BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF 1891 1.3.p.5L..i(>.i i).nj.i>. Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31 924031 221 1 65 TABLE OF CONTENTS. -CHAPTER I.— GENEEAL INSTRUCTIONS. Page Field Work 1 Care of Field Equipment 2 Field Notes 6 OfBce Work 12 CHAPTER II.— THE CHAIN AND TAPE. Units of Measure 13 Linear Measuring Instruments 14 Use of Chain and Tape 16 Perpendiculars 17 Parallels 18 Angles 19 Location of Points 19 Location of Objects 31 The Line Surveys 21 Ranging in Lines 21 Signals 33 Stakes and Stake Driving 23 Problem A 1. Length of Pace 34 A 2. Distances by Pacing 34 A 3. Axemen and Flagmen Practice 34 A 4. Range Pole Practice 34 A 5. Standardizing Chain or Tape 36 A 6. Distances with Surveyors' Chain 36 A 7. Distances with Engineers' Chain 37 A 8. Distances with 100-foot Steel Tape 38 A 9. Horizontal Distance on Slope 38 AlO. Angles of Triangle with Tape 30 All. Survey of Field with Tape 30 A 13. Area by Perpendiciilar Method 33 A13. Area by Three-Side Method 33 2 ix X TABLE OF CONTENTS. Pago A14. Area by Angle Method 33 A15. Area from Plat 34 A16. Survey of Field with Curved Boundary. . 33 A17. Area of Field with Curved Boundary. ... 33 A18. Area (of same) from Plat 36 A19. Passing an Obstacle with Tape 36 A30. Obstructed Distance with Tape 38 A31. Running in Curve with Tape 38 A33. Discussion of Errors of Chaining 40 A33. Testing Standard of Length 40 A34. Constants of Steel Tape 43 A35. Making a Standard Wire Tape 43 A36. Comparison of Chains and Tapes 43 CHAPTER III.— THE COMPASS. Types of Magnetic Compass 45 Declination of the Needle 46 Variation of the Declination 47 Local Attraction 48 The Vernier .' 49 Use of the Compass 49 Adjustments and Tests of Compass 50 Problem B 1. Declination of Needle 51 B 3. Angles of Triangle with Compass 52 B 3. Traverse of Field with Compass 53 B 4. Area of Field with Compass 54 B 5. Adjustment of Compass 56 B 6. Comparison of Compasses 56 CHAPTER IV.— THE LEVEL. Types of Level ' 57 The Telescope 58 Line of Collimatioh 58 Objective 5S Chromatic Aberration 58 Spherical Aberration 60 Eyepiece 60 Definition 61 Illumination 61 Aperture of Objective 61 TABLE OF CONTENTS. xi Page Size of Field 61 Mag-nifying Power 61 Parallax 61 Cross-Hairs 63 The Bubble Vial 63 Leveling Eods 64 Use of tte Level 65 Differential Leveling 66 Profile Leveling 68 Eeciprocal Leveling , 68 Contour Leveling 69 Grade Lines 69 Cross-sectioning 69 Running Lines 70 Practical Hints 70 Adjustment of Wye Level 72 Adjustment of Dumpy Level 75 Problem C 1. Differential Leveling with Hand Level. . . 76 C 3. Differential Leveling, Engineers' Level.. 77 C 3. Profile Leveling for Drain 77 C 4. Eailroad Profile Leveling 81 C 5. Vertical Curve 83 C 6. Establishing Grade Line : 83 C 7. Setting Slope Stakes ; 85 C 8. Calculation of Quantities 85 C 9. Staking Out a Borrow Pit 85 CIO. Levels for Street Paving : 86 Cll. Coiitour Leveling 88 C13. Use of Contour Map 1 89 C13. Eeciprocal Leveling '. 89 C14. Delicacy of Bubble Vial ; 90 C15. Comparison of Level Telescopes 91 C16. Tests of Wye Level 91 C17. Adjustment of Wye Level 93 C18. Sketching Wye Level 93 C19. Tests of Dumpy Level 93 C20. Adjustment of Dumpy Level 93 C21. Sketching Dumpy Level 93 C33. Stretching Cross-Hairs 93 C33. Error of Setting Level Target 94 C34. Making a Leveling Rod 95 C35. Comparison of Engineers' Levels 95 xii TABLE OF CONTENTS. Page CHAPTER v.— THE TRANSIT. Types of Transit 97 Use of the Transit 99 Prolongation of Lines 99 Horizontal Angles 100 Azimuth 100 Deflection 100 Vertical Angles 100 Traversing 1 00 Compass Bearings 101 Leveling with Transit 101 Grade Lines 101 Adjustment of Transit 102 Problem D 1. Angles of Triangle with Transit 106 D 2. Prolongation of Line with Transit 106 D 3. Intersection of Lines with Transit 108 D 4. Referencing Out a Point 109 D 5. Triangulation Across River 110 D 6. Passing Obstacle with Transit 110 D 7. Traverse of Field with Transit 113 D 8. Area of Field with Transit 113 D 9. Staking Out Building 114 DIO. Height of Tower with Transit 114 Dll. Survey of Line Shafting 116 D12. Survey of Race Track 117 D13. Angles of Triangle by Repetition 118 D14. True Meridian by Polaris at Elongation 119 D15. True Meridian by Polaris at Any Time.. 121 D16. True Meridian by Solar Transit 127 D17. True Meridian by Direct Observation ... 131 D18. Comparison of Transit Telescopes 132 D19. Test of a Transit 132 D20. Adjustment of a Transit 133 D21. Sketching a Transit 133 D22. Error of Setting Flag Pole 134 D33. Comparison of Engineer's Transits 135 CHAPTER VI.— TOPOGRAPHIC SURVEYING. Topography 137 The Stadia 139 The Plane Table 143 The Sextant 146 TABLE OF CONTENTS. xiii Problem E 1. Stadia Constants, with Fixed Hairs 148 E 2. Stadia Reduction Table 143 E 3. Azimutt Traverse witli Stadia 150 E 4. Plane Table Survey by Radiation 151 E 5. Plane Table Survey of Traversing 152 E 6. Plane Table Survey of Intersection 152 E 7. Three Point Problem with Plane Table 153 E 8. Angles of Triangle with Sextant 153 E 9. Coefficients of Standard Tape 153 ElO. Measurement of Base Line 155 Ell. Calculation of Triangulation System . . . 156 E13. Sketching Topography 156 E13. Topography with Transit and Stadia . . . 157 E14. Topography with Plane Table and Stadia 159 E15. Topographic Survey 159 CHAPTER VII.— LAND SURVEYING. Functions of a Surveyor 161 United States Rectangular System 163 Surveys by Metes and Bounds 173 Problem F 1. Investigation of Land Corner 173 F 3. Perpetuation of Land Corner 174 F 3. Reestablishing Quarter-Section Corner.. 175 F 4. Reestablishing Section Corner 176 F 5. Resurvey of Section 176 F 6. Resurvey of City Block 179 F 7. Resurvey by Metes and Bounds 179 F 8. Partition of Land 180 F 9. Design and Survey of Town Site 180 CHAPTER VIII.— RAILROAD SURVEYING. Organization 183 Transit Party 184 Level Party 191 Topography Party 194 Office Work 197 Cross-Sectioning Party 203 Land-Line Party 307 Bridge and Masonry Party 207 Resurvey Party 209 Problem G 1. Review of Instrumental Adjustments ... . 208 G 2. Use of Field Equipment 209 xiv TABLE OF CONTENTS. Page G 3. Preliminary Field Curve Practice 310 G 4. Indoor Curve Problems 310 CHAPTER IX.— EEEOES OF SURVEYING. Probable Error 811 Tests of Precision 315 Linear Errors 315 Angular Errors 316 Traverse Errors 316 Leveling Errors 333 CHAPTER X.— METHODS OF COMPUTING. Consistent Accuracy 333 Logarithmic Calculations 234 Arithmetical Calculations 335 Reckoning Tables 334 Computing Machines 334 CHAPTER XI.— TOPOGRAPHIC DRAWING AND FREE- HAND LETTERING. Practice Plates 237 Freehand Titles 245 Topographic Symbols 349 FIELD AND OFFICE TABLES. Table 1. Logarithms of Numbers 254 Table 2. Logarithmic Functions of Angles 277 Table 3. Natural Functions of Angles 323 Table 4. Squares, Cubes and Roots 367 Table 5. Trigonometric Functions 380 Explanation of Tables 382 Index 385 SPECIFICATIONS FOR A GOOD ENGINEER. " A good engineer must be of inflexible integrity, sober, truthful, accurate, resolute, discreet, of cool and sound judgment, must have command of his temper, must have courage to resist and repel attempts at intimidation, a firm- ness that is proof against solicitation, flattery or improper bias of any kind, must take an interest in his work, must be energetic, quick to decide, prompt to act, must be fair and impartial as a judge on the bench, must have experi- ence in his work and in dealing with men, vsrhich implies some maturity of years, must have business habits and knowledge of accounts. Men who combine these qualities are not to be picked up every day. Still they can be found. But they are greatly in demand, and when found, they are worth their price ; rather they are beyond price, and their value can not be estimated by dollars." — Chief Engineer Starling's Report to the Mississippi Levee Commissioners. " Be sure you are right, and then go ahead." — D. Crockett. CHAPTER I. GENERAL INSTRUCTIONS. FIELD WORK. Habitual Correctness. — Habitual correctness is a duty. Error should be looked upon as probable, and every precau- tion taken to verify data and results. Unchecked work may always be regarded as doubtful. A discrepancy which is found by the maker in time to be corrected by him before any damage is done is not necessarily discreditable, pro- vided the error is not repeated. However, habitual error is not only discreditable but dishonorable as well, and noth- ing except intentional dishonesty injures the reputation of the engineer more quickly or permanently. Consistent Accuracy. — The degree of precision sought in the field measurements should be governed strictly by the dictates of common sense and experience. Due considera- tion of the purposes of the survey and of the time available will enable one to avoid extreme precision when ordinary care would sufBce, or crudeness when exactness is required, or inconsistency between the degrees of precision observed in the several parts of the survey. It is a very common practice of beginners, and of many experienced engineers as well, to carry calculated results far beyond the consistent exactness. Speed. — Cultivate the habit of doing the field work quickly as well as accurately. True skill involves both quantity and quality of results. However, while the habit of rapid work can and should be acquired, the speed at- tempted in any given problem should never be such as to cast doubt upon the results. Slowness due to laziness is intolerable. Eamiliarity with Instructions. — The instructions for the day's work should be read over carefully, and prelim- inary steps, such as the preparation of field note forms, should be taken so as to save time and make the work in 2 GENERAL INSTEUCTIONS. the field as effective as possible The ability and also the desire to understand and obey instructions are as essential as the skill to execute them. Inferior Instruments. — Should a poor instrument or other equipment be assigned, a special eilort should be made to secure excellent results. In actual practice, beginners often have to work vifith defective instruments, but they should never seek, nor are they permitted, to justify poor results by the character of the field equipment. The stu- dent should therefore welcome an occasional opportunity to secure practice with poor instruments. Alternation of Duties. — The members of each party should alternate in discharging the several kinds of service involved in the field problems, unless otherwise instructed. Training in the subordinate positions is essential whether the beginner is to occupy them in actual practice or not, for intelligent direction of work demands thorough knowl- edge of all its details. Field Practice Decorum. — The decorum of surveying field practice should conform reasonably to that observed in other laboratory work. THE CAEE OF FIELD EQUIPMENT. RESPONSIBILITY. — The student is responsible for the proper use and safe return of all equipment. All cases of breakage, damage, loss or misplacement must be reported promptly. The equipment should be examined when as- signed and a report made at once of any injury or de- ficiency found, so that responsibility may be properly fixed. PRECAUTION'S.— Careful attention to the following practical suggestions will save needless wear to the equip- ment and reduce the danger of accidents to a minimum, besides adding to the quality and speed of the work. Tripod. — Inspect the tripod legs and shoes. The leg is of the proper tightness if, when lifted to an elevated posi- tion, it sinks gradually of its own weight. The tripod shoes should be tight and have reasonably sharp points. Setting' Up Indoors. — In setting up the instrument in- doors press the tripod shoes firmly into the fioor, prefer- ably with each point in a crack. Avoid disturbing other instruments in the room. Instrument Case. — Handle the instrument gently in re- moving it from and returning it to the case. It is always THE CAKE OF FIELD EQUIPMENT. 3 best to place the hands beneath the leveling base in hand- ling the detached instrument. Considerable patience is sometimes required to close the lid after returning the in- strument ; if properly placed the lid closes freely. Mounting the Instrument. — See that the instrument is securely attached to the tripod before shouldering it. Undue haste in this particular sometimes results in costly accidents. When screwing the instrument on the tripod head, it should be turned in a reverse direction until a slight jar is felt, indicating that the threads are properly engaged. Sunshade. — ^Always attach the sunshade regardless of the kind of weather. The sunshade is a part of the telescope tube and the adjustment of a delicate instrument naay sometimes be affected by its absence. In attaching or re- moving the sunshade or object glass cap, always hold the telescope tube firmly with one hand and with the other twist the shade or cap to the right to avoid unscrewing the object glass cell. Carrying the Instrument. — Do not carry the instru- ment on the shoulder in passing through doors or in climb- ing fences. Before shouldering the instrument, the prin- cipal motions should be slightly clamped ; with the transit, clamp the telescope on the line of centers ; and with the level, when the telescope is hanging down. In passing through timber with low branches, give special attention to the instrument. Before climbing a fence, set the instru- ment on the opposite side with tripod legs well spread. Setting XTp in the Field. — When setting up in the field, bring the tripod legs to a firm bearing with the plates ap- proximately level. Give the tripod legs additional spread in windy vsreather or in places where the instrument may be subject to vibration or other disturbance. On side-hill work place one leg up hill. With the level, place two tripod shoes in the general direction of the line of levels. Exposure of Instrument. — Do not expose the instru- ment to rain or dampness. In threatening weather the water proof bag should be taken to the field. Should the instrument get wet, wipe it thoroughly dry before return- ing it to the case. Protect the instrument from dust and dirt, and avoid undue exposure to the burning action of the sun. Avoid subjecting it to sudden changes of tempera- ture. In cold weather when bringing an instrument in- doors cover the instrument with the bag or return it to the case immediately to protect the lenses and graduations from condensed moisture. 4 GENERAL INSTRUCTIONS. Guarding tte Instrument. — ^Never leave an instrument unguarded in exposed situations such as in pastures, near driveways, or where blasting is in progress. Never leave an instrument standing on its tripod over night in a room. Manipulation of Instrument. — Cultivate from the very beginning the habit of delicate manipulation of the instru- ment. Many parts, when once impaired, can never be re- stored to their original condition. Rough and careless treatment of field instruments is characteristic of the un- skilled observer. Should any screw or other part of the in- strument work harshly, call immediate attention to it so that repairs may be made. Delay in such matters is very destructive to the instrument. Foot Screws. — In leveling the instrument, the foot screws should be brought just to a snug bearing. If the screws are too loose, the instrument rocks, and accurate work can not be done ; if too tight, the instrument is damaged, and the delicacy and accuracy of the observations are reduced. Much needless wear of the foot screws may be avoided if the plates are brought about level when the instrument is set up. With the level, a pair of foot screws should be shifted to the general direction of the back or fore sight before leveling up. Eyepiece. — Before beginning the observations, focus the eyepiece perfectly on the cross-hairs. This is best done by holding the note book page, handkerchief, or other white object a foot or so in front of the object glass so as to illum- inate the hairs ; and then, by means of the eyepiece slide, focus the microscope on a speck of dust on the cross-hairs near the middle of the field. To have the focusing true for natural vision, the eye should be momentarily closed sev- eral times between observations in order to allow the lenses of the eye to assume their normal condition. The omission of this precaution strains the eye and is quite cer- tain to cause parallax. After the eyepiece is focused on the cross-hairs, test for parallax by sighting at a well defined object and observing whether the cross-hairs seem to move as the eye is shifted slightly. Clamps. — Do not overstrain the clamps. In a well de- signed instrument the ears of the clamp screw are purpose- ly made small to prevent such abuse. Find by experiment just how tight to clamp the instrument in order to prevent slipping, and then clamp accordingly. Tangent Screws. — Use the tangent screws for slight motions only. To secure even wear the screws should THE CAKE OF FIELD EQUIPMENT. 5 be used equally in all parts of their length. The use of the wrong tangent movement is a fruitful source of error with beginners. Adjusting Scre'ws. — Unless the instrument is assigned expressly for adjustment, do not disturb the adjusting screws. XEagnetic Keedle. — Always lift the needle before should- ering the instrument. Do not permit tampering with the needle. If possible, avoid subjecting the needle to mag- netic influence, such as may exist on a trolley car. Should the needle become reversed in its polarity or require re- magnetization, it may be removed from the instrument and brought into the magnetic field of a dynamo or electric motor for several minutes, the needle being jarred slightly during the exposure; or a good horseshoe magnet may be used for the same purpose. The wire coil counterbalance on the needle will usually require shifting after the fore- going process. Lenses. — Do not remove or rub the lenses of the tele- scope. Should it be absolutely necessary to clean a lens, use a very soft rag with caution to avoid scratching or marring the polished surface. Protect the lenses from flying sand and dust, which in time seriously affect the definition of the telescope. Plumb Bob. — Do not abuse the point of the plumb bob and avoid needless knots in the plumb bob string. Cleaning Tripod Shoes. — Eemove the surplus soil from the tripod shoes before bringing the instrument indoors. Leveling Rods. — Leveling rods and stadia boards should not be leaned against trees or placed where they may fall. Avoid injury to the clamps, target and graduations. Do not mark the graduations with pencil or otherwise. Avoid needless exposure of the rod to moisture or to the sun. Flag Poles. — Flag poles should not be unduly strained and their points should be properly protected. Chains and Tapes. — Chains should not be jerked. Avoid kinks in steel tapes, especially during cool weather. When near driveways, in crowded streets, etc., use special care to protect the tape. Band tapes will be done up in 5-foot loops, figure 8 form, unless reels are provided. Etched tapes should be wiped clean and dry at the end of the day's work. Axes and Hatchets. — Axes and hatchets will be em- ployed for their legitimate purposes only. Their wanton use in clearing survey lines is forbidden, and their use at all, 6 GENERAL INSTEUCTIONS. for such purpose, on private premises must be governed strictly by the rights of the owner. Stakes. — The consumption of stakes should be controlled by reasonable economy, and surplus stakes returned to the general store. For the protection of mowing machines in meadows, etc., hub stakes should be driven flush with the surface of the ground, and other stakes should be left high enough to be visible. Whenever practicable, stakes which may endanger machines should be removed after serving the purpose for which they were set. FIELD NOTES. Scope of Field Notes. — The notes should be a complete record of each day's work in the field. In addition to the title of the problem and the record of the data observed, the field notes should include the date, weather, organiza- tion of party, equipment used, time devoted to the prob- lem, and any other information which is at all likely to be of service in connection with the problem. No item prop- erly belonging to the notes should be trusted to memory. Should the question arise as to the desirability of any item, it is always safe to include it. The habit of rigid self criti- cism of the field notes should be cultivated. Character of Notes. — The field notes should have char- acter and force. As a rule, the general character of the student's work can be judged with considerable certainty by the appearance of his field notes. A first-class page of field notes always commands respect, and tends to estab- lish and stimulate confidence in the recorder. The notes should be arranged systematically. Interpretation of Notes. — The field notes should have one and only one reasonable interpretation, and that the correct one. They should be perfectly legible and easily understood by anyone at all familiar with such matters. Original Notes. — Each student must keep complete notes of each problem. Field notes must not be taken on loose slips or sheets of paper or in other note books, but the original record must be put in the prescribed field note book during the progress of the field work. Field Note Book. — The field record raust be kept in the prescribed field note book. For ease of identification the name of the owner will be printed in bold letters at the top of the front cover of the field note book. FIELD NOTES. 7 Pencil. — To insure permanency all notes will be kept with a hard pencil, preferably a 4H. The pencil should be kept well sharpened and used with sufficient pressure to indent the surface of the paper somewhat. Title Page. — ^An appropriate title page will be printed on the iirst page of the field note book. Indexing and Cross Referencing. — A systematic index of the field notes will be kept on the four pages following the title page. Eelated notes on different pages will be lib- erally and plainly cross referenced. The pages of the note book will be numbered to facilitate indexing. Methods of Recording Field Notes. — There are three general methods of recording field notes, namely : ( 1 ) by sketch, (2) by description or narration, and 1[3) by tabula- tion. It is not uncommon to combine two or perhaps all three of these methods in the same problem or svirvey. Porm of notes. — All field notes must be recorded in a field note book ruled as shown below, except where cir- cumstances require modification. If no form is given, the student will devise one suited to the particular problem. Lettering. — Field notes will be printed habitually in the " Engineering News " style of freehand lettering, as treated in Eeinhardt's " Freehand Lettering." The body of the field notes will be recorded in the slanting letter and the head- ings will be made in the upright letter. The former slants to the right 1 : 2.5 and the so-called upright letter is made to slant to the left slightly, say 1 : 25. Lower case letters will be used in general, capitals being employed for initials and important words, as required. In the standard field note alphabet the height of lower case letters a, c, e, i, m, n, etc., is %o ™ch, and the height of lower case b, d, f, g, h, etc., and of all capital letters and all numerals is I^Q (1^) inch; lower case t is made four units (%o) inch high. This standard accords with best current practice and is based upon correct economic principles. Sample pages of field notes with letters and figures drawn full size are ' given on page 9. The student is expected to make the most of this opportunity to secure a liberal amount of practice in freehand lettering. Field Note Sketches. — Sketches will be used liberally in the notes and will be made in the flcU. If desired, a ruler may be used in drawing straight lines, but the student is urged to acquire skill at once in making good plain free- hand sketches. The field sketches should be bold and clear, in fair proportion, and of liberal size so as to avoid con- 8 GENERAL INSTKUCTIONS. fusion of detail. The exaggeration of certain details in a separate sketch sometimes adds greatly to the clearness of the notes. The sketches should be supplemented by de- scriptive statements when helpful, and important points of the sketch should be lettered for reference. The precise scaling of sketches in the field note book, while sometimes necessary is usually undesirable owing to the time con- sumed. It is also found that undue attention to the draft- ing of the sketch is very apt to occupy the mind and cause /• >! V J omissions of important numerical data. Since recorded figures and not the size of the field sketch itself must usual- ly be employed in the subsequent use of the notes, it is im- portant to review the record 'before leaving the field to detect omissions or inconsistencies. Making sketches on loose sheets or in other books and subsequently copying them into the regular field book is very objectionable practice and will not be permitted in the class work. Copies of field notes or sketches are never as trustworthy as the original record made (luring the progress of the field work. In very rapid surveys where legibility of the original record must perhaps suffer somewhat, it is excellent practice to tran- scribe the notes at once to a neighboring page, thus pre- serving the original rough notes for future reference. The original has more weight as evidence, but the neat copy FIELD NOTES. Station Value oF Anqle Amgles or Triangle 5-6-7 ^ndlieas. 88''5I' 4.7°4.7' Mean 88°50'50" 47°47W' 43'r3W m'00'30" (D/'FFerence in measurements not to exceed /') Left Hand Paqe. Observers, J. Doe & R. l?oe. With Engineers' TransIt. lioy./5J9l4, (2 hours). Warm and quiet Used He/lar& Brightly TrJnsit lioJO. Riqht Hand Vac\e. 10 GENERAL INSTRUCTIONS. made before the notes are " cold " is of great help in inter- preting them. Numerical Data. — The record of numerical data should be consistent with the precision of the survey. In obser- vations of the same class a uniform number of decimal places should be recorded. When the fraction in a result is exactly one-half the smallest unit or decimal place to be observed, record the even unit. Careful attention should be given to the IcgihUity of numerals. This is a matter in which the beginner is often very weak. This defect can be corrected best by giving studious attention and practice to both the form and vertical alinement of tabulated numerals. Erasures. — Erasures in the field notes should be avoided. In case a figure is incorrectly recorded, it should be crossed out and the correct entry made near by. The neat cancellation of an item in the notes inspires confidence, but evidence of an erasure or alteration easts doubt upon their genuineness. When a set of notes becomes so confused that erasure seems desirable, it should be tran- scribed, usually on another page. Rejection of a page of notes should be indicated by a neat cross mark, and cross reference should be made between the two places. Office Copies. — Office copies of field notes will be sub- mitted promptly, as required. These copies must be actiial transcripts from the original record contained in the field note book of the individual submitting the copy. When office copies are made, a memorandum of the fact should be entered on the page of the field note book. When so specified, the office copies will be executed in India ink. Criticism of Field Notes. — The field notes must be kept in shape for inspection at any time, and be submitted on call. All calculations and reductions must be kept up to date. The points to which chief attention should be di- rected in the criticism of the field notes are indicated in the following schedule. The student is expected to criticise his own notes and submit them ,in as perfect condition as pos- sible. For simplicity the criticisms will be indicated by stamping on the note book page the reference letters and numbers shown in the schedule. SCHEDULE OF POINTS. 11 SCHEDULE OF POINTS FOR THE CRITICISM OF FIELD NOTE BOOKS. A. SUBJECT MATTER. (1) General: (a) Descriptive title of problem. (b) Date. (c) Weather. (d) Organization of party. (e) Equipment used. (f ) Time devoted to the problem. (g) Indexing and cro.ss referencing. (h) Page numbering. ( i ) Title page. (j) Identification of field note book. (2) Becord of Data: (a) Accuracy. (b) Completeness. (c) Consistency. ( d) Arrangement. (e) Originality. B. EXECUTIDIT. (1) Lettering: (a) Style. ("Engineering News") (b) Size, (a, c, e, i, etc., %o in<^'^ high; b, d, f, g, etc., A, B, C, etc., and 1, 2, 3, etc., %o (%) "ich high; t, %o inch.) (c) Slant. (In body of notes, "slanting," 1:2.5 right; in headings, " upright," about 1 : 25 to left.) (d) Form. ( See Reinhardt's " Freehand Lettering." ) (e) Spacing. (Of letters in words; of numerals; of words; balancing in column or across page.) (f) Alinement. (Horizontal ; vertical.) (g) Permanency. (Use sharp hard pencil with pressure.) (2) Sketches. "(a) To be bold, clear and neat. (b) To be ample in amount. (c) To be of liberal size. (d) To be in fair proportion. (e) To be made freehand. ( f ) To be made in the field. 12 QENERAIi INSTRUCTIONS. OFFICE WORK. Importance of Office Work. — Capable office men are comparatively rare. Skill in drafting and computing is within the reach of most men who will devote proper time and effort to the work. Men who are skillful in both field and office work have the largest opportunity for advance- ment. Calculations. — All calculations and reductions of a per- manent character must be shown in the field note book in the specified form. Cross references between field data and calculations should be shown. Consistency between the precision of computed results and that of the observed data should be maintained. Computed results should be verified habitually, and the verified results indicated by a check mark. Since most computers are prone to repeat the same error, it is desirable in checking calculations to employ in- dependent methods and to follow a different order. A fruitful source of trouble is in the transcript of data, and this should be checked first when reviewing doubtful cal- culations. Skilled computers give much attention to methodical arrangement, and to contracted methods of computing and verifying results. Familiarity with the slide rule and other labor saving devices is important. (See Chapter X, Methods of Computing.) Drafting Boom Equipment. — The student is respon- sible for the proper use and care of drafting room furni- ture and equipment provided for his use. Drafting. — The standard of drafting is that indicated in Reinhardt's " Technic of Mechanical Drafting." Drafting Boom Decorum. — The decorum of the student in the drafting room will conform to that observed in first- class city drafting offices. CHAPTER II. THE CHAIN AND TAPE. METHODS OF FIELD WORK. Units of IVEeasure. — In the United States the foot is used by civil engineers in field measurements. Fractions of a foot are expressed decimally, the nearest 0.1 being taken in ordinary surveys, and the nearest 0.01 foot (say y^ inch) in more refined work. In railroad and similar " line " surveys by which a station stake is set every 100 feet, the unit of measure is really 100 feet instead of the foot. The term " station " was originally applied only to the actual point indicated by the numbered stake, but it is now universal practice in this country to use the word station in referring to either the point or the 100-foot unit distance. A fractional station is called a " plus " for the reason that a plus sign is used to mark the decimal point for the 100-foot unit, the common decimal point being reserved for fractions of a foot. The initial or starting stake of such a survey is numbered 0. The 100-foot chain is commonly called the " engineers' chain " to distinguish it from the 66-foot or lOO-link chain which is termed the " surveyors' chain " because of its special value in land surveys involving acreage. The latter is also called the Gunter chain after its inventor, and is otherwise known as the four-rod or four-pole chain. British engineers use the Gunter chain for both line and land surveys. The " surveyors' " or Gunter chain, while no longer used in actual surveying, is described in this book for the reason that the United States rectangular surveys were made throughout with the 66-foot chain. In the Spanish-American countries the vara is generally used in land surveys. The Castilian vara is 32.8748 inches long, but the state of California has adopted 32.372 inches, and Texas 331^ inches, as the legal length of the vara. While the metric system is used exclusively, or in part, in 13 14 THE CHAIN AND TAPE. each of the several United States government surveys, ex- cept those for public lands, little or no progress has been made towards its introduction in other than government surveys. Linear Measuring Instruments. — Two general types of linear measuring devices are used by surveyors, viz., the common chain and the tape. There are several kinds of each, according to the length, material, and method of graduation. Fig. 1. The common chain is made up of a series of links of wire having loops at the ends and connected by rings so as to afford flexibility. The engineers' chain is shown in (a), Fig. 1, the illustration being that of a 50-foot chain, or one- METHODS OF FIELD WOEK. 15 half the length generally used. The surveyors' or Gunter chain is shown in (b), Fig. 1. In the common chain the end graduation is the center of the cross bar of the handle, and every tenth foot or link is marked by a notched brass tag. In the 100-foot or 100-link chain the number of points on the tag indicates the multiple of ten units from the nearer end, and a circular tag marks the middle of the chain. The chain is done up hour-glass shape, as shown in the cut. Chaining pins made of steel wire are used in marking the end of the chain or tape in the usual process of linear measurement. A set of pins usually numbers eleven, as indicated at (c). Fig. 1. The pins are carried on a ring made of spring steel wire. The flat steel band, shown in (d) and (e), Fig. 1, is the best form of measuring device for most kinds of work. The band tape is usually 100 feet long. The end graduations of the band tape are usually indicated by brass shoiilders, w^hich "should point in the same direction, as shown in (f), Fig. 1. The 100-foot band tape is commonly graduated every foot of its length, and the end foot to every 0.1 foot, every fifth foot being numbered on a brass sleeve. Brass rivets are most commonly used in graduating this tape. The band tape may be rolled up on a special reel, as indi- cated in (d) and (e), although some engineers dispejise with the reel and do up the tape in the form of the figure 8 in loops of five feet or so. The steel tapes shown in (g) and (h) have etched gradu- ations. This style of tape is commonly graduated to 0.01 foot or yg inch. It is more fragile than the band tape and is commonly used on inore refined work. The form of the case shown in (h) has the advantage of allowing the tape to dry if wound up while damp. The " metallic " tape (i) , Fig. 1, is a woven linen line hav- ing fine brass wire in the warp. The steel tape is superior to the common chain chiefiy because of the permanency of its length. The smoothness and lightness of the steel tape are often important advan- tages, although the latter feature may be a serious draw- back at times. The tape is both easier to break and more difiicult to mend than the common chain. Tapes for measuring base lines with great precision have recently been made of Invar steel. Invar steel has a very small coefBcient of expansion. Invar steel tapes are very expensive. 16 THE CHAIN AND TAPE. Chaining. — In general, the horizontal distance is chained. Two persons, called head and rear chainmen, are required. The usual process is as follows : The line to be chained is first marked with range poles. The head chainman casts the chain out to the rear, and after setting one marking pin at the starting point and checking up the remaining ten pins on his ring, steps briskly to the front. The rear chainman allows the chain to pass through his hands to detect kinks and bent links. Just before the full length is drawn out, the rear chainman calls " halt," at which the head chainman turns, shakes out the chain and straiglitens It on the true line under the direction of the rear chainman. In order to allow a clear sight ahead, the front chainman should hold the chain handle with a pin in his right hand well away from his body, supporting the right elbow^ on the right knee, if de- sired. The rear chainman holds the handle in his left hand approximately at the starting point and motions with his right to the head chainman, his signals being distinct both as to direction and amount. Finally, when the straight and taut chain has been brought practically into the true line, the rear chainman, slipping the handle behind the pin at the starting point with his left hand, and steadying the top of the pin with his right, calls out " stick." The head chainman at this instant sets his pin in front of the chain handle and responds " stuck," at which signal and not before the rear chainman pulls the pin. Both now proceed, the rear chainman giving the prelim- inary " halt " signal as he approaches the pin just set by the head chainman. The chain is lined up, stretched, the front pin set, and the rear pin pulled on signal, as described for the first chain length. This process is repeated until the head chainman has set his tenth pin, when he calls " out " or " tally," at which the rear chainman walks ahead, counting his ping as he goes and, if there are ten, transfers them to the head chainman who also checks them up and replaces them on his ring. A similar check in the pins may be made at any time by remembering that the sum, omit- ting the one in the ground, should be ten. This safeguard should be taken often to detect loss of pins. The count of tallies should be carefully kept. When the end of the line is reached, the rear chainman steps ahead, and reads the fraction at the pin, noting the units with respect to the brass tags on the chain. The number of pins in the hand of the rear chainman indicates METHODS OF FIELD WORK. 17 the number of applications of the chain since the starting or last tally point. A like method is used in case inter- mediate points are to be noted along the line. On sloping ground the horizontal distance may be ob- tained either by leveling the chain and plumbing down from the elevated end, or by measuring on the slope and correcting for the inclination. In ordinary work the for- mer is preferred, owing to its simplicity. In " breaking chain " up or down a steep slope, the head chainman first carries the full chain ahead and places it carefully on the true line. A plumb bob, range pole or loaded chaining pin should be used in plumbing the points up or down. The segments of the chain should be in multiples of ten units, as a rule, and the breaking points should be " thumbed " by both chainmen to avoid blunders. Likewise, special cau- tion is required to avoid confusion in the count of pins dur- ing this process. The general method of measuring with the band tape is much the same as with the common chain. The chief dif- ference is due to the fact that the handle of the tape extends beyond the end graduation, so that it is more convenient for the head chainman to hold the handle in his left hand and rest his left elbowr on his left knee, setting the pin with his right hand. Another difEerence is in the method of reading fractions. It is best to read the fraction first 'by estimation, as with the chain, making sure of the feet; then shifting the tape along one foot, getting. an exact decimal record of the fraction by means of the end foot graduated to tenths ; the nearest 0.01 foot is estimated, or in especially refined work, read by scale. In railroad and similar line surveys, chaining pins are usually dispensed with and the ends of the chain are indi- cated by numbered stakes. The stake marked corre- sponds to the pin at the starting point, and the station stakes are marked thence according to the number of 100-foot units laid off. Perpendiculars. — Perpendiculars may be erected and let fall with the chain or tape by the following methods : (a) By the 3:4:5 method, shown in (a). Pig. 2, in which a triangle having sides in the ratio stated, is constructed. (b) By the chord bisection method, shown in (b), Fig. 3, in which a line is passed from the bisecting point of the chord to the center of the circle, or vice versa. 3 18 THE CHAIN AND TAPE. (c) By the semicircle method, shown in (c). Fig. 2, in which a semicircle is made to contain the required perpen- dicular. The first method corresponds to the use of the triangle in drafting. Good intersections are essential in the second and third methods. Eesults may be verified either by using another process, or by repeating the same method with the measurements or position reversed, as indicated in (d), rig. 2. (^) ^.^5 3 4- (b) — ^. — !d) \/ (e) ,< (A (b) Cc) / t \ / V (d) le)\ (s) Stake //ai w/t/>""' ""• Guard Stake Fig. 9. Signals. — There is little occasion for shouting in survey- ing field work if a proper system of sight signals is used. Each signal should have but one meaning and that a per- fectly distinct one. Signals indicating motion should at once show clearly both the direction and amount of motion desired. Some of the signals in common use are as follows : (a) " Eight " or " left," — the arm is extended distinctly in the desired direction and the motion of the forearm and hand is graduated to suit the lateral motion required. (b) " Up " or " down," — the arm is extended laterally and raised or lowered distinctly with motions to suit the magni- tude of the movement desired. Some levelers use the left arm for the " up " signal and the right for " down." (c) "Plumb the pole (or rod)," — If to the right, that arm is held vertically with hand extended and the entire body, arm included, is swung distinctly to the right, or vice versa. (d) "All right," — both arms are extended full length horizontally and waved vertically. METHODS or FIELD WORK. 23 (e) "Turning point" or "transit point," — the arm is swung slowly about the head. (f ) " Give line," — the flagman extends both arms upward, holding the flag pole horizontally, ending with the pole in its vertical position. If a. precise or tack point is meant, the signal is made quicker and sharper. (g) Numerals are usually made by counted vertical swings with the arm extended laterally. A station number is given with the right hand and the plus, if any, with the left ; or a rod reading in like manner. The successive counts are separated by a momentary pause, emphasized, if desired, by a slight swing with both hands. Stakes and Stake Driving. — ^A flat stake is used to mark the stations in a line survey, and a square stake or hub to mark transit stations, (a) and (b), Eig. 9. The station stake is numbered on the rear face, and the hub is witnessed by a flat guard stake driven slanting 10 inches or so to the left, Eig. 9. The numerals should be bold and distinct, and made with keel or waterproof crayon, pressed into the surface of the wood. Having located a point approximately vyith the flag pole, the stake should be driven truly plumb in order that the final point may fall near the center of its top. In driving a stake, the axeman should watch for signals. It is better to draw the stake by a slanting blow than to hammer the stake over after it is driven. Good stake drivers are scarce. PROBLEMS WITH THE CHAIN AND TAPE. General Statement. — Each problem is stated under the following heads : (a) Equipment. — In which are specified the articles and instruments assigned or required for the proper perform- ance of the problem. A copy of this manual and of the regulation field note book, with a hard pencil to keep the record, form part of the equipment for every problem as- signed. (b) Problem. — In which the problem is stated in general terms. The special assignments will be made by program. (c) Methods. — In which the methods to be used in the as- signed work are described more or less in detail. In some problems alternative methods are suggested, and in others the student is left to devise his own. 24 THE CHAIN AND TAPE. PKOBLEM Al. LENGTH OF PACE. (a) Equipment. — (No instrumental equipment required.) (b) Problem. — Investig-ate the length of pace as follows: (1) the natural pace; (2) an assumed pace of 3 feet; and (3) the effect of speed on the length of the pace. (c) Methods. — (1) On an assigned course of known length count the paces while walking at the natural rate. Observe the nearest 0.1 pace in the fraction at the end of the course. Secure ten consecutive results, with no rejections, varying not more than 3 per cent. (3) Repeat (1) for an assumed 3-foot pace. (3) Observe (in duplicate) time and paces for four or fi\e rates from very slow to very fast, with paces to nearest 0.1 and time to neare.st second. Record data and make reductions as in the form. PROBLEM A3. DISTANCES BY PACING. (a) Eqiiiiiiiieiit. — (No instrumental equipment required.) (b) Problem. — Pace the assigned distances. (c) Methods. — (1) Standardize the pace in duplicate on measui'ed base. (3) I'ace each line in duplicate, results dif- fering not more than 3 per cent. Record and reduce as in form. PROBLEM A3. AXEMAN AND FLAGMAN PRACTICE. (a) Equipment. — Flag pole, axe, 4 flat stakes, 1 hub, tacks. (b) Problem. — Practice the correct routine duties of axe- man and flagman. (c) Metliocls. — (1) Number three station stakes to indi- cate representative cases and drive them properly. (2) Drive a hub flush with ground and tack it ; number a wit- ness stake and drive it properly. (3) Arrange program of signals with partner, separate 1.000 feet or so and practice same. (4) Signal say flve station numbers to each other and afterwards compare notes. Make concise record of the foregoing steps. PROBLEM A4. RANGE POLE PRACTICE. (a) Equipment. — 4 flag poles. (b) Problem. — Given two hubs approximately 1,000 feet apart, interpolate a flag pole say 100 feet from one hub. PROBLEMS. 25 ^ 5spM3,^4,(S Jfrs-J CJeir and Cool ■ "^ lN\ ESTIS /^TION OF LEN6TH OF Pace • J-Doe, Surveyor- 1 Kind Races p< r 400 Ft. Lengfh Rem ;rks EFFECT OF SPEED Or LEN6TH OF PACE- | oF Pace Mean of Pace Sepm. ■Clear Kind Paces tn 400 Ff. Mean Time Speed of Hi Paces Paces Ft. Smooth ground' oF Pace Obscrv'd Mean Pace.P 40OFt Pacing, 5 ttltwsH J5gl> mtirt. eWfnd. Paces Paces Ft- Sec- FhptrSec Z 137-4 Ag. inst " Veryshvi 714-6 (s) (B) 3 159-0 ft 't tt >r 717-8 Zll-70 t-fS If7 Z-^O 4 137-1 n » Slow 16S-0 (k) (h) S 131-0 n f* n 167-S le7-7S 7-3i 111 3-60 ( 139-0 It Pf mara/ 139-4 ra re) 7 137-3 tt ti " 137-S 13S-4S Z-S9 71 S-63 ! J39-0 M n 3'Focf 133-3 I'd) 0J 9 J3S-0 f ff T, 1S3-6 133-4! 3-00 77 S-ZO 10 S-FictI 2 139-3 I3S-T( Z-S9 With tl A,, 'nsf- " Fast 174-7 lZS-3 IZS-OO fe) 3-ZO SS (e) e-90 (jl-e) J3J 1 I3Z-6 o:* } J 33-0 }* d ^ ■-t e 4 S 6 7 t 9 K I3i-^ 134-0 133-3 13Z-0 133-0 133-3 132->(S W ft tr b - \ m J. S^ «j ^ a / t M) 1 Z $ 4 5 e 7 1 II 133-0 133-02 3-01 " n Speed oF Wal ing, Ft- per Sec>;S* | (lO-l) y r DiSTA ICES BY ?A CINS Line Length iF Pace Len( th oF -ine J- Doe, Survey&j - No flirWOft Ft. Obarvtd Mean Length Sepl-14,' 14 fulfil Ts) Clear ^ Coo/' Paces, Paces Ft- / i44-0 z MZ-a 143-0 zsa i--# / 134-0 1 \ / "^ 11 z 1340 134-0 37S 1 \ e-9 1 Z17-0 1 \ / n 2 Zlg-S Z17S eio 1 \/ / z M/scot/nf- 1 mo tt 3 llS-5 Ug-Z 331 1 / \ HI 1 Z7a-o 1 / / / S » Z ZtO-E Z7S-g 7S5 ^■^y 1-9 t Z09-0 / / ""'^ / If Z ZIO-0 Z09-S saz 1 / ^' / s-s 1 zse-0 /--^' / If z ZMS Z87-Z g04- 9 / s-i 1 140-0 / " tt z 141-0 140-S 393 / / k t-l 1 ies-0 / tt z me 117- g sze / W- -E / i II 26 THE CHAIN AND TAPE. remove the distant pole, prolong the line by successive 100- foot sights and note the error at distant hub. Bepeat process for 200-foot and 300-foot sights. (c) Mctlwds. — (1) Set distant flag pole precisely behind hub and hold spike of pole on tack of near hub ; lying on ground back of near hub, line in pole 100 feet (paced) dis- tant ; remove pole from distant hub, and prolong by 100- foot sights up to distant hub, noting error to nearest 0.01 foot. (2) Eepeat in reverse direction, using 200-foot sights. (3) Eepeat with 300-foot sights. Avoid all bias. Record data in suitable form, describing steps concisely. PROBLEM A5. STANDARDIZING CHAIN OR TAPE. (a) Equipment. — Chain or tape assigned in any problem where standard length of chain may be of value. (b) ProMem. — Determine the length of the assigned chain or tape by comparison with the official standard under the conditions of actual use. (c) Metliods. — In standardizing tape, reproduce the con- ditions of actual use as regards tension, support, etc., bring one end graduation of chain or tape to coincide with one standard mark, and observe fraction at the other end with a scale. As a general rule, observe one more decimal place than is taken in the actual chaining. PROBLEM A6. DISTANCES WITH SURVEYORS' CHAIN. (a) Equipment. — Surveyors' chain, set of chaining pins, 2 plumb bobs, 2 flag poles (unless instructed otherwise). (b) Problem. — On an assigned chaining course about one mile long measure distances with the surveyors' chain to the nearest 0.1 link, and repeat the measurements in the opposite direction. (c) Methods. — (1) Standardize the chain before and after, as prescribed in A5. (3) Chain along the assigned course, noting the distances from the starting point to the several intermediate points and to the end station. Observe frac- tions to the nearest 0.1 link by estimation. (3) Repeat the chaining in the opposite direction, noting the distances from the end point, as before. The difference between the totals in the two directions should not exceed 1 : 3,000. Retain the same party organization throughout the problem. Record the data as in the prescribed form. PROBLEMS. 27 Line Chdin A-B A-C A-D A-B e-D B-C E-B B-A Note: Direction Cfiained Befhre After B- DISTjlMCES Obtrred Dif f- of Length Total Ch- Ch. W- The st/hs the 7■3^7 30-306 eO-3S7 79-g3S 19-473 4AS5I 7M06 73-133 svbsi ^uenf / Tea WITH Ratio l:d ^0-OOS have c 7t3 wL proble *7? I-ISS70 crc (3ee D. a^rsn^ f be i/sed irT3 In t?> 'scuss/pp -■/sj^n or c bs/mn ^- CoeF- C Lk- p-oe Surveyor's CiJai'n /fff3(f Ch3m/nsj?rJ-l oe^ SepfJS, 'l4-frffourt . Used &iipferCfr3i'ir\ff~^fO, haia Compared Ghain fv/ ^/f ' ''t before and Kiftercf^. bsfh Chained 3foi?0 Chan *in0 te^jj7/?m03f A on ffuard stake, of W' br/ck w^M curb J/he !ll-;thei?ce£'ly N'brfck side wa. es to nearest 0-1 marked B, C, from stsrtmff pi fnfA Chained same d/recfion carrying from Hub ^ fractions of^i/nk ruie tvas used in W-^ A B r/ ^earChalnman, R-Rae, Ciear^nd Cool- 1 J locker /f^35 off/ciaisfandsrd '^ chain ing. Course ''A", fvifh fack, market^ J, f0C3tetf3tS.-&^ on (freenSt- atf' •; Urbana^ ^lon^ said 5- line of 'kf obsen/^ing distanc- ik- to tacked hubs ■f the totai distance being noted- in therei^rse total distances of Oat. 'lews Ave- Dat'dE, were estimated- ibcht standardizing Chain- PROBLEM A7. DISTANCES WITH THE ENGINEERS' CHAIN. (a) Equipment. — Engineers' chain, set of chaining pins, 3 plumb bobs, 2 flag poles (unless instructed otherwise). (b) Problem. — On an assigned chaining course about one mile long measure distances with the engineers' chain to the nearest 0.1 foot, and repeat the measurements in the opposite direction. (c) Methods. — (1) Standardize the chain before and after, as prescribed in A5. (2) Chain along the assigned course, noting the distances from the starting point to the several intermediate points and to the end station. Observe frac- tions to the nearest 0.1 foot by estimation. (3) Repeat the chaining in the opposite direction, noting the distances from the end point, as before. The difference between the totals in the two directions should not exceed 1 : 3,000. Retain the same party organization throughout the problem. Re- cord the data as in the form. 28 THE CHAIN AND TAPE. PROBLEM A8. DISTANCE WITH 100-FOOT STEEL TAPE. (a) Equipment. — 100-foot steel band tape with end foot graduated to tenths, set of chaining- pins, 3 plumb bobs, 2 flag poles (unless instructed otherwise). (b) Problem. — On an assigned chaining course about one mile long measure distances with the 100-foot steel band tape to the nearest 0.01 foot, and repeat the measurements in the opposite direction. (c) Methods. — (1) Standardize before and after, as pre- scribed in A5. (2) Chain along the assigned course, noting the distances from the starting point to the several inter- mediate points and to the end station. In observing the fractions, first determine the foot units, then estimate the nearest 0.1 foot, then shift the tape along one foot and read the exact fraction on the end of the tape, estimating the nearest 0.01 foot. (3) llepeat the mea.surement in the op- posite direction, noting the distances from the end point, as before. The diiference between the totals in the two direc- tions should not exceed 1 : 5,000. Retain the same party organization. Record data as in the form. PROBLEM A9. HORIZONTAL DISTANCE ON SLOPE WITH STEEL TAPE. (a) Equipment. — 100-foot steel tape with etched gradua- tions to 0.01 foot, set of chaining pins, 3 plumb bobs, 3 flag poles, axe, supply of pegs, engineers' level and rod (unless otherwise instructed). (b) Problem. — Determine the horizontal distance between two assigned points on a steep slope, ( 1 ) by direct horizon- tal measurement, and (3) by measurement on the slope and reduction to the horizontal. (c) Methods. — (1) Standardize the tape for each method, as prescribed in A5, both before and after the day's chain- ing. (3) In chaining down hill, rear chainniaii lines in flag pole in hand of head chainman, then holds tape end to tack on hub ; flagman stands 50 feet or more from line opposite middle of tape and directs head chainman in leveling front end, then supports middle point of tape under direction of head chainman; head chainman, with spring balance at- tached to tape and using pole as help to steady pull, brings tension to 13 pounds ; recorder plumbs down front end, and sets pin slanting sidewise. After checking th? pin, proceed PROBLEMS. 29 DlST/ NCES WITI > Ensiheer's Chain- Line Direction Obstred DiFf-of Ratio CoeF- Held Chainman, R-Roe - Rear Chamman, JDk- Chained Length Total I'd C SepMS, '/4-CZ Hours) C/mdyS- Con/- Ft- Ft- Ft Used WO Ft- Chain ffSS, Locker miS- Cham Befire /iVx/0 Compared chain with official standard ti AFfer WO-IZ both before and after days chaining AS e- 4U0 Chained alonp ciiainlnff course "A", A-C tt ZODZ-Z beffinninff at hub with tack^ marked A-D »' 3Sg7-S A on^uardstake^ Iff cared 3tS- A-E ft 'J 5274-6 \ edge off/' brick waJk on Sreen St- ate- curt line ofNathewsAve-, e-D r 1/86-3 \ llrbana, Hi-; thence f 'iy alon0 said e-c ij il7Z-4 1 5-iine of It- brick walk, observing e-B rj 4730-Z 4 distances to nearest O-f ft- to e-A r> SZ74-3 I=/7SSP 0-04 tacked hubs S, C, D and E, the total 1=' n " e distances from starting point A SZ-743 e= cYlor ^=# being noted- (SeeDj igramy iJ Chained same course in tile reverse Note: T/ie s ^aye d ?/(? wL ( be ui ed in direction, carrying total distances 3 sub •^e^uer, tproi /er^end/cu!3re Length Sfandarel Length and shlftlni '3S required- 5ef /}e0s Ft- Ft- Ft- at points a. h and c- 5eptZS Measi/redanp fsASf, EBD and DBC m'fh Tape 99-99Z tape by cAof 1 metJiodflO^ft-j-adlc/s, and cheeked lymeasurj'n^ snffle be- Tape 99-SgO fweenASaadi dfjme CB profarfaed) AB i}6-8i -0-07 33e-7e 5ept-?6, '99 (ZM, ars) Prlzz'Ung ds are given 6x6'll'tl*l^ = 0-00lKZ95t96Ac 30Z-10 feet' Ac-'' ■he apf llcallo/. OF Multipli- cation FliLO 4m-t sesze 2S6 I4SI16 3S69B 133 'J 7 Square 4-3560 one ^ the i lethods opfios ''te Logar- ithms Z-6MIS Z-ZIOM Sept-Z7,14- Compuh r, A-B-tD-E, Perpendi Double Area^ 5q- Ft. 4-90S49 (81190) Z-71995 2-410IS 5-pOIO 2-18459 Z-5m9 5-Zt59S (J1450S) de by I elow chains- Ac- Data fromfff>- Transcript 81190 ■■hecked- (Hesult i t nearest 10 5^- Ft-) 186 ZSO 184 500 Z)45l 940 llTZSMIO_ B )37isi-B&l lijlnrm 11 1570-631 (Result 5-lllAc- Contract d Div'n Used ZZS9701 znBoo 8170 435& SB14- 34-85 323 305 Z4 Z£ J-Doe. ^ ULAR Method. Area- -Lab' '<7 nearest 0-001 Ac-} Contracted Mult'n 41191- 4513 Z034 113 15 S-I876 34 THE CHAIN AND TAPE. culations by logarithms, as a check. (4) Combine the checked results. Follow the form. PROBLEM A15. AEEA OF FIELD FROM PLAT. (a) Equipment. — Drafting- instruments, paper, etc., pla- nimeter (as assigned). (b) Prohlem. — Determine the area of the assigned field directly from the plat. (c) Methods. — (1) Make an accurate plat of the field from the notes secured in All, using a prescribed scale. (2) De- termine the area of the field by resolving the polygon into an equivalent triangle. (3) Determine the area from the plat by the polar planimeter and by one of the following " home-made " planimeters : " bird shot " planimeter, " jack knife " planimeter, cross-section paper, parallel strip, weighing, etc. (4) Prepare on the plat a tabulated com- parison of results secured by the several methods. (5) Finish the plat, as required. PROBLEM A16. SURVEY OF FIELD WITH CURVED BOUNDARY. (a) Equipment. — 100-foot tape, 50-foot metallic tape, set of chaining pins, 2 plumb bobs, 4 fiag poles. (b) Problem. — Make survey with tape of an assigned tract having a curved boundary, collecting all data required for plotting the field and calculating its area. (c) Methods. — (1) Standardize the tape once to nearest 0.01 foot. (2) Examine the tract carefully and plan the survey so as to secure a simple laj'out of base lines de- signed to give short offsets to the curved boundaries. (3) Locate the perpendiculars, if any, and chain all lines ; on the curved sides, take offsets so as to secure a definite loca- tion, and as a riile take equal intervals on the same line. Follow the form. PROBLEM A17. AREA OF FIELD WITH CURVED BOUNDARY. (a) Equipment. — (No instrumental equipment required). (b) Problem-. — Calculate the area of the assigned field with curved boundary by " Simpson's one-third rule," using the data collected in Problem A16. PROBLEMS. 35 Com Triangle ABB BOB BCP Data PUTAT Sid Line AB=3 SB'-b BA'C i) ON es Length Ft- 336-70 4?S-g4 ^41-14 F ARI ka»b«) Ft- 50Z-Zi 740-49 (93-00 Tra'nsc A OF (S-B) Ft- 10S-46 30J-SS ZZS-0S -!pt cl Field (s-b) Ft- 70-3S JZ3-90 Z09-94 ecked- Sept-2S A-B-C Ft Z0O-3S 3J4-05 ZSS-OS */4- Compute D-E, 3 SlDl Area oF Triangle r,J-Doe- ■-. Metho Areas 5-9t BC=3 CD^b Db=c 740-49 404-91 '4I3-7Z 431-01 I3!7-3I From f 093-00 a- S-3S447 -i-eisog-*- Z20J9O -■l- 43 300 0-71338 Triangle ABB Part CoMfUTATipH OP Value Ft- or" 330-70 423-14 34'33' BOB BCD AB-a BB^b ABB=e BE'a BD'b BBD-e SC=b pic=e 425-84 438-61 90°S9' Are>, Multip aSin-C 438-61 4l*-9l 64'39' OF ication abSin-C 16S5S ^02 IflELC Logar- itiitns 2-S273Z 2-02920 9-7B3S0 J703JZ 12773 3-^06 ZSB ± 4-9m7 (SI30II) 2-12923 2-64, 9-99994 5-21127 166 750 IS8SS2 23783 1586 357 2-64208 2-66743 9-936113 S-265S4 (114310) S-3S446 4-63909- Sepf-22, '14' A-B-C-D-E, Ahsle Double Areas Sq-Ft. Data From Transcript 81300 110 750 184 310 2)452 360 pp. checked- (Resultto ■-(■r 43500) (5-I92Ac) '»-, J-Doe- Method- Area^-^absin-C- nearest lOSij-Ft.) 5-192 Ac- (Result t J nearest 0-001 Ac-) 36 THE CHAIN AND TAPE. (c) Methods. — (1) Prepare form for calculation; tran- scribe data in convenient form for calculation, and carefully check copy. (2) Calculate the area of the polygon formed by the base lines, preferably by the perpendicular method. (3) Calculate the areas of the curved figures by " Simpson's one-third rule," which is as follows : " Divide the base line into an errii itumhrr of equal parts and erect ordinates at the points of division ; then add together the first and last ordinates, twice the sum of all the other odd ordinates, and four times the sum of all the even ordinates ; multiply the sum by one-third of the common distance between ordi- nates." (The field notes might have been taken with special reference to the rule, but it is better to take from the notes the largest cren number of equal segments, assuming the re- maining portion to be trapezoid or triangle.) (4) Give signs to the several results by reference to the field sketch, and combine them algebraically to get the net area, as shown in the accompanying form. PROBLEM A18. AKEA OP PTELD WITH CUEVED BOUNDARY FROM PLAT. (a) Equipment. — Drafting instruments, paper, etc., pla- nimeter (as assigned). (b) Problems. — Determine the area of the field with curved boundary directly from the plat. (c) Methods. — (1) Make an accurate plat of the field from the notes obtained in Al6, tising a prescribed scale. (3) Determine its area directly from plat by two methods men- tioned in (3) of A15, other than those used in that problem. (3) Prepare on the plat a tabulated comparison of the re- sults by the several methods. (4) Pinish the plat, as re- quired. PROBLEM A19. PASSING AN OBSTACLE WITH TAPE. (a) Equipment. — 100-foot steel tape, set of chaining pins, plumb bobs, 4 flag poles. (b) Problem. — Prolong an assigned line through an as- sumed obstacle by one method and prove by another, finally checking on a precise point previously established. (c) Methods. — Given two hubs, A and B, 200 feet apart prolong line and establish C 200 feet from B : (1) by con- structing a 200-foot square in one direction; and (2) by lay- PKOBLEMS. 37 c Su RVEY OF F\ :LD V ITH Curved Boundary Line- Offiiet L- Dist- OfFsd-l!' OffsetL Dist- Offset R HKit)Cl!smjmn,R-K(ie- Re^rChsinimn, J-Vae- Ft- ff Ft- Ff- Ft- Ft- Oct-2, '14. (3 Hours) Clear and y^arm- 26Z-S = d Tspe H136I, locker H^SS = lOO^OI ll-B Z4-0 Sketch shows obseri/ed lengths- Final 30-3 too es 309-1 area resuJt corrected For standard- 39-0 160 300 2-1 39- 1- IZO 2S0 !S il-S go 260 13-2 d IS- 6 40 240 14-7 ^^^lr~>,<'•»^_, ■0 220 lB-0 Line C CcfoJ. 200 ISO I4-S 10-0 # %^^ 4IS-4 = c 160 2-S 3-S 400 JS4-3 / .. V.'-'' // ?4-6 iS-4 39-3 360 3Z0 ZSO 7-2 IS-0 ■i9-7 MO 120 100 40-7 240 20-8 SO e 40-3 37-4 200 160 20-2 JS-4 60 40 "^ 30-1 no ja-3 20 »M <^^ ~~-kj 10 -g SO 40 LineD Cdtoe) /LineA(B09-ih]l, Line B = b I'll toe) Tepf Octl't'i 100-01 , Clear 'rWjrm a { ^ssdVp) gesdUp. \ \ Com >utation of Area of Field Data for Calculation of Areas Part 3be bee cde LineB LineC LIneP Chain TrueA Triangle, Base' 290-0, Alt ^ 145-3 •• '4IS-'4, i ^5^^5«)i ii^?, •9-1 ftoTTyxxtJi ^z- (.' K-6i'?0''l20'->\ _,--■ WO-'OI ■ea = Computed Area "-(H-O-OOOI) {l+0-00ai)'-(ltll-0ll02) Cnearly)- Oct-3, 14- Computer, J-Doi^- WITH Curved Boundary Indicated Calculations OafaFrompp- Transcr^t i[290-C ■-'-' i(4IS-4 i(404. \^[(0t9-S) < +2f2l-l-f37-4t4S-7t32-4) 1 ^■4(|0.8■^3e■l■^40^3t39-5^■24^ii ^^i(9-S>-l8-4) (^[(0tl3-S; { +2(31-8 i-39-O) \t 4(19-6 ■t39-4 -tSO-S)] ■ i(l3-S'-22-5) '^[(OtlS-0) t2(l8■4■^20■S) \f 4(10-3 +20-2 + 19-7)] i [20(15-0 +7-2) +(7-2*14-3)] m[(2-8+8-S) l+2(l4-8+l4-7) \+4(IO-0+IS-0+l3-2)J --i (2-8 "5-7) ^i 1(2-1'- 9-1) + 20(2-l +8-5) J {98352 -*> Chain Cor. K2000-0_ tAreas ■.ed- 21068 S6024 40328 (98352 ■ '. 1 2000-0 1487 8 116 II903I 20679 98351 -Areas II37S 87 6831 152 1961 273 2-?5l^ 38 THE CHAIN AND TAPE. ing off a 200-foot equilateral triangle on the opposite side using pins to mark points thus established. (3) Prolong the line by each method to the hub D, 200 feet from C, and record discrepancies in line. (4) Interpolate a point at G on true line between B and D, and note errors of prolonga- tion at G. Record as in the form. PEOBLEM A20. OBSTRUCTED DISTANCE WITH TAPE. (a) Equipment. — 100-foot steel tape, set of chaining pins, 2 plumb bobs, 4 flag poles. (b) Problem. — Determine the distance between two as- signed points through an assumed obstruction to both vision and measurement, using two independent methods, and finally chain the actual distance. (c) Methods. — (1) Standardize the tape. (2) Determine the distance between the assigned points by constructing a line parallel to the given line, and equal or bearing a known relation to it. (3) Secure a second result by running a random line from one hub past the other so that a per- pendicular less than 100 feet long may be let fall, measur- ing the two sides and calculating the hypothenuse. (4) After securing two results differing by not more than 1 : 1,000, chain the actual distance. Follow the form. PROBLEM A21. RUNNING IN CURVE WITH TAPE. (a) Equipment. — 100-foot steel tape, 50-foot metallic tape, set of chaining pins, 2 phimb bobs, 3 hubs, 6 flat stakes, marking crayon, tacks, five-place table of functions. (b) Prolyicm. — Lay out two lines making an assigned angle with each other, and connect them with a prescribed curve by the " chord offset " method. (c) Methods. — (1) Calculate the radius, R, for the given degree of curve, D. (2) Calculate the tangent distance, T, for the given radius, B, and angle of intersection, I. (3) Calculate the chord offset, d, and tangent offset, t, for the known radius, R, chord, c and degree, D. (4) At the given point intersection (P. I.), A, lay off the given angle, /, by the chord method. (5) Erom the P. I. lay off T along the two tangent lines and locate point tangent (P. T.) and point curve (P. G.), setting hubs at P. C. and P. T., with guard stake at each hub. (6) Run in the curve, by chord offsets, beginning at P. G. and checking at P. T. Calling P. PEOBLEMS. Passing am Obstacle Oct- 4,*!4 , [2 Hours) Chsr 3f7d W3rm- Tape ffo'iej, Uckffr /fo-3S, leir^fh ^ WO-OI • d/vjs/7 thref /ruts, Csef on true //he i>y transit), B ^Off/'t-frcnr A,3n'inj nearest C-Ifty and bisected CA at D snd CSatE- Chained DB- Then ca/cu/ated AB hy doubiihg fD- ^6o■yx^='^^i'4 ^an random line from A as c/ose as pract/cah/e to obstruct/'ojr so as to reduce , SF to a minimum • let fait perpendi'cuJar BF from B on random fine- iieasured Bf and fA to nearest O'lff- Calculat- ed hy pothenase AB ' AS' VS0-8^-tSl9-4^ = SZl-0 finally, after securing the above results, chained the actual distance AB- The three results are sumarized below Method- Obs-Dist. 5td-Cor. Red-Dist- By similar triangles By right triangle By actual measurement SZl-4 SZhO SZl-S -O-l -O-l -O-l SZl-3 SZO-9 5ZI-4 Total range = I tl040 - WITH Steel Tape . Chainmen, J-Poe ffnd ^'^off' pP\^BfHub) «A (Hub) 40 THE CHAIN AND TAPE. C. Station 0, establish Station 1 by laying off tangent offset, t, and chord, c Having one station on the curve, the next is located by prolonging the chord and forming an isosceles triangle having the chord offset as a base. Check on the P. T., noting the discrepancy of distance and line. Also establish the tangent again by tangent offset and observe the error of line. Follow the form. PROBLEM A23. DISCUSSION OF EREOES OF CHAINING (a) Equipment. — (No instrumental equipment, unless further data are desired, in which case Problems A6, A7 and A8 may be assigned again). (b) Prohlein. — Investigate the errors of linear measure- ment with the several kinds of chains and tape, with the view to determine practical working tests or coefficients of precision for actual use. (c) Methods. — Assume that the conditions in Problems A6, A7 and A8 are practically coniitant in the same problem, and that the actual differences between observed lengths of the several segments when chained in opposite dirc- tions, represent the normal errors with the particular chain and chainmen ; then tabulate: (1) the measured lengths of all possible segments of the chaining course, either from direct observation or by subtraction; (2) the actual errors or differences between the two results, giving signs; (3) the chaining ratios, I: d, and the decimal expressions of the same to six places; (4) the " coefficients of precision" for each case, calculated by formula, or more quickly, taken from the diagram in the chapter on errors of surveying ; (5) the mean decimal chaining ratio and its equivalent; and (6) the mean coefficient of precision. Follow the form. PEOBLEM A23. TESTING (OE ESTABLISHING) AN OF- FICIAL STANDARD OF LENGTH. (a) Equipment. — Standard tape (with certified length given), turnbuckle adjustments with bolts, spring balance, standard steel rule graduated to 0.01 inch, 2 thermometers, 2 microscopes, strips of wood, a watch. (b) Problem. — Make a series of ten observations with a standardized steel tape for the purpose of testing (or estab- lishing) an official standard of length, observing the near- est 0.0001 foot. (The Bureau of Standards, Washington, D. C, will standardize a tape for a small fee.) PEOBLEMS. 41 Location of Curve Dcf-e, '/4 ■ f3 tnurs) Clear an J ceo/- 100 ft- Steel Tape f/e-Bl/, IpeJier/fa-JS'/Hi'-ill ff/fen hub stA and a i/istant Ai/b B, l-a /ay off a h'ne A-C making an angle I of so' m'th BApro/onffed, and connect the tm lines mtli a ^O'ciirve, t/iatis, a curve having a central angle of ZO ' siiitended fy a W ft- chord, c- The radii/s was calculated thus : Since the chord of an arc Is tmce the sine of half the arc, chords ^rad-xsin-P 'chord . so ,a-F<, - rad-' __ ^ Calculated tangent distance thus .' In right triangle (O-P-C-Fl) Tan-Ast- - Mad-x tan -y / =ZS7'9xO-i39lff = Z4l'e Calculated chord offset d, and tangent off- set t, thus : By simitar triangles ^: c = '/''. ''•■%'=^,'34'-97,t'id=l7-^! (An approx- formula is d-I^D'3S;t-p'l7S) from A f/hlnt Intersection) laid off Tan-Pist- (T), locating /hint Curve Cf-C) and Piint Tangent (P-T-)- Began at PC- and ran ' in curve, asshoivn In sketch' Error of Closure at P-T- was O^'Z in line and. tl-'l ^ in distance- WITH Steel Ta Had- so- 000 34730 ISZ70 I3S91 1378 lie •A s '\ -A-if^ tn-9 0-I938-O ZiOii 164- ?S9 3 2'H-St Hd Chain, J-Doe- PE. P'r Chain , S-Koe- Axeman, B-f-Keen- flagman, fi-W-Sura- 'i-^/V Chd-Offset- 1/ moo 287-9 IIS2 211 202 9 Line A-B B-A A-C C-A A-D D-A A-e f-A B-C C-S B-P D-S B-f e-B C-D D-C c-e f-c D-£ E-D Direction Chained E- m e- w- £■ HI- E- w- E- W- E- W- E- W- E- W- E- W- E- Discu Observed Length Ft.. 41-f-SS 4l4--(l zm-79 3991-19 3991-74 S179-4I K79-S7 ISI9-II IS19-14 3SII2-II 3Sm3 4794-90 4794-96 1987-90 I987-S9 3Z7S-(9 327S-71 1217-79 I2S7-73 (L-ln m-ft- imits)\£, SSION Differ- ence, E f=t-. -0-03\ -0-06 -0-OS' -0.09 -0-03' -0.02 OF El.RORS Chaining Ratio l-.d ■f.f3< Hun ;-etT (Suilf. Coef' of Precision W,ft /■v l:i»QO a- 1:79130 i-.siseo i-irnms IU7S3S0 D-miii i-.niso i-tmos 1:01790 l:mi90 1:32920 1:41300 <>/■,«:■ ■j4^ 0-014 0-013 0-00/ 0-012 0-OOS 0-003 0-009 0-002 O-OII 0-OIS 0-OOS E n Oct-9, 14 - Computer, J -Doe • WITH Steel Tape. Pata from pp- Transcript 0-K- ABC D E O- ^ u o Distances by Subtraction* S-A 5279-37 e-B 4794-9G B-A 484-m A-C 2003-79 A-B S-C 1519-21 E-B 4794-96 E-B 1217-13 D-B }B07-I3 B-C 327S-72 E-D 1287-83 E-A 5279-57 EC 327S-72 C-A 2003-85 E-B ■4794-96 E-C 3275-72\ C-B ISI9-24i A-E 5279-4^ A-B 4£4-S8\ B-E 4794-90\ A-E S278-4l\ A-C 2003-79 C-E 3279-69 B-A 5279-57 E-D 1217-83 D-A 5991-74 A-D 3991-99 A-B 484-58 B-D 3507-11 A-D 3991-69 A-C 2003-79 C-D 1987-90 A-E 5279-48 A-D 3991-69 D-E 1217-79 D-C 1917-89 Designating Et and W- f4th Column) It Is seen that the returning results (except C-D) are greater- This Is explained by standard tape lengi-ha, vlz-f before =100-011, after '100-008, l-e- the tape gradually decreased In length, causing greater observed lengths- J 42 THE CHAIN AND TAPE. (c) Methods. — (If a new official standard is being estab- lished, one standard mark may be made permanent, and the precise distance taken to an approximate temporary point on the other bolt, the exact correction being applied after a sufficient number of results have been obtained. If the sun is shining, the tape should be protected by a wooden box or other covering throughout its length. Cloudy days or night time give best results. The observations should be made briskly so as to have slight range of temperature. Ccf/t! '14- Chiicly smf Cool- Test of 100-Ft- Standard Selec-f&d c/oi/c/y dsy ivifh s/i'ghf r3nge of Used Sfanc/srd Tspe Jio-417, msrkeii "US- 3t 62° F- mfh J2- lb- pull, tspe supporte\l, (-i'ltlt ^^5ffmz• 5 " 6 >• V 5^- <-/^ ^^. i>^ p?^i ::^^^ h J"* s \\ / ^ J _ ^ -ss ^ '. — ■ „..^ ^ -is=d L^ - ^ /V c k V^s; k^ 'S ^P l-Ko ^0,^ ^'^1 Maqnetic -^ •~~. s, s "-Sa ^^ ^f - Northern United States > ^^ 6' 5' 4' 3' Z' I' 0' I' t 5' 4' 5' 6' Fig. 11. (For additional data see bulletin of Department of Com- merce, U. S. Coast and Geodetic Survey, entitled " Principal Facts of the Earth's Magnetism.") MAGNETIC DECLINATION. 47 Variation of the Declination. — The declination of the needle is not a constant at any place. The change or fluctuation is called the variation of the declination. The variations of the magnetic needle are of several kinds: 48 THE COMPASS. secular, daily, annual, lunar, and irregular variations aueto magnetic storms. The most important of these is the secular variation which is illustrated in the upper diagram of Fig. 11 for a series of representative points in the United States. This diagram shows that the extreme range or swing of the needle is roughly 6° or 7°, and that the period of time between extreme positions is about a century and a half. Also that the wave of magnetic influence progresses across the continent alike in successive cycles. In 1900 the needle was at its extreme western position at Eastport, Me., and at its extreme eastern position at San Diego, Cal. The 3° East isogenic line passed through western Indiana, and was moving westward at the rate of about 4' per year. This rate of change was general throughout the central part of the United States, and is represented by the straight sections of the curve in the upper diagram of Fig. 11. The daily variation of the magnetic declination is shown graphically in the lower part of Fig. 11, the scale being greatly magnified laterally. It is seen that the needle un- dergoes each day a vibration similar in a general way to the grand swing of three centuries or so shown in the upper diagram. The magnitude of the daily movement in north- ern United States ranges from 5' in winter to nearly 12' in summer time. The needle is in its mean daily position between 10 and 11 a. m. for all seasons. The diagram rep- resents the normal magnetic day, of which there are per- haps five or six per month. Local Attraction. — The pointing of the needle is af- fected by the close proximity of magnetic substances, such 5_\ ^! a\ ^ PhleLevenhey,/_ J \ (C) Fig. 13. USE OF THE COMPASS. 