50927 0.50 882 0.50 837 -5 793 0.50 748 0.50 704 0.50 659 0.50615 9.98017 9.98013 9.98 009 9.98005 9.98001 9-97 997 9-97 993 9-97 989 9-97 986 4 4 4 4 4 4 4 3 49 48 47 46 45 44 43 42 4i 9 i 2 3 4 40-5 39-6 38-7 42 41 40 4.2 4.1 4.0 8.4 8.2 8.0 12.6 12-3 I2.O 1 6.8 16.4 16.0 20 9.47411 41 9-49 430 0.50570 9.97 982 4O g 25.2 24.6 24.0 21 22 23 24 25 26 27 28 29 9.47452 9.47 492 9-47 533 9-47 573 9.47613 9-47 6 54 9.47 694 9-47 734 9-47 774 40 4i 40 40 4i 40 40 40 9.49 474 9-495I9 9-49 5 6 3 9-49 6 7 9.49 652 9.49 696 9-49 740 9-49 784 9.49 828 45 44 44 45 44 44 44 44 0.50526 0.50481 0.50 437 0-50 393 0.50 348 0.50 304 0.50 260 0.50 216 0.^0 1/2 9.97978 9-97 974 9.97 970 9.97 966 9.97 962 9-97 95 s 9-97 954 9-9795 9-97 946 4 4 4 4 4 4 4 4 39 38 37 36 35 34 33 32 3i 9 i 2 29.4 28.7 28.0 33.6 32.8 32.0 37-8 36-9 36-0 39 5 4 3 3.9 0.5 0.4 0.3 7.8 i.o 0.8 0.6 3O 9.47814 9.49 872 0.50 128 9-97 942 3O 3 11.7 1.5 1.2 0.9 3i 32 33 34 P P 9-47 854 9.47 894 9-47 934 9-47 974 9.48014 9.48 054 9.48 094 948 133 40 40 40 40 40 40 39 9.49916 9.49 960 9.50 004 9.50 048 9.50092 9.50 136 9.50 1 80 9.50 223 44 44 44 44 44 44 43 0.50 084 0.50 040 0.49 996 0.49 952 0.49908 0.49 864 0.49 82O o-49 777 9-97 938 9-97 934 9-97 930 9.97 926 9.97922 9.97918 9.97914 9.97910 4 4 4 4 4 4 4 29 28 2 7 26 25 24 23 22 4 I 7 S 9 15.6 2.O 1.6 1.2 19.5 2.5 2.O 1.5 23.4 3.0 2.4 1.8 27.3 3.5 2.8 2.1 31.2 4.0 3.2 2.4 35.1 4.5 3.6 2.7 39 9-48I73 4 9.50 267 44 0-49 733 9-97 9o6 4 21 40 9.48213 9.50311 o. 4 y 689 9.97 902 20 4i 42 43 44 45 46 S 49 9.48 252 9.48 292 948 332 9.48371 9.48411 9.48 450 9.48 490 9.48 529 9.48 568 40 40 39 40 39 40 39 39 9-5 355 9.50398 9.50 442 9.50 485 9-5 529 9-5 572 9.50616 9-5 6 59 9-5 703 43 44 43 44 43 44 43 44 0.49 645 0.49 602 0.49 558 0-49 5 5 0.49471 0.49 428 0.49 384 0.49341 0.49 297 9-97 898 9-97 894 9-97 890 9.97 886 9.97 882 9.97878 9-97 874 9.97 870 9.97 866 4 4 4 4 4 4 4 4 19 18 '7 16 15 14 '3 12 II 2 3 4 5 544 43 45 44 4-3 5-6 5-5 12.9 16.9 16.5 21.5 28.1 27.5 30.1 39.4 38.5 38-7 - - 50 9.48 607 9.50 746 0.49 254 9.97 8(i IO 5i 52 53 54 P !i 59 9.48 647 9.48 686 9.48 725 9.48 764 9.48 803 9.48 842 9.48 881 9.48 920 9.48 959 39 39 39 39 39 39 39 39 ^Q 9.50789 9-5 833 9.50876 9.50919 9.50 962 9-5 l 5 9.51 048 9.51092 9-Si '35 44 43 43 43 43 43 44 43 0.49211 0.49 167 0.49 124 0.49081 0.49 038 0.48 995 0.48952 0.48 908 048865 9-97 857 9-97 853 9.97 849 9-97 845 9-97 841 9-97 837 9-97 833 9-97 829 9.97 823 4 4 4 4 4 4 4 4 9 8 7 6 5 4 3 2 I o i 2 3 4 433 43 45 44 5-4 7-5 7-3 I6.I 22-5 22.0 26.9 37-5 36-7 37-6 |60 9.48 998 9.51 178 0.48 822 9.97 821 O L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. d. ' P.P. 72 438 18 ' L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. p.p. 9.48 998 9.51 178 0.48822 9.97 821 6O I 9-49 037 9.51 221 0.48 779 9.97817 59 2 9-49 076 39 9-5 1 264 43 0.48 736 9.97812 5 S8 3 949"5 39 9.51 306 4 2 0.48 694 9.97 808 4 S7 43 42 41 4 5 9-49 153 9-49 192 ^ 39 9-51 349 9-5 ! 392 43 43 0.48651 0.48 608 9-97 804 9.97 800 4 4 56 55 I 2 4-3 4-2 4-1 8.6 8.4 8.2 6 9-49 231 39 9-5M35 43 0.48 565 9-97 796 54 3 12.9 12.6 12.3 7 8 9 10 9-49 269 9-49 308 9-49 347 3 s 39 39 38 9-5I478 9-51 520 9-51 5 6 3 9.51 606 43 42 43 43 0.48 522 0.48 480 0.48 437 9-97 792 9-97 788 9-97 784 4 4 4 5 53 S 2 50 4 i i 17.2 16.8 16.4 21.5 21.0 20.5 25.8 25.2 24.6 30.1 29.4 28.7 34.4 33.6 32.8 9-49 385 0.48 394 9.97 779 n 12 9-49 424 9-49 462 38 78 9.51 648 9.51 691 43 0.48 352 0.48 309 9-97 775 9.97771 4 49 48 9 38.7 37.8 36.9 13 9-49 5 9-5 ! 734 43 0.48 266 9-97 7 6 7 47 I 4 9-49 539 39 78 9.51 776 4 2 0.48 224 9-97 763 46 IS 9-49 577 78 9.51819 0.48 181 9-97 759 7 45 39 38 37 16 9.49615 9.51 861 0.48 139 9-97 754 3 44 i 3-9 3-8 3.7 17 18 9-49 654 9-49 692 38 78 9.51903 9.51946 43 0.48097 0.48 054 9-97 75 9-97 746 4 43 42 2 S 7-8 7-6 7-4 11.7 11.4 n. i 19 9-49 730 9.51 988 0.48012 9.97 742 4 15.6 15.2 14.8 20 9-49 7<>8 38 9.52031 42 o-47 969 9-97 738 40 5 19.5 19.0 18.5 21 9.49 806 78 9.52073 0-47 927 9-97 734 39 23.4 22.8 22.2 22 23 9-49 844 9.49 882 38 9-52115 9-52 157 42 0.47 883 0-47 843 9-97 729 9-97 725 5 4 38 37 7 8 27.3 26.6 25.9 31.2 30.4 29.6 24 9-49 920 ^ 78 9.52 200 43 0.47 800 9-97 72i 76 y 35- 1 34-2 33-3 III 9-49 95 s 9-49 996 38 9.52 242 9.52 284 42 o-47 758 0.47 716 9.97717 9.977U 4 35 34 11 29 9-5 34 9.50072 9.50 no 3 5 38 38 9-52 326 9-52 368 9.52410 4 2 42 42 42 o-47 6 74 o-47 632 0-47 590 9-97 78 9-97 74 9-97 700 b 4 4 33 32 i 36 5 4 3.6 0.5 0.4 30 9.50 148 9-5245 2 o-47 548 9.97696 i 7 30 3 1 32 ! 33 9-5 l8 5 9.50223 9.50 261 38 38 9-5 2 494 9-52 536 9.52 578 42 42 0.47 506 0-47 464 0.47 422 9.97691 9-97 687 9-97 68 3 3 4 4 2 9 28 27 4 5 6 14.4 2.0 1.6 1 8.0 2.5 2.0 21.6 3.0 2.4 ; 14 9.50 298 78 9.52 620 0.47 380 9-97 6 79 26 25.2 3.5 2.8 35 9-5 336 78 9.52 661 0-47 339 9-97 674 25 8 28.8 4.0 3.2 36 9-5 374 9.52 703 0.47 297 9-97 67 24 9 32.4 4.5 3.6 i ^7 9.50411 78 9-52 745 0-47 255 9.97 666 23 38 9-5 449 77 9-52 787 o-47 213 9.97662 i * 22 39 9.50486 9.52 829 4 1 0-47 I7 1 9-97657 i i 21 4O 9-50523 7,8 9.52870 42 0.47 130 9-97653 ! 20 41 9.50561 9.52912 0.47 088 9.97649 i 19 555 42 9-5 598 37 9-52953 4 2 0.47 047 9-97645 i e 18 43 42 41 43 9.50 635 ->8 9-52995 o-47 5 9.97640 3 1 7 44 9.50 673 9.50710 9-5 747 J 5 37 37 9-53037 9-53078 9-53 120 42 42 0.46 963 0.46 922 0.46 880 9-97 636 9-97 632 9.97 628 16 15 i 2 3 4-3 4-2 4-i 12.9 12.6 12.3 21.5 2I.O 20.5 47 ; 48 9.50 784 9.50821 37 37 9-53 161 9-53 202 41 4i 0.46 839 0.46 798 9-97 623 9.97 619 } 13 12 4 5 30.1 29.4 28.7 38-7 37-8 36-9 49 9.50 858 18 9-53 244 0.46 756 9.97613 II 5O 9.50 896 9-53 285 0.46715 9.97610 10 51 9-5 933 9-53 327 0.46 673 9.97606 ; ; 9 44. 4 5 2 9-5 97 9.53368 0.46 632 9.97 602 8 5S 9.51 007 9-53 409 4 1 0.46 591 9-97 597 5 7 43 42 41 54 SS 9.51043 9.5 1 080 .)6 37 9-5345 9-53492 4 2 0.46 550 0.46 508 9-97 593 9-97 589 4 4 6 S o i 5-4 5-2 5- 1 16.1 15.8 15.4 S6 9-5 1 "7 9-53533 4 1 0.46 467 9-97 584 4 ~ 26.9 26.2 25.6 9-5 1 J 54 37 9-53 574 0.46 426 9-97 580 T, 3 37.6 36.8 35.9 58 9.51 191 37 76 9-536I5 0-46 383 9-97 576 2 S9 9.51 227 77 9-53656 4 1 0.46 344 9-97 57 1 I 60 9.51 264 9-53697 0.46 303 9-97 5 6 7 L. Cos. d. L. Cot. |c. d. L. Tan. L. Sin. d. | ' P.P. 71 19 439 / L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. p. p. o 9.51 264 9-53697 ,, 0.46 303 9-97567 ' 60 I 9.51 301 9-53 738 0.46 262 9-97 563 S9 2 3 9oi 338 9-5 1 374 37 36 9-53 779 9.53 820 4 1 0.46 221 0.46 1 80 9-97 558 9-97 554 5 4 58 57 41 40 39 4 5 9.51411 37 36 9.53861 9-53 92 4 1 41 0.46 139 0.46 098 9-97 55 9-97 545 5 56 SS 2 4.1 4.0 3.9 8.2 8.0 7.8 6 9 9.51 484 9.51 520 9.51557 9-51 593 37 36 37 9-53 943 9-53 984 9-54025 9.54065 4 1 4i 4i 40 0.46 057 0.46016 0-45 975 0-45 935 9-97 54i 9-97 536 9-97 532 9-97 528 5 4 4 54 53 52 3 4 i 12-3 I2.O II-7 16.4 16.0 15.6 20.5 20.0 19.5 24.6 24.0 23.4 IO 9.51 629 37 9-54 106 ;, 0.45 894 9-97 523 50 8 32.8 32.0 31.2 ii 9.51 666 9-54 H7 0-45 853 9-97 5 f 9 49 9 36.9 36.0 35.1 12 9.51 702 36 9-54 187 o-45 813 9-97 5'5 48 1 3 9-51 738 9-54 228 o-45 77 2 9.97510 47 14 IS 9-51 774 9.51811 37 9-54 269 9-54 309 40 0-45 73i 0.45 691 9-97 5 6 9-97 5 01 5 46 4S 37 36 35 16 9-51 847 9-54 35 0.45 650 9-97497 44 I 3-7 3-6 3-5 17 18 19 20 9.51 883 9-5I9I9 9-5 1 955 I 36 9-54 390 9-54431 9-54471 40 4* 40 0.45 610 0.45 569 o-45 529 9-97492 9-97488 9-97 484 4 4 5 43 42 40 2 3 4 5 5 7-4 7-2 7- 1 1.1 10.8 10.5 14.8 14.4 14.0 18.5 18.0 17.5 9J>99J 9.54512 0.45 488 9-97 479 21 22 23 9.52027 9.52063 9.52099 36 36 9-54552 9-54 593 9-54 633 41 40 40 0-45 448 0.45 407 0.45 367 9-97475 7 9.97470 9-97 466 J 39 38 37 7 8 9 25.9 25.2 24.5 29.6 28 8 28.0 33-3 32.4 *i.c 24 9-52 135 j6 9-54673 41 o-45 327 9.97461 36 2 5 9.52171 36 9-54 7'4 40 0.45 286 9-97457 35 26 9.52 207 9-54 754 0.45 246 9-97 453 34 27 9.52 242 35 9-54 794 0.45 206 9-97 448 33 34 5 4 28 9-52278 36 9-54835 40 0.45 165 9-97 444 J 32 i 3.4 0.5 0.4 29 9-52 3 J 4 9.54 8/5 AO 0-45 125 9-97 439 ;| 3 1 2 6.8 i.o 0.8 30 9-52 35 9.54915 ^ 0.45 085 9-97435 P 30 3 10.2 1.5 1.2 32 9-52 385 9.52421 36 9-54 955 4 o 9-54995 Tn 0.45 045 0.45 005 9-9743 i 9-97 426 7 29 28 4 13.6 2.0 1.6 17.0 2.5 2.0 33 34 P 9-52456 9.52492 9-52527 9-52 563 36 35 36 9-55035 9.55075 9-55 "5 9-55 155 4 4O 4 0.44 965 0.44 925 0.44 885 0.44 845 9.97421 9.97417 9.97412 9.97 408 4 5 4 27 26 2 5 24 7 8 9 20.4 3-O 2.4 23-8 3-5 2.8 27.2 4.0 3.2 30.6 4.5 3.6 37 9-52598 35 9-55 195 0.44 805 9-97 403 5 23 ' 38 9.52 634 3*-* 9-55 2 35 0.44 765 9-97 399 22 39 9.52669 -6 40 0.44 725 9-97 394 ;| 21 4O 9.52 705 9-553I5 40 0.44 685 9-97390 r 20 41 9.52 740 9-55 355 0.44 645 9.97385 | 19 555 42 9-5 2 775 9-55 395 ! "g 0.44 605 9.97381 * 18 41 40 39 43 9.52 811 9-55 434 ~ 0.44 566 9-97 376 5 '7 o 44 45 46 9.52 846 9.52881 9.52916 35 35 35 9-55474 9-555*4 9-55 554 40 40 30 0.44526 0.44486 0.44 446 9-97 372 9-97 367 9-97 363 5 4 5 16 15 H 2 3 4.1 4.0 3.9 12-3 12.0 II-7 20-5 2O.O 19.5 28.7 28.0 27.3 47 9-5295I 9-55 593 "o 0.44 407 9-97 35 s 13 36.9 36.0 35.1 48 9.52986 35 9-55 633 40 0.4-4 367 9-97 353 12 3 49 9-53021 9-55 673 ^ 9 0.44327 9-97349 ( J II 50 9-53056 y.;^ 712 An 0.44 288 9j97344 4 10 5i 9-53 092 9.55 752 39 0.44 248 9-97 340 " 9 444 52 53 9.53126 9.53 161 35 9-55 79i 9.55831 40 0.44 209 0.44 169 9-97 335 4 9-97 33 I 8 7 41 40 39 54 9-53 196 35 9-55 870 39 0.44 130 9-97 326 5 6 5.1 5.0 4.9 55 35 9-55 9 10 0.44 090 9-97 322 5 1 15.4 15.0 14.6 5 6 9-53 266 9-55 949 0.44051 9.97317 4 25.6 25.0 24.4 57 9-53 301 33 9-55 989 40 0.44 01 1 9.97312 5 3 4 35-9 35- 34-1 5 8 35 9.56028 39 o-43 972 9.97 308 2 59 9-53 370 34 2C 9.56067 40 -43 933 9-97 303 5 4 I 60 9-53 405 9.56 107 1 0-43893 9-97 2 99 L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. | d. ' P.P. 70' 440 20' L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. p.p. I 2 3 4 1 9 IO ii 12 13 14 1 7 18 19 20 21 22 23 24 2 5 26 3 29 30 3 1 32 33 34 3 37 39 4O 42 43 44 :i 47 48 49 SO 5 2 53 54 55 5 6 57 58 59 60 9-53 405 35 35 34 34 35 34 35 34 35 34 34 34 34 35 34 34 34 34 34 34 34 34 34 34 34 34 9-56 107 39 39 40 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 38 39 39 39 38 39 39 38 39 38 39 39 38 39 $ 39 $ 39 38 38 38 38 38 11 37 38 0-43 893 9.97 299 5 5 4 4 5 5 4 5 5 4 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 5 4 5 5 4 r 4 5 5 5 4 5 5 5 5 4 5 5 5 4 5 5 5 5 4 5 5 5 5 60 H 57 56 55 54 53 5 2 5 1 50 49 48 47 46 45 44 43 42 4i 40 3 37 36 35 34 33 3i 30 29 28 2 7 26 2 5 ~4 23 22 21 2O 19 18 1 7 16 15 H '3 12 II IO 9 8 7 6 5 4 3 2 I 2 3 4 5 6 I 9 2 3 4 i i 9 i 2 3 4 I 9 40 39 38 4-0 3-9 3-8 8.0 7.8 7.6 12.0 11.7 11.4 16.0 15.6 15.2 20.0 19.5 19.0 24-0 23.4 22.8 28.O 27.3 26.6 32.0 31.2 30.4 36.0 35.1 34.2 37 35 34 3-7 3-5 3-4 7.4 7.0 6.8 1 1. 1 10.5 10.2 14.8 14.0 13.6 18.5 17.5 17.0 22.2 21.0 20-4 25.9 24.5 23.8 29.6 28.0 27.2 33-3 31-5 30.6 33 5 4 3.3 0.5 0.4 6.6 i.o 0.8 9-9 i-5 1-2 13.2 2.0 1.6 16.5 2.5 2.0 19.8 3.0 2.4 23.1 3.5 2.8 26.4 4.0 3.2 29.7 4.5 3.6 9-53 440 9-53475 9-53 59 9-53 544 9-53578 9-53613 9-53 647 9-53 682 9-537I6 9.56 146 9.56 185 9.56 224 9.56 264 9-56 303 9.56342 9.56 381 9.56 420 9-56459 9.56 498 9-56 537 9-56576 9-56615 9.56 654 9-5 6 693 9-56 732 9.56771 9.56 810 9.56 849 0.43 854 0.43 815 0.43 776 0-43 736 0.43 697 0.43 658 0.43 619 0.43 580 0.43 541 9.97 294 9-97 289 9-97 285 9.97 280 9-97 276 9.97271 9.97 266 9.97 262 9-97 2 57 9-53 75i 0.43 502 9.97252 9-53 785 9.53819 9-53 854 9.53888 9-53922 9-53957 9-53991 9.54025 9-54059 0-43 463 0.43 424 0-43 385 0-43 346 0-43 307 0.43 268 0.43 229 0.43 190 0.43151 9-97 248 9-97 2 43 9-97 238 9-97 2 34 9-97 229 9-97 224 9.97 220 9.97215 9.97 210 9-54 093 9.56 887 9.56 926 9-56 965 9.57004 9.57042 9.57081 9-57 120 9-57 5 8 9-57 r 97 9-57 235 0.43113 9.97 206 9-54 127 9-54 161 9-54 195 9-54 229 9-54 263 9-54 297 9-54331 9-54 365 9-54 399 o-43 074 0-43 035 0.42 996 0.42 958 0.42919 0.42 880 0.42 842 0.42 803 0.42 765 9-97 201 9-97 196 9-97 192 9-97 l8 7 9.97 182 9.97178 9-97 173 9.97 168 9-97 l6 3 9-54433 33 34 34 33 34 34 33 33 34 33 34 33 34 33 33 34 33 33 33 34 33 33 33 33 33 33 33 33 33 9-57 274 0.42 726 9-97 "59 9-54 466 9-54 5 9-54 534 9-54567 9.54601 9-54 635 9-54 668 9.54 702 9-54 735 9-57312 9-57 35 1 9-57 389 9.57428 9-57 466 9-57 54 9-57 543 9-57 5 81 9.57619 0.42 688 0.42 649 0.42 611 0.42572 0.42 534 0.42 496 0-42 457 0.42419 0.42 381 9-97 '54 9-97 49 9 97 J 45 9,97 140 9-97 !35 9-97 130 9.97 126 9.97 121 9.97116 2 3 4 5 2 3 4 5 555 40 39 38 4.0 3.9 3.8 12.0 II-7 II-4 2O.O 19.5 ig.O 28.0 27.3 26.6 36.0 35.1 34.2 544 9-54 769 9-57 658 0.42 342 9.97111 9-97 I0 7 9 97 102 9-97 97 9.97092 9.97087 9-97 083 9.97078 9-97073 9.97 068 9.97 063 9-97 59 9-97 S4 9-97 49 9-97 44 9-97 39 9-97 35 9.97030 9-97 02 5 9.97 020 9.54 802 9-54 836 9-54 869 9-54 903 9-54936 9-54 969 9-55003 9-55 036 9-55 069 9-57 696 9-57 734 9-57772 9.57810 9-57 849 9.57887 9-57 925 9-57 963 9.58001 "9.58 039 0.42 304 0.42 266 0.42 228 0.42 190 0.42 151 0.42113 0.42 075 0.42 037 0.41 999 9-55 I02 0.41 961 9-55 J 3 6 9-55 I6 9 9-55 202 9-55 235 9-55 268 9-55 3oi 9-55 334 9-55 367 9-55 400 9.58077 9.58"5 9-58 153 9.58 191 9.58 229 9.58 267 9.58 304 9-58 342 9.58 380 0.41 923 0.41 885 0.41 847 0.41 809 0.41 771 0.41 733 0.41 696 0.41 658 0.41 620 37 39 38 3.7 4.9 4.8 ii. i 14.6 14.2 18.5 24.4 23.8 25-9 34-i 33-2 33-3 9-55433 9.58418 0.41 582 9.97015 L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. d. ' P.P. 69' 21 441 L. Sin. d. L. Tan. c. d. L. Cot. L. 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Cos. d. L. Cot. Ic. d. L, Tan, L. Sin. i d. ' P.P. 68 442 22' L. Sin. d. L. Tan. c. d L. Cot. L. Cos. d. P. P. o 9-57358 31 9.60 641 36 o-39 359 9.96717 6 6O 2 3 4 1 9 9-57 389 9-57 420 9-57451 9.57 482 9-575*4 9-57 545 9-57 576 9-57 6 7 9-57 638 31 31 31 32 31 31 31 31 ^1 9.60677 9.60714 9.60 750 9.60 786 9.60 823 9.60 859 9.60895 9.60931 9.60 967 11 36 37 36 36 36 36 "?7 o-39 323 0.39 286 0.39 250 0.39 214 0.39177 0.39 141 0.39 105 0.39 069 0-39 033 9.96711 9.96 706 9.96 701 9.96 696 9.96 691 9.96 686 9.96681 9.96 676 9.96 670 5 5 5 5 5 5 5 6 59 58 57 56 55 54 53 5 2 2 3 4 5 6 7 37 36 35 3-7 3-6 3-5 7-4 7- 2 7-o 1 1. 1 10.8 10.5 14.8 14.4 14.0 18.5 18.0 17.5 22.2 21.6 21.0 25.9 25.2 24.5 IO 9-57 669 9.61 004 16 0.38 99') 9-96 665 SO ii 12 13 H 15 1 6 '7 18 19 9-57 7o 9-57 73i 9-57 762 9-57 793 9.57 824 9-57 855 9-57 885 9.57916 9-57 947 31 31 31 31 31 30 31 31 9.61 040 9.61 076 9.61 112 9.6l 148 9.6l 184 9.6l 22O 9.6l 256 9.6l 292 9.6l 328 36 36 36 36 36 36 36 3 -if) 0.38 960 0.38 924 0.38 888 0.38852 0.38816 0.38 780 0.38 744 0.38 708 0.38 672 9.96 660 9-96 655 9.96 650 9.96 645 9.96 640 9.96 634 9.96 629 9.96 624 9.96619 5 5 5 i 5 5 5 49 48 47 46 45 .44 43 42 4i i 2 3 4 5 33-3 3 2 -4 3 I -5 32 31 30 3-2 3-1 3-0 6.4 6.2 6.0 9-6 9-3 9-0 12.8 12.4 I2.O 16.0 15.5 15.0 2O 9-57978 ^O 9-6 1 364 -16 0.38 636 9.96 614 5 6 40 <> 19.2 18.6 18.0 21 22 23 24 3 27 28 29 9.58 008 9-58 039 9.58070 9.58 101 9-58131 9.58 162 9.58 192 9-58 223 9-58 253 31 31 31 30 31 3 31 30 31 9.61 400 9.61 436 9.61 472 9.61 508 9.6 1 544 9.61 579 9.61 615 9.61 651 9.61 687 36 36 36 36 35 36 36 36 5C 0.38 600 0.38 564 0.38 528 0.38 492 0.38 456 0.38421 0.3838? 