49 as iron ore, wire fences, railroad rails, etc. However, local attraction does not prevent correct work, provided back and fore sights are taken withont change of magnetic condi- tions. It is therefore especially important to avoid disturb- ances of the needle by the chain, axe, passing vehicles, elec- tric wires, etc., or by articles on the person of the observer, such as keys, knife, spectacle frame, wire in the hat rim, reading glass case, etc. Also the glass cover may become electrified by friction and attract the needle, in which case it may be discharged with the moistened finger, or by breathing on it. The Vernier. — The vernier is an auxiliary scale used to read fractional parts of the divisions of the main scale or limb. Verniers are retrograde or direct, according as the divisions on the vernier are larger or smaller than those on the limb. The vernier used on compasses for the setting ofE of the declination is direct, and is usually of the type shown in (c) of Fig. 13. In reading a vernier of any kind, blunders may be avoided by first estimating the fraction by eye be- fore noting the matched lines on the two scales. USE OF THE COMPASS. TJse. — The compass is used: (1)" to determine the bear- ings of lines ; (2) to measure the angle formed by two lines ; (3) to retrace old lines. The bearing of a line is the hori- zontal angle between the line and a meridian through one end of it. Bearings are measured from the north or south point 90° each way. The angle between two lines is the difEerence in their directions as indicated by the bearings. Having the true bearings of one side of a polygon, the true bearings of the others may be obtained by algebraic addi- tion of the angles ; or by using the declination vernier so as to read the true bearing direct on the fore sights. Practical Hints. — Point the north end of the compass box along the line and read the north end of the needle. Protect the pivot from needless wear by turning the needle in about the proper direction before releasing it. Always lift the needle before disturbing the compass. Habitually obtain duplicate needle readings on each sighting. Kead the needle by estimation to the nearest five minutes, that is, to the one-sixth part of one-half degree, which is the usual subdivision of the compass box. Care should be taken to avoid parallax in reading the needle. 5 50 THE COMPASS. ADJUSTMENTS AND TESTS. Elementary Lines. — The elementary Mnes of the compass, shown in (a) of Fig. 10, are : (1) the line of sight; (3) the vertical axis; (3) the plate level lines. The maker should see: (1) that the needle is strongly- magnetized; (3) that the magnetic axis corresponds with the line joining the two ends; (3) that the metal in the compass box is non-magnetic; (4) that the line of sights passes through the center of graduation; (5) that the plates are perpendicular to the vertical axis; (6) that the zero of the vernier coincides with the line of sights. The needle may be magnetized with a bar magnet or by putting it into the magnetic field of a dynamo. The metal of the compass box may be tested by reading the needle, then moving the vernier and noting if the needle has moved the same amount, this process being repeated at intervals around the full circle. The Principle of Reversion. — In adjusting surveying instruments, the presence, direction and amount of the er- ror are made evident by the method of reversions which doubles the apparent error. If there is no difEerence after reversion, there is no error. Plate Levels. — To make the plane of the plate level lines perpendieular to the vertical axis. — Level up the instrument by means of the plate levels and reverse the compass box in azimuth, that is, turn it through a horizontal angle of 180°. Correct one-half the error, if any, by means of the adjusting screws at the end of the level tube, and bring the bubble to the center by the ball and socket joint. The rea- sons for this process are shown in (a) of Eig. 13. Sights. — To make the plane of sights normal to the plane of the plate level lines. — With one sight removed and the instrument leveled, range in with the remaining sight two points as far apart vertically as possible, say on the side of a building. Eeverse in azimuth and bring the bottom of the sight in range with the lower point ; if the upper point is then in range, the sight is in adjustment. If not, correct one-half the error by putting paper under one side, or by filing oif the other side. Repeat process for the other sight. The Pivot. — To adjust the pirot to the center of the gradu- ated eircle. — Set the south end of the needle to read zero, and read the north end of the needle ; reverse the compass box in azimuth, repeat the observations, and correct one- half the difEerence between the two readings of the north PEOBLEMS. 51 end of the needle by bending the pivot, using the special wrench for the purpose. Turn the compass box 90° and repeat. See (b), Fig. 13. The Needle. — To straighten the needle. — Having adjusted the pivot, set the north end of the needle to read zero and bend the needle so that the south end reads zero also. Turn the compass box and test for other graduations. PEOBLEMS WITH THE COMPASS. PEOBLEM Bl. DECLINATION OP THE MAGNETIC NEEDLE. (a) Equipment. — Surveyors' compass, flag pole, reading glass. (b) Problem. — At a point on the true meridian determine the mean magnetic declination with the surveyors' compass. (c) Methods. — (1) Set the compass over one point and a, flag pole at another on the true meridian. (3) Lower the needle and sight at the flag pole carefully with the north end of the compass box to the front. (3) When the vibra- DCCLII ATION Hcedic Mean Undiaq nims'e- IHiS'E- 3 mtuid^as IVesff thr most p. vhabJe jtaf/ffn OF Time PM- Z'OS Z=/l Z-/S Z:ZZ Z:Z7 2:31 Z--3S Z:42 Z:4S Z.-S4 IHeedl Mean P-M- for da. '/y var^ '9f/ci1 Asstf/tf. 'Tff tfiai fhe m^nef/c ^I't/ejis sre nt\m3/ fqr f/7f ibf cer. fy PIffi rvjff of Oa/Iy V, *rf3t/OA a^detf 'e-aa tbee JfJ'Jl' va/vg o. fftvei ■ . fifr thi r part/iff lar Mia/rff~ Z--3C fo ■Jiff vJj- / esf/m ^t/on WITH SUBVEVOI^ OcflZ.'m.fZHours) i/sffef 0ur/ey Ccmpi reeejtf/y rema^nfi i Sef- cffmpass on true Unafi'ff/l S/ffi^af f/aypo/^ef a dManoe ofZPfiFf- neetffe ty Cdue s/xt/j part carefuJfy avoi'di't ^ paraJ/ax magnetic d/sforbi mces f/me fo nearesf PJsfvri'et^ neei^Je pi'vofaatf vei yi/fien osc//fafm -s rereatf fJie neeiflff Cffaf/at/ttt^ fhe prt utive reatfmpSf ra/T^e ofaofm irf'es-f were oht^n7ed- 's Compass- 'Jearanif Coo/- ■sH^Ze-ff/eeaJe I'zeo^f ant/ ^Vatc/r- mer/t/jan mff? i/ec- toread zero • on n7er/i//an sf" •f and read fo Sm/niftes 'oj?e-/i3/F decree) f and Observed m/nt/te • iy ///^ft'ny if from *; t/ren had ceased ^r/f 'ed slff/7f/n^; on/// /en consec- 'lavlng 3 majcfmum ^re f/jan ten min— 62 THE COMPASS. tions of the needle have ceased, move the vernier by means of the tangent screw so that the north end of the needle reads zero, and check the sighting of the compass. (4) Read the declination on the vernier to the nea,rest minute. (5) Lift the needle, verify the zero needle reading and the sighting, read the vernier and record; repeat the process until ten satisfactory consecutive values of the declination are obtained. Observe the time of each reading to the near- est minute. (6) Correct the mean of the ten values for daily variation by reference to the diagram. Fig. U, using the mean time. Record and reduce the data as in the form. ( Note that the values in the form were obtained by estimat- ing the nearest five minutes. Which is better? Try both if time allows.) PROBLEM B3. ANGLES OF TRIANGLE WITH COMPASS. (a) Equipment. — Survej'ors' compass, two flag poles, reading glass. (b) rrobtcni. — Measure the angles of a given triangle with the surveyors' compass. (c) Methods. — (1) Set the compass over one of the vertices of the triangle and a flag pole behind each of the other two. (2) Lower the needle and sight at one of the flag poles care- fully, with the north end of the box to the front. (3) AVhen the vibrations have ceased, read the north end of the needle to the nearest five minutes by estimation. (4) Lift the needle, verify the sighting and also the reading. (5) Turn the compass box to the other point and determine the bear- ing, as before. The required angle is the difference between the two bearings. (6) Measure the other two angles in like manner. The error of closure mvist not exceed 5 minutes. Follow the form. PROBLEM B3. TRAVERSE OF FIELD WITH COMPASS. (a) Equipment. — Surveyors' compass, 2 flag poles, engi- neers' chain, set of chaining pins. (b) I'rohiem. — Determine the bearings of the sides of an assigned field with the surveyors' compass and measure the lengths of the sides with an engineers' chain. (c) Mcthod-i. — (1) Set the compass over one of the corners of the fielfl which is free from local attraction, and set off the declination with the vernier. (2) Take back sight on the last point to the left and fore sight to the next point PROBLEMS. 53 A Statfon S 8 6 NSLI5 Line S-6 S-g S-5 e-6 6-g 6-S OF Tl ObMrvEd Bearing 5-g3'js')V hsWe V49'm M9°M'£ ilAMSL Needle Angle 77'3S' S4'4S' 47'4S' z 5-6 -8 WITH Surveyors Observerfi, R-Roe 0cf/3//4-fZ//our£^ Used hurley Comf £sch bearing w& (fup/j'cafe, the / turbed 3/?d tJn hettveen read/. (P/screpency not i Compass ■ fed/e being d/'S" JSD'HS' 9 ejtceed S m/nufes^ \ ^x — r X *.5 a / >, 7RAVE RSE • FlEU A-B-C D-E WITH Compass At D Chain- station Line Observed! Inten'or Adjusted Distance Observers : J- Doe & 'K-^ae- Bearing Angle Bearing Ft- 0ct-J6, '14. f 3 Naurs) Clears Windy A A-E 5-6s'sm 03'15' Used Si/r/eyCompa ^s, locker if^Z4- A-S i-3Z'4S'f- iJ2'45i 33e-£ Made needJeread .. era wiief7 poinfing B B-A mi'fsh /SdW trae fiar/h hysej fin0 off declination B-C V43'/Si' 54i'K'B 4e4-e m'fJj vernier an o iciinai-iar7 arc oF C C-B f43'^f'n SS'fS' Jf3'36'F- C-D ss/'j5'n w'si'n 4n-3 Read bean'nffs tvi th a- E/?d af{ffjnpaff& D D-C fs/is'e m'ss' . , toward tiie fanv ird station and p-e vzr^m m'm 6J6-0 read H- End oF Heedls- £ E-P iZZ^S'i S7'S0' £-A IfSO'jf^ mWe 241.6 N A S4S'PS' E?rti/^efrjgg = Oisfance "Cosine Bearing ■ '^"^^■'ho/ A N Depdrfure (pnyech'ononEandW line) ~ Disiance '^ Sine Bearing ■ \Mendian Disfance of a point is itb E distance Eor Wofan assumed reference meridian ■ ' Meridian d/sfance of a line is fhe Compass % Merid-Disf of ifs middle poinh (c) Methods. — (1) Prepare forms for calculations; tran- scribe data, and carefully verify copy. (2) Compute lati- tudes and departures by contracted multiplication, preserv- ing results to the nearest 0.1 foot. (3) Make the same cal- culations by logarithms, as a check. (4) Determine the ac- tual linear error of closure. (5) Determine the permissible error of closure (see chapter on errors of surveying). (6) If consistent, distribute the errors in proportion to the sev- eral latitudes and departures, respectively, repeating the additions as a check. (7) Transcribe field notes and ad- justed latitudes and departures, and verify transcript. (8) Calculate the meridian distances of the several stations and lines. (9) Calculate the latitude coordinates. (10) Calcu- late the partial trapezoidal areas by multiplying the merid- ian distances of the lines by the respective latitudes, pre- serving consistent accuracy, and observing algebraic signs. (11) Determine the area by taking the algebraic sum of the partial areas. Reduce to acres, and correct for standard. PROBLEMS. 55 ComiIass T(iaver4e Observed Distance Line Adjusted Bearing AB CD DE ff/inW £A H-eaWf. T4I-S Ft- 3}e-s 464-6 4S3-3 OF Compdtation Multipli ■ cation (Lat-Oisi HTlesyi 616-0 Distribution oF Error Line Lat. Dep< AB . -- • '- BC CO ne -h^h -^^ Field Logar- ithms xCos-Bg] 2-52$9i 9-$248l U5ISQ (2S5-0I) 2-B6708 9-86355 4-£3Ji e/e-0 123Z 370 \-B-C-D oF Lati Computed Latitude Ft. S-ZS3.0 S-330.3 2-S3KI (33$3i) 2-6S4Z2 9-ISZ4S zisin (6S-6S} 2-7S9SS i-7S64} (nasi) Z-3!3I0 ■M3« M71S3 Erro tudes' /■ 6S-6 S-6. s- o-t - oF C £rror fO-S' Oct-n, '/4 Compt/fer, J-Poe- P3t3 rrom pp- Jrsnscripf 0-K- E Latitudes and Departures- Adjusted Latitude Ft- s-^g^-g Multipli- cation [Deji-^Dist 5- 61-6 itni-i lf-M-4 tl.6)C-S 3-190-S Computation of Departures_ Logar- itnme KSinBs) Z-5Z191 9-733IS 2-?eoie (m-M) 2-66701 ■S34tl S5 Z31-S8 J4SO 217 Computed Departure Ft- E-|g^■0 2-50IS4 (3ms) 2-11422 9-39S57 2-17979 (471-40) 2-7!9Sg 9-S7SI4 2-3(472 (231-59) 2-3g3IO 9-93934 2-32244 m47!-4 W-231-6 £-70S-S »7IO-0 W- OS Adjusted Departure Fh E-lgZ-0 -0-2 W-47g-2 W-23/-S e-7e9-7 0709-7 (See Distrain) Pu-misasUe £rnr= -^S, 1/1 nnn -^ * ^'' Ocf- 77,14 ■ Compufer, J-Ooe. lljtj frompp- Tr3nscr/pf 0-K- C-D-E, Compass Traverse. 56 THE COMPASS. Follow the form. (13) Make plat of field, using total rect- angular coordinates, and checking by polar planimeter. PROBLEM B5. ADJUSTMENT OF THE COMPASS. (a) Equipment. — Surveyors' compass, adjusting pin, small screw driver. (b) Prolilem. — Make the necessary tests and adjustments of the surveyors' compass. (c) Methods. — Observe the following program: (1) test the magnetism of the needle; (2) test the metal of the compass box; (3) test and adjust the plate levels; (4) test the sights; (5) test the pivot; (6) test the needle. PROBLEM B6. COMPARISON OF DIFFERENT MAKES AND TYPES OP COMPASSES. (a) Equipment. — Department equipment, catalogs of rep- resentative makers of compasses. (b) Prohlem. — Make a critical comparison of the several types of compasses. (c) Methods. — Examine the department equipment and study the several catalogs carefully, noting the character- istic features, prices, etc. The following items, at least, should be included in the tabulated report : name of instru- ment, length of needle, length of alidade, vernier, tripod, weight, price, etc. CHAPTER IV. THE LEVEL. Description. — The engineers' level consists of a line of sight attached to a bubble vial and a vertical axis. Two types of level, the wye and dumpy, Fig. 14, are used by engi- neers. In the former the telescope rests in Y-shaped sup- ports, from which it may be removed. In the dumpy level the telescope is fixed. The dumpy is a favorite with IJritish Engineers' Wye Level. Fig. 14. Dumpy Level. Fig. 15. — Types of Levels. 57 58 THE LEVEL. and the wye level with American engineers. (The dumpy level with erecting eye-piece has been adopted as standard by the Division of Valuation, Interstate Commerce Com- mission.) The two types differ chiefly in the methods of adjustment. A third type, not shown in the cuts, is called the level of precision because of its use solely for work of extreme refinement. In Fig. 15 are shown: (a) an architects' or builders' level of the wye type; (b) a road builders' level of the dumpy type; (c) a reconnaissance level with a decimal scale for reading horizontal distances direct; (d) a water level some- times used in locating contours; (e) a Locke hand level; (f) a clinometer; (g) a binocular hand level. THE TELESCOPE. Principles. — The telescope used in the engineers' level and transit, shown in section in Figs. 16 and 23, consists of an objective or ohject glass which collects the light and forms an image in the plane of the cross-hairs, and an ocular or eyepiece which magnifies the image and cross-hairs. The cross-hairs are thus at the common focus of the oujective and eyepiece. The principle of this type of telescope, both optically and mechanically, may be illustrated by the photo- graphic camera if cross lines be ruled on the ground glass focusing plate and a microscope be used in viewing the image formed by the lens. Telescopes of the above class are called measuring telescopes, while those of the opera glass type are termed seeing telescopes. The latter have no real image formed between the object glass and eyepiece. Line of Colliniation. — The telescope of the level or tran- sit may be represented by a line, called the line of collima- tion, which joins the optical center of the objective and the intersection of the cross-hairs. The optical center is a point such that a ray of light passing through it emerges from the lens parallel to its original direction. The line of coUi- mation is independent of the eyepiece. Objective. — The objective is a double convex or plano- convex lens. In all good telescopes the objective is com- pound, that is, made up of two lenses, with the view to cor- rect two serious optical defects to which a simple lens is subject. These defects are called chromatic aberration and spherical aberration. Chromatic aberration is the separation, by the objective, of white light into its component colors. A lens which is Tangent Line of level Tube Optical Center : oF Objective Intersection of Cross Hairs^f (a) ObjeclCkss (forms imaqe in plane oFeross-hairs) Vertical Axisf. Clip^....,^ ..'rising Tangent to Bubble /Azimuth 5crew5 1^ I eye I Bar- (b) i''^l?imj5 fgual -'■""<{ Line oF 'WoliimationMxis oFW&derJ Bottom \ElementoFFin(f3 i Tangent Y ~VtoBubbIe "T l W Eyepiece'r (MagniFiesimaqe and cross-hairs) -""Vertical Axis Clip^-:.^jrl?inff WyeMs '^Altitude Screirs footScrem ^;_Bj.q^J ^«i- L>L (C) \ I ,_ True Line of Collimalion True Level Line from Target i- — Length oFBack Sight egaa/j True Level Line Through ' Bottom\FkmentoF things jf ^ Tanaeni:V ~\"to Bubble 1 f (d) j SI _ TrueJJnejf Collima2ion_ l^ toTargetiSase oFCone) ^■fo Length of foresight -— S\ Top oF Peg. (e) Correct Levels by Equal Sights. ^ True Line oFCollimation hi He t hod. 1 True Line oFCollimation. True Level line fnd-^ ^indfleiho'd. 60 THE LEVEL. free from this defect is called achromatic. A telescope is tested for the chromatic defect by focusing on a bright ob- ject, such as a piece of paper with the sun shining on it, and noting the colors on the edge of the object and es- pecially at the edge of the field of view as the focus is slightly deranged. Yellow and purple are the characteris- tic colors indicating good qualities in the lens. Spherical aberration is a defect which prevail? to a serious extent in a simple lens having spherical surfaces. It is due to a difference in the focal distance for different concentric or annular spaces of the objective, so that the plane of focus for rays passing through the outer edges of the lens is dif- ferent from that of the middle portion. A telescope is tested for this defect by focusing on a well defined object, such as a printed page, with the raj's of light cut off alter- nately from the middle and the edge of the lens. This is best done by means of a circular piece of paper with a small round hole in it. As a rule, the object glass in good levels and transits con- sists of a double convex lens of crown glass fitted to a con- cavo-convex or a plano-concave lens of flint glass, the former to the front. The defects described above are avoided through the different dispersive and refractive powers of the two kinds of glass, and by grinding the sur- faces of the two lenses to the proper curvatures. Eyepiece. — As in the camera, the image formed by the objective is inverted, so that if a simple microscope be used as an eyepiece, the observer sees objects inverted. Such an eyepiece is commonly used on the dumpy level, as shown in rig. 14. This form of eyepiece consists of two plano- convex lenses with their convex sides facing each other. The form of eyepiece most used in American instruments is the erecting eyepiece in which two plano-convex lenses re- place each of the two in the simpler form. The erecting eyepiece is much longer than the simple one, as may be seen at a glance in Fig. 14. While the simple eyepiece causes a little confusion at first, owing to the inversion of objects, it is much siiperior to the erecting eyepiece in the matter of clearness and illumination. The chief inherent defect in the eyepiece is a lade of flatness of the field. A single lens usually causes a distor- tion or curving of straight lines in the image, especially to- wards the edge of the field. A telescope is tested for this defect by observing a series of parallel right lines, prefer- THE TELESCOPE. 61 ably a series of concentric squares, which fill the entire field of view. In the best achromatic eyepieces, one or more of the sep- arate lenses may be compounded, the curvatures being sucli as to eliminate the color defect and give rectilinear qualities to the lens or combination of lenses. Definition. — The definition of a telescope depends upon the finish and also the accuracy of the grinding of the curved surfaces of the lenses. It may be tested by reading the time on a watch or a finely printed page at some dis- tance from the instrument. Illumination. — Illumination and definition are apt to be confused. Poor definition causes indefinite details, while poor illumination causes faintness in the image. The latter may be tested about dusk, or in a room which can be grad- ually darkened, and can be best appreciated if two tele- scopes of different illuminating qualities be compared. Aperture of Objective. — The aperture or effective di- ameter of the objective is determined by moving the end of a pencil slowly into the field and noting the point where it first appears to the eye when held say 8 or 10 inches back from the eyepiece. The process should be repeated in the reverse order. The annular space is deducted from the actual diameter to obtain the real aperture. Size of Field. — The field of the telescope is determined by noting the angle between the extreme rays of light which enter the effective aperture of the objective. With the tran- sit telescope, the limiting points may be marked on the side of a building and the angle measured directly with the plates ; or with either level or transit the angle may be cal- culated from the measured spread in a given distance. For simplicity, a distance of 57.3 feet may be taken, and the re- sult reduced to minutes. Magnifying Power. — The magnifying power of a tele- scope is expressed in diameters, or as the multiplication of linear dimension. It is determined most readily by making an observation with both eyes open, one looking through the telescope and the other by natural vision. The com- parison may be made by means of a leveling rod, or the courses of brick or weather-boarding on the side of a house may be used in like manner. Parallax. — Parallax is the apparent movement of the cross-hairs on the object with a slight movement of the ej'e, and is due to imperfect focusing of the eyepiece on the cross-hairs before focusing the objective. The eyepiece 62 THE LEVEL. should be focused tritli the eye normal, the cross-hairs being illuminated by holding the note book page or other white object a few inches in front of the objective. (/) (2) e© (5) f4) (b) Fig. 17. Cross-Hairs. — The cross-hairs are attached to a ring or reticule ■n'hich is held by two pairs of capstan headed screws. The hairs usually consist of spider lines, although some makers use platinum wires for the purpose. To re- move the reticule the eyepiece is taken out, one pair of screws is removed and a sharpened stick is inserted in a screw hole. The best spider lines are obtained from the spider's e.^^ nest. In Fig, 17, (a) shows the usual arrangement of the cross- hair ring and the method of attaching the hairs ; (b) shows the number and positions of hairs used, (1) being the most common, (2) the form for stadia work with the transit and also for estimating the lengths of sights with the level, (3) a form used by some makers with the level, and (4) a style found in English levels ; (c) shows the e^^ pod or case of the large brown spider (about half size) which yields the best lines for engineering instruments; (d) illustrates a convenient vest pocket outfit for replacing cross-hairs in the field, consisting of a supply of spider lines and some adhesive paper (bank note repair paper) each in a capsule or tin tube, and several sharpened sticks for stretching the hairs. Cross-hairs stretched in this manner may last indefi- nitely, or they may be fastened on permanently with shel- lac at the first opportunity. THE BUBBLE VIAL. Principle. — The spirit level consists of a sealed glass tube nearly filled ^^■ith ether or other liquid, and bent or ground so that the action of gravity on the liquid may indi- THE BUBBLE VIAL. 63 cate a level line by means of the bubble. The delicacy of the buble depends upon the radius of the curvature in a verti- cal plane, the greater the radius the more delicate the level. Thus, for example, a perfectly straight tube could not be used as a level. Curvature of Bubble Vials. — Good bubble vials are now- made by grinding or polishing the interior surface of a se- lected glass tvibe by revolution, as indicated in exaggerated form at (a) Pig. 18. As a general rule, only one side of the vial is actually used, it being customary to encase it in Tophnqent_ Line_ _ J Axh ofLevelTabe_ i\ b \\ (9> \ i r-4d tfsecfienk) Fig. 18. a brass tube having a slot or race on one side. However, both sides of the vial may be utilized, as in (b) and (c), Fig. 18, which show the reversion level adapted to the tran- sit and wye level, respectively. Bubble vials of several sizes are shown in (d), Fig. 18. It was formerly customary to grind out only a portion of the upper side of the glass tube, as shown at (e). The cheap vial, consisting merely of a bent tube, used mostly in carpenters' and masons' levels, is 64 THE LEVEL. shown at (f) ; and a method of increasing the precision of the bent tube by tilting it is indicated at (g), Fig. 18. Delicacy. — The delicacy of the bubble vial is designated either by the radius, usually in feet, or by the central angle in seconds corresponding to one division or one inch of the bubble scale. Two methods are employed to determine the delicacy of level vials, (1) by the optical method, as at (h), Fig. 18, where the radius is calculated from an observed tar- get movement at a given distance for an observed bubble movement, the two triangles being similar; and (2) by the level tester, as at (i), by means of which the angular move- ment is read from the micrometer head for a given move- ment of the bubble. The engineer usually employs the radial designation, while the maker expresses the delicacy in an- gular units. As shown at (h) and (i),Pig. 18, the radius in feet is equal to 17,189 divided by seconds per inch of bubble. Bubble Line. — The relations of the bubble to the other parts of the instrument are best understood by representing the vial by a line. This line may be either the axis of the surface of revolution in (a). Fig. 18, or to provide for either of the three forms of vial shown, it may be taken as the tangent line at the middle or top point. This tangent line will be meant hereafter in referring to the bubble line. LEVELING EODS. Types. — There are two classes or types of leveling rods ; (1) target rods, having, a sliding target which is brought into the line of sight by signals from the leveler ; and (2) aclf -reading or speaking rods which are read directly by the leveler. In Fig. 19, (a) is the Philadelphia rod ; (b) the New York rod; and (c) the Boston rod. The first is either a target or self-reading rod ; the second is a target rod, but may be read from the instrument when the rod is " short " ; the Boston rod is strictly a target rod. The Philadelphia rod is perhaps the favorite for most purposes, and the Boston rod is used least. A folding self-reading rod is shown at (d). Fig. 19 ; (e) is a woven pocket device which may be tacked to a strip of wood and used as a leveling rod; (f) is a rail- road contouring rod with an adjustable base ; (g) is a plain rod graduated to feet, for use with the water level. Targets. — The targets shown on the Philadelphia and Xew York rods, (a) and (b). Fig. 19, are called quadrant targets. That on the Boston rod, (c), is a modified form of USE OF THE LEVEL. 65 ^ 2 6. 4 do 6, ± .2. 4 6. PC 4 a pzi 4 6^ lS, 4 6. . .a. 4 6 a r/) D Pig. 19. the diamond target. A special form, called the corner tar- get, is bent to fit two sides of the rod to assist in plumb- ing it, and another target has two parallel planes for the same purpose. A detachable rod level is shown at (h). The target on rod (b), with the zero of the vernier 0.09 foot below the center of the target, frequently causes blunders. USE OF THE LEVEL. Use. — The engineers' level is used: (1) to determine dif- ferences of elevation; (2) to make profile surveys; (3) to locate contours; (4) to establish grade lines; (5) to cross section; (6) to run lines. 66 THE LEVEL. Differential Leveling. — Differential leveling consists of finding the difference of elevation between two or more points. In the simplest case the difference of elevation be- tween two points may be found from a single setting of the level, the leveling rod being used to determine the vertical distance from the plane of the instrument to each of the two points, and the difference between the rod read- ings taken. When the distance between the two points is too great, either vertically or horizontally, or both, to ad- mit of this simple process, two or more settings of the level are taken so as to secure a connected series of rod read- ings, the algebraic sum of which gives the desired differ- ence of elevation. This difference may be expressed either by the numerical result of the algebraic sum of the rod readings, or by assuming an elevation for the beginning point and calculating the elevation of the closing point by means of the observed rod readings. A haelc sight is a rod reading taken to determine the height of the instrument. A fore sight is a rod reading taken to de- termine the height of a point. A hench mark is a point se- lected or established for permanent reference in leveling operations. A turning point is a temporary reference point used in moving the instrument ahead to a new setting. The same point is often both a turning point and bench mark. The datum is the plane or surface of reference from which the elevations are reckoned ; it may be sea level, or an arbi- trary local datum. A level line is a line parallel to the sur- face of a smooth body of water. A horizontal line is tangent to a level line at any point. The curvature varies as the square of the distance from the point of tangeney, and is 0.001 foot in 304 feet, or 8 inches in one mile. In Fig. 19, (i) shows a metal and also a wooden peg com- monly used for turning points. Several forms of bench marks are shown in Fig. 19 ; ( j) is a mark on the corner of a stone water-table ; (k) a rivet leaded into a hole drilled in a stone slab ; (1) a railroad spike driven into a wooden post or telegraph pole ; (m) a projection cut on the root of a tree, preferably with a spike driven vertically into the top of the bench, and usually with a blaze above marked " B. M. No. — ." All bench marks and also turning points should be clearly described in the notes. Fig. 19a shows the essential details of differential level- ing. In practice the calculations are made mentally. Two chief essentials in correct differential leveling are : (1) that the 'bu'bl)le lie in exactly the same position (usu- USE OF THE LEVEL. 67 ally the middle) on hoth hack and fore sight; and (2) that the length of hack sight and fore sight, horizontally, shall be balanced. It is seen at (e), Fig. 16, that with the bubble always in the middle, the line of collimation generates a horizontal plane when in perfect adjustment, but a cone with axis vertical when out of adjustment; so that in tak- ing equal distances in the opposite directions, the base of the cone is used, this base being parallel to the true colli- ejf.l. level Line from B.M.Iio.I 5ta. B.5. H,l. F5. Elev. Di5t. Calculations Description of 6.(1.5 and 05. B.ni. mo.oo s.in 100.00 mi t4U B.i. CilylbtmmBo/i.H.mciin, mtertible, litM.BankBld'q Ad) ^■442 lOUl HO 10447 /f/ -1.16 F.5. ei -1.16 103.26 340 10320 01 H37.B.5. 107.63 /^Z Peq, KEcanJ. 6reen'3 lot. m) t4V ior.B m \ -3.551:5. 01 -}.U 104.01 1300 104.08 ez fi.9l B.5. 5i'c/em/k,F. qatepost.J.Doe. m) ■fISI lOSJ} ml 105.93 H3 -2.47) F.5. mi -Z.40 I03JS 300 103.59 mi n.W.bolt, (nickeJ)H3terplaa +10.70 ■ -7.11 100.00 940 S.E. car. Hiqh and East Sts . (Bal3ifcedB.5.andF.5 0/st. -7.11 <^'J-- ► t3.!9 \\ + -i.i3 -«-'";' m 0,ecJtedF/er.iyI(B.5.,F5.IJ Fig. 19a. — Details of Differential Leveling. mation plane. In the best leveling practice the instrument is adjusted as perfectly as possible and then used so that the residual errors balance each other. The three common styles of leveling rods may be read to 0.001 foot by vernier or by estimation on a scale to 0.005 foot. However, for most kinds of leveling, it is an absurd refinement to read the rod closer than 0.01 foot, especially with the usual maximum length of sight of 350 to 400 feet, and with the more or less sluggish bubbles supplied in the general run of leveling instruments. Furthermore, the horizontal hair usually covers 0.01 foot or so of the target at the maximum length of sight, that is, the target can move that amount without being noticed by the observer. 68 THE LEVEL. Profile Leveling. — Profile leveling consists of finding the relative elevations of a series of representative points along a surveyed line, for the purpose of constructing a pro- file or vertical section. The skeleton of profile leveling, that is, the precise bench marks and turning points with the successive heights of instrument, is identical with differen- tial leveling, already described. Having determined the height of instrument by taking a back sight on a bench mark of known or assumed, elevation, rod readings are taken at proper intervals along the measured and staked line. These readings are fore sights, but they are usually termed intermediate siplits to distinguish them from the more precise rod readings taken on turning points and bench marks. On railroad surveys intermediate sights are taken usually to the nearest 0.1 foot on the ground ; but in other cases, such as tile and sewer surveys, intermediates are often read to the nearest 0.01 foot on small pegs driven beside the station stakes flush with the surface of the ground. In railroad work, the benches, turning points, and intermediates of special importance are commonly read to 0.01 foot, although some engineers persist in the ques- tionable practice of taking the nearest 0.001. In drainage surveys the nearest 0.01 foot is usually taken on bench marks, although more carefully than on the intermediate peg points, and the nearest 0.1 foot is read on ground points. The errors of profile leveling are balanced on turning points by equal back and fore sights, as in differential lev- eling. If the instrument is seriously out of adjustment, an error is made in the case of odd bench marks with unbal- anced sights, and also on all intermediate sights. However, the error is usually unimportant when ground readings are taken to the nearest 0.1 foot. In important leveling, such as canal and drainage work, it is customary to run a line of check levels to prove benches, before construction begins. The profile is plotted to an exaggerated scale vertically on a special paper, called profile paper. Three kinds, known as plates A, B and C, are in general use. The most common is plate A, which is ruled in ^4"iiich squares with a further subdivision to %o inch vertically. In railroad profiles the scales most used are 400 feet to the inch horizontally and 20 feet vertically. A still greater exaggeration is generally used in drainage profiles. Reciprocal Leveling. — The application of differential leveling to the determination of the difference of elevation between two bench marks separated by a wide river or gorge USE OF THE LEVEL. 69 is termed reciprocal leveling. A setting of the level is taken on each side of the river, and the mean of the two re- sults is taken. The necessary unbalancing of distances in one setting is balanced \ip in the other. Each back or fore sight should be the mean of a series of careful observations. In best practice, simultaneous readings are taken with two levels. . Contour Leveling. — Contour leveling is an application of the methods of profile leveling to the location of contour lines, that is, lines having the same elevation. Two methods are employed: either (1) actually establishing points on the adopted contour planes on the ground and then locat- ing these points; or (2) taking random elevations at rep- resentative points and interpolating the contour lines from the plotted data. The latter is the more common. Tlie chief ptirpose of contour leveling is to make a contour map, and the process is essentially a part of topographic survey- ing, where it will be more fully considered. Grade Lines. — The establishment of grade lines is usu- ally the concluding part of profile leveling. After making the profile, the grade line is established by stretching a fine thread through the ruling points, taking into account the controlling conditions, such as maximum gradient or earth- work quantities on a railroad profile, the carrying capacity or the scour in the case of a ditch, etc. After laying the grade line on the profile, notes are made of the data, and the actual grade line is established. Two methods are used : (1) the height of instrument is determined as usual, and stakes are driven at measured intervals with their tops to match calculated rod readings; and (2) a limited number of ruling points are established by the first method or otherwise, and the remaining stakes are " shot in " by con- structing a line parallel to the ruling line used. The latter is more rapid, since a constant rod reading is used ; how- ever, the method is unreliable unless the foresight be checked frequently on a fixed target. Cross-Sectioning. — Cross-sectioning consists of staking out the limits of the transverse section of an excavation or embankment for the purpose of construction, and usually includes the collection of data for the calculation of the quantities. This may be done either with the engineers' level, rod and tape line, or with special rods called cross- section rods. The notes are taken as rectangular coordi- nates, usually with reference to the center of the finished 70 THE LEVEL. roadbed. The slope stakes are set where the side slope lines pierce the surface of the ground. Running Lines. — Lines are sometimes run with the en- gineers' level, provision being made in most good levels for the attachment of a plumb bob. A line may be prolonged by sighting in two points ahead. A clamp and tangent movement are necessary. Some builders' levels have a needle and also a roughly divided horizontal circle for use in staking out buildings. Practical Hints. — The following practical suggestions apply more or less directly to all kinds of leveling, and also in a general sense to transit work. Speed. — Cultivate the habit of briskness in all the de- tails of the work. While undue haste lowers the standard of the results, an effort should be made to gain speed steadily without sacrificing precision. Gain time for the more important details by moving rapidly from point to point. On rapid surveys both leveler and rodman often move in a trot. Neither rodman nor leveler should delay the other needlessly. Care of Instruments. — Do not carry the level on the shoul- der in climbing fences. Clamp the telescope slightly when hanging down Keep the tripod legs at the proper tight- ness, and avoid looseness in the tripod shoes. Avoid undue exposure to the elements, and guard the level from injury. Do not leave the instrument standing on the tripod in a room over night. Setting Up — In choosing a place to set the level up, con- sider visibility and elevation of back point and probable fore sight. Set up with plates about level. On side-hill ground place one leg up hill. In general, place two tripod shoes parallel to the general line of the levels. Leveling Up. — A pair of foot screws should be shifted to the general direction of the back or fore sight before level- ing up. Set the foot screws up just to a snug bearing and no tighter. If either pair of screws binds, loosen the other pair a little The bubble moves with the left thumb. Level up more precisely in the direction of the sight than trans- verse to it, but do not neglect the latter. Inspect the bubble squarely to avoid parallax, and also to prevent such blun- ders as reading the bubble iive spaces off center. Observations. — Adjust the eyepiece for parallax with the eye unstrained. It is much easier on the eye to observe with both eyes open. Read at the intersection of the cross- hairs, since the horizontal hair may be inclined. Set the USE OF THE LEVEL. 71 target approximately, check the bubble, and repeat the proc- ess several times before approving the sight. Be certain that the bubble is exactly in the middle at the instant of approving the target. If the level has horizontal stadia lines, beware of reading the wrong hair (the reticule may be rotated one-quarter so as to have the extra hairs vertical, or a filament may be attached to the middle horizontal hair to assist in identifying it) . Avoid disturbance of the tripod by stepping about the instrument. Assist the rodman in plumbing the rod. Let signals be perfectly definite both as to direction and amount, using the left hand for " up " and the right for " down," or vice versa. The leveler can work much more intelligently if he knows the space covered on the rod by one division of the bubble scale at the maximum length of sight, and also the space on the rod hidden by the cross hair. Adjustments. — Keep the instrument in good adjustment and then use it as though it were out of adjustment. Balancing Sights. — Balance the length of back sight and fore sight, and record the approximate distances. The dis- tances in the two directions may be made equal roughly by equality of focus, but it is better on careful work to pace the distances or determine them by means of the stadia lines in the level. If necessary to unbalance the sights, they should be balanced up at the first opportunity, and in general they should be in balance when closing on import- ant benches. When leveling up or down steep slopes, fol- low a zigzag course to avoid short sights. Take no sights longer than 350 or 400 feet. Leveling Rod. — The rod should be carefully plumbed, to accomplish which the rodman should stand squarely behind the rod and support it symmetrically between the tips of the extended fingers of the two hands. In precise work wave the rod to and fro towards the observer and take the minimum reading of the target. With " short " rods avoid the somewhat common blunder of 0.09 foot when the vernier slot is below the center of the target. With " long " rods, see that the target has not slipped from its true set- ting before reading the rod. Read the rod at least twice, and avoid blunders of 1 foot, 0.1 foot, etc. Careless rodinen sometimes invert the rod. Each rod reading on turning' points and bench marks should, when practicable, be read independently by both rodman and leveler. Bench Marks and Turning Points. — Wooden pegs or other substantial points shoiild be used to turn the instrument 72 THE LEVEL. on. Select bench marks with reference to ease of identifica- tion, the balancing of sights, freedom from disturbance, etc. As a rule, each bench mark should be used as a turning point so that the final closure of the circuit may prove the bench. Mark the benches and turning points and describe them in the notes so plainly that a stranger may readily find them. Green rodmen sometimes hammer at turning point pegs with the rod. When leveling near a still body of water, its surface may be used to save time and check the work. Record and Calculations. — Describe bench marks and turn- ing points clearly. It is good practice to apply algebraic signs to the back and fore sight rod readings. The eleva- tions should be calculated as fast as the rod readings are taken, and calculations on turning points should be made independently by leveler and rodman, and results compared at each point. The rodman may keep turning point notes in the form of a single column. The calculations should be further verified by adding up the columns of back sights and fore sights for each circuit, or page, or day's work, and the algebraic sum of the two compared with the difference between the initial and last calculated elevation. Error of Closure. — A circuit of levels run with a good level by careful men, observing all the foregoing pre- cautions, should check within 0.05 foot into the square root of the length of the circuit in miles (equivalent to 0.007 foot into the square root of the length of the circuit in 100-foot stations). In closing a circuit, the error should be care- fully determined, as above indicated, and the value of the coefficient of precision found. (See discussion of errors of leveling and precision diagrams in Chapter IX, Errors of Surveying.) ADJUSTMENT OE THE WYE LEVEL. Elementary Lines. — The principal elementary lines of the wje level, as shown in Fig. 16, are: (1) the line of col- limation ; (2) the bubble line; (3) the vertical axis. For the purpose of adjustment there should be added to these : (4) the axis of the rings; (5) the bottom element of the rings. The following relations should exist between these lines ; (a) the line of collimation and bubble line should be parallel ; (b) the bubble line should be perpendicular to the vertical axis. The first of these relations involves two steps, viz., (1) to make the bubble line parallel to the bot- ADJUSTMENT OF WYE LEVEL. 73 torn element of the rings, and (2) to make the line of col- limation coincide with the axis of the rings. The other relation involves the wye adjustment, and is similar to the plate level adjustment described in the chapter on the com- pass. Bubble. — To make the 'bubble line parallel to the bottom element of the rings. — Two steps are involved, (a) to place the bubble line in the same plane with the bottom element, and (b) to make the two lines parallel. Azimuth Screws. — To make the bubble line in the same plane with the bottom element of the rings. — Clamp the level over a pair of foot screws, loosen the wye clips, and level up ; rotate the telescope through a small angle, and if the bubble mov^s away from the middle, bring it back by means of the aximuth adjusting screws. Test by rotat- ing in the opposite direction. Leave the screws snug. Altitude Screws. — To make the bubble line and the bottom element of the rings parallel. — Jlake the element level with the foot screws and bring the bubble to the middle by means of the altitude adjusting screws. The element is made level by the method of reversions as follows : With the level clamped over a pair of foot screws, as above, lift the clips and level up precisely ; cautiously lift the tele- scope out of the wyes, turn it end for end, and very gently replace it in the wyes ; if the bubble moves, bring it half way back by means of the foot scretvs. Before disturbing adjusting screws make several reversals, and conclude the adjustment with screws snug. This end for end reversal is similar to that made with the carpenter's level, the straight edge of the level corresponding to the element of the rings. The lines involved are shown in Fig. 16. Line of CoUimation. — To make the line of collimation co- incide with the axis of the rings. — Loosen clips, sight on a point, say a nail head or the level target, more distant than the longest sight used in leveling; rotate the telescope half way and note the movement of the hair, if any. The line of collimation generates a cone, the axis of which is that of the rings, and the apex of which is at the optical center of the objective. Correct one-half the observed error by means of the capstan headed screws which hold the cross- hair ring. Gradually perfect the adjustment until the in- tersection of the cross-hairs remains fixed on the same point when reversed by rotation with reference to either hair. The adjustment should be concluded with the screws at a snug bearing. 74 THE LEVEL. After collimating the instrument for a long distance, the adjustment should be checked for a short distance, say 50 or 100 feet, so as to test the motion of the optical center of the objective. Bings. — The theory of the wye level demands perfect equality of the rings, that is, the parallelism of the axis and element, as in (c), Fig 16. Should the rings be unequal, either from poor workmanship or uneven wear in service, they form a cone instead of a cylinder, and the axis is not parallel to the element, as in (d), Fig. 16. Under the latter conditions, the principle of the wye level fails, and an in- dependent test is demanded. This is known as the two-peg test, the details of which are shown in (e) and (f). Fig. 16, and described in the adjustments of the dumpy level. If, after making the wye level adjustments above described, the two-peg test shows that the line of collimation and bubble line are not parallel, the rings are probably unequal and the instrument should thereafter be adjusted as a dumpy level. However, hasty conclusions should be guarded against. In case the instrument has a reversion level, shown at (c), Fig. 18, the equality of the rings may be tested by first adjusting the top tangent line of the bubble vial par- allel to the bottom element of the rings, and then after ro- tating the telescope half way round in the wyes, compare the bottom (now above) tangent line of the vial with the top (now below) element of the rings, all by the end for end reversion. However, the exact parallelism of the top and bottom tangent lines of the reversion level should first be proven by the two-peg method. Wyes. — To make bttihle line perpendicular to the vertical axis. — Make the vertical axis vertical and bring the bubble to the middle by means of the wye nuts. The vertical axis is made vertical by reversion thus : With clips pinned, level up ; reverse over the same pair of screws, and bring the bubble half way back with the foot screws. When adjusted, the bubble will remain in the middle during a complete rev- olution. This adjustment is identical in principle with the plate level adjustment of the compass and transit, illus- trated in (a). Pig. 13. The wye adjustment should follow the adjustment of the bubble line parallel to the element of the rings. The wye adjustment is a convenience, not a necessity. Centering the Eyepiece. — After collimating the level, the cross-hairs should appear in the center of the field. ADJUSTMENT OF DUMPY LEVEL. 75 The eyepiece is centered by moving its ring held by four screws. This adjustment is desirable, but not essential. ADJUSTMENT OP THE DUMPY LEVEL. Elementary Lines. — The principal elementary lines of the dumpy level are identical vvith those of the wye level (1) the line of coUimation; (2) the bubble line; (3) the vertical axis. As in the wye level, the bubble line should be (1) perpendicular to the vertical axis, and (2) parallel to the line of coUimation. However, owing to the difference in the construction of the two types of instrument, the auxiliary elementary lines are not recognized in the dumpy level. The transit with its attached level is identical in principle with the dumpy level. Bubble. — To make the iuhhle line perpendicular to the vertical axis. — Make the vertical axis vertical ty the method of reversions, and adjust the Jtuhhle to the middle. This adjustment is identical in principle with the plate level adjustment, shown in (a). Fig. 13. The bubble should re- main in the middle through a complete revolution. Line of CoUimation. — To make the line of coUimation parallel to the iuiile line. — Construct a level line, and ad- just the cross-hairs to agree with it. The level line is de- termined either by using the surface of a pond of water, or by driving two pegs at equal distances in opposite directions from the instrument, and taking careful rod readings on them with the bubble precisely in the middle, as shown at (e). Fig. 16. For simplicity, the two pegs may be driven to the same level, or two spikes may be driven at the same level in the sides of two fence posts, say 400 feet apart. Otherwise, determine the precise difference of elevation, as indicated in (e). Fig. 16. Then set the level almost over one of the pegs, level up, and as in the first method of (f). Fig. 16, set the target of the leveling rod at the line of col- limation, as indicated by the center of the object glass or eyepiece (this can be done more precisely than most levels will set the target at 400 feet distance) ; now with the rod on the other peg, sight at the target (shifted to allow for the difference if the two pegs are not on the same level) ; adjust the cross-hair to the level line so constructed. If preferred, the second method shown in (f). Fig. 16, mgy be used ; the level is set back of one peg, rod readings are taken on both pegs, allowance made for the difference in level of the two pegs, if any, the inclination of the line of 76 THE LEVEL. collimation determined, correction made for the small triangle from the level to the first peg, and finally the level line constructed by means of the calculated rod readings. The second method is simplified and made practically equivalent to the first by setting the level at minimum focusing distance from the first peg. The small corrective triangle is thus practically eliminated. Strictly speaking the rod readings should be corrected for the earth's curva- ture (0.001 foot in about 200 feet, or say 0.004 foot in 400 feet distance). However, the effect of curvature is reduced by atmospheric refraction ; and with errors of observation, sluggishness of bubble, etc., to contend with, the curvature correction should be ignored, especially when the rod is read to the nearest 0.01 foot. (The foregoing process is known as the "two-peg adjust- ment." Although exceedingly simple, this adjustment is commonly regarded as a " bug-bear " by many American engineers. But for it, the dumpy level would have the ex- tended use in this country which it merits. It is said that " the wye level is easy to adjust and usually needs adjust- ment." Many good levelers employ the " two-peg test " to prove the wye level adjustments. Time may be saved by establishing an adjusting base. The adjustments of a good dumpy level are very stable.) Uprights. — In some dumpy levels the uprights which connect the telescope with the level bar are adjustable, similar to the wyes of the wye level. This adjustment is designed to bring the bubble line perpendicular to the ver- tical axis in case the bubble is first adjusted parallel to the line of collimation. However, the best order is that already described, viz., first adjust the bubble line perpendicular to the vertical axis, and then the line of collimation par- allel to the bubble line, in which case the adjustable up- rights are unnecessary. PROBLEMS WITH THE LEVEL. PROBLEM CI. DIFFERENTIAL LEVFILING WITH THE HAND LEVEL (OR WATER LEVEL). (a) Eqvipwent. — Hand level (or water level), rod gradu- ated to feet. (b) ProMem. — Run an assigned level circuit with the hand level (or water level), observing the nearest 0.1 foot by estimation, and closing baclt on the starting point. PKOBLEMS. 7Y (c) Methods. — (1) Determine the correct position of the bubble of the hand level by sighting along a water table, or sill course of a building, or by the principles of the two- peg test. (If the water level is used, fill the tube so as to have a good exposure of the colored water in the glass up- rights.) (2) Take sights of 100 feet or so (paced), estimat- ing the rod reading to the nearest 0.1 foot; balance back and fore sights ; assume the elevation of the starting point, and keep the notes in a single column by addition and sub- traction, as in the 7th column. Fig. 19a. (3) Check back on the first point. Determine coefficient of precision. (The error of closure in feet should not exceed 0.5 Vdistance in miles.) PROBLEM C3. DIFFERENTIAL LEVELING WITH EN- GINEERS' LEVEL (OR TRA^'SIT WITH ATTACHED LEVEL). (a) Equipment. — Engineers' level (or transit with at- tached level), leveling rod, hatchet, pegs, spikes. (b) Problem. — Run the assigned level circuit, observing the nearest 0.01 foot, and closing back on the initial point. (c) Methods. — Follow the practical suggestions given at the conclusion of the " Use of the Level," giving special at- tention to the following points: (1) eliminate parallax of the eyepiece; (2) balance back and fore sight distances; (3) have the bubble precisely in the middle at the instant of sighting ; (4) both rodman and leveler read each rod and also make the calculations independently; (5) calculate ele- vations as rapidly as rod readings are obtained; (6) plumb the rod; (7) avoid blunders; (8) determine coefficient of precision; (9) no sights longer than 350 or 400 feet. Fol- low the first form shown to begin with, — ^the other after several circuits have been run. PROBLEM C3. PROFILE LEVELING FOR A DRAIN. (a) Equipment.— ^ngmeers' leveling instrument, leveling rod, 100-foot steel tape, stakes, pegs, axe. (b) Problem. — Make a survey, plat and profile, with esti- mate of cuts and quantities for a drain under assigned con- ditions. 78 THE LEVEL. (c) Methods. — (1) Examine the ground, determine the head and outlet of the drain, and select the general route. (3) Stake out the line, set stakes every 50 feet, or oftener if required to get a good profile, and drive a ground peg flush, say 2 feet to the right (or left) of each stake ; record data for mapping the line. (3) Starting with the assigned datum or bench mark, run levels over the line of the pro- posed drain, observing the nearest 0.01 foot both on turning points and ground pegs, the former somewhat more care- fully ; take rough ground levels, as required, to the nearest 0.1 foot; locate and determine the depth of intersecting drains or pipe lines, or other objects which may influence the grade line of the drain, and secure full data for placing the same on the profile ; observe due care with the back and Pig. 19b. fore sights, as in differential leveling, and conclude the leveling work with a line of check levels back to the initial bench mark ; a permanent bench mark should be established at each end of the drain, and if the length is considerable, at one or more intermediate points as well. (4) Make plat and profile of the drain line ; lay the grade line, taking into account all ruling points ; calculate the cuts, both to the nearest 0.01 foot, and also to the nearest 14"ii'ch; mark the latter on the stakes for the information of the ditcher, using waterproof keel and plain numerals ; make estimate of the quantity of drain pipe, and of the cost of the job. Follow the form and the profile in Eig. 19b. PROBLEMS. 79 ^, Levskr J-Doe ; Rodmi n, R-/fot. '^ Lev ;L Cm UIT, 5 :A LE /EL I •M- T( EN6S Hall B-M- 1 AND Return 5ta- P-S- H-I- F-S- Elev- Sights llct-ZH'/4- fZ/foi/rs) C/esr JP Warm- B-J' F-5 eXB-Pumpyleyef, . ir-Jfod, Uder IS- SLBM- 7ZM0. /*?, talt, e-end, S-Ctfing, '^■^nfrance^ Kll) ^■^■et> 7ZT-M 'ISO UnJversify Tfa/t- el -7se 7l!!-04 ZO0, weni/,/1- Coping, W entrance, Mi) *ZHff in-04 Z/0 TfafursJ Hisfory VaJ/- tHS-m -3-4Z 713-$Z ZSO^ B<>lt,lvend,l^SI-e. >,S- entrance. A 13) 1-S-S4- TI9-S» 771) ^ngineenng Hai BZ -0-30 719-Z6 so. Cament walk, ZO' li-e;H£nfrsnce, M4) 1-3-87 7ii-a -314 V9-39 ■SO Umvers/ty'/falf Baltf starflng point < yhove descritjed- HJM- H4-4I -I4-4Z I'M-S flfrmfssahla error ■■(IOI!7-/L=l>-OZ7et- H4-4I 7Z(I-I!S Actt/af srror of C/o S- Chsnt'lsb- - OOI (Ch !cJc) Um-v-Hal/ ^^'' i V MifHafHBll \ \ i 1 ersen U ! 5f. E-HB-M-, -*^^ In j- Hall ' ^ \ > ^ HH-Shermn.ltnltr. C-l-Bcytr, godman- ^ Leve .S FR >M k :YNOL [IS T( Lafayette, Ind- Sta. )f. H-1- F-S- Etev. lf(!vZ0,'ia.Clei/ily,Ciii!/-gSS-l>umpyLevel- BM-I 094-l>Z ~ ei9-!Z m flsnna-/7-g-£-M-,M-f^Z7,driven rail Clftls ei S-9S B9e-30 }-a 690-3S l(S, 740 Pep- direct from tide ffaae ^t- ez S70 Ml-JS 3-8Z 69Z-41 740. 740 •• Sandy Hock, /f-T-] 03 493 e9S-74 4-37 (93-H Z4t. 1(5 " {5ou//i alonff /fcnon X-X-) B4- 4Z3 693-7! 3-ZZ 69S-5Z zto ■Z40 ir 95 4Z4 t99-04 4-95 614--80 740. 740 n 66 4-73 099-33 4-44 694-60 740, Z40 n 67 4-44 m-61 4-ie 695-17 300. 340 » opposite eatliollc Cliureli,Reyni7ds- »S SIS t99-39 i-40 694^ZI m. m fi 99 4S4 197-ie (-07 (93-3Z 300, 300 " e side tr'k, iff men M- poles ZZ83-9e- ajff 411 e9S-iz 4-SS (93-31 300. 300 it »// 48Z 69S-86 4-08 694-04 300, TOO neiidstr'k,l,etivMn fefipole3ZZ56-57 BIfZ. 0-7& ((95-10. \ po. 3-tf-onoBk,MlpoleZZ9S, SO' e- track- Peg 3t tel.'pole Z30ff. BIZ 410 (91-00 4-96 693-90 300. 300 ai3 3-6S 69113 B-SZ 69Z48 3(0. 300 It ri 17 ft Z305- el4- Z-71 69Z-9Z S-9Z 690-ZI ilO, xo 11 eJS 3ZI 690-03 6-10 686-az 360 3(0 11 11 II /> Z313,e-side track- BM-3 Q-IJ) (6l(-Si. \ 'l5i H£-car.parapet ivall, Bridge //s ^7-4- »I6 4(S (90-13 3-lS 6S6-IS w W '^'nonJ^-K-B-7l-{'f^,-lfi-^^P''\ ^^^^ Peg \iO'e-trk;a—figg-IZ 8/r-4- t/7 4SI 691-21 &-Z9) 4-13 (eis-5i. (se-zo V 360. 3(0 til file (91-37 S-00 eie-zi 300, 3(0 II BI9 S-13 6lt-Z4 300 n at tell pole Z33a- J ss-7e 89-34 85-76 689-8Z =B-M-I .-3-58 'Check ^ -3-S8- _ 80 THE LEVEL. 5t3 Ml m ■tfP-S SMZi ■flM ■H41 m-B Mi tTO fi4 ■tS4( Survey for a Drain from Description Bed eF Btreafl? • SuitsUs eufUf-furdrsln, /f'fV- af tV-face of sUne 3rch lirfdge- Break of N-bankj Boneyard Branch' 5- edge oF 7' drive- Crosses drain from Con^erv^i-oty- 5 team pipe iine fo Conseri^^hiry- Cenfar J^drive fo ConsarvsfOry- }kaih of main track, U-^C-Sf-fiy- (Cafoff f/irouff/i University frOt/ntfe^ 1 Rslis ofsidefrackf Urt^ans and Cframpaign Fiectric Ry- firainis 2 L-Cf) of sfake- liims 5- in iV-parJeinfff Bcrrr/iiAve- 'iZ'Asfi tree 6' to JK- Cement watk ^-s/de Svrriil Avff- Crosses tf/Iiiary tiaii Steam Pipe Line- /point 3'W-SZ'3-of//WCor- of^ frt^lad- Line ri/ns firence W~ parallel to S 9'S-of S-Jine oP Spring field Ave- StaJcas are set Z'fo rig/it c/^ pirofosed frvttch for drain J wifii ieveiirfff pe^s flirsij with ground Beside sfakes- Ensineerins Laboratory- fiead Cfiainman, J-Doe^ Rear C//ainman, ^-Moe - Apr 2e, im. (Z Hours) CJoi/dy S Cod/- idCft- Stee/Tape, HiZ7S, ioclcerJf^3S- <4 Steam Pipe StRyCvtoff Springfieid A ve r leveier, R-Roe- Rodman, J-Poe- \ Ll VEL 1 lOTES FOR A Dl ;Am FROM ENSINEERINS LABORATORY- 5t8. B-5- H-I- F-5- Elev- loo-ao Srade Cut Oct 23, '14^ CZItrsj Clear and Cool- S-Endslvnesiif, IV-door, Fn^- Lab- ;r f-i-Z3 ICi-Zi Station stakes-are Z'R- proposed trench- iOS 9S-IS 9400 418 Peg drii/en flush mth gremd Besidesfake. t^O iZZ 0S-CI 93-20 481 n tS4i 3-Z SSC 93-13 4-9 Oroiiitd, 6'Sfeami'ipe,10'cBsmg,Jop 2!6 deep- t$i 3-Zi SSCH 9Z-94 S-16 Cement ivalkf P-side Burr/IiAve- 1 3-38 97-SS 92-40 S-4S Peg- tZi-B }-7Z nsi 9206 S4S Oralntums S-in IV- parking Surrii/Ave- tso 41S n-ss 91-eo 498 Peg z 5-71 9SSZ 90SO 4-72 tSO 6-3i 94-92 9000 492 }* etss -s-ss 9S-3S S9Se 552 {Turnrngfei/rj ff-Rai/, Main Track, l/-d£C-/?y. w tO-13 95-51 i ZHg 9343 8920 423 Peg tss 3-fB nci0 ISM see )7 Fauth Wye lei/e/- 4 4-3-I- 9117 S7-60 3S7 ?' Phils- Rod, Ikr-ZO- fSt s-so 9001 lllO 3-21 f» 5 SSZ 3999 » seoo 3-99 37 t4Zi »■! 19-4- SS91 3-5 Sr 718 10-3 713-6 ft » ?I9. 9-22 6 721-64 JI-4S 12-2 711-7 In Corn Field Stake, St3-219- Corn Fj'e/d- eftakt (712-41 2?0 713-0 221 ■4-4 717-2 '• 222 2-7 718-9 M 223 2-9 711-7 t« 224 2-3 719-3 t1 22S 3-4 718-2 Timber Pasture- t-m 12-4 709-2 Sully 226 ll-Z 710-4 r3S 6-0 71S-6 Break eF bank, Plum Rji/er- g-M-27 2-04 7/3-S2 7IS-33 10-/6 -tl-SS 1-18-05 6-0 Check {711-41 B-M-,nol-,U'elm,72'e., Sta-?2e-t6S- ,„, Cs,- station- - = fDre S/s/tt- K, J" . <■ ■ 6ack Sielit R- gtd (Umptilalw) i-SO 707-S +ia.os -4-S, ^ .-4-B, Pron/e of Locat/on Line. A.B.&.C.RR. -Blue. Water level; notes relative to same. ^lack. 5ui^race line, station nurr?erals, etc.- 210 ' ' SI-.. Z£0 '^roocx'F^z^d- 19c. PKOBLEMS. 83 cent. It is usual to give the alinement notes and prominent topography, as shown in Fig. 19c. (The complete series of steps involved in railroad and similar leveling for location and construction purposes is : (1) setting the station stakes ; (2) running the levels ; (3) making the profile; (4) laying the grade line on profile; (5) calculating vertical curves; (6) cross-sectioning for earthwork; (7) calculating earthwork quantities; (8) set- ting grade stakes.) PEOBLEM C5. VERTICAL CURVE. (a) Equipment. — Drafting instruments, profile paper. (b) ProMem. — Connect two grade lines by a parabolic curve, as assigned. (e) Methods. — (1) Plot the given grade lines, station numbers, etc., on a sheet of profile paper. (2) Pind the grade angle, i. e. the algebraic difference of the two rates of grade. (3) Determine the length of the vertical curve by dividing the grade angle by the assigned or adopted change of grade per station (notice the analogy to simple circular curves). (4) Calculate the apex correction. (5) Determine the corrections at the several stations or fractional stations (as assigned), and tabulate the stations and elevations. (6) Plot the vertical curve from the data so determined, as in Fig. 19d. (7) Also compute and plot the same curve by the method of chord gradients. PROBLEM C6. ESTABLISHING A GRADE LINE. (a) Equipment. — Leveling instrument, leveling rod, flag pole, 100-foot steel tape, stakes, axe. (b) Problem. — Establish an assigned grade line, (1) by measured distances and calculate rod readings, and (2) by " shooting in " the same line, for comparison.. (c) Methods. — (1) Stake oflE the distance between ruling points, and drive stakes to the required grade, or if desir- able, parallel to it, by dividing up the fall in proportion to the distance. (2) Set the level over one ruling point and determine the height from the point to the line of collima- tion by means of the leveling rod ; set the flag pole behind the other ruling point and establish a target, consisting of a rubber band holding a strip of paper wrapped about the 84 THE LEVEL. Vertfcd/ Curve. ■(J) ^ x Tangent Correct fans y. COMPARISOM OF RESULTS Elevation By Tanijer? rCorrfclitms By Chord Grad ients. Sbabion. oF Grade Tanqenb Curve Chord 6ra dienfcs. Curve Tanqent . Correction. Elevation. DifF. Oradient. Elevation. Fb. Fb. Fb. Percent. Percent. Fb. 84 108.00 f-I.OO) d5(P.Q 107.00 tO.OO 107.00 +0.10 -0.90 107.00 86 106.00 i-O.IO 106.10 +0.20 -0.70 106.10 87 10^.00 f0.40 105.40 fO.ZO -0.50 105.40 88 m.OB +0.90 104.90 +0.20 -O.iO 104.90 89 JOi.OO + 1.60 104.60 +0.20 -0.10 m.60 90(Apex) 102.00 +i.50 104.50 +0.20 +010 104.50 91 103.00 + 1.60 104.60 +0.20 +0.30 104.60 9Z I04.OO + 0.90 104.90 +0.20 +0.50 104.90 di 105.00 + 0.40 105.40 +0.20 +0.70 105.40 94 MOO + 0.10 106.10 + 0.20 ■ +0.90 106.10 95(P.T.) 96 107.00 708.00 tP.oo 107,00 +0.10 l+I.OO) 107.00 + 2:00=A Fig. 19d. pole at a height equal to the rod reading ; having thus con- structed a line parallel to the desired grade line, direct the telescope on the fore sight target, and with the same rod reading, " shoot in " the same stakes. Make careful record of data and comparative results. PEOBLEMS. 85 PROBLEM C7. SETTING SLOPE STAKES. (a) Equipment. — Leveling instrument, self-reading level- ing rod, 50-foot metallic tape, stakes, axe, marking crayon. (Or, instead of levelling instrument and rod, use special cross-sectioning rods, if assigned.) (b) Prohlcm. — Set slope stakes for the construction of a railroad, canal, etc., as assigned. (c) Methods. — (Follow the methods described in Chap- ter VTII, "Eailroad Surverying," under the head of " Cross- Sectioning.") PROBLEM C8. CALCLTjATION OF QUANTITIES. (a) Equipment. — (No' instrumental equipment imless pla- nimeter is assigned.) (b) ProMem.. — Compute the quantity of earthwork for an assigned set of cross-section notes. (c) Methods. — (1) Transcribe the notes and carefully verify the copy. (2) Calculate the sectional area for each station and intermediate in the notes, and prove the re- sults. ( 3 ) Calculate the volume by the " average end area " method, results to nearest 0.1 cubic yard, and check the same. (4) If so instructed, plot the notes on cross-section paper and determine the areas by means of the planimeter as a check. Record the results. PROBLEM C9. STAKING OUT A BORROW PIT. (a) Equipm,ent. — Engineers' level or transit with at- tached bubble, leveling rod tape, stakes, axe. (b) Problem. — Stake out a borrow pit and take notes re- quired for calculation of earthwork quantities. (c) Methods. — (1) Select a base line, preferably outside the limits of the proposed borrow pit, set substantial station stakes say 50 or 100 feet apart along this base ; designate these stakes A, B, C, etc. (2) Establish auxiliary refer- ence lines by erecting perpendiculars to the base line at the several stakes, driving temporary stakes for pegs at suit- able distances on these lines. (3) Establish a permanent bench mark and run levels, as in profile leveling, along lines starting at A, B, C, etc., noting elevations both at pegs and at marked intermediate changes of slope. (4) In 86 THE LEVEL. case actual construction is undertaken, repeat the levels along- the same auxiliary lines from time to timie and cal- culate the quantities. (5) Eecord complete data. r Leve Shtlen 4-K>t *76 3 7! t Z / f 7! 7! 7! .5 FDt B-S- JforfA / raperfy Center South Line, iYoper^ Hcrfh 3-Ii Ml 3-Qi PROF|iLE AMp QUAHTlTIEf T-P- H-l- ' fy. PfldofB 7li-0S line J / saly Si /{eely Line, 'dge South Ciil sF Br, t/trfh a operty Wne, S/ ren OSS 7/e-u 7Z0-/1 7?}- kmll. AO II* }S-i Srovnd' (All leve/s From si'/y/e- settih^-) AJ 14 S$i AH-4S Alt4S C46'») e-9 SS-4 A 1 2 3 4 fSO A Ai 9-7 }!-0 %, , /I'velK Ai Ai'SO K! S 47 27-0 34-4 4-0 4-1 $1-4 e S7 31-5 30-0 43 4-4 6I-B 1 \ .' 7 67 SS-S ZB-7 4-3 4-3 61-S J?:D='b:t I 17 40-Z ZI-0 4-4 4-7 ei-z ' R=^D' 8 77 ^0-0 Zl-Z 61-Z i \ \ Tarqet Movement (b) 7 67 35-S ZB-S 4-S 4-3 Bl-C at 100 ft- (D) e 57 21-1 iO-Z 4-4 47 ei-3 For Bubble Movement 5 4-7 76-7 341 4-4 4-4 SI-S \ >' (b) of 1 inch (0-013 ft) ■f 37 ??-3 3S-S 4-4 4-Z ei-1 r i No- Rods, (Ft.) b(Ft) 3 17 JSZ 4Z.S 4-J 4-1) 61-D / 4-03] 4-lSI O-04i z 17 J4-0 47-1 4-Z 4-3 U-l Z 4-SSt 4-346 0-040 1 (*Lsv Ten d 7 :! scala U TOHi visions 9-S inch) p micro. SIS Me3n = 4-5 4-4 61-0 -..on It 100 0.042 = in Ft. i 4 4-S4 4-S9. ' 4-S9I F 4-634 0-044 0-041 4-36 ZOtlls •rew C6 ei-i 'nch. 'respan Mem = 0-04Z JfolS Z seconds ofar r, .so Hiaf 'one division cf fca/e c ^■36 ■4secon is, ano one div 'sign on bubble lube (f nlhs oFffn inch) = i S secon dSf or ^Ssseon ^s per inch- Radiv 9 in fee . 17112 -.1-1 JS2 _ 19S feel. sees perinc i IS " J PKOBLEM C14. TEST OF DELICACY OF BUBBLE VIAL. (a) Equipment. — Engineers' leveling instrument, leveling rod, tape, level tester. (b) Prohlem. — Determine the radius of curvature of the assigned bubble vial. (1) by means of the optical test, and (3) by the level tester. (c) Methods. — (1) Measure off a base line say 100 feet long, set level at one end and hold rod on a peg driven at the other end ; note the target movement corresponding to a given bubble movement, both in the same linear unit ; cal- culate the radius by the method shown at (h). Fig. 18. (3) Set the level tester on a solid base and place the instru- ment on it, a? jjDd.icated at (i), Fig. 18; by means of the PROBLEMS. 91 micrometer head and known relations of the level tester, determine the angular equivalent in seconds for one divi- sion and also one inch movement of the bubble, from which calculate the radius of curvature of the vial in feet. Fol- low the form. PROBLEM C15. COMPARISON OF LEVEL TELESCOPES. (a) Equipment. — Five (or other specified number) engi- neers' levels (both wye and dumpy), leveling rod, metallic tape. (b) Prohlem. — ^Malce a critical examination and compari- son of the telescopes of the assigned instruments. (c) Methods. — Carefully read the discussion of the tele- scope in the text. Then compare the telescopes with refer- ence to : (1) magnifying power ; (2) chromatic aberration ; (3) spherical aberration ; (4) definition; (5) illumination; (6) flatness of fields; (7) angular width of field; (8) effec- tive aperture of objective. Make tabulated record of com- parisons, giving in separate columns; (a) locker number; (b) kind of level; (c) name of maker; (d) magnifying power, and so on for the other points examined. PROBLEM C16. TESTS OF THE WYE LEVEL. (a) Equipment. — ^Wye level, leveling rod, tape. (b) Problem. — Test the essential relations and adjust- ments of the wye level. (c) Methods. — Carefully note the construction of the as- signed level and the positions of the elementary lines. Then following the methods outlined in the text, test the fol- lowing adjustments (but do not disturb the adjusting screws) : (1) The bubble, both as to the azimuth and alti- tude movements ; find the position of the bubble when par- allel to the element of the rings. (2) The line of collima- tion ; its deviation from the axis in 400 feet. (3) The wyes ; finding the position of the bubble when the vertical axis is vertical. Keep a neat and systematic tabulated record of observed numerical data, with explanation of the several adjustments. 92 THE LEVEL. PEOBLEM C17. ADJUSTMENT OF THE WYE LEVEL. (a) Equipment. — Wye level (reserved expressly for ad- justment), leveling rod, tape, adjusting pin. (b) ProMem. — Make the full series of adjustments 'of the wye level. (c) Methods. — Follow the methods detailed in the text according to the following program: (1) Adjust the bubble line (a) into the same plane with the bottom element of the rings, and (b) parallel to that element. (3) Adjust the line of collimation to coincide with the axis of the rings, first on a long distance ; and then, to test the object glass slide, try it for a short distance ; if necessary, shift the reticule in rotation to make the horizontal hair horizontal, and also center the eyepiece. (3) Adjust the bubble line perpendicular to the vertical axis by means of the wye nuts. (4) Test the rings of the wye level by the two-peg test ; if the level has a reversion bubble, first test the paral- lelism of the top and bottom tangent lines, and then test the rings. Keep a, clear and systematic record. In each case, state (a) the desired relation, (b) the test, and (c) the adjustment. PEOBLEM C18. SKETCHING THE WYE LEVEL. (a) Equipment. — Wye level. (b) Pro6?e)».— Make a first-class freehand sketch of the assigned wye level. (c) MetlKidt^'. — The sketch should be correct in proportion and clear in detail. The essential parts should be desig- nated in neat and draftsmanlike form, and the elementary lines clearly indicated. PEOBLEM C19. TESTS OF THE DUMPY LEVEL. (a) Equipment. — Dumpj' level, leveling rod, tape. (b) Prohlem. — Test the essential relations and adjust- ments of the dumpy level. (c) Methods. — Carefully note the construction of the as- signed level and the position of the elementary lines. Then, following the methods outlined in the text, test the follow- ing adjustments: (1) the bubble line, whether perpendicu- lar to the vertical axis ; and if not, what is the angular inclination of the vertical axis when the bubble is in the PROBLEMS. 93 middle? (3) The line of collimation, whether parallel to the bubble line. Record the errors and observations sys- tematically. PROBLEM C30. ADJUSTMENT OP THE DUMPY LEVEL. (a) Equipment. — Dumpy level (reserved expressly for ad- justment), leveling- rod, tape, peg-s, axe, adjusting pin. (b) Pro6?c»).— Make the essential adjustments of the as- sig-ned dumpy level. (c) Methods. — (1) Adjust the bubble line perpendicular to the vertical axis. (2) Adjust the line of collimation par- allel to the bubble line by the two-peg method. In describ- ing the adjustments, the record should state (a) the desired relation, (b) the test, and (c) the adjustment. PROBLEM C21. SKETCHING THE DUMPY LEVEL. (See Problem C18.) PROBLEM C33. STRETCHING CROSS-HAIRS. (a) Equipment. — Engineers' level or transit (or cross- hair reticule), pocket cross-hair outfit, reading glass. (b) ProMem. — Renew the cross-hairs in a level or transit instrument by a method applicable to field use. (c) Methods. — (If instrument is provided, follow the complete program outlined below ; otherwise, merely stretch the lines on the reticule and test same.) (1) Remove the eyepiece, carefully preserving the screws from loss. (2) Remove one pair of the capstan headed reticule screws ; turn the ring edgewise and insert a sharpened stick in the exposed screw hole, take out the other two screws and re- move reticule from telescope tube. (3) Clean the cross-hair graduations, and support the reticule on a sharpened stick, or (if a transit) place it on the object glass with a piece of paper interposed to protect the lens. (4) Select from the capsule (see (d), Eig. 17) two spider lines 3 inches or more long, and fasten a stick to either end of each hair by means of glue from the adhesive paper. ( 5 ) Put the hairs in place, (with the bits of wood hanging loose), shifting them as desired with a pin point or knife blade. (6) Apply a bit of the moistened adhesive paper to the reticule over each hair. 94 THE LEVEL. and after a few minutes cut or break the sticks loose. (7) Test the hairs by blowing- on them full force. (8) If they stand this test, replace the reticule, and adjust the instru- ment. Make a record of the process. PROBLEM C33. ERROR OF SETTING A LEVEL TARGET. (a) Equipment. — Engineers' leveling instrument, leveling rod (preferably a New York or Boston rod), tape, pegs. (b) Protlem. — Determine the probable error of setting the level target at distances of 100 and 300 feet (or such other distances as may be assigned). >(7W7v Placed pair offa/t s of pegs and le, ' snug. Focused eyepj'eceon Fuiiy,hep/nf eye i 5et target ten time careFulJy veriFyio bubble each time be Peiernr/ined magnlF hy comparing 77.1/, . with one eye a/j, ' ifitliothereye. 28diameters. Found radius oFcur K=hD=MlxlOi t 0.0/13 Pidm.hor.h3ir,h= lelteredplace^measured anddr/JV ^peg. Same at 500 Ft. •reivs on general line 7,leamgscremjust =0.OO00ZH. crosshairs very care- normal condition. at each distance, 'Fyin y the posiiio/? oFche ne be 'ore apprm'n/i sight, ^gnit 'Jng power or telescope 0.1/, . on rod natural size 'dw^gnlFied l]y telescope roL idMag.Poivertolie 'atore oFdudb/e =/4S.'s r. ff.ff/xD.S '■" 400 0.1^024/n. rod-. (c) Methods. — (1) Determine the magnifying power of the telescope. .(2) Determine the radius of curvature of the level vial by the field method. (3) Determine the space on the rod covered by the diameter of the hair. (4) Drive a peg at 100 feet from the level, level up, and secure ten sat- isfactory consecutive rod readings with rod held truly plumb on the peg ; shift the target several inches between read- PEOBLEMS. 95 ings, and reset without bias ; reject no readings ; watch the bubble closely, but work briskly. (4) Repeat the series at 300 feet. (5) Determine for each distance the mean rod, the probable error of a single reading, and of the mean, as indicated in the form. PROBLEM C24. MAKING A LEVELING ROD. (a) Equipment. — Piece of straight dressed clear white pine of proper dimensions, steel tape graduated to 0.01 foot, carpenter's tri-square, paint, etc. (b) Proilem. — Make a self-reading leveling rod. (c) Methods. — (To be devised by the student. See Fig. 27 for suggested graduations.) PROBLEM C25. COMPARISON OE DIEEERENT MAKES AND TYPES OF ENGINEERS' LEVELS. (a) Equipment. — Department equipment, catalogs of rep- resentative engineering instrument makers. (b) Problem. — Make a critical comparison of the several types and makes of engineers' levels. (c) Methods. — Examine the department equipment and stiidy the several catalogs carefully, noting the usual and special features, prices, etc., and prepare a systematic sum- mary or digest of the same. Prepare brief specifications for a leveling instrument, and also suggest the preferred make. CHAPTER V. THE TRANSIT. Description. — The engineers' transit consists of an ali- clade, carrying the line of sight, attached to an inner verti- cal spindle (or upper motion) which turns in an outer an- nular spindle (or lower motion). The latter carries the horizontal graduated circle or limb, and is supported by the tripod head. The alidade includes the telescope, magnetic needle with its graduated circle, and the vernier ; it may be revolved while the graduated limb remains stationary. The horizontal limb is graduated to degrees and half degrees and sometimes to twenty minutes, and is numbered prefer- ably from zero to 360° in both directions. The complete transit differs from the plain transit, Fig. 20, in having a vertical arc and level bubble attached to the telescope. Complete Transit, Plain Transit. Fig. 30. 97 998 THE TRANSIT. (h) USE OF THE TRANSIT. 99 In Fig. 21 are shown: (a) the English theodolite; (b) the shifting plates and foot screws of a transit ; (c) the Saegmuller solar attachment to the transit; (d) the gra- dienter; (e) tripods ;•(£) reflectors; (g) reading glass ; (h) flagpoles; (i) plumb bobs; (j) the Brunton pocket transit. The Vernier. — The vernier is an auxiliary scale used to read fractional parts of the main graduated scale or limb. The least count of a direct vernier is found by dividing the value of one division of the limb by the number of divisions on the vernier. With a limb graduated to half degrees and a direct vernier reading to single minutes 30 divisions on the vernier cover 29 divisions on the limb. In reading a direct vernier observe the following rule : Bead from the zero of the limb to the zero of the vernier, then along on the vernier until coincident lines are found. Add the reading of the vernier to the reading of the limb. In setting the vernier to a given reading, as for example a zero reading for measuring an angle, the tangent move- ment should be given a quick short motion to secure the last reflnement, since a slow movement is not noticed by the eye. Notice adjacent and end graduations. In Pig. 23, (c) is a vernier reading to single minutes, (d) to half minutes (30"), and (e) to thirds of minutes (20"). The slant in the numerals on the limb corresponds with that on the vernier. USE OP THE TRANSIT. Use. — The complete transit is used: (1) to prolong lines; (2) to measure horizontal angles; (3) to measure vertical angles; (4) to run levels ; (5) to establish grade lines. The plain transit is conflned to the flrst two uses, unless it has a vertical clamp and tangent movement, when it may be used to " shoot in " grade lines. Prolongation of Lines. — If the instrument is in adjust- ment a line can be prolonged by sighting at the rear sta- tion and reversing the telescope in altitude. It is, however, not safe to depend on the adjustments of the transit, and important lines should always be prolonged by the method of " double sights," as given in Problem D2. Lines may be prolonged with the plates by sighting at the rear station with the A vernier reading 180°, reversing the alidade in azimuth and locating stations ahead with the A vernier reading zero. A third method employs two points ahead of the instrument. 100 THE TRANSIT. Measurement of Horizontal Angles. — Horizontal angles are measured as described in Problem Dl. If greater ac- curacy is required, angles may be measured by series or by repetition. By Series. — In measuring an angle by series all the angles around the point are read to the right, both verniers being read to eliminate eccentricity. The instrument is then reversed in altitude and azimuth and all the angles around the point are read to the left. The readings are checked by sighting back gn the first point in each case. These observations constitute one " set." The vernier is shifted between sets 360° divided by the number of sets. The arithmetical mean of the observed values is taken as the true value. By Repetition. — Angles are measured by repetition as described in Problem D13. This method is especially suited to the accurate measurement of angles with an ordinary transit, and is to be preferred to the series method, which is a favorite where precise instruments are used. In the repe- tition method all the instrumental errors are eliminated and the error of reading is very much reduced. It is doubt- ful if it is ever consistent to make more than 5 or 6 repe- titions. Azimuth.. — The azimuth of a line is the horizontal angle which it makes with a line of reference through one of its ends, the angles being measured to the right from 0° to 360°, as in (f) Fig. 23. It is usual to assume that the true meridian is the line of reference, the south point being taken as zero in common surveying. Deflection. — The deflection of a line is the angle that it makes with the preceding line produced, and is called de- flection right or left depending upon whether the angle is on the right or left side of the line produced, as in (h). Fig. 23. Vertical Angles. — Vertical angles are referred to the horizon determined by the plane of the level under the telescope, and are angles of depression or elevation relative to that plane. In measuring vertical angles the instrument should be leveled by means of the level under the telescope and correction should be made for index error of the ver- nier. With a transit having a complete vertical circle, the true vertical angle may be obtained by measuring the angle with the telescope normal and reversed and taking the mean. Traversing. — A traverse is a series of lines whose USE OF THE TEx\NSIT. 101 lengths and relative directions are known. Traverses are used in determining' areas, locating highways, railroads, etc. Azimuth Traverse. — In an azimuth traverse the azimuths of the lines are determined, nsiially passing around the field to the right. In orienting the transit at any station the A vernier is set to read the azimuth of the preceding cotirse, the telescope is reversed, directed towards the pre- ceding station and the lower motion clamped ; the telescope is then reversed in altitude. The reading of the A vernier with telescope normal will then give the azimuth of any line sighted on. If there is any error in collimation the transit may be oriented by sighting back ^vith the A vernier read- ing the back azimuth of the preceding course. In a closed traverse the last front azimuth should agree with the first back azimuth. The azimuth traverse is especially adapted to stadia and railroad work. Azimuths can be easily changed to bearings, if desired. Deflection Traverse. — In a deflection traverse the de- flection of each line is determined, usually passing around the fleld to the right. To avoid discrepancies due to error in collimation, the transit may be oriented by sighting at the preceding station with the A vernier set at 180°, the telescope being in its normal position, and the lower mo- tion clamped. The reading of the A vernier will then give the deflection of any line sighted on. Compass Bearings. — Compass bearings should always be read on an extended traverse as a check against such errors as using the wrong motion or an erroneous reading of the vernier. To guard against errors due to local attrac- tion, back and front bearing's should always be read, and the angle thus determined compared with the transit angle. Leveling ■with the Transit. — The transit with an at- tached level is the complete equivalent for the engineers' level. The instrument is leveled up with the plate levels first, after which the position of the attached bubble is con- trolled by means of the vertical tangent movement. Grade Lines. — Grade lines may be established with the transit either by means of known distances and calculated rod readings, or by " shooting in " a parallel line by means of the inclined telescope, as described under the use of the engineers' level. For the latter purpose the transit is rather more convenient than the level. Setting up the Transit. — To set the transit over a point, spread the legs so that they will make an angle of about 30°, place them symmetrically about the point with two legs 102 THE TRANSIT. down hill. Bring one plate level parallel to two of the legs, force these legs firmly into the ground and bring the plumb bob over the point and the plates approximately level with the third leg, changing the position of the plumb bob with a radial motion and leveling the plates with a circular mo- tion of the leg. Finish the centering with the shifting plates. In leveling up, the bubbles mo^'e with the left thumb. Use care to bring the foot screws to a proper bearing. Parallax. — Before beginning the observations the eye- piece should be carefully focused on the cross-hairs so as to prevent parallax. Back Sight With Transit. — Ahrays check the bacTc sight icfore moving the transit to see that the instrument has not been disturbed or that a wrong motion has not been used. Instrumental Errors. — The transit should be kept in as perfect adjustment as possible, and should be used habit- ually as though it were out of adjustment, that is, so that the instrumental errors will balance. No opportunity should be lost to test adjustments. ADJUSTJIENTS OF THE TRANSIT. Elementary Lines. — Fig. 22 shows the elementary lines of the transit, viz., (1) line of coUimation ; (2) horizontal axis; (3) vertical axis; (4) plate level lines; (5) attached level lines. These lines should have the following relations : (a) the plate levels should be perpendicular to the vertical axis ; (b) the line of collimation should be perpendicular to the horizontal axis; (c) the horizontal axis should be per- pendicular to the vertical axis; (d) the attached level line should be parallel to the line of collimation. The following additional relations should exist : (e) the vertical axes of the upper and lower motions should be coincident; (f) the optical center of the objective should be projected in the line of collimation ; (g) the center of the graduated circle should be the center of rotation, i. e., there should be no eccentricity. Plate Levels. — To make the plate levels perpendicular to the vertical axis. — Make the vertical axis vertical and ad- just the bubbles to the middle of their race. The vertical axis is made vertical by leveling up, reversing in azimuth, and if the bubbles move, bring them half way back with the foot screws. The adjustment is the same as for the compass, and the reasons are shown in (a). Fig. 13. ADJUSTMENTS OF THE TRANSIT. 103 After adjusting the plate levels with reference to say the upper motion, test them with the lower motion to prove the coincidence of the vertical axes. Op tied I Center (^^ oFObiective, fntersectior? oF Cro55-Hair5 line ofCollimatioi^'':., cc:i ^ J') ffrrkr liot . v exe^d /O At 9I1 OP Tr F Si'Sl' 47%7' 43'Z3' ANGLI: 5-«i-8 Mean Observers, J-Doe cF WITH Engineers' Nov-]B,imj(Z hears). Used Mellar^ Brightly The Jst- measuremeni on Sta-g y^ifh the phfes clamped di on Sfj-S with the re3dw0 the The sigi}tin0 on St3- Used trans/f poies them very mentS' Sketch siiows f plafi s f second measure, ffenf Transit- Warm and ijuiet. Transit No- 10- was made by siglitfng 'ower motion^ the xero} then sf^ht/n^ i^per matio/j, and fvas mads by ^nd then on Sta- 8- targets f piumbtn^ careA 'i/y over the monu- obsej '^ed angles- DOUBLE SIGHTINGS-^ Pbolonsation of Line-- i^JSh^/Sh I 'Setupaf-B "double sightedbF- /few tsck F is'o-OI left- of anginal f^ck- (Alhweble error is t'oiymSigM/ngs For 3aO's/0hts) I ^"'X syhted" toE. ^,'iK^a)fSef up 3fC, "double a \ a ^ I e/e' Did' A/ 'Set up sfB," double sighted to C/ ss FoUoiVsjfSee^ote-*-) (g) Back sighted on A (b)Plungedto c' ^(c) Rotated to A Cd) Plunged to c" fe) Bisected c'c" to ' l^ locffte tack C- {'Set up fft A; sighted on Flag 3t Ff dnve hub Bi removed Flag F- .(Brvxe MsA 3ndF Obsermsl-'"" "-'^-''f-C^hrs) Cool.doudy^ Vl-goe UsedK-S-F- Transit NS4- WITH Ensineers' Transit -Interpolation of Point. \Biseeted pp' St P- Set •K ■<: ^ •*, k,«.||l ■■^5S 1-^ ^^ up 3tA and checked Pf error, 0-02 to right- [Reversed in azimuth ? „e K C4) m shlFted transit so it ^ .- would again plunge *i / exactly on A and B ^ / Drove hub p" to bob- \i , 'Set up and shIFted P\0-I1 transit laterally ^. until it would \ plunge exactly ^. on A and B-(/eelfoh:)-^ \ Drove hub p* to plumb ii;b-^\ ^ {'Set Flags on tacks ah \ \ A and B, and determined point P- by lining in two ri poles- successively by eye-(See p-dS-J . Drove temporary peg- (Drove hubs A and B about 61X>' apart, (l)\ assumed to have hill between them^ I both visible From desired hub P- \ \ ohi\p" 'i HOTE. Watched plate leveis cioseiyj especially transverse bubble' ^J 108 THE TRANSIT. flag pole plumbed over tack in hub F, drive hub B about 300 feet from the transit and locate a tack in line very carefully. Eemove the flag pole from hub F. (3) Set the transit over hub B, back sight on hub A and clamp the ver- tical axis. (4) Reverse the telescope, drive hub C at a dis- tance of about 300 feet and mark line very carefully with a pencil. (5) Reverse the transit in azimuth, sight on hub A; reverse the telescope and locate a second point on hub C. Drive a tack midway between these two points. (6) Set the transit over the mean point on hub C, back sight on hub B, prolong 300 feet and set hub D by double sights. (7) Set over hub D, back sight on hub C, prolong, 300 feet and set hub E, as before. (8) Finally prolong from hub E, with back sight on D, and establish mean tack at terminal hub /''. Record the collimation errors at G, D, E, and the final error at F. Follow the form. PROBLEM D3. INTERSECTION OP LINES BY TRANSIT. (a) Equipment. — Transit, 3 flag poles, plumb bob string, axe, 6 hubs, 6 flat stakes, tacks, marking crayon. (b) Proileni. — Determine the intersection of the bisect- ing lines of two angles of a triangle and check by bisect- ing the third angle. (c) Methods. — (1) Drive and tack three hubs so as to form a triangle approximately equilateral and having sides about 400 feet long ; properly witness the hubs with guard stakes. (2) Set the transit over one of the vertices of the triangle, and measure the angle as in Problem Dl. (3) Set two hubs on the bisecting line, about 6 feet apart, so that the point of intersection of the bisecting lines will come between them, and mark the line by stretching a string be- tween the hubs. Check by measuring each half angle inde- pendently. (4) Set the transit over one of the other ver- tices of the triangle, measure the angle and determine the bisecting line as at the first point. (5) Drive a hub at the intersection of the two bisecting lines and mark the exact point with a tack ; check by measuring each half angle in- dependently. (6) Set the transit over the third vertex and determine the angular and linear error of intersection. (7) As a final check measure the angles around the point of in- tersection of the bisectors. The angular error of closure of any triangle should not exceed one minute. Follow the form. PROBLEMS. 109 static / 3 Z Station v_ Whole Angle ez'is' 73'm' 44'4S' An I-Q-i i-o-z Z-0-} Alliiwsli: ? INTEP L'HalF Angle ii'n'jo' zz'zzin' SE RHa:F Angle 3/WX' }6'Mk' ZZ'ziW ilhwah. Chick iiz'zi'm IZS'Sl'lO izMn m't. er/vr CT ON OlF LiH ■or Distance •dJ'-O Er Angle ■ OJ-'O :S 0-C3f1 WITH TRAN /t PEOBLEM D4. KEFEEENCING OUT A POINT. (a) Equipment. — Transit, 2 flag poles, 100-foot steel tape, axe, 6 hubs, 6 flat stakes, marking crayon, tacks. (b) Problem. — Eeference out a point with a transit and tape. 110 THE TKANSIT. (c) Methods. — (1) Drive two hubs about 500 feet apart and mark them with guard stakes. (2) Set the transit over one of the hubs and reference it out as shown in the diagram. All hubs should be driven flush with the ground, and the exact points should be marked by means of tacks driven into the tops of the hubs. Record in proper form. PROBLEM D5. TEIANGULATION ACROSS RIVER. (a) Equipment. — Transit, 3 flag poles, 100-foot steel tape, axe, 4 hubs, 4 flat stakes, tacks. (b) Proilem. — Determine the distance across an imag- inary river by triangulating with the transit arid check by direct measurement. Simpfe and Rapid Methods oFTrianquIation. l\\\\\\\ WWW! ///7g oF Survey Prolonged Across Fiver. .1 -m :^\\\v ..*-^"'* AB=;rT^^j=BC-Coseo5°U''=BCxl0.0l'(BCxH>hl^§Jr^x0.l) Sin544 il'l' 'Rule oFTen'.' (DWithtrsnsitatA, line in liubstBon opposite side of river. (ilTurnoFFangle 5°44'3ndwitlioneendoFtapeheldatB locate C by swinqinq on arc under direction oF transttman; IF the Front Flaqrnan be provid- ed with a metallic tape , he may locate C alone by hooking the ring oF the tape on 3 projecting tack in hul? B. The desired distance ABmaybe taken roughly as ten times the meas - ured distance BC. For greaterexact- nessj add 0. 1 Foot For each 100 Foot unit in the distance ABbs Found by the simple " rale oF ten "juststated. Leveling ^ Instrument AB:AD::BC:DF (c) Methods. — (To be devised by the student. Use this and the next problem to learn the relative merits of several good methods. The " rule of ten " method in the sketch be- low is very rapid and also quite accurate.) PROBLEM D6. PASSING OBSTACLE WITH TRANSIT. Ca) Equipment. — Transit, 100-feet steel tape, 2 flag poles, axe, hubs, flat stakes, tacks. PROBLEMS. Ill Tri.insuution 'Vcros ; a E iver station B C Distance Ft- Ill-tS UjB- D-B-C S-C-D Co leulatii n I Cxtar SO lo0-B-'D=lc}- ' - - 2. III s-.> 3-D='i<- Chalped dii tance Dift vrence Per. tiiessb. AnjJe Value 0°3O B-D '0- , ltf09+Itt-CIS39l SH-ffff* Ug- ti nSOJi ■ Z.r993 CM M/-06 J.W-t 1 tsulf- isd fj:d Sff'Jl ■Zli/0 /■f- /tl-9e mis Ft ft- Transit » R-Rae- i) CaJdiilCItar- lacker H^8; ■ert{233- B.sethabefD Li ckt WITH ENSINEERS' Observers - J- Dot Mciv-Z7,'r4 -fZJfi Used f3uth and Chai'mng . Kth transit ove.\ by "Method cF Double Sights', tvifh A asa ba Set /fab ate,. care, Checked - chaining 8'D- length cF Tape, il9-9SFt- ffisrrved distances ract rded' ^9°iO'(Complemait) ■ksight. iB-CwIth , andmeashred /. B<-D- computet' distance by Imaginary . r station A e D A-D A B C D A-D A P S A-e Pass Distance Ft. "E,u m-t>o ZOO-OS 101-03 199-93 "Ri^ ZO-00 ZOO-00 loose 200- OS izo-00 izo-00 ZOi-SS Z39-0S MS /N 08 5TACIE An]g lateral N-A-F A-e-D e-D-H B^f 'ectii n if-A-F a-F-6 F-S-H le Value Triang 60W bO'OO' S9°S9' Error Pish Ft. Meth :iht Andle Of fs|[t Metljod ff-A-B A-S-C B-C-D C-B-H 90 '00' 90'00' eo'oi' Met s'oo' lO'OO' s'oo' of Closure Line Ft. od -0-07 tO-OS lod" 0-09R- 0-tOL- 0-03R- WITH' EHSIHEERS' Observers :J-Dat Hovl7,i9/4, Used Soriey chaining Wth tlie'transit end a, in tbe line and prolonged tl lateral Triangle Angle OFFset "DtFlection f/V t, Transit S R-Roe- CZifo vrs) iVarm ff cloudy- ~, and. HS3Z- il ^■^M AssuiTied thatA-S was 3 frue jj7er/(/Zan^ B A lO'/l'L 5-33'ilSi C3refi/7/y checker f eachsngle^ c 4(4-9! S43iS'i joi^l S-/t!'/3'E Bahile dcwrr on 3. t sights- I C B B 4S3-7t IZ4i3'x 543^^'B SIJ'JB'H/ lU'sSR IflSln't £ei7fffh of Tspe - ^ fiec/aced measure. JOO^CIFf- 77enfs recordacf- D C 7f'llfji s^f/'Ssk A//cwab7e error o "closure = J'- e 116-Sl mi'zm 7e't^/i lfll7U3'£ L e D A Z42-14 Slili'lt HZl^s'lli SZ'33'R ss('3;'e i E i2 A r — r i70'J3' 370'JS' .*•'' jfn' (Cluck) IfW 1 / 1 \ 30W iS9'3S' Calcula ion oF Bearin gs— ' ' \ R AS swWe P-E HII)'43'e \ Y B JO'li'L e /??*> \ \ B-C SD'B'e e-A iH'3l'E A \ C im'si'r A W'JJ'R (Check, ) }'ji ^\ . \ D-e- M'4}'e >l. V /^ l1ov-ZI,}9I4 C0777pufer, J -Doe. ^\ Data frompp- Transcr/pt 0-JC- 1 TRAt SIT t RAVES SE, Fl :LD A B-C-D-I Latitudes and Departure?,- Line Adjostci) Eefced COMPl TATlOt OF L/ fit UDE3 COWh TATIOt OF DE PAR rUEES Bearmg Distance Multipli- Lo^ar- Computed Lat. Adjusted Multipli- Logar- ithms Computed Dep. Adjusted cation itlims Latitude Cor. Latitude cation )eparfure Cor. )eparture Ft. Ft. Ft- Ft- Ft. Ft. Ft. AB S-OCV 33S-05 S-33S-Ci ■10 5-334^93 OMO -00^ 00-00 SC 5-J0'l3'E 464-SB "'Vv 9-9330i 5-457-1} ■To 5-437-31 t«|?. Z-i(744 9-Z4m B-iZ-fl -f3 l-SZ-SO HM i-eeoso I-9I63Z je (4S7-6J) ■i- :ikk (SZ-47) *B7-iiie CD /t-6sisi/ 413.7Z ^S US459 9-eiill H-20I-Z4 ■OS H-Z0I-Z9 4eB-7Z Z-6S4SS S-9SS73 W43}-n ■18 m39-l9 ZSOZ^ 1-30370 (70I-Z4 ■f- 'II Z-64330 (439-87) ^ zoisW f-3S-S7S DE H-I0'43B ew-si ■",m Z-7199S 9-99Z}l 17-61378 ■IS H-60S-91 lil'/f. Z-7S99S S-Z6940 f-m-64 ■OS E-II46S 7-78731 Z-OSSiS ■'Sf (6BS-71) + TT?:i-fir (114-64) ■I- 61IS-777 EA 5-!6'3li. Z4Z-S4 '/5 7-3SS3Z 8-78361 5- 1476 ■00 5- 14-76 %'M Z-38S3Z 9-9S9Z0 BZ4Z-40 -10 £Z4Z-S0 H6S93 (J4-76) Z-384SZ F. /f-7se ^^Iff (Z4Z-40) 2143-n H-S07-OZ H-807-ZZ 5439-31 1-430,6} Error Actu Perm' op Clos .1 ure 55 ft. 6 Ft. Line AB ec CO 2 2 1 Cor -/O -/a ■OS 5-107-K ■40 S-807-2Z Line AS BC CO I 10 Cor. -00 •03 -11 W-43M1 ■36 W-43M3 5- 0-40 0-00 W- 0-36 0-00 (St Pisgrs «; PS £A } ■IS ■f- ,-i>t-^ 116 THE TRANSIT. the transit over the second hub, sight at the top of the tower and read the vertical angle, as before. (7) Eead the level rod on the base of the tower as before. Each angle and rod reading is to be based on duplicate readings. Fol- low the form. PROBLEM Dll. SURVEY OP LINE SHAFTING. (a) Equipment. — Engineers' transit with attached bubble, leveling rod (or instead of these engineers' instruments, a 16-foot metal-bovmd straight-edge with an adjustable bubble of say 20-foot radius, a long braided fishing line, and 3 long metal' suspenders made exactly alike, from which to sus- pend straight-edge from line of shafting), 2 good plumb bobs, 50-foot etched steel tape, copper tacks, hatchet. (b) Problem. — Make a survey of a line of shafting in a machine shop, and establish a true alinement for it, both vertically and transversely. Eesuryey oF North Line ShdFtinq, F/etal Shop. leveJs. :■■■? Line Hangers wmmmm4m (c) Methods. — (1) Establish a reference line for lateral deviations and carefully mark the same. (2) Select a suit- able permanent bench mark to which the levels may be re- ferred. (3) Determine the horizontal distance from the vertical reference plane to the line shafting at selected points, say at each hanger. (4) Determine the elevations of the same points by the methods of profile leveling. (5) Plot the data as suggested in the diagram. (6) Note the ruling points and permissible change both laterally and vertically at each hanger, and record the data. (7) Lay grade lines, and prepare data to shift the line shafting tp a ti'se position. (8) Make complete record of results. PROBLEMS. 117 PEOBLEM D12. SUEVEY OP EACE TEACK. Outfit for transit party (instrument ire, say No. 20, spring balance, ther- race track, as in- (a) Equipment. assig-ned, a long mometer, etc.). (b) Problem. — Make the survey for a structed. (c) Methods. — (1) Standardize steel tape, noting temper- ature and pull. (3) Make a careful examination of the tract of land with a view to secure the best location for the race Requlah'on One-Mile and Half-Mile Trottinq Tracks. I Grand Stand \ The standard distance kmeasured ona line 3 Feet From t/?e hub-board. The inner edge of tiie trsck is thus 2Tr-3=i8.85feeb shorter than the standard distance. The trac/c is banlted 'on curves Fron?l:iZtoi:i5, and, to provide drair7aqe, shouid be sioped one Foot on the straight stretches. The ends of curves are some - times Flattened. track as regards visibility, drainage, economy of construc- tion and maintenance, etc. (3) After fixing the ruling points, establish the principal axis of the track by locating the centers of the two semi-circles and the intersections of the axis with the curves ; also establish the ends of the curves, preferably on the true measured line (3 feet from the hub plank for a sulky track, and 18 inches from the inner edge for a bicycle track). (4) Eun in each quadrant, 118 THE TEANSIT. either by the deflection angle method, or, if trees or other obstructions do not prevent, by using the wire as a radius with observed pull ; set points 16 feet apart unless in- structed otherwise. (5) After locating the true line, check up the total distance very carefully. (6) Make plat and complete record of survey. PROBLEM D13. ANGLES OE TRIANGLE BY REPETITION. (a) Equipment. — Transit, reading glass, 3 chaining pins, 2 tripods with plumb bobs (if necessary). (b) Probletn. — Measure the angles of a prescribed tri- angle with transit by repetition. /' Observers : John Doe t ■ Richard Kos- \ At ISLES IF Tri ANSLE 5-6 -8 BY Repetition. Bi ■ffifBerger Transit''9-\ station BubDirec Object Vern-A- Vern-B- Mean Difference Angle Mean Angle Remarks ble tion Ae hm msh AS mh'u o'mW osW HovSD,'). tfZJfours) . 'ooI^^t/U /•• AS m'47h> '47'47'2II '47'ZI!" 47'47'ZD" Si'ng/e ss'si'zt za'si'n ' S6'20' ISI'SI'Z^' 47'47'I6" SXaps- lip UF) AS fWdn' mVis ' OO'M" AS *7'4T6l 'ZZf47'll) ' 47'0(l' 47'47'm" ..iffje l3Si(ifi SS'JlW ■ S^'4<)" 73S'S6'4I>" 47'47'Zll" 47'4:'lg" SKeps- AS a R AB oWss'' 7!th'm ' rn'os" J>ecI,'9S- ''3//mrs) I y^rm^iiv ref. AS 43'IZ'ZII mirit 'Z2V 4i'ZZ'Z0" 5/ngfc- 2ieViv 36i7'0 ■ sm' zis'sz'do" 43°ZZ'Z4" S£° S/ngJe it'siki iW'SI^ ' Sl'40' ZJ6'Sl'4l!" 43'ZZ'Z^'' 43'ZZ'ZZ" SXrps- A3 D R AS i^'tsii!'' rn'mit me^ Ae n'mii' m'siit ' so'zs' H'SI!'ZI>'' Single l4'lZ'0 'ze4V4i ' jz'4e> 444'JZ'4(>'' Sf'Sl'SZ' SReps- u L Ae mVit' t'ltio' u'mi' AS m'si'a 'sfy'zti ' Sl'Zl' s/'so'zo" Single Z64'/l'Zhri3,Dec-7,19/S S ZJ-Z Correction far Psi'lroad Tfme - 7-0 g-RTfmeV-C-PaJsnXDec.7,l9/S S ZD-Z kaducf-ion for Western flongatfon + 5 55'0 RslJroad Time i> a z''jS-o" Calculation of Azimutli of Polaris at Elon^'n Az!miifhPt>raris,Btons'f'n,Jsn-J,Wi l'Z9'9 CorrecHon far Pec- T, 1913 - 0-8 A:iimutj7 Polaris, flong't'n, Pec^T, 1913 l'Z9't Time of Elongation of Polaris' For Western flongsfion scfd S''SS'"- fa time U-C- Polaris • far Eastern flon^ation subtract S''SS'"- from time U-C- Observers, J-Oae ^ R-Roe- BY Ob's on Polaris at Elongatioh- Pec.7,l9ISfZ Hours), CIe.sr 3nc^ warm- duffSBer^er Transit lia-S, ZLanferns, Inbs, Zflatstalces,plank lS"t4''Z', 4- Sd nails, axe, watch settalceep Railroad time- Set transit over hvi> 3t i!40A'i^; si^iited at Polaris, depressed file teiescape and estabjisiied target about 500 ft- fn>m instrament- The pianic was placed af- r/^iit 3n0ies to Jine and fjai/ed to 3 Stake driven s--f-Z4^ff(^'"~Mean SoJarT/Jne, U-C-foJ3rJs CZ3^Z6^J T//7?e armament = /four An^/e of Pay of f/ie same cfafs-- Obs- on Polaris at any Time . Apr/IZI, J915 (2/fours) Clear 3? Warm. Buff 3? Ber^dr Transit If ^9j Z lanterns, hubs, Zf/at stakes, p/ank3e'x4'x 21 4~i(f-n3iJs, watch set to keep ^all" road or Standard T/mCj axe- Set transit aver hub at S'lS P-ff; sighted gt Polar/s, depressed teles-* cope and established target SOO ft' from Instrument, the plank was placed at right angles to line and nailed to a stake driven at each end- Set vertical hair on star, noted time, depressed the telescope and marked line en target with pencil- Apr 22, 19/5 (2 Ifrs^ f educed obsfra- -tlons using Azi- muth Tables' laid off Azimuth fa the fast and measured angle with the True tferld/an- Polaris at Time oF Observation Ir ^'6 fO-'» fO-'7 Setting fff'SF3 Hl'SSi tJ9'SS-'? m'iS'7 f/9'SS'6 W'JS-'S H3'i9-'Z Azi- riufh tZ'll' ZZ'IZ' Z2'ja' zz'ji ZZ'JI' zz'ts' z?'/o' zz'os' ZZ'll' ZZ'P9' ZZ'JO'i zzWi. o'm True Atiffiufh of Line MIoivabJe error 01'- frrorA Calculation of Setting fApf-Bedil 8!'30'"AM- LetltUife 40'(!6'tl; langifude SS'JS'W- App-Pecliffafion ffreenivlch Mean /foon. (etllT^ AM- ltere),l1syZt>,J90J, fJS'Si'S Correction for Z''Z3'"=Z-4i<)-S}, * Ol-'S tecJinat-Jon ofJunstf--3l>AM- = i-l9'£4'-S Xefracl-hn Cor^-3''30'ie/iremi'n-+ 0-7 Apii-Dccl-3tS:3Ci\M- - f/S'SS^S Apparent- Pec/- (^ett/n^) for f/je other times was eaJci/fsfeet in JiMe manner- WITH Solar Transit- Observers, J- Doe if B- Roe- MayZO,190/. C4 ffrs-) C/ear i^ warm- SuffifSergerTrans/f/i^S, ty/fh Saeg- muiler SffJarAttacIimentjItubs, ajie^ watch settolceep JSaJiroad Time, Se/ar fphemeris CMsndbmk for Enqineers, By 6eo./1.S3egr/7Uller, Bausch&Lomb OpticdICo. Rochester,/1.Y.) Tested Transitand SolarAttacbment and found both in perfect- adjustment. Set tran sit over hubj JeveJedup very carefui/y yvit/i ianf bt/bbJej found Index frror of Vert- Circle ='Zerff- Setoff -i9-°SS'S (-App-Pecl-) on Vertical circie-and Jeve/ed solar tet escape by means of its attached bubble- Set off t49'S4' Cfo-lafJ on Vertical circle-TeJescopff pointed S-bo/tr times- Set A vernierat zero and sighted af 5ta-3 with Jower rnotion- Unclamped upper motion, moved transif- on vertical axis andsoiar on its polar axis, and brought image of sun into center of solar = 3 x-iS-S" = l'47-4" N- Dedinafion at 9 A-M- = /f'47'43-SV. Average Vertical Angle liy Otservetfort 4e'o3'PO" (^rrecffon for PeFr3cf:or7 00'36' ' True Altifi/da Latitude of Obseri^atoryj U- of I- Station JOO'tl- Latitude of Stafian 4e'0Z'04" 40' 06' 00" 40' 06' 00" Co, J- P2S=. }/^'"i S- xSin-dS'PoleDist^ Cos-^PZS- \l 5A7. CoAlt-r. Sin- Ca-Laf- where S=Pole Plot. + Co-Alt--t Co-laf- Pole Dist- = 7/" 12' 16" Co-Alt = 43'STS6'' Co-Lat- = 49° £4' 00" S = ies°04'ie" is = SZ'32'06" Pole Diet- = 7J'IZ'IG" ^5- Pole Dist- = ll'J9'S0" Log-Sln-82'3Z'0e'= 3-99630 Log- Sin- II'I9'S0"= 9-29313 Co-lag- Sin- 43°S7'S6"= 0-ISSSO Co-log- Sin- 49'S4'00"=' 0-11638 2)l9-S6441 -20 Log-Cos-iPZS = 3-78224 iPZS= 32 '4-5 'IS" , PIS = IOS'26'36" Azimuth of Sun from the North Angle between Sun and Mark Observed Azimuth from Harth Station to Mark True Azimuth from Iforth Sfatj'on to Mark- Error PROBLEM D17. DETERMHSTATION OF TRUE MERIDIAN BY DIRECT OBSERVATION ON THE SUN. (a) Equipment. — Complete transit, reading glass, hub, axe, colored eyepiece or colored shade to fit over objective, good watch set to keep standard time, solar ephemeris. 132 THE TRANSIT. (b) Problem. — Determine a true meridian by a direct ob- servation on the svin with a transit. (c) Methods. — (1) Set the transit over a hub and level up very carefully with the attached bubble. (3) Test the ad- justments of the transit very carefully, and determine the index error of the vertical circle. (3) Sight on a horizontal mark and read the horizontal plates. (4) Sight at the sun directly, by the aid of the colored eyepiece or colored glass shade, and bring his image tangent to the horizontal and vertical wires. (5) Read vertical circle and horizontal plates. (6) Reverse the telescope and make a second ob- servation the same as the first except that the sun should be in the opposite quarter of the field of view. (7) The mean of the vertical and horizontal circle readings will give the apparent altitude and plate reading of the sun's center. (8) Observe the standard time of the observation and reduci to mean solar time by adding or subtracting 4 minutes for each degree that the place of observation and reduce to mean solar time by adding or subtracting 4 minutes for each degree that the place of observation is .>ast or west of the standard meridian. (9) Calculate the angle PZS in the P Z S triangle as shown in the accom- panying form. Refraction makes the sun appear too high and it should therefore be subtracted. (10) Determine the azimuth of the line from the hub to the mark and check the observed azimuth. (The data for this problem may be obtained from Saegmuller's " Solar Ephemeris and Refrac- tion Tables,'' or from the " Ephemeris of the Sun and Polaris, and Tables of Azimuths of Polaris," by the General Land Office, mentioned in Problem D16. Mean refraction of the sun for different altitudes is given in Table V.) (11) Where considerable accuracy is desired, make a second ob- servation when the sun is about the same distance on the opposite side of the meridian. The error of the determina- tion should not exceed 1 minute. PROBLEM D18. COMPARISON OF TRANSIT TELESCOPES. (a) Equipment. — Eive engineers' transits. (b) Prohlem. — Make a critical comparison of the tele- scopes of five engineers' transits. (c) Methods. — Follow the methods outlined in the com- parison of level telescopes. PROBLEMS. 133 PROBLEM D19. TEST OF A TRANSIT. (a) Equipment. — Transit, reading glass, leveling rod, chaining pins, foot rule. (b) Proilem. — Test the following adjustments of an as- signed transit: (1) Test the graduation for eccentricity. (2) Test the plate levels to see if they are perpendicular to the vertical axis. (3) Test the line of collimation to see if it is perpendicular to the horizontal axis. (4) Test the horizontal axis to see if it is perpendicular to the vertical axis. (5) Test the level under the telescope to see if the tangent to the tube at the center is parallel to the line of collimation. (6) Test the vertical circle to see if the vernier reads zero when the line of sight is horizontal. (c) Methods. — Make the tests as described in the first part of this chapter but do not make any of the adjust- ments or tamper with any of the parts of the instrument. Check each test. Make a careful record of the methods and errors, including a statement of the manner of doing cor- rect work with each adjustment out. PROBLEM D20. ADJUSTMENT OF A TRANSIT. (a) Equipment. — Transit, reading glass, leveling rod, chaining pins, adjusting pin, small screw driver. (c) Methods. — Make the following tests and adjustments of an assigned transit that has been thrown out of adjust- ment by the instructor: (1) Test the graduation for eccen- tricity. (2) Adjust the plate levels perpendicular to the vertical axis. (3) Adjust the line of collimation perpendicu- lar to the horizontal axis. (4) Adjust the horizontal axis perpendicular to the vertical axis. (5) Adjust the level nnder the telescope parallel to the line of collimation. (6) Adjust the zero of the vertical circle to read zero when the line of sight is horizontal. (7) Center the eyepiece. (c) Methods. — Make the tests and adjustments as de- scribed in the first part of this chapter. Use extreme care in manipulating the screws and if any of the parts stick or work harshly, call the instructor's attention before pro- ceeding. Repeat the tests and adjustments. Make a care- ful record of methods and errors. PROBLEM D21. SKETCHING A TRANSIT. (a) Equipment. — Engineers' transit. (b) Prohletn. — Make a first-class sketch of an engineers' transit. (c) Methods.- — (See similar problem with the level.) 134 THE TRANSIT. PROBLEM D33. ERROR OF SETTING FLAG POLE WITH TRANSIT. (a) Equipment. — Transit, iron flag pole, flat stake l"x 2"x 15", foot rule. (b) Prohlem. — Determine the probable error of setting a flag pole with the transit at a distance of 300 feet. Repeat for 600 feet. ^ Observers, J-Doe c R^Roe- ^ Err OR OF SET- IHG I UA6 'OLE WITH ENSINEER >' Transit- )Ht3nce No.oF Distance d d^ Dec-6,m4.(2hiiun 1 Cool and Quiet- Ft- StHlnj In- In- Used Suff S Seiyar Transit, LocJcer/io^^i 300 / I-/S 0-JS 0'03Z4 f/afsfake,/'"?' ^lS"and iron flagpole- Z J-3S ■02 ■0004 Sighted 3 f- /ran F/ag poie set- on stake 3 /■}0 ■06 ■0036 which had been } laced on ground -Zf=2rz 600 J (c) Methods. — (1) Set the transit up and sight at the flag pole plumbed near the middle of the stake at a distance of about 300 feet. (2) Measure the distance from the point of the flag pole to a mark on the stake. (3) Keep the vertical axis clamped, and move the pole to one side. (4) Set the pole with the transit, and measure the distance from the first line. (5) Repeat until at least ten consecutive satis- factory results are obtained. (6) Compute the probable error of a single observation and of the mean of all the observations (see chapter on errors of surveying), and re- duce the mean error to its angular value. (7) Repeat for 600 feet. Determine distances by pacing. Follow the form. PROBLEMS. 135 PROBLEM D23. REPORT ON DIFFERENT MAKES AND TYPES OP TRANSITS. (a) Equipment. — Department equipment, catalogs of the principal makers of engineers' transits. (b) Prolileni. — Make a critical comparison of the several types of transits made by the different makers. (c) Methods. — (See similar problem with the level.) CHAPTER VI. TOPOGRAPHIC SURVEYING. Topographic Map. — A topographic map is one which shows with practical accuracy all the drainage, culture, and relief features that the scale of the map will permit. These features may be grouped under three heads as follows : (1) the culture, or features constructed by man, as cities, villages, roads; (2) the hypsography, or relief of surface forms, as hills, valleys, plains; (3) the hydrography, or water features, as ponds, streams, lakes. The culture is usually represented by conventional symbols. The surface forms are shown by contours (lines of equal height), (a). Fig. 24, or hachures, (b), Fig. 24. The -water features are shown by soundings, conventional signs for bars, etc. Fig. 24. Topographic maps may be divided into two classes de- pending upon the scale of the map. Small scale topographic maps are made by the U. S. Coast and Geodetic Survey and the U. S. Geological Survey, and are drawn to a scale of 1 : 62,500, 1 : 125,000 or 1 : 250,000 with corresponding contour intervals of 5 to 50, 10 to 100, and 200 to 250 feet. These maps show the streams, highways, railroads, canals, etc., in 1.W 138 TOPOGRAPHIC SURVEYING. outline but do not show any features of a temporary char- acter. For topographic symbols, see Chapter XI. Large scale topographic maps are drawn to a scale of 400 feet to 1 inch ( 1 . 4800) , or greater, with contour intervals from 1 to 10 feet depending upon whether the ground is iiat or hilly Roads, streets, dwellings, streams, etc., are drawn to scale. Features too small to be properly represented when drawn to scale are drawn out of proportion to the scale of the map. Topographic Survey. — The object of a topographic sur- vey is the production of a topographic map, and hence neither time nor money should be wastefully expended in obtaining field data more refined than the needs of the map- ping demand. A topographic survey may be divided into three parts: (1) the reconnaissance; (2) the skeleton of the survey; (3) filling in the details. Reconnaissance. — The reconnaissance is a rapid prelim- inary survey to determine the best methods to use in mak- ing the survey and the location of the principal points of control. A careful reconnaissance enables the topographer to choose methods that are certain to result in a better map and a distinct saving of time. Skeleton. — There are three general methods of locating the skeleton of a topographic survey: (1) tie line survey with chain only, (2) traverse method with transit or com- pass; (3) triangulation system, (f), Fig. 30. The first method is used for the survey of small tracts. The second method, in which the distances are measured with the chain, tape, or stadia, is used on railroad and similar sur- veys. The third method, in which ■ triangulation stations are connected with each other and with a carefully meas- ured base line and base of verification, is used on surveys for small scale maps and on detailed or special surveys, such as surveys of cities and reservoir sites. Filling in Details. — There are three general methods em- ployed for filling in the details : ( 1 ) with transit or compass and chain; (2) with transit and stadia; (3) with plane table and stadia. The transit and stadia are used by the Mississippi and Missouri River Commissions. The plane table and stadia are used by the TJ. S. Coast and Geodetic and the U. S. Geological Surveys. Topographic City Survey. — A topographic city survey is one of the best examples of a survey for a large scale map. It is usually based on a system of triangulation executed with precision and connected with carefully measured base THE STADIA. 139 lines. The details of the survey are usually taken up in the following- order: (1) reconnaissance and location of trian- gulation stations ; (2) measurement of base line and base of verification; (3) measurement of angles by repetition ; (4) establishment of bench marks by running duplicate levels ; (5) adjustment of angles of triangulation system; (6) com- putation of sides, azimuths and coordinates; (7) filling in details, usually with transit and stadia; (8) plotting of triangulation and other important points on the map by rectangular coordinates; (9) plotting the details and com- pleting the map. The instructions given on the succeeding pages are for a survey of this type. Hydrographic Survey. — Hydrographic surveying is di- vided into river and marine. The first includes the location of bars and obstructions to navigation, and the determina- tion of the areas of cross-section, the amount of sediment carried, etc. The second includes the making of soundings, location of bars, ledges, buoys, etc. The depth of the water is determined by making soundings with a lead or rod, and the velocity is gaged by means of fioats or a current meter, (d). Fig. 31. Soundings are located: (1) by two angles read simulta- neously from both ends of a line on the shore, (f). Fig. 31; (2) by keeping the boat in line with two flags on shore, and determining the position on the line by means of an angle read on the shore, or by a time interval ; ( 3 ) by intersecting ranges, (g). Fig. 31 ; (4) by stretching a rope or wire across, the stream; (5) by measuring with a sextant in the boat at the instant that the sounding is taken two angles to three known points on the shore, (c). Fig. 31 ; the point is located by solving the three point problem graphically with the three arm protractor, (e). Fig. 31 ; (6) by locating the posi- tion of the boat at the instant that the soundings are taken with transit and stadia. The first three methods are used on small river or lake surveys. The fourth method is used where soundings are taken at frequent intervals. The fifth method has been used almost exclusively in locating sound- ings in harbors, lakes, and large rivers. The sixth method is rapidly coming into general use and promises to be the favorite method. THE STADIA. Description. — The stadia is a device for measuring dis- tances by reading an intercept on a, graduated rod. The stadia-hairs, shown in (g) , Fig. 27, are carried on the same 140 TOPOGEAPHIC SURVEYING. reticule as the cross-hairs and are placed equidistant from the horizontal hair. The stadia-hairs are sometimes placed on a separate reticule and made adjustable. It is, how- ever, considered better practice by most engineers to have the stadia-hairs fixed and use an interval factor, rather than try to space the hairs to suit a rod or to graduate a rod to suit an interval factor. Stadia Rods. — Stadia rods are always of the self reading type. In Fig. 27, (a) and (b) are the kind used on the U. S. Coast Survey; (c) on the U. S. Lake Survey; (d) and (c) by the U. S. Engineers. A target for marking on the rod the height of the horizontal axis of the transit above the station occupied is shown in (f), Fig. 27. Theory of the Stadia. — In Fig. 25, by the principles of optics, rays of light passing from points A and B on the rod through the objective so as to emerge parallel and pass through the stadia-hairs a and 6, respectively, must inter- Fig. 25. sect at the principal focal point (J in front of the objective ; therefore the rod intercept, s is proportional to the dis- tance, g from the principal focal point in front of the ob- jective. Stadia Formula For Horizontal Line of Sight and Ver- tical Rod. — In Fig. 25, from similar triangles we have From which :: i : f g^ rS = k. S (1) (2) and D = k. s + (c -f f ) (3) Stadia Formula For Inclined Line of Sight and Vertical Rod. — In Fig. 26 we have and but also THE STADIA. 141 BD=iAE. cosa (approx.) (4) D =k. s. cos a + (c + f ) (5) H ^ D. cos a k. s. cos2 a + (c + f ) cos a (7) = k. s — k. s. sin2 a -(- (c -|- f ) cos a (8) V = D. sin a (9) = k. s. sin a. cos a +(c + f) sin a (10) = l,^k. s. sin 2 a+(c+f) sin a (11) Use of the Stadia. — The transit is set up over a station of known elevation and with a given direction or azimuth to another visible station ; the height of the line of coUima- tion above the top of the station Is determined either by- holding the rod beside the instrument and setting the target, or preferably by graduating one leg of the tripod and using the plumb bob ; then with the transit oriented on a given line, " shots " are taken to representative points, and record made of the rod intercept, vertical angle and azi- muth. In reading the intercept the middle hair is first set roughly on the target, then one stadia-hair is set at the nearest foot-mark on the rod and the intercept read with the other stadia-hair, after which the precise vertical angle is taken, and tUe azimutli is read, 142 TOPOGRAPHIC SURVEYING. Beducing' the UTotes. — The notes may be reduced by means of tables, diagrams, or a special slide rule. The slide rule is the most rapid. There are several forms of stadia slide rule that are very accurate and are convenient for field use. < i < 4 i < 4 < (3J X X (b) > > CCJ (d) Fisr. 27. M M (6) r ^ (f) THE PLANE TABLE. Description.^The plane table consists of an alidade, car- rying a line of sight and a ruler with a fiducial edge. The alidade is free to move on a drawing board mounted on a tripod. The drawing board is leveled by means of plate levels. The line of sightf should make a fijced horizontal angle with the fiducial edge of the ruler. The complete plane table is a transit in which the horizontal limb has been replaced by a drawing board. There are three general types of plane tables: (1) the Coast Survey plane table, (a). Fig. 28; (2) the Johnson planB table, (b), Fig. 28; (3) the Gannet plane table, (d), Fig. 39. TTse of the Plane Table. — In making a survey with the plane table the angles are measured graphically and the THE PLANE TABLE. 143 lines and points are plotted in the field. The principal methods of making a survey with a plane table are: (1) radiation; (3) traversing; (3) intersection; (4) resection. Radiation. — In this method a convenient point on the Complete Plane Tables. Fig. 38. paper is set over a selected point in the field, and the table clamped. The line of sight is then directed towards each point to be located in turn and a line is drawn along the Eg. 39. 144 TOPOGKAPHIC SUKVEYING. fiducial edge of the ruler. The distances, which may be de- termined by measuring with chain, tape or stadia, are plotted to a convenient scale, (a). Fig. 30. Traversing. — This method is practically the same as traversing with a transit, (b). Fig. 30. Care should be used in orienting the plane table to get the point on the paper over the corresponding point on the ground as nearly as the character of the work requires. C m--: ^D A E f3) 'K' -y^R I r^j 3 --^bl Kg. 30. THE PLANE TABLE. 145 Intersection. — In this method the points are located by intersecting lines drawn from the ends of a measured base line, (c), Kg. 30. Resection. — In the resection method the plane table is set up at a random point and oriented with respect to either three or two given points, which gives rise to two methods known respectively as the three-point and two-point prob- lems. Three Point Problem. — Where three points are located on the map and are visible but inaccessible, the plane table is oriented by solving the " three point problem." There are several solutions, the best known of which are: (1) the mechanical solution; (3) the Coast Survey solution; (3) Bessel's solution; (4) algebraic solution. The problem is indeterminate if a circle can be passed through the four points. In the mechanical solution the two angles subtended by the three points are plotted graphically on a piece of trac- ing paper, and the point is located by placing the tracing paper over the plotted points. In Bessell's solution, (d), Fig. 30, a, 6, c are three points on the map corresponding to the three points, A, B, C on the ground, and D is the random point at the instrument whose location, d, it is desired to find on the map. Con- struct the angle 1 with vertex at point c as follows : Sight along the line ca at the point A, and clamp the vertical axis. Then center the alidade on c and sight at B by moving the alidade, and draw a line along the edge of the ruler. Con- struct the angle 3 with vertex at a in the same manner. The line joining 6 and e will pass through the point d required. Orient the board by sighting at B with the line of sight along the line e 6, and locate d by resection. Two Point Prohlem. — To orient the board when only two points are plotted, proceed as follows : Select a fourth point, C, that is visible, and with these two points as the ends of a base line, (e). Fig. 30, laid off to a convenient scale, locate two points a' and 6' on the map by intersec- tion. The error of orienting the board will be the angle between the lines o-6 and a'-h'. The table can now be oriented and the desired point located on the board by re- section. Adjustments. — The adjustments of the plane table are : (1) the plate levels; (3) the line of collimation; (3) the horizontal axis; (4) the attached level. These adjustments are practically the same as those for the transit. 11 146 TOPOGRAPHIC SUEVEYING. THE SEXTANT. Description. — The sextant consists of an arc of 60°, with each half degree numbered as a, whole degree, (a), Fig. 31, combined with mirrors so arranged that angles can be measured to 120°. Boat Boaf Boaf Boat (9) %^. ^>C-.viC<-.- Fig. 31. THE SEXTANT. 147 Theory. — The principle upon which the sextant is con- structed is that if a ray of light is reflected successively be- tween two plane mirrors, the angle between the first and last direction of the ray is twice the angle of the mirrors. In (b), Fig. 31, the angles of incidence and reflection are equal, i = r and i' :^r', and E = (i-|-r) _(i' + r')=2(r-r') C = (90° — i') _ (90° — r) = (r — r') and therefore E ^ 2 C but C = angle CIC, by geometry, since the mirrors are parallel for a zero reading. TTse of the Sextant. — To measure an angle between two objects with a sextant, bring its plane into the plane of the two objects ; sight at the fainter object with the tele- scope and bring the two images into coincidence. The reading is the angle sought. The angle will not be the true horizontal angle between the objects unless the objects are in the same level with the observer. Since the true vertex of the measured angle shifts for different angles, the sex- tant should not be used for measuring small angles be- tween objects near at hand. Adjustments, Index Glass. — To make the index glass, 1, perpendicular to the plane of the limb, bring the vernier to about the middle of the arc and examine the arc and its image in the index glass. If the glass is perpendicular to the plane of the limb, the image of the reflected and direct portions will form a continuous curve. Adjust the glass by means of the screws at the base. Horizon Glass. — To make the horizon glass, H, parallel to the index glass, I, for a zero reading. With the vernier set to read zero, sight at a star and note if the two images are in exact coincidence. If not, adjust the horizon glass until they are. If the horizon glass cannot be adjusted, bring the images into coincidence by moving the arm and read the vernier. This reading is the index error which must be applied with its proper sign to all the angles measured. Line of Collimation. — To make the line of collimation parallel to the limb. Place the sextant on a plane surface 148 TOPOGRAPHIC SURVEYING. and sight at a point about 20 feet away. Place two objects of equal height on the extreme ends of the limb, and note whether both lines of sight are parallel. If not, adjust the telescope by means of the screws in the ring that carries it. PROBLEMS IN TOPOGRAPHIC SURVEYING. PROBLEM El. DETERMINATION OP STADIA CON- STANTS OF TRANSIT WITH FIXED STADIA-HAIRS. (a) Equipment. — Complete transit, stadia rod, steel tape, set chaining pins, foot rule. (b) Prohlem. — Determine the stadia constants c, f and Ic for an assigned transit. (c) Methods. — (1) Set up the transit and set ten chaining pins in line about 100 feet apart on level ground. (2) Plumb the stadia rod by the side of the first pin. (3) Set the lower hair on an even foot or half foot mark keeping the telescope nearly level, and read the upper stadia-hair. (4) Record the intercept. (5) Read the intercept on the rod at the remaining pins. (6) Measure the distance from the center of the transit to each pin with the steel tape. (7) Focus the objective on a distant object, measure /' (the dis- tance from the plane of the cross-hairs to the center of the objective), and c (the distance from the center of the ob- jective to the center of the instrument). (8) Calculate the value of the stadia ratio, /r, for each distance by substitut- ing in the fundamental stadia formula. (9) Take the arith- metical mean of the ten determinations as the true value. (10) Compute the probable error of a single observation and of the mean of all the observations. The interval factor should be determined by the instrument man under the con- ditions of actual work. The determination should be checked at frequent intervals during the progress of the field work. Follow the prescribed form. PROBLEM E2. STADIA REDUCTION TABLE. (a) Equipment. — (No instrumental equipment required.) (b) ProTjlem. — Compute a stadia reduction table giving the horizontal distances from a point in front of the objec- tive equal to the principal focal distance for the stadia in- tervals from 0.01 feet to 10 feet, for the transit used in Problem El. PROBLEMS. 149 DlTERMIHATIO ) Ho. S Ft. IS/ • 2-70 • 3-SS : 4-)S ■' S-61 ' 6-Sll. ! 7-90 1 ill ieM-71 D Ft. m-4I Z6S-4I> iS5-3Z 4gZ-S0 sse-io e4i-ss 7S6-93 ■.7m t ' n-' m mz4 Z67-S3 iS4-15 399-7Z 4gMi SS5-J3 e4Z-4I 71S-76 m-Z3 m}-S4 = 0-6 = 0-e 0-47 f't- 0-70 1-17 OP k Ft. 39-lPZ Sg-96 91-0Z 93-71 S9-J/ 99-ZO 91S4 9S-47 9S-9! 99Ze 3TADl>fi i S9-1/4 ft- 0-0 I O-OS 0-n 0-3Z 0-07 o-je o-z:! M3 0-J3 0-ZZ 0(!M4 (/■0064 11.0144 0-J0Z4 M049 o-ozse 0-04C0 0-IS45 M163 II-0484 0-4443 0-lBff H-OSfr Cot|sTAHTS - Fixed Iairs- Obseri'ers, J-Poe ■ F ^-Rae- DecJ4, '14-CZHmn ■) Cau/ iS C/ni/dy- Used Buffs Berffs r 7r3nsifj Lacker J2, and Chaining Loc. -.er f{^3S' Ssf JO chaining /jfns m fins alra^f/ff^f/- apart on leveJ t around' Wjfh felescopff af 'fr,5i7sjf J7ear/y level and defff. •/Pined Intercepf "s"3Teach pm 4 'Seff/jj^ Joiver Jiair an a fotffar ha'/f fafft mark and reading i/p^ ^er hafr- Measured d/sfanci from cenfer af frans/f fo eacJ ' p/n tvifb sfeeS tape fa nearas/- d-07 ft- Wif/i objecfff/assj'ocusedo/jad/sfani' ohject defermin 'd c apd jf By measi/r//?^ d/sA 'nee frojn center of objecfj'ye fc center of the horJzonfaJ sx/i - -and tfiep/ane of the cross-wires respectively- i/etermined tl7e different vaiues of ic by sui>sfj formuJa D=icS ^uij/?ffij7 file ■tctf- Station Imt. Obj. A F 3 A C B D C E D F E A frroi • Allows AXIMITH TiAVERSE WITH hmv^ I6°B' 227'I6' 47°I6' O'Oi' ISOW 6'I4' ISe°J4' ze9°4e' no'oi' = o'or bte erro. M33- 6e3rin9 /l-4-'00% S-aVM- H-IZVC- H3'/ijV- usVe. H-4°0W- MWe- ti-z'zsi- M'ZlfW- i-is'is'n S4'0SB- Distance Ft. 43Z 6?t iZ4 499 7SS 7S6 eis 473 47S 434 Vertical Angle to'zo' -0'40' *0°3S' f-O'SO' -O'SO' -I'W tl'iz' -C'S6' i-0°S4' -0'S4' -o'zo' Elevation Ft. 7/g-OO (-7-Z) Error Alhnsifle Erroh 7I0-! 12:31 7IS-I 70Z-7 9-/0-I) 712-8 tlA 7za.z (-Z-B) 7/7-7 711-0 0-i S-Sft- ' Soar t Observers, J^Doe il? R- Transit and Pec-/5,/m,f3fiou Used BofF S Bergi and Stadia Stadia Constants 5i^/7tedat target i Angle. Oriented tiie transij 1 l^oe- > Stadia- <) Clear and Warm- Transit^ Locker /fo-lZ, '/lo-6- \tF=H7ft., k= 100-00 itatM-I-,for Vert- hyAzimuti? reversals- 150 TOPOGRAPHIC SURVEYING. (c) Methods. — (1) Prepare form for calculation. (3) Compute the horizontal distances by substituting the dif- ferent values of s in the stadia formula. Compute D' for values of s varying from 0.01 foot to 0.1 foot varying by 0.01 foot ; from 0.1 foot to 1 foot varying by 0.1 foot ; and from 1 foot to 10 feet varying by 1 foot. Stadia Reduction Table I (c+F)= 1-20 Feet- k=ll5-,75 D=kS + (c+F) = D'+Cc+F) 1 Stadia Reading Distance D'=kS Stadia Reading Distance D'=k.5 Stadia Reading »3 [)ist^nce D'=k.5 0-0/ l-Z 0-1 //■6 hO IIS-S ■OZ Z-i ■Z 25-Z z 2ih5 ■03 3-5 •3 34-7 3 347-Z ■04 4^6 ■4 46^3 4 463-0 ■05 5-8 ■B S7^3 S S78-8 •06 £■3 ■6 6S^4 6 634-5 ■07 8-1 ■7 Sl-0 7 SW-Z ■08 3-2 ■8 3Z^6 S 9Z6-0 ■09 I0^4 ■3 104-Z 3 1041 -S ■10 11-6 1-0 1I5-8 10 JI57-S (To use the table, take the sum of the values of D' cor- responding to the units, tenths and hundredths of s as given in the table. To the value of D' thus obtained add c plus /.) PROBLEM E3. AZIMUTH TRAVERSE WITH TRANSIT AND STADIA. (a) Equipment. — Complete transit, stadia rod, steel pocket tape. (b) Problem. — Make a traverse of the perimeter of an assigned field with a transit and stadia. (c) Metlwds. — (1) Set the transit over one corner of the field and set the A vernier to read the back azimuth of the preceding course. (2) Sight at a stadia rod held edgewise on the last station to the left with the telescope normal, and clamp the lower motion. (3) Read the intercept on the rod to the nearest 0.01 foot. (4) Sight at the target set at height of first station and read the vertical angle to the nearest minute. (The observer should measure the height of the horizontal axis above the station with the steel pocket tape, or one tripod leg may be graduated and the instrument height determined by swinging the plumb bob out against PROBLEMS. 161 tHe leg.) (5) Unclamp the upper motion, sight at the next station to the right and clamp the upper motion. (6) Read the A vernier, (this will be the azimuth of the course) . (7) Read the intercept on the rod. (8) Measure the vertical angle by sighting at the target set at the height of the hori- zontal axis as before. (9) Set the transit over the next station to the right and determine the intercepts and ver- tical angles as at the first station. (10) Determine the stadia intercepts and vertical angles at the remaining sta- tions, passing around the field to the right. (11) Reduce the intercepts to horizontal distances before recording. (12) Compute the vertical differences in elevation using mean distances and vertical angles. (13) Compute latitudes and departures to the nearest foot using a traverse diagram or traverse table. FoUow^ form B4. (14) Compute the per- missible error of closure of the traverse by means of Baker's formula (see Chapter IX "Errors of Surveying") ; using " a " equals one minute times square root of number of sides, and " 6 " equal 1 : 500. If consistent, distribute the errors in proportion to the several latitudes and departures, respectively. (15) Compute the area by means of latitudes and departures, and reduce to acres. PROBLEM E4. SURVEY OP FIELD WITH PLANE TABLE BY RADIATION. (a) Equipment. — Plane table, stadia rod, 2 flag poles, engineers' divided scale, drawing paper, 6H pencil. (b) Problem. — ^Make a survey of an assigned field by radiation with the plane table. (c) Methods. — (1) Set the plane table up at some conven- ient point in the field and select a point on the drawing board that will allow the entire field to be plotted on the paper. (2) Sight at one of the stations with the ruler cen- tered on the point on the paper. (3) Draw a line along the fiducial edge of the ruler towards the point. (4) Measure the distance to the point with the stadia. (5) Lay ofE the distance on the paper to the prescribed scale. (6) Locate the remaining points in the same manner. (7) Complete the map in pencil. The map should have a neat title, scale, meridian, etc. (8) Trace the map on tracing linen. (9) Compute the area by the perpendicular method, scaling the dimensions from the map. 152 TOPOGRAPHIC SURVEYING. PROBLEM E5. SURVEY OF A FIELD WITH PLANE TABLE BY TRAVERSING. (a) Equipment. — Plane table, stadia rod, 2 flag poles, engineers' divided scale, drawing paper, 6H pencil. (b) Prohlem. — Make a survey of an assigned field by tra- versing with the plane table. (c) Methods. — Follow the same general methods as those given for traversing with the transit. Adjust the plane table before beginning the problem. Complete the map and compute the area as in Problem E4. PROBLEM E6. SURVEY OF FIELB WITH PLANE TABLE BY INTERSECTION. (a) Equipment. — Plane table, 3 flag poles, engineers' di- vided scale, drawing paper, 6H pencil. (b) Prohlem. — Make a survey of an assigned field with the plane table by intersection. (c) Methods. — (1) Select and measure a base line having both ends visible from all the stations in the field. (3) Set the plane table over one end of the base line, sight at the other end of the base line and at each one of the stations of the field. (3) Se't the plane table over the other end of the base line, orient the instrument by sighting at the station first occupied and sight at all the stations in the field. (4) Complete map and compute area as in E4. PROBLEM E7. THREE POINT PROBLEM WITH PLANE TABLE. (a) Equipment. — Plane table, 2 flag poles, engineers' di- vided scale, 6H pencil. (b) Problem. — Having three points plotted on the map, required to locate a fourth point on the map by solving the " three point problem " with the plane table. (e) Methods.— (1) Use Bessell's solution. (2) Check by using the mechanical solution. PROBLEM E8. ANGLES OP TRIANGLES WITH SEXTANT. (a) Equipment. — Sextant, 3 flag poles. (b) Problem. — Measure the angles of an assigned tri- angle with the sextant. (c) Methods.— (To determine index error, sight at a d.is- PEOBLEMS. 153 < Angle 5 OF [■rian 3LE ) :-G-N WITH SEXT/ NT station Sextant Observed Index lorrected Mean Olfservers : J-Doe tP e-JSoe- Angle Error Angle Angle Ncli-ZSjm-C^ffoiiril CdaJ^C/ear. 6 II'M^C" £ncf 3'M' 3ZWM 7 To (fetermfnff I'ffdex ^rror; s/phfect ZS'X'JO' Inverted It }2W3t 32'JI>'!!!>' 3t f/s^ staff Im 'leawaysnd K IS'll'OO' frecf n es'46'M' made refjecfeifm ?ge traincide IS19'30' /averted „ mi'si' ef4S'4S' w/tf7 (f/recf j'magt ■ TJie reading H TB'Zt'OI" Brecf It 7}'OZ'0' of the verrt/er^a vearr index ISWKl" Inverted tt T}'/IZ'3S 73'tlZ'jS' error of tJ'Je' ■ SetfJa^potes ifacj m'ss'ii) orKandN- Actual t rroreF closure oz' tieJtt sexta/if over (fc s/^tt/-e'40' 245 24S i- 4'06' i- 17-6 65/./ 9 It ft JO 46'SS' 221 226 4- /'12' t- 6-0 6395 10 ^^ 11 11 4I°S1' 218 2/8 * 0'44' /■ 2-7 636.2 1/ 11 n K 34'00' 214 2/4 - /'20' - 5-0 628-5 /2 It n li IJ'SS' 217 2/0 - 9V -34-4 599-/ 15 11 11 14 4°/0' 221 221 - S'34' -37-S 596-0 /4 11 11 JS 15S'4S' 2S0 231 - //'4S' -45-9 £17-6 /S 11 11 16 3SZ'40' 212 136 - IS°I2' - S3'S sio.o /6 a il/y No-I n SW JSS 177 - //'SS - 37-4 ««•/ 17 It It 11 3l°2l' IS3 153 - 1/°00' - 2S-7 603-8 18 II It 13 4J°1S' 746 /42 - S'Z/' - 21-0 6/2-S /9 It It TO 67'SS' JS2 182 t Ao' <• 3-7 6i7-2 20 11 11 21 Sl'4l' 743 145 + O'ST f- 2-4 635-9 2/ l/yNc.Z 22 104'Sl' J24 J19 - /2'02 - 25./ 608-4 22 It II \2i ISi'SZ' /!0 1S3 -22'55 - 69.2 564-3 \23 " " (c) Methods. — (1) Set transit up over assigned triangu- lation or other point. (2) Orient instrument, i. e., set plates to given azimuth and sight at given back sight. (3) Measure height of axis above station hub with tape or by graduations on tripod leg, and set target to correspond. (4) Take shot on given back sight and reduce results as a check before proceeding. (The program for each shot is: (a) set middle hair roughly on target, then set one stadia 158 TOPOGRAPHIC SURVEYING. hair on nearest foot-mark and read intercept ; (b) set middle hair precisely on target and signal rodman " all right"; (c) read vertical angle; (d) read azimuth.) (5) Take side shots to representative points, keeping in mind the scale of the proposed map. Select points according to a systematic plan, following along ridges, gullies, etc. Con- tour points should be taken with reference to change of slope. (6) Reduce and plot the notes, and interpolate the contours, as in the accompanying diagram. (This topo- raphy sheet should be carefully preserved for use in Prob- lem E15.) (7) After completing the survey at the assigned station, move the instrument ahead to a new stadia station, taking both fore and back sights. (8) Lose no opportunity to take check sights at other triangulation stations, tra- verse points, etc. PROBLEMS. 159 PROBLEM E14. PILLING IN DETAILS WITH PLANE TABLE AND STADIA. (a) Equipment. — Complete plane table (preferably with prismatic eyepiece), 2 stadia rods, engineers' divided scale, drawing paper, 6H pencil, pocket tape. (b) Problem. — Locate the topographic details of an as- signed area with the plane table and stadia. (c) Methods. — Follow the same methods as in Problem E13 except that the notes are to be plotted on the drawing paper in place of being recorded m the field book. Mark the points by number and write the elevation of each point imder the number in the form of a fraction. Locate the contour points by interpolation on the map and connect the points by smooth curves. Complete the map in pencil and make a tracing if required. PROBLEM E15. TOPOGRAPHIC SURVEY. (a) Equipment. — Complete transit, 2 stadia rods, stakes, hubs, spring balance, pocket tape, stadia slide rule, seven- place logarithm table, (extra tripods, stadia reduction table, stadia reduction diagrams, etc., as required). (b) Problem. — Make a complete topographic survey of an assigned area and make a topographic map. (c) Methods. — (1) Make a reconnaissance and locate the triangulation stations. Care should be used to select the triangulation stations so that the sights will be clear and the triangles well formed. A system composed of quad- rilaterals or more complicated figures will give more con- ditions and checks than a simple string of triangles. A system composed of simple triangles is sufBcient for this survey. (2) Mark the triangulation stations with gas pipe 160 TOPOGRAPHIC SURVEYING. monuments about 4 feet long, the exact point being marked by a hole drilled in a bolt screwed into a cap on the top of the gas pipe. (3) Measure the base line and base of veri- fication as described in Problem ElO. (4) Measure the angles by repetition as described in Problem D13. (5) Cal- culate the skeleton as described in Problem Ell. (6) Es- tablish permanent bench marks and determine their eleva- tions and the elevation of the stations of the triangulation system by running duplicate levels with the engineers' level, reading the rod to 001 foot. (7) Fill in the details with either the transit and stadia or the plane table and stadia, or both, as described in Problems E13 and E14. (8) Com- plete the map in pencil on manila paper, and after it has been approved by the instructor trace it on tracing linen. The title, meridian, scale, lettering and border should re- ceive careful attention. CHAPTER VII. LAND SURVEYING. Kinds of Surveys. — Surveys of land are of two kinds : (a) original surveys ; (b) resurveys. Original Surveys. — An original survey is made for the purpose of establishing monuments, corners, lines, bound- aries, dividing land, etc. The survey of a townsite and the government survey of a section are examples of original surveys. Resurveys. — A resurvey is made for the purpose of iden- tifying and locating corners, monuments, lines and bound- aries that have been previously established. The resurvey of a city block, or a survey to relocate a section corner are examples of resurveys. Functions of a Surveyor. — In an original survey it is the function of the surveyor to make a perfect survey, es- tablish permanent monuments and true markings, and make a correct record of his work in the form of field notes and a plat. In a resurvey it is the function of the surveyor to find where the monuments, courses, lines and boundaries orig- inally were, and not where they ought to have been. Fail- ing in this it is his business to reestablish them as nearly as possible in the place they were originally placed. No reestablished monument, no matter how carefully relocated, will have the same weight as the original monument if the latter can be found. In making resurveys the surveyor has no ofHcial power to decide disputed points. He can act only as an expert witness. If the interested parties do not agree to accept his decision the question must be settled in the courts. Also see Problem F6, " Eesurvey of a City Block." Responsibility of the Stirveyor for the Correctness of His Survey. — An engineer in the discharge of his profes- sional duties requiring an exercise of judgment can be held liable only for failure to exercise reasonable care and skill, or :^or negligence or fraud. A surveyor is liable not only 161 162 LAND SURVEYING. for negligence or fraud but for want of skill. A surveyor agrees to not only do his work carefully, honestly, dili- gently, but skillfully as well. The precision required in making any particular survey in order to satisfy the re- quirement for skill will depend upon the conditions ; greater accuracy being required for making a survey of an ex- pensive city lot than for a survey of a farm. Surveying is a trade and the precision required in any particular case to show proper skill is a matter to be decided by the court after evidence has been submitted. Ownership of Surveyors' Notes. — Survey notes, data, maps, plats and records obtained by a surveyor while in the employ of a city, state, railroad or other corporation, or of a consulting or independent engineer belong to the employer. A city engineer or a county surveyor has no ownership rights in the notes, data, maps, plats and records which he prepares or obtains, or are prepared or obtained by him or by his assistants, in the exercise of the duties of his olBce as city engineer or county surve3'or. Survey notes, data, maps, plats and records obtained by a consulting or independent engineer in preparing a report or plans for a client, belong to the consulting or independent engineer. The client, whether it be an individual, city, state, or cor- poration, is entitled only to the finished report or plans, and is not entitled to the notes and data used in the prep- aration of the report or the plans. Bules for Besurveys. — The following rules may be safely observed in making resurveys. (1) The description of boundaries in a deed are to be taken as most strongly against the grantor. (2) A deed is to be construed so as to make it effectual rather than void. (3) The certain parts of a description are to prevail over the uncertain. (4) A conveyance by metes and bounds will convey all the land included within. (5) Monuments determine boundaries and transfer all the land included. (6) When a survey and a map disagree the survey pre- vails. (7) Marked lines and courses control courses and dis- tances. (8) The usual order of calls in a deed is; natural ob- jects, artificial objects, coiirse, distance, quantity. (9) A long established fence line is better evidence of SYSTEM OF PUBLIC LAND SURVEYS. 