0.38 349 0-38313 9.96 608 9.96 603 9.96 598 9-96 593 9.96 588 9.96 582 9.96577 9.96572 9.96 567 5 5 5 5 5 5 39 38 37 36 35 34 33 32 3' 7 S 9 2 22.4 21.7 21.0 25.6 24.8 24.0 28.8 27.9 27.0 29 6 5 2.9 0.6 0.5 5-8 1.2 1.0 87 i 8 ic 3O 9.58 284 9:61 722 l6 0.38 278 9.96 562 30 4 1 1. 6 2.4 2.0 3i 32 33 34 3 37 9.583'4 9-58 345 9-58 375 9.58406 9-58436 9.58467 9-58 497 31 30 31 30 31 3 9.61 758 9.61 794 9.61 830 9.61 865 9.61 901 9.61 936 9.61 972 36 36 35 36 35 36 0.38 242 0.38 206 0.38 170 0-38 135 0.38 099 0.38 064 0.38 028 9-96556 9-9655 1 9.96 546 9.96 541 9-96 535 9.96 530 9.96 523 5 5 5 6 5 5 29 28 27 26 25 24 21, 5 6 7 8 9 14.5 3-0 2.5 17.4 3.6 3.0 20.3 4-2 3-5 23.2 4.8 4.0 26.1 5.4 4.5 38 39 9.58 527 9-58 557 3 30 9.62 008 9.62 043 36 36 0.37 992 0-37957 9.96 520 9.96514 22 21 40 9.58 588 9.62 079 0.37921 9.96 509 2O 6 6 4i 42 43 44 2 47 48 49 9.58618 9.58 648 9.58 678 9.58 709 9-58 739 9.58 769 9-58 799 9.58829 9^8859 30 30 3i 30 30 30 30 3 9.62 114 9.62 150 9-62 185 9.62 221 9.62 256 9.62 292 9.62 327 9.62 362 9.62 398 35 36 35 36 3 35 Ii 0.37 886 0.37 850 o-37 815 o-37 779 o-37 744 o-37 708 o-37 673 0.37 638 0.37 602 9.96 504 9.96498 9-96493 9.96488 9-96483 9.96477 9.96472 9.96467 9.96 461 5 6 5 5 5 I 19 18 J 7 16 15 14 i3 12 II 36 35 o 3-o 2.9 9-o 8.8 , 15-0 14-6 2I.O 20.4 * 27.0 26.2 6 33-o 32.1 5O 9.58889 30 9-62 433 35 037567. 9.96456 j 5 10 5 1 5 2 53 54 55 5 6 P 59 9.58919 9-58 949 9.58979 9-59 009 9-59 039 9-59 069 9-59 098 9-59 128 9-59 158 30 30 30 30 3 29 3 30 30 9.62 468 9.62 504 9.62 539 9.62574 9.62 609 9.62 645 9.62 680 9.62715 9.62 750 36 35 35 H 35 35 35 0-37 532 0-37 496 0.37461 0.37 426 0-37391 o-37 355 0.37 320 o-37 285 o-37 250 9.96451 : 9-96 445 9.96 440 9-96435 9.96429 9.96 424 9.96419 9-964I3 9.96408 ; 6 5 5 6 5 5 6 5 9 8 7 6 5 4 3 2 I o i 2 3 4 5 555 37 36 35 3-7 3-6 3-5 1 1. 1 10.8 10.5 18.5 18.0 17.5 25.9 25.2 24.5 33-3 32-4 31-5 60 9-59 188 9.62 785 o-37 215 9.96 403 L. Cos. d. L. Cot. | c. d. L. Tan. L. Sin. d. / P. P. 67 23 44;! j ' L. Sin. d. L. Tan. c. d.j L. Cot. L. Cos. | d. P.P. 2 3 4 I 7 8 9 IO ii 12 13 15 17 IS 19 20 21 22 23 24 2 5 26 3 29 30 32 33 34 35 36 37 38 39 4O 41 42 43 44 49 SO 52 53 54 55 5 6 57 58 59 60 9-59 1 88 30 29 30 30 29 30 30 29 30 29 30 29 30 29 30 29 29 30 29 29 30 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 29 29 28 29 29 29 28 29 29 29 28 9.62 785 9.62 820 9.62 855 9.62 890 9.62 926 9.62 961 9.62 996 9.63 031 9.63 066 9.63 101 35 35 35 36 35 35 35 35 35 34 35 35 35 35 35 35 34 35 35 35 35 34 35 35 34 35 34 35 35 34 35 34 35 34 35 34 35 34 34 35 34 34 35 34 34 35 34 34 34 34 35 34 34 34 34 34 34 34 34 34 0-37 215 9.96 403 I 6 5 5 6 5 i 5 6 5 5 6 I 6 1 5 5 6 5 6 5 6 5 6 5 6 I 5 6 5 6 1 I 6 I 5 6 5 6 6 5 6 5 6 6 5 6 60 59 58 57 56 55 54 53 5 2 5 1 SO 49 48 47 46 45 44 43 42 4O 39 38 37 36 35 34 33 32 3 1 30 29 28 27 26 2 5 24 23 22 21 20 19 18 *7 16 '5 14 13 12 I I IO 9 8 7 6 5 4 3 2 36 35 34 i 3-6 3-5 3-4 2 7.2 7.0 6.8 3 10.8 10.5 10.2 4 14.4 14.0 13.6 5 1 8.0 17.5 17.0 6 21.6 21.0 20.4 7 25.2 24.5 23.8 8 28.8 28.0 27.2 9 32-4 31-5 30.6 30 29 28 ! 3.0 2.9 2.8 2 6.0 5.8 5.6 3 9.0 8.7 8.4 4 I2.O II. (> II. 2 5 15.0 14.5 14.0 6 18.0 17.4 16.8 7 | 21.0 20.3 19.6 S 24.0 23.2 22.4 , 9 27.0 26.1 25.2 6 5 i 1 0.6 0.5 2 1.2 1.0 3 1-8 1.5 4 2.4 2.0 5 3-0 2.5 6 3.6 3.0 7 4-2 3-5 8 4.8 4.0 9 5-4 4-5 9.59218 9-59 247 9-59 277 9-59 307 9-59 336 9-59 366 9-59 396 9o9425 9-59 484 0.37 1 80 -37 H5 0-37 n 0.37 074 0-37 039 0.37 004 0.36 969 0.36 934 0.36 899 9-96 397 9.96 392 9.96 387 9.96 381 9.96 376 9.96 370 9.96 365 9.96 360 9-96 354 9-63 135 0.36 865 9-96 349 9-59 5H 9-59 543 9-59573 9.59 602 9-59 632 9-59 66 1 9-59 690 9-59 720 9-59 749 9-63 170 9-63 205 9-63 240 9.63 275 9.63 310 9-63 345 9-63 379 9.63414 9-63 449 9.63 484 0.36 830 0-36 795 0.36 760 0.36 72? 0.36 690 0-36 655 0.36621 0.36 586 0.3655 1 9-96 343 9-96 338 9.96 333 9.96327 9.96 322 9.96316 9.96311 9.96 305 9.96 300 9-59 778 0.36516 9.96 294 9.96 289 9.96 284 9.96 278 9-96 273 9.96 267 9.96 262 9.96 256 9.96251 9.96 245 9.59 808 9-59 837 9-59 866 9-59 895 9.59 924 9-59 954 9.59983 9.60012 9.60041 9.60070 9.63519 9-63 553 9-63 588 9.63 623 9-63657 9.63 692 9.63 726 9.63 761 9.63 796 0.36481 0.36 447 0.36412 0-36 377 0.36 343 0.36 308 0.36 274 0.36 239 0.36 204 9.63 830 0.36 170 9.96 240 9.60 099 9.60128 9-6o 157 9.60 1 86 9.60 215 9.60244 9.60 273 9.60 302 9.60331 9.63 865 9.63 899 9-63 934 9-63 968 9.64 003 9.64 037 9.64072 9.64 106 9.64 140 0.36 135 0.36 101 0.36 066 0.36 032 0-35 997 0-35 963 0.35 928 0.35 894 0.35 860 9.96 234 9.96 229 9-96 223 9.96 218 9.96212 9.96 207 9.96 201 9.96 196 9.96 190 666 36 35 34 3.0 2.9 2.8 \ 9-0 8.8 8.5 15.0 14.6 14.2 J 21.0 2O-4 IU.X f 27.0 26.2 25.5 33-o 32-1 31-2 5 5 35 34 i 3-5 34 IO-5 10.2 17.5 17.0 ; 24.5 2 3-8 5 3'-5 30-6 9.60359 9-64I75 o-35 825 9^6 185 9.60 388 9.60417 9.60 446 9.60 474 9.60 503 9-60532 9.60 561 9.60 589 9.60618 9.64 209 9.64 243 9.64 278 9.64312 9.64 346 9.64 381 9.64415 9.64 449 9-64 483 o-35 79' 0-35 757 0-35 722 0.35 688 0-35 6 54 0-35 6l 9 0-35 585 0-35 55' 0.35517 9.96179 9-96I74 9.96 1 68 9.96 162 9-96 157 9.96151 9.96 146 9.96 140 9.96I3J 9.96 129 9.60 646 29 29 28 29 28 29 28 29 28 28 9.64517 o.35 483 9.60 675 9.60 704 9.60 732 9.60 761 9.60 789 9.60818 9.60 846 9-6o 875 9-6o 903 9.64 586 9.64 620 9.64 654 9.64 688 9.64 722 9.64 756 9.64 790 9.64 824 0.35 448 0.354H 0.35 38o 0.35 346 0.35312 0.35 278 0.35 244 -35 2I Q-35 '76 0.35 142 9.96 123 9.96 118 9.96 112 9.96 107 9.96 ioi 9.96 095 9.96 090 9.96 084 9.96079 9.96073 9.60931 9.64 858 L. Cos. d. L. Cot. jc. d. L. Tan. L. Sin. ! d. f P.P. 66 444 24 , L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. p.p. 9.60931 9-64 858 0.35 142 9.96073 6 60 I 2 3 4 9 9.60960 9.60 988 9.61 016 9.61 045 9.61 073 9.61 101 9.61 129 9.61 158 9.61 186 28 28 29 28 28 28 29 28 28 9.64 892 9.64 926 9.64 960 9-64 994 9.65028 9.65 062 9.65 096 9.65 130 9.65 164 34 34 34 34 34 34 34 34 0.35 108 0.35 074 0.35 040 0.35 006 0.34972 0-34 938 0.34 904 0.34 870 0.34 836 9.96 067 9.96 062 9.96 056 9.96050 9-96 045 9.96 039 9.96 034 9.96028 9.96 022 5 6 6 I 5 6 6 59 58 57 56 55 54 53 52 5i 34 33 i 34 3-3 2 6.8 6.6 3 10.2 9.9 4 13.6 13.2 5 17.0 16.5 to 9.61 214 28 9.65 197 0.34 803 9.96017 6 50 6 20.4 19.8 ii 12 13 14 15 16 17 '9 9.61 242 9.61 270 9.61 298 9.61 326 9-6i 354 9.61 382 9.61 411 9.61 438 9.61 466 28 28 28 28 28 29 2 7 28 28 9.65 231 9.65 265 9.65 299 9-65 333 9.65 366 9.65 400 9-65 434 9.65 467 9.65 501 34 34 34 33 34 34 33 34 0.34 769 0-34 735 0.34 701 0.34 667 0-34 634 0.34600 0.34 566 o-34 533 0.34 499 9.96011 9.96 005 9.96000 9-95 994 9.95 988 9-95 982 9-95 977 9-95 97 ' 9-95 965 6 5 6 6 6 5 6 6 49 48 47 46 45 44 43 42 4i 7 23.8 23.1 8 27.2 26.4 9 30.6 29.7 29 28 27 i 2.9 2.8 2.7 20 9.6 1 494 28 9-65 535 0.34 465 9.95 900 6 40 2 5-8 5-6 5-4 21 22 23 24 2 5 26 27 28 29 9.61 522 9-6i 550 9.61 578 9.61 606 9.6 1 634 9.61 662 9.61 689 9.61717 9-6i 745 28 28 28 28 28 2 7 28 28 28 9.65 568 9.65 602 9.65 636 9.65 669 9.65 703 9.65 736 9.65 770 9.65 803 9-65 837 34 34 33 34 33 34 33 34 0-34 432 0-34 398 0-34 364 0-34331 0.34 297 0.34 264 0.34 230 0-34 197 0.34 163 9-95 954 9-95 948 9-95 942 9-95 937 9-95 93i 9-95 925 9-95 920 9.95 914 9-95 908 6 6 5 6 6 5 6 6 6 3 37 36 35 34 33 32 3i 3 8.7 8.4 8.1 4 n.6 ii. 2 10.8 5 14.5 14.0 13.5 6 17.4 16.8 16.2 7 20.3 19.6 18.9 8 23.2 22.4 21.6 9 26.1 25.2 24.3 30 9.61 773 9.65 870 0.34 130 9-95 902 30 3i 32 33 34 35 36 3 39 9.61 800 9.61 828 9.61 856 9.61 883 9.61 911 9.61 939 9.61 966 9.61 994 9.62021 28 28 27 28 28 27 28 27 28 9.65 904 9-65 937 9.65971 9.66 004 9.66 038 9.66071 9.66 104 9.66138 9.66 171 33 34 33 34 33 33 34 33 0.34 096 0.34 063 0.34 029 o-33 996 0.33 962 0.33 929 0.33 896 0.33 862 0.33 829 9-95 897 9-95 8 9i 9-95 88? 9-95 879 9-95 873 9-95 868 9-95 862 9-95 856 9-95 850 6 6 6 6 5 6 6 6 6 29 28 2 7 26 25 24 23 22 21 6 5 i 0.6 0.5 2 1.2 1.0 3 1.8 1.5 4 2.4 2.0 5 3-o 2.5 6 3.6 3.0 7 4-2 3-5 8 4.8 4.0 40 9.62 049 9.66 204 0-33 796 9-95 844 2O 9 5-4 4-5 4i 42 43 44 9.62 076 9.62 104 9.62 131 9.62 159 28 27 28 27 9.66 238 9.66271 9.66 304 9-66 337 33 33 33 0.33 762 o-33 729 0.33 696 0-33 663 9-95 839 9-95 833 9-95 827 9.95821 6 6 6 6 19 18 1 7 16 4 4 i 47 48 49 9.62 186 9.62 214 9.62 241 9.62 268 9.62 296 28 27 27 28 9.66371 9.66 404 9.66437 9.66 470 9-66 503 33 33 33 33 0.33 629 o-33 596 o-33 563 o-33 53 0-33 497 9-95 815 9.95 810 9.95 804 9-95 798 9-95 792 5 6 6 6 fi 15 H 13 12 II 6 6 j^ 5O 9.62 323 9-66 537 0-33 463 9-95 786 6 IO 5i 5 2 53 54 55 56 P 59 9.62 350 9.62377 9.62 405 9.62 432 9.62459 9.62 486 9.62513 9.62 541 9.62 568 3 2 7 2 7 2 7 27 28 2 7 9-66 570 9.66 603 9.66 636 9.66 669 9.66 702 9-66 735 9.66 768 9.66801 9.66 834 33 33 33 33 33 33 33 33 0-33 430 0-33 397 0-33 364 0.33331 o-33 298 0.33 265 0.33 232 0-33 199 0.33 166 9-95 780 9-95 775 9-95 769 9-95 763 9-95 757 9-95 75 9-95 745 9-95 739 9-95 733 5 6 6 6- 6 6 6 6 9 8 7 6 5 4 3 2 I 2.8 2.8 3-4 8.5 8.2 10.2 14.2 13.8 17.0 > 19.8 19.2 23.8 J 25.5 24.8 30.6 5 31.2 30.2 60 9-62 595 9.66 867 o-33 133 9-95 728 L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. d. ' p.p. 65' 25 445 / L. Sin. d. L. Tan. c. d L. Cot. L. Cos. d. p. p. ! o 9.62 505 9.66 867 0-33 133 9-95 728 6 60 I z 3 4 1 7 8 9 9.62 622 9.62 649 9.62 676 9.62 703 9.62 730 9.62 757 9.62 784 9.62811 9.62 838 27 27 27 27 27 27 2 7 2 7 9.66 900 9-66933 9.66 966 9-66 999 9.67 032 9.67 065 9.67 098 9.67 131 9.67 163 33 33 33 33 33 33 33 32 0.33 loo 0.33 067 0-33 34 0.33001 0.32 968 0.32935 0.32 902 0.32 869 0-32837 9-95 722 9-95 7i6 9.95 710 9-95 74 9-95 698 9.95 692 9.95 686 9.95 680 9-95 6 74 6 6 6 6 6 6 6 6 5 59 58 57 56 55 54 53 52 Si 33 32 i 3-3 3-2 2 6.6 6.4 3 9-9 9-6 4 13-2 12.8 5 16.5 16.0 10 9.62 865 9.67 196 0.32 804 9-95 668 50 6 19.8 19.2 ii 12 3 H 16 !I 19 9.62892 9.62918 9.62 945 9.62972 9.62 999 9.63 026 9.63052 9.63079 9.63 106 26 2 7 27 2? 27 26 2 7 2 7 9.67 229 9.67 262 9.67 295 9.67 327 9.67 360 9-67 393 9.67426 9.67 458 9.67491 33 33 32 33 33 33 32 33 ?? 0.32771 0-32 738 0.32 705 0.32 673 0.32 640 0.32 607 0.32 574 0.32542 0.32 509 9-95 663 9-95 6 57 9-95651 9-95 64? 9-95 639 9-95 633 9-95 627 9-95 621 9-95615 5 6 6 6 6 6 6 6 6 6 49 48 47 46 45 44 43 42 4' 7 23.1 22.4 8 26.4 25.6 9 29.7 28.8 27 26 i 2.7 2.6 2 s 4 s 2 20 9-63 133 26 9.67 524 32 0.32476 9-95 609 6 40 3 8.1 7.8 21 22 23 24 25 26 27 28 29 9-63I59 9.63 186 9-63213 9-63 239 9.63 266 9.63 292 963319 9-63 345 9-63 372 27 2 7 26 27 26 27 26 z 9-67 556 9.67 589 9.67 622 9.67 654 9.67 687 9.67 719 9-6775 2 9.67 785 9.67817 33 33 32 33 3 2 33 33 32 3? 0.32 444 0.32411 0.32 378 0.32 346 0.32313 0.32 281 0.32 248 0.32215 0.32 183 9.95 603 9-95 597 9-95 59i 9-95 5 8 5 9-95 579 9-95 573 9-95 5 6 7 9-95 56i 9-95 55? 6 6 6 6 6 6 6 6 6 39 38 37 36 35 34 33 32 3i 4 10.8 10.4 5 '3-5 '3-0 6 16.2 15.6 7 18.9 18.2 8 21.6 20.8 9 24.3 23.4 30 9-63 398 9.67 850 0.32 150 9-95 549 6 30 765 3 1 32 33 34 P 37 38 39 9-63 425 9.63451 9.63 478 9.63 504 9-63 53i 9-63557 9-63 583 9.63 610 9.63 636 26 27 26 27 26 26 z ->6 9.67 882 9.67915 9.67 947 9.67 980 9.68012 9.68 044 9.68077 9.68 109 9.68 142 33 32 33 32 32 33 32 33 32 0.32 118 0.32 085 0.32053 0.32 020 0.31 988 0.31 956 0.31 923 0.31 891 0.31 858 9-95 543 9-95 537 9-95531 9-95 5 2 5 9-95 5 1 9 9-95513 9-95 57 9-95 5 9-95 494 6 6 6 6 6 6 7 6 6 29 28 27 26 25 2 4 23 22 21 i | 0.7 0.6 0.5 2 1 1.4 1.2 I.O 3 2.1 1.8 1.5 4 2.8 2.4 2.0 5 3-5 3-o 2.5 6 4.2 3.6 3.0 7 4-9 4-2 3-5 8 5.6 4.8 4.0 9 6.3 5.4 4.5 4O 9.63 662 27 9.68174 32 0.31 826 9-95 488 6 20 4i 42 4-2 9.63 689 9-637I5 n 6^ 741 26 26 9.68 206 9-68 239 9.68 271 33 32 0.31 794 0.31 761 0.31 729 9-95 482 9-95 476 9-95 47 6 6 '9 18 17 44 45 46 47 48 49 9.63 767 9-63 794 9.63 820 9.63 846 9.63 872 9.63 898 26 27 26 26 26 26 "6 9.68 303 9.68 336 9.68 368 9.68 400 9.68432 9-68465 32 33 32 32 32 33 32 0.31 697 0.31 664 0.31 632 0.31 600 , 0.31 568 0-3' 535 9-95 464 9-95 458 9-95 452 9-95 446 9-95 440 9-95 434 I 6 6 6 6 16 '5 14 13 12 II 766 32 32 33 50 9.63 924 "6 9-68 497 72 0.31 53 9-95 427 6 10 , 2.3 2.7 3.3 5' 52 53 54 II 57 58 59 9-63 95 9.63976 9.64 002 9.64028 9.64054 9.64 080 9.64 1 06 9.64 132 9.64 158 26 26 26 26 26 26 26 26 >6 9-68 529 9.68 561 9.68 593 9.68 626 9.68658 9.68 690 9.68 722 9-68 754 9.68 786 32 32 33 32 32 32 32 32 1.2 0-31 47 i 0.31 439 0.31 407 0-3' 374 0.31 342 0.31 310 0.31 278 0.31 246 0.31 214 9.95 421 9-954I5 9-95 409 9-95 403 9-95 397 9-95391 9-95 384 9-95 378 9-95 372 6 6 6 6 6 7 6 6 6 9 8 7 6 5 4 3 2 11.4 13.3 16.5 > 1 6.0 18.7 23.1 4 20.6 24.0 29.7 j ' * 9 - 3 - 60 9.64 184 9.68818 0.31 182 9-95 366 O L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. d. ' P.P. 64 446 26 1 , L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. p. p. o 9.64 184 26 9.68818 3 2 0.31 182 9-95 366 6 6O I 2 3 4 1 1 9 9.64 2IO 9.64 236 9.64 262 9.64 288 9-643I3 9-64 339 9.64 365 9.64 391 9.64417 26 26 26 25 26 26 26 26 9.68 850 9.68 882 9.68914 9.68 946 9.68 978 9.69010 9.69 042 9.69 074 9.69 106 32 32 32 32 3 2 32 32 32 0.31 150 0.31 1 18 0.31 086 0.31 054 0.31 022 0.30 990 0.30958 0.30 926 0.30 894 9-95 36o 9-95 354 9-95 348 9-95 34 i 9-95 335 9-95 329 9-95 323 9-953I7 9-95 3io 6 6 7 6 6 6 6 I 59 58 57 56 55 54 53 52 5i 32 31 i 3-2 3-1 2 6.4 6.2 3 9-6 9-3 4 i 12.8 12.4 5 16.0 15.5 ! 10 9.64 442 26 9.69 138 0.30 862 9-95 34 6 50 ii 12 13 H 15 16 \l 19 9.64 468 9.64 494 9.64 519 9-64 545 9-6457 1 9.64 596 9.64 622 9.64 647 9.64 673 26 25 26 26 25 26 2 5 26 9.69170 9.69 202 9.69 234 9.69 266 9.69 298 9.69 329 9.69 361 9-69 393 9.69 425 32 32 32 32 3i 32 32 32 0.30 830 0.30 798 0.30 766 0.30 734 0.30 702 0.30 671 0.30 639 0.30 607 -3 575 9.95 298 9-95 2 9 2 9.95 286 9-95 279 9-95 2 73 9.95 267 9-95 261 9-95 2 54 9-95 248 6 6 7 6 6 6 7 6 6 49 48 47 46 45 44 43 42 41 8 25.6 24.8 9 | 28.8 27.9 26 25 24 i | 2.6 2.5 2.4 2 5.2 5.0 4.8 20 9.64 698 26 9-69 457 0-30 543 9-95 242 5 40 3 7.8 7.5 7.2 21 22 23 24 3 3 29 9.64 724 9.64 749 9-64 775 9.64 800 9.64 826 9.64851 9.64 877 9.64 902 9.64 927 25 26 2 5 26 25 26 25 25 9.69 488 9.69 520 9-69 SS 2 9.69 584 9.69615 9.69 647 9.69 679 9.69 710 9.69 742 32 32 32 3i 32 32 3i 3 2 0.30512 0.30 480 0.30 448 0.30416 0.30 385 0.30 353 0.30321 0.30 290 0.30 258 9-95 236 9.95 229 9-95 223 9.95217 9.95211 9-95 204 9-95 "98 9-95 192 9-95 l8 5 \ 6 6 7 6 6 7 5 39 38 37 36 35 34 33 3 2 3i 4 10.4 10.0 9.6 5 13.0 12.5 12.0 6 15.6 15.0 14.4 7 18.2 17.5 16.8 8 20.8 20.0 19.2 9 23.4 22.5 21.6 30 9-64 953 9.69 774 3- ji 0.30 226 9-95 179 6 3O 7 6 3i 32 33 34 P 37 38 39 9.64 978 9.65 003 9.65 029 9-65 54 9.65 079 9.65 104 9-65 13 9-65 155 9.65 i So 11 25 25 2 5 26 25 25 9.69 805 9.69 837 9.69 868 9.69 900 9.69 932 9.69 963 9-69 995 9.70026 9.70058 32 3i 32 32 3i 32 3 1 32 M 0.30 193 0.30 163 0.30 132 0.30 100 0.30 068 0.30037 0.30 005 0.29 974 0.29 942 9-95 *73 9-95 l6 7 9-95 I6 9-95 J 54 9-95 H8 9-95 H 1 9-95 "35 9-95 I2 9 9-95 "2 6 7 6 6 I 7 6 29 28 2 7 26 2 5 24 23 22 21 i 0.7 0.6 2 1.4 1.2 3 2.1 1.8 4 2.8 2.4 5 3-5 3-o 6 4.2 3.6 7 4-9 4-2 8 5.6 4.8 9 6.3 5.4 4O 9.65 205 9.70 089 0.29911 9-95 II6 6 20 4i 42 9-65 230 9-65 255 11 9.70 121 9.70152 31 32 0.29 879 0.29 848 9-95 II0 9-95 I0 3 7 6 19 18 43 44 4 I 46 47 48 49 9.65 306 9-65 33' 9-65 35 6 9.65 381 9.65 406 9-6543I 2 5 2 5 2 5 25 25 25 9.70215 9.70 247 9.70 278 9.70 309 9.70341 9.70372 31 32 31 31 32 31 0.29 785 0.29 753 0.29 722 0.29 691 0.29 659 0.29 628 9-95 9Q 9-95 8 4 9.95 078 9.95071 9-95 o 6 ? 