163 actual boundaries than any survey made after the monu- ments of th^ original survey have disappeared. (10) A resurvey made after the monuments have disap- peared is to determine where the monuments were and not where they should have been. (11) All distances measiired between known monuments are to be pro rata or proportional distances. If the above rules do not cover the ease in question spe- cial court decisions on that particular point should be con- sulted. THE UNITED STATES EECTANGULAE SYSTEM OE PUBLIC LAND SUEVEYS. Historical. — The United States rectangular system of subdividing lands was adopted by congress May 20, 1785. The first public land surveys were made in the eastern part of the present state of Ohio under the direction of Capt. Thomas Hutchins,* Geographer of the United States, and were known as the " Seven Eanges." The townships were six miles square, and were laid out in ranges extending northward from the Ohio river ; the townships were num- bered from south to north, the ranges from east to west. In these initial surveys only the exterior lines of the town- ships were run, but mile corners were established on the township lines, and sections one mile square were marked on the plat and numbered from 1 to 36, commencing with section 1 in the southeast corner and running from south to north in each tier to 36 in the northwest section. The act of congress approved May 18, 1796, provided for the appointment of a surveyor general and changed the law relating to the surveys of public lands. Under this law the townships were subdivided into sections by running paral- lel lines two miles apart each way and setting a corner at the end of each mile. This law also provided that the sec- tions be numbered beginning with section 1 in the north- east corner of the township, thence west and east alter- nately to 36 in the southeast corner. This is the method of numbering still in use, shown in Figs. 33 and 34. * The earliest published reference to the rectangular sys- tem of land surveys is found in an appendix to " Bouquet's March," published in Philadelphia, 1764. Hutchins was en- gineer with this expedition to the forks of the Muskingum river, and wrote the appendix. (See reprint by Eobt. Clarke, Cincinnati.) 164 LAND SURVEYING. The act of congress approvefl May 10, 1800, required that townships be subdivided by running parallel lines through the same from east to west and from south to north at a distance of one mile from each other. Section corners and half section corners on the lines riinning from east to west were required to be set. The excess or deficiency was to be thrown into the north and west tiers of sections in the townships. Initial Point Standard P/ji First Standard Parallel North. T4N-, R IE. T3N-, RIE- T-2N-,1 IN-. IE. T4N-. R2E- T3N., R2E-, _ B| r T-?N-, ■ R£E- T- 1 N-, R.£E. T-4N-, R3E- I T- R- T-3N-, R-3E. , T-?N-, R-3E- 4N-, 4E. 3N-. 4E- At- ± T- I N-, R-3E- i T- 1K-. 4-E. Base Line rig. 33. The act of congress approved February 11, 1805, required that interior section lines be run every mile ; that corners be established every half mile on both township and sec- tion lines ; that discrepancies be thrown on the north and west sides of the township. This act of congress further provided " that all corners marked in the original surveys shall be established as the proper corners of sections, or subdivisions of sections ; and that corners of half and quarter sections not marked shall be placed as nearly as possible ' equidistant ' from those two corners which stand on the same line. The boundary lines actually run and marked shall be established as the proper boundary lines of the sections or subdivisions for which they were intended ; and the length of such lines as returned by the surveyor shall be held and considered as the true length thereof, and SYSTEM OF PUBLIC LAND SUEVEYS. 165 the boundary lines which shall not have been actually run and marked as aforesaid shall be ascertained by running' straight lines from the established corners to the opposite corresponding- corners." Under this law, which is still the established rule of procedure, each reported distance be- tween established monuments is an independnt unit of measure. The revised instructions issued in 1855 required that the sections be subdivided as shown in Fig. 33. The full lines representing " true " lines, are parallel to the east exterior line of the township, and the dotted lines, representing " random " lines, close on corners previously established. The order of the survey of the interior section lines is in- dicated by the small niimerals. Double corners on the north and west township lines, which were common in the earlier surveys, were thus avoided in the revised practice. Laws Inconsistent. — It is obviously impossible to pre- serve a true rectangular system on a spherical surface, ow- ing to the convergency of meridians.* To harmonize the methods of making surveys, the General Land Office has issued instructions for the survey of public lands from time to time. DETAILS OF SURVEY.— The details of the survey are taken up in the following order: (1) selection of initial points; (3) establishment of the base line; (3) establish- ment of the principal meridian; (4) running standard par- allels; (5) running the guide meridians; (6) running the township exteriors; (7) subdividing the township; (8) meandering lakes, rivers, streams, etc. See Figs. 33 and 33. Initial Points. — Initial points from which to start the survey are established whenever necessary under special instructions prescribed by the Commissioner of the General Land Office. Base Line. — The base line is extended east and west from the initial point on a parallel of latitude. The proper township, section and quarter comers are established and meander corners at the intersection of the line with all meanderable streams, lakes, or bayous. Two sets of chain- * The angular convergency, a, of two meridians is m. sin L, where m is the angular difEerence of longitude of meridians and L is the mean latitude of the two positions. The linear convergency, c, for a length, t, is t. sin a. Latitude 40°, the difference between the north and south sides of a town- ship is 0.60 chains. 166 LAND SURVEYING. men are employed and the mean of the two measurements is taken as the true value. When the transit is used, the base line — which is a small circle parallel to the equator — is run by making offsets from a tangent or secant line, the direction of the line being frequently checked by an obser- vation of Polaris. t 6 5^ i^zzlL Random K 1 ♦ 1 . ' r'43-^ 1 *!l * 1 ^ Random)^ SS-:^ Randon?!^ 9 ^ 4/ — >- Random^ 30-^ Random^ "A Random 8 >- " ^ -<—S5 k" ^ " h " h 15 si, 28— >- ■' A J7—>- 13 6 >- 1 " ^ •^49--^ " h 37—^ " A " \ " \ 24—^ 15— >■ ff z- — ^ -<--47 ^ " h " A " A 55 1 " h 35 1 1 rr 36 Fig. 33. Principal Meridian. — The principal meridian is extended either north or south, or in both directions from the initial point on a true meridian. The same precautions are ob- served as in the measurement of the base line. Standard Parallels. — Standard parallels, which are also called correction lines, are extended east and west from the principal meridian, at intervals of 24 miles north and south of the base line. They are surveyed like the base line. Guide Meridians. — Guide meridians are extended north from the base line, and standard parallels, at intervals of 24 miles east and west from the principal meridian, in the SYSTEM OF PUBLIC LAND SURVEYS. 167 manner prescribed for running the principal meridian. When existing conditions require that guide meridians shall be run south from the base or correction lines, they are initiated at properly established closing corners on such lines. Township Exteriors. — The township exteriors in a tract 24 miles square, bounded by standard lines, are surveyed successively through the block, beginning with the south- TowTUhip yo. 5 yorth, Ran&e Nil. 9 Weat, of a Principat Meridian Eatt I'.l "l« s '"tsef. 7 *!&,< . M__. i iiBiMj, \jj , .iBo:ao]; tjA t iii|ni; t^ « ijsoiooij ij^ « iiaotooi. ijj t iiw^ii i ^ 5-1* 67 ^ 3-fIO 'M St!p. St__ f^~63 ijH' ti ! Weat --^k'- I JVest oo West West g Sec It M "ft •) ' W fc- West a; West 00 ei 80|00 g SecllO S 6^0 [ West "mJoo I 3 "*io" Scd f 7__g g e^ 80)00 sojoo Sec 25 P See TFe^t 00 /A I iS West In-iT^ ^ ' Kr«( Standard Parailei Sees I £ce. £ I Sec. 4 | £ec. 5 «<0 3 West « g » ^3 To fi-c 3 rdo~ s Sec. 8 I Asc. r I Th« abovo plot represents a tfteoreticeU toton~hip tUth perfect subd-'.visfcne, eonliffwrus to the tiorHi side of a Stimd^ard .Parallel; ,fn atsumtC I ieiS'jr.. ^ Lm jUiide IQOOOO' W. of Or. Aria£S0ai.J6 A.' Fig. 34. western township. The meridional boundaries are run first from south to north on true meridians with permanent cor- ners at lawful distances ; the latitudinal boundaries are run from east to west on random or trial lines and corrected back on true lines. Allowance for the convergency of meridians is made whenever necessary. 168 LAND SURVEYING. Township Subdivisions. — A true meridian is established at the southeast corner of the township and the east and south boundaries of section 36 are retraced. Then begin- ning' at the corner to sections 35 and 36 on the southern boundary, a line is run north parallel to the township line, corners are established at a distance of 40 and 80 chains ; from the last named corner a random line is run eastward, parallel to the south boundary line of section 36, to its intersection with the east boundary of the township. A temporary corner is set at a distance of 40 chains, and a permanent corner is afterwards established midway be- Tti'04 to-ooi 40-00 — o o o o 1 1 ■--t 1 1 1 o o o o o o o 4 lzo-10 ro-oo. 40-00 1 rcj Zl-O0\ ZO-OO'. 40-00 s' I -^ — - f ^Z\-OV,Z0-0Oi 40-00 f<^J Fig. 35. tween the two permanent corners. The other corners are located in a similar manner, as shown in Pig. 33. The lines closing on the north and west boundary lines of the town- ship are made to close on the section corners already es- tablished. A theoretical township with perfect subdivisions is shown in Fig. 34. Meandering. — Navigable rivers and other streams hav- ing a width of three chains and upwards are meandered on both banks, at the ordinary high water line by taking the general course and distances of their sinuosities. The SYSTEM OF PUBLIC LAND SUEVEYS. 169 meanders of all lakes, navigable bayous, and deep ponds of the area of twenty-five acres and upwards are surveyed as directed for navigable streams. Meander corners are estab- lished where meander lines cross base lines, township lines, or section lines. , Subdivision of Sections. — In Kg. 35, (a) gives the sub- division of an interior section, (b) of section.2 on the north side, (c) of section 7 in the west tier, and (d) of section 6 in the northwest corner. Pig. 36. Description of Land. — Land is described in the rectan- gular system by giving its location in a civil township ; for example, in Kg. 36, the northeast quarter, containing 160 acres, would be described as: N E 14, Sec. 8, T 19 N, R 9 E, 3 P. M. The ten acre lot indicated in the northwest quarter would be described as: S E %, N W ^, N W %, Sec 8, T 19 N, R 9 E, 3 P. M. Corners. — The corner monuments may be as follows : (a) stone with pits and earthen mound; (b) stone with mound of stone ; (c) stone with bearing trees ; (e) post in mound of earth; (f) post in mound of stone ; (g) post with, bearing trees ; (h) simple mount of earth or stone ; (i) tree without bearing trees ; (j) tree with bearing trees ; (k) rock in place, etc. The trees on line are required to be blazed. The size, markings and proper corners to be used in any particular case and all other details are given in the 170 LAND SUEVEYING. " Manual of Surveying Instructions for the Survey of Pub- lic Lands of the United States," issued by the General Land Office, Washington, D. C. The last edition of the " Manual of Surveying Instruc- tions for the Survey of Public Lands " was issued in 1902 and may be obtained from the Superintendent of Docu- ments, Government Printing Office, Washington, D. C, price 75 cents per copy. A new edition of the Manual is prom- ised for 1915. The circular on the " Restoration of Lost and Obliterated Corners " mentioned in the next paragraph gives instructions for malting resurveys, and may be ob- tained free by addressing the Department of Interior, Gen- eral Land Office, Washington, D. C. Bestoration of Lost or Obliterated Corners.* — "An ob- literated corner is one where no visible evidence remains of the work of the original surveyor in establishing it. Its location maj', however, have been preserved beyond all question by acts of landowners, and by the memory of those who knew and recollect the true position of the original monument. In such cases it is not a lost corner. " A lost corner is one whose position can not be deter- mined beyond reasonable doubt, either from original marks or reliable external evidence." General Bales. — The following rules are derived from a brief synopsis of congressional legislation relating to sur- veys. " (1) The boundaries of the public lands established and returned by the duly appointed government surveyors, when approved by the surveyor general and accepted by the gov- ernment, are unchangeable. " (2) The original township, section, and quarter-section corners established by the government surveyors must stand as the true corners which they were intended to rep- resent, whether the corners be in place or not. " (3) Quarter-quarter corners not established by the gov- ernment surveyors shall be placed on the straight line joining the section and quarter-section corners and mid- way between them, except on the last half mile of section lines closing on the north and west boundaries of the townships, or on other lines between fractional sections. " (4) All subdivisional lines of a section running between corners established in the original survey of a township * Circular on the " Restoration of Lost and Obliterated Corners and Subdivision of sections," Department of In- terior, General Land Office, Washington, D. C. SYSTEM OF PUBLIC LAND SURVEYS. 171 must be straight lines, rtmning from the proper comer in one section line to its corresponding corner in the opposite section line. " (5) That in a fractional section where no opposite cor- responding corner has been or can be established, any re- quired subdivision line of such section must be run from the proper original corner in the boundary line due east and west, or north and south, as the case may be, to the water course, Indian reservation, or other boundary of such sec- tion, with due parallelism to section lines." " From the foregoing it will be plain that extinct cor- ners of the government surveys must be restored to their original locations, whenever it is possible to do so ; and hence resort should always be first had to the marks of the survey in the field. The locus of the missing corner should be first identified on the ground by the aid of the mound, pits, line trees, bearing trees, etc., described in the field notes of the original survey. " The identification of mounds, pits, buried memorials, witness trees, or other permanent objects noted in the field notes of survey, affords the best means of relocating the missing corner in its original position. If this can not be done, clear and convincing testimony of citizens as to the place it originally occupied should be taken, if such can be obtained. In any event, whether the locus of the corner be fixed by the one means or the other, such locus should always be tested and confirmed by measurements to known corners. No definite rule can be laid down as to what shall be sufficient evidence in such cases, and much must be left to the skill, fidelity, and good judgment of the surveyor in the performance of his work. " Actions or decisions by county surveyors which may re- sult in changes of boundaries of tracts of land and involve questions of ownership in connection therewith, are sub- ject to review by the local courts in proceedings instituted in accordance with the local statutes governing such matters." The pamphlet also contains much additional informa- tion of value. liOcations of Principal Meridians. — Principal meridians have been established as the needs of the surveys war- ranted. There are twenty-four principal meridians in all, the locations of which are given in the " Manual of In- structions," mentioned above. 172 LAND SURVEYING. Abridging Field Notes. — The government surveyors use the method of abridging field notes shovpn in Fig. 38. Cor- ners in the township boundary are referred to by letter; interior section corners are referred to by giving the num- bers of the sections meeting at the corner ; interior quarter section corners are referred to by giving the number on the section lines produced. OfFeEdDcCbBaA h - M m 6- 7- I \6 ^19--^ — F- I -31- 16 '--5- 8-- -^-9 's -^-16- -f 1^ ---II—- -4-- 4- zi- t \ I '-11- I -5--' -10- I 13 -"-15- ?-■ -"-1 II- — li- |3 -^-3'4- 13 ■14- 12 -26- I -+- ^-12- 1/ ^-15- I "2,5-^ ;/ -36- 1/ 1^ y N n o P p Q cj R Fig. 38. r 5 SURVEYS BY METES AND BOUNDS. That portion of the United States settled before the adop- tion of the rectangular system was surveyed by the method of metes and bounds. For the most part these surveys were very irregular and often involved complex and conflicting conditions. The entire eastern portion of the United States, and the state of Kentucky, were surveyed in this manner, PROBLEMS. 173 and further examples are found in tlie French, surveys in the states of IMichigan, Indiana, Illinois, Missouri, Louisiana, etc., and the Spanish surveys of Texas, California, etc. The general principles underlying the questions of ownership, priority of survey, the restoration of lost corners, etc., are identical whatever the system of survey used, PEOBLEMS IN LAND SURVEYING. PROBLEM Fl. INVESTIGATION OF A LAND CORNER. (a) Equipment. — Digging outfit, tape, etc., as required. (b) ProMem. — Collect complete evidence relative to an as- signed land corner, and after giving due w^eight to the same, laake a decision as to the true corner, (c) Methods. — (1) Make careful examination of the offi- cial field notes and records pertaining to the land corner in question and make extracts from the same for further ref- erence. (3) Seek oral evidence from those acquainted with the history of the corner. (3) Make a survey of fence lines and other physical evidence, such as witness trees or their stumps, etc., near the corner under investigation. (4) Make ? ^sAv / INVESTISATIOH OF S-W- CORNER., On'gingl UnifeiJ States Field Notes, on fl/e the S-W' Cor-, Sec-S, T-!9N.,R'9E; 3 P-M-_ Jocated on the Prairis remote from other three corners of the sect ton- On 0ct-tB,/g96, Col-S-T-Susey, when mvest/gat/on, stated that &bout 1850. ifvas then County Surveyor, was calfed the time mentioned the section lines fence' CohSusey says that hfs Fathe. surveyor near the fence corner evi the ar/0/ha/ U' 5- Survey comer- I^r- spot and found the decayed jooi'nt marked the true posit/on of the ^Po or more previous to Campbells resurve^ the boulder which was set in place t section come/} and that this monuir. pisced iy 3 much iar^er stone when iines' This stone stood /S^orso shove the ievel it was carefully towered by the Stree. Cify Engineer of Urban a- Resurveys that its present posit/on is fdenticai Conclusion- in view of Coi- Buseys other credible soi/rceSf and the enttn • character, it is conduced that the recognized is the true S'W- corner of Ca. npbej n ade i J-.Ooe. Survsyori > SECTIOH 8,T-19K.,R-9E-,3D:P-M- at Courfffousa &f Urbana, Hi-, describe as "^Post in Mound" the corner being the heavyxtimber which surrounds the Originai surxvey was made about JSZZ' ' for /nfo/:ai& tio/7 about the corner under when he vwa^ a boy, i^r- Campbeii, who on to re-est36. 'ish the SW- Cor, Sec-S- At near the cojux, " were occupied by rail fa pioneer setj Ver) pointed out to the ?s of a motrnd which he believed marked 'c/if the j9sarvi yor dug cere fully at the <7 sassafr^^s sft jJce which unquestiorrably •t in Plound **esti biished some TS years '- CohBusey sfai es that he himself carried / the County Surv eyor to pepetuate the 'nt was not cfistur bed until it was re- ''he roads wj^re op\ ''ned up on the section of the road . ^or ma. 7/ years- About IS94- ' Commission sr undei ' the direction of the since thet stone was /owered^ indicate with that previous -to the change • 'le statenoe m« with fh*. ' corroborat/on from abscence o,P conflicting evidence of any ihonument > tow and fof ft, any years so Sections, 7'i$H,,!i-$£^:$l P'M- • ^ : , ^ A > 174 LAND SURVEYING. careful examination of the site of the corner with the dig- ging outfit ; the digging should be done cautiously so as to avoid disturbance of existing stakes or other monuments. (5) If more than one monument be found, make due record of their character and positions, and make further inquiry- respecting them. (6) If no monument of any sort be found at first, continue the search diligently and do not give up finding the true corner as long as there is a remote chance of locating it. In any event, avoid wanton disturbance of any object or evidence that may have a bearing on the same. Keep a clear and concise record. PEOBLEM E3. PERPETUATION OF A LAND CORNER. (a) Equipment. — Digging outfit, a large boulder or other permanent monument, cold chisel, hatchet, plumib bob, string, stakes. (b) Problem. — Replace a temporary land corner by a per- manent monument. (c) Methods. — (1) Uncover the identified temporary mon- ument and carefully determine the true point with consist- Ar Survey of SeoH,T-2S-,R10W. C^/nmB/iced ef the 5B. cor. oF .^ec-/4: fcr the ccr. which //ujh f/^aftsr says //aeSf vnqbesfionedf as fhs cor- for otf mafije, Suis- d/sm.,S-4a'W., 77 Iks- Inirr ask J2Ins. cl/sm.,/f>f3'm,J?3 Iks J sef up a fall flag on the cor- and temporary stakes every JO chi ■ ^ sec- cor. lasf: i Intersected the W-lfee oFSec-14; 4Z correctpoint, Il-t6'£;l04 Iks bearing tree of- (/-$• $t/rveyj havl piece of steel T rail ^S /hs- long locust 16 ins- diam; 5-ZS'Mf iarroakIS " " ,N-7S'E. CHAINS 40-00 eO-24 4!K 60-1$ S-F-Kingsley, Head Cliainman' F-Hotigmanj Trausif/m C-Rowland . ^ear w 5-fom'/)gs,f/ajm3n. FOR. J-R- Comings ahd H- Rowland- Fi ^und apiece of strap railroad iron driven knows to have Ireen kept in the sama ■30 years' Ftsrkecl: d/st. disf. then ran W- or} random,var.Z°l5'e;setting in fine- Ran thence F- on corrected line Found cedar stake J ft- belowsurface Ho other evidence oF cor- to lie top of the stake For^ sec- cor.^ Planted granite toulder ZOi^Kxi cor., in true line hot ween qr- maple, IZins- diam., S-IS^F. ittrroak,l6 " it ll'34''F. 'ks' S- of the cor. Found rotten stake aF Fj om stump oF wh- oakj 24 ins- diam.f ^g surveyor^ mark distinct on it' Seta For cor. Marked: , lie Iks- disf. ISZ /« »» (lO!30A-M) at single sight will} transitfFrom con to for. l^rZ'^ST. oF road crossing and Zz Iks- 5 of line • . -ound' Put a piece oF T rail Z4" long on SBIks-SoFS. rail oF tt-e-k-B. Ifo tree near- ins, f with cross + mark For ^ quan^ec. Poland sec- cor. and marked: ' SS iks-dist. IIS n It . PKOBLEMS. 175 cut exactness. (2) Keference out the point by driving' two pairs of stakes with strings stretched so as intersect squarely over the corner. (3) After carefully checking the referencing, dig out the old monument to a depth suiKcient to receive the boulder and permit its top to set several inches beneath the natural surface if located in a road or where disturbance is probable. (4) Cut a plain cross mark on the top of the stone, and set it in place in the hole, packing the earth about it, testing the position of the mark by means of the reference stakes and strings and plumb bob ; finally leave the boulder set firmly in the cor- rect position. (5) Make reference measurements to suitable permanent points such as marks on curbing, gas pipes, wit- ness trees, etc., selected with respect to good intersections, and make reliable record of the witness notes after check- ing the same. (Other forms of permanent monuments are : gas pipe ; fish plate ; section of T-rail ; farm tile or vitrified pipe filled with cement mortar ;. post hole filled with mor- tar ; special solid monument burned like farm tile ; special casting similar to a gas main valve box, with hole in top to receive flag pole ; etc.) PROBLEM F3. REESTABLISHING A QUARTER-SECTION CORNER. (a) Equipment — Transit party outfit, digging tools, etc. (b) Problem. — Reestablish a quarter-section corner that has been obliterated or lost. (c) Methods. — (1) Collect and record all the available evidence which may assist in the discovery and identifica- tion of the corner. Examine the field notes of the original survey, the surveyors' plat book and the county atlas on file at the court house, and make diligent inquiry for credible and competent information, either written or oral as to the location of the corner. (3) Make a careful search for the monument. Trace all the lines of the original survey, pay- ing particular attention to bearing and sight trees. Dig in all the places indicated by the different lines and give up the search only after you have exhausted every possible clue. (3) If the corner cannot be found, reestablish it, giv- ing due weight to all the evidence. The surveyor should remember that the corner should be reestablished where it originally was and not where it ought to be. After having located a stake at the supposed location of the original monument, reference it out and renew the search. (4) 176 LAND SURVEYING. After the monument has been relocated, mark it in a per- manent manner as indicated in Problem F3, by a stone with a cross cut in its top or with a gas pipe well driven into the ground. Reference it out to at least two perma- nent objects selected with a view to securing a first class intersection. Make a careful record and preserve con- sistent accuracy in the work. PROBLEM r4. REESTABLISHING A SECTION CORNER. (a) Equipment. — Transit party outfit, digging tools, etc. (b) Problem. — Reestablish an obliterated or lost section corner. (c) Methods. — Follow the various methods described in Problem P3, giving special attention to the search for the original corner ; upon failing to find trace of it, run out lines with reference to the section, quarter, and quarter- quarter corners in the four directions, with linear measure- ments from the same and finally reach the most consistent decision with reference to such survey lines, ownership lines, fences, hedges, road centers, etc. (A fruitful cause of disturbance of section and other corners is careless use of road graders, or the failure to lower the corner sufB- ciently below the surface of the road.) PROBLEM F5. EESURVEY OF A SECTION. (a) Equipment. — Transit party outfit, digging tools, etc. (b) Problem. — Make a resurvey of an assigned section. (c) Methods. — (1) Make extracts from the field notes of the original survey and of all resurveys on file at the court house, and other notes that may be of value. Make dili- gent inquiry among the property owners for evidence as to the location of corners. (2) Retrace the lines, recording the location of old fences, timber markings and other evi- dences as to prior recognition of lines and corners. Use consistent accuracy. Record the original notes as given in the forms. Record the field notes in narrative style using the designation of corners as given in the resurvey plat in the form. Make a plat of the section in the manner pre- scribed by state law for a resurvey. PROBLEMS. 177 /^ iHVESTIGATIO^ OF lAND CORHERS ■COLLECTION OF EVIDENCE Extracts from Surveyors Plat Book Nov-Sf IS97, Found in the County Recorder's offi'ce fff UrBsna, J//., the '^Surveyors Plat Sook" containing plats offownsfiips showing exist- ing monaments and st/bdivisfons oF sections made by the County Surveyor, with cerf/F/'-^ - cafes oF various resurveyS' /fade ff?eFoffow~ /ng exfrscfs refating to Sec-8 , T'J9fii^-9F; ^HD- P'M' :- (From P-/S6) "l>sc-S,Jg7$, Surveyed at the reijuesf of F-Adams tlie east fine oFSec-B- Beginning atasf^ne prev/ousfy planted stJff cor' a/' said sectioHf and running thence S--to S'B' Cor' oFsame, wf)ere f Found a stone previous- ly set by Jofin Tfirasher and lewis Sommers, divided tfie distance pro rata ^ndsef Cor* sf/ffCar-oFS-f^: oFsame.'* • (Signed) T/?os-S-Xyfe Co' Surveyor. (From p' IS?) "Apr- if, IS84' Surveyed by reguesF oP 5-T-Susay the W- fines oF Sees- 8 and S •■ Seginnfng survey at S'W- Cffr- Sec- S wfyere Surveyor, J-Poe- OF 5EC-8,T.19N.,F- Apr.?5,IS99- oF Resurveys oF Ch a sfonq ispfanfedand running thence ff- to fiW-Cor-5ec-B, Found an excess oF40 Ilis., corrected back, came on fo^a stone planted by lewis Sommers at ^Sec-Can on fine be- fweei? Sees- Sand 6 • iafso pfanteda stone atSec.Cor'(S'e-7-S) and made theFoHow- ihg witnesses to tfie corner, VIZ-: Adoubfe burr oak, fS "dfam- bearing ft- $0;^"^, lOZiiks-jafsoa Wfi- Oak, f4"di3m., bear- ing ff-SSi^fSgfks. fafso set a stone aF them Cor. oFtbeSlV^ oFtfie5Wi,a'FSec.£* (Signed) Tffss-B'Kyfe Co-Surveyor' smpaign County- (portion oF Plat o/7 p-fSS, strewing exist- ing monuments-) 9E.,5rd. P-M- Stona 5fone ' 'Stcine T" stone Stone Stone Stone N 8 j^ r Surveyor, J-Doe- ^ INVESTISATIOH OF LANB CORNERS F Sec.8,T-I9N.,R-< iE.,3rdPM- COLLECTION OF EVIDENCE (Contmued) Aor. 2S, I8S9. Extracts prom FieM Motes of Origin; I Unitad States 5urv< >" Hov-^,1897, found in the CotinfyTressunsrs (5^c.6) ' (Sec £) ^ (Sec-4) Office It Urbsns, III-, tlie Pl^ Book contain - in} Plots sn(/ Abstracts of Field Hotss of 'l^^ ''" 00 Y'*^ \\ s '-'■*■ H Original United States Survey of Champaign Coonty, and made the following extracts v\ relating to Sec S, TISfi.,ll-SB, 3eo PM- .— DESCRIPTIONS OF ORISIHAL CORNERS (P-30) (Sec-7) 1 Se Xs /(Sec-9) s %e4 \^^ Corners WitnCK Trees Inches Courses Lints )Mi5i«tipn K 3d Diameter they Bear Distant % ^\^>= 5cc-Cor5- ^k 4,S,S,S p5A XBOak [W-Oak \w-Oak 24 14 S-S8°e- N-64'W M-U'E- is 230 (5ec-ll), 798 m mi , (Secie) Y^ (S"- ^ S,l,7,8 ZO H-16% zn DESCRIPTIONS OF "OBJE CTS OK THE LINE5"(P-75) WVt Pastil. Mound DESIGNATION DiSTAKCES PE5CRIPTI0K (fiWali \S-IVol. lit 24- n-sz'B- 44 Chs- Lies i,%ieji 'Ut Z4- S-IO'W- 42 H-betwimSSS ZS-00 Broak leading If- thence iSecdr- slang the channel of the HhYatS [am eim IZ s-ei'w- ?l> same 13 chs- then leaving S N-78'B- 30 it running B'ly. T'X«S [W-Oak \w-ll3k 6 MSB'S- Z3 SO- 19 Ash IZ'diam. 6 S-Wf- 20 e - S'i7 Z4-50 SmkSlks-rs-X-f'ly- R-CnS [Ash \eim n s 7'e- IB 3$-00 fnfered timber bs-HS5. 8 H-IO-E- 13 £■ ' S'S 4- 00 gntemit fimber Is-lf-fS. \^«B'S Pastil Mound 16-SO S/aokeoiks-ii- S'ly-J 13 178 LAND SURVEYlIsra. RESURVfiY OF 5E<;-t7,TltN-,R.l6W-,3D, CHAINS Se^an ■?/ 7' found sfske inpUee snd both bearing frets sfsndfng- Planted stoji£ ZS"^$"* 6'; marked-^ for oer* Thence 1} on random, var-Z^O'f-f setting temp, stakes every 10 etis- Intersected sec t/he T£Iks.W-af£. At S found rotten stake at correct point, 5-tS'W; eSlks- from stamp ofwti- oak, hearfng free of l/S-Survey • Prcve stake for con and put broken .earthenware and glass around ff' ftkd. wh-oak^/Z^dfam, tfS6^e.j i4Ztks; diso wh- oak /S'd/am-j if-S^'m, 63 Iks- from S ran B- on random, setting temp- stakes every 10 chs- Intersected sec- line 12 Iks- It- of Z- At Z Found earthen post in correct- position snd bearing trees of resurvey standing- Thence W- on corrected line. Set stake on true line- on neKtpage") to-n 39-9Z 0-98 (Cont V_ 5H-5mith, Head^ainmen' L-B-Brow/7, Axman- /■£■ Wilson, Pear tf ^-W-Smifh, Flagman. PM- FOR. THE Estate of Johh W. Smith. JuIyiZ, '^Z- Cloudy with showers- RESURVEY REFERENCE PLAT* e a F '' /• b h c 9 14 J /3 3 < Resurvey. Sec-H, Smith Estate (comtihued) CHAINS )M6 (Line S-Z cont'd} At to set stake with stones around it and marked : pine, IZ"diam.,ti46'W., 79 Iks. redo3k,Z4'di3m.,5:I$^°W; 7ZIks. ?3-}4- Set stake on true line- from 10 ran S- on random, vanZ'tS^B-, and set temp- stakes at Z0gnd40 chs- Then went fo &• found post 3nd 6eant\£ trees of resurvey sfandinq. Ran thence Wen random, var-Z'ZO'f. Z0-OZ Intersected random line from N- 6Jks. S- of temp- stake - 4C-IS Intersected random ^ line Slks-t^of temp- stske. gtl-04- Intersected sec- line 10 Iks- S- ofS- Cor- post dug out in road- Set Iron ptorr besm for con, 5-79'y/., 76 Iks., from bearing tree of U-S-Sur^^y. Thence B- on corrected line- !9-$3 At intersectton ofquarhsr ii>?ej set- ^_ post 1 1 1 1 1 II 1 1 1^ PROBLEMS. 179 PROBLEM F6. RESURVEY OP A CITY BLOCK. (a) Equipment. — Transit, 100-foot steel tape, chaining' pins, axe, hubs, stakes, 4 pieces one-inch gas pipe 2 feet long, notes of previous surveys, etc. (b) Problem. — Make a resurvey of an assigned city block. (c) Methods. — (1) Procure full notes of all the surveys and resurveys of the assigned block from the records at the court house and from any other source available. (2) Make a resurvey of the block, using the notes, and drive hubs for temporary corners. (3) Compute the latitudes and depar- tures of the courses, and if consistent balance the survey. (4) If the corners of the block as located are consistent with the existing property and street lines, drive gas pipes as permanent corners. (5) Subdivide the block into lots as shown in the notes. (6) Make a plat of the block on manila paper to the prescribed scale, showing block and lot lines, distances and angles obtained in making the survey, the names of the owners of the property and the names of the streets. Prepare a surveyors' certificate as provided by law. Trace the map if required. (The accuracy attained should be based on the valuation and other local conditions. Be- fore beginning the survey use every possible care to find the corners with reference to which the original survey was made. When lots are sold by number, the excess or de- ficiency should be divided pro rata. However, when lot lines have been long acquiesced in, it is doubtful if the courts will uphold the surveyor in interfering with the ancient lines of ownership. It then becomes necessary either to make a compromise survey that will be satisfactory to the owners, or to make a survey that is strictly according to the letter of the law, and submit the map and certificate to the courts for settlement. The surveyor should remember that he is simply an expert witness and that he had no final judicial powers.) PROBLEM F7. RESURVEY BY METES AND BOUNDS. (a) Equipment. — Transit party outfit, digging tools, etc. (b) Prohlem. — Make a resurvey of an assigned tract whose original survey was made by metes and bounds. (c) Methods. — (1) Collect full notes and data relating to the monuments, magnetic bearings, magnetic variation, date of survey, lengths of lines, etc. (2) Make a careful investigation of the lines and corners on the ground and 180 LAND SURVEYING. make notes of any evidence there found. (3) Locate and identify witli certainty as many as possible of the original monuments ; where double or contested corners exist, locate each definitely for further reference ; if corners are gen- erally lacking or doubtful, concentrate attention on at least two which give most promise of definite relocation, and re- establish these corners as carefully as possible. (4) Having at least two corners, retrace by random line the perimeter of the tract, according to the original description, begin- ning at one and closing on the other corner ; set temporary corner stakes at the several points ; note the linear and an- gular error of closure of the random traverse on the last monument. (5) Calculate the latitudes- and departures of the random survey, and determine the angular and linear relations between the random and the original survey ; also fix the position of the several random stakes relative to the supposed true positions of the respective corners. (6) Set stakes in the true positions, as calculated, reference them out, and renew the search for the original monu- ments. (7) Finally reestablish each corner in the most consistent position, put permanent corners in place, and take witness notes for each, making comiplete notes of the proceedings. Follow the form. PEOBLEM F8. PAETTTION OF LAND. (a) Equipment. — Transit party and digging outfits, etc. (b) ProMetn. — Make a partition of an assigned tract of land in accordance with instructions. (c) Methods. — (1) Make the necessary resurveys of the assigned tract, Identifying original monuments, and rees- tablishing lost corners as required. (2) JViake a plat of the partition. (3) Subdivide the land and set permanent cor- ners ; carefully establish witnesses to the corners and se- cure witness notes. (4) Prepare and file plat and descrip- tion as required by law. PEOBLEM FO. DESIGN AND SUEVEY OF A TOWN SITE (OE ADDITION). (a) Equipment. — Equipment for topographic survey for both field and office. (b) Problem. — Make a preliminary topographic survey of the proposed town site (or addition), design the plat, and make the surveys for blocks, lots, etc. PROBLEMS. 181 Resurvey of "Mission Rid6e" Consulted Cot/nty Records snd con Firmed Following Meander Notes fvr cenfer line oF highway ss descnbed in J-W-Msrt/n^s deed fo J-D-Clsrk- "H-eZ'B; 14-ch.; Il-43i% 8ch.; N-S'lV., 12 ch.; l1-7Zi'£;ll!-2Sch-; S-!2'W; e-43ch." Descriph'on referred fo sfones ef hegin-- ning and ending points- Fai'nd First- stone projecting above road, but could not locate last corner. Began at First monument ani^ ran on random according to meander notes, with Z'n'E' as magnetic declination- Drove temporary stake at each deFlect/on point and made careful search For monu- ments- Found no corners at infermediefz points, but identiFied marked boulder" as true corner at closing point SZ links due west of last sf alee oF r-andom • Made careFul calculation oF notes For shiFting over From random to true corners- (See plat opposite and cal- culations on next pair oF pa^es-) J-Doe, Surveyor- Mar- 10, 191S' Public Road for J-D-Clark- TransFerred corners sccording fo calculations and renetved searct? For original -monuments, keeping close watch For decayed stakes, but without success - Set stone at each true corner. Sta ('iandoi 1 A 1 A' B' C D' £'' F' Hate. CALClfLATIOfIS Dist- Ch. 14-00 8-00 l?-00 loss e-43 Line) H-ezio'e. H-43'il'B ns'm'w- H-nii'i- s-iik'if. al Sun irms ol H-ei'u'e- U-42'4t'B. lt-S'4Zk H-ll'4t'B- S-II'IS'W- The abi 0-ilch AF anc needle ey in Reau 13-SO 7-lS 11-8S 10-10 e-34 ve solu atFF AF' Lat- ch- H-6-S7 tl-S-gO H-11-9S 11-3-08 S-e-Z9 R^SURV^Y Dep. I1-Z7-40 S e-zs lt-Zl-11 ifvey) 11-6-33 N-S-7S 11-11-77 11-2 IB S- e-ii H-Z7-3i S- 6-?Z H-il-ll \on J3 Is due d englt corrections- Ch B-IMS E- S-Sl W- MS B- 978 W-1-34 E-?7-lS W- B-31 E-mO E- S3S Hi- 1-17 E- 9-0! W- 1-14- E-n-os W- Ml B-24-B4 \ . -<. I ased to dlFf^rence HAF OF Tot- Lat- ch- (N) 6-S7 li-ij 24-32 27-40 21-11 -/ / e-es lZ-41 24-18 27-ii 21-1! / :/ Data transcribed From pp- Copy OK- "MissioH Ridse" Road. Tot-Dep Ch-(E) 12-il 17-87 16-82 26-60 2S-26 12-10 17-4S 16-28 ZS-ll 24.64 / N kt?i f 24.64'— Notss For Shifting from Random to True Coi;pers Lat- Dep- Lks- Lks- SB' I1-6 W-26 CC' H-4 W-42 DD' S14 W-S4 £E' 5- 7 W-72 FF' W-62 Dist- Lks- 26-7 42-2 SS-'l 72-i 62-0 Bearing H-77'^Om H-84'3fk s-isifm S-84'27'U- w. sumption that the error oF closure oF > ; I'M' I'OO' I'OO' H45'4Sk mW 4 ;' 65 I^T. 64- o i-50 63 Pfe? S40' r40', 3'40' // 64*66.4 J. K. Brown +SII 62 2-40', I'OO' IW I'OQ' (1 1 Swampy^ m P.C. 040 oW OUO'f m4sk mm N\ ffeavyfimier 61 ^ 'fC.4°00'C.f. 6ltl0.4 60 60 IS a (Repeated from toppreceedinqpfqes) / '/ ^ 188 EAILROAD SURVEYING. of commencement and completion, etc., should be prepared. The notes will be kept in the prescribed form. The field notes are to be returned at the close of the day's work. All estimated data should be noted as such. Completeness and neatness of notes and records, facility and accuracy in handling- the instrument, and promptness in advancing the progress of the survey will count in the estimate of the work of the transitman. Head Chainman. — (Flag pole.) The progress of the chaining depends chiefly on the activity of the head chain- man. After setting a stake he should move off briskly (pre- ferably at a trot) and be prepared for the " halt " signal as he approaches the next station. When the full chain length is pulled out, the head chainman turns, holding the flag pole in one hand and the chain handle in the other, and sets the pole in line by signal from the rear chainman or transit- man. Much time can be saved in this process if the head chainman habitually walks about on line and if he sights back over the two stakes last set. If on curve location, he should line himself in on the prolongation of the preceding station chord, and then offset by. pacing or with flag pole a distance in feet equal to 1% times the degree of the curv& ; the calculation is made mentally and the pole can usually be set within a few inches of the correct position by the time a speedy transitman has the deflection angle set off. Having the line established, the pole is shifted to the correct distance, and the stake is driven plumb in the hole made by the flag pole spike. If the survey is a rapid preliminary line, the head chainman hastens ahead the in- stant the stake is started at the proper point, although in a more careful preliminary the chainmen check the dis- tance to the driven stake. On location surveys it is custo- mary for the chainmen to wait until the stake is driven and mark the exact distance on the top of the stake with the axe blade, and the exact line of signal from the transit- man. In this process the head chainman should keep in mind the convenience of the transitman, and in case the line is being run to a front flag, the chainman should be careful to clear the liMe frequently to allow check sights ahead. In breaking chain on steep slopes the full length of chain should usually be pulled out ahead and the chain thumbed at the breaking points so as to avoid blunders ; a plumb bob or flag pole should be iised in the process. In passing over fences it often saves time to drive a 10-d nail, with " butterfly " attached, in the top plank to serve as a TRANSIT PAETY. 189 check back sight from the next transit point. The chain- men should carefully avoid obstructing the transitman's view, to which end they should walk on the outside when locating curves. Bear Chainman. — (100-foot chain or tape, chaining pins (if allowed) , figuring pad or note book.) As the rear chain- man approaches the stake just set, he calls out " halt " and holds the end of the chain approximately over the stake, quickly lines in the flag pole in the hand of the head chain- man (or the pole is lined in by the transitman), the precise distance is given, and the chainmen move on briskly. As a rule, pluses should be read by the rear chainman, the front end being held at the point to be determined. Fractions will usually be taken to the nearest 0.1 foot, although 0.01 foot may at times be properly noted. It is the duty of the rear chainman to keep a record of pluses and topographic details when the transitman is not at hand. This record may be kept on a figuring pad and the memoranda handed at the first opportunity to the transitman, who transfers the data to his book and carefully preserves the slips for future reference. It is usually better, however, to keep the auxiliary notes in a memorandum book instead of on the loose slips. The chainmen should carefully avoid dis- turbing the transit legs. The responsibility for correct numbering of the station stakes rests chiefly on the rear chainman. It is his duty to remember the number of the previous station so as to catch blunders on the part of the stakeman. As he reaches the stake just driven, he mentally verifies its number and repeats it distinctly for the guidance of the stakeman in marking the stake to be driven ; the stakeman responds by calling the new number, and each repeats his number as a check before final approval. The rear chainman then charges his mind with the numbers and checks the newly set stake on reaching it. In case of dovibt he returns to the preceding stake and notes its numljer. Stakeman. — (Sack of flat and hub stakes, marking crayon, handaxe.) The stakeman with his supply of flat and hub stakes in a sack, should keep up with the head chainman and be standing, with stake and marking keel in hand, ready to number the new station stake on hearing the rear chainman call out the preceding station number ; the numbering is repeated, as already explained, before the .stake is driven. Chaining pins are not used, but their equivalent in checking tallies may be had by numbering the 190 RAILROAD SURVEYING. stakes ahead and tieing them up in sets of ten. By num- bering stakes at slack moments the stakeman gains time to assist the axeman in clearing the line, etc. However, special care should be taken to avoid omissions and dupli- cates. The stakeman should finish numbering the stake and hand it to the axeman by the time the head chainman has fixed the exact station point. The stakes should be numbered in a bold and legible manner, the keel being pressed into the wood for permanency. The number should read from the top of the stake downward. Stakes on an offsetted line should be so marked as 4'L or 3'R, beneath the station number. When survey lines are lettered, the serial letter should precede the station number. Guard stakes for P. I., P. C, P. T., reference points (R. P.), etc., should be clearly marked. The stakeman should assist the axeman in clearing the line and should drive stakes when the axeman is delayed. He should carefully avoid obstruct- ing the transitman's view. The stakeman is under the di- rection of the head chainman. Axeman. — (Axe, tacks, (and if so instructed) an extra sack of stakes with marking keel.) It is the duty of the axeman to drive stakes, remove underbrush from the line, clear an ample space about the transit station, etc. He is expressly warned, however, in student field practice, not to hack or cut trees or damage other property in any way, and in general, not to trespass on the rights of owners of premises entered in the progress of the survey. The flat station stakes are driven firmly crosswise to the line with the numbered face to the rear. Hubs are driven about flush and usually receive a tack ; they are properly witnessed by a flat guard stake driven 10 inches or so to the left, the marked face slanting towards the hub, as shown in Fig. 9, Chapter II. The axeman receives the marked stake from the stakeman and drives it plumb at the point marked by the spike of the flag pole. On location or careful preliminary surveys when the stakes are being lined in by transit, the axeman should stand on one side when driv- ing and keep a lookout for signals from the transitman. In shifting the stake as signaled he should use combined driving and drawing blows with the axe. When the precise point comes much to one side of the top of the hub, an- other hub should be driven alongside and the first one driven out of sight before the tack is set. The axeman should move ahead briskly and avoid delay to the chaining. The stakeman should, when necessary, drive the stake with LEVEL PAETY. 191 the spare handaxe. When the field force is scant, one man may serve in both capacities. The axeman is under the direct charge of the head chainman. Front Flagman. — (Flag pole, small supply of hubs and guard stakes in stake sack, handaxe, a few 10-d nails.) It is the duty of the front flagman to establish hub points ahead of the chaining party under the direction of the chief and transitman. In selecting transit stations he should keep in mind visibility and length of both fore sight and back sight, and to this end, points should be taken on ridge lines and where underbrush, etc., is least in the way. The practice of planting the flag pole behind the hub may be warranted occasionally, as for example, when the field party is shorthanded, but never when the regular flagman is not specially detailed for other duties. The front flag- man should keep close watch on the transitman and should habitually stand with the spike of the flag pole on the tack head and plumb the pole by standing squarely behind it and supporting it between the tips of the fingers of the two hands. Should the front flagman be flagging for an inter- polated point depending on a foresight which his pole would conceal, he should clear the line for a check sight by lean- ing the pole to one side. When crossing fences he should, when convenient, establish check sights on the top plank by driving a spike and attaching a " butterfly " Bear Flagman. — (Flag pole, hatchet, slips of paper.) The rear flagman gives back sight on the preceding transit station. The details of his duties are much the same as those of the front flagman. It is an excellent plan for him to cut a straight sapling or limb and plant it exactly be- hind the hub when signaled ahead. This picket pole is made more visible by splitting the top and inserting a slip of paper, to make a " butterfly." A series of such pickets on a long tangent line often afEords a flne check on the work when an elevated transit point is reached. LEVEL PARTY.— It is the purpose- of the level party to secure data concerning the elevations of the points along the line so that an accurate proflle may be made and the grade line established. The leveling party should be on the alert to detect errors in the work of the transit party, such as omitted or duplicated stations, etc. The party consists of two members: (1) leveler, (2) rodman. In very brushy country an axeman may be added, but this is usually un- necessary if the line cleared by the transit party is fol- lowed. 192 EAILROAD SUEVEYING. Cl EVEL Note 5 FO! . Railroad SuEvgy-) Hen, uveier/^ s t .7l — E o 0cfl3,llS3. Cocl. iwlft, Hodmsn- B-M 712-33 Spike inncfch si- r kI- oF Elm frss,eS'R- A ta-zi iis-eo. af S1S-15-I-4S, 2' S- sf r^iJ feiij:e- K S-4- 711-2 Grounc/ n 7-2 112-4 *t JS £■4 714.2 «] 13 6-4- 713-2 n zo 4-S 7IB-I » +30 ■2-1 717-S n 21 0-Z 7IS-4 ^1 s •3 -0-15 713-45 On hub ^l-SfJ-ZJ- K i-s-ss 7ZS-Zg 22 8-4- 71^-9 Oround +2S 6-6 721-7 -■> p-c- j'ao'c- IZ- 23 4-S 723-5 fi 24 3-S 724-S ti 2S 3-7 724-6 ') 26 IS 726-7 M OS-M- -J -17 726-71 Tap of grj/Jile hmS/sr, 74'X;Sh-26-H7- 7! ■/■S-32 7iS-63. 21 S-B 730-0 Grouncl tn 5-7 723-3 „ p-r. ?S 3-g 731-S n 7J 3-7 7il-S .1 50 4-3 731-3 n 31 S-2 730-f Profile 3bov&- fi (dheckec^ o's B-/i-'s srnf H-l-'i with - i-n 73S-63 712-33 -1-72 Ueck fej.Baok;). \ m-24 i-23-ZI- ^ Leveler. — (Level, adjusting pin, level note book.) The leveler should follow the most approved methods described under the head of differential and profile leveling' in Chap- ter IV. The nearest 0.01 foot should be observed on turn- ing points and bench mark rod readings and elevations and on occasional inaportant profile points. The fore sight rod readings on ground profile points are to be taken only to the nearest 0.1 foot and the nearest 0.1 foot in the height of instrument is to be used in calculating the elevation. (Be- ginners sometimes calculate elevations to 0.01 foot when the rod readings are taken only to the nearest 0.1 foot.) The leveler should be rax^id with his level as well as with figures. He should calculate elevations as fast as the rod readings are taken and should systematically cheek up the turning point and instrument heights as the work proceeds. As results are verified the same should be indicated by check marks. Each page of notes should be checked by summing up turning point back and fore sight rod readings, and com- paring -their difference with the difference between the first and last elevations or instrument heights, as the case may be, on the page. Follow the prescribed form. As far as LEVEL PARTY. 193 possible, bench, marks should be cheeked by including them in the circuit as turning points. Balance back and fore sight distances on turning points. Permanent bench marks should be established at least every 1500 feet, and located in places at once convenient and free from disturbance during construction. Later levels should check within 0.05 foot into the square root of the length of circuit in miles. When a discrepancy is found, a line of check levels must be run to fix responsibility for the error. In cross- ing streams, secure high water elevations, with dates, es- pecially of extraordinary floods, also low water level. In crossing highways obtain elevations each side for some distance with a view to avoid grade crossings. In going up or down steep slopes, gain all the vertical distance possible each setting, and follow a zig-zag course. The bottom of deep gullies may be determined by hand level. Assist the rodman in plumbing the rod, and on turning points and benches have the rod gently swung in a vertical plane to and from the instrument and take the minimum reading. The self-reading rod is to be preferred. Many levelers use the Philadelphia rod without target. If the target is used on turning points, the leveler should check the rod read- ing when practicable. Completeness, correctness and neatness of notes and rec- ords, and facility and accuracy in handling the level will be given chief weight in fixing the merit of the leveler's work. The level notes are to be returned at the end of the day's work. Biodman. — (Leveling rod, peg book, hatchet, turning point pegs, spikes, keel.) The rodman holds the rod at station stakes and at such plus points as may be required to make a representative profile. It is his duty to identify each station point and be on the lookout for duplicated or omitted stations. To this end he should habitually pace in each station, especially in grass or underbrush, and call out or signal the station number to the leveler. Should a blun- der in station numbering appear, he should positively con- firm the fact by retracing several stations, and then carry the corrected stationing ahead. The rod should be held truly plumb, which is best done by standing squarely be- hind the rod and supporting it with the tips of the fingers of both hands. On turning points, the rod should be waved gently in a vertical plane to and from the instrument. The rodman should pay special attention to placing the target right for long rods and examine it to note if it has slipped 194 RAILEOAD SURVEYING. before reading the rod. Errors of 1 foot, 0.1 foot, etc., should be carefully guarded against. Turning points should be selected with special reference to their solidity, and care should be taken not to disturb them. Station pegs and hubs are often used for turning points ; when so used, the precise fore sight to 0.01 foot should follow the usual ground rod reading to the nearest 0.1 foot. The rodman should use good judgment in selecting bench marks, locating them out of reach of probable disturbance during construction and describing them so as to be easily found. He should be ac- tive and do his best to keep close up with the transit party. The rodman should keep a peg book for recording turning points and instrument heights, and check his computations independently and compare results with the leveler. TOPOGKAPHY PARTY.— It is the purpose of the topography party to secure full data for mapping contours, property lines, buildings, roads, streams, and other import- ant topographic details. The width of territory to be em- braced in the survey depends on local conditions ; in places it may be as much as one-fourth or one-half mile from the line, although it is usually better to run alternate lines when the distance to be included becomes so great. The topog- raphy party often consists of only two men, but a party of four is much more efficient. Sometimes no regular topog- raphy party is provided, but after running a few miles of line ahead, the transit and level parties are formed into several parties to bring the topography up to the end of the preliminary line. For student practice the topography party will consist of four members: (1) topographer, (2) assistant topographer, (3) topography rodman, (4) tape- man. Topographer.- — (Topography board, topography sheet (or several sheets), hard pencil, compasses, eraser, etc.) The topography sheet should be prepared before going to the field, showing the alinement and other data needed from the transit notes, and elevations of all stations and pluses from the level notes. Cross-section paper is to be preferred. The center line may be plotted to one side of the center line of the sheet, when the topography is to be taken far- ther in one direction than the other. In order to secure full details, the scale of the field plat may well be double (or even more) that of the finished map. The topography sheet should show local conditions, such as gravel banks, rock ledges, etc., suitable for ballast or other constructive use ; out-croppings of rock or other material which may TOPOGKAPHY PAKTl. 195 affect the classification of the graduation; character of substrata at sites of bridge or other masonry work ; springs, wells, streams, etc., suitable for water supply ; approximate flood levels and other data relating to waterways or surface drainage ; location of streams, especially with reference to desirable crossings, freedom from probable change of chan- nel, etc. ; location of highways including elevations some distance either way with special reference to avoiding grade crossings ; other railroad lines, with the same point in view ; character and condition of crops and other farm improvements, names of owners, etc., — in short, any and all information that is at all likely to be of service in mapping the route, projecting the location, during construction, etc. In locating a group of buildings some distance from the line, fix the principal one by tie lines, by intersection or polar coordinates, and the others by measurement and sketch from it. Locate buildings near the line by rectangu- lar offsets, or by intersections of the principal outlines with the survey line. Contours are located by means of the hand level used by the assistant topographer. The con- tour interval should be five feet ordinarily, but niay be in- creased to ten or more feet on very steep slopes. The con- tour data should be selected with special reference to ridge and gully lines (see problem and plat on contour level- ing. Chapter IV). Ordinarily hand level lines may be run out at right angles ; angling lines along gulches and ridges may be located by estimation, pocket compass or tie lines. The plat is made by the topographer from data collected by the other members of the party. A common fault with the beginner in such work is the omission from the plat of im- portant numerical data, such as station numbers of land- line crossings, etc., owing to an undue attention to the minute details of the drafting work. A good topography record with contour notes on the left hand page and field sketch showing all numerical data on the right, is shown in the accompanying form. Assistant Topographer. — (Hand level, pocket compass, topography note book.) It is the duty of the assistant topographer to collect data for the use of the topographer in making the plat. He uses the hand level, notes station numbers, distances, bearings, etc., and makes such record of the same as may be required to fit local conditions. In contouring, a special rod with adjustable base (see Fig. 19, Chapter IV.), if available, may be used; otherwise, an or- dinary flag pole with alternate feet red and white is em- 196 EAILEOAB SURVEYING. r Sta. I3e 138 131 136 I3B J34- 133 I3S 131 130 129 128 (RAILROAD Left Contours 420 -tzs f J*'* ptf? y73 . 2S0 J8S tzi 'a' Center Elev. 43hl T0P(t)6RAPHY.) RightContours 43S 4^40 44S 107 eSB 369 42S 430 43S .£40 77 $7 Sa3 -406 ^■*- ^ 4io4ZS4S0^3S 420 4gS 430 43S ~SS'lt2 B13 309 420 ^?S 430 *SS SB loe J03 J¥S f 417-6 f 41SS ■I-30-7 "A 437' P 4f3-4 (Obsfrucied) 44s ^ ^^ 413 \ 4ZS430 435 440 S4 J03 175 ZaO \ Pasf-ura , /?5i i5i Owelltng I 13S Open Timber I ; I ; ' I / iPaature / ' / / / ' Fence BRW^ence.SKLini ^*' joFs^toSfane.mCar.Setlg* ' I I . AbBncfotiecf / / ^r^^i'ick Yard I « ' -S^ ^^""^ ■>"' Bniffr Timber \/ ^ ■n ■^ Si* '^ ill * \,'k\ 130/^ ' \ 1/ ployed. BeginniBg with the known profile elevation, as ex- tracted from the leveler's record, even five-foot contours are located, as a rule, nominally every 200 to 500 feet at right angles to the line, except as ruling ridges or gullies may suggest other directions. His record should be ample and legible, and include data and information which may not properly be placed on the plat. All estimated elevations, distances or dimensions should be noted as such. The as- sistant topographer works under the direction of the topog- rapher, but is expected to take the initiative in the collec- tion of data so as to permit his superior to devote proper attention to the field plat. Topography Bodman. — (Topography rod with adjust- able base (see (f). Fig. 19, Chapter IV.) or flag pole, hatchet.) It is the duty of the rodman to hold the topog- raphy rod as directed by the assistant topographer. He should be active and continually on the alert for informa- tion or data which the record book or sheet should contain. The rodman holds the zero end of the tape in measuring the distances. He should acquire skill in pacing on rough as well as smooth ground, and when sufficiently exact es- OFFICE WOKK. 197 pecially on ground remote from the surveyed line, lie should gain time by pacing in the distances to contour lines. Tapeman. — (Metallic (or band) tape, set of chaining pins, flag pole.) It is the duty of the tapeman to deter- mine distances with the help of the rodman. He should be vigilant in checking up tallies, reading fractions, level- ing the tape, breaking chain, plumbing down ends, etc., and should never be the cause of needless delay in the work. When required, he should measure angles, take tie lines, etc., with the tape. OFFICE WOBK.— The office work of each student in- cludes : (1) reconnaissance map, profile and report; (2) map showing preliminary lines with topography and pro- jected location lines; (3) preliminary profile with grade lines, approximate estimate of quantities, etc.; (4) final lo- cation map (traced from preliminary map) ; (5) location profile; (6) copies of field notes; (7) cross-section notes and estimate of graduation quantities; (8) estimate of cost of constrution ; (9) monthly estimates, progress pro- file, haul, prismoidal and curvature corrections, vouchers, etc., final estimate. B>econnaissance Report. — The reconnaissance map show- ing the area examined will be based upon such maps of the route as may be available. It should show the several ruling points and general routes selected for actual survey. The profile should be based upon barometric or hand level observations and distances scaled from the map or deter- mined roughly by pacing or otherwise on the ground. The report should refer to the map and profile and state the general scheme, the several ruling considerations or condi- tions, the details of the examination, a rough comparison of the several alternative routes, and a final summary and conclusion with definite recommendations. The report should be made in accordance with best usage as to form, composition, etc. (Considering the limited point of view of the beginner, the reconnaissance reports may not be required until the actual surveys are well along. In such case, however, the student is not to draw data from sources other than those above outlined.) Preliminary Hap. — The mapping should be the best product of the student's skill as a draftsman, and should conform closely to the department standards, which are based upon best current usage of leading American rail- roads. Unless otherwise instructed, the preliminary map 198 EAILEOAD SUKVEYIJNU. / PL ITTIN 3 Sh iET, ='REL1 UNARY Line "A 1 ^ Sta- DeFl- Cal-Brg- Dist- Lstl ■udes Deparlures Total Lat- Total Dep. { Ft. H- S- E W N- w. s ll-32'3^n m9-i 2097-7 1339-1 0-0 0-0 3 t Zff/}' JZVl.- H-UVW- 7S-S 33-6 71-0 20S7-7 1339-1 b cznas 71'IS'R- HJiVw 414-1 373-4 27S-6 2131-3 1410-S c dmsfi IO'3S'li- KS'4iW- 2tS-S ZOi-O 9S-I Z504-7 1686-4- d e K*S7l ZWR- im'n'w- 436-7 400-9 m-i T707-7 ns4-s e fum'- iiWr- lt-4'4l'>V- J64-1 164-2 13-B 310S-6 I9S7-6 F } ?w-' 4l'S9'L- li-siVw- JS7-9 94-S 119-9 3272-S 1971-1 9 h 4IHli 17'OSA- m'}!'w- 776-0 170-3 44-3 3167-6 2091-0 h iHIwi le'nx- tt-l'32'E- 3I0-I 209-9 S-3 3537-3 2I3B-3 1 j 44m Zl'27'l- ifn'ss'n/- IOS-9 93-6 36-1 3S47-S Z127-0 J k4efl!i i'n'L- KeVw- 307-1 276-2 J3S-3 3947-4 ZI63-0 k 1 «9nn IS'06'L- S-7l°4l'W- 331-9 63-0 32S-S 4223-6 2299-0 I mSr*43i 73 VR- KllK'lf- ZD2-7 l7!-9 9S-4 47S8-B 2624-S m ,iUt4li ZI'ZS'L- to'js'it- JS6-4 lOI-S 113-0 4337-5 2719-9 n SM2i iZ'SS'L- lt-ll'3t'lV- 332-6 41-4 329-1 4439-0 2838-9 p 5S'SS! 3i'S2'Jl- li-44'4iH. 301-7 2J9-2 217-4 44S7-4 3168-0 p q 6H43i ll°54'L- H-K'4t>k IZS-I 70-4 107-0 4706-6 3385-4 ? r I3f7d kWl s-iiWw- ZSI-S 39-4 24S-7 4777-0 3492-4 r sfffHi S0'I!3'R- K-4t'j7'n- 33-4-2 2J9-S 2S2-0 4737-6 3741-1 s t-eiwi 3B'/I'L- mViv- 266-9 27-3 Z6S-B 4957-1 3993-1 f u //*»-' Z'OB'R- m'sin- 317-0 43-! 313-S ■4914-4- 42BS-6 u V 7SHfi I'n'L- n-is^a'w- SS7S S-7 SS7-S BOZS-Z 4B72-4 ^ wumi 43'lS'R- H-4('lO'H/. 1067-6 739-4 770-1 SOii-9 ^£773-3 9 — Che 5130-2 fS900-3 w X 9167-1 5/77-7 104-4 ^104-4 S-3 5908-6 S-3 5900-3^ 5773-3 ;> J ^ Plo- From 5ta. Tsnger, A. L 26 4ftise- 6ffilSL 9}'h5 102*ilZ 143790 issmi 170'hf3 I8lt^jj. 114/iOi 193H9£ Z/3'm zzoTsci 23fii4i Z4Zt4Sl 2SZt3Sl TING To Sta- tMsin T 26 4f*isi 6M!-!- 93*65 102*511 143190 IBSfXT- I7M3 ISlihl l!47sOS. IS3fJfS. Tispe* zzoSoi zjofm 24z'f^» 252*322 \ Check.- Sheet Length Tangts Ft- Tr^ck.,. 1755-6 3196-9 4138-3 1466-3 320-9 1986-9 963-8 986-0 LOCATIO^ Length Curve Ft- t-e-j!-c 100-0 2500-0 18J2-B S86-7. 1186-7 1086-7 168-9 714-4 1195-0 517-1 /0938-0 Olieck- Angle I 7°3t}'R- zs'oo'e- 36°1S'R- 13'IS'l- 23°44'L zMr- 13'02'R- ll°4i'L- 23'S4'R 3g'47'L- I27J5R- lf3S'L 39'50'R- MS'lS'g 5-4'!S% SUR". EY, Belt R-R- Extension- ■\ Degrea Radius Tan- DM Dist- Calc. (Lat. \D.p. frot.Lat \Tot-I)ep P. R- T P-l-toPl Besring Ft. Ft. Ft- 5-3s'/5'e Ft. Ft, 0-0 7'30' 764-1 SO-} liZO-4 (s-z7isi 5. iW S730-0 1270-3 Pl-= 3363-7 s-z^fs ^ ZW 2865-0 937-8 P-l-= >l 4SS0-1 S-33'3llW s- l'30' 38Z0-0 445-4 PI- ;S-S! 5185-7 MO'lZ'W S' "C-s Z'OO' 2865-0 602-0 p-l-=. y if 2618-3 s-BVe- '^ ^ J. z'oo' 286S-0 550-0 p-r-= II 1307-3 SIS'lZ'W- 5. 1'30' 38Z0-0 436-4 P-I-' Z816-9 isMw- s f "S I'30' 3820-0 395-6 PI- !:^ 1963-8 s-nii'w- s- •^ -2'm' 2865-0 606-4 'p-l-^ IS6J-3 Hi'22'n 5. 4 (Check 7°30' 764-1 268-9 S-4'35'm (See p-l-= Tan^l-llaln Trk- f.-Y-lil-R-B- 5560-9 « 2 Z5617-S * III2I-8 last- (MT*lastT).' -313-0 1 I 1 check h5th column) totals byZ) J Z Tangenfs - IOS02-S I4II4-7 Check. 25617-5 OFFICE WOEK. 199 will be made on eggshell or paragon paper. There are three ways to plot the skeleton of the preliminary survey : (1) by laying ofE each successive deflection angle and dis- tance from the preceding line; (2) by laying ofE the suc- cessive calculated courses and distances from a precisely drawn meridian or other reference line; and (3) by rect- angular coordinates. The first method should not be used, since cumulative errors are probable. The second is rapid and free from serious objection ; if preferred, a modified base line may be assumed and the calculated bearings transferred to the same ; the angles may be laid ofE by means of scale and table of natural trigonometric functions from a precisely drawn base line and then transferred, as required, by parallel ruler or triangle ; this method is used most in practice. The third method is the most exact, and will be used by the student unless the second is specified. It involves the calculation of a plotting sheet, as shown in the accompanying form. The axis is usually a meridian line, but any line may be taken and the courses changed to suit. In making the plotting table, the data, calculated bearings, distances, etc., should be carefully checked through to the last point in the skeleton before the plotting is be- gun. Only one axis should be plotted, preferably the one having greater totals, so as to give short perpendiculars. Starting from the origin, 1000-foot points are pricked in along the axis to the specified scale, and marked 0, 10, 20, etc. ; the totals are interpolated on the axis and lettered ; exact perpendiculars about the right length are erected ; the second point is established by scaling the perpendicu- lar and the line is checked back on the preceding point ; if correct, the stations are pricked in and every fifth station and deflection points are enclosed in a small circle and neatly numbered ; the next course is so located and checked back by length of hypothenuse, the stations fixed and num- bered, and so on to the end of the line ; the courses should be taken in their order and none passed without checking satisfactorily. After the skeleton is completed, the topog- raphic details are penciled in, and the map finished and inked. The title, border, meridian (both true and mag- netic), etc., should be first-class in quality and in keeping with the rest of the map. Crude or careless lettering or other details of the map will cause its rejection. The title of the map, profile, etc., should be given in brief on the outside of the sheet or roll at each end. 200 KAILEOAD SURVEYING. Preliminary Profile. — Use Plate A profile paper in mak- ing the profiles. The level notes should first be carefully verified and then one person should read off while another plots the data. A hard pencil, 6H or 7H, sharpened to a long needle point should be used. The stations are first numbered along the bottom from left to right (or the re- verse, as prescribed) ; leaving six inches or so at the left for a title, and beginning at a prominent line with station 0, every tenth station is so numbered. The notes are exam- ined for lowest and highest elevation and a prominent line is assumed as an even 50 or 100-foot value relative to the datum. The horizontal scale is 400 feet and the vertical scale 20 feet to the inch. Points should be plotted no heavier than necessary, since the surface of profile paper will not permit much erasing. The surface line should be traced in close up to the plotted points, owing to the danger of overlooking abrupt breaks such as streams, ditches, etc. Pluses should be fixed by estimation. The surface line when completed should be inked with a ruling pen used freehand ; the weight of the line should be about the average of the ruled lines on the profile paper. (A special profiling" or contouring pen is much used for this purpose.) The profile should show the grade line, grade intersection, elevations and rates of grade in red ; water levels, and data relative to same in blue ; surface line, sta- tion numerals, etc., in black ; the alinement, important land- lines, streams, etc., should be shown at the bottom of the profile in black. The grade line should be laid nominally with a view to balance the cut and fill quantities, but this should be varied to suit local conditions, such as drainage, the elimination of grade crossings, classification of ma- terials, etc. The maximum gradients, the rate of compen- sation for curvature, etc., will be made to suit the specified conditions. The compensation for curvature will be al- lowed for on the preliminary profile by dropijing the grade line on maximum gradients at each deflection point. Grade intersection elevations and rates of grade will be given to the nearest 0.01 foot. Approximate Estimates. — Rapid estimates of earthwork quantities may be made direct from the profile either by reference to a table of level sections, or preferably by means of an earthwork scale. Estimates made in this way from the profile of a careful preliminary survey, often do not vary more than five per cent from the final construction quantities. OFFICE WORK. 201 Iiocation Map. — The location map may be traced from the preliminary map and should include the topography and such details as usually appear in the iinal record map of the located line. Contoiir lines may be traced in cad- mium yellow to insure satisfactory blue printing. Location Profile. — The location profile should be exe- cuted according to the standard specimen, and should in- clude estimates of earthwork as determined from the ac- tual cross-section notes, and quantities of other construc- tion materials. Curvature compensation will be shown on the location profile by reduced maximum gradients. Verti- cal curves will be calculated at a rate of change not to ex- ceed 0.05 foot per station, except at summits where it may be 0.10 foot or more per station. It should be prepared as the final record profile. Approximate profiles of projected lines, determined from the contour map, with rough esti- mates of quantities will also be prepared, as specified. Office Copies of Notes. — The complete level and transit notes, and topography notes as assigned, must be copied in the individual books by each student. These copies will be in pencil (or ink if so specified) and will be executed in a faithful and draftsmanlike manner according to the de- partment standards of lettering, etc. Estimates of Quantities. — The cross-section notes will be copied and the quantities of excavation and embankment calculated, as assigned. The cross-sectional areas will be calculated arithmetically and checked, especially on rough ground, by means of planimeter. The quantities will be calculated by average end areas, by tables, and by diagrams, so as to afford ample practice for the student in all the cur- rent methods. The estimate will also include all the other materials of construction. Bstlmate of Cost. — Each student will make a detailed summary of the quantities, fix prices, and estimate the probable total cost of the work, or of the assigned section. The prescribed form will be followed. The prices should be based on local conditions as far as possible. Construction Estimates. — Monthly estimates, estimates of haul, borrow^ pit estimates, classification, prismoidal and curvature corrections, progress profile, vouchers, force ac- count, etc., and final estimate will be prepared by each student in accordance with prescribed forms and standards. Right of Way Records. — Each student will be assigned a share of work in the preparation of right of way deeds and record maps. The following forms (from the " Engi- 202 RAILROAD SURVEYING. neering Rules and Instructions," Northern Pacific R. R.) will be used as models in preparing right of way descrip- tions. (Through government subdivisions) : " A strip, piece or parcel of land one hundred feet in width, situated in the northwest quarter of the northwest quarter of section ten, in township two north, range one west (S. 10, T. 2 N., R. 1 W.), Madison county, Montana, and having for its bound- aries two lines that are parallel with and equidistant from the center line of the railroad of the Railway Com- pany, as the same is now located (and constructed). For a more particular description, reference may be had to the plat drawn upon and made a part of this deed." (Lots in platted tracts) : "Lot seven (7), block six (6), in Smith's addition to Helena, Lewis and Clark county, Montana, according- to the recorded plat thereof." CROSS-SECTIONING PARTY.