9-95 59 7 6 6 7 6 6 1 7 16 J 5 H 13 12 II 776 32 31 32 SO 9-65 456 9.70 404 21 0.29 596 9-95 5 2 6 IO 2.3 2.2 2. 7 51 52 53 54 55 56 H 59 9.65 481 9.65 506 9-65 531 9.65 556 9.65 580 9.65 605 9.65 630 9-65 655 9.65 680 25 2 5 25 24 25 25 25 25 9.70435 9.70 466 9.70 498 9.70 529 9.70 560 9.70 592 9.70 623 9.70654 9.70 685 3i 32 3i 3 1 32 3 1 3i 3i 0.29 565 0.29 534 0.29 502 0.29471 0.29 440 0.29408 0.29 377 0.29 346 0.29315 9-95 46 9-95 39 9-95 33 9-95 02 7 9.95 020 9-95 OI 4 9-95 7 9-95 ooi 9-94 993 I 6 7 6 7 6 6 I 7 6 5 4 3 2 1 6.9 6.6 8.0 * 1 1.4 i i.i 13.3 1 6.0 15.5 18.7 * 20.6 19.9 24.0 5 25.1 24.4 29.3 29.7 28.8 - GO 9-65 75 9.70717 0.29 283 9-94 988 L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. d. ' P.P. 63 e 27" 447 / L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. p.p. 9.65 705 2 4 9.70717 0.29 283 9-94 988 6 60 2 3 4 I 7 8 9 9.65 729 9-65 754 9-65 779 9.65 804 9.65 828 9-65 853 9.65 878 9.65 902 9.65 927 25 2 5 25 24 25 2 5 24 2 5 9-7 748 9.70779 9.70810 9.70841 9.70873 9.70 904 9-70933 9.70966 9.70997 3i 3 1 3 1 32 0.29 252 0.29*21 0.29 190 0.29 I 59 0.29 127 O.29 096 O.29 065 0.29034 0.29 003 9.94982 9-94 975 9-94 969 9-94 962 9-94 95 6 9-94 949 9-94 943 9-94 936 9-94 93 7 6 7 6 7 6 7 6 59 58 57 56 55 54 53 52 5 1 32 31 30 i 32 3-1 3-o 2 6.4 6.2 '6.0 3 9-6 9-3 9-o 4 12.8 12.4 12.0 5 16.0 15.5 15.0 IO 9-65 952 9.71 028 0.28 972 9.94923 6 50 ii 12 '3 14 11 17 18 19 9.65 976 9.66 ooi 9.66025 9.66 050 9.66 075 9.66 099 9.66124 9.66 148 9.66 173 25 24 25 25 24 2 5 24 2 5 9-7 1 59 9.71090 9.71 121 9-7 1 X 53 9.71 184 9.71 215 9.71 246 9.71 277 9.71 308 3 1 32 3 1 0.28 94! 0.28910 0.28 879 0.28 847 0.28816 0.28 785 0.28 754 0.28 723 0.28 692 9.94917 9.94911 9-94 904 9-94 898 9.94891 9.94 885 9.94878 9.94871 9.94865 6 7 6 I 7 7 6 49 48 47 46 45 44 43 42 8 25.6 24.8 24.0 9 28.8 27.9 27.0 25 24 23 i 2.5 2.4 2.3 2 5.0 4.8 4.6 2O 9.66 197 2 4 9-7i 339 7T 0.28 661 9-94858 6 4O 3 7-5 7-2 6.9 21 22 23 24 3 27 28 29 9.66 221 9.66 246 9.66 270 9-66 295 9.66319 9-66 343 9.66 368 9.66 392 9.66416 2 5 24 25 24 24 2 5 24 24 9.71 370 9.71 401 9-71 43i 9.71 462 9-7 1 493 9.71 524 9-7i 555 9.71 586 9.71 617 30 31 3 1 3 1 3 1 0.28 630 0.28 599 0.28 569 0.28 538 0.28 507 0.28476 0.28 445 0.28414 0.28 383 9-94852 9-94 845 9-94 839 9-94 832 9.94826 9.94819 9.94813 9.94 806 9-94 799 7 6 7 6 7 6 7 7 5 P 37 36 35 34 33 32 5 12.5 12.0 11.5 6 15.0 14.4 13.8 7 17.5 16.8 16.1 8 20.0 19.2 18.4 9 22.5 21.6 20.7 3O 9.66441 9.71 648 0.28 352 9-94 793 3O 7 6 32 33 34 P 37 38 39 9.66 465 9.66489 9.66513 9.66 537 9.66 562 9.66 586 9.66610 9.66 634 9.66658 2 4 24 24 2 5 2 4 24 24 24 9.71 679 9.71 709 9.71 740 9.71 771 9.71 802 9-7 1 833 9.71 863 9.71 894 9.71925 30 30 0.28 321 0.28 291 0.28 260 0.28 229 0.28 198 0.28 167 0.28137 0.28 106 0.28075 9-94 786 9.94 780 9-94 773 9-94 7 6 7 9-94 760 9-94 753 9-94 747 9-94 740 9-94 734 6 7 6 7 7 6 I 29 28 27 26 25 24 23 22 21 i 0.7 0.6 2 1.4 1.2 3 2.1 1.8 4 2.8 2.4 5 3-5 3- 6 4.2 3.6 7 4.9 4.2 8 5.6 4.8 9 6.3 5.4 4O 9.66 682 9-7 r 955 0.28 045 9.94 727 20 42 9.66 706 9.66731 25 24 9.71986 9.72017 | 0.2,8014 0.27 983 9-94 7 20 9.94714 6 19 18 43 44 45 46 11 49 9- 755 9.66779 9.66 803 9.66827 9.66851 9.66875 9.66 899 24 24 24 24 24 24 9.72078 9.72 109 9.72 140 9.72 170 9.72 201 30 30 30 0.27 922 0.27 891 0.27 860 0.27 830 0.27 799 0.27 769 9-94 7 9-94 694 9-94 687 9.94 680 9-94 674 9-94 667 7 6 7 6 7 16 '5 '4 13 12 II 766 30 31 30 o 50 9.66 922 9.72 262 0.27 738 9.94 660 IO i *' f'| f'5 5 1 52 53 54 55 5 6 P 59 9.66 946 9.66970 9.66 994 9.67018 9.67 042 9.67066 9.67 090 9.67113 9.67 137 24 24 24 24 24 24 23 24 9.72 293 9-72354 9.72384 9.72415 9-7 2 445 9.72476 9.72 506 9-72 537 30 3 3 3' 30 30 0.27 707 0.27 677 0.27 646 0.27 616 0.27 585 0-27 553 0.27 524 0.27 494 0.27 463 9-94 6 54 9.94 647 9.94 640 9-94 634 9-94 627 9.94 620 9.94614 9-94 607 9.94 600 7 7 6 7 7 6 7 7 7 9 8 7 6 5 4 3 2 I 6.4 7.8 7.5 10.7 12.9 12.5 * 15.0 18.1 17.5 * 19.3 23.2 22.5 5 23.6 28.4 27.5 o 27.9 60 9.67 161 9.72567 0.27 433 9-94 593 o L. Cos. d. L. Cot. c. d | L.Tan. L, Sin. d. ' P.P. 62 448 28 C L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. p.p. o 9.67 161 9-72567 ! 0-27433 9-94 593 fi 6O I 9-67 185 9-72598 0.27 402 9-94 587 59 2 9.67 208 2 3 9.72 628 3 0-2737* 9-94 580 S8 3 9.67 232 9.72659 3 1 0.27 341 9-94 573 57 4 9.67 256 2 4 9.72 689 3 0.27311 9-94567' 31 30 29 9.67 280 9-67 303 24 23 9.72 720 9-72 75 3 1 30 0.27 280 0.27 250 9-94 560 9-94553 7 7 55 54 i 3.1 3.0 2.9 2 6.2 6.0 5.8 7 9-67 327 9.72 780 0.27 220 9.94 546 53 3 9.3 9.0 8.7 8 9-67 35 2 3 9.72 811 0.27 189 9-94 540 S2 4 12.4 12.0 1 1. 6 9 9-67 374 9.72841 0.27 159 9-94 533 5* 5 J5-5 J5-o H-5 IO 9.67 398 9.72872 0.27 128 9-94526 5O 6 18.6 18.0 17.4 ii 12 13 9.67421 9.67 445 9.67 468 24 23 9.72 902 9.72932 9-72 963 30 3 1 0.27 098 0.27068 9-94 5 '9 9-94 5 J 3 9.94 506 6 7 49 48 47 7 21.7 21.0 20.3 8 24.8 24.0 23.2 9 27.9 27.0 26.1 14 9.67 492 2 4 9.72993 3 0.27OO7 9.94 499 7 46 15 9.67515 2 3 9-73023 3 0.26977 9-94 492 7 45 16 9-67 539 973054 3 1 O.26 946 9-94 485 44 18 9.67 562 9.67 586 24 9.73084 9-73 "4 3 30 0.26916 0.26 886 9-94 479 9.94472 7 43 42 24 23 22 19 9.67 609 9-73 H4 3 0.26856 9-94 465 7 4 1 I 2. 4 2. 3 2.2 2O 21 22 23 24 3 9-67 633 23 24 23 2 3 24 23 9-73 175 9-73 205 9-73 235 9-73 265 9-73 295 9-73356 30 30 30 30 30 0.26 825 9-94 45 8 7 6 7 7 7 7 4O 39 38 37 36 35 34 2 4.8 4.6 4.4 3 7.2 6.9 6.6 4 9.6 9.2 8.8 5 12.0 11.5 n.o 6 14.4 13.8 13.2 7 16.8 16.1 15.4 8 19.2 18.4 17.6 9 21.6 20.7 19.8 9.67 656 9.67 680 9.67 703 9.67 726 9.67 730 9-67 773 0.26 795 0.26 765 0.26 735 0.26 703 0.26674 0.26644 9-94 45 * 9.94 443 9-94438 9-94431 9-94 424 9.94417 27 9.67 796 2 3 9-73 386 3 0.26614 9.94410 S3 28 9.67 820 2 4 9.73416 3 0.26 584 9-94 404 32 29 9-67 843 2 3 9-73446 3 0.26 554 9-94 397 3 1 3O 9.67 866 973476 0.26524 9-94 390 SO 31 9.67 890 9-73 57 0.26493 9-94 383 29 7 6 32 9.67913 2 3 9-73537 3 0.26 463 9-94 376 28 , 33 9.67 936 2 3 9-73 567 3 0.26433 9-94 369 7 27 2 1-4 1.2 34 9.67 959 2 3 9-73597 0.26403 9.94 362 26 3 2.1 1.8 35 9.67 982 2 3 9.73627 3 0.26373 9-94 355 g 25 4 2.8 2.4 36 9.68 006 9-73 657 0.26343 9-94 349 24 5 3-5 o-o 37 9.68 029 2 3 9-73 687 3 0.26313 9-94 342 7 23 6 4.2 3.6 9.68 052 2 3 9-737I7 3 0.26 283 9-94 335 7 22 7 4-9 4-2 39 9.68075 2 3 9-73 747 3 0.26 253 9-94 328 7 21 8 5.6 4.8 40 9.68 098 9-73 777 0.26 223 9 94 321 2O 9 6.3 5.4 41 9.68 121 9-73 807 0.26 193 9-943I4 19 42 9.68 144 2 3 9-73 837 3 0.26 163 9-94 307 7 18 43 9.68 167 2 3 9.73867 3 0.26 133 9-94 300 7 17 44 9.68 190 *3 9-73 897 3 0.26 103 9-94 293 i 16 45 9.68 213 2 3 9-73927 3 0.26073 9.94 286 15 46 9-68 237 9-73957 3 0.26043 9-94 279 47 9.68 260 2 3 973987 3 0.26013 9-94 273 J 3 48 9.68 283 2 3 9.74017 3 0.25 983 9.94 266 7 12 766 49 9-68 305 9.74047 3 0.25 953 9-94 259 7 ii 31 31 30 50 9.68 328 9.74077 0.25 923 9.94 252 10 , 51 52 53 54 55 56 9.68351 9-68 374 9-68 397 9.68 420 9 .68 443 9.68 466 23 23 23 23 23 9-74 107 9-74 137 9.74 1 66 9-74 196 9.74 226 9.74256 30 29 30 30 30 0.25 893 0.25 863 0.25 834 0.25 804 0-25 774 0.25 744 9-94 245 9-94 238 9.94 231 9-94 224 9.94217 9.94210 7 7 7 7 7 9 8 7 6 5 4 2.2 2.0 2.5 1 6.6 7.8 7.5 ii. i 12.9 12.5 15.5 18.1 17.5 J 19.9 23.2 22.5 5 24.4 28.4 27.5 6 28 . 8 _ _ I S7 9.68 489 2 3 9.74 286 3 0.25714 9 94 203 7 3 7 58 9.68512 9-74 3 l6 0.25 684 9.94 196 7 2 59 9-68 534 2 3 9-74 345 3 0-25 653 9-94 '89 7 I 60 9.68557 "7-1 575 0.25 623 9.94 182 L. Cos. d. L. Cot. |c. d. L. Tan. L. Sin. | d. ' P.P. 61 29 449 ' L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. p. p. ! 9-68557 2? 9-74 375 3O 0.25 625 9.94 182 60 I 2 3 4 I I 9 9.68 580 9.68 603 9.68 625 9.68 648 9.68671 9.68 694 9.68 716 9.68 739 9.68 762 23 22 23 23 23 22 23 23 9-74 405 9-74 435 9-74465 9-74 494 9-74 524 9-74 554 9-74583 9-74613 9-74 643 30 3 29 0.25 595 0-25 565 0-25 535 0.25 506 0.25 476 0.25 446 0.25417 0.25 387 9.94I75 9.94 1 68 9.94 161 9-94 154 9-94 H7 9.94 140 9-94 133 9.94126 9.94119 7 7 7 7 7 7 7 7 59 57 56 55 54 53 52 30 29 23 i 3.0 2.9 2.3 2 6.0 5.8 4.6 3 9.0 8.7 6.9 4 120 1 1. 6 9.2 5 15.0 14.5 11.5 iS.o 17.4 13.8 IO 9.68 784 9-74 673 0.25 327 9.94112 50 7 21.0 20.3 16.1 1 1 12 13 H 16 19 9.68 807 9.68 829 9.68852 9.68 875 9.68 897 9.68 920 9.68 942 9.68 965 9.68 987 22 23 23 22 23 22 23 22 2T. 9.74 702 9-74 732 9.74 762 9.74 791 9.74821 9-7485I 9.74880 9.74910 9-74 939 30 30 30 29 30 29 0.25 298 0.25 268 0.25 238 0.25 209 0.25 179 0.25 149 O.25 120 O.25 090 0.25 061 9.94 105 9.94 098 9.94 090 9-94 083 9.94076 9-94 069 9.94 062 9-94 05? 9-94 048 7 8 7 7 7 7 7 7 49 48 47 46 45 44 43 42 8 24.0 23.2 18.4 9 27.0 26.1 20.7 22 8 7 I 2.2 0.8 0.7 2 4.4 1.6 1.4 120 9.09010 22 9.74909 0.25031 9.94041 40 4 8.8 3.2 2.8 21 22 23 24 2 5 26 27 28 29 9.69 032 9.69 053 9.69 077 9.69 loo 9.69 122 9.69 144 9.69 167 9.69 189 9.69 212 23 22 23 22 22 23 22 23 9-74 998 9.75028 9-75058 9-75 087 9-75 "7 9-75 146 9.75 176 9-75 205 9-75 235 30 30 29 30 29 29 30 0.25 002 0.24 972 0.24 942 0.24913 0.24 883 0.24 854 0.24 824 0.24 795 0.24 765 9-94034 9-94 027 9.94 020 9.94012 9.94 005 9-93 998 9-93991 9-93 984 9-93 977 7 7 8 7 7 7 39 38 37 34 33 32 5 1 1.0 4.0 3.5 6 13.2 4.8 4.2 7 15.4 5.6 4.9 8 17.6 6.4 5.6 9 19.8 7.2 6.3 3O 9.69 234 22 9-75 264 29 0.24 736 9-93 970 30 32 34 P 37 39 9.69 256 9.69 279 9.69 301 9-69 323 9-69 345 9.69 368 9.69 390 9.69412 9-69 434 23 22 22 22 23 22 22 22 22 9-75 2 94 9-75 323 9-75 353 9-75 382 9-754" 9-75441 9-75 470 9.75 500 9-75 529 29 30 29 29 30 29 30 29 0.24 706 0.24677 0.24 647 0.24618 0.24 589 0.24 559 0.24 530 0.24 500 0.24471 9-93 963 9-93 955 9.93 948 9-93 94 ! 9-93 934 9-93 927 9-93 920 9.93912 9-93 905 7 7 7 7 8 7 29 28 27 26 25 24 23 22 21 8 8 30 29 1.9 1.8 \ 5-6 5-4 9-4 9-i : 13.1 12.7 40 9.69 456 9-75 558 0.24 442 9-93 898 20 : 16.9 16.3 4i 42 .43 44 45 46 47 49 9.69 479 9.69 501 9.69523 9-69 545 9.69 567 9.69 589 9.69 611 9-69 633 9-69 655 22 22 22 22 22 22 22 22 9-75 588 9.75617 9-75 6 47 9.75 676 9-75 705 9-75 735 9.75 764 9-75 793 9-75 822 29 30 29 29 30 29 29 29 0.24412 0.24 383 0-24 353 0.24 324 0.24 293 0.24 265 0.24 236 0.24 207 0.24 178 9-93 891 9-93 884 9-93 876 9.93 869 9-93 862 9-93 855 9-93 847 9.93 840 9-93 833 I 7 7 7 8 7 7 19 18 i? 16 '5 H 13 12 II 5 20.6 19.9 24.4 23.6 28.1 27.2 7 7 30 29 50 9.69 677 9.7^ 852 0.24 148 9-93 826 10 21 21 5i 52 53 54 H % 59 9.69721 9-69 743 9-69 765 9.69 787 9.69 809 9.69831 9.69 853 9.69875 22 22 22 22 22 22 22 22 9-75 881 9.75910 9-75 939 9-75 969 9-75 998 9.76027 9.76 056 9.76086 "7" "5 29 29 30 29 29 30 29 0.24 119 0.24090 0.24 061 0.24031 0.24 002 0.23973 0.23 944 0.23914 O.2.; 885 9.93819 9.93811 9-93 804 9-93 797 9-93 789 9.93 782 9-93 775 9-93 76^ 9 93 76o 8 7 7 8 7 7 7 8 9 8 7 6 5 4 3 2 I 1 6.4 6.2 10.7 10-4 6 15.0 14.5 J 19-3 18.6 1 23.6 22.8 7 27.9 26.9 60 9.69 897 9.76 144 0.23 856 9-93 753 L. Cos. d. L. Cot. c. d. L. Tan. L. Sin, d. ' P.P. K'M'I> SI.-KV. 29 60' 450 30' L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. p. p. , o 9.69897 22 9.76 144 20 0.23856 9-93 753 6O I 9-699I9 22 9-76 173 *y 0.23 827 9-93 746 59 2 9.69941 , 9.76 202 29 0.23 798 9-93 738 58 3 9.69963 ~ 2 * 9-76 231 29 0.23 769 9-93 73' 7 57 4 9-69 984 22 9.76 261 3 0-23 739 9-93 724 7 56 30 29 28 i 5 9.70006 9.76 290 2 9 0.23 710 9.937I7 I 55 i 3.0 2.9 2.8 6 9.70028 ^ 9.76319 29 0.23 68 1 9-93 79 54 2 6.0 5.8 5.6 7 8 9.70050 9.70072 9.76348 9.76377 2 9 29 0.23 652 0.23 623 9-93 72 9-93 695 7 7 g 53 52 3 4 9.0 8.7 8.4 \ 12.0 II. 6 II. 2 9 9.70093 9.76 406 2 9 2Q 0.23 594 9-93 687 5i 5 15.0 14.5 14.0 10 ii 12 13 9.70II5 9.70 137 9.70159 9.70 1 80 22 22 21 9-76435 9.76 464 9-76493 9.76522 *7 29 29 29 0.23 505 9.93 680 7 8 7 c 50 49 48 47 6 I 9 1 8.0 17.4 1 6.8 2I.O 2O-3 19.6 24.0 23.2 22.4 27.0 26.1 25.2 0.23 530 0.23 507 0.23 478 9.93 673 9-93 665 9.93 658 14 9.70 202 22 9.7655 1 29 2O 0.23 449 9-93 650 " 46 I^ 9.70 224 9-76 580 **y 0.23 420 9-93 643 / 45 16 9.70 245 9.76 609 2 9 0-23 391 9-93 636 7 44 '7 970 267 . 9.76 639 3 0.23361 9-93 628 43 22 21 IS 9.70 288 9.76 668 , 2 9 0.23 332 9-93621 7 42 i 2.2 2.1 19 9.70310 22 9.76697 28 0.23 303 9.93614 7 g 41 2 4-4 4.2 20 ~9-7 332 21 9.76 725 2Q 0-23 275 9.93 606 40 3 6.6 6.3 21 9.70353 22 9-76 754 ~y 0.23 246 9-93 599 g 39 4 8.8 8.4 22 9-70 375 9.76 783 2 9 0.23217 9-93591 38 5 1 1.0 10.5 23 9.70396 9.76812 2 9 0.23 1 88 9-93 584 7 37 6 13.2 12.6 24 2 5 26 9.70418 9-70439 9.70461 22 21 22 9.76841 9.76870 9.76 899 2 9 29 29 0.23 159 0.23 130 0.23 101 9-93577 9-93 569 9.93 562 7 8 7 36 35 34 9 15.4 14.7 17.6 16.8 19.8 18.9 27 9.70482 21 22 9.76928 2 9 20 0.23072 9-93 554 33 28 9.70 504 21 976957 *y 20 0.23 043 9-93 547 7 g 32 29 9-7 5 2 5 22 9.76 986 ^y 20 0.23014 9-93 539 3 30 9-70 547 21 9.77015 4-0 3-5 ) 4.8 4.2 37 9.70 697 22 9.77217 2 9 O.22 783 9.93480 7 23 ; 7 5-6 4-9 i 64 c 6 38 9.70718 21 9-77 246 2 \ 0.22 754 9-93472 ? 22 39 9-7 739 22 9-77 274 2Q 0.22 726 9-93 465 21 - ) 7.2 6.3 40 9.70 761 21 9-77 303 ^y 20 0.22 697 9-93457 20 4i 9.70 782 9-77 332 ^y 20 0.22 668 9-93 45 / 19 42 9.70 803 21 9.77361 ^y 29 0.22 639 9-93 442 7 18 43 44 9.70 824 9.70846 22 21 9-77 39 9.77418 28 2Q O.22 OIO 0.22 582 9-93 435 9.93427 8 1 7 16 45 9.70 867 9-77 447 *y 0.22553 9.93420 | ' '5 46 9.70888 21 9-77476 29 0.22 524 9-93412 | 4 47 9.70909 21 22 9-77 55 29 28 0.22495 9-93 405 7 8 13 777 48 9.70931 9-77 533 0.22467 9-93 397 12 an 29 as 49 9.70 952 21 9.77562 29 20 O.22 438 9-93 390 I II o 50 9.70973 21 9.77591 *y 28 O.22 409 9-93 .?X2 IO I 2.1 2.1 2.0 r . f, ~ ,~ 5 1 5 2 53 9.70 994 9.71015 9.71 036 21 21 22 9.77619 9.77 648 9.77677 29 29 0.22381 0.22 352 0.22323 9-93 375 9-93 367 9-93 36o 8 7 g 7 2 3 4 0.4 u.2 O.O [0.7 10.4 10.0 [5.0 14.5 14.0 19.3 18.6 iS.o 54 9.71 058 9.77 706 29 28 0.22 294 9-93 352 g 6 5 23.6 22.8 22.0 55 9.71 079 21 9-77 734 2Q O.22 266 9-93 344 7 5 6 27.9 26.9 26.0 56 9.71 loo 21 9-77 763 ^y 28 0.22237 9-93 337 / 8 4 7 57 9.71 121 21 9.77791 0.22 209 9-93 329 3 58 9.71 142 9.77 820 29 0.22 1 80 9-93 322 7 g 2 59 9-7' 163 21 9-77 849 29 28 0.22 151 3 3 1 4 7 I 60 9.71 184 9.77877 0.22 123 9-93 30? O L. Cos. d. L. Cot. c. d.| L. Tan. L. Sin. d. ' P.P. 59' 31 451 j ' L. Sin. d. L. Tan. c. d L. Cot. L. Cos. d. P. P. 9.71 184 9-77877 0.22 123 9-93 307 g 6O I 2 3 4 I I 9 9.71 205 9.71 220 9.71 247 9.-I 268 9.-I 289 9.71 310 9-7 1 33i 9- 7 1 352 9-7' 373 21 21 21 21 9.77906 9-77935 9-77 9<>3 9.77992 9.78020 9.78 049 9.78077 9.78 106 9-/S 135 29 28 2 9 28 29 28 29 29 28 0.22 094 0.22065 0.22037 0.22008 0.21 980 0.21951 0.21 923 ' 0.21 894 ! 0.21 865 9-93 299 9.93 291 9-93 284 9.93 276 9.93 269 9.93 26! 9-93 253 9.93 240 933 -.r s Q 8 7 8 7 8 s 59 58 57 56 55 54 53 5 2 5' 29 28 i I 2.9 2.8 2 5.8 5 .6 3 8.7 8.4 4 n. 6 n. 2 5 14-5 H-O 10 "71 393 9.78 163 j 0.21 8 37 9-93 230 5O 6 17.4 16.8 ii 12 13 14 15 16 1 7 18 19 9.71 414 9-7i 435 9-7i 45 6 9.71 477 9.71 498 9.71 519 9-7i 539 9.71 560 9-7i 581 21 21 21 21 21 20 21 21 9.78 192 9.78 220 9.78 249 9-78277 9.78 306 9-78 334 9-78 363 9.78391 9.78419 28 29 28 IS 29 28 28 0.21 808 0.21 780 0.21 751 0.21 723 0.21 694 0.21 666 0.21 637 0.21 609 0.21 1581 9-93 223 9-932I5 9-93 207 9.93 2CO 9.93 192 9-93 184 9-93I77 9.93 169 9-93 161 8 8 7 8 8 7 8 8 49 48 47 46 45 44 43 42 41 7 20.3 19.6 8 23.2 22.4 9 26.1 25.2 21 20 I 2.1 2.0 2 j 4.2 4.0 20 9.71 602 9.78 448 28 0.21 55 2 Q-93 154 3 40 3 | 6.3 6.0 21 22 23 24 3 2 7 ! 28 i 29 9.71 622 9-7 * 6 43 9.71 664 9.71 685 9-7i 705 9.71 726 9-7i 747 9.71 767 9.71 788 21 21 21 20 21 21 20 21 9.78476 9.78 505 9-78533 9.78 562 9.78 590 9.78618 9.78 647 9.78675 9.78 704 29 28 29 28 28 29 28 29 28 0.21 524 0.21 495 0.21 467 0.21 438 O.2I 4IO O.2I 382 0.21 353 0.21 325 0.21 296 9-93 M6 9-93 138 9-93 131 9-93I23 9-93II5 9-93 108 9-93 ioo 9-93 092 9-93 084 8 7 8 8 7 8 8 8 39 38 37 36 35 34 33 32 3i 4 8.4 8.0 5 10.5 10.0 6 12.6 12.0 7 14.7 14.0 8 1 6.8 1 6.0 9 18.9 18.0 3O 9.71 809 9-78732 28 0.21 268 9-93077 g 3O 8 7 3i 32 33 34 35 36 3 3 1 39 9.71 829 9.71 850 9.71 870 9.71 891 9.71911 9-7 1 932 9.71952 9-7i 973 9.71 994 21 20 21 20 21 20 21 21 9.78 760 9.78 789 9.78817 9.78 845 9.78874 9.78 902 9.78 930 9-78 959 9.78987 29 28 28 29 28 28 29 28 28 0.21 240 0.21 211 0.21 183 0.21 155 0.21 126 0.21 098 0.21 070 0.21 041 O.2I 013 9.93069 9.93061 9-93053 9-93 46 9-93 038 9-93 030 9-93 022 9.93014 9-93 007 8 O 7 Q 8 8 8 29 28 27 26 25 2 4 23 22 21 i ! 0.8 0.7 2 1.6 1.4 3 2.4 2.1 4 i 3.2 2.8 5 4-o 3-5 6 4.8 4.2 7 5.6 4.9 8 6.4 5.6 9 7-2 6.3 40 9.72014 9.79015 28 O.20 985 9.92 999 8 2O 4i 1 42 9.72034 9.72055 21 9-79043 9.79072 2 0.20957 0.20 928 9.92991 9.92983 8 19 18 43 44 45 46 47 48 49 9.72075 9.72 096 9.72116 9-72 137 9.72157 9.72177 9.72 198 21 20 21 20 20 21 2O 9.79 100 9.79128 9.79156 9-79 i &3 9.79213 9-79 241 9-79 269 X> 00 OO GO VG OOOOC 0.20 900 0.20872 0.20844 O.2O 815 0.2O 787 0.20 759 O.20 731 9.92976 9.92 968 9.92 960 9.92952 9.92 944 9.92 936 9.92 929 8 8 8 8 5 16 J5 M 13 12 II 888 30 29 28 1.9 1.8 1.8 |50 9.72 218 2O 9-79 297 0.20 703 9.92921 a, IO , 5- h 5-4 5- 2 5 1 52 53 54 !i H 59 9.72238 9.72259 9.72 279 9.72 299 9.72320 9.72 340 9.72 360 9.72381 9.72401 21 20 20 21 20 20 21 20 9-79 326 9-79354 9-79 382 9.79410 9-79438 9-79 466 9-79493 9-79523 9-7955' \OOOOOOCCCO ON 00 00 0< 0.20674 O.20 646 0.20618 O.2O 590 O.2O 562 0.20 534 0.20 505 0.20477 0.20449 9.92913 9.92 905 9.92 897 9.92 889 9.92881 9.92 874 9.92 866 9.92 858 9 92 850 8 8 8 8 7 8 8 8 8 9 8 6 5 4 3 2 -' I }.I 12.7 12.2 i 16.9 16.3 15.8 -* 2(>. > 19.9 19.2 24.4 23.6 22.8 ^ 28.1 27.2 26.2 60 9.72421 9-79 579 0.20421 u.uJ S,j O L. 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P.P. 9.72421 , n 9-79579 *8 0.20421 9-92 842 8 60 I 9.72441 9-79 607 28 0.20 393 9-92 834 g S9 2 9.72461 20 9-79 635 28 0.20 365 9.92 826 S8 3 9.72 482 21 9-79 663 0.20 337 9.92818 S7 4 9-7 2 502 20 9-79 691 28 O.20 309 9.92 810 56 29 28 27 j 5 9-72 522 9-79 7 1 9 28 0.20 28l 9.92 803 8 55 i 2.9 2.8 2.7 6 9.72 542 9-79 747 0.20 2 53 9.92 795 g 54 2 5-8 5-6 5-4 7 8 9 9.72 562 9.72582 9.72 602 2O 20 2O 9-79 776 9.79 804 9-79 832 2 9 28 28 28 O.2O 224 0.20 196 0.20 1 68 9.92 787 9.92 779 9.92 771 8 8 8 53 52 5i 3 4 I 8.7 8.4 8.1 11.6 11. 2 10.8 14.5 14.0 13.5 10 9.72 622 21 9.79 860 28 O.2O 140 9-92 763 8 50 7 20 3 19.6 18 9 ii 12 13 9.72 643 9.72663 9.72 683 20 20 9.79 888 9.79916 9.79 944 28 28 28 0. 2O 112 O.2OO84 0.20056 9-92 755 9.92 747 9-92 739 8 8 g 49 48 47 8 9 23.2 22.4 21.6 26.1 25.2 24.3 14 9.72 703 9.79972 28 0.2O O28 9.92 731 g 46 is 9.72 723 2O 20 9.80000 28 O.200OO 9.92 723 g 45 16 9-72 743 9.80028 0.19972 9.92 715 44 17 9.72 763 9.80056 ,0 0.19944 9.92 707 43 21 20 19 18 9-72 783 2O 9.80084 28 0.19916 9.92 699 42 i 2.1 2.0 1.9 19 9.72803 2O 9.80 112 "8 0.19888 9.92691 s 4 1 2 4.2 4.0 3.8 2O 9.72 823 9.80 140 "8 O.I9 860 9.92 683 s 4O 3 6.3 6.0 5.7 21 22 23 9.72 843 9.72863 9-72 883 2O 20 9.80 1 68 9-8o 195 9.80 223 Z 0.19832 O.19 805 0.19777 9-92675 9.92 667 9.92659 8 8 39 38 37 4 I 10.5 10.0 9.5 12.6 12.0 II.4 2 4 9.72 902 1 9 9.80 251 28 0.19 749 992651 Q 36 ^ 14.7 14.0 13.3 16 8 16 o 152 25 26 9.72922 9.72 942 20 9.80 279 9.80 307 28 0.19 721 0.19693 9-92 643 " 9-92 635 35 34 9 18.9 18.0 17.1 27 9.72 962 9-8o 335 -0 0.19665 9.92627 33 28 9.72982 9.80 363 -0 0.19637 9.92 619 52 29 9.73002 9.80 391 "S o. 1 9 609 9.92 611 S 31 30 9.73022 9.80419 "S 0.19 581 9.92 603 g 30 987 31 9.73041 9.80 447 0-19553 9-92 595 c 29 i 0.9 0.8 0.7 32 33 9.73061 9-73 081 20 20 9.80 474 9.80 502 28 0.19 526 0.19498 9-92587 1 o 9-92 579 g 28 27 2 3 1.8 1.6 1.4 2.7 2.4 2.1 34 9-73 ioi 9.80 530 ,0 0.19470 9-9257I 8 2(3 4 3.6 3.2 2.8 35 9.73121 9-80 558 0.19442 9-92563 i o 25 5 4-5 4-0 3-5 36 9-73 HO 1 9 9.80 586 ,0 0.19414 9-92 555 24 6 5-4 4-8 4-2 3 9-73 160 9-73 1 80 20 9.80 614 9.80 642 28 0.19 386 0.19358 9.92 546 ( 9-92 538 o 23 22 7 8 6-3 S- 6 4-9 7.2 6.4 5.6 39 9-73 200 9.80 669 27 28 0.19331 9-92530 i g 21 9 8.1 7.2 6.3 4O 9.73219 9.80 697 28 0.19303 9.92 522 g 20 41 9-73 239 9.80 725 28 0.19275 9-92 514 i o 19 42 9-73 259 9-8o 753 Q 80 78l 28 0.19247 9-92 56 g 18 43 44 9-73 298 20 9.80 808 27 ,0 0.19 192 9.92 490 8 1 7 16 45 9-733I8 9.80 836 ,0 0.19 164 9.92 482 15 46 9-73 337 1 9 9.80 864 28 0.19 136 9 g 47 9-73357 9.80 892 0.19 108 9.92 465 13 887 48 9-73377 9.80919 28 0.19081 9.92457 12 on 98 28 49 9-73 396 J 9 9.80 947 28 0.19053 9-92 449 g II 1 50 9-734I6 9.80 975 28 0.19025 9-92 441 8 IO 1.8 1.8 2.0 51 9-73435 9.81 003 0.18997 9-92 433 9 2 5.4 5.2 6.0 52 53 54 55 9-73455 9-73 474 9-73 494 9-73 5'3 19 20 19 9.81 030 9.81 058 9.81 086 9.81 113 28 28 27 28 0.18970 0.18942 0.18914 0.18887 9.92 425 9.92416 9.92 408 9.92 400 9 8 8 g 8 7 6 5 3 4 i 12.7 12.2 14.0 16.3 15.8 18.0 19.9 19.2 22.0 23.6 22.8 26.O 56 9-73533 9.81 141 28 0.18859 9.92 392 g 4 7 27.2 26.2 57 9-73552 9.81 169 0.18831 9.92 384 3 58 9-73572 9.81 196 2 7 0.18804 9-92 376 2 59 9-73591 9.81 224 28 0.18776 9.92 367 | I 6O 9.73611 9.81 252 o.i 8 748 9-92 359 L. Cos. d. L. Cot. |c. d.| L. Tan. L. Sin. d. > P.P. 57 33 453 / L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. p.p. o 9.73611 9.81 252 27 0.18748 9-92 359 8 6O I 2 3 4 I 7 8 9 9-73 630 9-73 650 9-73 669 9.73689 9-73 708 9-73 727 9-73 747 9-73 766 9-73 785 20 19 20 19 19 20 19 9 9.81 279 9-8 1 307 9-8i 335 9.81 362 9.81 390 9.81418 9.81 445 9-Si 473 9.81 500 28 28 2 7 28 28 27 28 2 7 28 0.18 721 o.i 8 693 0.18665 0.18638 0.18 610 0.18 582 0-18555 0.18 527 0.18500 9.92351 9-92 343 9-92 335 9.92 326 9.92318 9.92310 9.92 302 9.92 293 9.92 285 8 8 9 8 8 8 8 59 58 57 56 55 54 53 52 5 1 28 27 I 1 2.8 2. 7 2 | 5.6 5.4 3 8.4 8.1 4 1 1.2 10.8 5 '4-o 13-5 6 16 8 162 IO y-73 805 9.81 528 28 0.18472 9.92277 8 50 7 19.6 18.9 ii 12 13 14 5 16 18 19 9-73^24 9-73 843 9-73 863 9-73 882 9.73901 9.73921 9-73 940 9-73959 9-73978 19 20 19 19 20 19 19 19 19 9-8i 556 9.81 583 9.81 on 9.81 638 9.81 666 9.81 693 9.81 721 9.81 748 9.81 776 27 28 27 28 2 7 28 2 7 0.18444 0.18417 0.18389 0.18362 o.i 8 334 0.18307 0.18 279 0.18 252 o.i 8 224 9.92 269 9.92 260 9.92 252 9.92 244 9-92 235 9.92227 9.92219 9.92 211 9.92 2O2 ONOO 00 ONOO 00 OO ONOC 49 48 47 46 45 44 43 42 8 22.4 21.6 9 25.2 24.3 20 19 18 i 2.0 1.9 1.8 2 4.0 3.8 3.6 20 9-73 997 2O 9.81 803 28 0.18 197 9.92 194 8 40 3 6.0 5.7 5.4 21 22 23 24 a f 27 i 29 9.74017 9.74036 9-74055 9-74 074 9-74 093 9.74II3 9-74 132 9-74I5 1 9-74 17 19 19 19 19 20 19 19 19 iq 9.81 831 9.81 858 9.81 886 9.81913 9.81 941 9.81 968 9.81 996 9.82 023 9.82051 2 7 28 2 7 28 27 28 3 2 7 o.i 8 169 0.18142 0.18 114 0.18087 0.18059 0.18032 o. 1 8 004 0.17977 0.17949 9.92 186 9.92177 9.92 169 9.92 161 9.92 152 9.92 144 9.92 136 9.92 127 9.92 119 9 8 8 9 8 8 9 8 8 39 38 37 36 35 34 33 32 5 10.0 9.5 9.0 6 12.0 11.4 10.8 7 14.0 13.3 12.6 8 16.0 15.2 14.4 9 18.0 17.1 16.2 30 9-74 189 9.82078 28 0.17922 9.92 in 3O 31 32 33 34 35 i 36 39 9.74 208 9.74227 9-74 246 9-74 265 9-74 284 9-74 303 9-74 322 9-74341 9-74 360 19 19 J9 19 19 19 19 19 9.82 106 9.82 133 9.82 161 9.82 188 9.82 215 9.82 243 9.82 270 9.82 298 9.82 325 2 7 28 27 27 28 2 7 28 2 7 0.17894 0.17867 0.17839 0.17 812 0.17785 0.17757 0.17 730 0.17702 0.17675 9.92 IO2 9.92 094 9.92 086 9.92077 9.92 069 9.92 060 9.92052 9.92 044 9-92035 rN 00 00 ONOO ONOOOO ONO< 2 9 28 27 26 25 24 23 22 21 i 0.9 0.8 2 1.8 1.6 3 2.7 2.4 4 3-6 3-2 5 4-5 4-0 6 5.4 4.8 7 6.3 5.6 8 7.2 6.4 9 8.1 7.2 j 40 9-74 379 9-82 352 28 0.17648 9.92027 2O 41 42 43 9-74 398 9.74417 9.74 436 19 19 9.82 380 9.82 407 9.82 435 27 28 0.17 620 0-17593 0.17 565 9.92018 9.92 oio 9.92 OO2 00 00 << 19 18 17 44 45 46 47 48 49 9-74455 9-74474 9-74493 9-74 5 12 9-74531 9-74 549 19 19 19 19 19 18 9.82462 9.82 489 9.82517 9.82 544 9.82571 9.82 599 27 2 7 28 27 27 28 0.17538 0.17511 0.17483 0.17456 0.17429 0.17 401 9-9 1 993 9.91 985 9.91 976 9.91 968 9.91 959 9-9' 95 1 9 8 9 8 16 15 13 12 II 998 28 27 27 ? '.6 1-5 1-7 50 9-74 568 9.82 626 0-17374 9.91 942 g IO 2 4-7 4-5 5-i 51 52 53 54 II 59 9-74 S 8 ? 9.74606 9-74 625 9-74 644 9.74662 9.74 681 9-74 700 9.74719 9-74 737 19 19 19 18 19 19 19 18 9.82 653 9.82681 9.82 708 9-82 735 9.82 762 9.82 790 9.82817 9.82 844 9.82871 28 27 27 2 7 28 27 27 27 28 0.17347 0.17319 0.17 292 0.17 265 0.17 238 0.17 210 O.I? 183 0.17 156 0.17 129 9-91 934 9.91 925 9.91917 9.91 908 9.91 900 9.91 891 9.91 883 9.91 874 9.91 866 9 g 9 8 9 9 8 9 8 7 6 5 4 3 2 I 7.8 7.5 8.4 J 10.9 10.5 1 1.8 4 14.0 13.5 15.2 g 17.1 16.5 18.6 20.2 19.5 21-9 A 23.3 22.5 25.3 9 26 ' 4 25 ' 5 - 60 9-74 75 6 9.82 899 0.17 ioi 9.91 857 L. Cos. d. L. Cot. c. d, L. Tan. L. Sin. d. ' P.P. 56' 454 34 ' L. Sin. d. L. Tan. |c. d. L. Cot. L. Cos. d. P. P. o 9-74756 i In 9.82899 0.17 101 9.91 857 s 6O I 9-74 775 9.82926 1 0.17074 9-9'i 849 S9 2 9-74 794 18 9-82953 :: 0.17047 9.91 840 g S8 3 9.74812 9.82 980 28 0.17 020 9-91 832 57 4 9.74831 9.83 008 0.16992 9.91 823 g S6 28 27 26 5 6 9-74 850 9.74 868 1 9 18 9.83 035 9.83 062 27 2 7 0.16965 0.16938 9.91815 9.91 806 9 c 55 54 i 2.8 2.7 2.6 2 5.6 5.4 5.2 7 9-74 887 1 9 9.83 089 27 28 0.16911 9.91 798 S3 3 8.4 8.1 7.8 ! 8 9-74 906 18 9.83117 0.16883 9.91 789 g S2 4 1 1.2 10.8 10.4 9 IO 9-74 924 19 jg 9.83 144 2 7 27 0.16856 9.91 781 9 5 1 50 5 14.0 13.5 13.0 6 16.8 16.2 15.6 9-74 943 9.83 171 o.i 6 829 9.91 772 ii 12 13 9.74961 9.74 980 9-74 999 19 19 18 9.83 198 9.83 225 9.83 252 2 7 27 28 0.16802 0.16775 o.i 6 748 9.91 763 9-91 755 9.91 746 8 9 g 49 48 47 8 22.4 21.6 20.8 9 25.2 24.3 23.4 14 9.75017 9.83 280 o.i 6 720 9.91 738 46 15 9-75 36 1 9 9-83 307 2 7 o.i 6 693 9.91 729 4S 16 9-75 054 9-83 334 27 0.16666 9.91 720 * 44 i-j 9-75 73 1 9 18 9-83 36i 27 o.i 6 639 9.91 712 43 19 18 18 9.75091 9-83 388 2 0.16612 9.91 703 g 42 i 1.9 1.8 19 9-75 II0 18 9-834I5 ^ 0.16585 9.91 695 4 1 2 3.8 3.6 2O 9.75 128 9.83 442 0.16 558 9.91 686 ' 4O 3 5-7 5-4 21 22 23 9-75 M7 9-75 I6 5 9.75 184 18 19 9.83 470 9-83497 9.83 524 27 2 7 0.16 530 o.i 6 503 0.16476 9.91 677 g 9.91 669 9.91 660 9 39 38 37 4 7.6 7.2 5 9-5 9-o 6 11.4 10.8 24 2 5 9-75 2 2 9-75 221 19 9.8355I 9.83 578 27 2 7 0.16449 0.16422 9-91 651 j J 9.91 643 i 36 7 13.3 12.6 8 15.2 14.4 26 9-75 239 1 9.83 605 2 7 0.16395 9.91 634 34 9 7- 27 9-75 258 1 9 T o 9.83 632 27 0.16368 9.91 625 y 33 28 9-75 276 .0 9-83 659 0.16341 9.91617 32 29 9-75 294 19 9.83 686 2 7 0.16314 9.91 608 y 3O 9.75313 18 9-83 7' 3 0.16287 9-91 599 g 30 9 8 32 33 9-75331 9-75 35 9-75 368 19 18 T o 9.83 7.40 9-83 768 9-83 795 28 27 0.16260 0.16 232 o.i 6 205 9-9i 59i 9.91 582 9-9i 573 9 9 29 28 2 7 i 0.9 0.8 2 1.8 1.6 3 2.7 2.4 34 P 9-75 386 9-75 405 9-75 423 19 18 18 9.83 822 9-83 849 9.83 876 2 7 27 27 0.16 178 0.16 151 0.16 124 9.91 565 9.91 556 9-9i 547 9 26 25 2 4 4 3-6 3-2 5 4-5 4-o 6 5.4 4.8 37 9-75441 18 9-83 903 0.16097 9.91 538 o 23 8 7.2 6.4 9-75 459 9-83 930 o. 1 6 070 9-9i 530 22 9 8.1 7.2 39 9-75 478 11 9.83957 2 7 0.16043 9.91 521 9 21 40 9-75496 *Q 9.83 984 0.16016 0.91 512 Q 20 41 9-75 5*4 9.84011 0.15989 9.91 504 19 42 9-75533 il 9.84 038 0.15 962 9-9I495 9 18 43 9-75 55 1 9.84065 o.i5935 9.91 486 9 17 44 9-75 569 9-75 587 18 18 9.84092 9.84119 27 27 0.15 908 0.15 881 9.91 477 9.91 469 8 16 IS 46 9-75 605 9.84 146 0.15 854 9.91 460 9 14 9 8 8 47 48 9-75 624 9-75 642 18 18 9.84173 9.84 200 27 0.15 827 0.15 800 9.91451 9.91 442 9 13 12 28 28 27 49 9.75 660 9.84 227 27 0-15 773 9-9i 433 g II 1.6 1.8 1.7 5O 9-75 678 rS 9.84 254 "6 0.15 746 9.91 425 IO 4-7 5-2 5- 1 5 1 5 2 S3 9-75 696 9-75 7 J 4 9-75 733 18 19 9.84 280 9-84 307 9-84 334 27 2? 0.15 720 0.15693 0.15666 9.91 416 9.9: 407 9.91 398 9 9 7 7.8 8.8 8.4 J 10.9 12.2 1 1. 8 4 14.0 15.8 i;.2 54 55 56 9-75751 9-75 7 6 9 9.75 787 18 18 18 9.84 361 9.84 388 9.84415 27 27 0.15639 0.15 612 0-15 585 9.91 389 9.91 381 9.91 372 9 8 ! 9 6 5 4 g 17.1 19.2 18.6 20.2 22.8 2I.<) 5 23.3 26.2 25.3 26 4 1 57 9-75 805 18 9.84 442 0-15558 9-9i 363 3 9 S* 9-75 823 18 9.84 469 9-9i 354 2 59 9-75 841 iS 9.84 496 27 0.15 504 9-91 345 * I 60 9-75 859 9-84 523 - I 5477 9-9' 336 O L. Cos. d. L, Cot. c. d L. Tan. L. Sin. d. r P.P. 55 35 / L. Sin. d. L, Tan. c. d. L. Cot. L. Cos. d. P.P. o 9-75 8 59 18 9.84 523 0.15477 9-91 336 8 60 2 3 4 I 9 9-75877 9-75 895 9-759I3 9-75931 9-75 949 9-75 967 9-75 985 9.76003 9.76021 ooooooooooooooooa 9.84 ^50 9.84 576 9.84 603 9.84 630 9.84 657 9.84 684 9.84711 9.84 738 9.84 764 26 27 2 7 27 27 2 7 2 7 26 0.15450 0.15 424 -'5397 0.15370 0-15 343 0.15 316 0.15 289 0.15 262 0.15 236 9.91 328 9.91 319 9.91 310 9.91 301 9.91 292 9.91 283 9.91 274 9.91 266 9.91 257 9 9 9 9 9 9 8 9 59 58 57 56 55 54 53 5 2 5" I 2 3 4 5 6 I 27 26 2.7 2.6 5-4 S- 2 8.1 7.8 10.8 10.4 J 3-5 '3- 1 6.2 15.6 18.9 18.2 21.6 2O.8 IO 9.76039 18 9.84791 0.15 209 9.91 248 SO 9 2 4-3 23.4 1 1 12 13 14 \l 17 18 19 9.76057 9.76075 9.76093 9.76111 9.76 129 9.76 146 9.76 164 9.76 182 9.76 200 > oooooooo t--co oo oo o< 9.84818 9.84 845 9.84872 9.84 899 9-84 925 9.84952 9.84 979 9.85 006 9-85 033 27 27 2 7 26 2 7 27 2 7 2 7 26 0.15 182 0.15 155 0.15 128 0.15 101 0.15075 o. 1 5 048 O.I5 021 0.14994 0.14967 9.91 239 9.91 230 9.91 221 9.91 212 9.91 203 9.91 194 9.91 185 9.91 176 9.91 167 9 9 9 9 9 9 9 9 49 48 47 46 45 44 43 42 i 2 3 4 I I 18 17 1.8 1.7 3-6 3-4 5-4 5- 1 7.2 6.8 9-0 8.5 10.8 10.2 12.6 11.9 144 136 2O 9.76218 18 9-85 059 O.I494I 9.91 158 40 9 16.2 15.3 21 22 23 24 25 26 3 29 9.76 236 9.76 253 9.76271 9.76 289 9.76 307 9-76324 9.76 342 9.76 360 9.76 378 18 18 1 7 18 18 18 9.85 086 9-85 113 9.85 140 9.85 166 9-85 193 9.85 220 9.85 247 9-85 273 9.85 300 27 27 26 2 7 2 7 2 7 26 27 O.I49H 0.14887 0.14860 0.14834 0.14807 0.14 780 0-14753 0.14727 0.14 700 9.91 149 9.91 14! 9.91 132 9.91 123 9.91 114 9.91 105 9.91 096 9.91 087 9.91 078 8 9 9 9 9 9 9 9 39 38 37 36 35 34 33 32 2 3 4 5 6 7 10 9 8 !.0 O.Q 0.8 2.0 1.8 1.6 3.0 2.7 2.4 i 4.0 3.6 3.2 5.0 4.5 4.0 6.0 5.4 4.8 7-o 6.3 5.6 30 9-76395 18 9.85 327 0.14673 9.91 06 9 30 8 8.0 7.2 6.4 32 9-764I3 9-7643I 18 9-85 354 9-85 380 26 27 0.14 646 0.14 620 9.91 060 9.91051 9 9 2 9 28 9 9.0 8.1 7.2 33 34 35 36 37 38 39 9.76448 9.76466 9.76 484 9.76501 9.76519 9-7 6 554 00 00 f^OO OO t-0< 9.85 407 9-85 434 9.85 460 9.85 487 9-855H 9-85 540 9.85 567 27 26 2 7 2 7 26 27 - r 4 593 0.14 566 0.14540 0.14513 0.14486 0.14 460 Q.I443.S 9-91 033 9.91 023 9.91 014 9.91 005 9.90 996 9.90 987 9 IO 9 9 9 9 27 26 25 24 23 22 21 i 2 10 10 27 26 1.4 1.3 4-0 3-9 68 65 4O 9.76572 18 9.85 594 2 7 0.14406 9.90 y-S 20 3 9.4 9.1 42 43 44 47 48 49 9.76 590 9.76607 9.76625 9.76 642 9.76 660 9.76677 9.76 695 9.76712 9.76 730 '7 18 '7 18 1 7 18 18 9.85 620 9.85 647 9-85 674 9.85 700 9-85 727 9-85 754 9.85 780 9.85 807 9.85 834 27 2 7 26 27 27 26 27 2 7 26 0.14380 0-14353 0.14326 0.14300 0.14273 0.14246 O.I4 220 O.I 4 193 0.14 1 66 9.90 969 9.90 960 9.90951 9.90 942 9-90 933 9.90 924 9.90915 9.90 906 9.90 896 9 9 9 9 9 9 9 10 19 18 17 16 15 14 13 12 II 4 i 9 IO 12.2 II-7 14.8 14.3 17.6 16.9 20.2 19.5 23.0 22.1 25.6 24.7 9 9 50 9.76 747 18 9.85 860 0.14 140 9.90 SSj IO S 2 53 54 55 56 57 58 59 9-76 765 9.76782 9.76 800 9.76817 9-76835 9-76852 9.76870 9.76887 9.76 904 '7 18 1 7 18 18 1 7 17 18 9.85 887 9-859I3 9.85 940 9.85 967 9-85 993 9.86 020 9.86 046 9.86073 9.86 ioo 26 27 27 26 27 26 27 27 26 o.H"3 0.14087 0.14060 0.14033 0.14007 o. 1 3 980 o- I 3954 0.13927 o. 1 3 900 9.90 N;S 9.90 869 9.90 860 9.90851 9.90 842 9.90 832 9.90 823 9.1)0814 9.90 805 9 9 9 9 10 9 9 9 9 9 8 7 6 5 4 3 2 I 2 3 4 I 7 8 9 1.5 1.4 4-5 4-3 7-5 7-2 ! 