— It is the duty of the cross-sectioning party to set slope stakes for the proposed roadbed and to secure data for the calculation of earth- work quantities. The data should first be transcribed from the location level notes and profile into the cross-section book, including station numbers, surface and grade eleva- tions, rates of grade, bench mark record, etc. In order to avoid confusion in relation to directions right and left, the station numbers should run up the page, and plenty of space left for pluses in the notes, especially on rough ground. As shown in the form, the left hand page should be used for data and the other for the cross-section notes. The organization and equipment of the cross-sectioning party when using the engineers' level is: (1) recorder (note book), (3) leveler (engineer's level), (3) rodman (self-reading leveling rod, 50-foot tape), (4) axemen (axe, sack of flat stakes, marking keel). The usual routine is: (1) Determine height of instrument by back sight on iden- tified bench or turning point. (When a bench mark is re- mote and an original turning point can not be found, it may suffice in an emergency to check on the ground at several stations to the nearest 0.1 foot and use the mean height of instrument. Such places .should be verified later.) (2) Having the height of instrument, check the original eleva- tion of the station about to be cross-sectioned, reading the rod and checking off the elevation if it does not difl'er more than 0.1 foot or so ; in case of a new plus, take a rod read- ing and record the elevation. (3) Determine the "grade rod " for the station by subtracting the height of Instru- CEOSS-SECTIUJN IWU 203 Sta. 130 *40 129 HO +3i Mlfl9 T-e-fOS Kt-fSS it-SO fJ3 K7 tei *;i ■fSO -ts* KS TJl- Elev 74ZS 73es 732-3 72S9 72II-S 720-B 720-1 712-2 727g 73/-S 73e-S 739-2 741-4- 741-7 742-2 74i-l SratJe neso 736-SO 73M0 J3S-S0 736-SO 73(30 neso 731-30 w-so 731-30 731-30 731-SO 731-30 TSe-SO 736-SO 7X-S0 736-30 fORM Notes R sm%i i7-a J6-0 Renrar-kft (i-levei section //i cuf') (Level secf-f9n In cvf) (Orade point, L, Csmtl) {2 leveJ sect/on in Fill) {Levd sect/on in fill) li-and stringer, 8r.0J8- (Head of Dump) (Toe of Dump) Bridge its IS ]jtS*34 6,J4sp3/?3 \{B7+£0 (Head oF Dump) S-end sMngeri Sr-IKIS Ditch 2-'4'-4'T- 33'- (3 level section In Fill) (Srsde point- right) (Srade point center) (Srsde point leFt) (3 level section In ct/f) (level section in cut) (4 level section in cut) (S level section in cut) Cijk,tl',li:I, Fillsll',lhl-J Typical Cases L^vel Sections. -\{760) Grade PoinbCw/th Diaijonal Contour) Riqht. Center. Lefi^ 5iac-HiJI5ectioi}. FndofFillatTrestle. H.D. -.^ -H700)- 204 EAILEOAD SUEVEYING. Cross-SectionatSlatm /E7-f-53 headofDump TT — \ ? "k- -30.S ->^- Z^.6 —^. '"'7^^ Cros5-5ectio/?3l5tatio/7/E7 x^-d-^-IQ-*'/ ' OradeFoJnt 'Station ■T^Rav/ne \<:-/7.0—^ Srade FoM-^-S '^Jl£^ Station Z.O^,^i-^\' ; % IZ6f37 \-^II.O^^/0.0^ r='Z.O 126 /:s station '■Rra dp Point — (^> lZ6tl8 ^^---^fa^^ri-s OrossSection at Station iZ6^ ; \<-/0--^-iO-^, \ '^-/3.4--^—Z0.Z-—^ Cross-Section at Station J25 V-IO-^-IO-^ \ ■'<-i6.4--^6Z^. M Z9I- -->^-—/9.6- CEOSS-SECTIONING 205 ment from the grade elevation ; then note that cut or fill at any point of the cross-section is equal to surface rod minus grade rod (counting rods as minus when downward from the plane of the level dnd those upward as plus, this rule gives results always plus for cut and minus for fill, which agrees with the conception that cross-section notes are rectangular coordinates of the sectional area referred to the center of the finished roadbed as an origin) . (4) If the ground is level transversely, that is, does not vary more than 0.1 foot or so within the limits of the proposed grad- ing, then the distance from the center out to each side slope stake is half width of roadbed plus center cut or fill times rate of side slope; (thus for 20-foot roadbed, side slopes 1 to 1, and a cut of 18.6 feet, the distance out to slope stake on a level section would be 28.6 feet, or with a slope of 11.^ to 1, the distance out would be 10 plus 1% times 18.6, or 37.9 feet. Calculations of this sort should be done men- tally in an instant). (5) On three-level ground estimate the rise or fall of the surface from the center to about where the side slope stake should come, and add the same to, or subtract it from the center cut or fill, as the case may be ; compute the distance out to the point where the side slope line would pierce the ground surface and test the same with tape, rod and level by the foregoing rule for cut or fill ; continue to construct points on the side slope line until the common point is found. (6) The axeman marks " S. S." (slope stake) on one side of the stake with the cut or fill to the nearest 0.1 foot (as C 6.8 or F 10.2) and the station number on the other side ; the stake is driven slanting towards or away from the center line ac- cording as it is cut or fill. (7) On five-level ground or, in general, on ground involving any number of points or angles in the section, the cut or fill is taken at each break. (8) Should there appear to be danger of land slips, the cross-sectioning should be carried well beyond the limits of the slope stake points. (9) The cross-section notes are recorded as in the accompanying form, expressing the co- ordinates of each point in the form of a fraction, and dis- tinguishing the slope stake points by enclosure in a circle. (10) Having completed the cross-sectioning^ at the station, the same program is followed at the next point, first check- ing the elevation obtained in the original location levels ; the grade rod should be determined as before by subtract- ing the height of instrument from the grade elevation, and then checked by applying to the preceding grade rod th? 206 RAILEOAD SUEVEYING. rise or fall of grade from, the preceding point. (H) Cross-sections should be taken as a general rule at every station and at such intermediate points as will insure a reliable measurement of the earthwork quantities. It is not necessarily the lowest and highest points that are re- quired, but those points which, when joined by straight lines, will give the contents as nearly as possible equal to the true volume ; if the " average end areas " method is to be used in calculating the quantities, sections should be taken every 50 feet when the difference of center height is as much as 5 feet ; as a rule, slope stakes need not be set at cross-seclions taken between stations. (12) "Grade point" stakes (marked 0.0), should be set where the center line and each edge of the roadbed pierce the ground ; and also in side-hill sections in both cut and fill, where the road- bed plane cuts the ground line ; if the width of road- bed is different in cut and fill, the greater half-width is commonly used in locating the side grade point ; in the simplest case a contour line is perpendicular to the center line and the three grade points are at the same cross-sec- tion, forming two wedges ; in the more usual case the con- tour line is diagonal, and the three grade points are not in the same section, so that two pyramids are formed ; if the station numbers of the two side grade points differ by only a few feet, it is usual to simplify the record by taking the notes as for a wedge at the station number of the center grade point, although the side grade point stakes are set in their true positions ; as a rule, a complete cross- section is taken at each grade point. (13) In cross-section- ing for the end of an embankment at a wooden trestle the end slope is made the same as the side slope, and the end and side planes are joined by conical quadrants ; the dis- tance between " heads of dump " (H. D.) is usually 10 feet (5 feet at each end) less than the total length of stringers; a complete cross-section is taken at the " head of dump," and the "toe of dump" (T. D.) on each edge of the end slope is located and recorded ; on level ground the volume of the wedge-like solid so formed is found by dividing it into a triangular prism and two right conical quadrants ; on ground sloping transversely the end of dumip is made up of a middle prismoid and two conical quadrants, each of the latter being generated by a variable triangle revolved about a vertical axis through a corner of the top roadbed plane at " head of dump." The calculations in the foregoing method of cross-section- 207 ing may be simplified by preparing a table of distances out for the standard roadbed widths and slopes, or by using a special tape having the zero graduation at a distance from the end equal to the half-width of roadbed, and the re- maining graduations modified to suit the side slope ratio. The calculations may be further simplified by using a, spe- cial rod having an endless sliding tape graduation. The student will be given practice with these labor saving devices after he has first acquired familiarity with the principles of cross-sectioning without these aids. Cross-sectioning with rods alone is done in much the same manner as that described above. Two rods are used. The usual length of the rods is ten feet, and each is gradu- ated to tenths and has a bubble vial in one or both ends. The slope stake point is determined by leveling out from the ground at the center stake with reference to the center cut or fill, each rod being held alternately level and plumb. Other points in the cross-section, as well as grade points, etc., are determined in tEe same manner. The notes are kept as in the other method. On very rough ground, the rod method is usually the more rapid. Some engineers cross-section on rough ground by taking the elevation of each point and plotting the notes on cross-section paper, then using the planimeter to determine the areas. Borrow pits are often cross-sectioned by taking elevations at the intersections of two series of parallel lines forming squares. Laud-Line Party. — It is the duty of the right of way party to secure data for the preparation of right of way deeds. The party should consist of at least four: (1) re- corder, (2) transitman, (3) head chainman, (4) rear chain- man, (the chainmen also to serve as axemen and flagmen as required). Their equipment is the usual one of a transit party for such work. The party should secure ties with all section and other laud lines whenever crossed. The notes should show station numbers and angles of intersec- tion and distance along land line to the nearest identified land corner and also to important fences. As a rule, make the intersection by running through from one corner to the other. Where the line passes through a town, tie the cen- ter line to the plats, block lines, monuments, etc. Secure any records and make tracings of any plats, etc., at the recorder's office, that may be of service in preparing deeds. Bridge and Masonry Party. — The bridge and masonry survey party will determine drainage areas for culverts and other waterways, prospect for foundations, and stake out 208 KAILEOAD SUKVEYING. trestles, masonry work, etc. The usual organization will be four men : (1) recorder (in charge), (2) transitman or leveler, (3) chainman, rodman, flagman, etc., (4) chainman, axeman, flagman, etc., as the work assigned may demand. Besurvey Party. — The resurvey party will be assigned to such duties as the resurvey of yards, the collection of data for crossings frogs, running centers on old track, in- cluding spiraling, etc. It will usually be a, party of four. PROBLEMS IN RAILROAD SURVEYING. PROBLEM Gl. ADJUSTMENTS OF LEVEL AND TRANSIT. (a) Equipment. — Engineers' level and transit, adjusting pin. (b) Problem. — Test the essential adjustments of the as- signed instruments and correct any discrepancies found. (c) Methods. — This problem is designed to freshen the student's knowledg'e of the adjustments of the instruments, as well as to place the equipment in condition for accurate work. The adjustments will be made under the persona] direction of the instructor. The student should attempt to be speedy as well as accurate in testing and making the adjustments. PROBLEM G2. USE OF FIELD EQUIPilENT. (a) Equipment. — Complete equipment for railroad transit and level party, as specified in foregoing pages. (b) Prohlem. — Practice the detailed duties of each posi- tion in the transit and level party. (c) Methods. — This problem is designed as a "breaking in " exercise preparatory to engaging in the regular field work qf railroad location. With the manual in hand the duties of each position will be studied and practiced in turn. For example, each student will go through the following exercise with the transit as briskly as possible: (1) set transit over tack in hub, (2) level up, (3) set plate to zero, (4) reverse telescope and sight on back flag, (5) release needle, (6) phmge telescope, (7) read and record needle on back line prolonged, \8) sight at front flag pole, (9) read and record deflection angle right or left, (10) read and record needle on front line, (11) lift needle, (13) plunge telescope and check on back flag, (13) calculate needle PKOBLEMS. 209 angle and compare with plate reading, and if checked, shoulder transit; now repeat entire process at the same hub, more briskly than at first, if practicable, avoiding ref- erence to preceding record until the full series of steps is completed. Problem 2. Calculation* of Curve Elemen+s. DM'n'):sl337.6^ \^ (b) By Tails I'C. .,TandE. , tn'go. tan 3efo8.'s= o-seoes •exsec jd'o8.'s=0. /Sffjff Wf7'= eo°.S833-t- tl'n'= 4? 2833 + {Results to 0.0/-fti) Msthod. I4.07SS 776.71 109. IS D'lff. 776.77 209.17 Indicated WorK- Calculations. Leng th of Curve , i. , I — SO' 17' *-■" ■*V7' '"' = ^^. = (etszS^ eo'.2»33 _,^7Z^ f6) = 2T7\^c n)r4.0739 g-gy a*. 10^7 to 38 laoo 1799 loio Z330 ea!S33 )*.SB333 Tangent Pi'stanee . T. (at 7-= n tan-kZ — '337.e sx a.s8»se =<^ff.77) lb) r= 776.71 cH. Titecfie) = 33as.a ^Ko'/si = 33saj_ r, fea'n'j = 33s 7. /s )^se33(S) 2998.33 776.77 3Batz o,k. 2.9983 2899 2370 776.77 776.7/ O.06 Di-f^ due to approK. basis of method Cb), JS9 300 External Distance , £. -^^^^V9x'lO-^«"-3 The weights of these mean values vary inversely as the squares of the probable errors, or in this ease the weights are as — ^ to r-^ or as 13 to 5. The most probable value 4.0 D.o of the angle measured with the two transits will be the weighted mean. Z= 34° 55' + 33X12" + 36X5" 17 = 34° 55' 33". 9 The probable error of this result from (5) since Substituting r^'^i^-r^ we have iJ, = ± 4. "3 VTI = ± 3".6. 214 ERRORS IN SURVEYING. Eor other examples in the use of probable error see prob- able error of measuring a base line, probable error of set- ting a level target, probable error of setting a flag pole. Angle Measurement. — The measurement of an angle re- quires two pointings and two readings. If r^ and r., are the probable errors of reading and pointing, respectively ; the probable error of the measurement of an angle will from (5) be If i\ is the probable error of a single reading If the value of an angle is determined by n separate meas- urements the probable error due to reading will be nV2 If the value of an angle is determined by measuring the angle n times by repetition the probable error due to read- ing will be ni/2 It will thus be seen that the probable error due to reading is very much reduced by measuring an angle by the method of repetition. The errors of pointing, etc., however, make it doubtful whether it is ever advantageous to make n ex- ceed 5 or 6 with an engineers' transit. Angle Adjustment. — When the three angles of a triangle have been measured with equal care they should be adjusted by applying one-third of the error as a correction to each angle. When the interior angles of a polygon having n sides have been measured with equal care they should be adjusteJ by applying oiic-iith of the error as a correction to each angle. When n — 1 angles and their sum angle at a point have been measured with equal care they should be adjusted by applying one-nth part of the error as a correction to each angle. In a quadrilateral the triie values of the angles fulfil the following geometrical conditions : (1) the sum of the angles of each triangle is equal to 180° plus the spherical excess TESTS OF PEECISION. 215 (the spherical excess in seconds of arc is equal approxi- mately to the area in square miles divided by 78) ; (2) the computed length of any side when obtained from any other side through two independent sets of triangles is the same in both cases. When the angles of a quadrilateral have been measured, errors are certain to be present and the corrections that satisfy one of these conditions will not satisfy the other. The most probable values of the corrections to the angles are then determined by the Theory of Least Squares. TESTS OF PRECISION. Practical Tests. — In careful surveying where blunders are eliminated and the systematic and accidental errors are small and under control, it is found that the magnitude of the errors increases in close accord with the foregoing rational basis, tliat is, as the square root of the number of observations. The following practical tests of precision are based on this truth. Linear Errors. — Cumulative or systematic errors usually increase directly as the length of the line chained, while compensating or accidental errors vary about as the square root of the length. While both kinds of errors afEect all linear measurements, the former chiefly control the results of crude and the latter of accurate chaining. It is thus fairly consistent to express the precision of chaining in crude work in terms of the simple ratio of the length ; but as the chaining becomes more and more exact, the varia- tion of the differences between duplicate measurements approximates more and more closely to the law of square roots. Coefficients of precision derived from the latter relation may be based on either 100-foot units or foot units in the distance chained, as preferred. The former basis is used in the chaining diagram while the latter is found in the last paragraph of the explanatory matter on the second page referring to the precision of traverse surveys. The diagram of chaining errors shows chaining ratios by right lines radiating from the origin, and the law of square roots by means of parabolas. The coefficient of precision for a given observed difference between duplicate chainings is determined by inspection from the diagram, interpolat- ing between curves if an additional decimal place is desired in the result. In actual practice a pair of careful chain- 216 ERRORS IN SURVEY J JNU. men may determine the coefficient corresponding to a given degree of oare, and then vise this value either in testing their duplicate results, or in estimating the probable uncer- tainty of the lengths chained. For accurate chaining with the steel tape, duplicate measurements reduced for temperature, etc., or made under sensibly identical conditions, should not diifer more than 0.05 foot into the square root of the distance in 100-foot iniits. Careful work with the common chain- (estimating fractions to 0.1 foot) should not differ more than 0.1 foot into the square root of the distance in 100-foot units. Angular Errors. — In measuring deflection angles by alti- tude reversals, as in railroad traversing, there is, of course a cumulative discrepancy due to the collimation error, but generally speaking, careful angular measurements with good instruments are subject only to compensating or ac- cidental errors. Under the latter conditions the magnitude of the error of closure in a series of angles, either in a closed polygon or about a point, varies about as the square root of the number of angles. This relation is indicated graphically in the diagram of angular errors. In measuring angles with a transit reading to the nearest minute, the compensating uncertainty of a single reading is probably somewhat under 0.5 minute per angle, or about one minute for the closure of a triangle. If a reading glass be used and the vernier reads to the nearest half minute, the uncertainty is still further reduced. Again, in estimating the needle reading of a compass to the nearest 5 minutes (one-sixth part of a half-degree), the uncertainty of reading alone is perhaps 3 minutes, although this is increased by other conditions such as sluggishness of needle, etc., probably causing an uncertainty of as much as 5 minutes per angle, which later limit would produce an error of closure of a triangle of say 10 minutes, and of a, five-sided polygon of perhaps the same amount. (See dia- gram.) Traversing Errors. — The errors of traversing are made lip of the combined errors of linear and angular measure- ments. If the error of closure as determined from the lati- tudes and departures is large, the work should be scanned closely to detect blunders such as the substitution of sine for cosine, errors of 100 feet in chaining, misplacing deci- mal point, etc. After establishing the consistency of the residvial errors, they should be distributed either in propor- tion to the lengths of the several courses, as in the more TESTS OF PRECISION. 217 THE PRECISION OF CHAINING. 10 10 ^0 40 Lcn^h of Line Chained, l, in tOO' THE PRECISION OF ANGULAR MEASUREMENTS. "0 5 10 Number of An^Us in PoIy^^*^ °^ IS Series, W. to £5 16 21S ERIiOES IN SURVEYING. THE PRECISION OF TRAVERSE SURVEYS. The error of cfosure of a traverse /'s usually expressed as the ratio of the calculated linear error tt> the length of the perimeter of the fie/ol or polygon. The following table shows the h'mits prescribed by various author/ ties PrescHbed Limits For C/osure Of Traverses Authority. Conditions. Limits. Gillespie, (lassj. "Suri^eying,' p. 149. Compass Surveys. 1:300 to i:iooo A/sop. (I8S7). Compass Surveys. I.SOO "Surveying" p. 199. Transit Surveys. i.iooo to risoo Davi'es. (/S70>. "Surveying" p. 137. Farm Surveys- i:soo to I.IOOO Jordan. 0877). German Gov't Surveys. "Handbuch der Baden Instructions. /:400 Vermessungs- Prussian Instructions. 1:333 to l-.IOOO kunde;' Vol.1, p.a96. Stviss Gov't Surveys. Ordinary Country. 1:400 to 1:800 Mountainous Country, i:S67 to I: S3 3 Hodgnian. OS8SJ. "Surveying" p. 119. Compass Surveys. 1:300 to 1:1000 Johrjson. 0886). Farm Surveys. i:300 "Sur veyi'ng" p. 301. City Surveys. 1:1000 to ItSODO Baker. * (1888). "Engineers' Surveying /ns trum ents" p. S3. (See Foottiote). (See Footnote). Carhart. 0888). "Surveying' p. ISI. Ordinary Farm Surveys. i:Soo Level Ground. 1:1000 Rougit Ground. 1:200 to l:3O0 Average Transit Surveys. i:i200 Wood. (See Footnote). (See Footnote). (Roanoke, Va., 1692). _' Precise Traverses wit/A Repeated /Ingles. J 1:10 000 (Baltimore, Md-, 1394) 1:15 000 -^.04 Ft. Raymond. (/396J. "Surveying," p. 144. Ordinary Farm Surveys. nsoo Good Farm Surveys. 1:2000 Baker derives the fortnu/a E. = -/] where ' d^ ~^ /2 000 000 E IS the permissible /inear error of c/osure, P the /erjgth of the perimeter, I'd the ratio of the chaining error, and a the angular error of closure in minutes. A thorough te^t of this formula under a wide range of conditions proves if to be trustworthy' However, the use of a chaining rcrtio^ /:d, presumably of fixe'd value for the same chainmen, does not accord tv^th th& resu/ts of experience in careful ivarHj for it is found that the differences between duplicate chainings yary about as the square foot of the iength of fine. On the fo/low/n^ poge a sftnpfifred fhrmufa }s oisr^amed by as- suming the more cot7sistent re/a/ion Just stated for fhe chaitving errors. The resu/ts are about fhe^ame as thos^ obtained yv^ith Batter's formuta^ and the fbmt of the express iOf> is icfejrticaf tvith that used by iVood in the &t/titr?ore Surrey. TESTS OF PRECISION. 219 THE PRECISION OF TRAVERSE SURVEYS. The reasonable or perm/ssibte error of closure of a traverse Survey may he determined by the formula derived Leiotv, provided the errors of ff'e/d tvorft are under oorttrol and their magn/ttida is ftnotn/n, at /east apfsroxrmarely. Let P= length of perimeter. L= calculated error of latitudes. D~ calculated error of departures. E^ actual or calculared linear error of cfoSurG offravcr^c c = coefficient of precision of chaitring. C = linear error of closure due to chai/ilng errors. a= angular error of closure in miriutes. A •= //near error of closure due fo angular errors. Ef^ permissible or reasonable linear error of closure cfue fo errors of chaining and angle. In the triang/e of error the hypothenuse is y^="v/-*+D". In Dtagram A oe/otv lvalues of Eg may he read close enough for most cases. Diagram A may also serve as a crude grap/iical rrav~ erse table, and blunders in r/ye fie/d v^r/f may be /ocated by ir. /n careful chaining by men of some training, the error Marie's about as the sguare root of the distance, ff^c be the compensating error for the unit d/sfance, f/rei? C= cifp , The angu/ar error of closure in careful surveys prt^ai>/y occurs arrrong the sides in proportion to t/?eir /engths. Assuming this To be the case, the resulting linear error is A — aP.arc !=> .OOOSaP. In good worM the errors are snjalf in amount and egual/y If able to be plus and minds. Hence, the probable error of c/osarc due t-o the tirvo causes, i.e. thi> reasonali/e or pern?issib/e //near er- ror of closure is Ep=l/A'-*-C' —^/'.OOff'SaePJ'-t-c^P This formula may be much simp/if led by completing the sguarc and dropping rhe negative tern? under the radical, whence vvirh sufficient exactness, there resu/ts the genera/ formti/a Ep^.0003af*-^ I700c^ s • • • -fl) The very exact standard, P-^/SOOO-*:ad-ft.,used of Baltitporc, (see table_, preceding page), may be obtained from (O by tnatdng tt somewhat less than y- minute, and cs.oosft., these va/uas oeing chnsistent wit/^ the fie/d vnorH of that survey. The va/ue of c may be def-ermihed for the given ehoin/nen, or The chaining term of (I) may be taMen as fol/otvs:~ for heat tvarf^ (c^oos-ft.), .OSft} for dverage worH (c^.OIOft.),,Zft.; for fair worH CcK.O'SJ, ,•? ft.' and for poor nvorH (ci^.OZO), .8 ft. /n care*' ful traverse Surveys the angle ternf a/one affords a rigid test, so that formula (B) maybe used except vrhen a='0. Diagrcing 3 gives f£J for the genera/ run of traverse prob/err^s. Ep=.0003aP=.^sPg. f£, A. Actual Error. 0* S; 10' 15* 20" IV 30" 35' 1 ifi i^ ^^p 1 1 1 B. Permissible Error. Sse Formula (2) rpgro 8 9 10 Error of Deporture, O. tDOO SOW 3000 4000 5000 6000 7000 6000 9000 now Length of Perimeter. /? Feet (or LinKsJ 220 EREORS IN SURVEYING. THE PRECISION OF LEVEL CIRCUITS. (For Good Average Practice.) when the length of the level circuit is known in lOO-ft stations, or when merely the number of settings of the Instrument and the approx- imate average distarjce covered per setting are hnown, the following modlficatiofjs of the preceding lest are valuable. Let £= maximum permissible error of closure of level circuit. M = length of level circuit it) miles. L= lOD-ft. stations. L'~ approximate average tdisfartce covered per setting of the instrument in WO-ff: staflotis. 5 = number of instrumental settings in the circuit f^or ^ood average worH with the engmeers' level E = 0.05ft?fM from which E = 0.007 fhl/L and E = 0.007 fffES Substituting for 100 -ft. average sights, L'=8, E = O.OISS ft.VJ . 350— ■ - L'=7, E=O0lBZft.TlS • 300-- • ■ 11=6, E= 0.0163 fi.iS ■ SSO- ■ ■ L=S, E=0.0IS4ft.l/S For a very rapid approximate check under ordirtary conditions, it may be assumed that E^O.OlftYS. A graphical representation of these formulas is given belorv. Permissible Error of Closure of Level Circuits For Careful WorK with a Good Engineers' Level. Length of Circuit Given In Miles (Upper Curye); Or in the Number of Insfromental Settings fMialc/le Group of Curves); or in 100-Foot Units (Lower Carre in Diagram^. Length of Level Circuit, M, Miles. 5 10 15 20 035 0.30 iJo.JS 1 0.20 S0.I5 J 0.00 30 40 « 10 Length of EO 30 Level Circuit, L, 50 i : ; : :; :: -M M ;;;::;-;; : : :: :^ ::: : : :^ \ : 1 ^lili j ;[ iiMj|:j y 1 1 :: llllllmlllllliraaairfiliTtiJITfflTlilUI^ % & ilMIIMtHi 1 111 1 '. :: ;:g II lHjiLUiliJIll liWfi|Hr*Ki -U-U- ■ ■ i: ;Ji ■ ■■■■■■■■ fflB 1 \ ; : :::- 0.35 0.25 40 50 60 70 80 90 100 100-Foot 5tation5; or Number of Level Seftinq5,5- TESTS OF PRECISION. 221 THE PRECISION OF LEVEL CIRCUITS. The precision of spirit leveling is expressed by the formula Error of Closure =s Constant 1/ Length of Circuit In the fallonlnj summary of practice in representative surveys of The United States^ E is the majrimum limit of error of closure of a level circuit having a length of K kilometers or M miles. Precision of Leveling in Representative Surveys. MAXIMUM PERMISSIBLfi ERROR OF CLOSURE, Metric Unifi British Units. Coefficient to Coefficient to nearest nearest mm. O.OOIft. OiOlft. E=3mm?/K'=0.0ISftiM =0.om.'iM E= imm?/si<= 0.018 ft.iM\ Mississippi Piver Commission. (Ml). E= imm'SER-= 0.018 ft.T/M V= O.oiftiM Mississippi Kiver Com'nlBefore 1890. E= 5mm:>flf = O.OSI ff.W) United States Coast Survey. E= Smm^lZK = 0.0^9 ff.l/M -O.OiftM United States Lake Survey E=IOmm?[K = 0.012^.^^ =O.O^ft.iM Vnlted States Geological Survey. E= O.OSO ft.T/M = 0.05 ft.iM A simple practical test of the degree of precision attained in spirit leveling is found In the last column of the above table. This graduated scale of precision is given below graphically for distances to ten miles. NAME OF SURVEY. Chicaijo Sanitary District. Missouri River Commission. Precision Diagram for Level Circuits. I 2 3 4 Length (f Level Circuit M, Miles> 222 EEEORS IN SUEVEYING. common usage, or in the proportion of the respective lati- tudes and departures, as would seem to be more consistent. If the several courses have not been surveyed with like precision, weights should be assigned in distributing the errors. Absurd refinement should be avoided in making the distribution of errors. Leveling Errors. — Perhaps in no phase of surveying measurements is it more clearly established that accidental errors follow the law of square roots than in careful level- ing. The precision diagrams are based on best current usage. CHAPTER X. METHODS OF COMPUTING. Introduction. — To no one is the ability to make calcula- tions accurately and rapidly of more value than to the engi- neer. Many fail to appreciate the value of rapid methods of calculation, and have no conception of the amount of time that can be saved by the skillful use of arithmetic, logarithms, reckoning tables and computing machines. In the field the engineer has to depend upon the ordinary methods of arithmetic, or a table of logarithms for his results. The use of these aids should therefore receive special attention, for the engineer cannot afford to lose the time of his assistants while he makes unnecessary or ex- tended computations. In the ofBce tables of squares, reckoning tables, slide rules and computing machines can be used in many cases with profit. Consistent Accuracy. — It is safe to say that at least one- third of the time expended in making computations is wasted in trying to attain a higher degree of precision than the nature of the work requires. In making arithmetical computations where decimals are involved it is a common practice to carry the result out to its farthest limit and then drop a few figures at random. In using logarithms time and labor are lost by using tables that are more extensive than the data will warrant. The relative amount of work In using four, five', six and seven-place tables is about as 1, 2, 3 and 4. Besides the extra labor involved, the computer has u, result that is liable to give him an erroneous idea of the accuracy of his work. In making computations, in general, calculate the result to one more place than it is desired to retain. If several numbers are multiplied or divided, a given percentage of error in any one of them will produce the same percentage of error in the result. 223 224 METHODS OP COMPUTING. In taking the mean of a series of quantities it is consist- ent to retain one more place than is retained in the quan- tities themselves. In direct multiplication or division retain four places of significant figures in every factor for an accuracy of about one per cent ; retain five places of significant figures in every factor for an accuracy of about one-tenth of one per cent. LOGAEITHMIC CALCULATIONS. Iiogarithm Tables. — Logarithm tables contain the deci- mal part of the logarithm called the mantissa, the integral part called the characteristic is supplied by the computer. Four-place tables give the mantissa to four decimal places of numbers from 1 to 999, and by interpolation give the mantissa of numbers from 1 to 9,999. Four-place log- arithms should be used where four significant figures are sufficient, and should not be xised where an accuracy greater than one-half of one per cent is required. Five-place tables give the mantissa to five decimal places of numbers from 1 to 9,999, and by interpolation give the mantissa of numbers from 1 to 99,999. Five-place loga- rithms should be used where five significant figures are sufficient, and should not be used where an accuracy greater than one-twentieth of one per cent is required. Five-place tables are sufficiently accurate for most engineering work. Six-place tables give the mantissa to six decimal places of numbers from 1 to 9,999, and by interpolation give the mantissa of numbers from 1 to 99,999, the same as the five- place tables. Six-place tables give practically no gain in precision over fi.ve-place tables since the same numbers of significant figures are given in both tables, and in addition the labor of using a six- instead of a five-place table is about as 3 to 2, due to interpolation with larger diffier- ences. For the above reasons five-place tables have been selected for use in this book as being the most suitable tables for use in surveying. Seven-place tables give the mantissa to seven decimal places of numbers from 1 to 99,999, and by interpolation of numbers from 1 to 999,999. Seven place tables are rarely needed in engineering work, except in triangulation work where the angles are measured by repetition. ARITH^iIETICAL CALCULATIONS. 225 AEITHMETICAL CALCULATIONS. Requirements. — To become a rapid computer the follow- ing requirements are essential : (1) A good memory for retaining certain standard num.- bers for reference. (3) The power of performing the ordinary simple arith- metical operations of multiplication, division, etc., on num- bers with facility, quickness and accuracy. (3) The power of registration, i. e., of keeping a string of numbers in the mind and working accurately upon them. (4) The power of devising instantly the best method of performing a complicated problem as regards facility, quickness and certainty. It is obvious that all do not have the ability to become rapid computers, but even these can become fairly skillful by constant practice and perseverance. The ordinary pro- cesses of arithmetic should be performed with numbers in all possible positions. No more figures should be put down than necessary, and all operations should be performed mentally whenever possible. In the mental part the results should alone be stated, much time being lost by repeating each separate figure. Checks. — In order to check his work the computer should keep the following well known properties of numbers well fixed in his mind : (1). The sum or difference of two even or of two odd numbers is even. (3) The sum or difference of an even and odd number is odd. (3) The product of two even numbers is even. (4) The product of two odd numbers is odd. (5) The product of an even number and an odd number is even. (6) Checking results by the familiar operation of east- ing out the 9's depends upon the following properties of numbers : (a) A number divided by 9 leaves the same remainder as the sum of the digits divided by 9. For example : 4384 -H 9 = 487 -|- 1 (4-t-3H-8-l-4)^9 = 3-Fl (7)) The excess of 9's in the product equals the excess of 9's in the product of the excesses of the factors. 226 ilETHODS OF COMPUTING. 473,295 Excess = 3 4,235 Excess = 5 15 Excess = 6 2,004,404,325 Excess = Check (e) The excess of 9's in the dividend equals the excess of 9's in the product of the excesses in tlie di%'isor and quo- tient, plxis the excess in the remainder : 56)2443 Excess in divisor ^2 43 -)- 35 Excess in quotient = 7 Excess in remainder := 8 Excess in (2 X 7 + 8) =41 Excess in dividend —4j-^'^eck (7) Results should be checked by taking aliquot parts wherever possible, and by performing the operations in inverse order or performing inverse operations. Computa- tions performed by means of logarithms should be checked by making the computations roughly by means of arith- metic. Tlie prohahility of error should be recognized and precaution fallen, to verify results. ADBITIOUr. — Since the eye is accustomed to pass from left to right time can be saved, where the cohimns are not too long, by adding in the same way. The device of in- creasing or diminishing the numbers to make them mul- tiples of ten and then subtracting or adding to the result is very convenient, especially where several columns are added at one time. Ex. 1. — 96 47 143 212 69 32 87 331 49 380 The mental work in detail is as follows : 100 + 47 = 147 ; 147 — 4 = 143 ; 143 + 70 =: 213 ; 213 — 1 ^ 212; 212 + 30 + 90 = 332; 332 — 1 = 331; 331 + 50 = 381; 381 — 1=:380. Expert accountants use the method of adding columns in groups of 10, 20, 30, etc., small figures, indicating the mimlier of the group, being placed along the column at in- tervals depending upon the computer. This method is well MULTIPLICATION. 227 adapted to the addition of long columns where one is liable to be called away from his work. The progress of the work being then shown by the number of the group, plus the excess. MULTIPLICATIOUr. — In order to make the best use of the methods given, the computer should have perfect com- mand of the multiplication table as far as 20 at least. (1) When the tens differ by unity and the sum of the units equals 10, numbers may be multiplied by the follow- ing rule : Prom the squares of the tens of the larger number subtract the square of the units of the larger number. For the numbers may be represented by (a -\- i) and (a — 6), and the product will be (a + 6) {a — 6)^o^ — 6^ E.T. i.— (93 X87)=90= — 3= =8,100 — 9 = 8,091. (3) The product of composite numbers is best obtained mentally by resolving them into their factors and taking the products of the factors. ESB. 2.— 26 X 36 = 9 X 13X 8 — 936. Ex.3.— 48 X24=(24)^X 3 = 1,152. Multiples of 10. — To multiply by some number which is a factor of 10 or some multiple of 10, for example: Multi- ply CIO" A by B, where B = — — a Annex n ciphers to A, multiply by C and divide by d. Ex. i.— 4,324 X 625 = 4,334 ^ =(4,324,000 X 5)-H 8 = 3,702,500. Ex. 2.-7,924 X 25 = 792,400 H- 4 = 198,100. Squaring Small Numbers. — Numbers may be squared mentally by the following rule : Add to or subtract from one factor enough to make its units figure zero. Subtract from or add to the other factor tne same amount. Multiply together this sum and difEerence, and to the product add the square of the amount by which the factors were in- creased or diminished. Proof.— a^ — B^=(a-f6)(a— 6) a= = (a + 6)((i — 6)+6'. Ex. i.— (76) = = (73X80) + 4- = 5,776. 228 METHODS OF COMPUTING. Ex. 2.— (137) = = (124 X 130) + 3^ = 16,139. Ex. S.— ( 61/i) ^ = ( 6 X 6%) + (1/4) ^ = 39%e- Ex. J,.— (61^)^ = (6 X 7) + (1^)^ = 421/4. Ex.5.— (7.5)^ = (7x8) + (-5)' = 56.25. It will be seen that the process is very simple where the units place Is 5. (3) Having- the square of any number the square of the number next higher is obtained by the following rule : To the known square add the number and the next higher and the result will be the square of the next higher number. Ex.6.— (25)^=635. (26)^ = 635 + 35 + 36 = 676. (3) A very close approximation to the square of a quan- tity which is very near unity is obtained by adding algebra- ically two times the difference between the quantity and unity to the quantity. Proof. — (1 + 6)''= 1 + 36 + 6^ = 1 + 26, (approximate). Ex. 7.— (1.05) = = 1 + 2(1.05 — 1)=1+ 10=110. Ex. 8.— (.94)^=1 — 2(1 — .94)=1 — .12= 88. E.r. 9.— (2.034) = = 2=(1 + 2 X .017)= 4(1.034)= 4.136. Cross-Multiplication. — This consists in taking the prod- uct of each digit in the multiplicand by each digit in the multiplier and taking the sums, products of the same de- nomination being determined thus : units X units gives units ; tens X units and units X tens gives tens ; units X hundreds, tens X tens and hundreds X units give hundreds etc. All products are added mentally, only the final result being put down. Ex. i.— (2,347) = = 5,508,409 the final result being all that it is necessary to write down. The mental work is as follows, the figures in heavy t pe being figures in the prod- uct ; 7X7 = 49; 4 + 2(7X4)=60; 6 + 2(7X3) + 4= = 64; 6 + 3(3 X 7)+3(3 X 4)=58; 5 + 3(2 X 4) + 3= = 30; 3 + 2(3 X 2)= 15; 1 + 3==5. Ex. 2. — The product of any two numbers may be found in the same manner. 9,433 3,583 24,362,856 CEOSS-MULTIPLICATION. 229 The mental work is as follows :3X2 = 6;3X3 + 8X2 :=:25; 3 + 3X4 + 8X3 + 5X2 = 48; 4 + 3X9 + 8X4 + 5X3 + 2X3 = 82; 8 + 8X9 + 5X4 + 3X3 = 106; 10 + 5X9 + 3X4 = 63; 6 + 2X9 = 34. Ear.. 3. — The process of cross-multiplication may be sim- plified as follows : Eequired to multiply 4,338 by 736 ; write the multiplier on a slip of paper in inverse order and place it below the multiplicand with the left hand figure below the units place of the multiplicand thus : IMultiply together the figures in the same vertical column 6 X 8 ^ 48 ; set down the 8 and carry the 4 ; then move the slip one space to the left thus : 4,338 I ^37"! 8 Multiplying together the figures in the same vertical col- umns and taking the sum, 4 + 6X2 + 3 X8 = 40; set down the and carry the 4 ; then move the slip one space to the left, multiplying together the figures in the same vertical columns, adding, etc., we will finally have the work standing thus : 4,338 I 637 I 3,185,408 Removing the slip we have 4,328 736 3,185,408 The multiplier may be written on the bottom of a sheet in inverse order and placed above the multiplicand instead as above described. The work, however, is very much simplified by simply writing the multiplier in inverse order without using the slip : 4,328 637 3,185,408 230 ilETHODS OF GOAli'UTlJNCi. The mental work being as follows : 6X8^ 48; 4 + 6X 3 + 3X8 = 40; 4 + 6X3 + 3X3 + 7X8 =84 ; 8 + 6 X 4+3X3+7X3 = 55 ;5 + 3X4 + 7X3 =38 ; 3 + 7 X 4 = 31. It will be seen that this device removes most of the mental strain, there being no cross-products. CONTBACTED MULTIPLICATION.— In multiplying decimals, when the product is required to a few places of decimals, the work may be shortened as follows : Kequired a product correct to the nth decimal place. Write the multi- plier with its figures in reverse order, its units place under the nth decimal place of the multiplicand. Multiply the multiplicand by the figures in the multiplier, beginning with the right hand figure ; rejecting those figures in the multiplicand which are to the right of the figure used as a multiplier, increasing each product by as many units as would have been carried from the rejected part of the mul- tiplicand, taking the nearest unit in each case ; place the right hand figure of each partial product in the same col- umn, and add as in common multiplication. In most cases it is best to compute one more place than required. The following examples illustrate the process : Ex. 1. — The radius of a circle is 420.17 ft. What is its semicircumference to nearest 0.01 ft.? (vr^S. 14159265.) In the work below the partial products in the contracted multiplication are seen to correspond to the partials of the common method, taken in reverse order, the part to the right of the vertical line being rejected. The contracted multiplication is carried one more place than required. A dot is j)laced above each figure when it is rejected from the multiplicand. 4 2 0.1 7 O 4 3 0.1 7 5 6 2 9 5 1 4 1.3 S.1 4 1 5 9 3 !«0510 112 6051 42017 37 8153 16807 210|0 85 4 2 4 2 017 210 16 8 6 8 3 8 4 2 17 1 126051 I 1 3 2 0.0 O 3 1 3 2 0.0 3|1 3 8 1 Ex. 2. — The observed length of a line is 2231.63 ft. with a tape having a length of 100.018 ft. Required the reduced length of the line to the nearest 0.01 ft. CONTKACTED DIVISION. 231 Noting that each foot of the tape = 1.00018 ft. 2 2 3 1.6 3 2 2 3 1.6 3 8 1 0.1 1.0 1 8 223163 1785304 22 223163 18 - 223163000 2 2 3 2.0 3 2 2 3 2.0 3|1 6 9 3 4 Ex. 3. — Same observed length with a tape 99.982 ft. long. Required the reduced length. Each foot of the tape = 0.99983 =(1 — 0.00018) ft. 2 2 3 1.6 3 8 10 0.0- 22 18 — 0.4 2 3 3 1.6 3 0.9 9 8 3 4 4 6 3 2 6 1785304 200S467 2008467 2008467 2 2 3 1.2 3 223 1.2 283066 Ex. Jt. — To compare contracted multiplication with log- arithmic work, calculate 861.3 ft. X sin 17° 19' to the nearest 0.1 ft. log. 8 6 1.3 = 2.9 3 5 1 5 log. sin 17° 19' = 9.4 7 3 7 1 log. (2 5 6.4) =2.4 8 8 6 2 5 6.4 CONTBACTED DIVISION.— If the quotient is desired correct to the nth decimal place, the following method may be used : Find one-half of the desired figures in the quotient in the usual way and do not bring down a figure for the last remainder. Drop a figure from the right of the divisor and find another figure in the quotient. Then without bringing down any more figiires continue to discard figures from the divisor until the required places are obtained. Ex 1. — Divide 443.9425 by 24.311 to nearest hundredth. There will be four figures in the quotient, so we will find 8 6 1.3 5 6 7 9 2.0 1723 776 60 5 232 METHODS OF COMPUTING. the first two in the ordinary way. A dot is placed over each figure in the divisor when it is rejected. 2 4.3 2 ) 4 4 3.9 4 2 5 ( 1 8.2 5 2432 20074 10456 618 486 132 122 10 Divisor Near Unity. — '\A'hen the divisor is near unity a very close approximation is given by the method shown in the following problems : EJ!. i.— , „„^. ,, = 5(1 — .003554)= 5 X .996746 = 4.98373 1.003204 correct to within one unit in the fifth place. E^- 2.— -^=7(1+(1 — .9982))=7 X 1.0018 = 7.0126 correct to the last place. CONTBACTED SQTTAIIE ROOT. — A result correct to a required number of decimal places may be found by a process similar to the method employed for contracted divi- sion. Ex. 1. — Required the square root of 12,598.87325 correct to thousandths. We see by inspection that the root will contain six figures. Find in the ordinary way the first three figures. Form a new trial divisor in the usual way, 1 2 5 » S.S 7 3 2 5 ( 1 1 2.2 4 5 1 21)35 21 222 ) 498 444 224)548 448 100 89 11 11 CONTRACTED SQUARE ROOT. 233 and bring down only one figure for the dividend in place of two. Eind the remaining figures by contracted division. The last figure brought down is not increased whatever it may be followed by, since the contracted process tends to make the result a little too large. This method may be ap- plied to the extraction of cube roots, where it saves much work in finding long trial divisors. Square Koot of Small Numbers. — The approximate square roots of small numbers may be found by means of the following rule : Divide the given number by the number whose square is nearest the given number. The arith- metical mean of the quotient and divisor will be the ap- proximate square root of the number. The nearer the number is to a perfect square the less the error. For example, Ex. i.— V~35=(35/g -I- 6) -=- 3 = 5.93. Ex. 2.— V~8=(% + 3)-=-3 = 3.83. Ex 3.— V"^ =(7% -1-9)-:- 2 = 8.89. Ex. 4.— V128=(12%i + ll)-=-3n=11.31. Square B.oot by Subtraction. — ^While it possesses no points of merit in this connection, it would not be proper to pass the subject of square root without presenting the novel method of extracting square roots used with the Thomas Computing machine. The method depends upon the rela- tion existing between odd numbers and squares in the sys- tem of numbers having a radix ten. If we sum up the odd numbers, beginning at 1, we will observe the following relation : 1 = 1=; 1-1- 3 = 4 = 3=; 1-1- 3 -I- 5 = 9 = 3^; 1 -1-3 -f-S-f- 7 = 16 = 4", etc. It will be seen that the square root of the sum in each case is the number of the group. The method of extracting square roots is as follows : Point off in periods of two figures each. Subtract from the left hand period the odd numbers in order, beginning at unity, until a remainder is obtained less than the next odd number. Write for the first figure in the root the number which represents the number of subtractions made. Double the root already found and annex unity. Subtract as before, using for subtrahends the successive odd num- bers, the root figure being the number of subtractions made. 234 METHODS OF COMPUTING. Ex. 1. — Extract the square root of 53,824. r. 3834(232 _1 4 3 2 Hiibtractinns. 41)138 41 97 43 54 4 5 3 subtractions. 401)924 461 463 4 6 3 ... 2 subtractions. RECKONING TABLES. — Tables for use in computing are so numerous and well known that it would be useless to try to refer to them by name. Two valuable tables for obtaining products of numbers — which are well known in Germany, but comparatively unknown in this country — are, '■ Crelle's Eechentafeln," which gives the products of num- bers of three significant figures by three significant figures to 999 by 999 ; and " Zimmerman's Eechentafeln," which gives the products of numbers of two places of significant figures by numbers of three significant figures to 100 by 999. . COMPUTING MACHINES.— In Fig. 40, (a) is a Kutt- ner reckoning machine ; (b) a Thomas computing machine ; (c) a Fuller slide rule; (d) a Thacher slide rule; (e) an ordinary slide rule; (f) a Colby Stadia slide rule; (g) a Colby sewer slide rule; (h) a Grant calciilating machine; (i) a full circle protractor; (j) a Crozet protractor; (k) a protractor tee square ; (1) a polar planimeter ; (m) a " jack knife "' planimeter ; (n) a pantagraph ; (o) a, section liner ; (p) a spherical planimeter. In using the " jack knife " planimeter, the point is placed at the center of gravity, and the knife edge is placed on a line passing through the center of gravity of the figure. The point is then made to traverse the perimeter of the figure to be measured ; passing out to the perimeter and returning to the center of gravity of the figure on the same line. The distance from the final position of the knife edge to the line through the center of gravity, multiplied by the COMPUTING INSTRUMENTS. ra) 235 236 METHODS OF COMPUTING. length of the arm of the planimeter will give the area of the figure. The arm of the planimeter is usually made ten inches long and the distance measured in inches. The correct area may be obtained by means of the hatchet planimeter, without using the center of gravity of the figure, as follows: (1) Draw a tangent to the figure. (2) Trace the figure with the point starting with the hatchet on the tangent and the point at the point of tangency. (3) Trace the figure as before except that the point is to move around in the opposite direction. (4) The arithmetical mean of the two areas will be the true area. That this method is correct can be easily proved by the student. The other machines are described in the instructions ac- companying them when purchased. CHAPTER XI. TOPOGRAPHIC DRAWING AND LETTERING. LETTERING. — A magnified scale is used in the first six plates to giFB familiarity with form of letter and numeral, and also to produce freedom of hand motion. The six plates should first be made with a soft pencil sharpened to a needle point, and afterward with pen and india ink. In Plate 7 the height of letter is that prescribed in Chapter I. This standard size is not only well adapted to field notes and general drafting, but is economical of execution. The student should train the eye and acquire a " swing " of the hand by industrious practice in such exercises as the fol- lowing: (1) Pass a line freehand through two points; first sketch in the line roughly by a free swing of the forearm ; then partially erase and retrace ; finally test result with ruler. (2) Pass a circular arc through three points free- hand; follow sketch method just described and, after per- fecting the arc, sketch in the chords and locate the center freehand; test result mechanically. (3) Inscribe a circle in a square. (4) Inscribe an ellipse in a rectangle. (5) Inscribe an ellipse in an oblique parallelogram. (In the last three exercises give particular attention to points and lines of tangency and axes of symmetry.) After making the line or figure satisfactorily with pencil, it should be executed freehand in India ink. Practice should include spacing of letters and words, and for this purpose it is suggested that the student use the " specifications for a good engineer " following the preface. The student should not be content until he can make letters freehand so well that a close inspection is required to determine that they were not made mechanically. Freehand Titles. — Good freehand titles suffice for most drawings. In a good title consistent emphasis is given to the several parts, and the title as a whole accords with the purpose and character of the drawing. Elaborate and or- namental titles have a limited application, and should not be attempted at all unless the draftsman has special skill 237 238 TOPOGltiU'HIC DKAWING AND LETTEEING. FKEE HAND LETTEIUNG. 239 240 TOPOGKAPHIC DRAWING AND LETTERING. FKEE HAND LETTERING. 241 3^: iiiiiiii lllllll iiiiiiii iliiiliB iliUliil 17 242 TOPOGEAPHIC DEAWING AND LETTERING. s i -5^1 m m || li^i I I I w I i^^ H m m m m FKEE HAND LETTERING. 243 244 TOPOGRAPHIC DEAWING AJSfD LETTERING. / / / / 7 / 1 / / c [ 3 / 1 1 7 -' 1 / / TECzztmr II H 1 Ml )() P ^ ■i / // 7 7 1 / IM^UtEJl'Bi IK / fHI 10 P K 'i / ^ X ^^ ^^ / / TELfTEMlL IK // Ml i() ^t^ H •i // WY / / _ ^ cLinU'E.^'niii j-kt f fr,i ITt 77 7 I' ""f r 77 V 7 ^ 1 1 / ]_ / zAaH^^TfE^T- T't t ih( 777 77 ?7 -J-' T t TT" '/ / J whTLU-EJEirELTi frK 7 -rr);/ ) 1) 77 f?/ r 77 t W^ r "" -L d d T^ / y .( t^jit'Bznt ()\ / f- u ^ ^ 1 H 9 /; rV / / / I JZZlfEET^R ()\ f /' ^4 6^ IH ,y ^ / / - - ! SBLDIEEHI ,IK \ Mr \{) P (^ K S 1 11 V SEEDIEEII IK \ Ml \{) P Q i S 1 w y 1 5:3i:dei5hi IK \ Ml 10 F (^ { s 1 Y 7 — -1- — h: icdEtqin T"^ T THT 1 TT n r "S- T: TT T — ,r 1 nJJnBsJiqSi --\ T TTl 1 TT n r "=T t TT ■^r — \ zzi'icdErqiin: T"t^ T m 1 n T7" "n rr" ^ t W V — i 1 ; 115251122 ()<\ \ '( 'A s b » ^ h 135255155 \\h \ '( ' ^4 b f) ( H 4 h T 115255125 {)h 1 '( ' 4 b h / « <) . DRAWING PENS. 245 in sueli work. In designing titles, whether freehand or mechanical, skill in sketching in the outlines, guide lines, axes of symmetry, etc., is of much importance. On the following pages are a few examples of good titles. W Ei 1 ^ I :s a s z DBA WING PENS. — The following pens, arranged in order of fineness, will give sufficient variety for ordinary work. Gillott's 170, very fine, for very small lettering. Gillott's 303, extra fine, for small lettering. Gillott's 404, fine, for small lettering. Hunt 21, medium, for ordinary lettering. Hunt 513, Shot Point, for ordinary lettering. Leonardt 510, E. E. Ball Point, for large lettering and titles. Hunt 513, Round Point, for large lettering and titles. Leonardt 516, E. E. Ball Point, for large lettering and titles. Leonardt 516 E., Ball Point, for very large lettering and titles. Payzant Pens, K. & E. Co., Nos. 6, 5, 4, 3, 2, 1, for titles. The following rules should be observed in making letters on drawings free hand. Use the quill in inking the pen. Never dip the pen in the ink bottle. Keep the pen clean. Ink must not be allowed to dry on the pen and spread the points. Before rising a new pen moisten the points and wipe it dry to insure a free flow of ink. TOPOGRAPHIC SYMBOLS.— The standard symbols for topographic drawings adopted by the American Railway Engineering Association are given on pages 248 to 351. 246 TOPOGRAPHIC DEAWING AND LETTERING. Right-of-way Map liEwYoRK AMD Denver R.R. Shahion 551+55 to Station 54Z+75 Scale lin.=400 Ft. January 3, 1915 Of Fice 0? ChieF Engineer Denver,Colorado. Right°of=Way Map NewYork and Denver RR station 351+55 to Station 511+10 Scale 1 in =400 ft. January 30,1915 Office oF Chief Engineer Denuer, Colorado. •(oPOGRAPH/c Map OFTHE City OF Boulder,Colorado Surveyed by the Class in Topographic Surveying University of Colorado First Semester I9I4-I? Scale lin= 500 Ft. MAP TITLES DRAWING AND LETTEKING. 247 Right-of-WayMap flEWYORKAMDDEflVERR.R. Station 55k55 to Station 54^+75 ScaielinrMFL January3J9i5 ' Office of Chief fnqineer Denver, Coiorado. RmihitofWay Map New YORK& Denver KK St a tion 33U55 to Station 511 f 10 Scatelin-rdOOft. January 1,1915 Office of Chief Engineer Denver, Colorado. lojjographic Map ClTYOFBOULDER,COLORADO teucrsifg of (Hofora^o ¥ir$f0'tmtshr 1914-15 JcaU iMrSQOfJt. 248 TOPOGRAPHIC DRAWING AND LETTERING. HVDROfeRAPHY. Stream Springs and Sinks Lakes and Ponds Falls and Rapids Water Line Marsh Canals Ditches Contour System Sand Cliffs Cut Embanhment Top of Slope Bottom of Slope Name Relief. tuiiuiiijiiiiiiKliiuir: uuuimuiuiiiiiifiiii. TOPOGRAPHIC SYMBOLS. 249 -■^Railways (Topographical Maps.) Steam t — i — i — t-n — i — i — i — i — i — i Electric i i i i i i i i i i i Street Railways mimi »■ i « Railway Tracks (Track Maps.) Railway Track or Old Track to Remain — Old Track to he Taken up rz^-jin^-.^-.^-.^z Proposed Tracks — Proposed (Future) Tracks ~-:rz^z^rz^rL:n^rz _ . -^ , Color o ther than Reef or Foreign Tracks ~ ■^ dloc/i with Initials of Road Alinement rj"^^^^*^^^ I 12" ■ Left- ) 4'C.R. Z'C.L Boundary and Survey Lines. ( Political Divisions -, State, County Bethel T wp.-w.v ne Co.,Mich. 1 or Township Lines. pS^Tw7^TOcl!>d" J Government Surveys, Base, Meridian, sec i6.t. i zn.,r. i e..5"' pm^ "l Township,6ection or Harbor Line 5eoi3.T iiTT^ifir^p'M Street, Block or other Property Line Survey Lines -^ 4^^ -' Location oenrer l ineb if Monumented, Show Location and Proper Symbol Company Property Line _ State Kind and Height Fence (on Street Line ) ' ■ ' ■ '- _ , - -, , , State Hind and Height Fence (on Company Property Line) ■■ ' ■ •_--.-. 5H For Railway Trach and Yard Stvdiei Use Single or Double Linei^ 18 250 TOPOGRAPHIC DRAWING AND LETTERING. City !□□□□□! Village ■ Jr 1 • ir> City Lim its k^;^f^/^;^3;^3i Fire Limits \Ica}A/,/\£^Aa7\.£ZU. Section Corner 17 1 16 20 1 11 Section Center —-.^q.^- Triangulafion Station or Transit Point A Bench l^arl< B.M.Xl23H Stone Monument u Iron /Monument ■ Ml SCELLANEOUS. Pole Wire Lines Railway Tunnel Dimension Lines 'ndicaieNcofWim ^Ownership -^ — f — r =^ ^^ ~Blas,U True and Magnetic l^eridian I Graphic '■^ CO let/ 50' 100 TOPOGRAPHIC SYMBOLS. 251 Culverts, Sewers, etc. Masonry Arch or Flat Top Culvert \v.V.'.Z'.'~S~~"^ „. .,, I n V- , ^^ . JSrafe Kind and Lengihjjnd, Pipe or Wood Boj! Culvert or Dram \:-~:r---r::.:.":\ '^ ' Kind af Walls, If any.) ' Catch Basin D' C.B. Manhole -p- M. M Sump Qsump Water Supply and Pipe Lines. Give /f-v Water Tank o.STS^C^"'-^- OiveSite Water Column o 1 Track Pan u-uo-uoaxxaj Company Water Pipe 'Give'Siie Other Water Pipe ->-*->-->-->->-»-->->->-*-> ^, r Oive Size Steam or Oas - — Give Size Compreaed Air <■-■• ■■ •' '■-' ' • Highways and Crossings. Pub/ic and Mam Roada 7-/ Private and Secondary Roads Trails Street and Public Road Crossings - // //' Pnvate Road Crossing // Bridges. Girder ^ Truss ^— Trestle . )— 1— ' i SURVEYING MANUAL PAKT II FIELB AND OFFICE TABLES FOB TTSE IN SURVEYING. BY WILLIAM D. PENCE AND MILO S. KETCHUM Table 1. Logarithms of Numbers. Table 2. Logarithmic Functions of Angles. Table 3. Natural Functions of Angles. Table 4. Squares, Cubes, Square Roots, Cube Roots and Circles. Table 5. Trigonometric Functions. Explanation of Tables. The authors wish to thank the J. B. Lippincott Company for the use of Tables 1 and 2 taken from Suplee's " Five Place Logarithms," and Table 3 taken from Suplee's " Me- chanical Engineers' Reference Book " ; and the McGraw- Hill Book Company for the use of Tables 4 and 5, taken from Harger and Bonney's " Highway Engineers' Hand- book." All of the above tables are fully protected by copyright. 253 254 LOGARITHMS OF NUMBERS. Table 1. N um. 100 to 139. Log . 000 to 145. N L 0, 1 2 3 4 5 6 7 8 9 P. P. 100 00 000 043 087 130 173 217 260 303 346 389 44 43 101 432 475 518 661 604 647 689 732 776 817 1 2 4.4 8.8 4.3 8.6 102 860 903 945 988 *030 *072 *1]5 *157 *199 *242 103 01 284 326 368 410 462 494 536 578 620 662 3 13.2 12.9 104 703 745 787 828 870 912 953 995 *086 *078 4 5 17.6 22.0 17.2 21.5 105 02 119 160 202 243 284 325 366 407 449 490 6 7 8 26.4 30.8 85.2 25.8 30.1 34.4 106 531 572 612 663 694 735 776 816 857 898 107 938 979 *019 *060 *100 *141 *181 *222 *262 *302 9 89.6 38.7 108 03 342 383 423 463 503 543 583 623 663 703 42 41 109 743 782 822 862 902 941 981 *021 *060*100 110 04 139 179 218 258 297 336 376 416 454 493 1 2 4.2 8.4 4.1 8.2 111 532 571 610 660 689 727 766 805 844 883 3 12.6 12.3 112 922 961 999 *038 *077 *115 *154 *192 *231 *269 4 5 6 16.8 21.0 26.2 16.4 20.5 24.6 113 05 308 346 385 423 461 500 538 576 614 652 114 06 690 070 729 108 767 145 806 183 843 221 881 258 918 296 956 333 994 *032 371 408 7 8 9 29.4 33.6 37.8 28.7 32.8 36.9 116 446 483 521 558 595 633 670 707 744 781 Jf\ tn 117 819 856 893 930 967 *004 *041 *078 *115 *151 4U ov 118 07 188 225 262 298 835 372 408 415 482 518 1 4.0 3.9 119 555 591 628 664 700 787 773 809 846 882 2 3 8.0. 12.0 7.8 117 120 121 08 918 279 954 314 990 *027 *063 350 386 422 *099 *135 468 493 *171 *207 *243 529 565 600 4 5 6 16.0 20.0 24.0 16.6 19.5 23 4 122 636 672 707 743 778 814 849 884 920 955 7 28.0 27.3 123 991 *026 *061 *096 *132 *167 *202 *237 *272 *307 8 32.0 36.0 31.2 124 09 342 377 412 447 482 517 662 587 621 656 9 86.1 125 691 726 760 795 830 864 899 934 968 *003 38 37 126 10 037 072 106 140 176 209 243 278 812 346 1 3.8 3.7 127 380 415 449 483 617 551 585 619 653 687 2 7.6 7.4 128 721 755 789 823 857 890 924 968 992 *025 3 4 11.4 15.2 11.1 14.8 129 130 11 059 391 093 428 126 461 160 494 193 528 227 561 261 594 294 628 827 361 661 694 5 6 7 19.0 22.8 26.6 18.5 22.2 25.9 131 727 760 793 826 860 893 926 959 992 *024 8 30.4 29.6 132 12 057 090 123 166 189 222 254 287 320 352 9 34.2 33.3 133 385 418 450 483 516 548 581 613 646 678 36 35 134 710 743 775 808 840 872 905 937 969 *001 1 3 6 3.5 135 136 13 033 354 066 386 098 418 130 450 162 481 194 613 226 545 268 577 290 822 609 640 2 3 4 7!2 10.8 14 4 7!o 10.5 14 137 672 704 735 767 799 830 862 893 925 956 5 18.0 17.5 138 988 *019 *051 *082 *1U *146 *176 *208 *239 »270 6 21.6 21.0 139 14 301 333 364 396 426 467 489 520 551 582 7 8 25.2 28.8 24.5 28.0 140 613 644 675 706 737 768 799 829 860 891 9 32.4 31.5 N L 1 2 3 4 5 6 7 8 9 P. P. Table 1. LOGAEITHMS OF NUMBERS. 255 Num. 140 to 179. Log. 146 to 255. N L 1 2 3 4 5 6 7 8 9 P. P. 140 14 613 644 675 706 737 768 799 829 860 891 34 33 141 922 953 983 *014 *045 1 *076 *106 *137 *168 *198 1 2 3.4 6.8 3.3 6.6 142 15 229 259 290 320 351 381 412 442 473 503 143 534 564 594 625 655 685 715 746 776 806 3 10.2 9.9 144 836 866 897 927 957 987 *017 *047 *077 *107' 4 6 13.6 17.0 13.2 16.5 145 16 137 167 197 227 256 286 316 346 376 406 6 7 8 20.4 23.8 27.2 19.8 23.1 26.4 146 4S5 465 495 524 554 684 613 643 673 702 147 732 761 791 820 860 879 909 938 967 997 9 30.6 29.7 118 17 026 056 085 114 143 173 202 231 260 289 32 31 149 319 348 377 406 435 464 493 522 551 580 150 609 638 667 696 725 754 782 811 840 869 1 2 3.2 6.4 3.1 6.2 151 898 926 955 984 *013 *041 *070 *099 *127 *156 3 9.6 9.3 152 18 184 213 241 270 298 327 355 384 412 441 4 5 6 12.8 16.0 19.2 12.4 15.5 18.6 153 469 498 526 554 583 611 639 667 696 724 154 752 780 808 837 865 893 921 949 977 *005 7 8 22.4 26.6 21.7 24.8 155 19 033 061 089 U7 145 173 201 229 267 285 9 28.8 27.9 156 312 340 368 396 424 451 479 507 535 562 30 29 157 590 618 645 673 700 728 758 783 811 838 158 866 893 921 948 976 *003 *030 *058 *085 *112 1 3.0 2.9 159 20 140 167 194 222 249 276 303 330 358 385 2 3 6.0 9.0 5.8 8.7 160 412 439 466 493 520 548 675 602 629 656 4 5 6 12.0 15.0 18.0 11.6 14.5 161 683 710 737 763 790 817 844 871 898 926 17^4 162 952 978 *005 *032 *059 *086 *112 *139 *165 *192 7 21.0 20.3 163 21 219 245 272 299 325 352 378 405 431 458 8 9 24.0 27.0 23.2 26.1 164 484 511 537 564 590 617 643 669 696 722 165 748 775 801 827 854 880 906 932 958 986 28 27 166 22 Oil 037 063 089 115 141 167 194 220 246 1 2.8 2.7 167 272 298 324 350 376 401 427 453 479 505 2 3 5.6 8 4 5.4 8 1 168 531 557 583 608 634 660 686 712 737 763 4 ll!2 io!8 169 789 814 840 .866 891 917 943 968 994 *019 5 6 14.0 16.8 13.5 16.2 170 23 045 070 096 121 147 172 198 223 249 274 7 19.6 18.9 171 172 300 553 325 578 350 603 376 629 401 654 426 679 452 704 477 502 528 729 754 779 8 9 22.4 25.2 21.6 24.3 173 805 830 865 880 905 930 955 980 *005 *030 26 25 174 24 055 080 105 130 155 180 204 229 254 279 1 2.6 2.5 175 304 329 353 378 403 428 462 477 602 527 2 3 5.2 7.8 5.0 7 5 176 551 576 601 625 650 674 699 724 748 773 4 io!4 io!o 177 797 822 846 871 895 920 944 969 993 *018 5 13.0 12.5 178 25 042 066 091 115 139 164 188 212 237 261. 6 7 15.6 18 2 15.0 17.5 179 285 310 334 358 382 406 431 455 479 503 8 20.8 20.0 180 527 551 575 600 624 648 672 696 720 744 9 23.4 22.5 N L 1 2 3 4 5 6 7 8 9 1 P. P. 256 LOGAKTTHMS OF Nx^T^rp.ERS. Table 1. Num. 180 to 219. Log. 255 to 342. N L 1 2 3 4 S 6 7 8 9 P. P. ISO 25 527 551 575 600 624 648 672 696 720 744 24 181 768 792 816 840 864 888 912 935 959 983 1 2.4 ■ 182 26 007 031 055 079 102 126 150 174 198 221 2 4.8 183 245 269 293 316 340 364 387 411 435 458 3 7.2 184 482 505 529 553 576 600 623 647 670 694 4 5 9.6 12.0 185 717 741 764 788 811 834 858 881 905 928 6 7 14.4 16 8 186 951 975 988 *021 *045 *068 *091 *114 *138 *161 8 19 2 187 27 184 207 231 254 277 300 323 346 370 393 9 21.6 188 410 439 462 485 508 531 554 57Z, 600 623 23 189 646 669 692 715 738 761 784 807 830 852 IPO 875 898 921 944 967 989 *012 *035 *058 *081 1 2 4.6 191 28 103 126 149 171 194 217 240 262 285 307 3 6.9 192 330 353 375 398 421 443 466 488 511 533 4 5 9.2 11.5 193 556 578 601 6'23 616 668 691 713 735 758 6 13!8 194 780 803 825 847 870 892 914 937 959 981 7 8 16.1 18.4 195 29 003 026 048 070 092 115 137 169 181 203 9 20.7 196 226 248 270 292 314 336 358 380 403 425 22 197 447 469 491 513 535 557 579 601 623 045 198 667 688 710 732 7.51 776 798 820 842 863 1 2.2 199 885 907 929 951 973 994 *016 *038 *060 *081 2 3 4.4 6.6 200 30 103 125 146 168 190 211 233 265 276 298 4 5 8.8 11.0 201 320 811 363 384 406 428 449 471 49? 514 6 13^2 202 535 557 578 600 621 643 664 685 707 728 7 15.4 203 750 771 792 814 835 856 878 899 920 942 8 g 17.6 19.8 204 963 984 *006 *027 *048 *069 *091 *112 *133 *154 205 31 175 197 218 239 260 281 302 323 346 366 Zl 206 387 408 429 450 471 492 613 534 555 576 1 2.1 207 597 618 639 660 681 702 723 744 765 785 2 3 4.2 6.3 208 806 827 848 869 890 911 931 952 973 994 4 8!* 209 32 015 035 056 077 098 118 139 160 181 201 6 6 10.5 12.6 210 222 243 263 2K4 305 325 346 366 387 408 7 14.7 211 428 449 469 490 510 531 552 572 593 613 8 9 16.8 18.9 212 634 664 675 695 715 736 756 777 797 818 213 838 858 879 899 919 940 960 980 *001 *021 20 19 214 33 041 062 082 102 122 143 163 183 203 224 1 2.0 1.9 215 24-1 264 284 304 325 345 365 385 405 425 2 3 4 4.0 6.0 8.0 3.8 5.7 7.6 216 445 465 486 506 626 546 566 586 606 626 217 646 666 686 706 726 746 766 786 806 826 5 1 0.0 9.5 218 846 866 885. 905 925 945 965 985 *005 *025 6 1 7 1 8 1 2.0 4.0 6.0 11.4 13 3 15.2 219 34 044 064 084 104 124 143 163 183 203 223 220 212 262 282 301 321 341 361 380 400 420 9 1 8.0 17.1 N L 1 2 3 4 S 6 7 8 9 P. P. Table 1. LOGARITHMS OF JNUMBiiKa. vn Num. 220 to 259. Log. 342 to 414. N L 1 2 3 4 5 6 7 8 9 P. P. 220 34 242 262 282 301 321 341 361 380 400 420 221 439 459 479 498 518 537 657 677 596 616 20 222 635 655 674 694 713 733 753 772 792 811 1 2.0 223 830 850 869 889 908 928 947 967 986 *005 2 4.0 224 35 025 044 061 083 102 122 141 160 180 199 3 4 6.0 8.0 225 218 238 ^7 276 295 315 334 353 372 392 5 6 10.0 12 226 411 430 449 468 488 607 526 545 564 583 7 iiio 227 603 622 ■641 660 679 698 717 736 755 774 8 16.0 228 793 813 832 851 870 889 908 927 946 985 9 18.0 229 984 *003 *021 *04D *D59 *078 *097 *116 *135 *154 230 36 173 192 211 229 248 267 286 305 324 342 19 231 361 380 399 418 436 455 474 493 611 530 232 519 568 586 605 624 642 661 680 698 717 1 1.9 233 736 754 773 791 810 829 847 866 884 903 2 3 4 3.8 5.7 7.6 234 922 940 959 977 996 *014 *033 *051 *070 *088 235 37 107 125 144 162 181 199 218 236 254 273 5 6 9.5 11.4 236 291 310 328 316 365 383 401 420 438 457 7 13.3 237 475 493 511 530 548 566 585 603 621 639 8 9 15.2 17.1 238 658 676 694 712 731 749 767 786 803 822 239 840 858 876 894 912 931 949 967 985 *003 240 38 021 039 057 076 093 112 130 148 166 184 241 202 220 238 266 274 292 310 328 346 364 • 18 242 382 399 417 435 453 471 489 507 525 543 243 561 578 596 614 632 650 668 686 703 721 1 1.8 214 739 757 775 792 810 828 846 863 881 899 2 3 3.6 5.4 245 917 934 952 970 987 *005 *023 *041 *058 *076 1 5 7.2 9 246 39 094 111 129 146 161 182 199 217 235 262 6 io!8 247 270 287 305 322 340 358 375 393 410 428 7 12.6 248 445 463 480 498 515 533 650 668 685 602 8 9 14,4 Ifl 9. 249 620 637 655 672 690 707 724 742 759 777 250 794 811 829 846 863 881 898 915 933 960 251 967 985 *002 *019 *037 *064 *071 *088 *106 *123 262 40 140 157 175 192 209 226 243 261 278 295 253 312 329 346 361 381 398 415 432 449 466 17 254 483 600 518 535 652 569 586 603 620 637 1 1.7 255 654 671 688 705 722 739 756 773 790 807 2 3 3,4 5.1 266 824 841 858 875 892 909 926 943 960 976 4 6.8 257 993 *010 *027 *044 *061 *078 *095 *111 *128 *145 5 6 7 8.5 10.2 11.9 258 41 162 179 196 212 229 246 263 280 296 313 259 260 330 497 347 514 363 531 380 547 397 564 414 430 447 464 481 581 597 614 631 647 8 9 13.6 15.3 N L 1 2 3 4 S 6 7 8 9 P. P. 258 LOGARITHMS OP NUMBEliS. Table 1. Num 260 to 299. Log. 414 to 476. N L 1 2 3 4 5 6 7 8 9 P. P. 260 41 497 514 531 547 564 581 697 614 031 047 261 604 681 697 714 731 747 764 780 797 814 262 830 847 863 880 896 913 929 946 963 979 263 996 *012 *029 *045 *002 *078 *095 *111 *127 144 264 42 100 177 193 210 226 243 259 275 292 308 17 205 325 341 367 374 390 406 423 439 455 472 1 1 "7 260 488 504 521 537 653 570 580 602 619 635 2 3.4 207 651 667 684 700 716 732 749 766 781 797 3 5.1 208 813 830 846 862 878 894 911 927 943 959 4 6 6 6.8 8.5 10.2 269 975 991 *008 *024 *040 *056 *072 *088 *1(M *120 270 43 136 152 169 185 201 217 233 249 265 281 7 8 11.9 13.6 271 297 313 329 345 361 377 393 409 425 441 9 15.3 272 457 473 489 505 521 637 553 509 584 600 273 616 632 648 664 680 696 712 727 743 769 274 775 791 807 823 838 854 870 886 902 917 275 933 949 965 981 996 *012 *028 *044 *059 *075 270 44 091 107 122 138 154 170 185 201 217 232 16 277 248 264 279 295 311 326 342 358 373 389 278 404 420 436 451 467 483 498 514 529 545 1 1.6 279 560 576 592 607 623 638 654 669 685 700 2 3 3.2 4.8 280 7}6 731 747 762 778 793 809 824 840 855 4 5 6 6.4 8.0 9.6 281 871 886 902 917 932 948 963 979 994 *010 282 43 025 040 056 071 086 102 117 133 148 163 7 11.2 283 179 194 209 225 240 255 271 286 301 317 8 9 12.8 14.4 284 332 347 362 378 393 408 423 439 454 469 285 484 500 515 530 545 561 576 591 606 621 280 637 652 667 682 697 712 728 743 758 773 287 788 803 818 834 849 864 879 894 909 924 288 939 954 969 984 *000 *015 *030 *045 *060 *075 289 40 090 105 120 135 150 166 180 195 210 226 15 290 240 255 270 286 300 315 330 345 359 374 1 1.5 291 389 404 419 434 419 464 479 494 609 523 2 3 4 3.0 4.5 6.0 292 538 553 508 683 598 613 627 642 667 672 293 687 702 716 731 716 761 776 790 805 820 5 7.5 294 835 850 864 879 894 909 923 938 953 967 6 7 9.0 10.6 295 982 997 *012 *020 *041 *066 *070 *086 *100 *114 8 9 12.0 13.5 296 47 129 144 159 173 188 202 217 232 246 261 297 276 290 305 319 334 349 363 378 392 407 298 422 430 451 465 480 494 609 624 538 553 299 567 582 596 611 625 640 654 669 683 698 300 712 727 741 756 770 784 799 813 828 842 N L 1 2 3 4 S 6 7 8 9 P. P. Table 1. LOGARITHMS OF NUMBERS. 259 Num. 300 to 339. Log. 477 to 531. 1 8 P. P. 47 712 727 741 756 770 857 871 885 900 914 48 001 015 029 044 058 144 159 173 187 202 287 302 316 330 344 430 444 458 473 487 572 586 601 615 629 714 728 742 756 770 855 869 883 897 911 996 *010 *024 *038 *052 49 136 150 164 178 192 276 290 304 318 332 415 429 443 457 471 •554 568 582 596 610 693 707 721 734 748 831 845 859 872 886 969 98? 996 *010 *024 50 106 120 133 147 161 243 256 270 284 297 379 393 406 420 433 515 529 542 556 569 651 664 678 691 705 786 799 813 826 840 920 934 947 961 974 51 055 068 081 095 108 188 202 216 228 242 322 335 348 362 375 455 468 481 495 508 587 601 614 627 640 720 733 746 759 772 851 865 878 891 904 983 996 *009 *022 *035 52 114 127 140 153 166 244 257 270 284 297 375 388 401 414 427 504 617 530 543 556 634 647 660 673 686 763 776 789 802 815 892 905 917 930 943 53 020 033 046 058 071 148 161 173 186 199 784 799 813 828 842 929 943 958 972 986 073 087 101 116 130 216 230 244 259 273 359 373 387 401 416 501 515 530 544 558 643 657 671 686 700 785 799 813 827 841 926 940 964 968 982 *066 *080 *094 *108 *122 206 220 234 248 262 346 360 374 388 402 486 499 513 527 541 624 638 651 665 679 762 776 790 803 817 900 914 927 941 965 *037 *051 *066 *079 *092 174 188 202 215 229 311 326 338 362 365 447 461 474 488 501 583 596 610 623 637 718 732 746 759 772 853 866 880 893 907 987 *001 *014 *028 *041 121 136 148 162 175 255 268 282 295 308 388 402 415 428 441 521 534 548 561 574 654 667 680 693 706 786 799 812 825 838 917 930 943 957 970 *048 *a61 *075 *088 *101 179 192 205 218 231 310 323 336 349 362 440 463 466 479 492 569 582 595 608 621 699 711 724 737 750 827 840 853 866 879 956 969 982 994 *007 084 097 110 122 135 212 224 237 250 263 P. P. 260 LOGARITHMS OF NUMBERS. Table l. Num 340 to 379. Log 531 to 579. N L 1 2 3 4 5 6 7 8 9 p. p. 340 63 148 161 173 186 199 212 224 237 250 203 311 275 2XS 301 314 326 339 352 364 377 390 312 403 415 428 441 453 400 479 491 504 517 313 529 542 555 667 580 593 605 618 631 643 344 656 668 081 694 706 719 732 744 757 769 13 345 782 794 807 820 832 845 857 870 882 895 1 2 1.3 2.6 340 908 920 933 945 958 970 983 995 *008 *020 347 54 033 046 058 070 083 095 108 120 133 145 3 3.9 348 158 170 183 195 208 220 233 245 258 270 4 5 5.2 6 5 349 283 295 307 320 332 345 357 370 :«2 394 7^8 350 407 419 432 444 456 469 481 494 506 518 7 8 9.1 10.4 361 531 543 555 568 580 593 605 617 630 042 9 11.7 352 654 667 679 691 704 716 728 741 753 765 353 777 790 802 814 827 839 851 804 876 888 354 900 913 925 937 949 902 974 986 998 *011 365 55 023 036 047 060 072 084 096 108 121 133 356 145 157 169 182 194 206 218 230 242 255 12 367 267 279 291 303 315 328 340 352 364 376 358 388 400 413 425 437 449 401 473 485 497 1 1.2 359 509 522 534 546 558 570 582 594 606 618 2 3 2.4 3.0 360 630 642 654 666 678 691 703 715 727 739 4 5 6 4.8 6.0 7.2 361 761 763 775 787 799 811 823 835 847 859 362 871 883 895 907 919 931 943 955 967 979 7 8.4 363 991 *003 *015 *027 *038 *050 *062 *074 *080 *098 8 9 9.6 10.8 3G4 56 110 122 134 146 158 170 182 194 205 217 365 229 241 253 265 277 289 301 312 324 336 366 348 360 372 384 396 407 419 431 443 465 367 467 478 490 502 514 526 538 549 601 673 368 585 597 608 620 632 044 056 667 679 691 369 703 714 726 738 750 761 773 785 797 808 11 370 820 832 844 855 807 879 891 902 914 920 1 1.1 371 937 949 961 972 984 990 *008 *019 *031 *043 2 3 2.2 3.3 4.4 372 57 054 066 078 089 101 113 124 130 148 159 4 373 171 183 194 200 217 229 241 252 264 276 6 5.5 374 287 299 310 322 334 345 367 308 380 392 6 7 0.0 7.7 375 403 415 426 438 449 401 473 484 196 507 8 9 8.8 9.9 376 519 530 512 553 5(,5 570 ,688 000 Oil 623 377 634 646 057 669 080 692 703 715 726 738 378 749 761 772 784 795 807 818 830 841 852 379 864 875 887 898 910 921 933 944 955 967 380 978 990 »001 *013 *0a4 *035 *047 *058 TO70 *081 N L i 2 3 4 S 6 7 8 9 P. P. Table 1. LOGARITHMS OF NUMBERS. 2111 Num. 380 to 419. Log. 579 to 623. N L 1 2 3 4 S 6 7 8 9 P. P. 380 67 978 990 *001 *013 *024 *035 *047 *058 *070 *081 381 58 092 104 115 127 188 149 161 173 184 196 382 206 218 229 240 252 263 274 286 297 309. 383 320 331 343 354 365 377 388 399 410 422 384 433 444 456 467 478 490 501 612 524 535 1 1 385 546 557 569 580 591 602 614 625 636 647 386 659 670 681 692 704 715 726 737 749 760 1 ■* ■* 2 1.1 2.2 387 771 782 794 805 816 827 838 860 861 872 3 3.3 388 883 894 906 917 928 939 950 961 973 984 4 6 6 4.4 6.5 6.6 389 995 *006 *017 *028 *040 *051 *062 *073 *084 *095 390 59 106 118 129 140 151 162 173 184 196 207 7 8 7.7 8.8 391 218 229 240 251 262 273 284 295 306 318 9 9.9 392 329 340 351 362 ,373 384 395 406 417 428 393 439 450 461 472 483 494 606 517 628 539 394 550 561 572 583 594 605 616 627 638 649 395 660 671 682 693 704 715 726 787 748 759 396 770 780 791 802 813 824 835 846 857 868 10 397 879 890 901 912 923 934 945 956 966 977 398 988 999 *010 *021 *032 *043 *054 *066 *076 *086 1 1.0 399 60 097 108 119 130 141 152 163 173 184 195 2 3 2.0 3.0 400 206 217 228 239 249 260 271 282 293 304 4 6 6 4.0 6.0 6.0 401 314 325 336 347 358 369 379 390. 401 412 402 423 433 444 455 466 477 487 498 809 520 7 7.0 403 531 541 552 563 574 584 595 606 617 627 8 9 8.0 9.0 404 638 649 660 670 681 692 703 713 724 735 405 746 756 767 778 788 799 810 821 831 842 406 853 863 874 885 895 906 917 927 938 949 407 959 970 981 991 *002 *013 *023 *034 *045 *055 408 61 066 077 087 098 109 119 130 140 151 162 409 172 183 194 204 215 225 236 247 257 268 410 278 289 300 310 321 331 342 352 363 374 411 384 395 405 416 426 437 448 458 469 479 412 490 500 511 521 532 512 563 563 574 584 413 595 606 616 627 637 648 658 669 679 690 414 700 711 721 731 742 752 763 773 784 794 415 805 815 826 836 847 857 868 878 888 899 416 909 920 930 941 951 962 972 982 993 *003 417 62 014 024 034 045 055 066 076 086 097 107 418 118 128 138 149 169 170 180 190 201 211 419 221 232 242 262 263 273 284 294 304 315 420 325 335 346 356 366 377 387 397 408 418 N L 1 2 3 4 S 6 7 8 9 P. P. 262 LOGARITHMS OF NUMBERS. Table 1. Num . 420 to 459. Log . 623 to 662. N L 1 2 3 4 S 6 7 8 9 P. P. 420 62 325 335 346 356 366 377 387 397 408 418 421 428 439 449 459 469 480 490 500 511 521 422 531 542 552 562 572 583 593 603 613 624 423 634 644 655 665 675 685 696 706 716 726 424 737 747 757 767 778 788 798 808 818 829 425 839 849 859 870 880 890 900 910 921 931 426 941 951 961 972 982 992 *002 *012 *022 *033 427 63 043 053 063 073 083 094 104 114 124 134 428 144 155 165 175 185 195 205 215 225 236 10 429 246 256 266 276 286 296 306 317 327 337 430 347 357 367 377 387 397 407 417 428 438 1 ' ' 2 1.0 2.0 431 448 458 468 478 488 498 508 518 528 538 3 3.0 432 548 558 568 579 589 599 609 619 629 639 4 5 6 4.0 5.0 6.0 433 649 659 669 679 689 699 709 719 729 739 434 435 749 849 759 859 769 869 779 879 789 889 799 899 809 819 829 839 909 919 929 939 7 8 9 7.0 8.0 9.0 436 949 959 969 979 988 998 *008 *018 *028 *038 437 64 048 058 068 078 088 098 108 118 128 137 438 147 157 167 177 187 197 207 217 227 237 439 246 256 266 276 286 296 306 316 326 335 440 345 355 365 375 385 395 404 414 424 434 441 444 454 464 473 483 498 503 513 523 532 442 512 552 562 572 582 591 601 611 621 631 443 640 650 660 670 680 689 699 709 719 729 444 738 748 758 768 777 787 797 807 816 826 445 836 846 856 865 875 885 895 904 914 924 9 446 933 943 953 963 972 982 992 *002 *011 *021 1 0.9 447 65 031 040 050 060 070 079 089 099 108 118 2 3 4 1.8 2.7 3.6 448 128 137 147 157 167 176 186 196 205 215 449 4S0 225 321 234 331 244 341 254 350 263 360 273 369 283 292 302 312 379 389 398 408 5 6 7 4.5 6.4 6.3 451 418 427 437 447 456 466 475 485 495 504 8 9 7.2 8.1 452 514 523 533 543 552 562 571 581 591 600 453 610 619 629 639 648 .658 667 677 686 696 454 706 715 725 734 744 753 763 772 782 792 455 801 811 820 830 839 849 868 868 877 887 456 896 906 916 925 935 944 954 963 973 982 457 992 *001 *011 *020 *030 *039 *049 *058 *068 *077 458 66 087 096 106 115 124 134 143 153 162 172 459 181 191 200 210 219 229 238 247 257 266 460 276 285 295 304 314 323 332 342 351 361 N L i 2 3 4 S 6 7 8 9 P. P. Table 1. LOGARITHMS OF NUMBERS. 263 Num 460 to 499. Log. 662 to 698. N L 1 2 3 4 5 6 7 8 9 P. P. 460 66 276 285 295 304 314 323 332 342 351 361 461 370 380 389 398 408 417 427 436 445 455 462 464 474 483 492 502 511 521 530 539 549 463 558 567 577 586 596 605 614 624 633 642 464 652 661 671 680 689 699 708 717 727 736 465 745 755 764 773 783 792 801 811 820 829 466 839 848 857 867 876 885 894 904 913 922 467 932 941 950 960 969 978 987 997 *006 *015 468 67 025 034 043 052 062 071 080 089 099 108 10 469 117 127 136 145 154 164 173 182 191 201 470 210 219 228 237 247 256 265 274 284 293 1 2 1.0 2.0 471 302 311 321 330 339 348 357 367 376 385 3 3.0 472 394 403 413 422 431 440 449 459 468 477 4 5 6 4.0 5.0 6.0 473 486 495 504 514 523 532 511 550 560 569 474 475 578 669 587 679 596 688 605 697 614 706 624 715 633 724 642 651 660 733 742 752 7 8 9 7.0 8.0 9.0 476 761 770 779 788 797 806 815 825 834 843 477 852 861 870 879 888 897 906 916 925 934 478 943 952 961 970 979 988 997 *006 *015 *024 479 68 034 043 052 061 070 079 088 097 106 115 480 124 133 142 151 160 169 178 187 196 205 481 215 224 233 242 251 260 269 278 287 296 482 305 314 323 332 341 350 359 368 377 386 483 395 404 413 422 431 440 449 458 467 476 484 485 494 502 511 520 529 538 547 556 565 485 574 683 592 601 610 619 628 637 646 655 9 486 664 673 681 690 699 708 717 726 735 744 1 0.9 487 753 762 771 780 789 797 806 815 824 833 2 3 4 1.8 2.7 3.6 488 842 851 860 869 878 886 895 904 913 922 489 490 931 69 020 -7108 940 028 949 037 958 046 966 055 975 064 984 073 993 *002 *011 082 090 099 5 6 7 4.5 5.4 6.3 491 117 126 135 144 152 162 170 179 188 8 9 7.2 8.1 492 197 205 214 223 232 241 249 268 267 276 493 285 294 302 311 320 329 338 346 366 364 494 373 381 390 399 408 417 425 434 443 452 495 461 469 478 487 496 504 513 522 531 539 496 548 557 566 574 583 592 601 609 618 627 497 636 644 653 662 671 679 688 697 705 714 498 723 732 740 749 758 767 775 784 793 801 ' 499 810 819 827 836 845 854 862 871 880 888 SCO 897 906 914 923 932 940 949 958 966 975 N L 1 2 3 4 5 6 7 8 9 P. P. 264 LOGARITHMS OF NUMBERS. Table 1. Num . 500 to 539. Lo^ . 698 to 732. N L 1 2 3 4 5 6 7 8 9 P. P. soo 69 897 906 914 922 932 940 949 958 966 975 501 984 992 *001 *010 *018 *027 *036 *044 *053 *062 502 70 070 079 088 096 105 114 122 131 140 148 503 157 165 174 183 191 200 209 217 226 234 504 243 252 260 269 278 286 295 303 312 321 505 329 338 346 355 361 372 381 389 398 406 506 415 424 432 441 449 458 467 475 484 492 507 501 509 518 526 535 544 552 561 569 578 508 ^586 595 603 612 621 629 638 646 655 663 9 509 ' 672 680 689 697 706 714 723 731 740 749 510 757 766 774 783 791 800 808 817 825 834 1 2 0.9 1.8 511 842 851 859 868 876 885 893 902 910 919 3 2.7 512 927 935 944 952 961 969 978 986 995 *003 4 5 6 3.6 4.5 5.4 513 71 012 020 029 037 046 054 063 071 079 088 511 515 096 181 105 189 113 198 122 206 130 214 139 223 147 231 155 240 164 172 248 257 7 8 9 6.3 7.2 8.1 516 265 273 282 290 299 307 315 324 332 341 517 349 357 366 374 383 391 399 408 416 4'25 518 433 441 450 458 466 475 483 492 500 508 519 517 525 533 542 550 559 567 575 584 592 520 600 609 617 625 634 642 650 659 667 675 521 684 692 700 709 717 725 734 742 750 759 522 767 775 784 792 800 809 817 825 834 842 623 850 858 867 875 883 892 900 908 917 925 521 933 941 950 958 966 975 983 991 999 *008 525 72 016 0'24 032 041 049 057 066 074 082 090 8 526 099 107 115 123 132 140 148 156 165 173 1 0.8 527 181 189 198 206 214 222 230 239 247 265 2 3 4 1.6 2.4 3.2 528 263 272 280 288 296 304 313 321 329 337 529 530 346 428 351 436 362 444 370 452 378 460 387 469 395 477 403 485 411 419 '493 501 5 6 7 4.0 4.8 6.6 531 509 518 526 534 542 550 558 567 575 583 8 9 6.4 7.2 532 591 599 607 616 624 632 640 648 656 665 533 673 681 689 697 705 713 722 730 738 746 534 754 762 770 779 787 795 803 811 819 827 535 835 843 852 860 868 876 884 892 900 908 536 916 925 933 941 949 957 965 973 981 989 537 997 *006 *0U *D22 *030 *038 *046 *054 062 *070 538 73 078 086 094 102 HI 119 127 135 143 151 539 159 167 175 183 191 199 207 215 223 231 540 239 247 255 263 272 280 288 296 304 312 N L 1 2 3 4 5 6 7 8 9 P. P. Table 1. LOGARITHMS OF NUMBEKS. 265 Num. 540 to 579. Log. 732 to 763. N L 1 2 3 4 5 6 7 8 9 P. P. 540 73 239 247 255 263 272 280 288 296 304 312 541 ^20 328 336 344 352 360 368 376 384 392 542 400 408 416 424 432 440 448 456 464 472 543 480 488 496 504 512 520 528 536 544 552 544 560 568 576 584 592 600 608 616 624 632 545 640 648 656 664 672 679 687 695 703 711 546 719 727 785 743 751 759 767 775 783 791 547 799 807 815 823 830 838 846 854 862 870 548 878 886 894 902 910 918 926 933 941 949 8 -549 957 965 973 981 989 997 *005 *013 *020 *028 550 74 036 044 052 060 068 076 084 092 099 107 1 2 0.8 1.6 551 115 123 131 139 147 155 162 170 178 186 3 2.4 552 194 202 210 218 225 233 241 249 257 265 4 5 6 3.2 4.0 4.8 553 273 280 288 296 304 312 320 327 335 343 554 555 351 429 359 437 367 445 374 453 382 461 390 398 406 414 421 468 476 484, 492 600 7 8 9 5.6 6.4 7.2 556 507 515 523 531 539 547 554 562 570 578 557 586 593 601 609 617 024 632 610 648 656 558 663 671 679 687 695 702 710 718 726 733 559 741 749 757 764 772 780 788 796 803 811 560 819 827 834 842 850 858 865 873 881 889 561 896 904 912 920 927 935 943 950 958 966 562 974 981 989 997 *005 *012 *020 *028 *035 *043 563 75 051 059 066 074 082 089 097 105 113 120 564 128 136 143 151 159 166 174 182 189 197 565 205 213 220 228 236 243 251 259 266 274 7 566 282 289 297 305 312 320 328 335 343 351 1 0.7 567 358 366 374 381 389 397 404 412 420 427 2 3 4 1.4 2.1 2.8 568 435 442 450 458 465 473 481 488 496 504 569 511 519 526 534 542 549 557 565 572 580 5 6 7 3.5 4.2 4,9 570 587 595 603 610 618 626 633 641 648 656 571 664 671 679 686 694 702 709 717 724 732 8 9 5.6 6.3 572 740 747 755 762 770 778 785 793 800 808 573 815 823 831 838 846 853 861 868 876 884 574 891 899 906 914 921 929 937 944 952 959 575 967 974 982 989 997 *005 •012 *020 *027 *035 576 76 042 050 057 065 072 080 087 095 103 110 577 -118 125 133 140 148 155 163 170 178 185 578 193 200 208 215 223 230 238 245 253 260 579 268 275 283 290 298 305 313 320 328 335 580 343 350 358 365 373 380 388 395 403 410 N L J 2 3 4 5 6 7 8 9 P. P. 266 liOGAEITHMS OP NUMBERS. Table 1. Num. 580 to 619. Log;. 763 to 792. N L 1 2 3 4 S 6 7 8 9 P. P. 580 76 343 360 358 365 373 380 388 395 403 410 8 581 418 425 433 440 448 455 462 470 477 485 1 2 0.8 1.5 582 492 500 507 515 523 530 537 545 552 659 583 567 574 582 589 697 604 612 619 626 634 3 2.4 684 641 649 666 664 671 678 686 693 701 708 4 5 3.2 4 585 716 723 730 738 745 753 760 768 775 782 6 7 8 4.8 5.6 6.4 686 790 797 805 812 819 827 834 842 849 856 587 864 871 879 886 893 901 908 916 923 930 9 7.2 688 938 945 963 960 967 975 982 989 997 *004 589 77 012 019 026 034 041 048 056 063 070 078 590 085 093 100 107 115 122 129 137 144 151 591 159 166 173 181 188 196 203 210 217 226 592 232 240 247 264 262 269 276 283 291 298 593 305 313 320 327 335 342 349 367 364 371 594 379 386 393 401 408 415 422 430 437 444 595 452 469 466 474 481 488 495 503 610 517 596 525 532 539 646 654 561 568 576 583 590 697 697 605 612 619 627 634 641 648 656 663 7 598 670 677 685 692 699 706 714 721 728 735 599 743 760 767 76i 772 779 786 793 801 808 1 2 0.7 1.4 600 815 822 830 837 844 861 859 866 873 880 3 4 5 2!l 2.8 3.6 601 887 895 902 909 916 924 931 988 946 952 602 960 967 974 981 988 996 *003 *010 *017 *025 6 4.2 603 78 032 039 046 053 061 068 075 082 089 097 7 4.9 604 104 111 118 125 132 140 147 154 161 168 8 9 5.6 6.3 606 176 183 190 197 204 211 219 226 233 240 606 247 254 262 269 276 283 290 297 305 312 607 319 326 333 340 347 355 362 369 376 383 608 390 398 405 412 419 ,426 433 440 447 456 609 462 469 476 483 490 497 504 512 519 626 610 633 540 647 554 561 669 576 583 690 597 611 604 611 618 626 633 640 647 654 661 668 612 675 682 689 696 704 711 718 725 732 739 613 746 753 760 767 774 781 789 796 802 810 614 817 824 831 838 845 852 859 866 873 880 015 888 895 902 909 916 923 930 937 944 951 616 958 965 972 979 986 993 *000 *007 *014 *021 617 79 029 036 043 060 057 064 071 078 085 092 618 099 106 113 120 127 134 141 148 155 162 619 169 176 183 190 197 204 211 218 226 232 620 239 246 253 260 267 274 281 288 295 302 N L 1 2 3 4 5 6 7 8 9 P. P. Table 1. LOGAEITHMS OP NUMBEKS. . 267 Num. 620 to 659. Log;. 792 to 819. N L 1 2 3 4 5 6 7 8 9 P. P. 620 79 239 246 253 260 267 274 281 288 295 302 621 309 316 323 330 337 344 351 358 365 372 622 379 386 393 400 407 414 421 428 435 442 623 449 456 463 470 477 484 491 498 505 511 624 518 525 532 539 546 553 560 567 574 581 625 588 595 602 609 616 623 630 637 644 660 626 657 664 671 678 685 692 699 706 713 720 627 727 734 741 748 754 761 768 775 782 789 628 796 803 810 817 824 831 837 844 851 858 629 865 872 879 886 893 900 906 913 920 927 630 934 941 948 955 962 969 975 982 989 996 631 80 003 010 017 024 030 037 044 051 058 065 632 072 079 085 092 099 106 113 120 127 134 633 140 147 154 161 168 175 182 188 195 202 634 209 216 223 229 236 243 250 257 264 271 635 277 284 291 298 305 312 318 325 332 339 636 346 353 359 366 373 380 387 393 400 407 7 637 414 421 428 434 441 448 455 462 468 475 638 482 489 496 502 509 516 523 530 536 543 1 0.7 639 550 557 564 570 577 584 591 598 604 611 2 3 1.4 2.1 640 618 625 632 638 645 652 659 665 672 679 4 5 6 2.8 3.5 4.2 641 686 693 699 706 713 720 726 733 740 747 642 754 760 767 774 781 787 794 801 808 814 7 4.9 643 821 828 835 841 848 855 862 868 875 882 8 9 5.6 6.3 644 889 895 902 909 916 922 929 936 943 949 645 956 963 969 976 983 990 996 *003 *010 *017 646 81 023 030 037 043 050 057 064 070 077 084 647 090 097 104 111 117 124 131 137 144 151 648 158 164 171 178 184 191 198 204 211 218 649 224 231 238 245 251 258 265 271 278 285 650 291 298 305 311 318 325 331 338 345 351 651 358 365 371 378 385 391 398 405 411 418 652 425 431 438 445 451 458 465 471 478 485 653 491 498 505 511 518 525 531 538 544 551 654 558 564 571 578^584 591 598 604 611 617 655 624 631 637 644 651 657 664 671 677 684 656 690 697 704 710 717 723 730 737 743 750 657 757 763 770 776 783 790 796 803 809 816 658 823 829 836 842 849 856 862 869 875 882 659 889 895 902 908 915 921 928 935 941 948 660 954 961 968 974 981 987 994 *000 *007 *014 N L I 2 3 4 5 6 7 8 9 P. P. 268 LOGARITHMS OP NUMHilJltS. ^um . 660 to 699. Log . 819 to 845. N L 1 2 3 4 5 6 7 8 9 P. P. 660 81 954 961 968 974 981 987 994 *000 *007 *014 7 661 82 020 027 033 040 046 053 060 066 073 079 662 086 092 099 105 112 119 125 132 138 146 1 u. / 1.4 663 151 158 164 171 178 184 191 197 204 210 3 2.1 664 217 223 230 236 243 249 256 263 269 276 4 6 2.8 3.5 665 282 289 295 302 308 315 321 328 334 311 6 7 8 4.2 4.9 5.6 666 347 354 360 367 373 380 387 393 400 406 667 413 419 426 432 439 445 452 458 465 471 9 6.3 668 478 484 491 497 504 510 517 523 530 536 669 543 549 556 562 569 575 682 588 596 601 670 607 614 620 627 633 640 646 653 659 666 671 672 679 685 692 698 705 711 718 724 730 672 737 743 750 756 763 769 776 782 78? 795 673 802 808 814 821 827 834 840 847 853' 860 674 866 872 879 886 892 898 905 911 918 924 675 930 937 943 950 956 963 969 975 982 988 676 995 *001 *008 *014> *020 *027 *033 *040 *046 *052 677 83 059 065 072 078 085^ 091 097 104 110 117 6 678 123 129 136 142 149 155 161 168 174 181 679 187 193 200 206 213 219 225 232 238 215 1 ; 0.6 2l 1.2 680 251 257 264 270 276 283 289 296 302 308 3 1.8 4 2.4 5 3.0 681 315 321 327 334 340 347 353 369 366 372 682 378 385 391 398 404 4le> 417 423 429 436 6 3.6 683 442 448 455 461 467 474' 480 487 493 499 7 4.2 8 ! 4.8 9 1 5.4 684 506 512 518 625 531 637 .544 650 556 563 685 569 675 582 588 694 601 607 613 620 626 686 632 639 645 651 658 664 670 677 683 689 687 696 702 708 715 721 727 734 740 746 753 688 769 765 771 778 7Si 790 797 803 809 816 689 822 828 835 841 847 853 860 866 872 879 690 885 891 897 904 910 916 923 929 935 942 691 948 954 960 967 973 979 985 992 998 *004 692 84 Oil 017 023 029 036 042 048 055 061 067 693 073 080 086 092 098 105 111 117 123 130 694 136 142 148 155 161 167 173 180 186 192 695 198 205 211 217 223 230 236 242 218 255 696 261 267 273 280 286 292 298 305 311 317 697 323 330 336 342 348 354 361 367 373 379 698 386 392 398 404 410 417 423 429 435 442 699 448 451 460 466 473 479 485 491 497 504 700 510 516 522 528 636 541 647 653 559 566 9 N L 1 2 3 4 S 6 7 8 P. P. Table 1. LOGARITHMS OF NUMBERS. 269 Num. 700 to 739. Log. 845 to 869. 85 510 516 522 528 535 572 578 584 590 597 634 640 6^16 652 658 696 702 708 714 720 757 763 770 776 782 819 825 831 837 814 880 887 893 899 905 942 948 954 960 967 003 009 016 022 028 065 071 077 083 089 126 132 138 144 150 187 193 199 205 211 248 254 260 266 272 309 315 321 327 333 370 376 382 388 394 431 437 443 449 455 491 497 503 509 516 552 558 664 570 576 612 618 625 631 637 673 679 685 691 697 733 739 745 751 757 794 800 806 812 818 854 860 866 872 878 914 920 926 932 938 974 980 986 992 998 86 034 040 046 052 058 094 100 106 112 118 153 159 165 171 177 213 219 225 231 237 273 279 285 291 297 332 338 344 350 356 392 398 404 410 415 451 457 463 469 475 510 516 522 528 534 570 576 581 587 593 629 635 611 646 652 688 694 700 705 711 747 753 759 764 770 806 812 817 823 829 864 870 876 882 888 923 929 935 941 947 L 1 2 3 4 541 547 553 559 666 603 609 615 621 628 665 671 677 6S3 689 726 733 739 715 751 788 794 800 807 813 860 856 862 868 874 911 917 924 930 936 973 979 985 991 997 034 040 046 052 058 095 101 107 114 120 166 163 169 175 ISl 217 224 230 236 242 278 285 291 297 303 339 345 362 358 364 400 406 412 418 425 461 467 473 479 485 522 528 634 540 546 582 588 594 600 600 643 649 655 661 667 703 709 715 721 727 763 769 775 781 788 824 830 836 842 848 884 890 896 902 908 944 950 956 962 968 *004 *010 *016 *022 *028 064 070 076 082 088 124 130 136 141 147 183 189 195 201 207 243 249 255 261 267 303 308 314 320 326 362 368 374 380 386 421 427 433 439 445 481 487 493 499 504 540 546 562 558 564 599 605 611 617 623 658 664 670 676 682 717 723 729 735 741 776 782 788 794 800 835 841 847 853 859 894 900 906 911 917 953 968 964 970 976 P. P. 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 P. P. 270 LOGARITHMS OP NUMBERS. Table 1. Num . 740 to 779. Log . 869 to 892 N L ( 2 3 4 S 6 7 8 9 P. P. 740 8ti 923 929 935 941 947 953 958 964 970 976 741 982 988 994 999 *005 *011 *017 *023 *029 *035 742 87 040 046 052 058 064 070 075 081 087 093 743 099 105 111 116 122 128 134 140 146 151 744 157 163 169 175 181 186 192 198 204 210 745 216 221 227 233 239 245 251 256 262 268 746 274 280 286 291 297 303 309 315 320 326 747 332 338 344 349 855 361 367 373 379 384 748 390 396 402 408 413 419 425 431 437 442 749 448 454 460 466 471 477 483 489 495 500 7S0 506 512 518 523 529 535 541 547 552 568 751 564 570 576 581 687 593 599 604 610 616 752 622 628 633 639 645 651 656 662 668 674 753 679 685 691 697 703 708 714 720 726 731 754 737 743 749 754 760 766 772 777 783 789 755 795 800 806 812 818 823 829 835 841 846 756 852 858 864 869 876 881 887 892 898 904 757 910 915 921 927 933 938 944 960 965 961 6 758 967 973 978 984 990 996 *001 *007 *013 *018 759 88 024 030 036 041 047 053 058 064 070 076 1 " '^ 2 u.o 1.2 760 081 087 093 098 104 110 116 121 127 133 3 4 1.8 2.4 761 138 144 150 156 161 167 173 178 184 190 5 3.0 762 195 201 207 213 218 224 230 235 241 247 6 7 8 3.6 4.2 4.8 763 252 268 264 270 275 281 287 292 298 304 764 309 315 321 326 332 338 343 349 356 360 9 5.4 765 366 372 377 383 389 395 400 406 412 417 766 423 429 434 440 446 451 457 463 468 474 767 480 485 491 497 502 508 513 519 625 530 768 536 542 547 553 559 564 570 576 581 687 769 593 598 604 610 615 621 627 632 638 643 770 649 655 660 666 672 677 683 689 694 700 771 705 711 717 722 728 734 739 745 750 756 772 762 767 773 779 784 790 795 801 807 812 773 818 824 829 835 840 846 852 857 863 868 774 874 880 885 891 897 902 908 913 919 925 775 930 936 941 947 953 958 964 969 975 981 776 986 992 997 *003 *009 *014 *020 *025 *031 *037 777 89 042 048 053 059 064 070 076 081 087 092 778 098 104 109 115 120 126 131 137 143 148 779 154 159 165 170 176 182 187 193 198 204 780 209 215 221 226 232 237 243 248 254 260 N L 1 2 3 4 5 6 7 8 9 P. P. Table 1. LOGAEITHMS OF NUMBERS. 271 Num 780 to 819. Log 892 to 913. N L 1 2 3 4 S 6 7 8 9 P. P. 780 89 209 215 221 226 232 237 243 248 254 260 781 265 271 276 282 287 293 298 304 310 315 782 321 326 332 337 343 348 354 360 365 371 783 376 382 887 393 398 404 409 415 421 426 784 432 437 443 448 454 459 465 470 476 481 785 487 492 498 504 509 515 620 526 531 537 786 542 548 553 559 564 570 575 581 586 592 787 597 603 609 614 620 625 631 636 642 647 788 653 658 664 669 675 680 686 691 697 702 789 708 713 719 724 730 735 741 746 752 757 790 763 768 774 779 785 790 796 801 807 812 791 818 823 829 834 840 845 851 856 862 867 792 873 878 883 889 894 900 905 911 916 922 793 927 933 938 944 949 955 960 966 971 977 794 982 988 993 998 *004 *009 *015 *020 *026 *031 795 90 037 042 048 053 059 064 069 075 080 086 796 091 097 102 108 113 119 124 129 135 140 797 146 151 157 162 168 173 179 184 189 195 5 798 200 206 211 217 222 227 233 238 244 249 1 2 0.5 1.0 799 255 260 266 271 276 282 287 293 298 304 800 309 314 320 325 331 336 342 347 352 358 3 4 1.5 2.0 801 363 369 374 380 385 390 396 401 407 412 5 2.5 802 417 423 428 434 439 445 450 455 461 466 6 7 8 3.0 3.5 4.0 803 472 477 482 488 493 499 504 509 515 520 804 526 531 536 542 547 553 558 563 569 574 9 4.5 805 580 585 590 596 601 607 612 617 623 628 806 634 639 644 650 655 660 666 671 677 682 807 687 693 698 703 709 714 720 725 730 736 808 741 747 752 757 763 768 773 779 784 789 809 795 800 806 811 816 822 827 832 838 843 810 849 854 859 865 870 875 881 886 891 897 811 902 907 913 918 924 929 934 940 945 950 812 956 961 966 972 977 982 988 993 998 *004 813 91 009 014 020 025 030 036 041 046 052 057 814 062 068 073 078 084 089 094 100 105 110 815 116 121 126 132 137 142 148 153 158 164 816 169 174 180 185 190 196 201 206 212 217 817 222 228 233 238 243 249 254 259 265 270 818 275 281 286 291 297 302 307 312 318 323 819 328 334 339 344 350 355 360 365 371 376 820 381 387 392 397 403 408 413 418 424 429 N L 1 2 3 4 S 6 7 8 9 P. P. 272 LOGARITHMS OF NUMBERS. Table 1. Num. 820 to 859. Log. 91-3 to 934. 8 P. P. 93 381 387 392 397 403 431 440 445 450 455 487 4'.I2 498 503 508 540 545 531 556 561 593 598 603 609 614 645 651 656 661 666 698 703 709 714 719 751 756 761 766 772 803 808 814 819 824 855 861 866 871 876 908 913 918 924 929 960 965 971 976 981 012 018 023 028 033 065 070 075 080 085 117 122 127 132 137 169 174 179 184 189 221 226 231 236 241 273 278 283 288 293 324 330 335 340 345 376 381 387 392 397 428 433 438 443 449 480 485 490 495 500 531 536 542 547 552 583 588 593 598 603 634 639 645 650 655 686 691 696 701 706 737 742 747 752 758 788 793 799 804 809 840 845 850 855 860 891 896 901 906 911 942 947 952 957 962 993 998 *003 *008 *013 044 049 054 059 064 095 100 105 110 115 146 151 156 161 166 197 202 207 212 217 247 252 258 263 268 298 303 308 313 318 349 354 359 364 369 399 404 409 414 420 450 455 460 465 470 408 413 418 424 429 461 466 471 477 482 514 519 524 529 535 566 572 577 582 587 619 624 630 635 640 672 677 6S2 687 693 724 730 735 740 745 777 782 787 793 798 829 834 840 845 850 882 887 892 897 903 934 939 944 950 955 986 991 997 *002 *007 038 044 049 054 059 091 096 101 106 111 143 148 153 158 163 195 200 205 210 215 247 252 257 262 267 298 304 309 314 319 350 355 361 366 371 402 407 412 418 423 454 459 464 469 474 505 511 516 521 526 557 562 567 572 578 609 614 619 624 629 660 665 670 675 681 711 716 722 727 732 763 768 773 778 783 814 819 824 829 834 865 870 875 881 886 916 921 927 932 937 967 973 978 983 988 *018 *024 *029 *034 *039 069 075 080 085 090 120 125 131 136 141 171 176 181 186 192 222 227 232 237 242 273 278 283 288 293 323 328 334 339 344 374 379 384 389 394 425 430 435 440 445 475 480 485 490 495 1 0.5 2 1.0 3 1.5 4 2.0 5 2.5 6 3.0 7 3.5 8 4.0 9 4.5 P. p. Table 1. LOGAEITHMS 0¥ NUMBERS. 273 Num. 860 to 899. Log. 934 to 954. N L 1 2 3 4 5 6 7 8 9 P. P. 860 93 450 455 460 465 470 475 480 485 490 495 861 500 505 510 515 520 526 531 536 541 546 862 551 556 561 566 571 576 581 586 591 696 863 601 606 611 616 621 626 631 636 641 646 864 651 656 661 666 671 676 682 687 692 697 865 702 707 712 717 722 727 732 737 742 747 866 752 757 762 767 772 777 782 787 792 797 867 802 807 812 817 822 827 832 837 842 847 868 852 857 862 867 872 877 882 887 892 897 869 902 907 912 917 922 927 932 937 942 947 870 952 957 962 967 972 977 982 987 992 997 871 94 002 007 012 017 022 027 032 037 042 047 872 052 057 062 067 072 077 082 086 091 096 873 101 106 111 116 121 126 131 136 141 146 874 151 156 161 166 171 176 181 186 191 196 875 201 206 211 216 221 226 231 236 240 245 876 250 255 260 265 270 275 280 285 290 295 877 300 305 310 315 320 325 330 335 340 345 S 878 349 354 359 364 369 374 379 384 389 394 1 2 0.5 1.0 879 399 404 409 414 419 424 429 433 438 443 880 448 453 458 463 468 473 478 483 488 493 3 4 1.5 2.0 881 498 503 507 512 517 522 527 532 537 542 5 2.5 882 547 552 557 562 567 671 576 581 586 591 6 7 8 3.0 3.5 4.0 883 596 601 606 611 616 621 626 630 635 640 884 645 650 655 660 665 670 675 680 685 689 9 4.6 885 694 699 704 709 714 719 724 729 734 738 886 743 748 753 758 763 768 773 778 783 787 887 792 797 802 807 812 817 822 827 832 836 888 841 846 851 856 861 866 871 876 880 885 889 890 895 900 905 910 915 919 924 929 934 890 939 944 949 954 959 963 968 973 978 983 891 988 993 998 *002 *007 *012 *017 *022 *027 *032 892 95 036 041 046 051 066 061 066 071 075 080 893 085 090 095 100 105 109 114 119 124 129 894 134 139 143 148 153 158 163 168 173 177 895 182 187 192 197 202 207 211 216 221 226 896 231 236 240 245 260 255 260 265 270 274 897 279 284 289' 294 299 303 308 313 318 323 898 328 332 337 342 347 352 357 361 366 371 899 376 381 386 390 395 400 405 410 415 419 900 424 429 434 439 444 448 463 458 463 468 N L 1 2 3 4 5 6 7 8 9 P. P. 19 274 LOGARITHMS OF NUMBERS. Table 1. Num 900 to 939. Log 954 to 973. N L i 2 3 4 5 6 7 8 9 P. P. 900 95 424 429 434 439 444 448 463 458 463 468 901 472 477 4S2 487 492 497 501 506 511 616 902 521 5'26 530 535 540 645 650 654 669 564 903' 569 571 578 583 588 693 698 602 607 612 904 617 622 626 631 636 641 646 650 655 660 905 665 670 674 679 684 689 694 698 703 708 906 713 718 722 727 732 737 742 746 751 756 907 761 766 770 775 780 785 789 794 799 801 908 809 813 818 823 828 832 837 842 847 852 909 856 861 866 871 875 880 886 890 896 899 910 904 909 914 918 923 928 933 938 942 947 911 952 957 961 966 971 976 980 985 990 995 912 999 *004 *009 *014 *019 *023 *028 *033 *038 *042 913 96 047 052 057 061 066 071 076 080 085 090 914 095 099 104 109 114 118 123 128 133 137 915 142 147 152 156 161 166 171 175 180 185 916 190 194 199 204 209 213 218 223 227 232 917 237 242 246 251 256 261 265 270 275 280 S 918 284 289 294 298 303 308 313 317 322 327 1 2 0.5 1.0 919 332 336 341 346 350- 355 360 365 369 374 920 379 384 388 393 398 402 407 412 417 421 3 4 1.5 2.0 921 426 431 43ft 440 445 450 454 459 464 468 5 2.5 922 473 478 483 487 492 497 601 506 511 615 6 7 8 3.0 3.5 4.0 923 520 525 530 534 539 544 548 553 658 562 924 667 572 677 681 586 591 695 600 606 609 9 4.5 925 614 619 624 628 633 638 612 647 652 656 926 661 666 670 675 680 686 689 694 699 703 927 708 713 717 722 727 731 736 741 745 750 928 765 759 764 769 774 778 783 788 792 797 929 802 806 811 816 820 825 830 834 839 844 930 848 853 858 862 867 872 876 881 886 890 931 896 900 904 909 914 918 923 928 932 937 932 942 946 951 956 960 965 970 974 979 981 933 988 993 997 *002 *007 *011 *016 *021 *026 *030 934 97 035 039 044 049 053 058 063 067 072 077 935 081 086 090 095 100 104 109 114 118 123 936 128 132 137 142 146 151 165 160 165 169 937 174 179 183 188 192 197 202 206 211 216 938 220 225 230 234 239 243 248 263 257 262 939 267 271 276 280 285 290 294 299 304 308 940 313 Sl7 322 327 331 336 340 345 850 354 N L 1 2 3 4 S 6 7 8 9 P. P. Table 1. LOGARITHMS OF NUMBERS. 275 Num. 940 to 979. Log. 973 to 991. P. P. 313 317 322 327 331 359 364 368 373 377 405 410 414 419 424 451 456 460 465 470 497 502 506 511 516 548 548 552 557 562 589 594 598 603 607 635 640 644 649 653 681 685 690 695 699 727 731 736 740 745 772 777 782 786 791 818 823 827 832 836 864 868 873 877 882 909 914 918 923 928 955 959 964 968 973 000 005 009 014 019 046 050 055 059 064 091 096 100 105 109 137 141 146 150 155 182 186 191 195 200 227 232 236 241 245 272 277 281 286 290 318 322 327 331 336 363 367 372 376 381 408 412 417 421 426 453 457 462 466 471 498 502 507 511 516 543 547 552 556 561 588 592 597 601 605 632 637 641 646 650 677 682 686 691 695 722 726 731 735 740 767 771 776 780 784 811 816 820 825 829 856 860 865 869 874 900 905 909 914 918 945 949 954 958 963 989 994 998 *003 *007 034 038 043 mj 052 078 083 087 092 096 123 127 131 136 140 336 340 345 350 354 382 387 391 396 400 428 433 437 442 447 474 479 483 488 493 520 525 529 534 539 566 571 575 580 585 612 617 621 626 630 658 663 667 672 676 704 708 713 717 722 749 754 759 763 768 795 800 804 809 813 841 845 850 855 859 886 891 896 900 905 932 937 941 946 950 978 982 987 991 996 023 028 032 037 041 068 073 078 082 087 114 118 123 127 132 159 164 168 173 177 204 209 214 218 223 250 254 259 263 268 295 299 304 308 313 340 345 349 354 358 385 390 394 399 403 430 435 439 444 448 475 480 484 489 493 520 525 529 534 538 565 570 574 579 583 610 614 619 623 628 655 659 664 668 673 700 704 709 713 717 744 749 753 758 762 789 793 798 ■ 802 807 834 838 843 847 851 878 883 887 892 896 923 927 932 936 941 967 972 976 981 985 *012 *016 *021 *025 *029 056 061 065 069 074 100 105 109 114 118 145 149 154 158 162 P. P. 276 LOGARITHMS OP NUMBERS. Table 1. N um. 980 to 1 000. Log 99 to 999. N L 1 2 3 4 5 6 7 8 9 P. P. 980 99 123 127 131 136 140 145 149 154 158 162 981 167 171 176 180 185 189 193 198 202 207 982 211 216 220 224 229 233 2:B8 242 247 251 983 255 260 261 269 273 277 282 286 291 295 984 300 304 308 313 317 322 326 330 335 339 985 344 348 352 357 361 366 370 374 379 383 986 388 392 396 401 405 410 414 419 423 427 987 432 436 411 445 449 454 458 463 467 471 988 476 480 481 489 493 498 602 506 611 516 989 520 624 528 533 537 642 646 550 555 569 4 990 564 568 572 577 581 685 590 594 599 603 1 0.4 991 992 607 651 612 656 616 660 621 664 625 669 629 673 634 677 638 682 642 686 647 691 2 3 4 0.8 1.2 1.6 993 695 699 704 708 712 717 721 726 730 734 6 2.0 994 739 743 747 752 756 760 765 769 774 778 6 7 2.4 2.8 995 782 787 791 795 800 804 808 813 817 822 8 9 3.2 3.6 995 826 830 835 839 843 848 862 856 861 865 997 870 874 878 883 887 891 896 900 904 909 998 913 917 922 926 930 935 939 944 948 952 999 957 961 965 970 974 978 983 987 991 996 1000 000 000 043 087 130 174 217 260 304 347 391 N L 1 2 3 4 S 6 7 8 9 P. P. Logarithms of Im portant Numbers. Number. Logarithm. TT == 3.141 .593 0.497 150 i^ = 4.188 790 0.622 089 i^ = 0..523 599 1.718 999 1 TT = 0.318 310 1.502 850 TT- = 9.869 604 0.994 300 1 TT- = 0.101 321 1.005 700 )n « 1.772 4M 0.248 675 \n = 0.564 190 T.751 425 r"; =- 1.464 592 0.165 717 fn = 0.682 784 1.834 283 »/6 A'1 = 1.240 701 0.093 667 Table 2. LOGAEITHMIC ANGULAR FUNCTIONS. 277 0° Logarithms. 179° M. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. M. Inf. Neg. Infinite. Inf. Neg. Infinite. 10.00000 10.00000 60 1 6.46373 13.53627 6.46373 13.53627 00000 00000 59 2 76476 23524 76476 23524 00000 00000 58 3 94085 05915 94085 05915 00000 00000 57 4 7.06579 12.93421 7.06579 12.93421 00000 00000 56 5 7.16270 12.83730 7.16270 12.83730 10.00000 10.00000 55 6 24188 75812 24188 75812 00000 OuOOO 54 7 30882 69118 30882 69118 00000 00000 53 8 36682 63318 36682 63.318 00000 00000 52 9 41797 58203 41797 58203 00000 00000 51 10 7.46373 12.53627 7.46373 12.53627 10.00000 10.00000 50 11 50512 4948S 50512 49488 00000 00000 49 12 54291 45709 54291 45709 ■ 00000 00000 48 13 57767 42233 57767 42233 00000 00000 47 14 60985 39015 60986 39014 00000 00000 46 15 7.63982 12.36018 7.63982 12.36018 10.00000 10.00000 45 16 66784 33216 66785 33215 00000 00000 44 17 69417 30583 69418 30582 00001 9.99999 43 18 71900 28100 71900 28100 00001 99999 42 19 74248 25752 74248 25752 00001 99999 41 20 7.76475 12.23525 7.76476 12.23524 10.00001 9.99999 40 21 78594 21406 78595 21405 00001 99999 39 22 80615 19385 80615 19385 00001 99999 38 23 82545 17455 82546 17454 00001 99999 37 24 81393 15607 84394 15606 00001 99999 36 25 7.86166 12.13834 7.86167 12.13833 10.00001 9.99999 35 26 87870 12130 87871 • 12129 00001. 99999 34 27 89509 10491 89510 10490 00001 99999 33 28 91088 08912 91089 08911 00001 99999 32 29 92612 07388 92613 07387 00002 99998 31 30 7.94084 12.05916 7.94086 12.05914 10.00002 9.99998 30 31 95508 04192 95510 04490 . 00002 99998 29 32 96887 03113 96889 03111 00002 99998 28 33 98223 01777 98225 01775 00002 99998 27 34 99520 00480 99522 00478 00002 99998 26 35 8.00779 11.99221 8.00781 11.99219 10.00002 9.99998 25 36 02002 97998 02004 97996 00002 99998 24 37 03192 96808 03194 96806 00003 99997 23 38 04350 95650 04353 95647 00003 99997 22 39 05478 94522 05481 94519 00003 99997 21 40 8.06578 11.93422 8.06581 11.93419 10.00003 9.99997 20 41 07650 92350 07653 92347 00003 99997 19 42 08696 91304 08700 91300 00003 99997 18 43 09718 90282 09722 90278 00003 99997 17 44 10717 89283 10720 89280 00004 99996 16 45 8.11693 11.88307 8.11696 11.88304 10.00004 9.99996 15 40 12647 87353 12651 87349 00004 99996 14 47 13581 86419 13585 86415 00004 99996 13 48 14495 85505 14500 85500 00004 99996 12 49 15.391 84609 15395 84605 00004 99996 11 50 8.16268 11.83732 8.16273 11.83727 10.00005 9.99995 10 61 17128 82872 17133 82867 00005 99995 9 52 17971 82029 17976 82024 00005 99995 8 53 18798 81202 18804 81196 00005 99995 7 54 19610 80390 19616 80384 00005 99995 6 55 8.20407 11.79593 8.20413 11,79587 10.00006 9.99994 5 66 21189 78811 21195 78805 00006 99994 4 57 21958 78042 21964 78036 00006 99994 3 58 22713 77287 22720 77280 00006 99994 2 59 23456 76544 23462 76538 00006 99994 1 60 24186 75814 24192 75808 00007 99993 M. Cosine. Secant. Cotangent Tangent. Cosecant. Sine. M. 90° 89° 278 LOGARITHMIC ANGULAR FUNCTIONS. TaUe 2. 1° Logarithms. 178= M. Sine. (!osi_'<"int. Tangent. Cotani^ent. Secant. 10.00007 Co.sine. 9.99993 M. 8.24186 11.76814 8.24192 11.75808 60 1 24903 75097 24910 75090 00007 99993 59 2 25609 74391 25616 74384 00007 99993 58 3 26304 73696 26312 73688 00007 99993 57 4 26988 73012 26996 73004 00008 99992 56 5 8.27661 11.72339 8.27669 11.72331 10.00008 9.99992 55 6 28324 71676 28332 71668 00008 99992 54 7 28977 71023 28986 71014 00008 99992 53 8 29621 70379 29629 70371 00008 99992 52 9 30265 69745 30263 69737 00009 99991 51 10 8.30879 11.69121 8.30888 11.69112 10.00009 9.99991 50 11 31495 68506 31605 68498 00009 99991 49 12 32103 67897 32112 67888 00010 99990 48 13 32702 67298 32711 67289 00010 '99990 47 14 33292 66708 33302 66698 00010 99990 46 15 8.33875 11.66125 8.33886 11.66114 10.00010 9.99990 45 16 34450 65550 34461 65839 00011 99989 44 17 36018 64982 . 36029 64971 00011 99989 43 18 35678 64422 35590 64410 00011 99989 42 19 36131 63869 36143 63857 00011 99989 41 20 8.36678 11.63322 8.36689 11.63311 10.00012 9.99988 40 21 37217 62783 37229 62771 00012 99988 39 22 37760 62250 37762 62238 00012 99988 38 23 38276 61724 38289 61711 00013 99987 37 24 38796 61204 38809 61191 00013 99987 36 25 8.39310 11.60690 8.39323 11.60677 10.00013 9.99987 35 26 39818 60182 39832 60168 00014 99986 84 27 40320 59680 40334 59666 00014 99986 33 28 40816 59184 40830 69170 00014 99986 32 29 41307 58693 41321 58679 00015 99985 31 30 8.41792 11.58208 8.41807 11.58193 10.00015 9.99985 30 31 42272 57728 . 42287 67713 00015 99985 29 32 42746 57254 42762 57238 00016 99984 28 33 43216 56784 43232 56768 00016 99984 27 34 43680 56320 43696 56304 00016 99984 26 35 8.44139 11.55861 8.4415G 11.55844 10.00017 9.99983 25 36 44594 65406 44611 56389 00017 99983 24 37 45044 54956 45061 54939 00017 99983 23 38 48489 54511 46507 54493 00018 99982 22 39 45930 54070 46948 54052 00018 99982 21 40 8.46366 11.53634 8.46385 11.83616 10.00018 9.99982 20 41 46799 53201 46817 63183 00019 99981 19 42 47226 52774 47245 52755 00019 99981 18 43 47660 52350 47669 52331 00019 99981 17 44 48069 51931 48089 51911 00020 99980 16 45 8.48486 11.61515 8.48505 11.51495 10.00020 9.99980 15 46 48896 51104 48917 51083 00021 99979 14 47 49304 60696 49326 80675 00021 99979 13 48 49708 60292 49729 80271 00021 99979 12 49 60108 49892 50130 49870 00022 99978 11 50 8.60604 11.49496 8.60627 11.49473 10.00022 9.99978 10 51 50897 49103 50920 49080 00023 99977 9 62 51287 48713 51310 48690 00023 99977 8 63 51673 i 48327 51696 48304 00023 99977 - 7 64 52055 47945 52079 47921' 00024 99976 6 55 8.62434 11.47566 8.52459 11.47541 10.00024 9.99976 5 56 52810 47190 6'2835 47165 00025 99975 4 67 53183 46817 63208 46792 00026 99976 3 68 53552 46448 63578 46422 00026 99974 2 59 53919 46081 53945 46056 00026 99974 1 60 54282 46718 64308 46692 00026 99974 M. Cosine. Secant. Cotangent. Tangent Cosecant. Sine. M. 91° 88=" Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 279 2= Logarithms. 77° M. Sine. Cosecant. Tangent. Cotangent. Recant. Cosine. M. 8.54282 11.45718 8.54308 11.46692 10.00026 9.99974 60 1 54642 45358 64669 45331 00027 99973 59 2 54999 45001 55027 44973 00027 99973 58 3 55354 44646 65382 44618 00028 99972 57 4 55705 44295 65734 44266 00028 99972 56 5 8.56054 11.43946 8.56083 11.43917 10.00029 9.99971 55 6 56400 43600 56429 43571 00029 99971 64 7 56743 43257 56773 43227 00030 99970 53 8 57084 42916 57114 42886 00030 99970 52 9 57421 42579 67462 42548 00031 99969 61 10 8.57757 11.42243 8.57788 11.42212 10.00031 9.99969 60 11 58089 41911 58121 41879 00032 99968 49 12 58419 41581 6S451 41549 00032 99968 48 13 58747 41253 68779 41221 00033 99967 47 14 59072 40928 59105 40895 00033 99967 46 15 8.59395 11.40605 8.59428 11.40572 10.00033 9.99967 45 16 59715 40285 59749 40251 00034 99966 44 17 60033 39967 60068 3993,2 00034 99966 43 18 60349 39651 60384 39616 00035 99965 42 19 60662 39338 60698 39302 00036 99964 41 20 8.60973 11.39027 8.61009 11.38991 10.00036 9.99964 40 21 61282 38718 61319 38681 00037 99963 39 22 61589 38411 61626 38374 00037 99963 38 23 61894 38106 61931 38069 00038 99962 37 24 62196 37804 62234 37766 00038 99962 36 25 8.62497 11.37503 8.62535 11.37466 10.00039 9.99961 35 26 62795 37205 62834 37166 00039 99961 34 27 63091 36909 63131 36869 00040 99960 33 28 63385 36615 63426 36574 00040 99960 32 29 63678 36322 63718 36282 00041 99959 31 30 8.63968 11.36032 8.64009 11.35991 10.00041 9.99959 30 31 64256 35744 64298 36702 00042 99958 29 32 64543 35457 64685 35416 00042 99958 28 33 64827 36173 64870 351.S0 00043 99957 27 34 65110 34890 65154 34846 00044 99956 26 35 8.66391 11.34609 8.65435 11.34565 10.00044 9.99966 25 36 65670 34330 65715 34286 00045 99955 24 37 65947 34053 66993 34007 00045 99955 23 38 66223 33777 66269 33731 00046 99954 22 39 66497 33603 66543 33467 00046 99954 21 40 8.66769 11.33231 8.66816 11.33184 10.00047 9.99953 20 41 67039 32961 67087 32913 00048 99952 19 42 67308 32692 67356 32644 00048 999.52 18 43 67575 32426 67624 32376 00049 99951 17 44 67841 32159 67890 32110 00049 99951 16 45 8.68104 11.31896 8.68154 11.31846 10.00050 9.99960 15 46 68367 31633 68417 31583 00051 99949 14 47 08627 31373 68678 31322 00051 99949 13 48 68886 31114 68938 31062 00062 99948 12 49 69144 30856 69196 30804 00062 99948 11 50 8.69400 11.30600 8.69453 11.30547 10.00053 9.99947 10 51 69654 30346 69708 30292 00054 99946 9 52 69907 30093 69962 30038 00054 99946 8 53 70159 29841 70214 29786 00065 99945 1 54 70409 29591 70466 29636 00056 99944 6 55 8.70658 11.29342 8.70714 11.29286 10.00056 9.99944 5 56 70905 29095 70962 29038 00057 99943 4 57 71151 28849 71208 28792 00068 99942 3 58 71395 28605 71453 28647 00058 99942 2 59 71638 28362 71697 28303 00059 99941 1 60 71880 28120 71940 28060 ooOeo 99940 M. Cosine. Secant. Cotangent, Tangent. Cosecant. Sine. M. 87° 280 LOGARTTHMTC ANGULAR FUNCTIONS. Table 2. 3° Logarithms. 176° M. Sine. Cosecant. Tangent. Cotangent Secant. Ctirtine. M. 8.71880 11.28120 8.71940 11,28060 10.00060 9.99940 60 1 72120 27880 72181 27819 00060 99940 59 2 72359 27641 72420 27580 00061 99939 58 3 72597 27403 72659 27341 00062 99938 57 4 72834 27166 72896 27104 00062 99938 66 5 8.73069 11.26931 8,73132 11,26868 10.00063 9.99937 55 6 73303 26697 73366 2B634 00064 99936 54 7 73635 26465 73600 26400 00064 99936 53 8 73767 26233 73832 26168 00066 99935 52 9 73997 26003 74063 25937 00066 99934 51 10 8.74226 11.25774 8,74292 11.25708 10.00066 9.99934 50 11 74454 25546 74521 25479 00067 99933 49 12 74680 2,5320 74748 25252 00068 99932 48 13 74906 26094 74974 26026 00068 99932 47 14 75130 24870 76199 24801 00069 99931 46 15 8.75353 11.24647 8.75423 11.24577 10,00070 9.99930 45 15 75576 24426 75645 24355 00071 99929 44 17 76796 24206 75867 24133 00071 99929 43 18 76015 23985 76087 23913 00072 99928 42 19 76234 23766 76306 23694 00073 99927 41 20 8.76451 11.23549 8.76626 11,23475 10.00074 9,99926 40 21 76667 23333 76742 23258 00074 99926 39 22 76883 23117 76958 23042 00075 99925 38 23 77097 22903 77173 22827 00076 99924 37 24 77310 22690 77387 22613 00077 99923 36 25 8.77522 11.22478 8.77600 11.22400 10,00077 9,99923 36 26 77733 22267 77811 22189 00078 99922 34 27 77943 22057 78022 21978 00079 99921 33 28 78152 21848 78232 21768 00080 99920 32 29 78360 21640 78441 215,59 00080 99920 31 30 8.78568 11,21432 8.7,8649 11,21351 10.00081 9,99919 30 31 78774 21226 78855 21145 00082 99918 29 32 78979 21021 79061 20939 00083 99917 28 33 79183 20817 79266 20734 00083 99917 27 34 79386 20614 79470 20530 00084 99916 26 35 8.79588 11.20412 8.79673 11,20327 10.00085 9,99915 25 36 79789 20211 79875 20125 00086 99914 24 37 79990 20010 80076 19924 00087 99913 23 38 80189 19811 80277 19723 00087 99913 22 39 80388 19612 80476 19524 00088 99912 21 40 8.80585 11.19415 8,80674 11.19326 10,00089 9.99911 20 41 80782 19218 80872 19128 00090 99910 19 42 80978 19022 81068 18932 00091 99909 18 43 81173 18827 81264 18736 00091 99909 17 44 81367 18633 81459 18641 00092 99908 16 45 8.81660 11.18440 8,81653 11,18347 10,00093 9.99907 16 46 81752 18248 81846 18154 00094 99906 14 47 81944 18056 82038 17962 00095 99905 13 48 82134 17866 82230 17770 00096 99904 12 49 82324 17676 82420 17580 00096 99904 11 60 8.82613 11.17487 8.82610 11.17390 10.00097 9.99903 10 61 82701 17299 82799 17201 00098 99902 9 52 82888 17112 82987 17013 00099 99901 8 53 83075 16925 83176 16825 00100 99900 7 54 83261 16739 83361 16639 00101 99899 6 56 8.83446 11.16554 8.83647 11,16453 10.00102 9.99898 5 56 83630 16370 83732 16268 00102 99898 4 57 83813 16187 83916 16084 00103 99897 3 58 83996 16004 84100 16900 00104 99896 2 59 84177 1.5823 84282 16718 00105 99895 1 60 84358 15642 84464 15636 00106 99894 mTI' Cosine. Secant. Cotangent. Tangent, Cosecant. Sine. M. 86" ■Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 281 4° Logarithms. 175° M. Sine. Cosecant. Tangent. Cotangent. Secant. Coeine. M. 8.84358 11.15642 8.84464 11.15536 10.00106 9.99894 60 1 84539 15461 84646 15S54 00107 99893 59 2 84718 15282 84826 15174 00108 99892 58 3 84897 15103 86006 14994 00109 99891 57 4 85075 14925 85185 14815 00109 99891 56 5 8.85262 11.14748 8.85363 11.14637 10.00110 9.99890 55 6 85429 14571 86540 14460 00111 99889 54 7 85605 14395 85717 14283 00112 99888 63 8 85780 .., 14220 85893 14107 00113 99887 52 9 85955 ■ ' 14045 86069 13931 00114 99886 51 10 8.86128 11.13872 8.86243 11.13757 10.00115 9.99885 50 H 86301 13699 86417 13583 00116 99884 49 12 86474 13526 86591 13409 00117 99883 48 13 86645 13355 86763 13237 00118 99882 47 14 86816 13184 86935 13065 00119 99881 46 15 8.86987 11.13013 8.87106 11.12894 10.00120 9.99880 45 16 87156 12844 87277 12723 00121 99879 44 17 87325 12675 87447 12553 00121 99879 43 18 87494 12506 87616 12384 00122 . 99878 42 19 87661 12339 87785 12215 00123 99877 41 20 8.87829 11.12171 8.87953 11.12047 10.00124 9.99876 40 21 87995 12005 88120 11880 00125 99875 39 22 88161 11839 88287 11713 00126 99874 33 23 88326 11674 88453 11547 00127 99873 37 24 88490 11510 88618 11382 00128 99872 36 25 8.88654 11.11346 8.88783 11.11217 10.00129 9.99871 35 26 88817 11183 88948 11052 00130 99870 34 27 88980 11020 89111 10889 00131 99869 33 28 89142 10858 89274 10726 00132 99868 32 29 89304 10696 89437 10563 00133 99867 31 30 8.89464 11.10536 8.89598 11.10402 10.00134 9.99866 30 31 89625 10375 89760 10240 00135 99865 29 32 89784 10216 89920 10080 00136 99864 28 33 89943 10057 90080 09920 00137 99863 27 34 90102 09898 90240 09760 00138 99862 26 35 8.90260 11.09740 8.90399 11.09601 10.00139 9.99861 25 36 90417 09583 90557 09443 00140 99860 24 37 90574 09426 90715 09285 00141 99859 23 38 90730 09270 90872 09128 00142 99858 22 39 90885 09115 91029 08971 00143 99857 21 40 8.91040 11.08960 8.91185 11.08815 10.00144 9.99856 20 41 91195 08805 91340 08660 00145 99855 19 42 91349 08651 91495 08505 00146 99854 18 43 91502 08498 91650 08350 00147 99853 17 44 91655 08345 91803 08197 00148 99852 16 45 8.91807 11.08193 8.91957 11.08043 10,00149 9.99861 15 46 91959 08041 92110 07890 00150 99850 14 47 98110 07890 92262 07738 00152 99848 13 48 92261 07739 92414 07586 00153 99847 12 49 92411 07589 92565 07435 00164 99846 11 50 8.92561 11.07439 8.92716 11.07284 10.00155 9.99845 10 •51 92710 07290 92866 07134 ■ 00156 99844 9 52 ' 92859 07141 93016 06984 00157 99843 8 53 93007 06993 93165 06835 00158 99842 7 5J 93154 06846 93313 06687 00159 99841 6 55 8.93301 11.06699 8.93462 11.06538 10.00160 9.99840 5 56 93448 06552 93609 06391 00161 99839 4 57 93594 06406 93756 06244 00162 99838 3 58 93740 06260 93903 06097 00163 99837 2 59 93885 06115 94049 05951 00164 99836 1 60 94030 05970 94195 05805 00166 99834 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 20 282 LOGARITHMIC ANGULAK FUNCTIONS. Table 3. 5° Logarithms. 74° M. Sine. Cosecant. Tangent. Cotangent, Scran t. Cosine. M. 8.94030 11.06970 8.94195 11.05805 10.00166 9.99834 60 1 94174 06826 94340 05660 00167 99833 69 2 94317 05683 94486 06516 00168 99832 58 3 944G1 05539 94630 06370 00169 99831 57 4 94603 05397 94773 05227 00170 99830 56 5 8.94746 11.06254 8.94917 11.06083 10.00171 9.99829 66 6 94887 06113 95060 04940 00172 99828 64 7 96029 04971 96202 04798 00173 99827 63 8 95170 04830 95344 04656 00175 99825 52 9 9.5310 04690 95486 04614 00176 99824 51 10 8.95450 11.04650 8.95627 11.04373 10.00177 9.99823 50 11 95589 04411 95767 04233 00178 99822 49 12 95728 04272 95908 04092 00179 99821 48 13 95867 04133 96047 03953 00180 99820 47 14 96005 03995 96187 03813 00181 99819 46 16 8.96143 11.03857 8.96326 11.03676 10.00183 9.99817 45 16 96280 03720 96464 03536 00184 99816 44 17 96417 03583 96602 03398 00185 99815 43 18 965.53 . 03447 96739 03261 00186 99814 42 19 96689 03311 9(i»77 03123 00187 99813 41 20 8.96825 11.03175 8.97013 11.02987 10.00188 9.99812 40 21 96960 03040 97150 02850 00190 99810 39 22 97095 02905 97285 02715 00191 99809 38 23 97229 02771 97421 02679 00192 99808 37 24 97363 02637 97556 02444 00193 99807 36 25 8.97496 11.0'2501 8.97691 11.02309 10.00194 9.99806 35 26 97629 02371 97825 02175 00196 99804 34 27 97762 02238 97959 02041 00197 99803 33 28 97894 02106 98092 01908 00198 99802 32 29 98026 01974 98225 01775 00199 9^801 31 30 8.98157 11.01843 8.98358 11.01642 10.00200 9.99800 30 31 98288 01712 98490 01510 00202 99798 29 32 98419 01681 98622 01378 00203 99797 28 33 98649 01451 98753 01247 00204 99796 27 34 98679 01321 98884 01116 00205 99795 26 35 8.98808 11.01192 8.99015 11.00985 10.00207 9.99793 25 36 98937 01063 99145 00855 00208 99792 24 37 99066 00934 99275 00726 00209 99791 23 38 99194 00806 99405 00596 00210 99790 22 39 99322 00678 99534 00466 00212 99788 21 40 8.99450 11.00550 8.99662 11.00338 10.00213 9.99787 20 41 99577 00423 99791 00209 00214 99786 19 42 99704 00296 99919 00081 00215 99785 18 43 99830 00170 9.00046 10.99954 00217 99783 17 44 99966 00044 00174 99826 00218 99782 16 45 9.00082 10.99918 9.00301 10.99699 10.00219 9.99781 15 46 00207 99793 00427 99573 00220 99780 14 47 00332 99668 00553 99447 00222 99778 13 48 00456 99544 00679 99321 00223 99777 12 49 00681 99419 00805 99196 00224 99776 11 50 9.00704 10.99296 9.00930 10.99070 10.00225 9.99775 10 51 00828 99172 01055 98945 00227 99773 9 62 00951 99049 01179 98821 00228 99772 8 53 01074 98926 01303 98697 00229 99771 7 54 01196 98804 01427 98673 00231 99769 6 55 9.01318 10.98682 9.01650 10.984.50 10.00232 9.99768 5 56 01440 98560 01673 98327 00233 99767 4 57 01661 98439 01796 98204 00236 99765 3 68 01682 98318 01918 98082 00236 99764 2 69 01803 98197 02040 97960 00237 99763 1 60 01923 98077 02162 97838 00239 99761 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 84° Table 2. LOGAEITHMIC ANGULAR FUNCTIONS. 283 6° Logarithms. 173° M. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. M. 9.01923 10.98077 9.02162 10.97838 10.00239 9.99761 60 1 02043 97957 02283 97717 00240 99760 59 2 02163 97837 02404 97596 00241 99759 58 3 02283 97717 02525 97475 00243 99757 57 4 02402 97598 02645 97355 00244 99756 56 5 9.02520 10.97480 9.02766 10.97234 10.00248 9.99755 55 6 02639 97361 02885 97115 00247 99753 54 7 02757 97243 03005 96995 00248 99752 53 8 02874 97126 03124 96876 00249 99751 52 9 02992 97008 03242 96758 00261 99749 51 10 9.03109 10.96891 9.03361 10.96639 10.00252 9.99748 50 U 03226 96774 03479 96621 00253 99747 49 12 03342 96668 03597 96403 00285 99745 48 13 03458 96542 03714 96286 00266 99744 47 14 03674 96426 03832 96168 00268 99742 46 15 9.03690 10.96310 9.03948 10.96052 10.00259 9.99741 45 16 03805 96198 04065 95935 00260 99740 44 17 03920 96080 04181 95819 00262 99738 43 18 04034 95966 04297 96703 00263 99737 42 19 04149 95861 04413 95587 00264 99736 41 20 9.04262 10.95738 9.04528 10.95472 10.00266 9.99734 40 21 04376 96624 04643 95357 00267 99733 39 22 04490 9.5510 04768 95242 00269 99731 38 23 04603 95397 04873 95127 00270 99730 37 24 04715 9.5286 04987 95013 00272 99728 36 25 9.04828 10.95172 9.05101 10.94899 10.00273 9.99727 35 26 04940 9.5060 05214 94786 00274 99726 31 27 05052 94948 05328 94672 0)270 99724 33 28 05164 94836 05441 94559 00277 99723 32 29 05275 94725 05653 . 94447 00279 99721 31 30 9.05386 10.94614 9.05666 10.94334 10.00280 9.99720 30 31 05497 94503 05778 91222 00282 99718 29 32 05607 94393 05890 94110 00283 99717 28 33 05717 94283 06002 93998 00284 99716 27 34 05827 94173 06113 93887 00286 99714 26 35 9.05937 10.94063 9.06224 10.93776 10.00287 9.99713 25 36 06046 93954 06335 93665 00289 99711 24 37 06155 93845 06445 93565 00290 99710 23 38 06264 93736 06566 93144 00292 99708 22 39 06372 93628 06666 93334 00293 99707 21 40 9.06481 10.93519 9.06775 10.93225 10.00295 9.99705 20 41 06589 93411 06885 93115 00296 99704 19 42 06696 93304 06994 93006 00298 99702 18 43 06804 93196 07103 92897 00299 99701 17 44 06911 93089 07211 92789 00301 99699 16 45 9.07018 10.92982 9.07320 10.92680 10.00302 9.99698 • 15 46 07124 92876 07428 92572 00304 99696 14 47 07231 92769 07536 92464 00305 99695 13 48 07337 92663 07643 92357 00307 99693 12 49 07442 92568 07751 92249 00308 99692 11 50 9.07548 10.92452 9.07858 10.92142 10.00310 9.99690 10 51 07653 92347 07964 92036 00311 99689 9 52 07768 92242 08071 91929 00313 99687 8 53 07863 92137 08177 91823 00314 99686 7 54 07968 92032 08283 91717 00316 99684 6 55 9.08072 10.91928 9.08389 10.91611 10.00317 9.99683 5 66 08176 91821 08495 91505 00319 99681 4 57 08280 91720 08600 91400 00320 99680 3 58 08383 91617 08705 91295 00322 99678 2 59 08486 91514 08810 91190 00323 99677 1 60 08689 91411 08914 91086 00325 99675 M. Cosine. Secant. C^5 tangent. Tangent. Cosecant. Sine. M. 83° 284 LOGARITHMIC ANGULAR FUNCTIONS. Tables. 7° Logarithms. 72° M. Sine. 9.0S5S9 Cosocant. Tangent. Cotangent. Recant. Cosine. M. 10.91411 9.08914 10.91086 10.00325 9.99675 60 1 08692 91308 09019 90981 00326 99674 69 2 08795 91205 09123 90877 00.328 99672 58 3 08897 91103 09227 90773 00330 99670 57 4 08999 91001 09330 90670 00331 99669 ,56 6 9.09101 10.90899 9.09434 10.90666 10.00333 9.99667 ,55 e 09202 90798 09537 90463 00334 99666 54 7 09304 90696 09610 90360 00336 99664 53 8 09105 90595 09742 90258 00337 99663 52 9 09506 90494 09845 90155 00339 99661 51 10 9.09606 10.90394 9.09947 10.900.53 10.00341 9.99659 ■50 H 09707 90293 10049 89961 00342 99658 49 12 09807 90193 10150 89850 00344 99656 48 13 09907 90093 10252 89748 00345 99655 47 1-1 10006 89994 10353 89647 00347 99663 46 15 9.10106 10.89894 9.10464 10.89,546 10.00349 9.99651 45 16 10205 89795 10565 89445 00350 99650 44 17 10304 89696 10656 89344 003.52 99648 43 18 10402 89598 10756 89244 00353 99647 42 19 10501 89499 108.56 89141 00355 99646 41 20 9.10.599 10.89401 9.10966 10.89044 10.00.357 9.99643 40 21 10697 89303 11066 88944 00368 99642 39 22 10795 89205 11155 88845 00360 99640 38 23 10893 89107 11254 88746 00362 99638 37 24 10990 89010 11353 88647 00363 99637 36 25 9A1087 10.88913 9.114.52 10.88548 10.00365 9.99635 35 26 11184 88816 11.561 88449 00367 99633 34 27 11281 88719 11649 88351 00368 99632 33 28 11377 88623 11747 88253 00370 99630 32 29 11474 88526 11845 88155 00371 99629 31 30 9.11570 10.88430 9.11943 10.88067 10.00373 9.99627 30 31 11666 88334 12040 87960 00375 99625 29 32 11761 88239 12138 87862 00376 99624 28 33 11857 88143 12235 87765 00378 99622 27 34 11952 88048 12332 87668 00380 99620 26 35 9.12047 10.87953 9.12428 10.87572 10.00382 9.99618 25 36 12142 87858 1'2525 87475 00383 99617 24 37 12236 87764 12621 87379 00386 99615 23 38 12331 87669 12717 87283 00387 99613 22 39 12425 87575 12813 87187 00388 99612 21 40 9.12619 10.87481 9.12909 10.87091 10.00390 9.99610 20 41 12612 .S73SM 13004 86996 00392 99608 19 42 12706 87294 13099 86901 00393 99607 18 43 12799 87201 13194 86806 00395 99605 17 44 12892 87108 13289 86711 00397 99603 16 45 9.12985 10.87015 9.13384 10.86616 10.00399 9.99601 15 46 13078 86922 13478 86522 00400 99600 14 47 13171 86829 l;5573 86127 00402 99598 13 48 13263 86737 13667 86333 00404 99596 12 49 13355 86645 13761 86239 00405 99595 11 60 9.13447 10.86553 9.13864 10.86146 10.00407 9.99593 10 51 13539 86461 13948 86052 00409 99.591 9 52 13630 86370 11041 8.5959 00411 99.589 8 53 13722 86278 141.34 85866 00412 99.588 7 54 13813 86187 14227 85773 00414 996S6 6 55 9.13904 10.86096 9.14320 10.a5680 10.00416 9.99684 5 56 13994 86006 14412 85588 00418 99,-82 4 57 14085 85915 14504 85496 00419 99581 3 58 14175 85825 14597 86403 00421 99579 2 69 14266 8.5734 14688 8.5312 00423 99577 1 60 14356 85644 14780 85220 00425 99575 sr. Cosine. Secant. Cotangent Tangent. Cosecant. Sine. M. Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 285 8° Logarithms. 171° M. Sine. Cosecant. Tangent. Cotangent. Secant. Co.siue. M. 9.14356 10.85644 9.14780 10.86220 10.00425 9.99675 60 1 14445 85655 14872 85128 00426 99574 59 2 14535 85465 14963 85037 00428 99572 68 3 14624 85376 16054 849-16 00430 99570 ,57 4 14714 85286 15145 84866 00432 99568 56 5 9.14803 10.85197 9.16236 10.84764 10.004.34 9.99566 55 6 14891 85109 16327 84673 00435 99565 64 7 14980 85020 15417 84583 00437 99563 53 8 15069 84931 16508 84492 00439 99561 52 9 15157 84813 1.5698 84402 00441 99559 51 10 9.15245 10.84765 9.15688 10.84312 10.00443 9.99657 50 11 15333 84667 15777 84223 00444 99556 49 12 15421 84579 15867 84133 00446 99554 48 13 15608 84492 15966 84044 00448 99552 47 14 15596 84404 16046 83954 004,50 99550 46 15 9.15683 10.84317 9.16136 10.83865 10.00462 9.99548 45 16 15770 84230 16224 83776 004.51 99546 44 17 15857 84143 16312 83688 00465 99545 43 18 15944 84056 16401 83599 00457 99513 42 19 16030 83970 16489 83611 00459 99641 41 20 9.16116 10.83884 9.16577 10.83423 10.00461 9.99539 40 21 16203 83797 16665 83336 00463 99537 39 22 16289 83711 16753 83217 00465 99536 38 23 16374 83626 16841 83169 00467 99633 37 24 16460 83640 16928 83072 00468 99632 36 25 9.16545 10.83466 9.17016 10.82984 10.00470 9.99530 35 26 16631 83369 17103 82897 C0472 99528 34 27 16716 83284 17190 82810 00474 99526 33 28 16801 83199 17277 82723 00476 99524 32 29 16886 83114 17.363 82637 00478 99522 31 30 9.16970 10.83030 9.17450 10.82550 10.00480 9.99520 30 31 17055 82945 17536 82464 00482 99518 29 32 17139 82861 17622 82378 00483 99617 28 33 17223 82777 17708 82292 00485 99515 27 3-t 17307 82693 17794 822C6 00487 99613 26 35 9.17391 10.82609 9.17880 10.82120 10.00489 9.99511 25 36 17474 82626 17965 82035 00491 99509 24 a^7 17558 82442 18051 81949 00493 99507 23 ^8 17641 82359 18136 81864 00495 99505 22 39 17724 82276 18221 81779 00497 99503 21 40 9.17807 10.82193 9.18306 10.81694 10.00499 9.99501 20 41 17890 82110 18391 81609 00501 99499 19 42 17973 82027 18475 81525 00503 99497 18 43 18065 81945 18560 81440 00605 99495 17 44 18137 81863 186U J1356 00506 99494 16 45 9.18220 10.81780 9.18728 10,81272 10.00.508 9.99492 15 46 18302 81698 18812 81188 00510 99490 14 47 18383 81617 18896 81104 00512 99488 13 48 18465 81535 18979 81021 00614 99486 12 49 18547 81453 19063 80937 00516 99484 11 50 9.18628 10.81372 9.19146 10.80851 10.00518 9.99482 10 51 18709 81291 19229 80771 00520 99480 9 52 18790 81210 19312 80688 00522 99478 8 53 18871 81129 19395 80605 00524 99476 7 54 18952 81048 19478 80622 00526 99474 6 55 9.19033 10.80967 9.19661 10.80439 10.00628 9.99472 5 56 19113 80887 19643 803,57 00630 99470 4 57 19193 80807 19725 80276 00532 99468 3 58 19273 80727 19807 80193 00534 99466 2 59 19353 80647 19889 80111 . 00.536 99464 1 60 19433 80567 19971 80029 00538 99462 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 81° ^86 LOGARITHMIC ANGULAR FUNCTIONS. Table 3. 9° Logarithms. 170° M. Sine. Cosecant. Tangent. Cotangent Secant. Cosine. M. 9.19433 10.80507 9.19971 10.80029 10.00538 9.99462 60 1 19513 804S7 20053 79947 00540 99460 59 2 19592 80408 20134 79866 00542 99468 58 3 19672 80328 20216 79784 00544 99456 67 i 19751 80249 20297 79703 00546 994.54 56 5 9.19830 10.80170 9.20378 10.79622 10.00548 9.99452 65 6 19909 80091 20459 79541 00550 99450 54 7 19988 80012 20540 79460 C0552 99448 63 8 20067 79933 20621 79379 00554 99446 52 9 20145 79855 20701 79299 00666 99444 51 10 9.20223 10.79777 9.20782 10.79218 10.00558 9.99442 50 11 20302 79698 20862 79138 00560 99440 49 12 20380 79620 20912 79058 00562 99438 48 13 20458 79542 21022 78978 00564 99436 47 14 20535 79465 21102 78898 00666 99434 46 15 9.20613 10.79,387 9.21182 10.78818 10.00568 9.99432 45 16 20691 79309 21261 78739 00571 99429 44 17 20768 79232 21341 78659 00573 99427 43 18 20845 79155 21420 78580 00575 99425 42 19 20922 79078 21499 78501 00677 99423 41 20 9.20999 10.79001 9.21578 10.78422 10.00579 9.99421 40 21 21076 78924 21657 78343 00581 99419 39 22 21153 78847 21736 78264 00583 99417 38 23 21229 78771 21814 78186 00585 99415 37 24 21306 78694 21893 78107 00587 99413 36 25 9.21382 10.78618 9.21971 10.78029 10.00589 9.99411 35 26 21458 78642 22049 77951 00691 99409 34 27 21534 78466 22127 77873 00593 99407 33 28 21610 78390 22205 77795 00596 99404 32 29 21685 78315 22283 77717 00598 99402 31 30 9.21761 10.78239 9.22361 10.77639 10.00600 9.99400 30 31 21836 78164 22438 77562 00602 99398 29 32 21912 78088 22516 77484 00604 99396 28 33 21987 78013 22593 77407 00600 99394 27 34 22062 77938 22670 77330 00608 99392 26 35 9.22137 10.77863 9.22747 10.77253 10.00610 9.99390 25 36 22211 77789 22824 77176 00612 99388 24 37 22286 77714 22901 77099 00615 99385 23 38 22361 77639 22977 77023 00617 99383 39 22435 77665 23054 76946 00619 99381 21 40 9.22509 10.77491 9.23130 10.76870 10.00621 9.99379 20 41 22583 77417 23206 76794 00623 99377 19 42 22667 77343 23283 76717 00625 99375 18 43 22731 77269 23359 76641 00628 99372 17 44 22805 77195 23435 76565 00630 99370 16 45 9.22878 10.77122 9.23510 10.76490 10.00632 9.99368 15 46 22952 77048 23586 76414 00634 99366 14 47 23025 76975 23661 76339 00636 99364 13 48 23098 76902 23737 76263 00638 99362 12 49 23171 76829 23812 76188 00641 99359 11 50 9.23244 10.76756 9.23887 10.76113 10.00643 9.99357 10 51 23317 76683 23902 76038 00645 99355 9 52 23390 76610 24037 75963 00647 99353 8 53 23462 76538 24112 75888 00649 99361 7 54 23535 76465 2J1S0 75814 00652 99348 6 55 9.23607 10.76393 9.24261 10.7.5739 10.00654 9.99346 5 56 23679 76321 24335 75665 00656 99344 4 57 23752 76248 24410 75590 00658 99342 3 58 23823 76177 •2US-1 75516 00660 99340 2 59 23895 76105 24 .WS 75442 00663 99337 1 CO 23967 76833 24632 75368 00665 99335 M. Cosine. Seciint. Cotangent. Tangent. Cosecant. Sine. M. pp. 80° Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 287 10° Logarithms. J 69° M, Sine. Cosecant. Tangent. 9.24632 Cotangent. 10.75368 Secant. Cosine. M. 9.23967 10.76033 10.0066.5 9.99336 60 1 24039 75961 24706 75294 00667 99333 69 2 24110 76890 24779 76221 00669 99331 58 3 21181 75819 24853 75147 00672 99328 57 4 24253 75747 24926 76074 00674 99326 56 5 9.24324 10.75676 9.26000 10.76000 10.00676 9.99324 56 6 24395 75605 25073 74927 00678 99322 54 7 24466 7.5534 25146 74854 00681 99319 53 8 21536 75464 25219 74781 00683 99317 52 9 24607 75393 25292 74708 00685 99315 51 10 9.24677 10.75323 9.26305 10.74635 10.00687 9.99313 50 11 24748 76252 25437 74563 00690 99310 49 12 24818 76182 25510 74490 00692 99308 48 13 21888 76112 25682 74418 00694 99306 47 14 24958 75042 25665 74345 00696 99304 46 15 9.25028 10.74972 9.25727 10.74273 10.00699 9.99301 46 16 25098 74902 25799 74201 00701 99299 44 17 25168 74832 26871 74129 00703 99297 43 18 25237 74763 26943 74057 00706 99294 42 19 25307 74693 26016 73985 00708 99292 41 20 9.25376 10.74624 9.26086 10.73914 10.00710 9.99290 40 21 25445 74566 26168 73842 00712 99288 39 22 25514 74486 26229 73771 00715 99285 38 23 25583 74417 26301 73699 00717 99283 37 24 25652 74348 26372 73628 00719 99281 36 25 9.25721 10.74279 9.26443 10.73557 10.00722 9.99278 35 26 25790 74210 26514 73486 00724 99276 34 27 26858 74142 26585 73115 00726 99274 33 28 25927 74073 26655 73345 00729 99271 32 29 25995 74006 26726 73274 00731 99269 31 30 9.26063 10.73937 9.26797 10.73203 10.00733 9.99267 30 31 26131 73869 26867 731.33 00736 99264 29 32 26199 73801 26937 73063 00738 99262 28 33 26267 73733 27008 72992 00740 99260 27 34 26335 73666 27078 72922 00743 99257 26 35 9.26403 10.73597 9.27148 10.72852 10.00745 9.99255 25 36 26470 73530 27218 72782 00748 99252 24 37 26638 73462 27288 72712 00760 99250 23 38 26605 73395 27357 72643 00762 99248 22 39 26672 73328 27427 72573 00755 99245 21 40 9.26739 10.73261 9.27496 10.72504 10.00757 9.99243 20 41 26806 73194 27566 72434 00759 99241 19 42 26873 73127 27635 72365 00762 99238 18 43 26940 73060 27704 72296 00764 99236 17 44 27007 72998 27773 72227 00767 99233 16 45 9.27073 10.72927 9.27842 10.72168 10.00769 9.99231 15 46 27140 72860 27911 72089 00771 99229 14 47 27206 72794 27980 72020 00774 99226 13 48 27273 72727 28049 71951 00776 99224 12 49 27339 72661 28117 71883 00779 99221 11 50 9.27405 10.72596 9.28186 10.71814 10.00781 9.99219 10 51 27471 72529 28254 71746 00783 99217 9 52 27537 72463 28323 71677 00786 99214 8 53 27602 72398 28391 71609 00788 99212 7 64 27668 72332 28459 71641 00791 99209 6 55 9.27734 10.72266 9.28527 10.71473 10.00793 9.99207 5 56 27799 72201 28595 71405 00796 99204 4 57 27864 72136 28662 71338 00798 99202 3 58 27930 72070 28730 71270 00800 99200 2 59 27995 72005 28798 71202 00803 99197 1 60 28060 71940 28865 71135 00806 99195 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 100° 79° 288 LOGARITHMIC ANGULAR FUNCTIONS. Table 3. 11° Logarithms. 68° M. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. M. 9.28060 10.71940 9.28865 10.711.36 10.00806 9.99195 60 1 28125 71875 28933 71067 00808 99192 69 2 28190 71810 29000 71000 00810 99190 58 3 28254 71746 29067 70933 00813 99187 57 4 28319 71681 29134 70866 00816 99185 56 5 9.28384 10.71616 9.29201 10.70799 10.00818 9.99182 55 6 28448 71652 29268 70732 00820 99180 64 7 28512 71488 29335 70665 00823 99177 53 8 28677 71423 29402 70598 00826 99175 52 9 28641 71359 29468 70532 00828 99172 51 10 9.28705 10.71295 9.29536 10.70405 10.00830 9.99170 60 n 28769 71231 29601 70399 00833 99167 49 12 28833 71167 29668 70332 00835 99165 48 13 28896 71104 29734 70266 00838 99162 47 14 28960 71040 29800 70200 00840 99160 46 15 9.29024 10.70976 9.29866 10.70134 10.00843 9.99157 45 16 29087 70913 29932 70068 00845 99156 44 17 29150 70850 29998 70002 00848 99152 43 18 29214 70786 30064 69936 00850 99160 42 19 • 29277 70723 30130 69870 00853 99147 41 20 9.29340 10.70660 9.30196 10.69805 10.00856 9.99145 40 21 29403 70597 30261 69739 00858 99142 39 22 29466 70534 30326 69674 00860 99140 38 23 29529 70471 30391 69609 00863 99137 37 24 29591 70409 30457 69643 00865 99135 36 25 9.29654 10.70346 9.30522 10.69478 10.00868 9.99132 35 26 29716 70284 30587 6941.3 00870 99130 34 27 29779 70221 30&52 69348 00873 99127 33 28 29841 70169 30717 69283 00876 99124 32 29 29903 70097 30782 69218 00878 99122 31 80 9.29966 10.70034 9.30846 10.69154 10.00881 9.99119 30 31 30028 69972 30911 69089 00883 99117 29 32 30090 69910 30976 69026 00886 99114 28 33 30151 69849 31040 68960 00888 99112 27 34 30213 69787 31104 68896 00891 99109 28 35 9.30275 10.6972.5 9.31168 10.68832 10.00894 9.99106 25 36 30336 69664 31233 68767 00896 99104 24 37 30398 69602 31297 68703 00899 99101 23 38 30459 69541 31361 68639 00901 99099 22 39 30621 69479 31425 68675 00904 99096 21 40 9.30582 10.69418 9.31489 10.68511 10.00907 9.99093 20 41 30643 69357 31552 68448 00909 99091 19 42 30704 69296 31616 CS3M4 00912 99088 18 43 30766 69235 31679 68321 00914 99086 17 44 30826 69174 31743 68267 00917 99083 16 45 9.30887 10.69113 9.31806 10.68194 10.00920 9.99080 15 46 30947 69053 31870 68130 00922 99078 14 47 31008 68992 31933 68067 00925 99076 13 48 31068 68932 31996 68004 00928 99072 12 49 31129 68871 32059 679J1 00930 99070 11 50 9.31189 10.68811 9.32122 10.67878 10.00933 9.99067 10 51 31250 68750 32185 67815 00936 99064 9 52 31310 68690 32248 67752 00938 99062 8 53 31370 68630 32311 67689 00941 99059 7 54 31430 68570 32373 67627 00944 99056 6 65 9.31490 10.68510 9.32436 10.67564 10.00946 9.99064 5 56 31649 68451 32498 67502 00949 99061 4 57 31609 68391 32561 67439 00952 99048 3 58 31669 68331 32623 67377 00954 99046 2 59 31728 68272 32685 67315 00957 99043 1 60 31788 68212 32747 67253 00960 99040 M. Coeiiie. Secant. Cotangent, Tangent. Cosecant. Sine. M. 10i° 78° Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 289 12° Logarithms. 167° M-. Sine. CofMicant. Tangent. Cotangent Secant. Cosine. M. 9.31788 10.68212 9.32747 10.67253 10.00960 9.99040 60 1 31847 68153 32810 67190 00962 99038 59 2 31907 68093 32872 67128 00965 99035 58 3 31966 68034 32933 67067 00968 99032 57 4 32025 67975 32995 67005 00970 99030 56 5 9.32084 10.67916 9.33057 10.66943 10.00973 9.99027 56 6 32143 67857 83119 66881 00976 99024 54 7 32202 67798 33180 66820 00978 99022 53 8 32261 67739 33242 66758 00981 99019 52 9 32319 67681 38303 66697 00984 99016 51 10 9.32378 10.67622 9.33365 10.66635 10.00987 9.99013 50 11 32437 67563 83426 66574 00989 99011 49 12 32495 67505 33487 66513 00992 99008 48 13 32553 67447 33548 66452 00995 99005 47 14 32612 67388 33609 66391 00998 99002 46 15 9.32670 10.67330 9.33670 10.66330 10.01000 9.99000 45 16 32728 67272 33731 66269 01003 98997 44 17 32786 67214 33792 66208 01006 98994 43 18 32844 67156 33853 66147 01009 98991 42 19 32902 67098 33913 66087 01011 98989 41 20 9.32960 10.67040 9.33974 10.66026 10.01014 9.98986 40 21 83018 66982 34034 65966 01017 98983 89 22 33075 66925 84095 65905 01020 98980 38 23 33133 66867 341,55 65845 01022 98978 37 24 33190 66810 34215 65785 01025 98975 86 25 9.33248 10.667.52 9.34276 10.65724 10.01028 9.98972 35 26 33305 66695 34336 65664 01031 98969 34 27 33362 66638 34.396 65604 01083 98967 38 28 33420 66580 34456 65544 01036 98964 32 29 33477 66523 34516 65484 01039 98961 31 3D 9.33534 10.66466 9.34576 10.65424 10.01042 9.98958 30 31 33.591 66409 34635 65365 01045 98955 29 32 33647 66353 34695 65305 01047 98953 28 33 33704 66296 34755 6.5245 01060 98950 27 34 33761 66239 34814 65186 01063 98947 26 35 9.33818 10.66182 9.34874 10.65126 10.01056 9.98944 25 36 33874 66126 34933 65067 01059 98941 24 37 339.31 66069 34992 66008 01062 98938 23 38 33987 66013 35051 64949 01064 98936 22 39 34043 65957 35111 64889 01067 98933 21 40 9.34100 10.65900 9.35170 10.64830 10.01070 9.98980 20 41 34156 65844 35229 64771 01073 98927 19 42 34212 65788 35288 64712 01076 98924 18 43 34268 65732 35347 64663 01079 98921 17 44 34324 6.5876 35405 64696 01081 98919 16 45 9.34380 10.65620 9.35464 10.64536 10.01084 9.98916 15 46 34436 65564. 35523 64477 01087 98913 14 47 34491 65509 35581 64419 0109U 98910 13 48 34547 65453 ,35640 64360 01093 98907 12 49 34602 65398 3.5698 64302 01096 98904 11 50 9.34658 10.65342 9.35757 10.64243 10.01099 9.98901 10 51 34713 65287 35815 64185 01102 98898 9 52 34769 65231 35873 64127 01104 98896 8 53 34824 66176 35931 64069 01107 98893 7 54 34879 65121 35989 64011 OHIO 98890 6 55 9.34934 10.65066 9.36047 10.63953 10.01113 9.98887 5 56 34989 65011 36105 63895 01116 98884 4 57 35044 64956 36163 63837 01119 98881 3 58 35099 64901 36221 63779 01122 98878 2 59 351.54 64846 36279 63721 01125 98875 1 60 35209 64791 36336 63664 01128 98872 M. Cosine. Secant. Cotangent Tangent. Cosecant. Sine. M. 102° 77° 290 LOGARITHMIC ANGULAR FUNCTIONS. Table 3. 13° Logarithms. 166° M, Sine. Cosecant. Tangent. Cotangent, Secant. Cosine. M, 9.35209 10.64791 9.36336 10.63664 10.01128 9.98872 60 1 35263 64737 36394 63606 01131 98869 69 2 35318 64682 36452 63548 01133 98867 68 3 35373 64627 36509 63491 01136 98864 57 4 36427 64573 36566 63434 01139 98861 66 5 9.36481 10.64519 9.36624 10.63376 10.01142 9.98858 55 6 36536 64464 36681 63319 01115 98855 54 7 36590 61410 36738 63262 01148 98852 63 8 35644 04356 36795 63205 01161 98849 52 9 35698 61302 36852 63148 01154 98846 51 10 9.35752 10.64248 9.36909 10.63091 10,01157 9.98843 50 11 35806 64194 36966 63034 01160 98840 49 12 35860 64140 37023 62977 01163 98837 48 13 35914 64086 .37080 62920 01166 98834 47 14 35968 64032 37137 62863 01169 98831 46 15 9.36022 10.63978 9.37193 10.62807 10,01172 9.98828 45 16 36075 03925 37250 62750 01175 98825 44 17 30129 63871 37306 62694 01178 98822 43 18 36182 63818 37363 62637 01181 98819 42 19 36236 63764 37419 62581 01184 98816 41 20 9.36289 10.63711 9.37476 10.6'2521 10.01187 9.98813 40 21 36342 63658 37532 62468 01190 '98810 39 22 36395 63606 37688 62412 01193 98807 38 23 36449 63551 37644 62356 01196 98804 37 24 36502 63498 37700 62300 01199 98801 36 26 9.36555 10.63445 9.37766 10.62214 10.01202 9.98798 36 26 36608 63392 37812 62188 01205 98795 34 27 36660 63340 37868 62132 01208 98792 33 28 36713 63287 37924 62076 01211 98789 32 29 36766 63234 .37980 62020 01214 98786 31 30 9.36819 10.63181 9.38035 10.61965 10.01217 9.98783 3D 81 36871 63129 38091 61909 01220 98780 29 32 36924 63076 38147 61853 01223 98777 28 33 36976 63024 38202 61798 01226 98774 27 34 37028 62972 38257 61743 01229 98771 26 35 9.37081 10.62919 9.38313 10.61687 10.01232 9.98768 25 36 37133 62867 38368 01632 01235 98765 24 37 37185 62815 38423 61677 01238 98762 23 38 37237 62763 38479 61521 01241 98759 22 39 37289 62711 38534 61466 01244 98756 21 40 9.37341 10.62659 9.38589 10.61411 10.01247 ?, 98753 ■20 41 37393 62607 38644 61356 01250 98760 19 42 37445 6-2555 38699 61301 01254 98746 18 43 37497 6'2603 38754 61246 01257 98743 17 44 37549 6'2451 38808 61192 01260 98740 16 45 9,37600 10.62400 9.38863 10.61137 10.01263 9,98737 15 46 37652 62348 38918 61082 01266 98734 14 47 37703 62297 38972 61028 01269 98731 13 48 37755 62245 39027 60973 01272 98728 12 49 37806 62194 39082 60918 01275 98725 11 60 9.37858 10.62142 9.39136 10,60864 10.01278 9,98722 10 51 37909 62091 39190 60810 01281 98719 9 62 37960 62040 39245 607.55 01285 98715 8 63 38011 61989 39299 60701 01288 98712 7 54 38062 61938 39353 60647 01291 98709 6 55 9.38113 10.61887 9.39407 10.60593 10.01294 9,98706 5 56 38164 61836 39461 60.539 01297 98703 4 57 38215 61785 39516 60485 01300 98700 3 58 38266 61734 39569 60431 01303 98697 2 69 38317 61683 39623 60377 01306 98694 1 60 38368 61632 39677 60323 01310 98690 mT CoBiue. Secant. Cotangent. Tangent. Cosecant. Sine, M. 103° 76° Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 291 J4° Logarithms. 