10.^ 10.1 '3-5 ! 3- 16.5 15.9 19.5 18.8 22.5 21.7 25.5 24.6 60 9.76 022 9.86 126 0.13874 9.90 796 L. Cos, d. L, Cot, c. d L. Tan. L. Sin, d. ' - - P. P. 54 456 36 L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d, P. P. o 9.76 922 17 9.86 126 27 0.13874 9.90 796 60 I 2 3 4 5 6 7 8 9 9-76 939 9.76957 9.76 974 9.76991 9.77009 9.77 026 9-77 043 9.77061 9.77078 18 7 17 18 7 7 18 7 17 9-86 153 9.86179 9.86 206 9.86 232 9.86 259 9.86 285 9.86312 9-86 338 9-86 365 26 27 26 3 27 26 2 7 0.13847 0.13821 0.13794 0.13 768 0.13 741 O.I37I5 0.13688 0.13 662 0-13635 9.90 787 9.90 777 9.90 768 9-90 759 9.90 750 9.90 741 9.90 731 9.90 722 9.90713 10 9 9 9 9 10 9 9 59 58 57 56 55 54 53 52 5 1 I 2 3 4 27 26 2.7 2.6 5-4 5-2 8.1 7.8 10.8 10.4 13-5 13-0 10 9-77095 17 9-86 392 26 o. 1 3 608 9.90 704 50 6 1 6.2 15.6 ii 12 13 H 15 16 !I 19 9.77 112 9-77 13 9-77 M7 9-77 164 9-77 181 9.77 199 9.77216 9-77 233 9-77 250 18 '7 17 17 18 >7 17 [2 9.86418 9-86 445 9.86471 9.86 498 9-86 524 9-86551 9.86577 9.86 603 9.86 630 2 7 26 27 26 27 26 26 27 26 0.13 582 0-13555 0.13529 0.13 502 0.13476 0.13449 0.13423 0-13397 0.13370 9.90 694 9.90 685 9.90 676 9.90 667 9.90657 9.90 648 9-90 639 9.90 630 9.90 620 9 9 9 10 9 9 9 10 49 48 47 46 45 44 43 42 4i i 9 i 2 18.9 18.2 21.6 20.8 24.3 23.4 18 17 16 1.8 1.7 1.6 3.6 3.4 3.2 20 9.77 268 17 9.86 656 0-13 344 9.90611 40 3 5-4 5-i 4-8 21 22 23 24 25 26 3 29 9-77 283 9-77 32 9-773I9 9-7-7 336 9-77 353 9-77 37 9-77 387 9-77 405 9.77422 17 17 '7 17 '7 17 18 J 7 9.86 683 9.86 709 9-86 736 9.86 762 9.86 789 9.86815 9.86 842 9.86 868 9.86 894 26 2 7 26 2 7 26 27 26 26 O.I33I7 0.13 291 0.13 264 0.13 238 0.13211 0.13 185 0.13 158 0.13132 0.13 106 9.90 602 9.90 592 9.90 583 9-90574 9.90 565 9.90555 9.90 546 9.90537 9.90 527 10 9 9 9 10 9 9 10 39 38 37 36 35 34 33 32 3 1 4 6 i 7 i 8 i 9 i 7.2 6.8 6.4 ?.o 8.5 8.0 3.8 IO.2 9.6 2.6 11.9 1 1.2 1-4 13.6 12.8 5.2 15.3 14.4 30 9-77 439 9.86921 26 0.13079 9.90518 30 10 9 3i 32 33 34 35 36 i 39 9-77 45 6 9-77473 9-77 490 9-77 57 9-77 524 9-77541 9-77558 9-77575 9-77 592 17 *7 17 17 17 17 17 l? 9-86 947 9-86 974 9.87 ooo 9.87027 9.87053 9.87 079 9.87 106 9-87 132 9.87158 2 7 26 27 26 26 27 26 26 0.13053 0.13026 0.13000 0.12973 0.12947 0.12921 0. 1 2 894 0.12868 0.12842 9.90 509 9.90499 9.90490 9.90 480 9.90471 9.90 462 9.90452 9.90 443 9-90 434 10 9 10 9 9 10 9 9 29 28 27 26 25 24 23 22 21 i 2 3 4 I I 9 i.o 0.9 2.0 1.8 3-o 2.7 4.0 3.6 5- 4-5 6.0 5.4 7-0 6.3 8.0 7.2 9.0 8.1 40 9-77 6o9 9.87 1 8? 26 0.12 815 9.90 424 2O 4i 42 9.77 626 9-77 643 i? 9.87 211 9.87 2 3 8 27 26 0.12789 0.12 762 9.90415 9.90 405 10 19 18 43 44 45 46 3 49 9.77 660 9.77677 9-77 6 94 9.77711 9-77 728 9-77 744 9.77761 i? J 7 r 7 7 16 17 17 9.87 264 9.87 290 9-873I7 9-87 343 9-87 369 9.87 396 9.87422 26 27 26 26 3 ->6 0.12736 O.I2 710 0.12683 O.I2 657 O.I2 631 0. 1 2 604 O.I25 7 8 9.90 396 9.90 386 9-90 377 9.90368 9.90358 9-90 349 9-90 339 10 9 9 10 9 10 9 17 16 15 M 13 12 II 9 9 27 26 1.5 1.4 50 9.77778 17 9.87 448 27 0.12552 9-90 330 10 IO 2 4-5 4-3 5i 52 53 54 11 H 59 9-77 795 9.77812 9-77 829 9-77846 9.77862 9-77 8 79 9.77 896 9-779I3 9-77 930 '7 '7 I? 16 17 '7 17 :z 9.87475 9.87 501 9.87527 9-87554 9-87 580 9.87 606 9-87 633 9.87 659 9-87 685 26 26 27 26 26 27 26 26 26 0.12525 0.12499 0.12473 0. 1 2 446 O.I2420 0.12394 0.12367 O.I234I O.I23I5 9.90 320 9.90311 9.90 301 9.90 292 9.90 282 9.90 273 9.90 263 9.90 254 9.90 244 9 10 9 10 9 10 9 10 9 7 6 5 4 3 2 3 4 6 9 10.5 10.1 13-5 '3-o 16.5 15.9 19.5 1 8.8 22.5 21.7 25.5 24.6 60 9.77946 9.87711 0.12 289 9.90 235 O L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. d. ' P. P. 53 37 457 / L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. 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P.P. o 9.79 887 16 9.90 837 26 0.09 163 9.89 050 6O 2 3 4 5 6 9 9-79 903 9.79918 9-79 934 9-79 95 9-79 965 9-79 98i 9.79 996 9.80012 9.80027 15 16 16 !i 15 16 15 16 9.90 863 9.90 889 9.90914 9.90 940 9.90 966 9.90 992 9.91 018 9.91 043 9.91 069 26 2 5 26 26 26 26 2 5 26 "6 0.09 137 0.09 1 1 1 0.09 086 0.09 060 0.09 034 0.09 008 0.08 982 0.08957 0.08931 9.89 040 9.89 030 9.89 020 9.89 009 9.88 999 9.88 989 9.88 978 9.88 968 9.88958 10 IO II 10 10 II 10 10 IO 59 58 57 56 55 54 53 5 2 5i 2 3 4 I 26 25 2.6 2.5 5.2 5.0 7-8 7-5 10.4 10.0 13.0 12.5 ic 6 no 10 9.80 043 9.91 095 26 0.08 905 9.88 948 5O 7 : 18.2 17.5 ii 12 13 14 15 3 19 9.80 058 9.80 074 9.80 089 9.80 105 9.80 120 9.80 136 9.80I5I 9.80 1 66 9.80 182 16 15 16 II 15 15 16 i ^ 9.91 121 9-91 H7 9.91 172 9.91 198 9.91 224 9.91 250 9.91 276 9.91 301 9.91 327 26 2 5 26 26 26 26 2 5 26 "6 0.08 879 0.08 853 0.08 828 0.08 802 0.08 776 0.08 750 0.08 724 0.08 699 0.08 673 9.88937 9.88 927 9.88917 9.88 906 9.88 896 9.88 886 9.88 875 9.88 865 9.88855 IO IO II 10 10 II IO IO 1 1 49 48 47 46 45 44 43 42 4i 8 9 2 3 20.8 20.0 23.4 22.5 16 15 1.6 1.5 3-2 3-o 4-8 4-5 2O 9.80 197 16 9-9i 353 "6 0.08 647 9.88 844 IO 4O 4 6.4 6.0 21 22 23 24 25 26 3 29 9.80213 9.80 228 9.80 244 9.80 259 9.80 274 9.80 290 9.80 305 9.80 320 9.80 336 15 16 1 S 15 16 15 :i 9-9i 379 9.91 404 9.91 430 9.91 456 9.91 482 9.91 507 9-9i 533 9-9i 559 9-9 1 585" 25 26 26 26 25 26 26 26 0.08 621 0.08 596 0.08 570 0.08 544 0.08518 0.08 493 0.08 467 0.08441 0.08415 9.88834 9.88 824 9.88813 9.88 803 9-88 793 9.88 782 9.88 772 9.88 761 9.88751 IO II 10 10 II 10 II 10 10 39 38 37 36 35 34 33 3 2 3 1 5 6 I 9 8.0 7.5 9.6 9.0 II. 2 10.5 12.8 12.0 I 4 .4 13.5 11 10 30 9.80351 9.91 610 26 0.08 390 9.88 741 3O 3i 32 33 34 35 36 37 38 39 9.80 366 9.80 382 9.80 397 9.80412 9.80 428 9.80 443 9.80458 9.80 473 9.80 489 16 !5 IS 16 15 15 !i 9.91 636 9.91 662 9.91 688 9-9i 713 9-9i 739 9.91 765 9.91 791 9.91 816 9.91 842 26 26 2 5 26 26 26 2 5 26 26 0.08 364 0.08 338 0.08 3 1 2 0.08 187 0.08 261 0.08 235 0.08 209 0.08 184 0.08 158 9-88 730 9.88 720 9.88 709 9.88 699 9.88 688 9.88 678 9.88 668 9-88 657 9.88 647 10 II IO II 10 10 II IO 29 28 2 7 26 25 24 23 22 21 3 4 I 9 3-3 3-o 4.4 4.0 1:1 IS 7-7 7- 8.8 8.0 9-9 9-0 4O 9.80 504 I c 9.91 868 2; 0.08 132 9.88 636 IO 20 4i 42 43 44 45 46 47 48 49 9.80519 9.80 534 9.80550 9-80 565 9.80 580 9-8o 595 9.80610 9.80 625 9.80 641 15 16 J 5 15 X 5 15 ;i 9.91 893 9.91919 9-91 945 9.91 971 9.91 996 9.92022 9.92 048 9.92073 9.92 099 26 26 26 2 5 26 26 % 26 0.08 107 0.08081 0.08 05 5 0.08 029 0.08 004 0.07 978 0.07 952 0.07 927 0.07 901 9.88 626 9.88615 9.88 605 9.88 594 9.88 584 9.88 573 9.88 563 9.88552 o.SX -,42 II 10 II IO II 10 10 19 18 7 16 15 H 13 12 II i 11 11 26 25 1.2 I.I 3-5 3-4 5-9 5-7 5O 9.80 656 9.92125 0.07 875 9.88531 IO 3 8.3 8.0 5 1 5 2 53 54 II 57 5 59 9.80671 9.80 686 9.80 701 9.80 716 9.80 731 9.80 746 9.80 762 9.80 777 9.80 792 15 15 '5 1 S '5 16 i5 15 9.92 150 9.92 176 9.92 202 9.92 227 9.92 253 9.92 279 9.92 304 9-92 330 9.92 356 2 5 26 26 25 26 26 25 26 26 0.07 S^o 0.07 S.?4 0.07 798 0.07 773 0.07 747 0.07 721 0.07 696 0.07 670 0.07 644 9.88 =;-! 9.88510 9.88 499 9.88 489 9.SS 478 y.SS 408 9.88457 9.88 447 9.88 436 1 1 1 1 10 II 10 II IO II 1 1 9 8 7 6 5 4 3 2 9 10 ii IO.6 IO.2 13.0 12.5 15.4 14. S 20.1 19.3 22-5 21.0 24.8 23.9 60 9.80807 9 .92 3 8l 0.07 619 9.88425 L_ L. Cos. d. L, Cot, c. d. L. Tan. L. Sin. d. ' P.P. 50' 460 40' / L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. P.P. I o 9.80807 I c 9.92 381 ?6 0.07 619 9-88 425 IO 60 I 2 3 4 I 9 9.80 822 9.80837 9.80 852 9.80 867 9.80 882 9.80 897 9.80912 9.80927 9.80 942 15 15 15 15 15 15 '5 9.92 407 9-92 433 9.92 458 9.92 484 9.92510 9-92 535 9.92561 9.92 587 9.92 612 26 25 26 26 2 5 26 26 3 0-07 593 0.07 567 0.07 542 0.07 516 0.07 490 0.07 465 0.07 439 0.07413 0.07 388 9.88415 9.88 404 9-88 394 9-88 383 9-88 372 9.88 362 9.88351 9.88 340 9-88 330 ii IO II II IO II II 10 P 57 56 55 54 53 5 2 2 3 4 5 6 26 25 2.6 2.5 5- 2 5- 7-8 7-5 10.4 i o.o 13.0 12.5 15.6 15.0 10 9.80957 9.92 638 0.07 362 9.88319 50 7 g 18.2 17.5 ii 12 13 IS 3 19 9.80 972 9.80 987 9.81 002 9.8l 017 9.8l 032 9.8l 047 9.81 061 9.81 076 9.81 091 15 15 15 15 15 '5 15 9.92 663 9.92 689 9.92715 9.92 740 9.92 766 9.92 792 9.92817 9.92 843 9.92 868 26 26 25 26 26 25 26 3 0-07 337 0.07 311 0.07 285 0.07 260 0.07 234 0.07 208 0.07 183 0.07 157 0.07 132 9.88 308 9.88 298 9.88 287 9.88 276 9.88 266 9-88 255 9.88 244 9-88 234 9.88 223 10 II II IO II II IO II 49 48 47 46 45 44 43 42 9 i 2 3 23.4 22.5 15 14 1.5 1.4 3.0 2.8 4-5 4-2 60 56 20 9.81 106 1C 9.92 894 26 0.07 1 06 9.88212 40 7-5 7- 21 22 23 24 3 27 28 29 9.81 121 9.81 136 9.81 151 9.81 166 9.81 1 80 9.81 195 9.8l 210 9 .8l 225 9.8l 240 15 15 15 H 15 15 9.92 920 9.92 945 9.92971 9.92 996 9-93 022 9-93 048 9-93 073 9-93 099 9.93124 25 26 25 26 26 25 26 25 0.07 080 0.07 055 0.07 029 0.07 004 0.06 978 0.06952 0.06 927 0.06 901 0.06 876 9.88 201 9.88 191 9.88 1 80 9.88 169 9.88 158 9.88 148 9-88 137 9.88 126 9.88115 IO II II II 10 II II II 39 38 37 36 35 34 33 32 7 8 9 i 9.0 8.4 10.5 9.8 I2.O II. 2 13.5 12.6 11 10 I.I 1.0 30 9.8l 254 9.93 150 0.06 850 9.88 105 30 2 2.2 2.0 32 33 34 35 36 3 39 9.8l 269 9.8l 284 9.8l 299 9.8l 314 9 .8l 328 9-8 1 343 9.81 358 9.81 372 9.81 387 15 15 15 H 15 15 15 IT 9-93 175 9-93 201 9-93 227 9-93 252 9-93 278 9-93 303 9-93 329 9-93 354 9-93 38o 26 26 25 26 25 26 n 26 0.06 825 0.06 799 0.06 773 0.06 748 0.06 722 0.06 697 0.06671 0.06 646 0.06 620 9.88 094 9.88083 9.88 072 9.88061 9.88051 9.88 040 9.88029 9.88018 9.88 007 II II II IO II II II II II 29 28 27 26 25 24 23 22 21 3 4 I I 9 3-3 3-o 44 4.0 5-5 5-o 6.6 6.0 7-7 7- 8.8 8.0 9-9 9-o 40 9.81 402 9.93 406 0.06 594 9.87 996 20 41 42 43 44 47 48 49 9.81 417 9.81 431 9.81 446 9.81 461 9-8i 475 9.81 490 9.81 505 9.81 519 9-8 1 534 14 15 '5 14 15 15 15 1C 9-93431 9-93457 9-93 482 9-93 58 9-93 533 9-93 559 9-93 584 9.93610 9-93 636 25 26 25 26 3 2 5 26 26 0.06 569 0.06 543 0.06 518 0.06 492 0.06 467 0.06 441 0.06416 0.06 390 0.06 364 9.87 985 9-87 975 9.87 964 9-87 953 9.87 942 9-87931 9.87 920 9.87 909 9.87 898 IO II II II II II II II II 19 18 17 16 '5 14 13 12 II 2 3 11 10 10 26 26 25 1.2 1.3 1.2 3-5 3-9 3-8 5-9 6.5 6.2 SO 9.81 549 14 9-93 66 1 26 0.06 339 9.87 887 IO 10 4 8.3 9.1 8.8 5 2 53 54 55 56 H 59 9.81 563 9.81 578 9.81 592 9.81 607 9.81 622 9.81 636 9.81 651 9.81 665 9.81 680 15 15 15 IS 14 9-93 687 9-93 7 12 9-93 738 9-93 763 9-93 789 9.93814 9.93 840 9-93 865 9-93 891 25 26 25 26 2 5 26 0.06313 0.06 288 0.06 262 0.06 237 O.06 211 0.06 1 86 0.06 1 60 0.06 135 0.06 109 9.87 877 9.87 866 9-87 855 9.87 844 9-87 833 9.87 822 9.87811 9.87 800 9.87 789 II II II II II II II II 1 1 9 8 7 6 5 4 3 2 I O \O 00-vJ ON<-" 3.0 14.3 13.8 5.4 16.9 16.2 7.7 19.5 18.8 !O.I 22.1 21.2 >2-5 24.7 23.8 !4.8 60 9.81 694 9.93916 0.06 084 9.87 778 L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. d. t P. P. 49 C 41 461 , / L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. p. P. o 9.81 694 9.93916 26 0.06 084 9-87 778 6O I 2 3 4 I 7 8 9 9.81 709 9.81 723 9.81 738 9.81 752 9.81 767 9.81 781 9.81 796 9.81 810 9.81 825 >4 15 H 15 H 15 9-93 942 9-93 967 9-93 993 9.94018 9.94 044 9.94 069 9-94 095 9.94120 9.94 146 25 26 25 26 25 26 3 2s 0.06 058 0.06 033 0.06 007 0.05 982 0.05 956 0.05931 0.05 905 0.05 880 0.05 854 9.87 767 9.87 756 9-87 745 9-8? 734 9.87 723 9.87712 9.87 701 9.87 690 9-87 679 ii ii ii ii n ii ii 1 1 1 1 59 58 57 56 55 54 53 52 5' 2 3 4 26 25 2.6 2.5 5.2 5.0 7-8 7-5 10.4 10.0 13.0 12.5 15.6 15.0 IO 9.81 839 9.94171 26 0.05 829 9.87 bbS 50 7 18.2 17.5 1 1 12 13 14 15 17 19 9.81 854 9.81 868 9.81 882 9.81 897 9.81 911 9.81 926 9.81 940 9.81955 9.81 969 15 '4 '5 14 14 9-94 197 9-94 222 9-94 2 4 8 9-94 273 9.94 299 9-94 324 9-94 35 9-94 375 9.94 401 25 26 25 26 25 26 25 26 0.05 803 0.05 778 0.05 752 0.05 727 0.05 701 0.05 676 0.05 650 0.05 625 0-05 599 9-87 657 9.87 646 9.87 635 9.87 624 9.87613 9.87 601 9.87 590 9-87 579 9.87 568 n ii ii ii 12 II 1 1 II 49 48 47 46 45 44 43 42 41 8 9 I 2 3 20.8 20.0 23.4 22-5 15 14 1.5 1.4 3.0 2.8 4-5 4-2 20 9.81 983 9-94 426 26 0.05 574 9-87 557 II 4O 4 6.0 5.6 21 22 23 i 24 2 5 26 3 29 9.81 998 9.82012 9.82 026 9.82041 9-82 055 9.82 069 9 82 084 9.82 098 9.82 112 14 15 14 '5 H 14 9-9445 2 9-94477 9-94 53 9-94 528 9-94 554 9-94 579 9-94 604 9-94 630 9-94 655 25 26 25 26 25 2 5 26 2 5 0.05 548 0-05 523 0.05 497 0.05 472 0.05 446 0.05 421 0.05 396 0.05 370 0-05 345 9.87 546 9-87 535 9.87 524 9-87 5 J 3 9.87 501 9.87 490 9.87 479 9.87 468 9-87457 II 11 II 12 II II II II 39 38 37 36 35 34 33 32 I 9 7-5 7- 9.0 8.4 10.5 9.8 12.0 I 1.2 13.5 12.6 12 11 3O 9.82 126 9.94681 0.05 319 9.87 446 30 j 1.2 I.I 32 33 34 P IS 39 9.82 141 9.82 153 9.82 169 9.82 184 9.82 198 9.82 212 9.82 226 9.82 240 9.82 255 '4 H 15 14 H 15 9.94 706 9-94 732 9-94 757 9-94 783 9.94 808 9.94 834 9.94 859 9-94 884 9.94910 26 25 26 3 25 2 5 26 2C 0.05 294 0.05 268 0.05 243 0.05 217 0.05 192 0.05 1 66 0.05 141 0.05 116 0.05 090 9-87 434 9-87423 9.87412 9.87 401 9.87 390 9-87 378 9.87 367 9-87 356 9-87 345 II II II II 12 II I I II 1 1 29 28 27 26 25 24 23 22 21 2 4 5 6 7 8 9 2.4 2.2 3-6 3-3 4-8 4-4 6.0 5.5 7.2 6.6 8-4 7-7 9.6 8.8 10.8 9.9 4O 9.82 269 14 9-94 935 ?6 0.05 065 9-87 334 12 20 42 43 44 45 46 47 48 49 9.82 283 9.82 297 9.823II 9.82 326 9.82 340 9-82 354 9.82 368 9.82 382 9.82 396 4 15 14 14 9.94961 9.94 986 9.95012 9-95 37 9-95 62 9.95 088 9-95 "3 9-95 39 9-95 l6 4 2 5 3 25 26 2 5 26 0.05 039 0.05 014 0.04 988 0.04963 0.04 938 0.04912 0.04 887 0.04 861 0.04 836 9.87 322 9-87311 9.87 300 9.87 288 9.87 277 9.87 266 9-87 255 9.87 243 9.87 232 II II 12 II II II 12 II II in '7 16 15 14 13 12 II o 2 3 12 12 11 26 25 25 I.I I.O I.I 3-2 3-i 3-4 5-4 5- 2 5-7 7-6 7-3 8.0 50 9.82410 14 9-95 ! 9 0.04810 9.87 221 12 IO e 9.8 9.4 IO.2 51 52 53 54 55 56 P 59 9.82 424 9.82439 9-82453 9.82 467 9.82481 9.82 493 9.82 509 9.82 523 9-82537 15 H H H 14 9-952I5 9.95 240 9-95 266 9-95 291 9-953I7 9-95 342 9-95 368 9-95 393 9.95 418 3 25 26 2 5 26 25 0.04 783 0.04 760 0.04 734 0.04 709 0.04 683 0.04 658 0.04 632 0.04 607 0.04 582 9.87 209 9.87 198 9.87 187 9-87 *75 9.87 164 9-87 153 9.87 141 9.87 130 9.87119 II II 12 II 11 12 II II I 2 7 6 5 4 3 2 10 \ :;> 1-9 II.3 12.5 4.1 13.5 14.8 6.2 15.6 17.0 8-4 7-7 '9-3 0.6 19.8 21.6 2.8 21.9 23.9 4.9 24.0 - 60 9-82551 9-95 444 0.04 556 9.87 I0 7 L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. d. 1 P. P. 48 462 42 1 L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. p.p. o 9-82551 9-95 444 0.04 556 9.87 107 1 1 60 2 3 4 I 9 9.82 565 9.82 579 9.82 593 9.82 607 9.82 621 9.82 635 9.82 649 9.82 663 9.82 677 H 14 H 4 H 14 H '4 9-95 4t>9 9-95 495 9-95 520 9-95 545 9-95 57i 9-95 596 9.95 622 9-95 6 47 9-95 6 7 2 26 2 5 2 5 26 25 26 2 5 2 5 0.04 531 0.04 505 0.04 480 0.04 455 0.04 429 0.04 404 0.04 378 0.04 353 0.04 328 9.87 090 9.87 085 9-87073 9.87 062 9.87050 9.87 039 9.87 028 9.87016 9.87 005 ii 12 II 12 II II 12 II 59 58 57 56 55 54 53 52 5 1 I 2 3 4 26 25 2.6 2.5 S- 2 5-o 7-8 7-5 10.4 10.0 13.0 12.5 15.6 15.0 10 9.82091 9-95 6 98 0.04 302 9.86 993 5O 7 18.2 17.5 ii 12 13 14 15 16 1 7 18 19 9.82 705 9.82719 9-82 733 9.82 747 9.82761 9-82 775 9.82 788 9.82 802 9.82816 U 14 H 14 H 13 H H 9-95 723 9-95 748 9-95 774 9-95 799 9-95 825 9-95 850 9-95 875 9.95 901 9-95 926 2 5 26 2 5 26 25 2 5 26 2 5 0.04 277 0.04 252 0.04 226 O.04 201 0.04175 0.04 150 0.04 1 25 0.04 099 0.04 074 9.86982 9.86 970 9.86 959 9-86 947 9.86 936 9.86 924 9.86913 9.86 902 9.86 890 12 II 12 II 12 II II 12 49 48 47 46 45 44 43 42 4i 8 9 i 2 3 20.8 20.0 23.4 22.5 14 13 1.4 1.3 2.8 2.6 4-2 3-9 2O 9.82 830 9-95 95 2 0.04 048 9.86879 40 4 5.6 5.2 21 22 23 24 3 27 28 29 9.82 844 9.82 858 9.82872 9.82 885 9.82 899 9.82913 9.82927 9.82941 9-82955 14 14 13 14 14 H 14 H 9-95977 9.96 002 9.96028 9.96053 9.96078 9.96 104 9.96 129 9-96 155 9.96 1 80 2 5 26 25 2 5 26 25 26 25 0.04 023 0.03 998 0.03 972 0.03 947 0.03 922 0.03 896 0.03 871 0.03 845 0.03 820 9.86 867 9.86855 9.86 844 9.86 832 9.86 821 9.86 809 9.86 798 9.86 786 9-86 775 12 II 12 II 12 II 12 II 12 39 38 37 36 35 34 33 32 3i 6 1 7 8 9 i 8.4 7.8 9.8 9.1 I 1. 2 IO-4 12.6 11.7 12 11 1.2 I.I 3O 9.82 968 9.96 205 26 0-03 795 9-86 763 II 30 2 2-4 2.2 3 1 32 33 34 ii IS 39 9.82 982 9.82 996 9.83010 9.83 023 9-83 037 9.8305 1 9.83 065 9.83 078 9.83 092 H M 13 H H H 13 14 14 9.96 231 9.96 256 9.96 281 9.96 307 9-96332 9.96357 9-96 383 9.96408 9-96433 25 2 5 26 2 5 2 5 26 2 5 3 0.03 769 0.03 744 0.03719 0.03 693 0.03 668 0.03 643 0.03617 0.03 592 0.03 567 9-86 752 9.86 740 9.86 728 9.86717 9.86 705 9.86 694 9.86682 9.86 670 9.86 659 12 12 II 12 II 12 12 II 12 29 28 27 26 25 24 23 22 21 3 4 i 9 3-6 3-3 4.8 4.4 6 - 5-5 7.2 6.6 8.4 7-7 9.6 8.8 10.8 9.9 4O 9.83 106 9-96 459 0.03 541 9.86 647 20 41 42 43 44 $ % 49 9.83 120 9.83 133 9.83 147 9.83 161 9-83 174 9.83 1 88 9.83 202 9-832I5 9.83 229 13 H H 13 H 14 13 14 9-96484 9.96510 9-96 535 9-96 560 9.96 586 9.96611 9.96 636 9.96 662 9.96 687 26 25 25 26 25 25 26 25 0.03 5 1 6 0.03 490 0.03 465 0.03 440 0.03414 0.03 389 0.03 364 0.03 338 0.03313 9.86635 9.86 624 9.86612 9.86600 9.86 589 9.86577 9-86 565 9-86554 9.86 542 II 12 12 II 12 12 II 12 19 18 17 16 15 H 13 12 II o i 2 3 * 12 11 11 26 26 25 I.I 1.2 I.I 3-2 3-5 3-4 5-4 5-9 5-7 7.6 8.3 8.0 50 9.83 2 4 2 9.96712 26 0.03 288 9.86 530 IO |l 1.9 13.0 12.5 5i 52 53 54 I 58 59 9.83 2 5 6 9.83 2 7 9.83 283 9.83 297 9-833'0 9-83 324 9-83 338 9.8335I 9-83 365 H '3 H 13 H 14 13 H 9.96 738 9.96 763 9.96 788 9.96814 9.96 839 9.96 864 9.96 890 9.96915 9.96 940 25 25 26 25 25 26 25 25 0.03 262 0.03 237 0.03 212 0.03 1 86 0.03 161 0.03 136 0.03 1 10 0.03 085 0.03 060 9.86518 9-86 507 9.86 495 9.86 483 9.86 472 9.86 460 9.86 448 9-86436 9-86425 II 12 12 II 12 12 12 II 9 8 7 6 5 4 3 2 I i c I 1 ^.i 15.4 14.8 5.2 17.7 17.0 B.4 20.1 19.3 D.6 22.5 21.6 2.8 24.8 23.9 t-9 6O 9-83 378 9.96 966 0.03 034 9.86413 O L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. d. ' P.P. 47 43 C 4(J3 1 r L. Sin. | d. L. Tan. c. d. L. Cot. L. Cos. d. p. p. o I 2 3 4 I 7 8 9 IO ii 12 13 H !i 17 19 9-83 378 14 13 H i3 14 13 H 13 H 9.96 966 25 11 2 5 11 2 5 2 5 % 2 5 11 2 5 2 5 26 25 11 25 25 26 25 2 5 3 2 5 11 25 2 5 26 25 25 25 26 2 5 2 5 26 25 25 26 25 2 5 3 25 3 25 25 25 26 25 M 25 2 5 25 26 0.03 034 0.03 009 O.O2 984 0.02 958 0.02 933 0.02 908 0.02 882 0.02857 O.O2 832 O.O2 807 9.86413 12 12 12 II 12 12 12 12 12 II 12 12 12 12 12 12 12 " 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 13 12 12 12 12 12 12 13 12 12 12 \2 13 60 59 58 57 56 55 54 53 52 5i 50 49 48 47 46 45 44 43 42 4i 40 39 38 37 36 35 34 33 3 2 3i 30 29 28 27 26 2 5 24 23 22 21 20 19 18 17 16 15 H U 12 II IO 9 8 7 6 5 4 3 2 O 2 3 4 I I 9 i 2 3 4 i I 9 i 2 3 4 9 26 25 2.6 2.5 S- 2 5- 7-8 7-5 10.4 i o.o 13.0 12.5 15.6 15.0 18.2 17.5 20.8 2O.O 23.4 22.5 14 13 1.4 1.3 2.8 2.6 4-2 3-9 5- 6 5-2 7-o 6.5 8.4 7.8 9.8 9.1 1 1. 2 IO-4 12.6 11.7 12 11 1.2 I.I 2.4 2.2 3-6 3-3 4.8 4.4 6.0 5.5 7.2 6.6 8.4 7.7 9.6 8.8 10.8 9.9 9-83 392 9-83 405 9.83419 9-83 432 9.83 446 9-83 459 9-83 473 9.83 486 9.83 500 9-835U 9-83 S 2 ? 9.83 540 9-S3 554 9-83 5 6 7 9.83 581 9-83 594 9.83 608 9.83 621 9-83 634 9.96991 9.97016 9-97 042 9-97 6 7 9-97 092 9.97118 9-97 43 9.97 168 9-97 193 9.86 401 9.86 389 9.86377 9.86 366 9-86 354 9-86 342 9-86 330 9.86318 9.86 306 9-86 295 9.97 219 O.O2 781 13 H 13 H J 3 14 i3 i3 H 13 13 14 13 H U 13 H 13 ! 3 H 13 13 13 H 13 13 i3 H 13 13 13 14 13 13 13 13 H 13 13 13 13 13 13 H 13 3 '3 i3 U 9-97 244 9-97 269 9.97 295 9-97 320 9-97 345 9-97371 9-97 396 9.97421 9-97 447 9-97 472 9-97 497 9-97 523 9-97 548 9-97 573 9-97 598 9-97 624 9-97 649 9-97 6 74 9.97 700 O.O2 756 0.02 731 0.02 705 0.02 680 O.O2 655 O.O2 629 O.02 604 0.02 579 0.02 553 9.86 283 9.86271 9-86 259 9.86 247 9.86235 9.86 223 9.86211 9.86 200 9.86 188 20 9.83 648 O.O2 528 9.86 176 21 22 23 24 s 27 28 29 3O 3 1 32 33 34 3 37 3* 39 40 41 42 43 44 J2 47 48 49 50 5 1 52 53 54 H H 59 60 9.83 661 9.83 674 9.83 688 9.83 701 9-83715 9.83 728 9-83 741 9-83 755 9.83 768 0.02 503 0.02 477 0.02452 O.O2 427 O.O2 4O2 O.02 376 0.02351 O.02 326 0.02 30O 9.86 164 9.86 152 9.86 140 9.86 128 9.86 116 9.86 104 9.86 092 9.86 080 9.86 068 9.83 781 9-97 725 9-97 75 9-97 77 6 9.97 801 9.97 826 9.97851 9-97 877 9-97 902 9-97 927 9-97953 O.O2 275 9.86 056 9-83 795 9.83 808 9.83 821 9-83 834 9.83 848 9.83 861 9.83 874 9.83 887 9-8390I 9.83914 O.02 250 O.O2 224 O.O2 199 O.02 174 O.O2 149 O.02 123 O.O2 098 O.02 073 0.02 047 9.86 044 9.86032 9-86 020 9-86 008 9.85 996 9.85 984 9-85 97 2 9.85 9 60 9.85 948 9-97 978 O.02 022 9.85 936 13 13 12 26 25 25 1.0 I.O I.O 2 3-0 2.9 3.1 5.0 4.8 5-2 7.0 6.7 7.3 * 9-0 8.7 9.4 ? i i.o 10.6 n,j 13.0 12.5 13.5 7 15.0 14.4 15.6 17.0 16.3 17.7 jj 19.0 18.3 19.8 2I.O 20.2 21.9 23.0 22.1 24.0 1* 25.0 24.0 9.83927 9.83 940 9-83 954 9.83 967 9.83 980 9-83 993 9.84 006 9.84 020 9-84033 9.98 003 9.98029 9.98054 9.98079 9.98 104 9.98 130 9-98 155 9.98 1 80 9.98 206 o.o i 997 o.oi 971 o.o i 946 o.oi 921 o.oi 896 o.oi 870 o.oi 843 o.oi 820 o.oi 794 9.85 924 9.85912 9.85 900 9.85 888 9.85 876 9.85 864 9.85851 9.85 839 9.85 827 9.84 046 9.84059 9.84072 9.84085 9.84098. 9.84 112 9.84 125 9.84 I 3 8 9.84 151 9.84 164 "9-84I77 9-98231 o.oi 769 9-85815 9.98 256 9.98 281 9.98 307 9-98 332 9-9-S 3S7 9.98 383 9.98 408 9-98 433 9-98 458 9.98 484 o.oi 744 o.oi 719 o.oi 693 o.oi 668 o.oi 643 o.oi 617 o.oi 592 o.oi 567 o.oi ; }_' o.oi S"' 9.85 803 9-85 79i 9-85 779 9.85 766 9-85 754 9.85 742 9-85 730 9.85718 9.85 706 ~9-85 693 L. Cos. d. L. Cot. c. d.| L. Tan. L. Sin. d. ' p.p. 46 464 44' L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. P.P. O 9.84 177 9.98 484 o.oi 516 9.85 693 60 I 2 3 4 5 6 7 8 9 9.84 190 9.84 203 9.84 216 9.84 229 9.84 242 9-^4 255 9.84 269 9.84 282 9.84 295 13 13 13 13 13 H 13 i3 9.98 509 9-98 534 9.98 560 9.98 585 9.98 610 9.98 635 9.98 661 9.98 686 9.98711 3 25 2 5 2 5 26 25 2 5 o.o i 491 o.oi 466 o.oi 440 o.oi 415 o.oi 390 o.oi 365 o.oi 339 o.oi 314 o.oi 289 9.85 68 1 9.85 669 9-85 657 9-85 645 9.85 632 9.85 620 9.85 608 9.85 596 9.85 583 12 12 12 13 12 12 12 59 58 57 56 55 54 53 5 2 5i I 2 3 4 i I ! 7 i 26 25 14 2.6 2.5 1.4 5.2 5.0 2.8 7-8 7-5 4-2 0.4 10.0 5.6 3.0 12.5 7.0 5.6 15.0 8.4 8.2 17.5 9.8 IO 9.84 308 9-98 737 o.oi 263 9.85 57i 5O ii 12 13 H !i 3 19 9.84321 9-84 334 9-84 347 9.84 360 9-84 373 9.84 385 9.84 398 9.84411 9.84 424 13 i3 13 13 12 13 13 U 9.98 762 9.98 787 9.98 812 9.98 838 9.98 863 9.98 888 9.98913 9-98 939 9.98 964 25 25 26 25 2 5 2 5 26 25 o.oi 238 o.oi 213 o.oi 1 88 o.oi 162 o.oi 137 O.OI 112 o.oi 087 o.oi 06 1 o.oi 036 9-85 559 9-85 547 9-85 534 9.85 522 9.85510 9-85 497 9-85 485 9-85 473 9.85 460 12 13 12 12 13 12 12 13 49 48 47 46 45 44 43 42 4i I 2 3 4 i 13 12 1.3 1.2 2.6 2.4 3-9 3-6 5.2 4.8 6.5 6.0 7.8 7.2 20 9-84 437 9.98 989 26 O.OI OI I 9.85 448 4O 7 9.1 8.4 21 22 23 24 9.84 450 9.84 463 9.84 476 9-84489 13 13 13 n 9.99015 9.99 040 9-99 065 9.99 090 25 2 5 25 26 o.oo 985 o.oo 960 o.oo 933 0.00910 9.85 436 9.85 423 9.85411 9-85 399 13 12 12 13 39 38 37 36 8 9 10.4 9.6 11.7 10.8 2 26 3 29 9.84 502 9.84515 9-84528 9-84 54 9.84553 13 J 3 12 13 9-99 Hi 9.99 1 66 9.99 191 9-99217 25 2 5 2 5 26 o.oo 859 o.oo 834 o.oo 809 o.oo 783 9.55 350 9-85 374 9.85 361 9-85 349 9.85 337 12 13 12 12 35 34 33 32 3i 13 13 26 25 30 9.84 566 9.99 242 o.oo 758 9-85 324 30 o I O I.O 3i 32 33 34 35 36 ii 39 9-84579 9.84 592 9.84 605 9.84618 9.84 630 9.84 643 9-84 656 9.84 669 9.84 682 3 13 13 12 13 13 13 13 9-99 267 9.99 293 9-99 3i8 9-99 343 9-99 368 9-99 394 9.99419 9-99 444 9.99 469 26 25 2 5 2 5 26 25 25 3 o.oo 733 o.oo 707 o.oo 682 0.00657 o.oo 632 o.oo 606 o.oo 581 o.oo 556 o.oo 531 9.85312 9.85 299 9.85 287 9-85 274 9.85 262 9.85 250 9.85 237 9.85 225 9.85 212 13 12 13 12 12 13 12 13 29 28 27 26 2 5 2 4 23 22 21 2 3 4 7 8 9 10 3.0 2.9 5.0 4.8 7.0 6.7 9.0 8.7 i i.o 10.6 13.0 12.5 15.0 14.4 17.0 16.3 19.0 18.3 40 9.84 694 9-99 495 o.oo 505 9.85 2OO 2O ii 4i 42 43 44 45 46 47 48 49 9.84 707 9.84 720 9-84 733 9.84 745 9.84 758 9.84771 9.84 784 9.84 796 9.84809 13 13 12 13 13 13 12 13 I 7 9-99 520 9-99 545 9-99 57 9.99 596 9.99 621 9.99 646 9-99 672 9.99 697 9-99 722 25 25 26 25 25 26 25 25 o.oo 480 o.oo 453 o.oo 430 o.oo 404 o.oo 379 o.oo 354 o.oo 328 o.oo 303 o.oo 278 9.85 I8 7 9.85175 9.85 l62 9.85 I^O 9.85 137 9.85 123 9.85 112 9.85 too 9.85 087 12 13 12 13 12 13 12 13 19 18 J 7 16 15 H 13 12 n 12 3 O I 2 25.0 24.0 12 12 26 25 I.I I.O 3-2 3-i 5.4 5.2 5O 9.84 822 1 -1 9-99 747 "A o.oo 253 9.85 074 10 3 7-6 7-3 5i 52 53 54 55 5 6 P 59 9.84 835 9.84 847 9.84 860 9.84 873 9.84 885 9.84 898 9.84911 9.84 923 9.84 936 12 3 13 12 13 13 12 13 I? 9-99 773 9-99 798 9-99 823 9-99 848 9-99 874 9.99 899 9-99 924 9.99 949 9-99 975 25 25 25 26 25 25 3 o.oo 227 O.OO 2O2 o.oo 177 o.oo 152 o.oo 126 O.OO 101 0.00076 o.oo 05 1 o.oo 025 9.85 062 9.85 049 9.85 037 9.85 024 9.85012 9-84 999 9.84 986 9.84 974 9.84961 13 12 13 12 13 13 12 13 9 8 7 6 5 4 3 2 I 5 6 I 9 10 ii 12 9.8 9.4 11.9 11.5 14.1 13.5 16.2 15.6 18.4 17-7 20.6 19.8 22.8 21-9 24.9 24.0 60 9.84 949 o.oo ooo 0.00 OOO 9.84 949 O L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. d. ' P.P. 45' TABLES XVII., XVIII. NATURAL TRIGONOMETRIC FUNCTIONS. K'M'H st'Rv. 30 465 466 TABLE XVII. NATURAL SIXKS AND COSINKS. Natural Sines. Prop. Angle. 0' 10' 20' 30' 40' 50' 60' Angle. Parts forV. .000000 .002909 .0058 1 8 .0087 27 .011635 .0145 44 .017452 89 2-9 i .017452 .0203 6 .0232 7 .0261 8 .0290 8 .03199 .0349 o 88 2.9 2 .0349 o .0378 i .0407 i .0436 2 .0465 3 .0494 3 05234 87 2.9 3 0523 4 05524 .0581 4 .0610 5 .0639 5 .0668 5 .0697 6 86 2.9 4 .0697 6 .0726 6 0755 6 .0784 6 .0813 6 .0842 6 .0871 6 85 2.9 5 .0871 6 .09005 .09295 09585 .0987 4 .10164 1045 3 84 2.9 6 10453 .10742 .1103 i .11320 .11609 .11898 .12187 83 2.9 7 .12187 .12476 .12764 1305 3 1334 1363 .1392 82 2.9 8 .1392 .1421 .1449 .1478 1507 1536 .1564 81 2.9 9 .1564 1593 .1622 .1650 .1679 .1708 1736 80 2-9 IO .1736 .1765 1794 .1822 .1851 .1880 .1908 79 2-9 ii .1908 1937 .1965 .1994 .2022 .2051 .2079 78 2.9 12 .2079 .2108 .2136 .2164 2193 .2221 .2250 77 2.8 13 .2250 .2278 .2306 2334 2363 .2391 .2419 76 2.8 H .2419 .2447 .2476 .2504 2532 .2560 .2588 75 2.8 15 .2588 .2616 .2644 .2672 .2700 .2728 .2756 74 2.8 16 .2756 .2784 .2812 .2840 .2868 .2896 .2924 73 2.8 X 7 .2924 .2952 2979 .3007 .3035 .3062 .3090 7 2 2.8 18 .3090 .3118 3H5 3 J 73 3201 .3228 3256 7i 2.8 19 .3256 3283 33" .3338 3365 3393 .3420 70 2.7 20 3420 3448 3475 .3502 3529 3557 3584 69 2.7 21 .3584 .3611 3638 3665 .3692 3719 3746 68 2-7 22 3746 3773 .3800 .3827 3854 3881 3907 67 2.7 23 3907 3934 .3961 3987 .4014 .4041 .4067 66 2-7 24 .4067 .4094 .4120 .4147 4173 .4200 .4226 65 2-7 25 .4226 4253 4279 4305 4331 4358 4384 64 2.6 26 .4384 .4410 4436 .4462 .4488 45 H 454 63 2.6 2 7 4540 .4566 4592 .4617 4643 .4669 4695 62 2.6 28 4695 .4720 .4746 .4772 4797 .4823 .4848 61 2.6 29 .4848 .4874 .4899 .4924 495 4975 .5000 60 2-5 3O .5000 5025 55 5075 .5100 5125 5150 59 2-5 3i S^o S 1 7S .5200 5225 5 2 50 5275 5299 58 2-5 32 5299 5324 5348 5373 5398 .5422 5446 57 2-5 33 .5446 5471 5495 55i9 5544 .5568 5592 56 2.4 34 5592 .5616 .5640 .5664 .5688 5712 5736 55 2.4 35 5736 .5760 .5783 .5807 5831 5854 .5878 54 2.4 36 .5878 .5901 5925 .5948 5972 5995 .6018 53 2-3 37 .6018 .6041 .6065 .6088 .6m .6134 6i57 52 2-3 38 6i57 .6180 .6202 .6225 .6248 .6271 .6293 5 1 2-3 39 .6293 .6316 6338 .6361 6383 .6406 .6428 50 2.3 4O .6428 .6450 .6472 .6494 .6517 6539 .6561 49 2.2 4i .6561 .6583 .6604 .6626 .6648 .6670 .6691 48 2.2 42 .6691 6713 6734 .6756 .6777 .6799 .6820 47 2.2 43 .6820 .6841 .6862 .6884 .6905 .6926 .6947 46 2.1 44 .6947 .6967 .6988 .7009 .7030 .7050 .7071 45 2.1 60' 50' 40' 30' 20' 10' 0' Angle. Natural Cosines. KAI, SINKS AM> COSINKS (conti Natural Sines. 467 Prop. Angle 0' 10' 20' 30' 40' 50' 60' Angle. Parts forV. 45 .7071 .7092 .7112 7133 .7153 7173 7'93 44 2.O 46 7'93 .7214 7234 7254 .7274 7294 73H 43 2.O 47 73M 7333 7353 7373 7392 .7412 7431 42 2.O 48 7431 7451 .7470 .7490 .7509 7528 7547 4i 1-9 49 7547 .7566 7585 .7604 .7623 .7642 .7660 40 1-9 5O .7660 .7679 .7698 .7716 7735 7753 .7771 39 9 5 1 .7771 .7790 .7808 .7826 .7844 .7862 .7880 38 .8 52 .7880 .7898 .7916 7934 795i .7969 .7986 37 .8 53 .7986 .8004 .8021 .8039 .8056 8073 .8090 36 7 54 .8090 .8107 .8124 .8141 .8158 8i75 .8192 35 7 55 .8192 .8208 .8225 .8241 .8258 .8274 .8290 34 1.6 56 .8290 8307 8323 8339 8355 8371 8387 33 1.6 57 .8387 .8403 .8418 8434 .8450 .8465 .8480 32 1.6 58 .8480 .8496 .8511 .8526 .8542 8557 8572 3i i-5 59 8572 .8587 .8601 .8616 8631 .8646 .8660 30 1-5 60 .8660 8675 .8689 .8704 .8718 8732 .8746 29 1.4 61 .8746 .8760 .8774 .8788 .8802 .8816 .8829 28 1.4 62 .8829 8843 8857 .8870 .8884 .8897 .8910 27 1.4 63 .8910 .8923 .8936 8949 .8962 8975 .8988 26 1-3 64 .8988 .9001 .9013 .9026 .9038 .9051 .9063 25 i-3 65 .9063 9075 .9088 .9100 .9112 .9124 9135 24 1.2 66 9 '35 .9147 9i59 .9171 .9182 .9194 .9205 23 1.2 67 .9205 .9216 .9228 9239 .9250 .9261 .9272 22 I.I 68 .9272 .9283 9293 9304 9315 9325 9336 21 I.I 69 9336 9346 9356 9367 9377 .9387 9397 20 I.O 7O 9397 .9407 .9417 .9426 9436 .9446 9455 19 I.O 7' 9455 9465 9474 9483 9492 .9502 95" 18 0.9 72 95" .9520 9528 9537 9546 9555 95 6 3 17 0.9 73 9563 9572 .9580 .9588 9596 .9605 .9613 16 0.8 74 .9613 .9621 .9628 9636 .9644 .9652 .9659 1 5 0.8 75 9659 .9667 .9674 .9681 .9689 .9696 973 14 0.7 76 .9703 .9710 .9717 .9724 973 9737 9744 13 0.7 77 9744 9750 9757 9763 .9769 9775 .9781 12 0.6 78 .9781 .9787 9793 9799 .9805 .9811 .9816 II 0.6 79 .9816 .9822 .9827 9833 .9838 9843 .9848 IO o-5 80 .9848 9853 .9858 9863 .9868 .9872 .9877 9 o-5 81 9877 .9881 .9886 .9890 .9894 .9899 9903 8 0.4 82 9903 .9907 .9911 .9914 .9918 .9922 9925 7 0.4 83 9925 .9929 9932 9936 9939 9942 9945 6 o-3 84 9945 .9948 995 * 9954 9957 9959 .9962 ' 5 0-3 85 .9962 .9964 .9967 .9969 997 1 9974 .9976 4 O.2 86 .9976 .9978 .9980 .9981 .9983 .9985 .9986 3 0.2 87 .9986 .9988 9989 .9990 .9992 9993 9994 2 O.I 88 9994 9995 .9996 9997 9997 9998 9998 I O.J 89 .9998 9999 9999 I.OOOO I.OOOO I.OOOO I.OOOO O o.o 60' 50' 40' 30' 20' 10' 0' Angle. Natural Cosines. 468 TABLE XVIII. XATI-KAI. TAN<;I:MS AXD COTAXCJKNTS. Natural Tangents. Prop. Angle 0' 10' 20' 30' 40' 50' 60' Angle Parts for 1'. .00000 .0029 i .0058 2 .00873 .01164- 0145 5 .01746 89 2-9 i .01746 .0203 6 .0232 8 .0261 9 .0291 o .0320 i .0349 2 88 2.9 2 .0349 2 0378 3 .0407 5 .0436 6 .0465 8 .0494 9 .0524 I 87 2.9 3 .0524 I 0553 3 .0582 4 .061 1 6 .0640 8 .0670 o .0699 3 86 2-9 4 .0699 3 .0728 5 0757 8 .0787 o .08163 .0845 6 .0874 9 85 2.9 5 .0874 9 .0904 2 0933 5 .0962 9 .09923 .1021 6 .1051 o 84 2.9 6 .1051 o .10805 .11099 "394 .11688 .11983 .12278 83 2.9 7 .12278 12574 .12869 13165 .1346 1376 .1405 82 3-o 8 .1405 H35 .1465 H95 1524 1554 .1584 81 3-o 9 .1584 .1614 .1644 1673 1703 1733 '7 6 3 80 3-o 10 !7 6 3 1793 .1823 1853 .1883 .1914 .1944 79 3-o ii .1944 .1974 .2004 .2035 2065 .2095 .2126 78 3-o 12 .2126 .2156 .2186 .2217 2247 .2278 .2309 77 3-i '3 .2309 2339 .2370 .2401 .2432 .2462 2493 76 3-i H 2493 2524 2555 .2586 .2617 .2648 .2679 75 3-i 15 .2679 .2711 .2742 2773 .2805 .2836 .2867 74 3-i 1 6 .2867 .2899 .2931 -.2962 .2994 .3026 3057 73 3-2 i? 3057 .3089 .3121 3153 3185 3217 3249 72 3-2 18 3249 .3281 3314 3346 3378 34" 3443 7* 3-2 19 3443 347 6 .3508 3541 3574 3607 .3640 70 3-3 20 .3640 3673 .3706 3739 3772 .3805 3839 69 3-3 21 3839 .3872 .3906 3939 3973 .4006 .4040 68 3-4 22 .4040 .4074 .4108 .4142 .4176 .4210 4245 67 3-4 23 4245 4279 43H 4348 4383 .4417 4452 66 3-5 24 .4452 .4487 .4522 4557 4592 .4628 4663 65 3-5 25 .4663 .4699 4734 477 .4806 .4841 .4877 64 3-6 26 4877 .4913 495 .4986 .5022 5059 595 63 3-6 27 595 5132 .5169 5206 5243 .5280 5317 62 3-7 28 5317 5354 5392 543 5467 555 5543 61 3-8 29 5543 558i .5619 .5658 .5696 5735 5774 60 3-8 30 5774 .5812 5851 .5890 5930 5969 .6009 59 3-9 3i .6009 .6048 .6088 .6128 .6168 .6208 .6249 58 4.0 32 .6249 .6289 633 6371 .6412 6453 6494 57 4.1 33 .6494 .6536 6577 .6619 .6661 .6703 6745 56 4.2 34 6745 .6787 .6830 6.873 .6916 6959 .7002 55 4-3 33 .7002 .7046 .7089 7*33 .7177 .7221 .7265 54 4-4 36 .7265 7310 7355 .7400 7445 .7490 7536 53 4-5 37 7536 758i .7627 7673 .7720 .7766 7813 52 4.6 38 78i3 .7860 .7907 7954 .8002 .8050 .8098 5i 4-7 39 .8098 .8146 8i95 8243 .8292 .8342 8391 5 4-9 4O 8391 .8441 .8491 .8541 8591 .8642 8693 49 5- 4i .8693 .8744 .8796 .8847 .8899 8952 .9004 48 5-2 42 .9004 .9057 .9110 9163 .9217 .9271 9325 47 5-4 43 9325 .9380 9435 .9490 9545 .9601 9657 46 5'5 44 .9657 9713 .9770 9827 .9884 9942 I.OOOO 45 5-7 60' 50' 40' 30' 20' 10' 0' Angle. Natural Cotangents. NATCKAI. TANUKMS AND COTAXGKNTS (continued). Natural Tangents. 409 Prop. Angle. 0' 10' 20' 30' 40' 50' 60' Angle. Parts for 1'. 45 I.OOOO 1.0058 1.0117 1.0176 1.0235 1.0295 '0355 44 5-9 46 r -355 1.0416 1.0477 1.0538 1-0599 1. 0661 1.0724 43 6.1 47 1.0724 1.0786 1.0850 1.0913 1.0977 .1041 1.1106 42 6.4 48 1.1106 1.1171 1.1237 1.1303 1.1369 .1436 1.1504 41 6.6 49 1.1504 I-I57I 1.1640 1.1708 1.1778 .1847 1.1918 40 6-9 5O 1.1918 1.1988 1.2059 1.2131 1.2203 .2276 1-2349 39 7.2 5* 1.2349 1.2423 1.2497 1.2572 1.2647 2723 1.2799 38 7-5 52 1.2799 1.2876 1.2954 1.3032 1.3111 1.3190 1.3270 37 7-9 53 1.3270 I-335 1 1-3432 L35H '3597 1.3680 1.3764 36 8.2 54 I-3764 1-3848 1-3934 1.4019 1.4106 I-4I93 1.4281 35 8.6 55 1.4281 1-437 1.4460 1-4550 1.4641 1-4733 1.4826 34 9-i 56 1.4826 1.4919 L50I3 1.5108 1.5204 1.5301 1-5399 33 9-6 57 1-5399 1-5497 1-5597 1.5697 I.5798 1.5900 1.6003 32 10. 1 58 1.6003 1.6107 1.6212 1.6319 1.6426 1-6534 1-6643 3i 10.7 59 1.6643 I-6753 1.6864 1.6977 1.7090 1.7205 1.7321 30 "3 6O 1.7321 L7437 I.7556 I -7 6 75 1.7796 1.7917 1.8040 29 I2.O 61 i .8040 1.8165 1.8291 1.8418 1.8546 1.8676 1.8807 28 12.8 62 1.8807 1.8940 1.9074 1.9210 1-9347 1.9486 1.9626 2 7 I 3 .6 63 1.9626 1.9768 1.9912 2.0057 2.0204 2-0353 2.0503 26 14.6 64 2.0503 2.0655 2.0809 2.0965 2.1123 2.1283 2.1445 2 5 15-7 65 2.1445 2.1609 2-1775 2.1943 2.2113 2.2286 2.2460 24 16.9 66 2.2460 2.2637 2.2817 2.2998 2.3183 2-3369 2-3559 23 18.3 67 2-3559 2.3750 2-3945 2.4142 2-4342 2-4545 2-475 i 22 19.9 68 2.4751 2.4960 2.5172 2.5386 2-5605 2.5826 2.6051 21 21.7 69 2.6051 2.6279 2.6511 2.6746 2.6985 2.7228 2 -7475 2O 23-7 70 2-7475 2-7725 2.7980 2.8239 2.8502 2.8770 2.9042 19 7i 2.9042 2.9319 2.9600 2.9887 3-0178 3-0475 3.0777 18 72 3-0777 3.1084 3-1397 3.1716 3.2041 3-2371 3.2709 J 7 73 3.2709 3-3052 3-3402 3-3759 3-4124 3-4495 34874 16 74 3-4874 3.5261 3-5656 3-6059 3.6470 3-6891 3-7321 '5 75 3-7321 3.7760 3.8208 3-8667 3-9I36 3-9617 4.0108 14 76 4.0108 4.0611 4.1126 4-1653 4-2193 4.2747 4-3315 3 77 4.3315 4-3897 4-4494 4-5107 4.5736 4.6382 4.7046 12 78 4.7046 4-7729 4.8430 4.9152 4.9894 5.0658 5.1446 II 79 5.1446 5-2257 5-3093 5-3955 54845 5-5764 5-6713 IO 8O 5-67I3 5-7694 5.8708 5-9758 6.0844 6.1970 6.3138 9 Si 6.3138 6.4348 6.5606 6.6912 6.8269 6.9682 7.1154 8 82 7-"54 7.2687 74287 7-5958 77704 7-9530 8.1443 7 83 8.1443 8.3450 8-5555 8.7769 9.0098 9-2553 9-5*44 6 84 9-5'44 9.7882 10.0780 10.3854 10.7119 11.0594 11.4301 5 85 11.4301 11.8262 12.2505 12.7062 13.1969 13.7267 14.3007 4 86 14.3007 14.9244 15.6048 16.3499 17.1693 18.0750 19.0811 3 87 19.0811 20.2056 21.4704 22.9038 24.5418 26.4316 28.6363 2 88 28.6363 31.2416 34-3678 38.1885 42.9641 49.1039 57.2900 I 89 57.2900 68.7501 85.9398 114.5887 171.8854 343-7737 00 O 60' 50' 40' 30' 20' 10' 0' Angle. Natural Cotangents. PLATE I PLATE I Itutial Meridian - Madrid PLATE III. , - V VERTICAL SECTION EAST AND WEST. PLATE IV. VERTICAL SECTION NORTH AND SOUTH. Plan and Vertical Sections of the UNDERGROUND WORKS CORA BLANCA MINE NEW ALMADEN California Scale in feet 100 ISO 100 April, 1896 Tlu Pifura in braetttt, tkut (610), inditat, t* wrticat dinne 4./o. th, datum poit, a m,num.,,l o* 1*. ,.mmit of Mi,, Hill flu tkaft* . Inclined line, defined, 10. Inclined stadia readings, 132. Index glass of sextant, 295. Indian meridian, 360. Instruments for surveying : Chains, 13. Tapes, 15, 17. Measuring rods, 17. Pins, 18. Range poles, 18. Vernier, 31. INDEX. 477 Instruments for surveying : Level bubble, 35. Level, 40. Telescope, 43. Leveling rods, 47. Locke hand level, 72, 73. Abney level and clinometer, 72, 73. Gurley's monocular and binocular hand levels, 73. Barometer, 74, 76. Compass, 77. Special forms of compass, 94. Transit, 95. Tachymeter, 99, 311. Solar transit, 116. Saegmuller solar attachment, 123. Stadia, 127, 131. Slide rule, 136, 179. Planimeter, 172. Mannheim rule, 181, 198. Thacher rule, 196. Fuller's slide rule, 197. Transit for city surveying, 231. Protractors, 263. Colby's protractor, 265. Ockerson's protractor, 266. Plane table, 268. Lead, 289. Sextant, 294. Current meter, 208. . Rod floats, 303. German dial, 313. Hanging clinometer, 313. Lamps, 314. Intersections, Method of, 272. Introduction, 9-12. Preliminary conceptions, 9. Surveying defined, 11. Irregular areas, 155, 161. Islands, in streams, Ownership of, 348. in Great Lakes, 349. Isogonic line, 87. Jacob staff, 94. Johnson, J. B., References to, 25, 263. Johnson, W. D., Leveling head in- vented by, 269. Judicial functions of surveyors, 341- 350. Imperfect training, 341. Duties, regarding location of mon- uments, 342, 344. Relocation of extinct corntrs, :}(:}. Judicial functions of surveyors : Corners not to be established, 343. Must regard occupation and claims, 344. Must ascertain facts, 346. Surplus and deficiency in appor- tionment, 346. Difficulties with meander lines, 34(5. Extension of water fronts, 348. Bed ownership on water fronts, 348. Riparian rights on lakes, 349. Key to topographical symbols on niups, 267. Lakes, Riparian rights in, 217, 349. Meandering of, 228. Dredging of, 275. Land, Map of, 11. Division of, 163. Title to, 343. Land survey computations, 141-198. Considerations and definitions, 141. Error of closure, 143. Balancing the survey, 144-148. Supplying omissions, 149-152. Areas, 152-156. Coordinates, 156-162. Dividing land, 163-165. Model examples, 165-172. The planimeter, 172-179. The slide rule, 179-198. Land surveys, 201-237. Obstacles and problems, 201-203. Two common problems, 204. Surveying with chain alone, 204- 208. Farm surveys, 208-219. United States public land surveys, 219-229. City surveying, 230-237. Lateral adjustment, 69. Latitude, Parallels of, 10. Observation for, 121. of a line and point, defined, 142. differences, 142, 147-152, 155, 162, 168, 204. Errors in, 146, 148. and longitude plotting, 209, 210, 261, 301, 306. Table of lengths of, 372. Formula for, 364. Laying out curves, 240. 478 INDEX. Lead used for soundings, 288, 289. Least squares, formula from Treatise on, 30. Reference to demonstration of rule for balancing survey by, 147. Method of, mentioned, 303. Legal requirements for public land surveys, 222. Length, Standard of, 231. Lesser Bear, Constellation of, 90. Lettering on maps, 210, 267. Level and horizontal lines, Measure- ment of, 13-30. Level line, defined, 10. Line to be measured, 13. Instruments used, 13-18. Methods, 18-22. Classes of errors, 22. Causes of errors, 23. Temperature, 24. Sag and pull, 25, 26. Alignment, 27. Slope, 27. Precision to be obtained, 29. Level bubble, 35-38. Level, term in mining, defined, 308. Level tube for chaining, 27. Level under telescope, 114. Level, Use and adjustment of, 50-71. Adjustment and setting up, 50. Differential leveling, 51. Profile leveling, 53. Making the profile, 58. Leveling over an area, 60. Errors, 60. Curvature and refraction, 62. Reciprocal leveling, 63. Focusing, 63. Adjustments named, 64. Collimation adjustment, 64. Adjustment of objective slide, 65. Bubble adjustment, 67. Peg method, 68. Lateral adjustment, 69. Y adjustment, 70. Adjustment of dumpy level, 70. Adjustment of any level on a metal base, 71. Hand, 72, 73. Level vial, 36, 37. Leveling or measuring differences of altitude, 40-76, 308. General principle, 40. Leveling or measuring differences of altitude, Instruments for, 40-50. Use of level, 50-63. Adjustments of level, 63-71. Minor instruments, 72, 73. Leveling with barometers, 74-76. in underground surveys, 308. Leveling rods, 47-50. Limb of transit, 96. Line of collimation, defined, 46. in transit, 108. Revolution of, 109. Linear transformations, Table of, 377. Linen tapes, 17. Local attraction of magnetic needle, 93. Local mean times of culmination and elongation of Polaris, 365. Locating arcs by chain, 242. Location surveys, 208, 219. Locke hand level, 72, 73. Lode claim, 306. Lodes, Coloring of, on maps, 320. Logarithms, Computations by, in land surveys, 165. Scale of, defined, 180. Use of scale, 181. Infinite extent of scale, 184. Scale of logarithmic sines and tan- gents, 193, 194. Table of numbers, 397-418. Table of trigonometric functions, 419-464. Longitude, Meridians of, 10. of a line and point, defined, 142. Differences of, 142, 147-1. r >0, 151, 153, 162, 204. Errors in, 148. Double, 152, 153, 155, 18. Table of lengths of, 372. Lot, Description of, 233. Louisiana meridian, 359. Lunar variation of declination, 86. Magnetic bearings, 212. Magnetic declination, 86-94, 368-:57<). defined, 86. Variations of, 86. Determination of declination, 87. Determination of true meridian, 89. Azimuth at elongation, 91. Local attraction, 93. Special forms of compasses, 94. INDEX. 479 Magnetic declination, Table of, 368- 370. Magnetic meridian, 78, 86, 103, 107, 211. Magnetic needle, 77, 78, 364. Mannheim rule, 181, 198. "Manual of Surveying Instructions," 221, 228, 365. Map lettering, 210, 267. Mapping, 261-268, 320, 321. Triangles, 261. Outline of method for topography, 261. Stadia line, 262. Side shots, 263. Colby's protractor, 265. Ockerson's protractor, 266. Finishing the map, 267. Requirements for maps, 267. Scale, 268. for metal and coal mines, 320. Scale in mines, 321. Problems, 321. Maps, defined, 11. of farm surveys, 209. of city surveys, 237. Topographical, 244. Requirements for, 267, 354-356. Contour for grading, 281, 282. of harbor surveys, 294. of mines, 320. for public records, 353. Maximum velocity of streams, 298. Mean surface of earth, 9. Meander corners, 228. Meander lines, 346, 347, 349. Meandering a stream, 227. Measuring differences of altitude. See "Leveling." Measuring velocity discharge, 298-304. Position of maximum velocity, 298. Current meters, 298. Use of meter, 300. Rating the meter, 300. Rod floats, 303. Discharge, 304. Mercurial barometer, Leveling with, 74, 76. Meridians, of longitude, 10. Magnetic and true, 86, 89, 225. Determination of, 105. Principal, in U. S. land surveys, 219, 223, 357-360. Angular convergence of, 224, 371. Meridians, Guide, 224. of reference, 261. Table of distance between, 371. Meridian and time, by transit and sun, 125, 126. Metal base, Adjustment of level on, 71. Metal mines, 306, 320. Metallic tapes, 17. "Metes and bounds," Description by, 216, 235. Methods for earthwork computations, 275-281. Occurrence of problem, 275. Prisms, 275. Prismoids, 276. Prismoidal formula, 277. Approximations, 277. Area grading, 279. Street grading, 280. Excavation under water, 281. Methods of measuring level and hori- zontal lines, 18-22. Preliminary statement, 18. Chaining, 18. Hints, 21. Chaining on slopes, 21. Meters changed to feet, 379. Michigan meridian, 358. Michigan, Survey of lands in, 341. Micrometer screw, 36. Mine surveying, 305-321. Surface surveys, 305-308. Underground surveys, 308-316. Connecting surface and under- ground work, 316-320. Mapping the survey, 320, 321. Mining claim, Form of, 306. Minus sights, 51, 56, 57, 60. Model examples. See "Problems." Modulus of elasticity of chains, 25. Monocular hand level, 73. Montana meridian, 360. Monuments for surveys, 209, 211, 216, 218, 234, 237, 253, 307, 341, 344, 345, 351, 355. Mount Diablo meridian, 369. Multiplication by slide rule, 182. Myers's, John H., problems in coordi- nates, 328. Natural scale of maps, 268. Natural sines and cosines, Table of, 466, 407. 480 INDEX. Natural tangents and cotangents, Table of, 468, 469. "Nautical Almanac," 120, 126. Needle and pivot of compass, 80. Needle checks on azimuths, 107. New Almaden mine, 310. New Mexico meridian, 359. New York leveling rod, 47, 48. North Star, Observation on, 90. Notes, in profile leveling, 55. in traversing, 84. in stadia measurements, 140. for land surveys, 169, 208, 228, 229. for topography, 250, 252. in triangulation, 256. for mine surveys, 307, 320. for underground work, 315. Private, 351, 364. Objective of telescope, 43, 45, 46. Objective slide, Adjustment of, 65. Oblate spheroid of revolution, 9. < )bstacles in land surveys, 201. Ockerson's protractor, 266. Offsets, 155, 156, 161. Omissions, supplied in land survey computations, 149-152, 171. Necessity for, 149. General discussion, 149. Cases I. -VI., 150. Algebraic solution, 151. Ordinates, 156, 160, 161. Ore, Mining of, 308. Orientation, of transit, 104, 261, 273. of plane table, 273. Origin of coordinates, 156. Original land surveys, 208, 209-211. Ormsbee, J. J., on three-tripod system, 315. Oughtred, inventor of sliding scales, 180. Ownership of surveys, 351-356. Parallelepiped, Truncated, 276. Parallels, of latitude, 10. Standard, 223. Park drive curves, 238. Partridge's improvement of slide rule, 180. Patenting mining claims, 307. Peg, a turning point, 57. Peg method of adjusting telescopes, 68, 71, 274. Philadelphia leveling rod, 47, 48, 49. Pins used in chaining, 18. Pitch, defined, 309. Pivot of compass, 80. Placer claim, 306. Plane survey, denned, 12. Plane table, 268-274, 294. Description, 268. Use, 270. Three-point and two- point prob- lems, 272. Adjustments, 274. used in harbor surveys, 294. Planimeter, 172-179, 270, 284. Description, 172. Use, 173. Theory, 174. Zero circumference, 178. Circumference of wheel, 178. Length of arm, 179. used in measuring areas, 270, 284. Plate bubbles, of compass, 79. of transit, 108. Plotting, of maps, 209. by latitudes and longitudes, 209, 210, 261, 301, 306. of corners, 210. of points on maps, 292. Plumb line, a vertical line, 10. Use of, 22, 90. Plumbing, 253. Plummet lamps, 314. Plummets for mine surveys, 318, 319. Plus sights, 61, 60. Polar axis of solar transit, 123. Polar coordinates, 249. Polar distance of Polaris, 364. Polar planimeter, 172, 173. Polaris, Observations on, 90. Elongation and culmination of, 90, 106, 107, 120, 223, 365. Declination of, 92. Polar distance of, 92, 364. Porro, inventor of stadia rod, 127. Positive coordinates, 160. Precise level, 43, 47, 61. Precision necessary, in measurement, 29. in city surveying, 230. Price, W. G., Observations by, 301, 339. Principal meridians, 219, 223, 357-360. Prisms, 60, 275-277. Prismatic compass, 94. INDEX. 481 Prismoids, 276. Prismoidal formula, 276, 277, 279, 283, 284. Private notes of survey, 351, 354. Problems and examples, 157-159, 165- 172, 186, 201-204, 321, 322-340. in coordinates, 157-159, 328. in land surveys, 165-172, 201-204. with slide rule, 186. in mine surveying, 321. on Chapter 1.^ 322. on Chapters III. and IV., 324. on Chapters V. and VI., 326. on Chapter VIII., 335. on Chapter IX., 336. on Chapter X., 338. on Chapter XI., 339. on Chapter XII., 340. Profile leveling, 53-60. Profile paper, 58, 59. Profiles, Cross, 280. Proportion by slide rule, 183. Protractor, Land survey drawing made by, 141. Vernier, 263. Colby's, 265. Ockerson's, 266. Three-arm, 292, 293. Plotting with, 306. Public surveys, in Michigan, 341. Base lines governing, 357-360. See also "U. S. public land sur- veys." Pull, Formula for, 25. Elimination of, 26. element noted, 254. Pull scale, 231. Pyramid, 276, 277. Quarter sections, 220. Rack and pinion movement of tele- scope, 44. Radiation, Method of, 270. Railroads, Surveys of, 238, 268. Curves on, 238. Random lines, 203, 225, 227. Random surveys, 214, 215. Range lines, 220, 225, 226, 227, 291. Range poles, 18. Raymond's (R. W.) " Glossary of Min- ing and Metallurgic Terms," 308. Reading glasses, 98. H'M'U SURV. ;J1 Reciprocal leveling, 63. Reed's "Topographical Drawing and Sketching," 267. Reels, 15-17. Reference lines, 223. Reflector in telescope, 106. Refraction, Effect of, 63, 139. Corrections for, 121, 125, 126. Table of corrections for, 364, 366, 367. Repetition measurement of angles, 255, 256. Report of survey, 214. Report of U. S. Coast and Geodetic Survey, 75, 86, 87, 368. Reservoir, Building of, 275. Capacity of, 281. in embankment and excavation, 285, 286. Resurveys, 208, 211-218. Riparian rights, 217, 347, 349. Ritchie and Haskell's direction meter, 304. River, Dredging of, 275. River, Survey of, 294. Roads, Survey of, 238. Rod floats, Velocities of, 303. Rod level, 61. Rods, Wooden and metallic, 13, 17. Gas-pipe, 289. for stadia measurements, 131, 136- 138. Rolling planimeter, 172, 173. Runner, in slide rule, 180, 185, 191. Saegmuller solar attachment, 123. Sag, Formula for, 25. Elimination of, 26. Saint Helena meridian, 359. Saint Stephen's meridian, 358. Salt Lake meridian, 359. Salt River meridian, 360. San Bernardino meridian, 360. Scale, of slide rule, 179. Logarithmic, 180-195. . of logarithmic sines and tangents, 193, 194. of topographical maps, 268. of mine maps, 321. Schott's formulas for declination, 87. Scow, Displacement of, 281. Sea level, 246. Secant, External, 239. 482 INDEX. Sections, 220. Secular variations of declination, 86. Sequence of figures, in operations with slide rule, 185. Seven Ranges, 221. Severn tunnel, 320. Sextant, 77, 294-298. use mentioned, 77. Description of, 294. Use of, 295. Theory of, 296. Adjustments of, 297. Other forms of, 298. Shafts, defined, 308. Mine entered by, 317, 318. Shifting center of transit, 99. Side shots, 249, 263. Sights of compass, 80. Similar figures, Areas of, 277. Similar triangles, 128. Simple triangulation, 253-261. when used, 253. Measuring base line, 253. Measuring angles, 254. Notes, 256. Adjusting triangles, 259. Computing triangles, 260. Use of triangles, 260. Sines and cosines, 165, 193. Table of logarithmic, 419-464. Table of natural, 466, 467. Sketch of tract to be surveyed, 205. for topography, 251. Slide direct, 186. inverted, 189. reversed, 193. Slide rule, 134, 136, 140, 179-198, 253, 281, 380. used in stadia readings, 134, 136, 140, 380. for differences of elevation, 136. described, 179. Historical note of, 180. Construction of scales, 180. Use of scales, 181. Mannheim rule, 181. Use of rule, 182. Extent of logarithmic scale, 184. Sequence of figures, 185. Shifting the slide, 185. Use of runner, 185. Squares and square roots, 185. Statement of problems, 186, Slide rule, Slide direct, 186. Slide inverted, 189. Use of runner in complicated ex- pressions, 191. Gage points, 192. Extraction of cube roots, 192. Slide reversed, 193. Scale of logarithmic sines, 193. Scale of logarithmic tangents, 194. The Thacher rule, 196. Settings for Thacher rule, 197. Fuller's slide rule, 197. Mannheim rule, 198. used in volumetric computations, 281. Slopes, Chaining on, 27. Formula for measuring on, 28. in mines, 309. Smith's observations for stadia meas- urements, 139. Smithsonian geographical tables, 372. Solar attachment of transit, 122. Solar compass, 77, 116. Solar transit, 1 16-125. See " Transit." Soundings, 288-294. in hydrographic surveying, 288. Making soundings, 289. Locating soundings, 290. Occurrence of methods, 293. Survey of harbor, river, etc., 294. Sounding lead, 288. Spacing of stadia wires, 129. Spanish vara, 221. Spider lines, 44. Spirit level, 35. jV;. Squares and square roots by slide rule, 185. Stadia line, Plotting of, 261, 262, 263. Stadia measurements, 127-140. defined, 127. Method explained, 128. Spacing of the wires, 129. To approximate the value of /, 130. The value of c, 130. The value of |and (/ + c), 130. The rod, 131. Inclined readings, 132. Table of, 133. Difference of elevation, 134. Diagram, 134. Slide rule, 136. Graduating a stadia, 136, INDEX. 483 Stadia measurements, Smith's obser- vations, 139. Notes, 140. Stadia method of topography, 249. Stadia reduction table, 380-382. Stadia surveys in coal mines, 306. Stadia traverse, 261. Stadia wires of tachy meter, 99. Standard length, 231. Standard parallels, 223. Stanley's " Surveying and Leveling Instruments," 71. Station marks in mines, 309, 310. Statutes for surveyors, Need of, 356. Steel, Coefficient of expansion of, 24. Steel tapes, 15, 24, 313. Slope, denned, 309. Streams, Meandering of, 227. Cross sections of, 288. Street grading, 280. Striding level, 71, 109. Strike, defined, 309. Structures, Volume of, 285. Stiibben, J., Paper by, 237. Subcurrents, Direction of, 288, 304. Subdivision in land surveys, 219. of townships, 226. Sun, Declination of, 117, 125, 126. Surface currents, Direction of, 288, 304. Surface form, Representation of, 244. Surface grading, 281, 283. Surface monuments of mines, 307. Surface of earth, 9, 11. Surface surveys, 305-308. Coal mines, 305. Metal mines, 306. Form of mining claim, 306. Surveying the claim, 307. Surface monuments, 307. Surveying, defined, 11, 351. Surveyors, Difference of opinion among, 351. Laxity and dishonesty of, 352. Imperfect maps of, 353, 354. Duties regarding maps, 355, 356. Surveys, Public, in Michigan, 341. See "U. S. public land surveys." Suspended planimeter, 172, 173. Tables, 361-469 : I. Correction to 100 units measured along slopes given, 361. II. Correction coefficient for temperature and hygro- metric conditions, 3(51. III. Barometric elevations, 362. IV. Polar distance of Polaris, 364. V. Daily variation of magnetic needle, 364. VI. Approximate local mean times of elongation and culmination of Polaris, 365. VII. Refraction corrections, 366. VIII. Magnetic declination formu- las, 368. IX. Angular convergences and distances between merid- ians, 371. X. Length of one minute of lati- tude and longitude, 372. XI. Trigonometric functions and formulas, 373. XII. Lengths of circular arcs, 376. XIII. Linear transformations, 377. XIV. Horizontal distances and dif- ferences of elevation, 380. XV. Common logarithms of num- bers, 397-418. XVI. Logarithms of trigonomet- ric functions, 419-4(54. XVII. Natural sines and cosines, 466, 467. XVIII. Natural tangents and co- tangents, 468, 469. Tacheometer, 127. Tachymeter, described, 99. term for complete transit equipped for stadia work, 127. used in mine surveys, 311, 312. Tallahassee meridian, 358. Tangents, Logarithmic scale of, 194. Intersection of, 239. Table of natural, 468, 469. Tapes, used in measuring, 13. Steel, 15. Ribbon, 16. Linen, 17. Errors in, 22, 23. Expansion of, 24. used in land surveys, 204. used in city surveys, 231. used in measuring base line, 254. used in mine surveys, 313, 317. 484 INDEX. Target lamp, 314. Targets of leveling rods, 47-49. Telescope, of level, 43. Spider lines of, 44. Field of view of, 45. Eyepiece of, 46. Spherical aberration of, 46. Axis of revolution of, 114. Magnifying power of, 230. pointing in triangulation, 256. Temperature correction of tapes and chains, 24. Temperature element noted, 254. Temperature correction, Table of, 361. Texas, Public lands of, 221. Thacher rule, 182, 196, 197. Theodolite, 123. Theory of planimeter, 174-178. Three-arm protractor, 292, 293. Three-point problem, 272, 273, 292. Three-tripod system, 316. Time, by transit and sun, 125, 126. Topographical surveying, 11, 127, 244- 274, 305. denned, 11. Use of stadia in, 127. Topography, 244-253. Simple triangulation, 253-261. Mapping, 261-268. The plane table, 268-274. in mine surveying, 305. Topography, 244-253, 288. Methods of representing surface form, 244. Field methods for small area, 247. Contour map, 248. Field methods for large area, 248. Transit and stadia method, 249. of bed of water, 288. Township, a subdivision of public land, 220. Subdivisions of, 226. Theoretical, 226. Township exteriors, 225. "Transactions American Society Civil Engineers," 237. Transformations, Linear, 377-379. Gunter's chains to feet, 377. Gunter's chains to meters, 378. Feet to meters, 379. Transit, 77,95-126, 148, 205, 213, 231, 249, 255, 256, 261, 311, 312, 317-319. Transit, Description, 77, 95. Tachymeter, 99. Carrying transit, 100. Setting up, 101. To produce a straight line with, 101. Measuring angles with, 102. Azimuth, 102. Traversing, 103. used instead of compass, 104. Determining meridian, 105. Needle checks on azimuths, 107. Requirements for adjusted, 108. Plate bubbles, 108. Line of collimation, 109. Level under telescope, 114. Vertical circle, 115. Eyepiece, 115. Eccentricity, 116. Solar transit explained, 116. Fundamental conception for solar, 116. Description of solar, 118. Method of use of solar, 119. Limitations of solar, 120. Latitude, 121. Refraction, 121. Adjustments of solar, named, 122. Lines of collimation of solar, 122. Declination vernier of solar, 122. Polar axis of solar, 123. Engineer's, 123, 255, 256. Saegmuller's solar attachment, de- scribed, 123. Adjustments of Saegmuller's, 125. used in land surveys, 148, 213. used with chain, 205. City surveyor's, 231. Orientation of, 261. used in mine surveys, 311, 312, 317- 319. Transit and stadia method of topogra- phy, 249. Traverse, Running of, 83. Traverse tables, 165. Traversing, with compass, 83. with transit, 103. with plane table, 270. in underground surveys, 308. Triangles, Similar, 128. Division of fields into, 206. Adjustment of, 259. Computing, 260. INDEX. 485 Triangles, used in mapping, 261. Solution of, 373-376. Triangulation, Simple, 253. Stations, 254, 260, 272, 294. System in topography, 272. System in coal mine surveys, 305. Trigonometer, 306. Trigonometric functions, Signs of, 162. Formulas of, 373-376. Tables of, 419-469. Tropic of Cancer, Extension of, 117. Tropic of Capricorn, Extension of, 117. Tunnel, defined, 308. Mine entered by, 316. Turning points in leveling, 63. Two-point problem, 272, 273. Underground surveys, 308-316. General statement, 308. Definitions, 308. Location and form of station marks, 309. Instruments used, 310. Devices for making stations visible, 313. Notes, 315. United States Coast and Geodetic Sur- vey, References to, 75, 86, 231, 268, 270, 361, 362, 365, 368. United States deputy surveyors, 219. United States Geological Survey, Ref- erences to, 268, 270. United States public land surveys, 219-229. Character of work, 219. Scheme of subdivision, 219. Historical note, 221. Legal requirements, inconsistent, 222. Principal reference lines, 223. Standard parallels, 223. Guide meridians, 224. Angular convergence of two merid- ians, 224. Township exteriors, 225. Subdivisions of townships, 226. Meandering a stream, 227. Corners, 228. Notes, 228. United States surveyor general, 221. Vara, Spanish measure, 221. Variations of declination, secular, an- nual, lunar, and diurnal, 86. Vega logarithmic tables, 165. Veins in mines, 306. Velocity of flow of stream, 288. Vernier and level bubble, 31-39. Vernier, defined, 31. Direct, 31, 32. Retrograde, 33. Double, 34. Declination, 122. Vernier protractor, 263. Vertical angle, line, and plane, defined, 10. Vertical circle of transit, 115. Volumes estimated from a map,281-286. A reservoir, 281. Application to surface grading, 283. Application to structures, 286. Volume of earth, measured, 275. in street grading, 280. in mines, 308. See also "Earthwork computa- tions." Wagon roads, Survey of, 238. Washington meridian, 359. Water boundaries, Statutes regarding, 216, 217. Water, Excavation under, 281. Wedges in earthwork, 276, 277. Weighting courses, 148. Weir measurements, 288. Willamette meridian, 360. Winslow's stadia reduction table, 380. Winze, defined, 309. Wires, Illumination of, 106, 311. Witness points, 209, 228. Wright's "Adjustment of Observa- tions," 147. Y adjustment, 64, 70. Y level, 43, 47, 61. Zenith-pole arc, 92. Zenith-star arc, 92. Zero azimuth, 162. Zero circumference, 178. Zeta Ursse Majoris, 90. Typography by J. S. Gushing & Co., Norwood, Mm. . UNIVERSITY OF CALIFORNIA AT LOS ANGELES THE UNIVERSITY LIBRARY This book is DUE on the last date stamped below WAR 7 19601 "91979 17 J938 ,JUfl? MAR 3 1942 JUN 14 W p SEP 101948 JUN 19 1952! JUN23RECD NOV 31952 Form L-9-15m-3,'34 QLAPR 4 C'DYRL FEB,2,2t)0' UNIVERSITY of CALIFORNIA AT LOS ANGELES LIBRARY UCLA-Young Research Library TA545 .R21t L 009 586 018 5 UC SOUTHERN REGIONAL LIBRARY FACILITY AA 001 270148 8