165° M. Sine. Cosecant. Tangent. Cotangent, Secant. Cosine. M. 9.38368 10.61632 9.39677 10.60323 10.01310 9.98690 60 1 38418 61582 39731 60269 01313 98687 59 2 38469 01531 39785 60215 01316 98684 58 3 38519 61481 39838 60162 01319 ■98681 57 4 38570 61430 39892 60108 01322 98678 56 5 9.38620 10.61380 9.39945 10.60055 10.01325 9.98675 55 6 38670 61330 39999 60001 01329 98671 54 7 38721 61279 40052 59948 01332 98668 53 8 38771 61229 40106 59894 01335 98665 52 9 38821 61179 40159 59841 01338 98662 51 10 9.;^8871 10.61129 9.40212 10.59788 10.01341 9.98659 50 11 38921 61079 40266 59734 01344 98656 49 12 38971 61029 40319 59681 01348 98652 48 13 39021 60979 40372 59628 01351 98649 47 H 39071 60929 40425 59575 01354 98646 46 15 9.39121 10.60879 9.40478 10.59522 10.01357 9.98643 45 16 39170 60830 40531 59469 01360 98640 44 17 39220 60780 40584 59416 01864 98636 43 18 39270 60730 40636 59364 01367 986.33 42 19 39319 60681 40689 59311 01370 98630 41 20 9.39369 10.60631 9.40742 10.59258 10.01373 9.98627 40 21 39118 60582 40795 59205 01377 98623 39 22 39467 60533 40847 59163 01380 98620 38 23 39517 60483 40900 59100 01383 98617 37 24 39566 60434 40952 59048 01386 98614 36 25 9.39615 10.60385 9.41005 10.58995 10.01390 9.98610 35 26 39664 60336 41057 58943 01393 98607 34 27 39713 60287 41109 58891 01396 98604 33 28 39762 60238 41161 58839 01399 98601 32 29 39811 60189 41214 58786 01403 98597 31 30 9.39860 10.60140 9.41266 10.58734 10.01406 9.98594 30 31 39909 60091 41318 58682 01409 98591 29 32 39958 60042 41370 58630 01412 98588 28 33 40006 59994 41422 58578 01416 98584 27 34 40055 59945 41474 58526 01419 98581 26 35 9.40103 10.59897 9.41526 10.58474 10.01422 9.98578 25 36 40152 59848 41578 58422 01426 98574 24 37 40200 59800 41629 58371 01429 98571 23 38 40249 59751 41681 58319 01432 98568 22 39 40297 59703 41733 58267 01435 98565 21 40 9.40346 10.59654 9.41784 10.58216 10.01439 9.98561 20 41 40394 59606 41836 58164 01442 98558 19 42 40442 59658 41887 58113 01445 98555 18 43 40490 59510 41939 58061 01449 98551 17 44 40538 59462 41990 58010 01452 98548 16 45 9.40586 10.59414 9.42041 10.57959 10.01455 9.98545 15 46 40634 59366 42093 57907 01459 98541 14 47 40682 59318 42144 57856 01462 98538 13 48 40730 59270 42195 57805* 01465 98536 12 49 40778 59222 42246 57754 01469 98531 11 50 9.40825 10.59175 9.42297 10.57703 10.01472 9.98528 10 51 40873 59127 42348 57652 01475 98525 9 52 40921 59079 42399 57601 01479 98521 ■• 8 53 40968 59032 42450 57550 01482 98518 7 54 41016 58984 42501 57499 01485 98515 6 55 9.41063 10.58937 9.42552 10.57448 10.01489 9.98511 5 56 41111 58889 42603 67397 01492 98508 4 57 41158 58842 42653 57347 01495 98505 3 58 41205 58795 42704 57296 01499 98501 2 59 41252 58748 42755 57245 01502 98498 1 60 41300 58700 42805 57195 01506 98494 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 104° 7S° 292 LOGARITHMIC ANGULAR FUNCTIONS. Table 2. IS': Logarithms. J 64° M. Sine. Cosecant. Tangent. Cotangent 1 Secant. Cosine. M. 9.41300 10.58700 9.42805 10.57195 10.01506 9.98494 60 1 41347 58653 42855 57144 01509 98491 59 2 41394 58606 42906 57094 01512 98488 58 3 41441 58559 42957 67043 01516 98484 57 4 41488 58512 43007 56993 01619 98481 56 6 9.41535 10.58465 9.43057 10.56943 10.01523 9.98477 55 6 41,=.82 .584 1,S 43108 56892 01526 98474 .54 7 41(iiS .58372 43158 56842 01529 98471 63 8 JlC.To 58325 43208 56792 01533 98467 52 9 41722 58278 43268 56742 01536 98464 51 10 9.417I-.S 10.58232 9.43308 10.56692 10.01540 9.98460 60 11 41815 .58185 43.3.58 56642 01543 98457 49 12 41861 58139 43408 66592 01547 98463 48 13 41908 .58092 43458 56542 01650 98450 47 14 419.54 58046 43508 56492 01553 98447 46 15 9.42001 10.57999 9.43.558 10.56442 10.01557 9.98443 45 16 42047 57953 43607 56393 01560 98440 44 17 42093 57907 43657 56343 01564 98436 43 18 42140 57860 43707 56293 01667 98433 42 19 42186 57814 43756 66244 01571 98429 41 20 9.42232 10..57768 9.43806 10.56194 10.01674 9.98426 40 21 42278 57722 43855 66145 01578 98422 39 22 42324 57676 43906 56095 01581 98419 38 23 42370 57630 43964 56046 01586 98415 37 24 42416 57584 44004 55996 01688 98412 36 25 9.42461 10.57539 9.44063 '10.56947 10.01691 9.98409 35 26 42507 57493 44102 ,55898 01.595 98405 34 27 42553 57447 44151 55849 01598 98402 33 28 42599 67401 44201 65799 01602 98398 32 29 42644 57356 44250 55760 01605 98396 31 30 9.42690 10.57310 9.44299 10..55701 10.01609 9.98391 30 31 42735 57265 44348 55652 01612 98388 29 32 42781 57219 44397 66603 01616 98384 28 33 42826 57174 44446 .55654 01619 98381 27 34 42872 57128 44495 56505 01623 98377 26 35 9.42917 10.57083 9.44.544 10.56456 10.01627 9.98373 25 3G 42962 57038 44592 55408 01630 98370 24 37 43008 56992 44641 66359 01634 98366 23 38 43053 56947 44690 55310 016.37 98363 22 39 43098 56902 44738 6.5262 01641 98369 21 40 9.43143 10.56857 9.44787 10.55213 10.01644 9.98356 20 41 43188 56812 44836 55164 01648 98352 19 42 43233 56767 44884 55116 01661 98349 18 43 43278 56722 44933 55067 01655 98345 17 44 43323 56677 44981 55019 01658 98342 16 45 9.43367 10.56033 9.45029 10.54971 10.01662 9.98338 15 46 43412 56.588 45078 54922 01666 98334 14 47 43457 66543 45126 54874 01669 98831 13 48 43502 66498 46174 54826 01673 98327 12 49 43546 564,54 45222 54778 01676 98324 11 60 9.43591 10.56409 9.46271 10.54729 10.01680 9.98320 10 51 43635 66365 45319 54681 01683 98317 9 52 43680 56320 45367 54633 01687 98313 8 53 43724 56276 45415 54.585 01691 98309 7 64 43769 56231 45463 54537 01694 98306 6 55 9.43813 10.56187 9.45511 10.,54489 10.01698 9.98302 5 56 43857 56143 45559 64441 01701 98299 4 57 43901 56099 45606 54394 01705 98295 3 58 43946 56054 45654 54346 01709 98291 2 59 43990 56010 4.5702 54298 01712 98288 1 60 44034 55966 45750 54250 01716 98284 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 105° 74° Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 293 16° Logarithms. 163° M. Sine. Cosecant. Tangent. Cotangent.! Secant. Ci>.''Ule. M. 9.44034 10.56966 9.45750 10.54250 10.01716 9.98284 60 1 44078 55922 45797 54203 01719 982S1 59 2 44122 55878 45845 541.55 01723 98277 68 3 44166 55834 45892 54108 01727 98273 57 4 44210 55790 46940 54060 01730 98270 56 5 9.44253 10.55747 9.45987 10.54013 10.01734 9.98266 55 6 44297 55703 46035 53965 01738 98262 54 7 44341 55659 46082 53918 01741 98269 53 8 44385 55615 46130 53870 01745 982.65 52 9 44428 55572 46177 53823 0]74'9 98251 51 10 9.44472 10.55528 9.46224 10.53776 10.01752 9.98248 50 11 44516 55484 46271 53729 01756 98244 49 12 44559 55441 46319 53681 01760 98240 48 13 44602 55398 46366 63634 01763 98237 47 14 44646 55354 46413 5.3587 01767 98233 46 15 9.44689 10.55311 9.46460 10.5.3540 10.01771 9.98229 45 16 44733 55267 46507 63493 01774 98226 44 17 44776 55224 46554 53446 01778 98222 43 18 44819 55181 46601 53399 01782 98218 42 19 44862 55138 46648 53352 01785 98215 41 20 9.44905 10.55095 9.46694 10.53306 10.01789 9.98211 40 21 44948 55052 46741 53259 01793 98207 39 2^^ 44992 55008 46788 63212 01796 98204 38 23 45035 54965 46835 63165 01800 98200 37 24 45077 54923 46881 53119 01804 98196 36 25 9.45120 10.54880 9.46928 10.53072 10.01808 9.98192 35 26 45163 54837 46975 53025 01811 98189 34 27 45206 54794 47021 52979 01815 98185 33 28 45249 54751 47068 62932 01819 98181 32 29 45292 54708 47114 52886 01823 98177 31 30 9.45334 10.54666 9.47160 10.52840 10.01826 9.98174 30 31 45377 54623 47207 52793 01830 98170 29 32 45419 54581 47263 52747 01834 98166 28 33 45462 54538 47299 52701 01838 98162 27 34 45504 54496 47346 52654 01841 98159 26 35 9.45547 10.54453 9.47392 10.52608 10.01845 9.98165 25 36 45589 54411 47438 52562 01849 98151 24 37 45632 54368 47484 52516 01853 98147 23 38 45674 54326 47530 52470 01856 98144 22 39 45716 54284 47576 62424 01860 98140 21 40 9.45758 10.54242 9.47622 10.52378 10.01864 9.98136 20 41 45801 54199 47668 52332 01868 98132 19 42 45843 54157 47714 52286 01871 98129 18 43 45885 64115 47760 62240 01875 98125 17 44 45927 54073 47806 52194 01879 98121 16 45 9.45969 10.54031 9.47852 10:52148 10.01883 9.98117 15 46 46011 53989 47897 52103 01887 98113 U 47 46053 53947 47943 52057 01890 98110 13 48 46095 53905 47989 52011 01894 98106 12 49 46136 53864 48035 ■ 51965 03898 98102 11 50 9.46178 10.53822 9.48080 10.51920 10.01902 9.98098 10 51 46220 53780 48126 61874 01906 98094 9 52 46262 53738 48171 51829 01910 98090 8 53 46303 53697 48217 51783 01913 98087 7 54 46345 53655 48262 61738 01917 98083 6 55 9.46386 10.53614 9.48307 10.51693 10.01921 9.98079 5 56 46428 53672 48353 51647 01925 98075 4 57 46469 53531 48398 51602 01929 98071 3 58 46511 53489 48443 51557 01933 98067 2 59 46552 53448 48489 51511 01937 98063 1 60 46594 53406 48534 51466 01940 98060 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 106° 73° 294 LOGARITHMIC ANGULAR FUNCTIONS. Table 2. Logarithms. 162° M. Sine. Coaecant.. Tangent. Cotangent. Secant. Cosine. M. 9.46594 10.53406 9.48534 10.51466 10.01940 9.98060 60 1 46635 53365 48579 51421 01944 98056 59 2 46676 53.324 48624 51376 01948 98052 68 3 46717 53283 48669 61331 01952 98048 57 4 46758 53242 48714 61286 01956 98044 56 5 9.41K00 10.53200 9.48759 10.51241 10.01960 9.98040 55 6 iimi 53159 48804 51196 01964 98036 54 7 46,H82 53118 48849 51151 01968 98032 63 8 46923 53077 48894 51106 01971 98029 52 9 46964 53036 48939 51061 01975 98025 51 10 9.47005 10.52995 9.48984 10.61016 10.01979 9.98021 50 11 47045 52955 49029 60971 01983 98017 49 12 47086 52914 49073 50927 01987 98013 48 13 47127 52873 49118 50882 01991 98009 47 14 47168 5'2832 49163 60,837 01996 98005 46 15 9.47209 10.52791 9.49207 10.60793 10.01999 9.98001 45 16 47249 52751 49252 50748 02003 97997 44 17 47290 52710 49296 50704 02007 97993 43 18 47330 52670 49341 50659 02011 97989 42 19 47371 52629 49385 60615 02014 97986 41 20 9.47411 10.52589 9.49430 10.50570 10.02018 9.97982 40 21 47452 52548 49474 50626 02022 97978 39 22 47492 52508 49.519 60481 02026 97974 38 23 47533 52467 49563 50437 02030 97970 37 24 47573 52427 49607 50393 02034 97966 36 25 9.47613 10.52387 9.49652 10..50348 10.02038 9.97962 35 26 47654 62346 49696 60304 02042 97958 34 27 47694 52306 49740 50260 02046 97964 33 28 47734 52266 49784 50216 02050 97960 32 29 47774 52226 49828 50172 02054 97946 31 30 9.47814 10.52186 9.49872 10..50128 10.02058 9.97942 30 31 47854 52146 49916 50084 02062 97938 29 32 47894 52106 49960 50040 02066 97934 28 33 47934 52066 50004 49996 02070 97930 27 34 47974 52026 50048 49952 02074 97926 26 35 9.48014 10.51986 9.50092 10.49908 10.02078 9.97922 25 36 48054 51946 50136 49864 02082 97918 24 37 48094 51906 50180 49820 02086 97914 23 38 48133 51867 50223 49777 02090 97910 22 39 48173 51827 50267 49733 02094 97906 21 40 9.48213 10.51787 9.50311 10.49689 10.02098 9.97902 20 41 48252 51748 50355 49645 02102 97898 19 42 48292 51708 50398 49602 02106 97894 18 43 48332 51668 50442 49558 02110 97890 17 44 48371 51629 50485 49515 02114 97886 16 45 9.48411 10.51589 9.50529 10.49471 10.02118 9.97882 15 46 48450 51550 ,50572 49428 02122 97878 14 47 48490 51510 50616 49384 02126 97874 13 48 48529 51471 60659 49341 02130 97870 12 49 48568 51432 50703 49297 02134 97866 11 50 9.48607 10.51393 9.60746 10.49254 10.02139 9.97861 10 51 48647 51353 60789 49211 02148 97857 9 52 48686 51314 60833 49167 02147 97853 8 53 48725 51275 50876 49124 02151 97849 7 54 48764 51236 50919 49081 02155 97845 6 55 9.48803 10.51197 9.50962 10.49038 10.021,59 9.97841 5 56 48842 51158 51006 48995 02163 97837 4 57 48881 51119 ,51048 48962 02167 97833 3 58 48920 51080 51092 48908 02171 97829 2 59 48959 51041 51136 48865 02175 97826 1 60 48938 51002 51178 48822 02179 97821 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 107° Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 295 18° Logarithms. 161° M. Sine. CoBGcant. Tangent. Cotangent. Secant. Cosine. M. 9.48998 10.51002 9.51178 10.48822 10.02179 9.97821 60 1 49037 50963 51221 48779 02183 97817 59 2 49076 50924 51264 48736 02188 97812 68 3 49115 60885 51306 48694 02192 97808 57 4 49153 50847 51349 48651 02196 • 97804 56 5 9.49192 10.50808 9.51392 10.48608 10.02200 9.97800 56 6 49231 50769 51435 48565 02204 97796 54 7 49269 50731 51478 48522 02208 97792 53 8 49308 50692 51620 48480 02212 97788 52 9 49347 50653 51563 48437 02216 97784 51 10 9.49385 10..50615 9.51606 10.48394 10.02221 9.97779 50 11 49424 50576 51648 48352 02226 97775 49 12 49462 50538 61691 48309 02229 97771 48 13 49500 50500 51734 48266 02233 97767 47 14 49539 50461 61776 48224 02237 97763 46 15 9.49577 10.50423 9.51819 10.48181 10.02241 9.97769 45 16 49615 50385 51861 48139 02246 97754 44 17 49654 50346 61903 48097 02250 97750 43 18 49692 60308 61946 48064 02254 97746 42 19 49730 50270 51988 48012 02258 97742 41 20 9.49768 10.60232 9.62031 10.47969 10.02262 9.97738 40 21 49806 50194 62073 47927 02266 97734 39 22 49844 50156 52115 47886 02271 97729 38 23 49882 50118 52157 47843 02275 97725 37 24 49920 50080 62200 47800 02279 97721 36 25 9.49958 10.50042 9.52242 10.47758 10.02283 9.97717 35 26 49996 50004 52284 47716 02287 97713 34 27 50034 49966 52326 47674 02292 97708 33 28 50072 49928 52368 47632 02296 97704 32 29 50110 49890 52410 47590 02300 97700 31 30 9.50148 10.49852 9.52452 10.47548 10.02304 9.97696 30 31 50185 49815 52494 47506 02309 97691 29 32 50223 49777 52536 47464 02313 97687 28 83 50261 49739 52678 47422 02317 97683 27 34 50298 49702 52620 47380 02321 97679 26 35 9.50336 10.49664 9.62661 10.47339 10.02326 9.97674 25 86 50374 49626 52703 47297 02330 97670 24 37 50411 49589 52745 47255 02334 97666 23 38 50449 49551 52787 47213 02338 97662 22 39 50486 49514 62829 47171 02343 97657 21 40 9.50523 10.49477 9.62870 10.47130 10.02347 9.97663 20 41 50561 49439 52912 47088 02351 97649 19 42 50598 49402 52953 47047 02355 97645 18 43 50635 49365 52995 47005 02:360 97640 17 44 50673 49327 53037 46963 02364 97636 16 45 9.50710 10.49290 9.53078 10.46922 10.02368 9,97632 16 46 50747 49253 53120 46880 02372 97628 14 47 50784 49216 53161 46839 02377 97623 IS 48 50821 49179 63202 46798 02381 97619 12 49 50858 49142 53244 46756 02385 97616 11 50 9.50896 10.49104 9.53285 10.46715 10.02390 9.97610 10 51 50933 49067 53327 46673 02394 97606 9 52 50970 49030 53368 46632 02398 97602 8 53 51007 48993 53409 46591 02403 97697 7 54 51043 48957 53450 46560 02407 97593 6 55 9.51080 10.48920 9.63492 10.46508 10.02411 9.97689 5 56 51117 48883 53633 46467 02416 97684 4 57 51154 48846 53574 46426 02420 97580 3 58 51191 48809 53615 46385 02424 97576 2 59 51227 48773 53666 46344 02429 97571 1 60 51264 48736 53697 46303 02433 97567 M. Cosine. Secant. Cotangent. Tangent. | Cosecant. Sine. M. 71° 296 LOGAEITHMIC ANGULAR FUNCTIONS. Table 2. 19° Logar thms. Si'cant. 10.024:53 60° M. Sine. CnSCCilllt. i 10.48730 Tiingent. 9.,53697 Cotangent.! 10.46303 Cosine. 9,97567 M. 9.51264 60 1 51301 48699 53738 46262 024:17 97563 59 2 51338 48602 53779 46221 024 12 97558 53 3 51374 48626 53820 46180 0214(1 97584 57 i 51411 18,689 5;3861 46139 02 160 97,550 56 5 9.51447 10.486,63 9..63902 10.46098 10.024:66 9.97545 55 6 51484 4861li 53943 46057 02459 97,641 54 7 51620 484S0 53984 46016 02464 97536 53 8 .^1557 4.S4-13 .64025 4.5975 02468 97632 52 9 51593 48407 54065 45935 01^172 97528 51 10 9..51(i29 10.48371 9.64106 10.45894 10.02477 9.97523 50 11 51666 483: M 54147 4,5853 02481 97519 49 12 51702 48298 ,64187 45813 02485 97515 48 13 51738 482C.2 ,54228 45772 02490 97510 47 14 51774 48226 ,64269 45731 02494 97506 46 15 9..51S11 10.48189 9.,>4309 10.45691 10.02499 9.97501 45 16 51847 48163 54350 45650 0'2503 97497 44 17 51883 48117 54390 45610 02508 97492 43 18 51919 48U81 64431 45569 02512 97488 42 19 51955 48046 ,54471 4.5529 02516 97484 41 20 9.61991 10.48009 9.64612 10.4,648S 10.0'2521 9.97479 40 21 .62027 47973 54552 4:6418 02626 97475 39 22 62063 47937 54593 4.6407 02630 97470 38 23 .62099 47901 54633 46:167 02634 97466 37 24 ,62136 47865 ,54673 46:127 02,639 97461 36 25 9..62171 10.47829 9,54714 10. 1.6286 10.02,643 9.974,57 35 26 ."v>2()7 17793 64764 4.6246 02647 974.63 34 27 .62242 477,68 64794 462011 0'2552 97448 33 28 .62278 47722 64H36 45166 02556 97444 32 29 ,62314 4768i; ,61876 45126 02.561 97439 31 30 9.62360 10.476,60 9..14916 10.4.6086 10.0'2565 9.97435 30 31 52385 47615 54966 46046 02570 97430 29 32 52421 47579 54996 4.6006 0'2574 974-26 28 33 52456 47,644 . .65035 44966 02579 97421 27 34 52492 47508 55075 44926 02583 97417 26 35 9..62527 10.47473 9.,65115 10.44886 10.02588 9.97412 25 36 .626Ci:! 47437 551,65 44846 02592 97408 24 37 52598 47402 55195 44,S06 0-2697 97403 23 38 52634 47366 ,65235 44766 02601 97399 22 39 52669 47331 55275 44725 02606 97394 21 40 9.52705 10.47295 9.,65315 10.446,S6 10.02610 9.97390 20 41 52740 47260 65355 44645 02615 97385 19 42 52775 47225 55395 44605 02619 97381 18 43 52811 47189 55434 44566 02624 97376 17 44 62846 47154 56474 44526 02628 97372 16 45 9.52881 10.47119 9.,55,514 10.44486 10.02633 9.97367 15 46 52916 47084 56554 44446 02637 97363 14 47 52951 47049 55593 44407 02642 97358 13 48 52986 47014 55633 44:l{;7 02647 97353 12 49 63021 46979 ,66673 44:127 02651 97,349 11 50 9..63056 10.46944 9,55712 ]0.442,S8 10.02656 9.97344 10 51 53092 46908 55752 44248 02660 97340 9 52 53126 46874 ,55791 44209 02665 97335 8 53 53161 46839 55831 44169 02669 97331 7 54 53196 46804 ,65870 44130 02674 97:326 6 55 9.53231 10.46769 9.56910 10.44090 10.02678 9.97322 5 56 ,53266 46734 5.6949 440.51 02683 97317 4 57 53301 46699 55989 44011 02688 97312 3 58 ,53336 46664 56028 43972 02692 97308 2 59 53370 46630 56067 43933 02697 97303 i 60 53405 46,595 56107 43893 02701 97299 M. Cosine. Set-ant. Colaiipent. Tangent. Cosecant. Sine, M. 109° 70° Table 2. LOGAKITHMIC ANGULAR FUNCTIONS. 297 20° Log:arithins. 159° M. Sine. Cosecant. Tangent. Cotangent Secant. Cosine. M. 9.53405 10.46595 9.56107 10.43893 10.02701 9.97299 00 1 53440 46560 56146 43854 02706 97294 59 2 53475 46525 56186 43815 02711 97289 58 3 53509 46491 56224 43776 02716 97285 57 4 53544 46456 66264 43736 02720 97'280 56 5 9.53578 10.46422 9.56303 10.43697 10.02724 9.97276 55 6 53613 46387 56342 43658 02729 97271 54 7 53647 46353 56381 43619 02734 97266 53 8 53682 46318 56420 43580 02738 97262 62 9 53716 46284 66159 43541 02743 97257 51 10 9.53751 10.46249 9.56498 10.43502 10.02748 9.97252 50 11 53785 46215 66537 43463 02762 97248 49 12 53819 46181 56676 43424 02757 97'243 48 13 53854 46146 56615 43385 02762 97238 47 14 53888 46112 66654 43346 02766 97234 46 15 9.53922 10.46078 9.56693 10.43307 10.02771 9.97229 45 16 53957 46043 56732 43268 02776 97224 44 17 53991 46009 66771 43229 02780 97220 43 18 54026 45975 66810 43190 02785 97216 42 19 .54059 45941 56849 43151 02790 97210 41 20 9.54093 10.45907 9.56887 10.43113 10.02794 9.97206 40 21 54127 45873 56926 43074 02799 97201 39 22 54161 45839 56965 43035 02804 97196 38 23 54195 45805 57004 42996 02808 97192 37 24 54229 45771 57042 42958 02813 97187 36 25 9.54263 10.45737 9.67081 10.42919 10.02818 9.97182 85 26 54297 43703 67120 42880 02822 97178 34 27 54331 45669 57168 42842 02827 97173 33 28 51365 4.5635 57197 42803 02832 97168 32 29 54399 45601 67236 42765 02837 97163 31 30 9.54433 10.45567 9.57274 10.42726 10.02841 9.97159 30 31 54466 45534 57312 42688 02846 97164 29 32 54500 45500 57361 42649 02851 97149 28 33 54534 45466 57389 42611 0'2855 97145 27 34 54567 4.W33 57428 42572 02860 97140 26 35 9.54601 10.45399 9.57466 10.42584 10.02865 9.97136 25 36 54635 45365 57504 42496 02870 97130 24 37 54668 45332 57543 42457 02874 97126 23 38 54702 45298 57581 42419 02879 97121 22 39 54735 45265 57619 42381 02884 97116 21 40 9.54769 10.45231 9.57658 10.42342 10.02889 9.97111 20 41 54802 45198 57696 42304 02893 97107 19 42 54836 45164 57734 42266 02898 97102 18 43 54869 45131 67772 42228 02903 97097 17 44 54903 45097 57810 42190 02908 97092 16 45 9.54936 10.45064 9.67849 10.42151 10.02913 9.97087 15 46 54969 45031 67887 42113 02917 97083 14 47 55003 44997 57925 42075 02922 97078 13 48 55036 44964 57963 42037 02927 97073 12 49 55069 44931 58001 41999 02932 97068 11 50 9.55102 10.44898 9.58039 10.41961 10.02937 9.97063 10 51 55136 44864 58077 41923 02941 97059 9 52 55169 44831 68115 41885 02946 97064 8 53 55202 44798 58153 41847 02961 97049 7 54 55235 44765 68191 41809 02966 97044 6 55 9.55268 10.44732 9.58229 10.41771 10.02961 9.97039 5 56 55301 44699 58267 41733 02965 97035 4 57 55334 44666 58304 41696 02970 97030 3 58 55367 44633 58342 41658 02975 97026 2 59 55400 44600 58380 41620 02980 97020 1 60 55433 44567 58418 41582 02985 97015 31. CosJDe. Secant. Cotangent. Tangent. Cosecant. Sine. M. 110° 69° 298 LOGARITHMIC ANGULAR FUNCTIONS. Table 2. 21° Logarithms. 158° M. Sine. CoHecunt. Tangent. Cotangent. Secant. Cosine. M. 9.55433 10.44567 9.58418 10.41682 10.02985 9.97015 60 1 55466 44534 58456 41645 02990 97010 59 2 55499 44501 58493 41607 02995 97005 58 3 55532 44468 58531 41469 02999 97001 57 4 65564 44436 68569 41431 03004 96996 56 5 9.65697 10.44403 9.68606 10.41394 10.03009 9.96991 .55 6 65630 44370 58644 41356 03014 96986 64 7 55663 44337 58681 41319 03019 96981 53 8 55695 44306 58719 41281 03024 96976 52 9 55728 44272 68767 41243 03029 96971 61 10 9.55761 10.44239 9.58794 10.41206 10.03034 9.96966 50 11 65793 44207 58832 41168 03038 96962 49 12 65826 44174 58869 41131 03043 96967 48 13 55858 44142 68907 41093 03048 96952 47 14 65891 44109 58944 41056 03053 96947 46 15 9.56923 10.44077 9..58981 10.41019 10.03068 9.96942 46 16 66956 44044 69019 40981 03063 96937 44 17 56988 44012 .59056 40944 03068 96932 43 18 56021 43979 69094 40906 03073 96927 42 19 56053 43947 59131 40869 03078 96922 41 20 9.56086 10.43915 9.59168 10.40832 10.03083 9.96917 40 21 56118 43882 69206 40795 03088 96912 39 22 66160 43850 59243 40757 03093 96907 38 23 56182 43818 59280 40720 03097 96903 37 24 56215 43785 69317 40683 03102 96898 36 25 9.56247 10.43753 9.69354 10.40646 10.03107 9.96893 85 26 56279 43721 59391 40609 03112 96888 34 27 66311 43689 59429 40571 03117 90883 33 2« 66343 43657 59466 40534 03122 96878 32 29 56375 43626 59603 40497 03127 96873 31 30 9.56408 10.43592 9..595-10 10.40460 10.03132 9.96868 30 31 56440 43660 59577 40423 03137 96863 29 32 56472 43528 69614 40386 03142 96858 '28 33 66604 4;i490 69651 40349 03147 96858 27 34 66536 4:3464 69688 40312 03152 96848 26 35 9.56568 10.43432 9.59725 10.40276 10.03157 9.96843 25 36 56599 43401 69762 40238 03162 96838 24 37 56631 43369 59799 40201 03167 96833 23 38 66663 43337 69836 40165 03172 96828 22 39 66695 43305 59872 40128 03177 96823 21 40 9.66727 10.43273 9.59909 10.40091 10.03182 9.96818 20 41 ,56759 43241 59946 40054 03187 96813 19 42 56790 43210 59983 40017 03192 96808 18 43 56822 43178 60019 39981 03197 96803 17 44 56864 43146 60056 39944 03202 96798 16 45 9.66886 10.43114 9.60093 10.39907 10.03207 9.96793 15 46 56917 43083 60130 39870 03212 96788 14 47 56949 43051 60166 39834 03217 96783 13 48 56980 43020 60203 39797 03222 96778 12 49 67012 42988 60240 39760 03228 96772 11 50 9.67044 10.42966 9.60276 10.39724 10.03233 9.96767 10 51 67076 42925 60313 39687 03238 96762 9 52 57107 4'2893 60349 396.51 03243 96757 8 53 67138 42862 60386 39614 03248 96762 7 54 57169 4'2831 60422 39578 03253 96747 6 65 9.57201 10.42799 9.60469 10.39541 10.03258 9.96742 5 56 57232 42768 60495 39505 03263 96737 4 57 67264 42736 60532 39468 03268 96732 3 58 67296 42706 60568 39432 03273 96727 2 59 67326 42674 60605 39396 03278 96722 1 60 57368 42642 60641 39369 03283 96717 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 111° 68° Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 299 22° Logarithms. IS?"^ M. Sine. Cosecant. Tangent. Cotangent. Secant, Cosine, M. 9.57358 10.42642 9.60641 10.39359 10,03283 9.96717 60 1 57389 42611 60677 39323 03289 96711 59 2 57420 42580 60714 39286 03294 96706 88 3 57451 42549 60750 39250 03299 96701 57 4 57482 42518 60786 39214 03304 96696 56 6 9,57514 10.42486 9.60823 10.39177 10,03309 9,96691 55 6 57545 4'2455 60859 39141 03314 96686 54 7 57576 42424 60895 39105 03319 96681 53 8 57607 42393 60931 39069 08324 96676 52 9 57638 42362 60967 39033 03380 96670 61 10 9.57669 10.42331 9.61004 10.38996 10,03335 9,96665 50 11 57700 42300 61040 38960 03340 96660 49 12 57731 42269 61076 38924 03345 96655 48 13 57762 42238 61112 38888 03350 96650 47 14 57793 42207 61148 38852 03355 96645 46 15 9.57824 10.42176 9.61184 10.38816 10,03360 9,96640 45 16 57855 42145 61220 38780 03366 96634 44 17 57885 42115 61266 38744 03371 96629 43 18 57916 42084 61292 38708 03376 96624 42 19 57947 42053 61328 38672 03381 96619 41 20 9.57978 10.42022 9.61364 10.38636 10,03386 9,96614 40 21 58008 41992 61400 38600 03392 96608 39 22 58039 41961 61436 38564 03397 96603 38 23 58070 41930 61472 38528 03402 96598 37 24 58101 41899 61508 38492 03407 96593 36 25 9.58131 10.41869 9.61544 10.38456 10,03412 9,96588 35 26 58162 41838 61579 38421 03418 96582 34 27 58192 41808 61615 38385 03423 96577 33 28 .58223 41777 61651 38349 03428 96572 32 29 58253 41747 61687 38313 03433 96667 31 30 9.58284 10.41716 9.61722 10.38278 10,03438 9.96562 30 31 58314 41686 61758 38242 03444 96556 29 32 58345 41655 61794 38206 03449 96551 28 33 58375 41625 61830 38170 03454 96546 27 34 58406 41594 61865 38136 03159 96541 26 35 9.58436 10.41564 9.61901 10.38099 10,03465 9.96536 26 36 58467 41533 61936 38064 03470 96530 24 37 58497 41503 61972 38028 03475 96525 23 38 58527 41473 62008 37992 03480 96520 22 39 58557 41443 62043 37957 03486 96514 21 40 9.58588 10.41412 9.62079 10.37921 10,03491 9.96509 20 41 58618 41382 62114 37886 03496 96504 19 42 58648 41352 62150 37850 03502 96498 18 43 58678 41322 62185 37815 03507 96493 17 44 58709 41291 62221 37779 03512 96488 16 45 9.58739 10.41261 9.62266 10.37744 10,03517 9.96483 15 46 58769 41231 62292 37708 03523 96477 14 47 58799 41201 62327 37673 03528 96472 13 48 58829 41171 62362 37638 03533 96467 12 49 58859 41141 62398 37602 03539 96461 11 50 9.58889 10.41111 9.62433 10.37567 10,03544 9.96466 10 51 58919 41081 62468 37532 03549 96451 9 52 58949 41051 62604 37496 03555 96445 8 53 58979 41021 62539 37461 03560 96440 7 54 59009 40991 62574 37426 03565 96435 6 55 9.59039 10.40961 9.62609 10.37391 10,03571 9.96429 5 56 59069 40931 62645 37355 03576 96424 4 57 59098 40902 62680 37320 03581 96419 3 58 59128 40872 62715 37285 03587 96413 2 59 59158 40842 62750 37250 03592 96408 1 60 59188 40812 62785 37215 03597 96403 M. Cosine. Secant. Cotangent. Tangent, Cosecant, Sine. M. J 12° 67° 300 LOGAEITHMIC ANGULAE FUNCTIONS. Table 2. 23° Logarithms. 1S6° M. Sine. CuStTilllt. 10.40812 Tangent. Cotangent Secant. Cosine. M. 9.59188 9.62785 10.37215 10.03597 9.96403 60 1 59218 40782 628'20 37180 03603 96397 59 2 59247 40753 62855 87145 03608 96392 58 3 59277 40723 62890 37110 03613 96387 57 4 59307 40693 62926 37074 03619 96381 56 5 y..'.933ll 10.40664 9.62961 10.37039 10.03624 9.96376 66 6 59361! 40634 62996 37004 03630 96370 64 7 .■.93911 40604 63031 36969 03636 96365 53 8 59-l-i'i 40575 63066 369.34 03640 96360 52 9 59455 40M6 63101 36899 03646 96364 51 10 »..59484 10.40516 9.63135 10.36865 10.03651 9.96349 50 11 59514 40486 63170 36830 03657 96343 49 12 59M3 40457 63205 36795 03662 96338 48 13 59573 40427 63240 36760 03667 96333 47 1-1 69602 40398 63275 30725 03673 96327 46 15 9.59632 10.40368 9.63310 10.30690 10.03678 9.96322 45 16 59661 40339 63345 36655 03684 96316 44 17 59690 40310 6.3379 36621 03689 96311 43 18 59720 40280 63414 36686 03695 96305 42 19 59749 40261 63449 36551 03700 96300 41 20 9.5977S 10.40222 9.63484 10.36516 10.03706 9.96294 40 21 59h08 40192 63619 36481 03711 96289 39 22 59837 40163 63653 36447 03716 96284 38 23 5986(1 40134 63588 36412 03722 96278 37 24 59895 40105 63623 36377 03727 96273 36 25 9.59924 10.40076 9.63657 10.36343 10.03733 9.96267 35 26 59964 40046 63692 36308 03738 96262 34 27 59983 40017 63726 36274 03744 96256 33 28 60012 39988 63761 36239 03749 96261 32 29 60041 39959 63796 36204 03755 96246 31 30 9.60070 10.39930 9.63830 10.36170 10.03760 9.96240 30 31 60090 39901 63865 36135 03766 96234 29 32 60128 39872 63899 36101 03771 96229 28 33 60157 39843 63934 36066 03777 96223 27 34 60186 39814 63968 36032 03782 96218 26 35 9.60215 10.39785 9.64003 10.36997 10.03788 9.96212 25 36 (10244 39766 64037 35963 03793 96207 24 37 60273 39727 64072 35928 03799 96201 23 38 60302 39698 64106 85894 03804 96196 22 39 60331 39669 64140 35860 03810 96190 21 40 9,60359 10.39641 9.64175 10.35825 10.03815 9.96185 20 41 60388 39612 64209 36791 03821 96179 19 42 60417 39583 64243 36757 03826 96174 18 43 60446 396.54 64278 35722 03832 96168 17 44 60474 39626 64312 35688 03838 96162 16 46 9.60503 10.39497 9.64346 10.35654 10.03843 9.96157 15 46 60532 39468 64381 36619 03849 96151 14 47 60581 39439 64416 35585 o:«54 96146 13 48 60589 39411 64449 35551 03860 96140 12 49 60618 39382 64483 35517 03865 96136 11 50 9.60646 10.39354 9.64517 10.35483 10.03871 9.96129 10 51 60675 393'26 64.552 35448 03877 96123 9 52 60704 39296 64586 35414 03882 96118 8 53 60732 392 J8 64620 35380 03888 96112 7 64 60761 39239 64664 35346 o;«93 96107 6 65 9.60789 10.39L'l 9.64688 10.36312 10.03899 9.96101 5 56 60818 39182 64722 36278 03905 96096 4 57 60846 39154 64756 35244 03910 96090 3 58 60875 39125 64790 35210 03916 96084 2 59 60903 39097 64824 35176 03921 96079 1 60 60931 33069 P4858 35142 03927 96073 M. Cosine. Secant. Cotangent. Tangent. Cosenint. Sine. M. U3° 66° Table 3. LOGARITHMIC ANGULAR FUNCTIONS. 301 24° Logarithms. 1SS° M. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. M. 9.60931 10.39069 9.64858 10.35142 10.03927 9.96073 60 1 60960 39040 64892 35108 03933 96067 69 2 60988 39012 64926 35074 03938 96062 58 3 61016 38984 64960 35040 03944 96056 67 4 61045 38955 64994 35006 03950 96050 56 5 9.61073 10.38927 9.65028 10.34972 10.03955 9.96045 56 6 61101 38899 65062 34938 03961 96039 54 7 61129 38871 65096 34904 03966 96034 63 8 61188 38842 66130 34870 03972 96028 62 9 61186 38814 65164 34836 03978 96022 51 10 9.61214 10.38786 9.65197 10.34803 10.03983 9.96017 50 11 61242 38768 G5231 34769 03989 960H 49 12 61270 38730 66265 34735 03995 96005 48 13 61298 38702 65299 34701 04000 96000 47 14 61326 38674 65333 34667 04006 95994 46 15 9.61364 10.38646 9.66366 10.34634 10.04012 9.95988 46 16 61382 38618 65400 34600 04018 95982 44 17 61411 38589 65434 34566 04023 95977 43 18 61438 38562 65467 34533 04029 95971 42 19 61466 38684 65601 34499 04035 95965 41 20 9.61494 10.38606 9.65636 10.34466 10.04040 9.95960 40 21 61522 38478 65568 S4432 04046 96954 39 22 61560 38460 66602 34398 04052 95948 38 23 61578 38422 65636 34364 04058 96942 37 24 61606 38394 65669 34331 04063 96937 36 25 9.61634 10.38366 9.65703 10.34297 10.04069 9.95931 35 26 61662 38338 66736 34264 04075 95925 34 27 61689 38311 66770 34230 04080 95920 33 28 61717 38283 65803 34197 04086 95914 32 29 61745 38255 66837 34163 04092 95908 31 30 9.61773 10.38227 9.66870 10.34130 10.04098 9.95902 30 31 61800 38200 66904 34096 04103 96897 29 32 61828 38172 65937 34063 04109 96891 28 33 61856 38144 65971 34029 04115 96886 27 34 61883 38117 66004 33996 04121 95879 26 35 9.61911 10.38089 9.66038 10.33962 10.04127 9.95873 25 36 61939 38061 66071 33929 04132 96868 24 37 61966 38034 66104 33896 04138 96862 23 38 61994 38006 66138 33862 04144 95856 22 39 62021 37979 66171 33829 04150 95850 21 40 9.62049 10.37951 9.66204 10.33796 10.04156 9.95844 20 41 62076 37924 66238 33762 04161 95839 19 42 62104 37896 66271 33729 04167 96833 18 43 62131 37869 66304 33696 04173 96827 17 44 62169 37841 66337 33663 04179 95821 16 45 9.62186 10.37814 9.66371 10.33629 10.04185 9.95815 15 46 62214 37786 66404 33596 04190 95810 14 47 62241 37769 66437 33563 04196 95804 13 48 62268 37732 66470 33630 04202 95798 12 49 62296 37704 66503 33497 04208 95792 11 50 9.62323 10.37677 9.66637 10.33463 10.04214 9.95786 10 51 62360 37650 66570 33430 04220 96780 9 52 62377 37623 66603 33397 04226 96776 8 63 62405 37595 66636 33364 04231 95769 7 54 62432 37668 66669 33331 04237 95763 6 65 9.62469 10.37541 9.66702 10.33298 10.04243 9.95767 6 56 62486 37514 66735 33266 04249 95751 4 57 62613 37487 66768 33232 04255 96746 3 58 62541 37459 66801 33199 04261 96739 2 69 6'2668 37432 66834 33166 04267 9.5733 1 60 62596 37405 66867 33133 04272 95728 M. Cosiue. Secant. Cotangent. Tangent. Cosecant. Sine. M. 114° 302 LOGARITHMIC ANGULAE FUNCTIONS. Table 2. 25° Logarithms. 154° M. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. M. 9.62595 10.37405 9.66867 10.33138 10.04272 9.95728 60 1 62622 37378 66900 33100 04-278 95722 69 2 62649 37351 66938 83067 04284 95716 58 3 62676 37324 66966 33034 04290 95710 57 4 6-2703 37297 66999 38001 04296 95704 56 5 9.62730 10.37270 9.67032 10.32968 10.04302 9.96698 55 6 62757 37243 67065 32985 04308 95692 54 7 62784 87216 67098 32902 04314 95686 53 8 62811 37189 671.81 82869 04320 95680 52 9 62838 37162 67163 32837 04326 96674 51 10 9.62865 10.37135 9.67196 10.32804 10.04332 9.95668 50 11 62892 37108 67229 32771 04337 95663 49 12 62918 37082 67262 3-2788 04343 98657 48 13 62945 .37055 67295 32705 M349 95661 47 14 62972 37028 67827 8-2673 04355 95645 46 15 9.62999 10.37001 9.67360 10.32640 10.04361 9.95639 45 16 68026 36974 67393 32607 04367 95633 44 17 63052 36948 67426 32574 04378 95627 43 18 63079 36921 67458 3-2542 04379 956-21 42 19 63106 36894 67491 32509 04385 95615 41 20 9.63138 10.36867 9.67524 10.32476 10.04391 9.95609 40 21 63159 36841 67,556 32444 04397 95603 39 22 63)86 86814. 67689 32411 04403 96597 38 23 68213 86787 67622 3'2378 04409 95.591 37 24 68239 36761 67654 32346 04415 95585 36 25 9.68266 10.36734 9.67687 10.32313 10.04421 9.95579 35 26 68292 ^ 36708 67719 8-2281 04427 96578 34 27 63319 A 36681 67752 32-248 04433 96567 33 28 63345 36655 67785 32215 04439 96661 82 29 63372 36628 67817 82183 04445 95565 31 30 9.68398 10.36602 9.67850 10.32150 10.04451 9.95549 30 81 68426 36575 67882 3'2118 04457 95543 29 32 63451 36.549 67915 32085 04463 96537 28 33 63478 36522 67947 32053 04469 96531 •27 34 63504 36496 67980 32020 04476 955-25 26 35 9.63531 10..36469 9.68012 10.81988 10.04481 9.95519 25 36 63557 36448 68044 81956 04487 95518 24 37 63588 86417 68077 81928 04493 95507 23 88 63610 36890 68109 81891 04500 96500 22- 39 63636 36864 68142 31858 04506 95494 21 40 9.63662 10.86838 9.68174 10.31826 10.04512 9.9.5488 20 41 63689 86811 68206 31794 04518 9.5482 19 42 63715 36285 68-289 31761 04524 95476 18 43 63741 86259 68-271 31729 04530 95470 17 44 63767 86233 68303 31697 04536 96464 16 45 9.63794 10.86206 9.68386 10.31664 10.04542 9.95458 15 46 63820 86180 68368 81682 04548 95462 14 47 63846 86154 68400 81600 04554 95446 13 48 63872 86128 68432 81568 04560 95440 12 49 63898 86102 68465 31535 04566 96484 11 50 9.63924 10.36076 9.68497 10.31503 10.04573 9.96427 10 51 63960 36050 685-29 31471 ' 04579 95421 9 52 63976 36024 68.561 81439 04585 95415 8 53 64002 85998 68598 81407 04591 95409 7 54 64028 35972 68626 31374 04597 95403 6 55 9.64054 10.85946 9.68658 10.31342 10.04603 9.95397 5 56 64080 85920 68690 81310 04609 95391 4 57 64106 35894 68722 31-278 04616 95384 3 58 64132 35868 68754 31246 04622 95378 2 59 64158 35842 68786 31214 04628 95372 i 60 64184 35816 68818 31182 04634 95366 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 115° Table 2. LOGARITHMIC ANGULAK FUNCTIONS. 303 26° Logarithms. 153° M. Sine. Cosecant. Tangent. Cotangent, Secant, Cosine. M, 9.64184 10.35816 9.68818 10.31182 10.04634 9.95366 60 1 64210 35790 68850 311.50 04640 95360 59 2 64236 35764 68882 31118 04646 95354 58 3 64262 35738 68914 ■ 81086 04652 95348 57 4 64288 35712 68946 31054 046,59 95341 56 5 9.64313 10.35687 9.68978 10.31022 10,04665 9.9.5335 55 6 64339 35661 69010 30990 04671 95329 54 7 64365 35635 69042 .30958 04677 95323 53 8 64391 35609 69074 30926 04683 9.5317 .52 9 64417 35583 69106 ,30894 04690 95310 51 10 9.64442 10.35558 9.69138 10.30862 10,04696 9.95.304 50 11 64468 35532 69170 30830 04702 95298 49 12 64494 35506 69202 30798 04708 95292 48 13 64519 35481 69234 30766 04714 95286 47 14 61545 35455 69266 30734 04721 95279 46 15 9.64571 10.35429 9.69298 10.30702 10.04727 9.9.5273 45 16 64596 35404 69329 30671 04733 95267 44 17 64622 35378 69361 30639 047.39 95261 43 18 61647 35353 69393 30607 04746 95254 42 19 64673 35327 69125 30.575 04752 95248 41 20 9.64698 10.35302 9.69457 10.30.543 10.047.58 9.95242 40 21 64724 35276 69488 30512 04764 95236 39 22 64749 35251 69520 30480 04771 95229 38 23 64775 35225 69552 30448 04777 95223 37 24 64800 35200 69584 80416 04783 95217 36 25 9.64826 10.35174 9.69615 10.30385 10,04789 9.95211 35 26 64851 35149 69647 30353 04796 95204 34 27 64877 35123 69679 30321 04802 95198 33 28 64902 35098 69710 30290 04808 95192 32 29 64927 35073 69742 30258 04815 95185 31 30 9.64953 10.35047 9.69774 10.30226 10,04821 9.95179 30 31 64978 35022 69805 30195 04827 95173 29 32 65003 34997 69837 30163 04833 95167 28 33 65029 34971 69868 30132 04840 95160 27 34 65054 34946 69900 30100 04846 95154 26 35 9.65079 10.34921 9,69932 10.30068 10,04852 9,95148 25 36 65104 34896 69963 30037 04859 95141 24 37- - '>6.5130 34870 69995 30005 01865 95135 23 38 65155 34845 70026 29974 04871 95129 22 39 65180 34820 70058 29942 04878 95122 21 40 9.65205 10.34795 9.70089 • 10.29911 10,04884 9,95116 20 41 65230 34770 70121 29879 04890 95110 19 42 65255 34745 70152 29848 04897 95103 18 43 65281 34719 70184 29816 04903 9.5097 17 44 65306 34694 70215 29785 04910 95090 16 45 9.65331 10.34669 9.70247 10,29753 10,04916 9,95084 15 46 65356 34644 70278 29722 04922 95078 14 47 65381 34619 70309 29691 04929 95071 13 48 65406 34594 70341 29659 04935 95065 12 49 65431 34569 70372 29628 04941 95059 11 50 9.65456 10.34544 9.70404 10.29596 10,04948 9,95052 10 51 65481 34519 70435 29565 049.54 95046 9 52 65506 34494 70466 29.534 04961 95039 8 53 65531 34469 70498 29502 04967 9.5033 7 54 65556 34444 70529 29471 04973 95027 6 55 9.65580 10.34420 9,70.560 10,2944r 10.0498f^ 9.95020 5 56 65605 34395 70592 29408 04986 95014 4 57 65630 34370 70623 29377 04993 95007 3 58 65665 34345 70654 29346 04999 95001 2 59 65680 34320 70685 29315 05005 94995 1 60 65705 34295 70717 29283 05012 94988 M. Cosine. Secant. Cotangent Tangent, Cosecant. Sine, M, 116° 63° 304 LOGAETTHMTC ANGULAR FUNCTIONS. Table 2. 27° Logarithms. 52° M, Sine. Cosecant. Tangent. Cotangent, Secant. Cosine. M. 9.65705 10.34295 9.70717 10,29283 10.05012 9.94988 60 1 65729 34271 70748 29252 06018 94982 59 2 65754 34246 70779 29221 0,5026 94975 58 3 65779 34221 70810 29190 05031 94969 57 4 6.M14 34196 70841 29159 05038 94962 66 5 9,li5SiH 10.34172 9.70873 10.29127 10,05044 9.94956 55 6 li.iSS;! 31147 70904 29096 05051 94949 54 7 65878 34122 70935 29065 05057 94943 53 8 65902 34098 70966 29034 05064 94936 52 9 65927 34073 70997 29003 06070 94930 51 10 9.65952 10.34048 9.71028 10.28972 10,06077 9.91923 50 11 65976 34024 71059 28941 05083 94917 49 12 66001 33999 71090 28910 05089 94911 48 13 66025 33975 71121 28879 05096 94904 47 14 660.10 33950 71153 28847 05102 94898 46 15 9.66075 10.33925 9.71184 10.28816 10,05109 9.94891 45 16 66099 33901 71215 28785 05115 94885 44 17 66124 33876 71216 28754 05122 94878 43 18 66148 33852 71277 28723 05129 94871 42 19 66173 33827 71308 28692 05135 94865 41 20 9.66197 10.33803 9.71339 10.28(i61 10.06142 9.94868 40 21 66221 33779 71370 28630 05148 94852 39 22 66216 33754 71401 28599 05155 94845 38 23 66270 33730 71431 28569 05161 94839 37 24 66295 33705 71462 2,S53,H 05168 94832 36 26 9.66319 10.33681 9.71193 10.28507 10.05174 9.94826 35 26 66343 33657 71524 28476 05181 94819 34 27 66368 33632 715.55 28146 05187 94813 33 28 66392 33608 71586 28414 05194 94806 32 29 66416 S3584 71617 28383 06201 94799 31 30 9.66441 10.335.59 9.71648 10.28352 10.05207 9.94793 30 31 66465 33536 71679 28321 05214 94786 29 32 66489 33511 71709 28291 05220 94780 28 33 66513 33487 71740 28260 05227 94773 27 34 66537 33463 71771 28229 062:B 94767 26 35 9.66.562 10.33438 9.71802 10.28198 10.06240 9.94760 25 36 66586 33414 71833 28167 06247 94753 24 37 66610 33390 71863 28137 05253 94747 23 38 66634 33366 71894 28106 05260 94740 22 39 66658 3,3342 71926 28075 05266 94734 21 40 9.66682 10.33318 9.71955 10.2.><0J5 10.05273 9.94727 20 41 66706 33294 71986 28014 05280 94720 19 42 66731 33269 72017 27983 05286 94714 18 43 66765 33245 72048 27952 05293 94707 17 44 66779 33221 72078 27922 05300 94700 16 45 9.66S03 10.33197 9.72109 10.27891 10.05306 9.94694 15 46 66827 33173 72140 27860 06313 94687 14 47 66861 33149 72170 27830 05320 94680 13 48 66875 331'25 72201 27799 0,5326 94674 12 49 66899 33101 72231 27769 0,5333 94667 11 50 9.66922 10.33078 9.72262 10,27738 10.05340 9.94660 10 51 66946 33054 72293 27707 06346 94654 9 52 66970 33030 72323 27677 05353 94647 8 53 66994 33006 723,54 27646 06360 94640 7 54 67018 32982 72384 27616 05366 94634 6 55 9.67042 10.32958 9.72415 10.27,585 10.05373 9.94627 5 56 67066 32934 72445 275,55 05380 94620 4 57 67090 32910 72476 27524 08386 94614 3 58 67113 32887 725Uli 27191 06393 94607 2 59 67137 32863 72537 27463 0.5400 94600 1 60 67161 32839 72567 27433 06407 91593 M. Cosine. Secant. Cotangent, Tangent. Cosecant. Sine. M. Table 3. LOGARITHMIC ANGULAR FUNCTIONS. 305 28° Logarithms. 1S1° M. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. M. 9.67161 10.32839 9.72567 10.27433 10.05407 9.94593 60 1 67185 32815 72598 27402 05413 94587 59 2 67208 32792 72628 27372 05420 94580 58 3 67232 32768 72659 27341 05427 91573 57 4 67256 32744 72689 27311 06433 94567 56 5 9.67280 10.32720 9.72720 10.27280 10.06440 9.94860 55 6 67303 32697 72760 27260 05447 94553 54 7 67327 32673 72780 27220 05454 94516 63 8 67350 32650 72811 27189 05460 94640 52 9 67374 32626 72841 27169 05467 94533 51 10 9.67398 10.32602 9.72872 10.27128 10.05474 9.94526 60 11 67421 32679 72902 27098 06481 94519 49 12 67445 32655 72932 27068 05487 94513 48 13 67468 32532 72963 27037 05494 94506 47 14 67492 32508 72993 27007 05601 94499 46 15 9.67515 10.32486 9.73023 10.26977 10.05508 9.94492 45 16 67539 32461 73054 26946 06515 94485 41 17 67562 32438 73084 26916 05521 94479 43 18 67586 32414 73114 26886 06528 94472 42 19 67609 32391 73144 26866 05535 94465 41 20 9.67633 10.32367 9.73175 10.26826 10.05642 9.94468 40 21 67666 32344 73205 26795 06549 94451 39 22 67680 32320 73235 26765 06556 94446 38 23 67703 32297 73265 26735 05562 94438 37 24 67726 32274 73295 26705 05669 94431 36 25 9.67750 10.32250 9.73326 10.26674 10.05576 9.94424 35 26 67773 32227 73356 26644 05583 94417 34 27 67796 32204 73386 26614 06590 94410 33 28 67820 32180 73416 26584 05596 94404 32 29 67843 32157 73446 26554 05603 94397 31 30 9.67866 10.32134 9.73476 10.26524 10.05610 9.94390 30 31 67890 32110 73507 26493 05617 94383 29 32 67913 32087 73637 26463 05624 94376 28 33 67936 32064 73567 26433 05631 94369 27 34 67959 32041 73597 26403 05638 94362 26 35 9.67982 10.32018 9.73627 10.26373 10.0O646 9.9ii355 26 36 68006 31994 73657 26343 05651 94349 24 37 68029 31971 73687 26313 0o658 9*342 23 38 68052 31948 73717 26283 05666 94335 22 39 68075 31925 73747 26263 0o672 9*328 21 40 9.68098 10.31902 9.73777 10.26223 10.0O679 9.94321 20 41 68121 31879 73807 26193 O0686 94314 19 42 68144 31856 73837 26163 0o693 94307 18 43 68167 31833 73867 26133 06700 94300 17 44 68190 31810 73897 26103 06707 94293 16 45 9.68213 10.31787 9.73927 10.26073 10.0D714 9.94286 15 46 68237 31763 73957 26043 05721 94279 14 47 68260 31740 73987 26013 05727 94273 13 48 68283 31717 74017 25983 05734 94266 12 49 68305 31695 74047 25953 06741 94259 11 50 9.68328 10.31672 9.74077 10.26923 10.06748 9.94252 10 51 68351 31649 74107 26893 05755 94246 9 52 68374 31626 74137 25863 05762 94238 8 53 68397 31603 74166 26834 05769 94231 7 54 68420 31580 74196 26804 05776 94224 6 55 9.68443 10.31.557 9.74226 10.25774 10.06783 9.94217 5 56 68466 31534 74256 25744 05790 94210 4 57 68489 31511 74286 25714 05797 94203 3 58 68512 31488 74316 25684 05804 94196 2 59 68534 31466 74345 26666 05811 94189 1 60 68557 31443 74375 26626 05818 94182 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 118° 21 61° 306 LOGARITHMIC ANGULAR FUNCTIONS. Table 3. 29° Logarithms. 150° M. Sine. ("'itKLTailt. Tangent. Cotangent. Secant. CoBJne. M. 9.68557 10.31443 9.74375 10.25625 10.0,5818 9.94182 60 1 e.'^.wo 31420 74406 2.5595 05825 94175 59 2 fisco:) 31397 74435 25565 05832 94168 58 3 6S625 :!i:i75 7-1465 25535 05839 94161 57 4 fism.s 3l:l.'-i2 74494 2.5506 0.5846 94154 56 5 9.liSli71 10.:n;)29 9.74524 10.25476 10.05863 9.94147 55 6 (■„Sli94 ;-ii;;o« 74554 25446 05860 94140 54 7 twn; 31284 74683 25417 0.5867 94133 53 8 (;s7n9 31261 74613 26387 05874 94126 52 9 (iS7l'i2 31238 74643 25357 05881 94119 61 10 O.CWSl 10.31216 9.74673 10.25327 10.05888 9.94112 50 11 6.SS07 31193 74702 2.5298 05895 94105 49 12 C,SX29 31171 74732 25268 06902 94098 48 13 iisK-a 31148 74762 25238 05910 94090 47 14 6S,S7.') 31125 74791 26209 05917 94083 46 15 9.6S.S97 10.31103 9.74821 10.25179 10.05924 9.94076 45 16 |;,S920 31080 74851 25149 0,6931 94069 44 17 68942 310.68 74880 25120 05938 94062 43 18 68965 31035 74910 25090 0,5945 94055 42 19 68987 31013 74939 25061 05952 94048 41 20 9.69010 10.30990 9.74969 10.25031 10.05959 9.94041 40 21 69032 30968 74998 25002 05966 94034 39 22 69055 30945 75028 24972 05973 94027 38 23 69077 30923 750.58 24942 06980 94020 37 24 69100 30900 76087 24913 06988 94012 36 25 9.69122 10.30878 9.75117 10.24883 10.05996 9.94005 35 26 69144 308.56 75146 248,54 06002 93998 34 27 69167 30833 75176 24824 06009 93991 33 28 69189 30811 75205 24796 06016 93984 32 29 69212 30788 75235 24765 06023 93977 31 30 9.69234 10.30766 9.75264 10.24736 10.06030 9.93970 30 31 69266 30744 75294 24706 06037 93963 29 32 69279 30721 75323 24677 06045 93955 28 33 69301 30699 75353 24647 06052 93948 27 34 69323 30677 7,5382 24618 06059 93941 26 35 9.69345 10.30655 9.754n 10.24589 10.06066 9.93934 25 36 69368 30632 75441 24.559 06073 93927 24 37 69390 30610 75470 24530 06080 93920 23 88 69412 30588 75500 24500 06088 93912 22 39 69434 30566 75529 24471 06095 93905 21 40 9.69466 10.30.544 9.75558 10.24442 10.06102 9.93898 20 41 69479 30521 75588 24412 06109 93891 19 42 69501 30499 75617 24383 06116 93884 18 43 69523 30477 75647 24:!63 00124 93876 17 44 69545 30455 75676 24324 06131 93869 16 45 9.69567 10.30433 9.75705 10.24295 10.06138 9.9.3862 15 46 69.589 30411 75735 24265 06145 93855 14 47 69611 30389 75764 24236 08153 93847 13 48 69633 30367 75793 24207 06160 93840 12 49 69655 30345 75822 24178 06167 93833 11 50 9.69677 10.30323 9.75862 10.24148 10.06174 9.93826 10 51 69699 30301 76881 24119 06181 93819 9 52 C9721 30279 76910 24090 06189 93811 8 53 69743 30257 76939 24061 06196 93804 7 54 69705 30236 76969 24031 06203 93797 6 55 9.69787 10.30213 9.76998 10.24002 10.06211 9.93789 5 56 69809 30191 76027 23973 06218 93782, 4 57 69831 30169 76056 23944 06225 93776 3 58 69S53 30147 76086 23914 06232 93768 2 59 69875 30125 76116 23,SS5 06240 93760 1 60 C9897 30103 76144 23856 06247 93753 M. t'l>MJH\ Secant. Cotangent. Tansent. 1 Cosecant. Sine. M. 119° 60° Table 2. LOGAKITHMIC ANGULAR FUNCTIONS. 307 30° Logarithms. 49° M. Sine. Cosecant. Tangent. Cotangent, Secant. Cosine. M. 9.69897 10.30103 9.76144 10.23856 10.06247 9.93763 60 1 B9919 30081 76173 23827 06254 93746 59 2 69941 30059 76202 23798 06262 93738 58 3 69963 30037 76231 23769 06269 93731 67 4 69984 30016 76261 23739 . 06276 93724 66 5 9.70006 10.29994 9.76290 10.23710 10.06283 9.93717 55 6 70028 29972 76319 23681 06291 93709 64 7 70050 29950 76348 23652 06298 93702 53 8 70072 29928 76377 23623 06305 93695 52 9 70093 29907 76406 23594 06313 93687 61 10 9.70115 10.29885 9.76435 10.23565 10.06320 9.93680 60 11 70137 29863 76464 23536 06327 93673 49 12 70159 29841 76493 23507 06335 93665 48 13 70180 29820 76522 23478 06342 93668 47 14 70202 29798 76551 23149 06350 93650 46 15 9.70224 10.29776 9.76580 10.23420 10.06357 9.93643 45 16 70245 29755 76609 23391 06364 93636 44 17 70267 29733 76639 23361 06372 93628 43 18 70288 29712 76668 23332 06379 93621 42 19 70310 29690 76697 23303 06386 93614 41 20 9.70332 10.29668 9.76725 10.23275 10.06394 9.93606 40 21 70353 29647 76754 23246 06401 93599 39 22 70375 29625 76783 23217 06409 93591 38 23 70396 29604 76812 23188 06416 93584 37 24 70418 29582 76841 2.3159 06423 93577 36 25 9.70439 10.29561 9.76870 10.23130 10.06431 9.93669 35 26 70461 29539 76899 23101 06438 93562 34 27 70482 29518 76928 23072 06446 93554 33 28 70504 29496 76957 ■ 23043 06453 93547 32 29 70525 29475 76986 23014 06461 93539 31 30 9.70547 10.29453 9.77015 10.22985 10.06468 9.93532 30 31 70568 29432 77044 22956 06475 93525 29 32 70590 29410 77073 22927 06483 93517 28 33 70611 29389 77101 22899 06490 93510 27 34 70633 29367 77130 22870 06498 93502 26 35 9.70654 10.29346 9.77159 10.22841 10.06505 9.93495 25 36 70675 29325 77188 22812 06513 93487 24 37 70697 29303 77217 22783 06520 93480 23 38 70718 29282 77246 22764 06528 93172 22 39 70739 29261 77274 22726 06535 93465 21 40 9.70761 10.29239 9.77303 10.22697 10.06543 9.93467 20 41 70782 29218 77332 22668 06550 93450 19 42 70803 29197 77361 22639 06558 93442 18 43 70824 29176 77390 22610 06565 93435 17 44 70846 29154 77418 22582 06573 93427 16 45 9.70867 10.29133 9.77447 10.22553 10.06680 9.93420 15 46 70888 29112 77476 22524 06588 93412 14 47 70909 29091 77505 22495 06695 93405 13 48 70931 29069 77533 22467 06603 93397 12 49 70952 29048 77562 22438 06610 93390 11 50 9.70973 10.29027 9.77591 10.22409 10.06618 9.93382 10 51 70994 29006 77619 22381 06625 93375 9 52 71015 28985 77648 22352 06633 93367 8 53 71036 28964 77677 22323 06640 93360 7 54 71058 28942 77706 22294 06648 93352 6 55 9.71079 10.28921 9.77734 10.22266 10.06656 9.93,344 5 56 71100 28900 77763 22237 06663 93337 4 57 71121 28879 77791 22209 06671 93329 3 68 71142 28858 77820 22180 06678 93322 2 59 71163 28837 77849 22161 06686 93314 1 60 71184 28816 77877 22123 06693 93307 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 120° 59° 308 LOGAEITHMIC ANGULAR FUNCTIONS. Table 2. 31° Logarithms. J48° M. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. M. 9.71184 10.28816 9.77877 10.22123 10.06693 9.93307 60 1 71205 28795 77906 22094 06701 93299 69 2 71226 28774 77935 22065 06709 93291 58 3 71247 28753 77963 22037 06716 93284 57 4 71268 28732 77992 22008 06724 93276 66 5 9.71289 10.28711 9.78020 10.21980 10.06731 9.93269 55 6 71310 28690 78049 21951 06739 93261 54 7 71331 28669 78077 21923 06747 93253 53 8 71352 28648 78106 21894 06754 93246 52 9 71373 28627 78135 21865 06762 93238 51 10 9.71393 10.28607 9.78163 10.21837 10.06770 9.93230 50 11 71414 28586 78192 21808 06777 93223 49 12 71435 28565 78220 21780 06785 93216 48 13 71456 28544 78249 21751 06793 93207 47 14 71477 28523 78277 21723 06800 93200 46 15 9.71498 10.28502 9.78306 10.21694 10.06808 9.93192 46 16 71519 28481 78334 21666 06816 93184 44 17 71539 28461 78363 21637 06823 93177 43 18 71560 28440 78391 21609 06831 93169 42 19 71581 28419 78419 21581 06839 93161 41 20 9.71602 10.28398 9.78448 10.21562 10.06846 9.93154 40 21 71622 28378 78476 21524 06854 93146 39 22 71643 28367 78505 21495 06862 93138 38 28 71664 28336 78533 21467 06869 93131 37 24 71685 28315 78562 21438 06877 93123 36 25 9.71705 10.28295 9.78590 10.21410 10.06885 9.93115 35 26 71726 28274 78618 21382 06892 93108 34 27 71747 28253 78647 21353 06900 93100 33 28 71767 28233 78675 21325 06908 93092 32 29 71788 28212 78704 21296 06916 93084 31 30 9.71809 10.28191 9.78732 10.21268 10.06923 9.93077 30 31 71829 28171 78760 21240 06931 93069 29 32 71860 28150 78789 21211 06939 93061 28 33 71870 28130 78817 21183 06947 93063 27 34 71891 28109 78845 21165 06954 93046 26 35 9.71911 10.28089 9.78874 10.21126 10.06962 9.93038 25 36 71932 28068 78902 21098 06970 93030 24 37 71952 28048 78930 21070 06978 93022 23 38 71973 28027 78959 21041 06986 93014 22 39 71994 28006 78987 21013 06993 93007 21 40 9.72014 10.27986 9.79015 10.20985 10.07001 9.92999 20 41 72034 27966 79043 20967 07009 92991 19 42 72065 27945 79072 20928 07017 92983 18 43 72075 27925 79100 20900 07024 92976 17 44 72096 27904 79128 20872 07032 92968 16 45 9.72116 10.27884 9.79156 10.20844 10.07040 9.92960 15 46 72137 27863 79185 20815 07018 92952 14 47 72157 27843 79213 20787 07066 92944 13 48 72177 27823 79241 20769 07064 92936 12 49 72198 27802 79269 20731 07071 92929 11 50 9.72218 10.27782 9.79297 10.20703 10.07079 9.92921 10 51 72238 27762 79326 20674 07087 92913 9 52 72259 27741 79354 20646 07095 92906 8 53 72279 27721 79382 20618 07103 92897 7 54 72299 27701 79410 20690 07111 9'2889 6 65 9.72320 10.27680 9.79438 10.20562 10.07119 9.92881 5 56 72340 27660 79466 20534 07126 92874 4 57 72360 27640 79495 20505 07134 92866 3 58 72381 27619 79523 20477 07142 92858 2 59 72401 27599 79551 20449 07150 92850 1 60 72421 27579 79579 20421 07158 92842 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 121° 58° Table 3. LOGARITHMIC ANGULAR FUNCTIONS. 309 32° Logarithms. 147° M. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. M. 9.72421 10.27579 9.79579 10.20421 10.07158 9.92842 60 1 72441 27559 79607 20393 07166 92834 59 2 72461 27539 79635 20365 07174 92826 58 3 72482 27518 79663 20337 07182 92818 57 4 72502 27498 79691 20309 07190 92810 56 5 9.72522 10.27478 9.79719 10.20281 10.07197 9.92803 56 6 72642 27458 79747 20253 07205 92795 54 7 72562 27438 79776 20224 07213 92787 53 8 72582 27418 79804 20196 07221 92779 52 9 72602 27398 79832 20168 07229 92771 51 10 9.72622 10.27378 9.79860 10.20140 10.07237 9.92763 50 11 72643 27357 79888 20112 07245 92765 49 12 72663 27337 79916 20084 07253 92747 48 13 72683 27317 79944 20056 07261 92739 47 14 72703 27297 79972 20028 07269 92731 46 15 9.72723 10.27277 9.80000 10.20000 10.07277 9.92723 45 16 72743 27257 80028 19972 07285 92715 44 17 72763 27237 80056 19944 07293 92707 43 18 72783 27217 80084 19916 07301 92699 42 19 72803 27197 80112 19888 07309 92691 41 20 9.72823 10.27177 9.80140 10.19860 10.07317 9.92683 40 21 72843 27157 80168 19832 07325 92675 39 22 72863 27137 80195 19805 07333 92667 38 23 72883 27117 80223 19777 07341 92659 37 24 72902 27098 80251 19749 07349 92661 36 25 9.72922 10.27078 9.80279 10.19721 10.07357 9.92643 35 26 72942 2'7058 80307 19693 07365 92635 34 27 72962 27038 80335 19665 07373 92627 33 28 72982 27018 80363 19637 07381 92619 32 29 73002 20998 80391 19609 07389 92611 31 30 9.73022 10.26978 9.80419 10.19581 10.07397 9.92603 30 31 73041 26959 80447 19553 07405 92595 29 32 73061 26939 80474 19526 07413 92587 28 83 73081 26919 80502 19498 07421 92579 27 84 73101 26899 80530 19470 07429 92571 26 35 9.73121 10.26879 9.80558 10.19442 10.07437 9.92563 25 36 73140 26860 80586 19414 07445 92555 24 37 73160 26840 80614 19386 07454 92546 23 88 73180 • 26820 80642 19358 07462 92538 22 39 73200 26800 80669 19331 07470 92530 21 40 9.73219 10.26781 9.80697 10.19303 10,07478 9.92622 20 41 73239 26761 80725 19275 07486 92614 19 42 73259 26741 80753 19247 07494 92506 18 43 73278 26722 80781 19219 07502 92498 17 44 7329S 26702 80808 19192 07510 92490 16 45 9.73318 10.26682 9.80836 10.19164 10.07518 9.92482 15 46 73337 26663 80864 19136 07527 92473 14 47 73357 26643 80892 19108 07535 92465 13 48 73377 26623 80919 19081 07543 92467 12 49 73396 26604 80947 19053 07551 92449 11 50 9.73416 10.26584 9.80975 10.19025 10.07559 9.92441 10 51 73435 26565 81003 18997 07567 92433 9 52 73455 26545 81030 18970 07575 92425 8 53 73474 26526 81058 18942 07584 92416 7 54 73494 26506 81086 18914 07592 92408 6 55 9.73513 10.26487 9.81113 10.18887 10.07600 9.92400 5 56 73533 26167 81141 18859 07608 92392 4 57 73652 26448 81169 18831 07616 92384 3 58 73572 26428 81196 18804 07624 92376 2 59 73591 26409 81224 18776 07633 92367 1 60 73611 26389 81252 18748 07641 92359 M. CosinG. Secant. Cotangent. Tangent. CoBecant. Sine. M. 122° 57° 110 LOGARITHMIC ANGULAR FUNCTIONS. Table 2. 3° Logarithms. 146° 9. Sine. Cosecant. Tangent. Cotangent, Secant. Cosine. M. 9.73611 10.26389 9.81252 10.18748 10.07641 9.92359 60 1 73630 26370 81279 18721 07649 92351 59 2 73650 26360 81307 18693 07657 92343 58 3 73069 26331 813.35 18665 07665 92335 57 4 73689 26311 81362 18638 07674 92326 56 5 9.73708 10.26292 9.81390 10.18610 10.07682 9.92:318 55 6 73727 26273 81418 ia5S2 07690 9-2310 54 7 73747 26253 81445 18556 07698 92302 53 8 73766 26234 81473 18627 07707 92293 52 9 73785 26215 81500 18500 07715 92285 51 10 9.73805 10.26196 9.81628 10.18472 10.07723 9.922/7 50 LI 73824 20176 81656 18444 07731 92269 49 L2 73843 26167 81683 18417 07740 922b0 48 13 73863 26137 81611 18389 07748 92262 47 14 73882 26118 81638 18362 07756 92244 46 15 9.73901 10.26099 9.81666 10.18334 10.07765 9.92236 45 16 73921 26079 81693 18307 07773 92227 44 17 73940 26060 81721 18279 07781 92219 43 18 73959 26041 S1748 18262 07789 92211 42 19 73978 26022 81776 18224 07798 92202 41 20 9.73997 10.26003 9.81803 10.18197 10.07806 9.92194 40 21 74017 25983 81831 18169 07814 92186 39 22 74036 25964 81868 18142 07823 92177 38 23 74055 25945 81886 18114 07831 92169 37 24 74074 25926 81913 18087 07839 92161 36 25 9.74093 10.25907 9.81941 10.18059 10,07848 9.92152 35 26 74113 25887 81968 18032 07856 92144 34 27 74132 25868 81996 18004 07864 92136 33 28 74161 25849 82023 17977 07873 92127 32 29 74170 25830 82051 17949 07881 92119 31 3D 9.74189 10.25811 9.82078 10.17922 10.07889 9.92111 30 31 74208 25792 82106 17894 07898 92102 29 32 74227 25773 82133 17867 07906 92094 28 33 74246 25754 82161 17839 07914 92086 27 34 74265 25735 82188 17812 07923 92077 26 35 9.74284 10.25716 9.82216 10.17785 10.07931 9.92069 25 36 74303 25697 82243 17757 07940 92060 24 37 74322 26678 82270 17730 07948 92052 23 38 74341 25659 82298 17702 07956 92044 22 39 74360 26640 82325 17676 07965 92035 21 10 9.74379 10.26621 9.82352 10.17648 10.07973 9.92027 20 41 74398 25602 82380 17620 07982 92018 19 12 74417 25583 82407 17593 07990 92010 18 13 74436 25564 82436 17566 07998 92002 17 14 74455 25645 82462 17538 08007 91993 16 15 9.74474 10.25526 9.82489 10.17511 10.08015 9.91985 15 16 74493 25607 82517 17483 08024 91976 14 17 74512 25488 82544 17456 08032 91968 13 18 74531 26469 82671 17429 08041 91969 12 19 74649 25451 82699 17401 08049 91951 11 50 9.74568 10.25432 9.82626 10.17374 10.08068 9.91942 10 51 74587 26413 82653 17347 08066 91934 9 52 74606 25394 82681 17319 08076 91925 8 53 74625 25375 82708 17292 08083 91917 7 54 74644 26356 82735 17266 08092 91908 6 55 9.74662 10.25338 9.82762 10.17238 10.08100 9.91900 6 56 74681 25319 82790 17210 08109 91891 4 57 74700 25300 82817 17183 08117 91883 3 58 74719 25281 82844 17156 08126 91874 2 59 74737 25263 82871 17129 08134 91866 1 60 74756 26244 82899 17101 08143 91857 M. Cosine. Secant. Cotan{?ent. Tangent. Cosecant. Sine. M: 23° Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 311 54° Logarithms. 145° M. Sine. Cosecant. Tangent. Cotangent. Secant, Cosine. M. 9.74756 10.25244 9.82899 10.17101 10.08143 9.91857 60 1 74775 25225 82926 17074 08151 91849 59 2 74794 25206 82953 17047 08160 91840 58 3 74812 25188 82980 17020 08168 91832 57 4 74831 25169 83008 16992 08177 91823 66 5 9.74850 10.25150 9.83035 10.16965 10.08185 9.91815 55 6 74868 25132 83062 16938 08194 91806 54 7 74887 25113 83089 16911 08202 91798 53 8 74906 25094 83117 16883 08211 91789 52 9 74924 25076 83144 16856 08219 91781 51 10 9.74943 10.25057 9.83171 10.16829 10.08228 9.91772 50 11 74961 25039 83198 16802 08237 91763 49 12 74980 25020 83225 16775 08245 91755 48 13 74999 25001 83252 16748 08254 91746 47 14 75017 24983 83280 16720 08262 91738 46 15 9.75036 10.24964 9.83307 10.16693 10.08271 9.91729 45 16 75054 24946 83334 16666 08280 91720 44 17 75073 24927 83361 16639 08288 91712 ■ 43 18 75091 24909 83388 16612 08297 91703 42 19 75110 24890 83415 16585 08305 91695 41 20 9.75128 10.24872 9.83442 10.16558 10.08314 9.91686 40 21 75147 24853 83470 16530 08323 91677 39 22 75165 24835 83497 16503 08331 91669 38 23 75184 21816 83524 16476 08340 91660 37 24 75202 24798 83551 16449 08349 91651 36 25 9.75221 10.24779 9.83578 10.16422 10.08357 9.91643 35 26 75239 24761 83605 16395 08366 91634 34 27 75258 24742 83632 16368 08375 91625 33 28 75276 21724 83659 16341 08383 91617 32 29 75294 24706 83686 16314 08392 91608 31 30 9.75313 10.24687 9.83713 10.16287 10.08401 9.91599 30 31 75331 24669 83740 16260 08409 91691 29 32 75350 24650 83768 16232 08418 91682 28 33 75368 24632 83795 16205 08427 91573 27 34 75386 24614 83822 16178 08435 91665 26 85 9.75405 10.21595 9.83849 10.16151 10.08444 9.91656 25 36 75423 24577 83876 16124 08453 91547 24 37 75441 24559 83903 16097 08462 91538 23 38 75459 24541 83930 16070 08470 91530 22 39 75178 24522 83957 16043 08479 91521 21 40 9.75496 10.24504 9.83984 10.16016 10.08488 9.91612 20 41 75514 24486 84011 15989 08496 91604 19 42 75533 24467 84038 15962 08505 91495 18 43 75551 24449 84065 15935 08514 91486 17 44 75569 24431 84092 15908 08523 91477 16 45 9.75587 10.21413 9.84119 10.15881 10.08531 9.91469 15 46 75605 24395 84146 15854 08540 91460 14 47 75624 24376 84173 15827 08549 91451 13 48 75642 24358 84200 15800 08558 91442 12 49 75660 24340 84227 15773 08567 91433 11 50 9.75678 10.24322 9.84254 10.15746 10.08575 9.91425 10 51 75696 24304 84280 15720 08584 91416 9 52 75714 24286 84307 15693 08593 91407 8 53 75733 24267 84334 15666 08602 91398 7 54 75751 24249 84361 15639 08611 91389 6 55 9.75769 10.24231 9.84388 10.15612 10.08619 9.91381 5 56 75787 24213 84415 15585 08628 91372 4 57 75805 24195 84442 15568 08637 91363 3 5S 75823 24177 84469 15531 08646 91354 2 59 76841 24159 84496 15504 ■08655 91345 1 60 75859 24141 84523 15477 08664 91336 M. Cosine. Secant. Cotangent Tangent. Cosecant. Sine. M. 124° J12 LOGARITHMIC ANGULAR FUNCTIONS. Table 2. (5° Logarithms. 144° M. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. M. 9.75859 10.24141 9.84523 10.15477 10.08664 9.91336 60 1 75877 24123 84650 15460 08672 91328 59 2 75895 21105 84576 15424 08681 91319 58 3 75913 24087 84603 15397 08690 91310 57 4 75931 24069 84630 15370 08699 91301 56 5 9.75949 10.24051 9.84657 10.15343 10.08708 9.91292 65 6 75967 24033 84684 15316 08717 91283 54 7 75985 24015 84711 15289 08726 91274 53, 8 76003 23997 84738 15262 08734 91266 52 9 76021 23979 84764 15236 08743 91257 51 10 9.76039 10.23961 9.84791 10.15209 10.08752 9.91248 50 11 76057 23943 84818 15182 08761 91239 49 n 76076. . 23925 84845 15155 08770 91230 48 13 76093 23907 84872 15128 08779 91221 47 14 76111 23889 84899 15101 08788 91212 46 15 9.76129 10.23871 9.8-1925 10.15075 10.08797 9.91203 .45 16 76146 23854 84952 15048 08806 91194 44 17 76164 23836 84979 16021 08815 91185 43 18 76182 23818 86006 14994 08824 91176 42 19 76200 23800 86033 14967 08833 91167 41 20 9.76218 10.23782 9.85059 10.14941 10.08842 9.91158 40 21 76236 23764 85086 14914 08851 91149 39 22 76253 23747 85113 14887 08859 91141 38 23 76271 23729 8.5140 14860 08868 91132 37 24 76289 23711 85166 14834 08877 911^^ 36 25 9.76307 10.23693 9.85193 10.14807 10.08886 9.91114 35 26 76324 23676 85220 14780 08895 91105 34 27 76342 23658 85247 14753 08904 91096 33 28 76360 23640 85273 14727 08913 91087 32 29 76378 23622 85300 14700 08922 91078 31 )0 9.76395 10.23606 9.86327 10.14673 10.08931 9.91069 30 Jl 76413 23587 85364 14646 08940 91060 29 !2 76431 2:3569 86380 14620 08949 91051 28 i3 76448 23652 86407 14693 08958 91042 27 !4 76466 23534 85434 14566 08967 91033 26 !5 9.76484 10.23516 9.85460 10.14540 10.08977 9.91023 25 56 76501 23499 85487 14513 08986 91014 24 57 76519 23481 86514 14486 08995 91005 23 !8 76537 23463 85540 14460 09004 90996 22 i9 76554 23446 85567 14433 09013 90987 21 10 9.76672 10.23428 9.85594 10.14406 10.09022 9.90978 20 11 76690 23410 86620 14380 09031 90969 19 12 76607 23393 85647 14.363 09040 90960 18 13 76625 23375 85674 14326 09049 90961 17 14 76642 23368 85700 14300 09058 90942 16 15 9.76660 10.23340 9.85727 10.14273 10.09067 9.90933 15 16 76677 23323 86754 14246 09076 90924 14 17 76695 23305 86780 14220 09085 90915 13 18 76712 23288 85807 14193 09094 90906 12 19 767.30 23270 85834 14166 09104 90896 11 lO 9.76747 10.23253 9.85860 10.14140 10.09113 9.90887 10 p1 76765 23235 85887 14113 09122 90878 9 i2 76782 23218 85913 14087 09131 90869 8 .3 76800 23200 85940 14060 09140 90860 7 .4 76817 23183 86967 14033 09149 90851 6 .5 9.76835 10.23165 9.85993 10.14007 10.09158 9.90842 5 i6 76852 23148 86020 13980 09168 90832 4 i7 76870 23130 86046 13954 09177 90823 3 iS 76887 23113 86073 13927 09186 90814 2 i9 76904 23096 86100 13900 09195 90805 1 iO 76922 23078 86126 13874 09204 90796 I. Cosine. Secjlnt. Cofcmgent. Tangent. Cosecant. Sine. M. 25° Table 2. LOGAEIJHMIC ANGULAR FUNCTIONS. 313 36° Logarithms. 143° M. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. M. 9.76922 10.23078 9.86126 10.13874 10.09204 9.90796 60 1 76939 23061 86153 13847 09213 90787 59 2 76957 23043 86179 13821 09223 90777 58 3 76974 23026 86206 13794 09232 90768 67 4 76991 23009 86232 13768 09241 90769 66 5 9.77009 10.22991 9.86259 10.13741 10.09250 9.90750 55 6 77026 22974 86285 13716 09269 90741 54 7 77043 22957 86312 13688 09269 90731 63 8 77061 22939 86338 13662 09278 90722 52 9 77078 22922 86365 13635 09287 90713 51 10 9.77095 10.22905 9.86392 10.13608 10.09296 9.90704 50 11 77112 22888 86418 13,582 09306 90694 49 12 77130 22870 86445 13555 09315 90685 48 13 77147 22853 86471 13529 09324 90676 47 14 77164 22836 86498 13502 09333 90667 46 16 9.77181 10.22819 9.86524 10.13476 10.09343 9.90667 45 16 77199 22801 86551 13449 09352 90648 44 17 77216 22784 86577 13423 09361 90639 43 18 77233 22767 86603 13397 09370 90630 42 19 77250 22750 86630 13370 09380 90620 41 20 9.77268 10.22732 9.86656 10.13344 10.09389 9.90611 40 21 77285 22715 86683 13317 09398 90602 39 22 77302 22698 86709 13291 09408 90592 38 23 77319 22681 86736 13264 09417 90683 37 24 77336 22664 86762 13238 09426 90574 36 25 9.77353 10.22647 9.86789 10.13211 10.09435 9.90665 35 26 77370 22630 86815 13185 09445 90556 34 27 77387 22613 86842 13168 09464 90546 33 28 77405 22595 86868 13132 09463 90537 32 29 77422 22578 86894 13106 09473 90627 31 30 9.77439 10.22561 9.86921 10.13079 10.09482 9.90618 30 31 77456 22544 86947 13053 09491 90509 29 32 77473 22527 86974 13026 09501 90499 28 33 77490 22510 87000 13000 09610 90490 27 34 77507 22493 87027 12973 09520 90480 26 35 9.77524 10.22476 9.87053 10.12947 10.09529 9.90471 25 36 77541 22459 87079 12921 09538 90462 24 37 77558 22442 87106 12894 09548 90452 23 38 77575 22425 87132 12868 09557 90443 22 39 77592 22408 87158 12842 09666 90434 21 40 9.77609 10.22391 9.87185 10.12815 10.09676 9.90424 20 41 77626 22374 87211 12789 09685 90415 19 42 77643 22357 87238 12762 09595 90405 18 43 77660 22340 87264 12736 09604 90396 17 44 77677 22323 87290 12710 09614 90386 16 45 9.77694 10.22306 9.87317 10.12683 10.09623 9.90377 15 46 77711 22289 87343 12657 09632 90368 14 47 77728 22272 87369 12631 09642 90358 13 48 77744 22-256 87396 12604 09651 90349 12 49 77761 22239 87422 12678 09661 90339 11 50 9.77778 10.22222 9.87448 10.12652 10.09670 9.90330 10 51 77795 22205 87475 12525 09680 90320 9 52 77812 22188 87501 12499 09689 90311 8 53 77829 22171 87527 12473 09699 90301 7 54 77846 22154 87564 12446 09708 90292 6 55 9.77862 10.22138 9.87680 10.12420 10.09718 9.90282 5 56 77879 22121 87606 12394 09727 90273 4 57 77896 22104- 87633 12367 09737 90263 3 58 77913 22087 87669 12341 09746 90254 2 59 77930 22070 87685 12315 09766 90244 1 60 77946 22054 87711 12289 09765 90235 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 126° 22 53° 514 LOGARITHMIC ANGULAR FUNCTIONS. Table 3. Logarithms. 142° M. Sine. Cosecant. Tangent. Cotangent, Secant. Cosine. M. 9.77946 10.22054 9.87711 10.12289 10.09765 9.90235 60 1 77963 22037 87738 12262 09776 902'2,5 59 2 77980 22020 87764 12236 09784 90216 58 3 77997 22003 87790 12210 09794 90200 57 4 78013 21987 87817 12183 09803 90197 66 6 9.78030 10.21970 9.87843 10.12167 10.09813 9.90187 55 6 78047 21953 87869 12131 09822 90178 54 7 78063 21937 87895 12106 09832 90168 53 8 78080 . 21920 87922 12078 09841 90169 52 9 78097 21903 87948 12052 09851 90149 61 10 9.78113 10.21887 9.87974 10.12026 10.09861 9.90139 50 11 78130 21870 88000 12000 09870 90130 49 12 78147 21853 88027 11973 09880 90120 48 13 78163 21837 88053 11947 09889 90111 47 14 78180 21820 88079 11921 09899 90101 46 15 9.78197 10.21803 9.88105 10.11896 10.09909 9.90091 45 16 78213 21787 88131 11869 09918 90082 44 17 78230 21770 88158 11842 09928 90072 43 18 78246 21764 88184 11816 09937 90063 42 19 7«263 21737 88210 11790 09947 90063 41 20 9.78280 10.21720 9.88236 10.11764 10.09957 9.90043 40 21 78296 21704 88262 11738 09966 90034 39 22 78313 21687 88289 11711 09976 90024 38 23 78329 21671 88315 11686 09986 90014 37 24 78346 21654 88341 11659 09995 90005 36 25 9.78362 10.21638 9.88367 10.11633 10.10005 9.89995 36 26 78379 21621 88393 11607 10015 89985 34 27 78395 21606 88420 11680 10024 89976 33 28 78412 21588 88446 11664 10034 89966 32 29 78428 21572 88172 11628 10044 89956 31 30 9.78445 10.21555 9.88498 10.11.602 10.10063 9.89947 30 31 78461 21539 88524 11476 10063 89937 29 32 78478 21522 88660 11460 10073 89927 28 33 78494 21506 88677 11423 10082 89918 27 34 78510 21490 88603 11397 10092 89908 26 35 9.78527 10.21473 9.88629 10.11371 10.10102 9.89898 26 36 78543 21457 88655 11346 10112 89888 24 37 78560 21440 88681 11319 10121 89879 23 38 78576 21424 88707 11293 10131 89869 22 39 78592 21408 88733 11267 10141 89859 21 40 9.78609 10.21391 9.88759 10.11241 10.10151 9.89849 20 11 78625 21375 88780 11214 10160 89840 19 12 78642 21358 88812 11188 10170 89830 18 13 78658 21342 88838 11162 10180 89820 17 14 78674 21326 88864 11136 10190 89810 16 16 9.78691 10.21309 9.88890 10-11110 10.10199 9.89801 16 16 78707 21293 88916 11084 10209 89791 14 17 78723 21277 88942 11058 10219 89781 13 18 78739 21261 88968 11032 10229 89771 12 19 78756 21244 88994 11006 10239 89761 11 )0 9.78772 10.21228 9.89020 10.10980 10.10248 9.89752 10 )1 78788 21212 89046 10954 10258 89742 9 )2 78805 21195 89073 10927 10268 89732 8 )3 78821 21179 89099 10901 10278 89722 7 )4 78837 21163 89125 10875 10288 89712 6 )5 9.78853 10.21147 9.891.51 10.10849 10.10298 9.89702 5 )6 78869 21131 89177 10823 10307 89693 4 i7 78886 21114 89203 10797 10317 89683 3 i8 78902 21098 89229 10771 ■ 10327 89673 2 .9 78918 21082 89255 10746 10337 89663 1 ;o 78934 21066 StTunt. 89281 10719 10347 89653 I. Cosine. Cotangent. Tangent. | Cosecant. Sine. M. 27° 52° Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 315 38° Logarithms. 141° M. Sine. CoBecaut. Tangent. Cotangent. Secant. Cosine. M. 9.78934 10.21066 9.89281 10.10719 10.10347 9.89653 60 1 78950 21050 89307 10693 10357 89643 59 2 78967 21033 89333 10667 10367 89633 58 3 78983 21017 89359 10641 10376 89624 57 4 78999 21001 89385 10615 10386 89614 56 5 9.79015 10.20985 9.89411 10.10589 10.10396 9.89604 56 6 79031 20969 89437 10563 10406 89594 54 7 79017 20953 89463 10537 10416 89584 53 8 79063 20937 89489 10511 10426 89574 52 9 79079 20921 89515 10485 10436 89564 51 10 9.79095 10.20905 9.89541 10.10459 10.10446 9.89554 50 U 79111 20889 89567 10433 10466 89544 49 12 79128 20872 89593 10407 10466 89534 48 13 79144 20856 89619 10381 10476 89624 47 14 79160 20840 89515 10365 10486 89614 46 15 9.79176 10.20824 9.89671 10.10329 10.10496 9.89504 45 16 79192 20808 89697 10303 10505 89495 44 17 79208 20792 89723 10277 10515 89486 43 18 79224 20776 89749 10251 105'2.5 89476 42 19 79240 20760 89775 10225 10536 89465 41 20 9.79256 10.20744 9.89801 10.10199 10.10546 9.89455 40 21 79272 20728 89827 10173 10655 89445 39 22 79288 20712 89853 10147 10665 89435 38 23 79304 20696 89879 10121 10575 89426 37 24 79319 20681 89905 10095 10585 89416 36 25 9.79335 10.20665 9.89931 10.10069 10.10595 9.89406 35 26 79351 20649 89957 10043 10606 89395 34 27 79367 20633 89983 10017 10616 89385 33 28 79383 20617 90009 09991 10625 89375 32 29 79399 20601 90035 09965 10636 89364 31 30 9.79415 10.20585 9.90061 10.09939 10.10646 9.89354 30 31 79431 20569 90086 09914 10656 89344 29 32 79447 20553 90112 09888 10666 89334 28 33 79463 20537 90138 09862 10676 89324 27 34 79478 20522 90164 09836 10686 89314 26 35 9.79494 10.20506 9.90190 10.09810 10.10696 9.89304 25 36 79510 20490 90216 09784 10706 89294 24 » 79526 20474 90242 09758 10716 89284 23 38 79542 20458 90268 09732 10726 89274 22 39 79558 20442 90294 09706 10736 89264 21 ■40 9.79573 10.20427 9.90320 10.09680 10.10746 9.89254 20 41 79589 20111 90346 09664 10756 89244 19 42 79605 20395 90371 09629 10767 89233 18 43 79621 20379 90397 09603 10777 89223 17 44 79636 20364 90423 09577 10787 89213 16 45 9.79652 10.20348 9.90449 10.09551 10.10797 9.89203 16 46 79668 20332 90475 09526 10807 89193 14 47 79684 20316 90501 09499 10817 89183 13 48 79699 20301 90527 09473 10827 89173 12 49 79715 20285 90553 09447 10838 89162 11 50 9.79731 10.20269 9.90578 10.09422 10.10848 9.89152 10 51 79746 20254 90604 09396 10858 89142 9 52 79762 20238 90630 09370 10868 89132 8 53 79778 20222 90656 09344 10878 89122 7 64 79793 20207 90682 09318 10888 89112 6 55 9.79809 10.20191 9.90708 10.09292 10.10899 9.89101 5 56 79825 20175 90784 09266 10909 89091 4 57 79840 20160 90759 09241 10919 89081 3 58 79856 20144 90785 09215 10929 89071 2 59 79872 20128 90811 09189 10940' 89060 1 60 79887 20113 90837 09163 10950 89060 M. CoBioe. Secant. Cotangent. Tangent, Cosecant. Sine. , M. 128° 51° ;16 LOGARITHMIC ANGULAR FUNCTIONS. Table 3. 9° Logar thms. 140° H. Sine. Cosecant. Tangent. Cotangent. Secant. 10.10950 Cosine. M. 9.79887 10.20113 9.90837 10.09163 9.89050 60 1 79903 20097 90863 09137 10960 89040 59 2 79918 20082 90889 09111 10970 89030 58 3 79934 20066 90914 09086 10980 89020 67 4 79950 20050 90940 09060 10991 89009 56 5 9.79965 10.20035 9.90966 10.09034 10.11001 9.88999 55 6 79981 20019 90992 09008 11011 88989 64 7 79995 20004 91018 08982 11022 88978 53 8 80012 19988 91043 08957 110.32 88968 52 9 80027 19973 91069 08931 11042 889.58 51 10 9.80043 10.19957 9.91095 10.08905 10.110.52 9.88948 .60 11 80058 19942 91121 08879 11063 88937 49 12 80074 19926 91147 08853 11073 88927 48 13 80089 19911 91172 08828 11083 88917 47 14 80105 19895 91198 08802 11094 88906 46 15 9.80120 10.19880 9.91224 10.08776 10.11104 9.88896 45 Ifi 80136 19864 91250 08750 11114 88886 44 17 80151 19849 91276 08724 11125 88875 43 18 80166 19834 91301 08699 11135 88865 42 19 80182 19818 91327 08673 11145 88855 41 20 9.80197 10.19803 9.91353 10.08647 10.111.56 9.88844 40 21 80213 19787 91.379 08621 11166 88834 39 22 80228 19772 91404 08696 11176 88824 38 23 80244 19756 91430 08570 11187 88813 37 24 80259 19741 91456 08.544 11197 88803 36 25 9.80274 10.19726 9.91482 10.08518 10.11207 9.88793 35 26 80290 19710 91507 08493 11218 88782 34 27 80305 19695 91533 08467 11228 88772 33 28 80320 19680 91559 08441 11239 88761 32 29 80336 19664 91585 08416 11249 88761 31 30 9.80351 10.19649 9.91610 10.08390 10.11259 9.88741 30 31 80366 19634 91636 08364 11270 88730 29 32 80382 19618 91662 08338 11280 88720 28 33 80397 19603 91688 08312 11291 88709 27 34 80412 19588 91713 08287 11301 88699 26 35 9.80428 10.19572 9.91739 10.08261 10.11312 9.88688 25 36 80443 19.157 91765 08235 11322 88678 24 37 80458 19.V12 91791 08209 11332 88668 23 38 80473 19527 91816 08184 11343 88657 22 39 80489 19511 91842 08158 11353 88647 21 40 9.80504 10.19496 9.91868 10.08132 10.11364 9.88636 20 41 80519 19481 91893 08107 11374 88626 19 42 80534 19466 91919 08081 11385 88616 18 43 80550 19450 91945 08056 11395 88605 17 44 80565 19435 91971 08029 11406 88594 16 45 9.80580 10.19420 9.91996 10.08004 10.11416 9.88584 15 46 80595 19405 92022 07978 11427 88573 14 47 80610 19390 92048 07952 11437 88563 13 48 80626 19375 92073 07927 11448 88552 12 49 80641 19359 92099 07901 114.58 88542 11 50 9.80656 10.19344 9.92126 10.07875 10.11469 9.88531 10 61 80671 19329 92150 07850 11479 88521 9 52 80686 19314 92176 07824 11490 88510 8 53 80701 19299 92202 07798 11501 88499 7 54 80716 19284 92227 07773 11.511 88489 6 55 9.80731 10.19269 9.92253 10.07747 10.11522 9.88478 5 56 80746 19254 92279 07721 11532 88468 4 57 80762 19238 92304 07696 11543 88467 3 58 80777 19223 92330 07670 11553 88447 2 59 80792 19208 92356 07644 11564 &84a6 1 60 80807 19193 92381 07619 11575 88426 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 29° 50° Table 2. LOGARITHMIC ANGULAE FUNCTIONS. 317 40° Logarithms. 39° M. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. M. 9.80807 10.19193 9.92381 10.07619 10.11575 9.88425 60 1 80822 19178 92407 07593 11585 88415 59 2 80837 19163 92433 07667 11596 88404 58 3 80a62 19148 92158 07542 11606 88394 57 4 80867 19133 92484 07516 11617 88383 66 5 9.80882 10.19118 9.92510 10.07490 10.11628 9.88372 66 6 80897 19103 92635 07466 11638 88362 54 7 80912 19088 92561 07439 11649 88361 53 8 80927 19073 92687 07413 11660 88340 62 9 80942 19058 92612 07388 11670 88330 51 10 9.80957 10.19043 9.92638 10.07362 10.11681 9.88319 50 11 80972 19028 92663 07337 11692 88308 49 12 80987 19013 92689 07311 11702 88298 48 13 81002 18998 92716 07285 11713 88287 47 14 81017 18983 92740 07260 11724 88276 46 15 9.81032 10.18968 9.92766 10.07234 10.11734 9.88266 45 16 81047 18953 92792 07208 11745 88266 44 17 81061 18939 92817 07183 11766 88244 43 18 81076 18924 92843 07167 11766 88234 42 19 81091 18909 92868 07132 11777 88223 41 20 9.81106 10.18894 9.92894 10.07106 10.11788 9.88212 40 21 81121 18879 92920 07080 11799 88201 39 22 81136 18864 92945 07065 11809 88191 38 23 81161 18849 92971 07029 11820 88180 37 24 81166 18834 92996 07004 11831 88169 36 25 9.81180 10.18820 9.93022 10.06978 10.11842 9.88158 36 26 81195 18805 98048 06952 11852 88148 34 27 81210 18790 93073 06927 11863 88137 33 28 81226 18775 93099 06901 11874 88126 32 29 81240 18760 93124 06876 11886 88115 31 30 9.81254 10.18746 9.93160 10.06850 10.11896 9.88106 30 31 81269 18731 93175 06825 11906 88094 29 32 81284 18716 93201 06799 11917 88083 28 33 81299 18701 93227 06773 11928 88072 27 34 81314 . 18686 93262 06748 11939 88061 26 35 9.81328 10.18672 9.93278 10.06722 10.11949 9.88061 26 36 81343 18657 93303 06697 11960 88040 24 37 81358 18642 93329 06671 11971 88029 23 38 81372 18628 93354 06646 11982 88018 22 39 81387 18613 93380 06620 11993 88007 21 40 9.81402 10.18598 9.93406 10.06594 10.12004 9.87996 20 41 81417 18583 93481 06569 12015 87986 19 42 81431 18569 93467 06543 12025 87976 18 43 81446 18554 93482 06618 12036 87964 17 44 81461 18639 93508 06492 12047 87963 16 46 9.81475 10.18526 9.93633 10.06467 10.12068 9.87942 16 46 81490 18610 93559 06441 12069 87931 14 47 81506 18495 93584 06416 12080 87920 13 48 81519 18481 93610 06390 12091 87909 12 49 81534 18466 93636 06364 12102 87898 11 50 9.81549 10.18451 9.93661 10.06339 10.12113 9.87887 10 51 81563 18437 93687 06313 12123 87877 9 52 81578 18422 93712 06288 12134 87866 8 53 81692 18408 93738 06262 12145 87866 7 54 81607 18393 93763 06237 12166 87844 6 55 9.81622 10.18378 9.93789 10.06211 10.12167 9.87833 6 66 81636 18364 93814 06186 12178 87822 4 57 81661 18349 93840 06160 12189 87811 3 58 81665 18335 93865 06135 12200 87800 2 59 81680 18320 93891 06109 12211 87789 1 60 81694 18306 93916 06084 12222 87778 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 130° 49° !18 LOGARITHMIC ANGULAR FUNCTIONS. Table 2. i° Logarithms. 38° M. Sine. Cosecant. Tangent. Cotangent. ■Secant, Cosine, M. 9.81694 10.18306 9.93916 10.06084 10,12222 9,87778 60 1 81709 18291 93942 06058 12233 87767 59 2 81723 18277 93967 06033 12244 87766 58 3 81738 18262 93993 06007 12255 87746 S' 4 81752 18248 94018 05982 12266 87734 56 5 9.81767 10.18233 9.94044 10.0,5966 10,12277 9,87723 65 6 81781 18219 94069 05931 12288 87712 54 -7 S1796 18204 94095 05906 12299 87701 53 8 81810 18190 94120 06880 12310 87690 52 9 8182.5 18175 94146 osse-i 12321 87679 51 10 9.81839 10.18161 9.94171 10.06829 10,12332 9.87608 50 11 81854 18146 94197 05803 12343 87657 49 12 81868 18132 94222 05778 12354 87640 48 13 81882 18118 94248 05752 12365 87635 47 14 81897 18103 94273 06727 12376 87624 46 15 9.81911 10.18089 9.94299 10,05701 10,12387 9,87613 45 16 81926 18074 94324 05676 12399 87601 44 17 81940 V 18060 94350 06650 12-110 87590 43 18 81955 18045 94375 0,5625 12421 87579 42 19 81969 18031 94401 05699 12432 87668 41 20 9.81983 10.18017 9.94426 10,0,5574 10,12443 9,87667 40 21 81998 18002 94462 05.548 124.54 87546 39 22 82012 17988 94477 05523 12465 87535 38 23 82026 17974 94503 05497 12476 87624 37 24 82041 17959 94.528 0.5472 12487 87513 36 25 9.82055 10.17945 9.94,564 10,05446 10.12499 9,87501 35 26 82069 17931 94579 05421 12.510 87490 34 27 82084 17916 94604 05396 12521 87479 33 28 82098 17902 94630 05370 12532 87468 32 29 82112 17888 94G55 05345 12513 874,57 31 30 9,82126 10.17874 9.94681 10,05319 10.12.554 9,87446 30 SI 82141 17859 94706 05294 125ri6 87434 29 32 82155 17845 94732 0.5268 12577 87423 28 33 82169 17831 94757 05'243 12588 87412 27 34 S21S1 17816 94783 0,5217 12599 87401 26 36 9.K219X 10.17802 9.94808 10,05192 10,12610 9.87390 25 36 .S2212 17788 94834 05166 12622 87378 24 37 82226 17774 94859 05141 121 ;33 87367 23 38 82240 17760 94884 06116 12644 87356 22 39 82255 17745 94910 05090 126.55 87346 21 10 9.82269 10.17731 9.94935 10,06065 10,12666 9.87334 20 41 82283 17717 94961 0,5039 12678 87322 19 42 82297 17703 94986 05014 12689 87311 18 43 82311 17689 95012 04988 12700 87300 17 44 82326 17674 95037 04963 12712 87288 16 45 9.8'2340 10.17660 9.95062 10,04938 10,12723 9-87277 15 le 82354 17646 96088 04912 12734 87266 14 17 82368 17632 95113 04887 12745 87255 13 18 82382 17618 95139 04861 12757 87243 12 19 82396 17604 95164 04836 12768 87232 11 50 9.82410 10.17690 9,95190 10,04810 10,12779 9,87221 10 51 82424 17576 95216 04786 12791 87209 9 52 8'2439 17561 95240 04760 12802 87198 8 53 82453 17647 95266 04734 12813 87187 7 54 82467 17533 95291 04709 12825 87175 6 55 9.82481 10.17519 9.95317 10,04683 10,12836 9,87164 5 56 82495 17505 9.5342 04658 12847 87153 4 57 8'2509 17491 95368 04632 12859 87141 3 58 82523 17477 95393 04607 12870' 87130 2 59 82637 17463 95418 04.582 12881 87119 1 30 82651 17449 95444 04556 12893 87107 H. Coaine. Secant. Cotangent. Tangent, Cosecant, Sine, M, 31° 48° ile 2. LOGAEITHMIC ANGULAR FUNCTIONS. 319 Logarithms. 137° Sine. Cosecant. Tangent. Cotangent, Secant. Cosine. M, 9.82551 10.17449 9.95444 10.04556 10.12893 9.87107 60 82565 17435 95469 04531 12904 87096 89 82579 17421 95495 04505 12915 87085 58 82593 17407 95520 04480 12927 87073 57 82607 17393 95545 04455 12938 87062 56 9.82621 10.17379 9.95571 10.04429 10.12950 9 87050 55 82635 17365 95596 04404 12961 87039 54 82649 17351 95622 04378 12972 87028 53 82663 17337 95647 04353 12984 87016 52 82677 17323 95672 04328 12995 87005 51 9.82691 10.17309 9.95698 10.04302 10.13007 9.86993 50 82705 17295 95723 04277 13018 86982 49 82719 17281 95748 04252 13030 86970 48 82733 17267 95774 04226 13041 86959 47 82747 17253 95799 04201 13053 86947 46 9.82761 10.17239 9.95826 10.04175 10.13064 9.86936 45 82775 17225 95850 04150 13076 86924 44 82788 17212 95875 04125 13087 86913 43 82802 17198 95901 01099 13098 86902 42 82816 17184 95926 04074 13110 86890 41 9.82830 10.17170 9.95952 10.04048 10.13121 9.86879 40 82844 17166 95977 04023 13133 86867 39 82858 17142 96002 03998 13145 86855 38 82872 17128 96028 03972 13156 86844 37 82885 17115 96053 03947 13168 86832 36 9.82899 10.17101 9.96078 10.03922 10.13179 9.86821 35 82913 17087 96104 03896 13191 86809 34 82927 17073 96129 03871 13202 86798 S3 82941 17059 96155 03845 13214 86786 32 82955 17045 96180 03820 13225 86775 31 9.82968 10.17032 9.96205 10.03795 10.13237 9.86763 30 82982 17018 96231 03769 13248 86752 29 82996 17004 96256 03744 13260 86740 28 83010 16990 96281 03719 13272 86728 27 83023 16977 96307 03693 13283 86717 26 9.83037 10.16963 9.96332 10.03668 10.13295 9.86705 25 83051 16949 96357 03643 13306 86694 24 83065 16935 96383 03617 13318 86682 23 83078 16922 96408 03592 13330 86670 22 83092 16908 96433 03567 13341 86669 21 9.83106 10.16894 9.96459 10.03541 10.13353 9.86647 20 83120 16880 96484 03516 13365 86635 19 83133 16867 96510 03490 13376 86624 18 83147 16853 96535 03465 13388 86612 -, 17 83161 16839 96560 03440 13400 86600 16 9.83174 10.16826 9.96586 10.08414 10.13411 9.86589 15 83188 16812 96611 03389 13423 86577 14 83202 16798 96636 03364 13435 86665 13 83215 16785 96662 03338 13446 86554 12 83229 16771 96687 03313 13458 86542 11 9.83242 10.16758 9.96712 10.03288 10.13470 9.86530 10 83256 16744 98738 03262 13482 86518 9 83270 16730 96763 03237 13493 86507 8 83283 16717 96788 03212 13505 86495 7 83297 16703 96814 03186 13517 86483 6 9.83310 10.16690 9.96839 10.03161 10.13528 9.86472 5 83324 16676 96864 031.36 13.540 86460 4 83338 16662 96890 03110 13552 86448 3 83351 16649 96915 03085 13564 86436 2 83365 16635 96940 03060 13575 86425 1 83378 16622 96966 03034 13587 86413 Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 47° 320 LOGARITHMIC ANGULAR FUNCTIONS. Table 3. 43° Logarithms. 1 36° M. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. M. 9.83378 10.16622 9.96966 10.03031 10.13587 9.86413 60 1 83392 16608 96991 03009 13599 86401 59 2 83405 16595 97016 02981 13611 86389 58 3 83419 16581 97012 02958 13623 86377 57 4 83432 16568 97067 02933 13634 86366 50 5 9.83446 10.16551 9.97092 10.02908 10.13646 9.86354 .56 6 83459 16511 97118 02882 13658 86342 54 7 83173 16527 97143 02857 13670 86330 53 8 83486 16511 97168 02832 13682 86318 52 9 83500 16500 97193 02807 13694 85306 51 10 9.83513 10.16187 9.97219 10.02781 10.13705 9.86295 50 11 83527 16173 97211 02756 13717 86283 49 12 83510 16160 97269 02731 13729 86271 48 13 83554 16116 97295 02705 13741 86259 47 U 83567 16133 97320 02680 13753 86247 16 15 9.83581 10.16119 9.97315 10.02655 10.13765 9.86235 45 16 83591 16406 97371 02629 13777 86223 14 17 83608 16392 97896 02601 13789 86211 43 18 83621 16379 97121 02679 13800 86200 42 19 83634 16366 97117 02553 13812 86188 41 20 9.83648 10.16352 9.97172 10.02528 10.13824 9.86176 40 21 83661 16339 97197 02503 13836 86164 39 22 83674 16320 97523 02177 13848 86162 38 23 83688 16312 97518 02452 13860 86140 37 24 83701 16299 97573 02127 13872 86128 36 25 9.83715 10.16285 9.97598 10.02102 10.13884 9.86116 35 26 83728 16272 97621 02376 13896 86101 34 27 83741 16259 97649 02351 13908 86092 33 28 83755 16245 97674 02326 13920 86080 32 29 83768 16232 97700 02300 13932 86068 31 30 9.83781 10.16219 9.97725 10.02275 10.13944 9.86056 30 31 83795 16205 97750 02250 13966 86044 29 32 83808 16192 97776 02221 13968 86032 28 33 83821 16179 97801 02199 13980 86020 27 34 83831 16166 97826 02171 13992 86008 26 35 9.83848 10.16152 9.97851 10.02119 10.14004 9.86996 25 36 83861 16139 97877 02123 14016 85984 24 37 83874 16126 97902 02098 14028 85972 23 38 83887 16113 97927 02073 14040 86960 22 39 83901 16099 97953 02047 11052 85948 21 40 9.83914 10.16086 9.97978 10.02022 10.14064 9.85936 20 41 83927 16073 98003 01997 14076 85924 19 42 83940 16060 98029 01971 14088 85912 18 43 83951 16046 98054 01946 14100 85900 17 44 83967 16033 98079 01921 14112 85888 16 45 9.83980 10.16020 9.98101 10.01896 10.14124 9.85876 16 46 83993 16007 98130 01870 11136 85864 14 47 84006 15994 98155 01815 11119 85851 13 48 84020 15980 98180 01820 11161 85839 12 49 84033 15967 98206 01794 11173 85827 11 50 9.84046 10.15954 9.98231 10.01769 10.11185 9.85815 10 51 84059 15911 98256 01744 11197 85803 9 52 84072 15928 98281 01719 11209 85791 8 53 84086 15915 98307 01693 11221 85779 7 54 81098 15902 98332 01668 14234 85766 6 55 9.81112 10.15888 9.98357 10.01643 10.14246 9.85754 5 56 81125 15875 98383 01617 14268 85742 4 57 81138 15862 98108 01592 14270 86730 3 58 81151 15819 98133 01567 14'282 85718 2 59 81164 15836 9S4.')S 01542 14294 85706 1 60 81177 15823 984 Si Cotan^^ent. 01516 14307 85693 M. CoBine. Secant. Tangent. Cosecant. Sine. M. Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 321 44° Logarithms. 135° M. ■ Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. M. 9.84177 10.15823 9.98484 10.01616 10.14307 9.85693 60 1 81190 35810 98609 01491 14319 85681 59 2 84203 15797 98534 01466 14331 85669 58 3 84216 1,5784 98560 01440 14343 85667 67 4 84229 15771 98585 01415 14356 85645 56 5 9.84242 10.15758 9.98610 10.01390 10.14368 9.85632 56 6 84255 15745 98635 01365 14380 85620 54 7 84269 15731 98661 01339 t 14392 8.5608 53 8 84282 15718 98686 01314 14404 85696 62 9 84295 15705 98711 01289 14417 85583 61 10 9.84308 10.15692 9.98737 10.01263 10.14429 9.85571 50 11 84321 15679 98762 01238 14441 86559 49 12 84334 15666 98787 01213 14463 86547 48 13 84347 15653 98812 01188 14466 85534 47 14 84360 15640 98838 01162 14478 86522 46 15 9.84373 10.15627 9.98863 10.01137 10.14490 9.85510 46 16 84385 15615 98888 01112 14503 85497 44 17 84398 15602 9S913 01087 14515 86485 43 18 84411 15589 98939 01061 14527 85473 42 19 84424 15576 98964 01036 145-10 85460 41 20 9.84437 10.15563 9.98989 10.01011 10.14652 9.85448 40 21 84450 15550 99016 00985 14564 85436 39 22 84463 16537 99010 00960 14577 85423 38 23 84476 16524 99065 00935 14589 85411 37 24 84489 16511 99090 00910 14601 85399 36 25 9.84502 10.15498 9.99116 10.00884 10.14614 9.85386 35 26 84515 15485 99141 00859 14626 85374 34 27 84528 15472 99166 00834 14639 85361 33 28 84540 16460 99191 00809 14651 85349 32 29 84553 16447 99217 00783 14663 86337 31 3D 9.84566 10.16434 9.99'242 10.00768 10.14676 9.86324 30 31 &lo79 15421 99267 00733 14688 86312 29 32 84592 15408 99293 00707 14701 85299 28 33 84605 15395 99318 00682 14713 86287 27 34 84618 15382 99343 00667 14726 85274 26 35 9.84630 10.15370 9.99368 10.00632 10.14738 9.85262 26 36 84643 15357 99394 00606 14750 85250 24 37 84656 15344 99419 00581 14763 85237 23 38 84669 15331 99444 00656 14776 85225 22 39 84682 15318 99469 00531 14788 86212 21 40 9.84694 10.15306 9.99495 10.00505 10.14800 9.86200 20 41 84707 16293 99620 00480 14813 85187 19 42 84720 16280 99545 00155 14825 85175 18 43 84733 15267 99570 00430 14838 86162 17 44 84745 1.5255 99696 00404 14850 86150 -16 45 9.84758 10.15242 9.99621 10.00379 10.14863 9.86137 16 46 84771 16229 99646 00354 14875 86125 14 47 84784 16216 99672 00328 14888 85112 13 48 84796 15204 99697 00303 14900 86100 12 49 84809 1.5191 99722 00278 14913 85087 11 50 9.84822 10.15178 9.99747 10.00263 10.14926 9.85074 10 61 84835 15165 99773 00227 14938 85062 9 52 84847 15153 99798 00202 14951 85049 8 53 84860 15140 99823 00177 14963 86037 7 54 84873 16127 99848 00152 14976 86024 6 55 9.84885 10.15116 9.99874 10.00126 10.14988 9.85012 5 56 84898 15102 99899 00101 15001 84999 4 57 84911 15089 99924 00076 16014 84986 3 68 84923 15077 99949 00051 16026 84974 2 59 84936 15064 99975 00025 15039 84961 1 60 84949 16051 10.00000 00000 15051 84949 M. Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. M. 134° 122 NATURAL FUNCTIONS. Table 3. Natural Trigonometrical Functions. 179° rl. Sine. Vrs. cos. Cosec'nt Tang. Co tang. Secant. Vrs. ein. Cosine, M. .00000 1.0000 Infinite, .00000 Infinite. 1.0000 .00000 1,0000 60 1 . 0029 .99971 3437,7 . 0029 3437,7 .0000 . 0000 ,0000 59 2 . 0058 . 9942 1718.9 . 0058 1718.9 .0000 . 0000 .0000 58 3 . 0087 . 9913 1145.9 . 0087 1146.9 .0000 . 0000 .0000 57 4 . 0116 . 9884 859.44 . 0116 869,44 .0000 . 0000 .0000 66 5 .00145 .99854 687.55 .00145 687.55 1,0000 .00000 1.0000 55 6 . 0174 . 9826 572.96 . 0174 572.96 .0000 . 0000 .0000 54 7 . 0204 . 9796 491.11 . 0204 491.11 .0000 . 0000 .0000 53 8 . 0233 . 9767 429.72 . 0233 429.72 .0000 . 0000 .0000 62 9 . 0262 . 9738 381.97 . 0262 381.97 .0000 . 0000 .0000 51 .00291 .99709 313.77 .00291 343.77 1.0000 .00000 .99999 50 1 . 0320 . 9680 312.52 . 0320 312.52 .0000 . 0000 . 9999 49 2 . 0349 . 9651 286,48 . 0349 286.48 .0000 . 0001 . 9999 48 3 . 0378 . 9622 64,14 . 0378 64.44 .0000 . 0001 . 9999 47 4 . 0407 . 9593 45.55 . 0107 45.65 .0000 . 0001 . 9999 46 5 .00436 .99564 229.18 .00436 229,18 1,0000 .00001 .99999 45 6 . 0465 . 9534 14.86 . 0465 11,86 .0000 . 0001 . 9999 44 7 . 0194 . 9505 02.22 . 0494 02,22 .0000 . 0001 . 9999 43 8 . 0524 . 9476 190.99 . 0524 190.98 .0000 . 0001 . 9999 42 9 . 0553 . 9447 80.93 . 0553 80.93 .0000 . 0001 . 9998 41 .00582 .99418 171.89 .00582 171.88 1.0000 .00002 .99998 40 1 . OGll . 9389 63.70 . 0611 63.70 .0000 . 0002 . 9998 39 2 . 0640 . 9360 56.26 . 0640 56.26 .0000 . 0002 . 9998 38 3 . 0669 . 9331 49.47 . 0669 49.46 .0000 . 0002 . 9998 37 i . 0098 . 9302 43.24 . 0698 43.24 .0000 . 0002 . 9997 36 5 .00727 .99273 137.51 .00727 137,51 1.0000 .00003 .99997 36 6 . 0756 . 9244 32.22 . 0756 32,22 .0000 . 0003 . 9997 34 7 . 0785 . 9215 27.32 . 0785 27,32 .0000 . 0003 . 9997 33 8 . 0814 . 9185 22.78 . 0814 22.77 .0000 . 0003 . 9997 32 9 . 0843 . 9156 18.54 . 0844 18.64 .0000 . 0003 . 9996 31 .00873 .99127 114,59 .00873 114.59 1.0000 .00004 .99996 30 1 . 0902 . 9098 10,90 . 0902 10,89 .0000 . 0004 . 9996 29 2 . 0931 . 9069 07,43 . 0931 07,43 .0000 . 0004 . 9996 28 3 . 0960 . 9040 04.17 . 0960 04,17 .0000 . 0005 . 9995 27 ■1 . 0989 . 9011 01,11 . 0989 01.11 .0000 . 0005 . 9995 26 5 .01018 .98982 98.223 .01018 98.218 1.0000 .00005 .99995 25 6 . 1047 . 8953 5.495 . 1047 6.489 .0000 . 0005 . 9994 24 7 . 1076 . 8924 2.914 . 1076 2.908 .0000 . 0006 . 9994 23 8 . 1105 . 8895 0.469 . 1105 0.463 .0001 . 0006 . 9994 22 9 . 1134 . 8865 88,149 . 1134 88.143 .0001 . 0006 . 9993 21 .01163 .98836 85.946 .01164 85.940 1,0001 .00007 .99993 20 1 . 1193 . 8807 3.849 . 1193 3.843 .0001 . 0007 . 9993 19 2 . 1222 . 8778 1.853 . 1222 1.847 .0001 . 0007 . 9992 18 3 . 1251 . 8749 79.950 . 1261 79.943 .0001 . 0008 . 9992 17 i . 1280 . 8720 8.133 . 1280 8.126 .0001 . 0008 . 9992 16 5 .01309 .98691 76.396 .01309 76.390 1.0001 .00008 .99991 15 6 . 1338 . 8662 4.736 . 1338 4.729 .0001 . 0009 . 9991 14 7 . 1367 . 8633 3.146 . 1367 3,139 .0001 . 0009 . 9991 13 8 . 1396 . sr,04 1.622 . 1396 1.615 .0001 . 0010 . 9990 12 9 . 1425 . S575 0.160 . 1125 0.153 .0001 . 0010 . 9990 11 .01454 .9X546 68.757 .01454 68.750 1.0001 .00010 .99989 10 1 . 1183 . 8.)16 7.409 . 1184 7,102 .0001 . 0011 . 9989 9 2 . 1512 . S4N7 6.113 . 1513 6,105 .0001 . 0011 . 9988 8 3 . 1512 . 8458 4,866 . 1542 4,858 .0001 . 0012 . 9988 7 i . 1571 . 8429 3.664 . 1571 3,657 .0001 . 0012 . 9988 6 5 .01600 .98400 62.507 .01600 62,499 1.0001 .00013 .99987 5 6 . 1629 . 8371 1,891 . 1629 1,383 .0001 . 0013 . 9987 4 7 . 1658 . 8342 0,314 . 1668 0,306 .0001 . 0014 . 9987 3 3 . 1687 . 8313 59.274 . 1687 59,266 .0001 . 0014 . 9986 2 9 . 1716 . 8284 8.270 . 1716 8,261 .0001 . 0015 . 9985 1 . 1745 . 8265 7.299 . 1745 7,290 .0001 . 0015 . 9985 I. Cosine. Vrs. sin. Secant. Cotang. Tang, Cosec'nt Vrs. COS. Sine. M. Table 3. NATURAL FUNCTIONS. 323 1° Natural Trigonometrical Functions. 178° M. Sine. Vrs. COS. Cosec'nt Tang. Cotang. Secant, Vra. sin. Cosine. M. .01745 .98255 57.299 .01745 57.290 1.0001 .00015 .99985 60 1 . 1774 . 8226 6.359 . 1775 6.350 .0001 . 0016 . 9984 59 2 . 1803 . 8196 5.450 . 1804 5.441 .0001 . 0016 . 9984 58 3 . 1832 . 8167 4.570 . 1833 4.561 .0002 . 0017 . 9983 67 i . 1861 . 8138 3.718 . 1862 3.708 .0002 . 0017 . 9983 56 5 .01891 .98109 52.891 .01891 52.882 1.0002 .00018 .99982 55 6 . 1920 . 8080 2.090 . 1920 2.081 .0002 . 0018 . 9981 54 7 . 1949 . 8051 1.313 . 1949 1.303 .0002 . 0019 . 9981 63 8 . 1978 . 8022 0.568 . 1978 0.548 .0002 . 0019 . 9980 62 9 . 2007 . 7993 49.826 . 2007 49.816 .0002 . 0020 . 9980 51 10 .02036 .97964 49.114 .02036 49.104 1.0002 .00021 .99979 60 11 . 2065 . 7935 8.422 . 2066 8.412 .0002 . 0021 . 9979 49 12 . 2094 . 7906 .7.750 . 2095 7.739 .0002 . 0022 . 9978 48 13 . 2123 . 7877 7.096 . 2124 7.085 .0002 . 0022 . 9977 47 14 . 2152 . 7847 6.460 . 2163 6.449 .0002 , .,0023 . 9977 46 15 .02181 .97818 46.840 .02182 45.829 1.0002 .00024 .99976 45 16 . 2210 . 7789 5.237 . 2211 5.226 .0002 . 0024 . 9975 44 17 . 2240 . 7760 4.650 . 2240 4.638 .0002 . 0026 . 9975 43 18 . 2269 . 7731 4.077 . 2269 4.066 .0002 . 0026 . 9974 42 19 . 2298 . 7702 3.520 . 2298 3.608 .0003 . 0026 . 9974 41 20 .02327 .97673 42.976 .02327 42.964 1.0003 .00027 .99973 40 21 . 2356 . 7644 2.445 . 2367 2.433 .0003 . 0028 . 9972 39 22 . 2385 . 7615 1.928 . 2386 1.916 .0003 . 0028 . 9971 38 23 . 2414 . 7586 1.423 . 2415 1.410 .0003 . 0029 . 9971 .37 24 . 2443 . 7557 0.930 . 2444 0.917 .0003 . 0030 . 9970 36 25 .02472 .97528 40.448 .02473 40.436 1.0003 .00030 .99969 35 26 . 2501 . 7499 39.978 . 2502 39.966 .0003 . 0031 . 9969 34 27 . 2530 . 7469 9.518 . 2531 9.506 .0003 . 0032 . 9968 33 28 . 2559 . 7440 9.069 . 2560 9.057 .0003 . 0033 . 9967 32 29 . 2589 . 7411 8.631 . 2589 8.618 .0003 . 0033 . 9966 31 30 .02618 .97382 38.201 .02618 38.188 1.0003 .00034 .99966 30 31 . 2647 . 7353 7.782 . 2648 7.769 .0003 . 0036 .. 9966 29 32 . 2676 . 7324 7.371 . 2677 7.358 .0003 . 0036 . 9964 28 33 . 2705 . 7295 6.969 . 2706 6.966 .0004 . 0036 . 9963 27 34 . 2734 . 7266 6.676 . 2736 6.663 .0004 . 0037 9963 26 35 .02763 .97237 36.191 .02764 36.177 1.0004 .00038 .99962 25 36 . 2792 . 7208 5.814 . 2793 5.800 .0004 . 0039 . 9961 24 37 . 2821 . 7179 5.445 . 2822 5.431 .0004 . 0040 9960 23 38 . 2850 . 7150 5.084 . 2851 5.069 .0004 . 0041 . 9959 22 39 . 2879 . 7121 4.729 . 2880 4.715 .0004 . 0041 . 9958 21 40 .02908 .97091 34.382 .02910 34.368 1.0004 .00042 .99958 20 41 . 2937 . 7062 4.042 . 2939 4.027 .0004 . 0043 . 9957 19 42 . 2967 . 7033 3.708 . 2968 3.693 .0004 . 0044 . 9966 18 43 . 2996 . 7004 3.381 . 2997 3.366 .0004 . 0046 . 9955 17 44 . 3025 . 6975 3.060 . 3026 3.046 .0004 . 0046 . 9954 16 45 .03054 .96946 32.746 .03055 32.730 1.0005 .00046 .99963 15 46 . 3083 . 6917 2.437 . 3084 2.421 .0005 . 0047 . 9962 14 47 . 3112 . 6888 2.134 . 3113 2.118 .0005 . 0048 . 9961 13 48 . 3141 . 6869 1.836 . 3143 1.820 .0005 . 0049 . 9951 12 49 . 3170 . 6830 1.544 . 3172 1.528 .0005 . 0050 . 9950 11 50 .03199 .96801 31.267 .03201 31.241 1.0005 .00051 .99949 10 61 . 3228 . 6772 • 0.976 . 3230 0.960 .0005 . 0052 . 9948 9 52 . 3267 . 6743 0.699 . 3259 0.683 .0005 . 0053 . 9947 8 53 . 3286 . 6713 0.428 . 3288 0.411 .0005 . 0054 . 9946 7 54 . 3315 . 6684 0.161 . 3317 0.145 .0005 . 0065 . 9945 6 55 .03344 .96665 29.899 .03346 29.882 1.0005 .00056 .99944 5 56 . 3374 . 6626 9.641 . 3375 9.624 .0006 . 0057 . 9943 4 57 . 3403 . 6597 9.388 . 3405 9.371 .0006 . 0058 . 9942 3 58 . 3432 . 6668 9.139 . 3434 9.122 .0006 . 0069 . 9941 2 69 . 3461 . 6539 8.894 . 3463 8.877 .0006 . 0060 . 9940 1 60 . 3490 . 6510 8.664 . 3492 8.636 .0006 . 0061 . 9939 M. Cosine. Vrs. sin. Secant. Cotong. Tang. Cosec'nt Vrs. cos. Sine. M. 91° 88° 324 NATURAL FUNCTIONS. Table 3. 2° Natural Trigonometrical Functions. 177° M. Sine. Vrs. COS. Cosec'nt Tang. Cotang. Secant. Vra. sin. Cosine. M. .03490 .96510 28.654 .03492 28.636 1.0006 .00061 .99939 60 1 . 3519 . 6481 8.417 . 3521 8.399 .0006 . 0062 . 9938 69 2 . 3548 . 6452 8.184 . 3550 8.166 .0006 . 0063 . 9937 58 3 . 3577 . 6423 7.955 . 3579 7.937 .0006 . 0064 . 9936 57 4 . 3606 . 6394 7.730 . 8608 7.712 .0006 . 0065 . 9935 56 5 .03635 .96365 27.508 .03638 27.490 1.0007 .00066 .99934 55 C . 3664 . 6336 7.290 . 3667 7.271 .0007 . 0067 . 9933 54 7 . 3693 . 6306 7.075 . 3696 7.066 .0007 . 0068 . 9932 53 8 .3722 . 6277 6.864 . 3725 6.845 .0007 . 0069 . 9931 52 9 . 3751 . 6248 6.655 . 3754 6.637 .0007 . 0070 . 9930 51 10 .03781 .96219 26.450 .03783 26.432 1.0007 .00071 .99928 50 11 . 3810 . 6190 6.249 . 3812 6.230 .0007 . 0073 . 9927 49 12 . 3839 . 6161 6.050 . 3842 6.031 .0007 . 0074 . 9926 48 13 . 3868 . 6132 5.354 . 3871 5.835 .0007 . 0075 . 9925 47 14 . 3897 . 6103 5.661 . 3900 ■ 6.642 .0008 . 0076 . 9924 46 15 .03926 .96074 25.471 .03929 25.452 1.0008 .00077 .99923 45 16 . 3955 . 6045 6.284 . 3968 5.264 .0008 . 0078 . 9922 44 17 . 3984 . 6016 8.100 . 3987 5.080 .0008 . 0079 . 9921 43 18 . 4013 . 5987 4.918 . 4016 4.898 .0008 . 0080 . 9919 42 19 . 4042 . 5968 4.739 . 4046 4.718 .0008 . 0082 . 9918 41 20 .04071 .95929 24.562 .04075 24.642 1.0008 .00083 .99917 40 21 . 4100 . 5900 4.388 . 4104 4.367 .0008 . 0084 . 9916 39 22 . 4129 . 5870 4.216 . 4133 4.196 .0008 . 0085 . 9915 38 23 . 4158 . 5841 4.047 . 4162 4.026 .0009 . 0086 . 9913 37 24 . 4187 . 5812 3.880 . 4191 3.859 .0009 . 0088 . 9912 36 25 .04217 .95783 23.716 .04220 23.694 1.0009 .00089 .99911 35 26 . 4246 . 5754 3.553 . 4249 3.532 .0009 . 0090 . 9910 34 27 . 4275 . 5725 3.393 . 4279 3.372 .0009 . 0091 . 9908 33 28 . 4304 . 5696 3.235 . 4308 3.214 .0009 . 0093 . 9907 32 29 . 4333 . 5667 3.079 . 4337 3.068 .0009 . 0094 . 9906 31 30 .04362 .95638 22.925 .04366 22.904 1.0009 .00095 .99905 30 31 . 4391 . 5609 2.774 . 4395 2.752 .0010 . 0096 . 9903 29 32 . 4420 . 6580 2.624 . 4424 2.602 .0010 . 0098 . 9902 28 33 . 4449 . 5551 2.476 . 4453 2.454 .0010 . 0099 . 9901 27 34 . 4478 . 5622 2.330 . 4483 3ie08 .0010 . 0100 . 9900 26 35 .04507 .95493 22.186 .04512 22.164 1.0010 .00102 .99898 25 30 . 4536 . 5464 2.044 . 4541 2.022 .0010 . 0103 . 9897 24 37 . 4565 . 5435 1.904 . 4570 1.881 .0010 . 0104 . 9896 23 38 . 4594 . 5405 1.765 . 4599 1.742 .0010 . 0106 . 9894 22 39 . 4623 . 6376 1.629 . 4628 1.606 .0011 . 0107 . 9893 21 40 .04652 .96347 21.494 .04657 21.470 1.0011 .00108 .99892 20 41- . 4681 . 6318 1.360 . 4687 1.337 .0011 . 0110 . 9890 19 42 . 4711 . 5289 1.228 . 4716 1.205 .0011 . 0111 . 9889 18 43 . 4740 . 5260 1.098 . 4745 1.075 .0011 . 0112 . 9888 17 44 . 4769 . 5231 0.970 . 4774 0.946 .0011 . 0114 . 98S6 16 45 .04798 .95202 20.843 .04803 20.819 1.0011 .00115 .99885 15 46 . 4827 . 5173 0.717 . 4832 0.693 .0012 . 0116 . 9883 14 47 . 4856 . 5144 0.593 . 4862 0.569 .0012 . 0118 . 9882 13 48 . 4885 . 5115 0.471 . 4891 0.446 .0012 . 0119 . 9881 12 49 . 4914 . 6086 0.350 . 4920 0.325 .0012 . 0121 . 9879 11 50 .04943 .96057 20.230 .04949 20.205 1.0012 .00122 .99878 10 51 . 4972 . 6028 0.112 . 4978 0.087 .0012 . 0124 . 9876 9 62 . 5001 . 4999 19.995 . 5007 19.970 .0012 . 0125 . 9875 8 53 5030 4970 9.880 . 5037 9.854 .0013 . 0127 . 9873 7 54 . 6059 . 4941 9.766 . 5066 9.740 .0013 . 0128 . 9872 G 65 .05088 .94912 19.653 .05095 19.627 1.0013 .00129 .99870 5 66 . 5117 . 4883 9.541 . 5124 9.515 .0013 . 0131 . 9869 4 57 . 5146 . 4853 9.431 . 6153 9.405 .0013 . 0132 . 9867 3 58 . 5175 . 4824 9.322 . 6182 9.296 .0013 . 0134 . 9866 2 59 . 5204 . 4795 9.214 . 6212 9.188 .0013 . 0135 . 9864 1 60 . 5234 . 4766 9.107 . 5241 9.081 .0014 . 0137 . 9863 M. Cosine, Vrs. sin. Secant. Co tang. Tang. Cosec'nt AVs. cos. Sine. M. 92° 87° lies. NATUEAL FUNCTIUJNS. 325 Natural Trigonometrical P|fnctions. 176° Sine. Yra. COB. Cosec'nt Tang. Co tang. Secant. Vrs. sin. Cosine. M. .05234 .94766 19.107 .05241 19.081 1.0014 .00137 .99863 60 . 5263 . 4737 9.002 . 5270 8.975 .0014 . 0138 . 9861 59 . 6292 . 4708 8.897 . 5299 8.871 .0014 . 0140 . 9860 58 . 5321 . 4679 8.794 . 5328 8.768 .0014 . 0142 . 9868 57 . 5350 . 4650 8.692 . 5357 8.665 .0014 . 0143 . 9857 56 .05379 .94621 18.591 .05387 18.564 1.0014 .00145 .99865 55 . 5408 . 4592 8.491 . 5416 8.464 .0016 . 0146 . 9854 64 . 5437 . 4563 8.393 . 5445 8.365 .0015 . 0148 . 9852 53 . 5466 . 4534 8.295 . 5474 8.268 .0016 . 0149 . 9850 62 . 5495 . 4505 8.198 . 5503 8.171 .0015 . 0151 . 9849 51 .05524 .94476 18.103 .05532 18.075 1.0015 .00153 .99847 50 . 5553 . 4447 8.008 . 5562 7.980 .0015 . 0154 . 9846 49 . 5582 . 4418 7.914 . 5591 7.886 .0016 . 0156 . 9844 48 . 5611 . 4389 7.821 . 5620 7.793 .0016 . 0157 . 9842 47 . 5640 . 4360 7.730 . 5649 7.701 .0016 . 0159 . 9841 46 .05669 .94331 17.639 .05678 17.610 1.0016 .00161 .99839 45 . 5698 . 4302 7.549 . 5707 7.520 .0016 . 0162 . 9837 44 . 5727 . 4273 7.460 . 5737 7.431 .0016 . 0164 . 9836 43 . 5756 .4244 7.372 . 5766 7.343 .0017 . 0166 . 9834 42 . 6785 . 4214 7.285 . 5795 7.256 .0017 . 0167 . 9832 41 .05814 .94185 17.198 .05824 17.169 1.0017 .00169 .99831 40 . 5843 . 4156 7.113 . 5853 7.084 .0017 . 0171 . 9829 39 . 4127 7.028 . 5883 6.999 .0017 . 0172 . 9827 38 5^02 . 4098 6.944 . 5912 6.915 .001? . 0174 . 9826 37 ! 5931 . 4069 6.861 . 5941 6.832 .0018 . 0176 . 9824 36 .05960 .94040 16.779 .05970 16.750 1.0018 .00178 .99822 35 . 5989 . 4011 6.698 . 5999 6.668 .0018 . 0179 . 9820 34 . 6018 . 3982 6.617 . 6029 6.587 .0018 . 0181 . 9819 33 . 6047 . 3953 6.538 . 6053 6.507 .0018 . 0183 . 9817 32 . 6076 . 3924 6.459 . 6087 6.428 .0018 . 0186 . 9815 31 .06105 .93895 16.380 .06116 16.350 1.0019 .00186 .99813 30 . 6134 . 3866 6.303 . 6145 6.272 .0019 . 0188 . 9812 29 . 6163 . 3837 6.226 . 6175 6.195 .0019 . 0190 . 9810 28 . 6192 . 3808 6.150 . 6204 6.119 .0019 . 0192 . 9808 27 . 6221 . 3777 6.075 . 6233 6.043 .0019 . 0194 . 9806 26 .06250 .93750 16.000 .06262 15.969 1.0019 .00196 .99804 25 . 6279 . 3721 5.926 . 6291 6.894 .0020 . 0197 . 9803 24 . 6308 . 3692 5.853 . 6321 6.821 .0020 . 0199 . 9801 23 . 6337 . 3663 5.780 . 6350 6.748 .0020 . 0201 . 9799 22 . 6366 . 3634 5.708 . 6379 5.676 .0020 . 0203 . 9797 21 .06395 .93605 15.637 .06408 15.605 1.0020 .00205 .99795 20 . 6424 . 3576 5.566 . 6437 5.534 .0021 . 0206 . 9793 19 . 6453 . 3547 5.496 . 6467 6.464 .0021 . 0208 . 9791 18 . 6482 . 3518 5.427 . 6496 5.394 .0021 . 0210 . 9790 17 . 6511 . 3489 5.358 . 6525 5.325 .0021 . 0212 . 9788 16 .06540 .93460 15.290 .06554 16.267 1.0021 .00214 .99786 15 . 6569 . 3431 5.222 . 6583 5.189 .0022 . 0216 . 9784 14 . 6598 . 3402 5.155 . 6613 5.122 .0022 . 0218 . 9782 13 . 6627 . 3373 5.089 . 6642 5.066 .0022 . 0220 . 9780 12 . 6656 . 3343 5.023 . 6671 4.990 .0022 . 0222 . 9778 11 .06685 .93314 14.958 .06700 14.924 1.0022 .00224 .99776 10 . 6714 . 3285 4.893 . 6730 4.860 .0023 . 0226 . 9774 9 . 6743 . 3256 4.829 . 6759 4.795 .0023 . 0228 . 9772 8 . 6772 . 3227 4.765 . 6788 4.732 .0023 . 0230 . 9770 7 . 6801 . 3198 4.702 . 6817 4.668 .0023 . 0231 . 9768 6 .06830 .93169 14.640 .06846 14.606 1.0023 .00233 .99766 5 . 6859 . 3140 4.578 . 6876 4.644 .0024 . 0235 . 9764 4 . 6888 . 3111 4.517 . 6905 4.482 .0024 . 0237 . 9762 3 . 6918 .3082 4.456 . 6934 4.421 .0024 . 0239 . 9760 2 . 6947 . 3053 4.395 . 6963 4.361 .0024 . 0241 . 9758 1 . 6976 . 3024 4.335 . 6993 4.301 .0024 . 0243 . 9766 Cosine. Vrs. sin. Secant. Cotang. Tang. CoBBC'nt Vrs. cos. Sine. 86° 326 NATURAL FUNCTIONS. Table 3. 4° Natural Trigonometrical Functions. 175° M. Sine. Vrs. C08. Cosec'nt Tang. Cotang. Secant. Vrs. sin. Cosine. M. .06976 .93024 14.335 .06993 14.301 1.0024 .00243 .99756 60 1 . 7005 . 2995 4.276 . 7022 4.241 .00'25 . 0246 . 9754 59 2 . 7034 . 2966 4.217 . 7051 4.182 .0025 . 02'18 . 9752 58 3 . 7053 . 2937 4.159 . 7080 4.123 .0026 . 0250 . 9750 57 4 . 7092 . 2908 4.101 . 7110 4.065 .0025 . 0252 . 9748 66 5 .07121 .92879 14.043 .07139 14.008 1.0025 .00254 .99746 65 6 . 7150 . 2850 3.986 . 7168 3.961 .0026 . 0256 . 9744 54 7 . 7179 . 2821 3.930 . 7197 3.894 .0026 . 0268 . 9742 53 8 . 7208 . 2792 3.874 . 7226 3.838 .0026 . 0260 . 9740 ,52 9 . 7237 . 2763 3.818 . 7256 3.782 .0026 . 0262 . 9738 61 10 .07266 .92734 13.763 .07285 13.727 1.0026 .00264 .99736 50 11 . 7295 . 2705 3.708 . 7314 3.672 .0027 . 0266 . 9733 49 12 . 7324 . 2676 3.654 . 7343 3.617 .0027 . 0268 . 9731 48 13 . 7353 . 2647 3.600 . 7373 3.563 .0027 . 0271 . 9729 47 l-l . 7382 . 2618 3.547 . 7402 3.510 .0027 . 0273 . 9727 46 15 .07411 .92589 13.494 .07431 13.457 1.0027 .00276 .99725 45 16 . 7440 . 2560 3.441 . 7460 3.404 .0028 . 0277 . 9723 44 17 . 7469 . 2.531 8.389 . 7490 3.351 .0028 . 0279 . 9721 43 18 . 7498 . 2502 3.337 . 7.519 3.299 .0028 . 0281 . 9718 42 19 . 7527 . 2473 3.286 . 7648 3.248 .0028 . 0284 . 9716 41 20 .07556 .92444 13.235 .07.577 13.197 1.0029 .00286 .99714 40 21 . 7585 . 2415 3.184 . 7607 3.146 .0029 . 0288 . 9712 39 22 . 7614 . 2386 8.134 . 7636 3.096 .0029 . 0290 . 9710 38 23 . 7643 . 2357 3.084 . 7665 3.046 .0029 . 0292 . 9707 37 24 7672 . 2328 3.034 . 7694 2.996 .0029 . 0295 . 9705 36 2.>) .07701 .92299 12.985 .07724 12.947 1.0030 .00297 .99703 35 26 7730 . 2270 2.937 . 7763 2.898 .0030 . 0299 . 9701 34 27 . 7759 . 2241 2.888 . 7782 2.849 .0030 . 0301 . 9698 33 28 . 7788 . 2212 2.840 . 7812 2.801 .0030 . 0304 . 9696 32 29 7817 . 2183 2.793 . 7841 2.764 .0031 . 0306 . 9694 31 30 .07846 .92154 12.745 .07870 12.706 1.0031 .00308 .99692 30 31 . 7875 . 2125 2.698 . 7899 2.659 .0031 . 0310 . 9689 29 32 . 7904 . 2096 2.052 . 7929 2.612 .0031 . 0313 . 9687 28 83 . 7933 . 2067 2.006 . 7968 2.566 .0032 . 0315 . 9685 27 34 . 7962 . 2038 2.560 . 7987 2.520 .0032 . 0317 . 9682 26 35 .07991 .92009 12.614 .08016 12.474 1.0032 .00320 .99680 25 36 . 802O . 1980 2.469 . 8046 2.429 .0032 . 0322 . 9678 24 37 . 8049 . 1951 2.424 . 8075 2.384 .0032 . 0324 . 9675 23 38 . 8078 . 1922 2.379 . 8104 2.339 .0033 . 0327 . 9673 22 39 . 8107 . 1893 2.335 . 8134 2.295 .0033 . 0329 . 9671 21 40 .08136 .91864 12.291' .08163 12.250 1.0033 .00331 .99668 20 41 . 8165 . 1835 2.248 . 8192 2.207 .0033 . 0334 . 9666 19 42 . 8194 . 1806 2.204 . 8221 2.163 .0034 . 0336 . 9664 18 43 . 8223 . 1777 2.161 . 8251 2.120 .0034 . 0339 . 9661 17 44 . 8282 . 1748 2.118 . 8280 2.077 .0034 . 0341 . 9659 16 45 .08281 .91719 12.076 .08309 12.035 1.0034 .00343 .99656 15 46 . 8310 . 1690 2.034 . 8339 1.992 .0035 . 0346 . 9654 14 47 . 8339 . 1661 1.992 . 8368 1.950 .0035 . 0348 . 9652 13 48 . 8368 . 1632 1.960 . 8397 1.909 .0035 . 0351 . 9649 12 49 . 8397 . 1603 1.909 . 8426 1.867 .0035 . 0353 . 9647 H 50 .08426 .91574 11.868 .08466 11.826 1.0036 .00356 .99644 10 51 . 8455 1545 1.828 . 8485 1.785 .0036 . 0358 . 9642 9 52 . 8484 . 1516 1.787 . 85i4 1.746 .0036 . 0360 . 9639 8 53 . 8513 . 1487 1.747 . 8544 1.704 .0036 . 0363 . 9637 7 54 . 8542 1468 1.707 . 8.573 1.664 .0037 . 0365 . 9634 6 55 .08571 .91429 11.668 .08602 11.625 1.0037 .00368 .99632 5 66 . 8600 . 1400 1.628 . 8632 1.685 .0037 . 0370 . 9629 4 67 . 8629 1371 1.589 . 8661 1.546 .0037 . 0373 . 9627 3 58 . 8658 1342 1.560 . 8690 1.507 .0038 . 0376 . 9624 2 69 . 8687 . 1313 1.512 . 8719 1.468 .0038 . 0378 . 9622 1 60 . 8715 . 1284 1474 . 8719 1.430 .0038 . 0380 . 9619 M. CuBilin. Vrs. sin. Secant. Cotang. Tang. Cosec'nt Vra. COS. Sine. M. 40 85° Table 3. NATURAL FUNCTIONS. 327 s° Naturcl Trigonometrical Functions, 174° M, Sino. Vra. COS. CoBec'nt Tang. Co tang. Secant. Vrs. sin. Cosine. M. .08715 .91284 11.474 .08749 11.430 1.0038 .00380 .99019 60 1 . 8744 . 1255 1.436 . 8778 1.392 .0038 . 0383 . 9017 59 2 . 8773 . 1226 1.398 . 8807 1.354 .0039 . 0386 . 9614 68 3 . 8802 . 1197 1.360 . 8837 1.316 .0039 . 0388 . 9612 57 4 . 8831 . 1168 1.323 . 8866 1.279 .0039 . 0391 . 9609 56 5 .08800 .91139 11.286 .08895 11.242 1.0039 .00393 .99607 55 6 . 8889 . 1110 1.249 . 8925 1.205 .0010 . 0396 . 9604 54 7 . 8918 . 1082 1.213 . 8954 1.1G8 .0040 . 0398 . 9601 53 8 . 8947 . 1053 1.176 . 8983 1.132 . .0040 . 0401 . 9599 52 9 . 8976 . 1024 1.140 . 9013 1.095 .0040 . 0404 . 9596 51 10 .09005 .90995 11.104 .09042 11.059 1.0041 .00106 .99594 50 11 . 9031 . 09G6 1.069 . 9071 1.024 .0011 . 0109 . 9591 19 12 . 9063 . 0937 1.033 . 9101 0.988 .0011 . 0111 . 9588 18 13 . 9092 . 0908 0.998 . 9130 0.953 .0041 . 0111 . 9586 17 14 . 9121 . 0879 0.963 . 9159 0.918 .0042 . 0117 . 9583 16 15 .09150 .90850 10.929 .09189 10.883 1.0042 .00119 .99580 15 16 . 9179 . 0821 0.894 . 9218 0.848 .0012 . 0122 . 9578 14 17 . 9208 . 0792 0.860 . 9247 0.814 .0013 . 0125 . 9575 13 18 . 9237 . 0763 0.826 . 9277 0.780 .0013 . 0127 . 9572 42 19 . 9266 . 0734 0.792 . 9306 0.746 .0013 . 0130 . 9570 11 20 .09295 .90705 10.758 .09335 10.712 1.0013 .00133 .99567 40 21 . 9324 . 0676 0.725 . 9365 0.678 .0014 . 0436 . 9564 39 22 . 9353 . 0647 0.692 . 9394 0.645 .0011 . 0138 . 9562 38 23 . 9382 . 0618 0.659 . 9423 0.612 .0011 . 0111 . 9559 37 24 . 9411 . 0589 0.626 . 94.53 0.579 .0011 . 0144 . 9556 36 25 .09440 .90560 10.593 .09482 10.546 1.0045 .00416 .99553 35 26 . 9469 . 0531 0.561 . 9511 0.514 .0045 . 0149 . 9551 34 27 . 9498 . 0502 0.529 . 9541 0.481 .0045 . 0152 . 9548 33 28 . 9527 . 0473 0.497 . 9570 0.449 .0046 . 0155 . 9545 32 29 . 9556 . 0444 0.465 . 9599 0.417 .0046 . 0158 . 9542 31 80 .09584 .90415 10.433 .09629 10.385 1.0046 .00160 .99M0 30 81 . 9613 . 0386 0.402 . 9658 0.354 .0046 . 0163 . 9537 29 32 . 9642 . 0357 0.371 . 9088 0.322 .0047 . 0466 . 9534 28 33 . 9671 . 0328 0.340 . 9717 0.291 .0047 . 0169 . 9531 27 34 . 9700 . 0300 0.309 . 9746 0.260 .0017 . 0472 . 9528 26 35 .09729 .90271 10.278 .09776 10.229 1.0048 .00171 .99525 25 86 . 9758 . 0242 0.248 . 9805 0.199 .0048 . 0177 . 9523 24 37 . 9787 . 0213 0.217 . 9834 0.168 .0048 . 0180 . 9520 23 38 . 9816 . 0184 0.187 . 9864 0,138 .0048 . 0183 . 9517 22 39 . 9845 . 0155 0.157 . 9893 0.108 .0049 . 0486 . 9514 21 40 .09874 .90126 10.127 .09922 10.078 1.0049 .00489 .99511 20 41 . 9903 . 0097 0.098 . 9952 0.048 .0049 . 0191 . 9508 19 42 . 9932 . 0068 0.068 . 9981 0.019 .0050 . 0494 . 9505 18 43 . 9961 . 0039 0.039 .10011 9.9893 .0050 . 0197 . 9503 17 44 . 9990 . 0010 0.010 . 0010 .9601 .0050 . 0500 . 9500 16 45 .10019 .89981 9.9812 .10069 9.9310 1.0050 .00503 .99497 15 46 . 0048 . 9952 .9525 . 0099 .9021 .0051 . 0506 . 9494 14 47 . 0077 . 9923 .9239 . 0128 .8734 .0051 . 0509 . 9191 13 48 . 0106 . 9894 .8955 . 0158 .8448 .0051 . 0512 . 9188 12 49 . 0134 . 9865 .8672 . 0187 .8164 .0052 . 0515 . 9185 11 50 .10163 .89836 9.8391 .10216 9.7882 1.0052 .00518 .99182 10 51 . 0192 . 9807 -..8112 . 0246 .7601 -.0052 . 0521 . 9179 9 52 . 0221 ■. 9779 .7834 . 0275 .7322 .0053 . 0524 . 9176 8 53 . 0250 . 9750 .7558 . 0305 .7044 .0053 . 0527 . 9473 7 54 . 0279 . 9721 .7283 . 0334 .6768 .0053 . 0530 . 9470 6 55 .10308 .89692 9.7010 .10363 9.6493 1.0053 .00533 .99467 5 56 . 0337 . 9663 .6739 . 0393 .6220 .0051 . 0536 . 9461 4 57 . 0366 . 9634 .6469 . 0422 .5949 .0054 . 0539 . 9161 3 58 . 0395 . 9605 .6200 . 0452 .5679 .0054 . 0542 . 9458 2 59 . 0424 . 9576 .5933 . 0481 .5411 .0055 . 0545 . 9455 1 60 . 0453 . 9547 .5668 . 0510 .5144 .0055 . 0548 . 9152 M. Coaine. Vre. sin. Secant. Cotang. Tang. Cosec'nt Vrs. COS. Sine. M. 95° 84° 328 NATURAL FUNCTIONS. Table a. 6° Natural Trigonometrical Functions. 173° M. Sine. Vrs. COS. Cosec'nt Tang. Co tang. Secant. Vro. sin. Cosine. 31. .10453 .89547 9.5668 .10510 9.5144 1.0056 .00548 .99452 60 1 . 0482 . 9518 .5404 . 0540 .4878 .0055 . 0561 . 9449 69 2 . 0511 . 9489 .5141 . 0569 .4614 .0056 . 0554 . 9446 58 3 . 0540 . 9460 .4880 . 0599 .4351 .0056 . 0557 . 9443 57 4 . 0568 . 9431 .4020 . 0628 .4090 .0056 . 0560 . 9440 56 5 .10597 .89402 9.4362 .10657 9.3831 1.0057 .00563 .99437 55 6 . 0626 . 9373 .4105 . 0687 .3572 .0057 . 0666 . 9434 64 7 . 0655 . 9345 .3850 . 0716 .3315 .0057 . 0569 . 9431 53 8 . 0684 . 9316 .3596 . 0746 .3060 .0057 . 0.i72 . 9428 52 9 . 0713 . 9287 .3343 . 0775 .2806 .0058 . 0575 . 9424 51 10 .10742 .89258 9.3092 .10805 9.2553 1.0058 .00579 .99421 50 11 . 0771 . 9229 .2842 . 0834 .2302 .0068 . 0582 . 9418 49 12 . 0800 . 9200 .2593 . 0863 .2051 .0059 . 0585 . 9415 48 13 . 0829 . 9171 .2346 . 0893 .1803 .0059 . 0588 . 9412 47 14 . 0858 . 9142 .2100 . 0922 .1655 .0069 . 0591 . 9409 46 15 .10887 .89113 9.1855 .10952 9.1309 1.0060 .00594 .99406 45 16 . 0916 . 9084 .1612 . 0981 .1064 .0060 . 0597 . 9402 44 17 . 0944 . 9055 .1370 . 1011 .0821 .0060 . 0601 . 9399 43 18 . 0973 . 9026 .1129 . 1040 .0579 .0061 . 0604 . 9396 42 19 . 1002 . 8998 .0890 . 1069 .0338 .0061 . 0607 . 9393 41 20 .11031 .88969 9.0651 .11099 9.0098 1.0061 .00610 .99390 40 21 . 1060 . 8940 .0414 . 1128 8.9860 .0062 . 0613 . 9386 39 22 . 1089 . 8911 .0179 . 1158 .9623 .0062 . 0617 . 9383 38 23 . 1118 . 8882 8.9944 . 1187 .9387 .0062 . 0620 . 9380 37 2i . 1147 . 8853 .9711 . 1217 .9152 .0063 . 0623 . 9377 36 25 .11176 .88824 8.9479 .11246 8.8918 1.0063 .00626 .99373 35 26 . 1205 . 8795 .9248 . 1276 .8686 .0063 . 0630 . 9370 34 27 . 1234 . 8766 .9018 . 1305 .8455 .0064 . 0633 . 9367 33 28 . 1262 . 8737 .8790 . 1335 .8225 .0064 . 0636 . 9364 32 29 . 1291 . 8708 .8663 . 1364 .7996 .0064 . 0639 . 9360 31 30 .11320 .88680 8.8337 .11393 8.7769 1.0065 .00643 .99357 30 31 . 1349 . 8651 .8112 . 1423 .7542 .0005 . 0646 . 9354 29 32 . 1378 . 8622 .7888 . 14.52 .7317 .0065 . 0649 . 9350 28 33 . 1407 . 8593 .7665 . 1482 .7093 .0066 . 0653 . 9347 27 34 . 1436 . 8564 .7414 . 1511 .6870 .0066 . 0656 . 9344 26 35 .11465 .88535 8.7223 .11641 8.6648 1.0066 .00659 .99341 25 36 . 1494 . 8506 .7004 . 1570 .6427 .0067 . 0663 . 9337 24 37 . 1523 . 8477 .6786 . 1600 .6208 .0067 . 0666 . 9334 23 38 . 1551 . 8448 .6569 . 1629 .5989 .0067 . 0669 . 9330 22 39 . 1580 . 8420 .6353 . 1659 .5772 .0068 . 0673 . 9327 21 40 .11609 .88391 8.6138 .11688 8.6555 1.0068 .00676 .99324 20 41 . 1638 . 8362 .5924 . 1718 .5340 .0068 . 0679 . 9320 19 42 . 1667 . 8333 .5711 . 1747 .5126 .0069 . 0683 . 9317 18 43 . 1696 . 8304 .5499 . 1777 .4913 .0069 . 0686 . 9314 17 44 . 1725 . 8272 .5289 . 1806 .4701 .0069 . 0690 . 9310 16 45 .11754 .88246 8.5079 .11836 8.4489 1.0070 .00693 .99307 15 46 . 1783 . 8217 .4871 . 1865 .4279 .0070 . 0696 . 9303 14 47 . 1811 . 8188 .4603 . 1895 .4070 .0070 . 0700 . 9300 IS 48 . 1840 . 8160 .4457 . 1924 .3862 .0071 . 0703 . 9296 12 49 . 1869 . 8131 .4251 . 1954 .3655 .0071 . 0707 . 9293 11 50 .11898 .88102 8.4046 .11983 8.3449 1.0071 .00710 .99290 10 51 . 1927 . 8073 .3843 . 2013 .3244 .0U72 . 0714 . 9286 9 52 . 1956 . 8044 .3640 . 2042 .3040 .0072 . 0717 . 9'283 8 53 . 1985 . 8015 .3139 . 2072 .2837 .0073 . 0721 . 9279 7 54 . 2014 . 7986 .3238 . 2101 .2635 .0073 . 0724 . 9276 6 65 .12042 .87957 8.3039 .12131 8.2434 1.0073 .00728 .99272 5 56 . 2071 . 7928 .2840 . 2160 .2234 .0074 . 0731 . 9269 4 67 . 2100 . 7900 .2642 . 2190 .2035 .0074 . 0735 . 9265 3 58 . 2129 . 7871 .2446 . 2219 .1837 .0074 . 0738 . 9262 2 69 . 2158 . 7842 .2250 . 2249 .1640 .0075 . 0742 . 9258 1 .60 2187 . 7813 .20.55 . 2278 .1443 .0075 . 0745 . 9265 M. Cosine. Vrs. sin. Secant. Co tang. Tang. Cosec'nt Vrs. COB. Sine. M. 96° 83° Table 3. NATUKAL FUNCTIONS. 329 7° Natural Trigonometrical Functions. 172° M. Sine. Vre. COS. Cosec'nt Tang. Co tang. Secant. Vrs. siu. Cosine. M. .12187 .87813 8.2055 .12278 8.1443 1.0075 .00745 .99255 60 1 . 2216 . 7787 .1861 . 2308 .1248 .0075 . 0749 .9251 59 2 . 2245 . 7755 .1668 . 2337 .1053 .0076 . 0752 . 9247 58 3 . 2273 .7726 .1476 . 2367 .0860 .0076 . 0756 . 9244 .57 4 . 2302 . 7697 .1285 . 2396 .0667 .0076 . 0760 . 9240 56 5 .12331 .87669 8.1094 .12426 8.0476 1.0077 .00763 .99237 55 6 . 2360 . 7640 .0905 . 2456 .0285 .0077 . 0767 . 9233 .54 7 . 2389 . 7611 .0717 . 2485 .0095 .0078 . 0770 . 9229 53 8 . 2418 . 7582 .0529 . 2515 7.9906 .0078 . 0774 . 9226 52 9 . 2447 7553 : .0342 . 2544 .9717 .0078 . 0778 . 9222 51 10 .12476 .87524 8.0156 .12574 7.9.130 1.0079 .00781 .99219 50 11 . 2504 . 7495 7.9971 . 2603 .9.344 .0079 . 0785 . 9215 49 12 . 2533 . 7467 .9787 . 2633 .9158 .0079 . 0788 . 9211 48 13 . 2662 . 74.38 .9604 . 2662 .8973 .0080 . 0792 . 9208 47 14 . 2591 . 7409 .9421 . 2692 .8789 .0080 . 07% . 9204 46 15 .12620 .87:380 7.9240 .12722 7.8606 1.0080 .00799 .99200 45 16 . 2G49 . 7351 .9059 . 2751 .8424 .0081 . 0803 . 9197 44 17 . 2678 . 7322 .8879 . 2781 .8243 .0081 . 0807 . 9193 43 18 . 2706 . 7293 .8700 . 2810 .8062 .0082 . 0810 . 9189 42 19 . 2735 . 7265 .8522 . 2840 .7882 .0082 . 0814 . 9186 41 20 .12764 .87236 7.8344 .12869 7.7703 1.0082 .00818 .99182 40 21 . 2793 . 7207 .8168 . 2899 .7525 .0083 . 0822 . 9178 39 22 . 2822 . 7178 .7992 . 2928 .7348 .0083 . 0825 . 9174 38 23 . 2851 . 7149 .7817 . 2958 .7171 .0084 . 0829 . 9171 37 24 . 2879 7120 .7642 . 2988 .6996 .0084 . oass . 9467 36 25 .12908 .87091 7.7469 .13017 7.6821 1.0084 .00837 .99163 35 26 . 2937 . 7063 .7296 . 3047 .6646 .0085 . 0840 . 9160 34 27 . 2966 . 7034 .7124 . 3076 .6473 .0085 . 0844 . 9156 33 28 . 2995 . 7005 .6953 . 3100 .6300 .0085 . 0848 . 9152 32 29 . 3024 . 6976 .6783 . 3136 .6129 .0086 . 0852 . 9148 31 30 .13053 .86947 7.6613 .13165 7..5957 1.0086 .00855 .99144 30 31 . 3081 . 6918 .6414 . 3195 .5787 .0087 . 0859 . 9141 29 32 . 3110 . 6890 .6276 . 3224 .5617 .0087 . 0863 . 9137 28 33 . 3139 . 6861 .6108 . 3254 .5449 .0087 . 0867 . 9133 27 34 . 3168 . 6832 .5942 . 3284 .5280 .0088 . 0871 9129 26 35 .13197 .86803 7.5776 .13313 7.5113 1.0088 .00875 .99125 25 36 . 3226 . 6774 ..5611 . 3343 .4946 .0089 . 0878 . 9121 24 37 . 3254 . 6745 .5446 . 3372 .4780 .0089 . 0882 . 9118 23 38 . 3283 . 6717 .5282 . 3402 .4615 .0089 . 0886 . 9114 22 39 . 3312 . C688 .5119 . 3432 .4451 .0090 . 0890 . 9110 21 40 .13341 .86659 7.4957 .13461 7.4287 1.0090 .00894 .99106 20 41 . 3370 . G630 .4795 . 3491 .4124 .0090 . 0898 . 9102 19 42 . 3399 . 6601 .4634 . 3520 .3961 .0091 . 0902 . 9098 18 43 . 3427 . 6572 .4474 . 3550 ;3800 .0091 . 0905 . 9094 17 44 . 3456 . 6544 .4315 . 3580 .3639 .0092 . 0909 . 9090 16 45 .13485 .86515 7.4156 .13609 7.3479 1.0092 .00913 .99086 15 46 . 3514 . 6486 .3998 . 3639 .3319 .0092 . 0917 . 9083 14 47 . 3543 . 6457 .3840 . 3669 .3160 .0093 . 0921 . 9079 13 48 . 3571 . 6428 .3683 . 3698 .3002 .0093 . 0925 . 9075 12 49 . 3600 . 6400 .3527 . 3728 .2844 .0094 . 0929 . 9070 11 50 .13629 .86371 7.3372 .13757 7.2687 1.0094 .00933 .99067 10 51 . 3658 . 6342 .3217 . 3787 .2531 .0094 . 0937 . 9063 9 52 . 3687 . 6313 .3063 . 3817 .2375 .0095 . 0941 ■ . 9059 8 53 . 3716 . 6284 .2909 . 3846 .2220 .0095 . 0945 . 9055 7 54 . 3744 . 6255 .2757 . 3876 .2066 .0096 . 0949 . 9051 6 55 .13773 .86227 7.2604 .13906 7.1912 1.0096 .00953 .99047 5 56 . 3802 . 6198 .2453 . 3935 .1759 .0097 . 0957 . 9043 4 57 . 3831 . 6169 .2302 . 3965 .1607 .0097 . 0961 . 9039 3 58 . 3860 . 6140 .2152 . 3995 .1455 .0097 . 0965 . 9035 2 59 . 3888 . 6111 .2002 . 4024 .1304 .0098 . 0969 . 9031 1 60 . 3917 . 6083 .1853 . 4054 .1154 .0098 . 0973 . 9027 M. Cosine. Vrs. Bin. Secant. Cotang. Tang. CoBec'nt Vrs. COS. Sine. M. P7° 82° 330 NATURAL FUNCTIONS. Table 3. 8° Natural Trigonometrical Functions. 171° M. Sine. Vrs. COS. Cosec'nt Tiing. Cotang. Secant. Vrs. Bin. Cosine. M. .13917 .86083 7.1853 .14054 7.1154 1.0098 .00973 .99027 60 1 . 3946 . 6054 .1704 . 4084 .1004 .0099 . 0977 . 9023 59 2 . 3975 . 6025 .1557 . 4113 .0854 .0099 . 0981 . 9019 58 3 . 4004 . 5996 .1409 . 4143 .0706 .0099 . 0985 . 9015 .57 4 . 4032 . 5967 .1263 . 4173 .0558 .0100 . 0989 . 9010 56 5 .14001 .85939 7.1117 .14202 7.0410 1.0100 .00993 .99006 55 6 . 4090 . 5910 .0972 . 4232 .0264 .0101 . 0998 . 9002 54 7 . 4119 . 5881 .0827 . 4262 .0117 .0101 . 1002 . 8998 63 8 . 4148 . 5852 .0683 . 4291 6.9972 .0102 . 1006 . 8994 62 9 . 4176 . 5823 .0539 . 4321 .9827 .0102 . 1010 . 8990 51 10 .14205 .85795 7.0396 .14351 6.9682 1.0102 .01014 .98986 50 11 . 4234 . 5766 .0254 . 4;wo .9538 .0103 . 1018 . 8982 49 12 . 4263 . 5737 .0112 . 4410 .9395 .0103 . 1022 . 8978 48 13 . 4292 . 5708 6.9971 . 4440 .9252 .0104 . 1026 . 8973 47 14 . 4320 . 5679 .9830 . 4470 .9110 .0104 . 1031 . 8969 46 15 .14349 .85651 6.9690 .14499 6.8969 1.0104 .01035 .98965 45 16 . 4378 . 5622 .9550 . 4529 .8828 .0105 . 1039 . 8961 44 17 . 4407 . 5593 .9411 . 4.559 .8687 .0105 . 1043 . 8957 43 18 . 4436 . 5564 .9273 . 4588 .8547 .0106 . 1047 . 8952 42 19 . 4464 . 5536 .9135 . 4618 .8408 .0106 . 1052 . 8948 41 20 .14493 .85507 6.8998 .14048 6.8269 1.0107 .01056 .98944 40 21 . 4522 . 5478 .8861 . 4677 .8131 .0107 . 1060 . 8940 39 22 . 4551 . 5449 .8725 . 4707 .7993 .0107 . 1064 . 8936 38 23 . 4579 . 5420 .8589 . 4737 .7856 .0108 . 1068 . 8931 37 24 . 4608 . 5392 .8454 . 4767 .7720 .0108 . 1073 . 8927 36 25 .14637 .85363 6.8320 .14796 6.7584 1.0109 .01077 .98923 35 26 . 4666 . 5334 .8185 . 4826 .7448 .0109 . 1081 . 8919 34 27 . 4695 . 5305 .8052 . 4856 .7313 .0130 . 1085 . 8914 33 28 . 4723 . 5277 .7919 . 4886 .7179 .0110 . 1090 . 8910 32 29 . 4752 . 5248 .7787 . 4915 .7045 .0111 . 1094 . 8906 31 SO .14781 .85219 6.7655 .14945 6.6911 1.0111 .01098 .98901 30 31 . 4810 . 5190 .7523 . 4975 .6779 .0111 . 1103 . 8897 29 32 . 4838 .5161 .7392 . 5004 .6646 .0112 . 1107 . 8893 28 33 . 4867 . 5133 .7262 . 5034 .6514 .0112 . nil . 8889 27 34 . 4896 . 5104 .7132 . 5064 .6383 .0113 . 1116 . 8884 28 35 .14925 .85075 6.7003 .15094 6.0262 1.0113 .01120 .98880 25 36 . 4953 . 5046 .6874 . 5123 .6122 .0114 . 1124 . 8876 24 37 . 4982 . 6018 .6745 . 5153 .5992 .0114 . 1129 . 8871 23 38 . 5011 . 4989 .6617 . 5183 .6863 .0115 . 1133 . 8867 22 39 . 5040 . 4960 .6490 . 5213 .5734 .0116 . 1137 . 8862 21 40 .15068 .84931 6.6363 .15243 6.5605 1.0U6 .01142 .98858 20 41 . 5097 . 4903 .6237 . 5272 .5478 .0116 . 1146 . 8854 19 42 . 5126 . 4874 .6111 . 5302 .5350 .0116 . 1151 . 8849 18 43 . 5155 . 4845 .5985 . 5332 .5223 .0117 . 1155 . 8845 17 44 . 5183 . 4816 .5860 . 5362 .5097 .0117 . 1159 . 8840 16 45 .15212 .84788 6,5736 .15391 6.4971 1.0118 .01164 .98836 15 46 . 5241 . 4759 .6612 . 5421 .4845 .0118 . 1168 . 8832 14 47 . 5270 . 4730 .5488 . 5451 .4720 .0119 . 1173 . 8827 13 48 . 5298 . 4701 .5365 .5481 .4696 .0119 . 1177 . 8823 12 49 . 5328 . 4672 .5243 . 5511 .4472 .0119 . 1182 . 8818 11 50 .15356 .84644 6.6121 .15540 6.4348 1.012U .01186 .98814 10 51 . 5385 . 4615 .4999 . 5570 .4225 .0120 . 1190 . 8809 9 52 . 5413 . 4586 .4878 . 5600 .4103 .0121 . 1195 . 8805 8 53 . 5442 . 4558 .4757 . 5630 .3980 .0121 . 1199 . 8800 7 54 . 5471 . 4529 .4637 . 5659 .3859 .0122 . 1204 . 8796 6 55 .15500 .84500 6.4517 .15689 6.3737 1.0122 .01208 .98791 5 66 . 5528 . 4471 .4398 . 5719 .3616 .0123 . 1213 . 8787 4 57 . 6557 . 4443 .4279 . 5749 .3496 .0123 . 1217 . 8782 3 58 . 5586 . 4414 .4160 . 5779 .3376 .0124 . 1222 . 8778 2 59 . 5615 . 4385 .4042 . 5809 .3257 .0124 . 1227 . 8773 1 60 . 5643 . 4366 .3924 . 6838 .3137 .0125 . 1231 . 8769 M. Cosine. Vrs. sin. Secant. Cotang. Tang. Cosec'nt Sine. Vrs. COS. M. Table 3. NATURAL FUNCTIONS. 331 90 Natural Trigonometrical Functions. 170° M^ Sine. Vrs. COS. Cosec'nt Tang. Cotang. Secant. Vrs. Bin. Cosine. M. .15643 .84356 6.3924 .15838 6.3137 1.0125 .01231 .98769 60 1 . 5672 . 4328 .3807 . 5868 .3019 .0125 . 1236 . 8764 59 2 . 5701 . 4299 .3690 . 5898 - .2901 .0125 . 1240 . 8760 58 3 . 5730 . 4270 .3574 . 5928 .2783 .0126 . 1245 . 8755 57 4 . 5758 . 4242 .3458 . 5958 .2665 .0126 . 1249 . 8750 56 5 .15787 .84213 6.3343 .15987 6.2548 1.0127 .01254 .98716 56 6 . 5816 . 4184 .3228 . 6017 .2432 .0127 . 1259 . 8741 54 7 . 5844 . 4155 .3113 . 6047 .2316 .0128 . 1263 . 8737 53 8 . 5873 . 4127 .2999 . 6077 .2200 .0128 . 1268 . 8732 52 9 . 5902 . 4098 .2885 . 6107 .2085 .0129 . 1272 . 8727 51 10 .15931 .84069 6.2772 .16137 6.1970 1.0129 .01277 .98723 50 11 . 6959 . 4041 .2659 . 6167 .1856 .0130 . 12S2 . 8718 49 12 . 5988 . 4012 .2546 . 6196 .1742 .0130 . 1286 ■ . 8714 48 IS . 6017 . 3983 .2434 . 6226 .1628 .0131 . 1291 . 8709 47 14 . 6045 . 3954 .2322 . 6256 .1515 .0131 . 1296 . 8704 46 15 .16074 .83926 6.2211 .16286 6.1402 1.0132 .01300 .98700 46 16 . 6103 . 3897 .2100 . 6316 .1290 .0132 . 1305 . 8695 44 17 . 6132 . 3868 .1990 . 6346 .1178 .0133 . 1310 . 8690 43 18 . 6160 . 3840 .1880 . 6376 .1066 .0133 . 1314 . 8685 42 19 . 6189 . 3811 .1770 . 6405 .0955 .0134 . 1319 . 8681 41 20 .16218 .83782 6.1661 .16435 6.0844 1.0134 .01324 .98676 40 21 . 6246 . 3753 .15.52 . 6465 .0734 .0135 . 1328 . 8671 39 22 . 6275 . 3725 .1443 . 6495 .0624 .0135 . 1333 . 8lili7 38 23 . 6304 . 3696 .1335 . 6525 .0514 .0136 . 1338 . 8i;(i2 37 24 . 6333 . 3667 .1227 . 6555 .0405 .0136 . 1343 . 8657 36 26 .16361 .83639 6.1120 .16585 6.0296 1.0136 .01347 .9Si;52 35 26 . 6390 . 3610 .1013 . 6615 .0188 .0137 . 1352 . 8648 34 27 . 6419 . 3581 ■ .0906 . 6644 .0080 .0137 . 1357 . 8643 33 28 . 6447 . 3553 .0800 . 6674 5.9972 .0138 . 1362 . 8638 32 29 . 6476 . 3524 .0694 . 6704 .9865 .0138 . 1367 . 8633 31 30 .16505 .83495 6.0588 .16734 5.9758 1.0139 .01371 .98628 30 31 . 6533 . 3466 .0483 . 6764 .9651 .0139 . 1376 . 8624 29 32 . 6562 . 3438 .0379 . 6794 .9545 .0140 . 1381 . 8619 28 33 . 6591 . 3409 .0274 . 6824 .9439 .0140 . 1386 . 8614 27 34 . 6619 . 3380 .0170 . 6854 .9333 .0141 . 1391 . 8609 26 35 .16648 .83852 6.0066 .16884 5.9228 1.0141 .01395 .98604 25 36 . 6677 . 3323 5.9963 . 6911 .9123 .0142 . 1400 . 8600 24 37 . 6706 . 3294 .9860 . 6944 .9019 .0142 . 1405 . 8595 23 38 . 6734 . 3266 .9758 . 6973 .8915 .0143 . 1410 . 8590 22 39 . 6763 . 3237 .9655 . 7003 .8811 .0143 . 1415 . 8585 21 40 .16791 .83208 5.9554 .17033 5.8708 1.0144 .01420 .98580 20 41 . 6820 . 3180 .9452 . 7063 .8605 .0144 . 1425 . 8575 19 42 . 6849 . 3151 .9351 . 7093 .8602 .0145 . 1430 . 8570 18 43 . 6878 . 3122 .9250 . 7123 .8400 .0145 . 1434 . 8566 17 44 . 6906 . 3094 .9150 . 7153 .8298 .0146 . 1439 . 8560 16 45 .16935 .83065 5.9049 .17183 5.8196 1.0146 .01444 .98556 15 46 . 6964 . 3036 .8950 . 7213 .8095 .0147 . 1449 . 8551 14 47 . 6992 . 3008 .8850 . 7243 .7994 .0147 . 1454 . 8546 13 48 . 7021 . 2979 .8751 . 7273 .7894 .0148 . 1459 . 8541 12 49 . 7050 . 2950 .8652 . 7803 .7793 .0148 . 1464 . 8536 11 50 .17078 .82922 5.8554 .17333 5.7694 1.0149 .01469 .98531 10 61 . 7107 . 2893 .8456 . 7363 .7594 .0150 . 1474 . 8526 9 52 . 7136 . 2864 .8358 . 7393 .7495 .0150 . 1479 . 8521 8 53 . 7164 . 2836 .8201 . 7423 .7396 .0151 . 1484 . 8516 7 54 . 7193 . 2807 .8163 . 7463 .7297 .0151 . 1489 . 8511 6 55 .17221 .82778 5.8067 .17483 5.7199 1.0152 .01494 .98506 6 56 . 7250 . 2750 .7970 . 7513 .7101 .0152 . 1499 . 8501 4 67 . 7279 . 2721 • .7874 . 7543 .7004 .0153 . 1604 . 8496 3 58 . 7307 . 2692 .7778 . 7573 .6906 .0153 . 1509 . 8491 2 59 . 7336 . 2664 .7683 . 7603 .6809 .0154 . 1514 . 8486 1 60 . 7365 . 2635 .7588 . 7633 .6713 .0154 . 1519 . 8481 M. Cosine. Vra. ein. Secant. Cotang. Tang. Cosec'nt Vrs. COS. Sine. M. 332 NATURAL FUNCTIONS. Table 3. 10° Natural Trigonometrical Functions. «69<^ M. Sine. Vra. C03. Cosec'nt Tang. Cotang. Secant. Vrs. sin. Cosine. M. .17365 .82635 5.7588 .17633 5.6713 1.0164 .01519 .98481 60 1 . 7393 . 2606 .7193 . 7663 .6616 .0155 . 1524 . 8476 59 2 . 7422 . 2578 .7398 . 7693 .6520 .0155 . 1529 . 8471 58 3 . 7451 . 2549 .7304 . 7723 .6126 .0156 . 1534 . 8465 57 4 . 7479 . 2521 .7210 . 7753 .6329 .0156 . 1539 . 8460 56 5 .17508 .82192 6.7117 .17783 5.6234 1.0157 .01544 .98455 55 6 . 7537 . 2463 .7023 . 7813 .6140. .0167 . 1550 . 8450 54 7 . 7565 . 2435 .6930 . 7813 .6045 .0158 . 1656 . 8445 53 8 . 7594 . 2106 .6838 . 7873 .5951 .0158 . 1.560 . 8140 52 9 . 7622 . 2377 .6745 . 7903 .6867 .0169 . 1566 . 8435 51 10 .17651 .82349 5.6653 .17933 5.6764 1.0169 .01570 .98130 50 11 . 7680 . 2320 .6561 . 7963 .5670 .0160 . 1575 . 8425 49 12 . 7708 . 2291 .6470 . 7993 .5578 .0160 . 1680 . 8419 48 13 . 7737 . 2263 .6379 . 8023 .5186 .0161 . 1686 . 8414 47 14 . 7766 . 2234 .6288 . 8063 .6393 .0162 . 1591 . 8109 46 15 .17794 .82206 5.6197 .18083 5.5301 1.0162 .01596 .98404 46 16 . 7823 . 2177 .6107 . 8113 .6209 .0163 . 1601 . 8399 44 17 . 7852 . 2148 .6017 . 8113 .5117 .0163 . 1C06 . 8394 43 18 . 7880 . 2120 .6928 . 8173 .5026 .0161 . 1611 . 8388 42 19 . 7909 . 2091 .5838 . 8203 .4936 .0164 . 1617 . 8383 41 20 .17937 .82062 5.5719 .18233 5.1845 1.0166 .01622 .98378 40 21 . 7966 . 2031 .6660 . 8^63 .4755 .0165 . 1627 . 8373 39 22 . 7995 . 2005 .5672 . 8293 .1665 .0166 . 1632 . 8368 38 23 . 8023 . 1977 .5184 . 8323 .4575 .0166 . 1638 . 8362 37 24 . 8052 . 1948 .5396 . 8363 .4186 .0107 . 1613 . 8367 36 25 .18080 .81919 6.5308 .18383 5.1396 1.0167 .01618 .98352 36 26 . 8109 . 1891 ■ .6221 . 8413 .4308 .0168 . 1653 . 8347 34 27 . 8138 . 1862 .5134 . 8144 .1219 .0169 ■ 1659 . 8341 33 28 . 8166 . 1834 .5017 . 8474 .1131 .0169 . 1661 . 8336 32 29 . 8195 . 1805 .1960 . 8501 .4043 .0170 . 1669 . 8331 31 30 .18223 .81776 6.1874 .18531 6.3955 1.0170 .01674 .98326 30 31 . 8252 . 1748 .1788 . 8561 .3868 .0171 . 1680 . 8320 29 32 . 8281 . 1719 .1702 . 8591 .3780 .0171 . 1685 . 8315 28 33 . 8309 . 1691 .1617 . 8624 .3691 .0172 . 1690 . 8309 27 34 . 8338 . 1662 .1532 . 8654 .3607 .0172 . 1696 . 8304 26 35 .18366 .81633 6.4117 .18684 5.3521 1.0173 .01701 .98299 26 36 . 8395 . 1605 .4362 . 8714 .3134 .0174 . 1706 . 8293 24 37 . 8424 . 1576 .4278 . 8745 .3349 .0174 . 1712 . 8288 23 38 . 8452 . 1518 .1194 . 8775 .3263 .0176 . 1717 . 8283 22 39 . 8481 . 1519 .4110 . 8805 .3178 .0175 . 1722 . 8277 21 40 .18509 .81190 5.4026 .18836 5.3093 1.0176 .01728 .98272 20 41 . 8538 . 1162 .3943 . 8866 .3008 .0176 . 1733 . 8267 19 42 . 8567 . 1133 .3860 . 8896 .2923 .0177 . 1739 . 8261 18 43 . 8595 . 1405 .3777 . 8926 .2839 .0177 . 1714 . 8266 17 44 . 8624 . 1376 .3695 . 8955 .2755 .0178 . 1719 . 8250 16 45 .18652 .81318 6.3612 .18985 5.2671 1.0179 .01756 .98245 15 46 . 8681 . 1319 .3530 . 9016 .2588 .0179 . 1700 . 8240 14 47 . 8709 . 1290 .3449 . 9046 .2606 .0180 . 1766 . 8234 13 48 . 8738 . 1262 .3367 . 9076 .2422 .0180 . 1771 . 8229 12 49 . 8767 . 1233 .3286 . 9106 .2339 .0181 . 1777 . 8223 11 50 .18795 .81205 5.3205 .19136 6.2'257 1.0181 .01782 .98218 10 51 . 8824 . 1176 .3124 . 9166 .2174 .0182 . 1788 . 8212 9 52 . 8852 . 1117 .3044 . 9197 .2092 .0182 . 1793 . 8207 8 53 . 8881 . 1119 .2963 . 9227 .2011 .0183 . 1799 . 8201 7 54 . 8909 . 1090 .2883 . 9257 .1929 .0181 . 1804 . 8196 6 55 .18938 .81062 5.2803 .19287 5.1818 1.0184 .01810 .98190 5 56 . 8967 . 1033 .2721 . 9317 .1767 .0185 . 1815 . 8185 4 57 . 8995 . 1005 .2615 . 9347 .1686 .018S" . 1821 . 8179 3 58 . 9024 . 0976 .2566 . 9378 .1606 .0186 . 1826 . 8174 2 59 . 9052 . 0918 .2487 . 9108 .1525 .0186 . 1832 . 8168 1 00 . 9081 . 0919 .2108 . 9138 .1145 .0187 . 1837 . 8163 M. CoBiue. Vrs. sin. Secant. Cotaug. Tang. Cosec'nt Vrs. COS. Sine. M. 100° 79° Table 3. NATTJRAL FUNCTIONS. 333 11° Natural Trigonometrical Functions, 168° mT Sine. Vrs. COS. Cosec'nt Tang. Cotang. Secant. Vrs. Bill. Cosine. M. .19081 .80919 6.2408 .19438 5.1445 1.0187 .01837 .98163 60 1 . 9109 . 0890 .2330 . 9468 .1366 .0188 . 1843 . 8157 59 2 . 9138 . 0862 .2262 . 9498 .1286 .0188 . 1848 . 8152 58 3 . 9166 .0833 .2174 . 9629 .1207 .0189 . 1854 . 8146 57 4 . 9195 . 0805 .2097 . 9559 .1128 .0189 . 1859 . 8140 56 5 .19224 .80776 5.2019 .19589 5.1049 1.0190 .01866 .98135 55 6 ■. 9252 . 0748 .1942 . 9619 .0970 .0191 . 1871 . 8129 54 7 . 9281 .0719 .1866 •. 9649 .0892 .0191 . 1876 . 8124 53 8 . 9309 . 0691 .1788 . 9680 .0814 .0192 . 1882 . 8118 52 9 . 9338 . 0662 .1712 . 9710 .0736 .0192 . 1887 . 8112 61 10 .19366 .80634 5.1636 .19740 5.0658 1.0193 .01893 .98107 50 11 . 9395 . 0605 .1560 . 9770 .0581 .0193 . 1899 . 8101 49 12 . 9423 . 0576 .1484 . 9800 .0504 .0194 . 1904 . 8096 48 13 . 9452 . 0548 .1409 . 9831 .0427 .0196 . 1910 . 8090 47 14 . 9480 . 0519 .1333 . 9861 .0350 .0196 . 1916 . 8084 46 15 .19509 .80491 5.1268 .19891 5.0273 1.0196 .01921 .98078 45 16 . 9637 . 0462 .1183 . 9921 .0197 .0196 . 1927 . 8073 44 17 . 9566 . 0434 .1109 . 9952 .0121 .0197 . 1933 . 8067 43 18 . 9595 . 0406 .1034 . 9982 .0045 .0198 . 1938 . 8061 42 19 . 9623 . 0377 .0960 .20012 4.9969 .0198 . 1944 . 8066 41 20 .19652 .80348 5.0886 .20042 4.9894 1.0199 .01950 .98060 40 21 . 9680 . 0320 .0812 . 0073 .9819 .0199 . 1956 . 8044 39 22 . 97C9 . 0291 .0739 . 0103 .9744 .0200 . 1961 . 8039 38 23 . 9737 . 0263 .0666 . 0133 .9669 .0201 . 1967 . 8033 37 24 . 9766 . 0234 .0593 . 0163 .9694 .0201 . 1973 . 8027 36 25 .19794 .80206 5.0520 .20194 4.9520 1.0202 .01979 .98021 35 26 . 9823 . 0177 .0447 . 0224 .9446 .0202 . 1984 . 8016 34 27 . 9861 . 0149 .0375 . 0254 .9372 .0203 . 1990 . 8010 33 28 . 9880 . 0120 .0302 . 0285 .9298 .0204 . 1996 . 8004 32 29 . 9908 . 0092 .0230 . 0315 .9225 .0204 . 2002 . 7998 31 30 .19937 .80063 5.0158 .20345 4.9151 1.0206 .02007 .97992 30 31 . 9965 . 0035 .0087 . 0375 .9078 .0205 . 2013 . 7987 29 32 . 9994 . 0006 .0015 . 0406 .9006 .0206 . 2019 . 7981 28 83 .20022 .79978 4.9944 . 0436 .8933 .0207 . 2025 . 7975 27 34 . 0051 . 9949 .9873 . 0466 .8860 .0207 . 2031 . 7969 26 35 .20079 .79921 4.9802 .20497 4.8788 1.0208 .02037 .97963 25 36 . 0108 . 9892 .9732 . 0527 .8716 .0208 . 2042 . 7957 24 37 . 013B . 9863 .9661 . 05C7 .8644 .0209 . 2048 . 7952 23 38 . 0165 . 9835 .9591 . 0688 .8573 .0210 . 2054 . 7946 22 39 . 0193 . 9807 .9621 . 0618 .8501 .0210 . 2060 . 7940 21 40 .20222 .79778 4.9452 .20648 4.8430 1.0211 .02066 .97934 20 41 . 0250 . 9760 .9382 . 0679 .8359 .0211 . 2072 . 7928 19 42 . 0279 . 9721 .9313 . 0709 .8288 .0212 . 2078 . 7922 18 43 . 0307 . 9693 .9243 . 0739 .8217 .0213 . 2084 . 7916 17 44 . 0336 . 9664 .9175 . 0770 .8147 .0213 . 2089 . 7910 16 45 .20364 .79636 4.9106 .20800 4.8077 1.0214 .02095 .97904 15 46 . 0393 . 9607 .9037 . 0830 .8007 .0215 . 2101 . 7899 14 47 . 0421 . 9679 .8969 . 0861 .7937 .0216 . 2107 . 7893 13 48 . 0450 . 9660 .8901 . 0891 .7867 .0216 . 2113 . 7887 12 49 . 0478 . 9622 .8833 . 0921 .7798 .0216 . 2119 . 7881 11 50 .20506 .79493 4.8765 .20952 4.7728 1.0217 .02125 .97875 10 61 . 0535 . 9466 .8697 . 0982 .7659 .0218 . 2131 . 7869 9 52 . 0563 . 9436 .8630 . 1012 .7691 .0218 . 2137 . 7863 8 53 . 0592 . 9408 .8563 . 1043 ,7522 .0219 . 2143 . 7857 7 64 .0620 . 9379 .8496 . 1073 .7453 .0220 . 2149 . 7851 6 55 .20649 .79361 4.8429 .21104 4.7385 1.0220 .02165 .97845 6 66 . 0677 . 9323 .8362 . 1134 .7317 .0221 . 2161 . 7839 4 57 . 0706 . 9294 .8296 . 1164 .7249 .0221 . 2167 . 7833 3 58 . 0734 . 9266 .8229 . 1196 .7181 .0222 . 2173 . 7827 2 69 . 0763 . 9237 .8163 . 1226 .7114 .0223 . 2179 . 7821 1 60 . 0791 . 9209 .8097 . 1256 .7046 .0223 . 2185 . 7815 m7 Cosine. Vra. an. Secant. Cotang. Tang. Cosec'nt Vrs. COS. Sine. M. 101° 78° 334 NATURAL FUNCTIONS. Table b. 12 D Natural Trigonometrical Functions. 167° M. Sine. Vrs. COS. Cosec'nt Tang. Co tang. Secant. Vrs. sin. Cosine. M. .20791 .79209 4.8097 .21256 4.7046 1.0223 .02185 .97815 60 1 . 0820 . 9180 .8032 . 1286 .6979 .0224 . 2191 . 7809 69 2 . 0848 . 9152 .7966 . 1316 .6912 .0225 . 2197 . 7803 68 3 . 0876 . 9123 .7901 . 1347 .6845 .0225 . 2203 . 7806 57 4 . 0905 . 9105 .7835 . 1377 .6778 .0226 . 2209 . 7790 56 5 .20933 .79006 4.7770 .21408 4.0712 1.0226 .02215 .97784 55 6 . 0962 . 9038 .7706 . 1438 .6646 .0227 . 2222 . 7778 54 7 . 0990 . 9010 .7641 . 1468 .6580 .0228 . 2228 . 7772 53 8 . 1019 . 8981 .7576 . 1499 .6514 .0228 . 2234 . 7766 62 9 . 1047 . 8953 .7512 . 1529 .6448 .0229 . 2240 . 7760 51 10 .21076 .78924 4.7448 .21560 4.6382 1.0230 .02246 .97754 50 U . 1104 . 8896 .7384 . 1590 .6317 .0230 . 2252 . 7748 49 12 . 1132 . 8867 .7320 . 1621 .6252 .0231 . 2258 . 7741 48 13 . 1161 . 8839 .7257 . 1651 .6187 .0232 . 2264 . 7735 47 14 . 1189 . 8811 .7193 . 1682 .6122 .0232 . 2271 . 7729 46 15 .21218 .78782 4.7130 .21712 4.6057 1.0233 .02277 .97723 45 16 . 1246 . 8754 .7067 . 1742 .5993 .0234 . 2283 . 7717 44 17 . 1275 . 8726 .7004 . 1773 .5928 .0234 . 2289 . 7711 43 18 . 1303 . 8697 .6942 . 1803 .5864 .0235 . 2295 . 7704 42 19 . 1331 . 8668 .6879 . 1834 .5800 .0235 . 2302 . 7698 41 20 .21360 .78640 4.6817 .21864 4.5736 1.0236 .02308 .97692 40 21 . 1388 . 8612 .6754 . 1895 .6673 .0237 . 2314 . 7686 39 22 . 1117 . 8583 .6692 . 1925 .6609 .0237 . 2320 . 7680 38 23 . 1445 . 8555 .6631 . 1956 .6546 .0238 . 2326 . 7673 37 24 . 1473 . 8526 .6569 . 1986 .5483 .0239 . 2333 . 7667 36 25 .21502 .78508 4.6507 .22017 4.5420 1.0239 .02339 .97661 35 26 . 1530 . 8470 .6446 . 2047 .5357 .0240 . 2345 . 7655 34 27 . 1.559 . 8441 .6385 . 2078 .5294 .0241 . 2351 . 7648 33 28 . 1587 . 8413 .6324 . 2108 .5232 ,0241 . 2358 . 7642 32 29 . 1615 . 8384 .6263 . 2139 .5169 .0242 . 2364 . 7636 31 30 .21644 .78356 4.6202 .22169 4.5107 1.0243 .02370 .97630 30 31 . 1672 . 8328 .6142 . 2200 .5045 .0243 . 2377 . 7623 29 32 . 1701 . 8299 .6081 . 2230 .4983 .0244 . 2383 . 7617 28 33 . 1729 . 8271 .6021 . 2261 .4921 .0245 . 2389 . 7611 27 34 . 1757 . 8242 .5961 . 2291 .4860 .0245 . 2396 . 7604 26 35 .21786 .78214 4.5901 .22322 4.4799 1.0246 .02402 .97598 25 36 . 1814 . 8186 .5841 . 2353 .4737 .0247 . 2408 . 7592 24 37 . 1843 . 8154 .5782 . 2383 .4676 .0247 . 2415 . 7585 23 38 . 1871 . 8129 .5722 . 2414 .4615 .0248 . 2421 . 7.579 22 39 . 1899 . 8100 .5663 . 2444 .4555 .0249 . 2427 . 7573 21 40 .21928 .78072 4.5C04 .22475 4.4494 1.0249 .02434 .97566 20 41 . 1956 . 8043 .55J5 . 2505 .4434 .0250 . 2440 . 7560 19 42 . 1985 . 8015 .5486 . 2536 .4373 .0251 . 2446 . 7553 18 43 . 2013 . 7987 .5428 . 2566 .4313 .0251 . 24.53 . 7547 17 44 . 2041 . 7959 .5369 . 2597 .4263 .0252 . 2459 . 7541 16 45 .22070 .77930 4.5311 .22628 4.4194 1.0253 .02466 .97634 15 46 . 2098 . 7902 .5253 . 2658 .4134 .0253 . 2472 . 7528 14 47 . 2126 . 7873 .5195 . 2689 .4074 .0254 . 2479 . 7521 13 48 . 2155 . 7845 .5137 . 2719 .4015 .0265 . 2485 . 7515 12 49 . 2183 . 7817 .5079 . 2750 .3956 .0255 . 2491 7608 11 50 .22211 .77788 4.5021 .22781 4.3897 1.0256 .02498 .97502 10 51 . 2240 . 7760 .4964 . 2811 .3838 .0257 . 2504 . 7495 9 52 . 2268 . 7732 .4907 . 2842 .3779 .0257 . 2511 . 7489 8 63 . 2297 . 7703 .4850 . 2872 .3721 .0268 . 2517 . 7483 7 64 . 2325 . 7675 .4793 . 2903 .3662 .0259 . 2524 . 7476 6 55 .22353 .77647 4.4736 .22934 4.3604 1.0260 .02530 .97470 5 56 . 2382 . 7618 .4679 . 2964 .3646 .0260 . 2537 . 7463 4 57 . 2-110 7590 .4623 . 2995 .3488 .0261 . 2543 . 7457 3 58 2438 . 7561 .4566 . 3026 .3430 .0262 . 2550 . 7450 2 59 . 2467 . 7533 .4510 . 3056 •.3372 .0262 . 2556 . 7443 1 60 . 2495 . 7505 .4454 . 3087 .3315 .0263 . 2563 . 7437 C M. Cosine. Vrs. sin. Secant. Co tang. Tang. Cosec'ntl Vrs. COB. Sine. M. 102° 77" Table 3. NATUEAL FUNCTIONS. 335 13° Natural Trigonometrical Functions. 166° M. Sine. Vrs. COS. Cosec'nt Tang. Cotang. Secant. Vrs. sin. Cosine. M. .22495 .77505 4.4454 .23087 4.3315 1.0263 .02563 .97437 60 1 . 2523 . 7476 .4398 . 3137. .3257 .0264 . 2569 . 7430 59 2 . 2552 . 7448 .4312 . 3118 .3200 .0264 . 2576 . 7424 58 3 . 2580 . 7420 .4287 .3179 .3143 .0265 . 2.583 . 7417 57 4 . 2608 . 7391 .4231 . 3209 .3086 .0266 . 2589 . 7411 66 5 .22637 .77363 4.4176 .23240 4.3029 1.0266 .02596 .97404 65 6 . 2665 . 7335 .4121 . 3270 .2972 .0267 . 2602 . 7398 54 7 . 2693 . 7306 .4065 . 3301 .2916 .0268 . 2609 . 7391 63 8 . 2722 . 7278 .4011 . 3332- .2859 .0268 . 2616 . 7384 52 9 . 2750 . 7250 .3956 . .3363 .2803 .0269 . 2622 . 7378 51 10 .22778 .77221 4.3901 .23393 4.2747 1.0270 .02629 .97371 50 11 . 2807 . 7193 .3847 . 3424 .2691 .0271 . 2635 . 7364 49 12 . 2835 . 7165 .3792 . 3455 .2635 .0271 . 2642 . 7368 48 13 . 2863 . 7136 .3738 . 3485 .2579 .0272 . 2649 . 7351 47 14 . 2892 . 7108 .3684 . 3516 .2524 .0273 . 2655 . 7344 46 15 .22920 .77080 4.3630 .23547 4.2468 1.0273 .02662 .97338 45 16 . 2948 . 7052 .3676 . 3577 .2413 .0274 . 2669 . 7331 44 17 . 2977 . 7023 .3522 . 3608 .2358 .0275 . 2675 . 7324 43 18 . 3005 . 6995 .3469 . 3639 .2303 .0276 . 2682 . 7318 42 19 . 3033 . 6967 .3415 . 3670 .2218 .0276 . 2689 . 7311 41 20 .23061 .76938 4.3362 .23700 4.2193 1.0277 .02695 .97304 40 21 . 3090 . 6910 .3309 . 3731 .2139 .0278 . 2702 . 7298 39 22 . 3118 . 6882 .3256 . 3762 .2084 .0278 . 2709 . 7291 38 23 . 3146 ■ . 6853 .3203 . 3793 .2030 .0279 . 2716 . 7284 37 24 . 3175 . 6825 .3150 . 3823 .1976 .0280 . 2722 . 7277 36 25 .23203 .76797 4.3098 .23854 4.1921 1.0280 .02729 .97271 35 26 . 3231 . 6769 .3045 . 3885 .1867 .0281 . 2736 . 7264 34 27 . 3260 . 6740 .2993 . 3916 .1814 .0282 . 2743 . 7257 33 28 . 3288 . 6712 .2941 . 3946 .1760 .0283 . 2749 . 7250 32 29 . 3316 . 6684 .2888 . 3977 .1706 .0283 . 2756 . 7244 31 30 .23344 .76655 4.2836 .24008 4.1663 1.0284 .02763 .97237 30 31 . 3373 . 6627 .2785 . 4039 .1600 .0285 . 2770 . 7230 29 32 . 3401 . 6599 .2733 . 4069 .1516 .0285 . 2777 . 7223 28 33 . 3429 . 6571 .2681 . 4100 .1493 .0286 . 2783 . 7216 27 34 . 3458 . 6542 .2630 . 4131 .1440 .0287 . 2790 7210 26 35 .23486 .76514 4.2579 .24162 4.1388 1.0288 .02797 .97203 25 36 . 3514 . 6486 .2527 . 4192 .1335 .0288 . 2804 . 7196 ■24 37 . 3542 . 6457 .2476 . 4223 .1282. .0289 . 2811 . 7189 23 38 . 3571 . 6129 .2425 . 4254 .1230 .0290 . 2818 . 7182 22 39 . 3599 . 6401 .2375 . 4285 .1178 .0291 . 2824 . 7175 21 40 .23627 .76373 4.2324 .24316 4.1126 1.0291 .02831 .97169 20 41 . 3655 . 6344 .2273 . 4346 .1073 .0292 . 2838 . 7162 19 42 . 3684 . 6316 .2223 . 4377 .1022 .0293 . 2846 . 7165 18 43 . 3712 . 6288 .2173 . 4408 .0970 .0293 . 2852 . 7148 17 44 . 3740 . 6260 .2122 . 4439 .0918 .0294 . 2869 . 7141 16 45 .23768 .76231 4.2072 .24470 4.0867 1.0295 .02866 .97134 15 46 . 3797 . 6203 .2022 . 4501 .0815 .0296 . 2873 . 7127 14 47 . 3825 . 6175 .1972 . 4531 .0764 .0296 . 2880 . 7120 13 48 . 3853 . 6147 .1923 . 4562 .0713 .0297 . 2886 . 7113 12 49 . 3881 . 6118 .1873 . 4693 .0662 .0298 . 2893 . 7106 11 50 .23910 .76090 4.1824 .24624 4.0611 1.0299 .02900 .97099 10 51 . 3938 . 6062 .1774 .4655 .0560 .0299 . 2907 . 7092 9 52 . 3966 . 6034 .1725 .4686 .0509 .0300 . 2914 . 7086 8 5S . 3994 . 6005 .1676 . 4717 .0458 .0301 . 2921 . 7079 7 54 . 4023 . 5977 .1627 . 4747 .0408 .0302 . 2928 . 7072 6 55 .24051 .75949 4.1578 .24778 4.0368 1.0302 .02935 .97065 5 66 . 4079 . 5921 .1529 . 4809 .0307 .0303 . 2942 . 7058 4 57 . 4107 . 5892 .1481 . 4840 .0257 .0304 . 2949 . 7051 3 58 . 4136 . 5864 .1432 . 4871 .0207 .0305 . 2956 . 7044 2 59 . 4164 . 5836 .1384 . 4902 .0157 .0305 . 2963 . 7037 1 60 . 4192 . 5808 .1336 . 4933 .0108 .0306 . 2970 . 7029 M. CosiDe. Vrs. sin. Secant. Cotang. Tang. Cosec'nt Vrs. COS. Sine. M. 103° 76° 336 NATURAL FUNCTIONS. Table 3. 14° Natural T rigonometrical Functions. 1 55° 5L Sine. Vrs. COS. Cosec'nt Tang. Cotang, Secant. "Vrs. sin. Cosine. m! .24192 .75808 4.1336 .24933 4.0108 1.0306 .02970 .97029 60 1 . 4220 . 5779 .1287 . 4964 .0058 .0307 . 2977 . 7022 59 2 . 4249 . 5751 .1239 . 4995 .0009 .0308 . 2984 . 7015 58 3 . 4277 . 5723 .1191 . 5025 3.9959 .0308 . 2991 . 7008 57 4 . 4305 . 5695 .1144 . 5056 .9910 .0309 . 2999 . 7001 56 5 .24333 .75667 4.1096 .25087 3.9861 1.0310 .03006 .96994 55 6 . 4361 . 5638 .1048 . 5118 .9812 .0311 . 3013 . 6987 54 7 . 4390 . 5610 .1001 . 5149 .9763 .0311 . 3020 . 6980 53 8 . 4418 . 5582 .0953 . 5180 .9714 .0.312 . 3027 . 6973 .52 9 . 4146 . 5564 .0906 . 5211 .9665 .0313 . 3034 . 6966 51 10 .24474 .75526 4.0859 .25242 3.9616 1.0314 .03041 .96959 50 11 . 4602 . 5497 .0812 . 6273 .9.668 .0314 . 3048 . 6952 49 12 . 4531 . 5469 .0765 . 6304 .9520 .0315 . 3055 . 6944 48 13 . 4559 . 5441 .0718 . 5336 .9471 .0316 . 3063 . 6937 47 14 . 4587 . 5413 .0672 . 5366 .9423 .0317 . 3070 . 6930 46 15 .24615 .75385 4.0625 .25397 8.9375 1.0317 .0.3077 .96923 45 16 . 4643 . 6366 .0579 . 5128 .9327 .0318 . 3084 . 6916 44 17 . 4672 . 6328 .0532 . 6459 .9279 .0319 . 3091 . 6909 43 18 . 4700 . 6300 .0486 . 5490 .9231 .0320 . 3098 . 6901 42 19 . 4728 . 5272 .0440 . 5521 .9184 .0320 . 3106 . 6894 41 20 .24756 .75244 4.0394 .25562 3.9136 1.0321 .03113 .96887 40 21 . 4784 . 5215 .0348 . 5583 .9089 .0322 . 3120 . 6880 39 22 . 4813 . 6187 .0302 . 5614 .9042 .0323 . 3127 . 6873 38 23 . 4841 . 5159 .0266 . 5645 .8994 .0323 . 3134 . 6865 37 24 . 4869 . 5131 .0211 . 5676 .8947 .0324 . 3142 . 6858 36 26 .24897 .75103 4.0165 .25707 3.8900 1.0.326 .03149 .96851 35 26 . 4925 . 5075 .0120 . 5738 .8853 .0326 . 3166 . 6844 34 27 . 4963 . 5046 .0074 . 5769 .8807 .0327 . 3163 . 6836 33 28 4982 . 6018 .0029 . 5800 .8760 .0327 . 3171 . 6829 32 29 . 5010 . 4990 3.9984 . 5831 .8713 .0328 . 3178 . 6822 31 30 .25038 .74962 3.9939 .25862 3.8667 1.0329 .03185 .96815 30 31 5066 . 4934 .9894 . 5893 .8621 .0330 . 3192 . 6807 29 32 . 6094 . 4906 .9850 . 5924 .8574 .0330 . 3200 . 6800 28 33 . 5122 . 4877 .9805 . 5965 .8528 .0331 . 3207 . 6793 27 34 . 5151 . 4849 .9760 . 5986 .8482 .0332 . 3214 . 6785 26 35 .25179 .74821 3.9716 .26017 3.8436 1.0333 .03222 .96778 26 36 . 5207 . 4793 .9672 . 6048 .8390 .0334 . 3229 . 6771 ■24 37 . 5235 . 4765 .9627 . 6079 .8345 .0334 . 3236 . 6763 23 38 . 5263 . 4737 .9583 . 6110 .8299 .0335 . 3244 . 6766 22 39 . 5291 . 4709 .9539 . 6141 .8254 .0336 . 3251 . 6749 21 40 .25319 .74680 3.9495 .26172 3.8208 1.0337 .03258 .96741 20 41 . 5348 . 4652 .9451 . 6203 .8163 .0338 . 3266 . 6734 19 42 . 5376 . 4624 .9408 . 6234 .8118 .0338 . 3273 . 6727 18 43 . 6404 . 4596 .9364 . 6266 .8073 .0339 . 3281 . 6719 17 44 . 6432 . 4568 .9320 . 6297 .8027 .0340 . 3'288 . 6712 16 45 .26460 .74540 3.9277 .26328 3.7983 1.0341 .03295 .96704 15 46 . 5488 . 4612 .9234 . 6369 .7938 .0341 . 3303 . 6697 14 47 . 5516 . 4483 .9190 . 6390 .7893 .0342 . 3310 . 6690 13 48 . 5544 . 4465 .9147 . 6421 .7848 .0343 . 8318 . 6682 12 49 . 5573 . 4427 .9104 . 6462 .7804 .0344 . 3325 . 6675 11 50 .25601 .74399 3.9061 .26483 3.7759 1.0345 .03332 .96667 10 51 . 5629 . 4371 .9018 . 6514 .7715 .0345 . 3340 . 6660 9 52 . 5657 . 4344 .8976 . 6546 .7671 .0346 . 3347 . 6652 8 53 . 6685 . 4315 .8933 . 6577 .7027 .0347 . 3355 . 6645 7 54 . 5713 . 4287 .8890 . 6608 .7583 .0348 . 3362 . 6638 6 55 .25741 .74269 3.8848 .26639 3.7539 1.0349 .03370 .96630 6 56 . 5769 . 4230 .8805 . 0670 .7495 .0349 . 3377 . 6623 4 57 . 6798 . 4202 .8763 . 6701 .7461 .0360 . 3385 . 6615 3 58 . 6826 . 4174 .8721 . 6732 .7407 .0361 . 3392 . 6608 2 59 . 5864 . 4146 .8679 . 6764 .7364 .0352 . 3400 . 6600 1 60 . 5882 . 4118 .8637 . 6796 .7320 .0353 . 3407 . 6592 U. Cosine. Vrs. siu. Secant. Co tang. Taug. Cosec'ntI Vre. COS. Sine. M. 104° 75° Table 3. NATURAL FUNCTIONS. 337 15 3 Natural Trigonometrical Functions. 164° m7 Sine. Vts. cos. Ooeec'nt Tang. Ootting. Secant. ■Vrs. Bin. Cosine. M. .25882 .74118 3.8637 .26795 3.7820 1.0353 .03407 .96592 60 1 . 5910 . 4090 .8595 . 6826 .7277 .0353 . 3115 . 6585 59 2 . 5938 . 4062 .8553 6857 .7234 .0354 . 3422 . 6577 58 3 . 5966 . 4034 .8512 . to88 .7191 .0355 . 3430 . 6570 57 4 . 5994 . 4006 .8-170 . 6920 .7117 .0356 . 3438 . 6562 56 5 .26022 .73978 3.8428 .26951 3.7104 1.0367 .03445 .96.555 55 6 . 6050 . 3949 .WS7 . 6982 .7062 .0368 . 3453 . 6.547 54 7 . 6078 . 3921 .8346 . 7013 .7019 .0358 . 3460 . 6540 53 8 . 6107 . 3893 .8304 . 7044 .6976 .0359 . 3468 .6532 52 9 . 6135 . 3865 .8263 . 7076 .6933 .0360 . 3475 . 6524 51 10 .26163 .73837 3.8222 .27107 3.6891 1.0361 .03483 .96517 50 U . 6191 . 3809 .8181 . 7138 .6848 .0362 . 3491 . 6609 49 12 . 6219 . 3781 .8140 . 7169 .6806 .0362 . 3498 . 6502 48 13 . 6247 . 3753 .8100 . 7201 .6764 .0363 . 3506 . 6494 47 M . 6275 . 3725 .8059 . 7232 .6722 .0364 . 3514 . 6486 46 15 .26303 .73697 3.8018 .27263 3.6679 1.0365 .03521 .96479 45 16 . 6331 . 3669 .7978 . 7294 .6637 .0366 . 3529 . 6471 44 17 . 6359 . 3641 .7937 . 7326 .6596 .0367 . 3536 . 6463 43 18 . 6387 . 3613 .7897 . 7357 .6554 .0367 . 3544 . 6456 42 19 . 6415 . 3585 .7857 . 7388 .6512 .0368 . 3552 . 6448 41 20 .26443 .73556 3.7816 .27419 3.6470 1.0369 .03560 .96440 40 21 . 6471 . 3528 .7776 . 7451 .6429 .0370 . 3567 . 6433 39 22 . 6499 . 3500 .7736 . 7482 .6387 .0371 . 3575 . 6425 38 23 . 6527 . 3472 .7697 . 7513 .6346 .0371 . 3583 . 6117 37 24 . 6556 . 3444 .7657 . 7544 .6305 .0372 . 3590 . 6409 36 25 .26584 .73416 3.7617 .27576 3.6263 1.0373 .03598 .96402 35 26 . 6612 . 3388 .7577 . 7607 .6222 .0374 . 3606 . 6394 34 27 . 6640 . 3360 .7538 .7638 .6181 .0375 . 3614 . 6386 33 28 . 6668 . 3332 .7498 . 7670 .6140 .0376 . 3621 . 6378 32 29 . 6696 . &304 .7459 . 7701 .6100 .0376 . 3629 . 6371 31 30 .26724 .73276 3.7420 .27732 3.6059 1.0377 .03637 .96363 30 31 . 6752 . 3248 .7380 . 7764 .6018 .0378 . 3645 . 6355 29 32 . 6780 . 3220 .7341 . 7795 .5977 .0379 . 3652 . 6347 28 83 . 6808 . 3192 .7302 . 7826 .5937 .0380 . 3660 . 6340 27 34 . 6835 . 3164 .7263 . 7858 .6896 .0381 . 3668 . 6332 26 35 .26864 .73136 3.7224 .27889 3.5856 1.0382 .03676 .96324 25 36 . 6892 . 3108 .7186 . 7920 .5816 .0382 . 3684 . 6316 24 37 . 6920 . 3080 .7147 . 7952 .5776 .0383 . 3691 . 6308 23 38 . 6948 . 3052 .7108 . 7983 .5736 .0384 . 3699 . 6301 22 39 . 6976 . 3024 .7070 . 8014 .5696 .0385 . 3707 . 6293 21 40 .27004 .72996 3.7031 .28046 3.5656 1.0386 .03715 .96285 20 41 . 7032 . 2968 .6993 . 8077 .5616 .0387 . 3723 . 6277 19 42 . 7060 . 2940 .6955 . 8109 .5576 .0387 . 3731 . 6269 18 43 . 7088 . 2912 .6917 . 8140 .5536 .0388 . 3739 . 6261 17 44 . 7116 . 2884 .0878 . 8171 .5497 .0389 . 3746 . 6253 16 45 .27114 .72856 3.6810 .28203 3.5457 1.0390 .03754 .96245 15 46 . 7172 . 2828 .6802 . 8234 .5418 .0391 . 3762 . 6238 14 47 . 7200 . 2800 .6765 . 8266 .5378 .0392 . 3770 . 6230 13 48 . 7228 . 2772 .6727 . 8297 .5339 .0393 . 3778 . 6222 12 49 . 7256 . 2744 .6689 . 8328 .5300 .0393 . 3786 . 6214 11 50 .27284 .72716 3.6651 .28360 3.5261 1.0394 .03794 .96206 10 51 . 7312 . 2688 .6614 . 8391 .5222 .0395 . 3802 . 6198 9 52 . 7340 . 2660 .6576 . 8423 .5183 .0396 . 3810 . 6190 8 53 . 7368 . 2632 .6539 . 84S4 .5144 .0397 . 3818 . 6182 7 54 . 7396 . 2604 .6502 . 8486 .5105 .0398 . 3826 . 6174 6 55 .27424 .72576 3.6464 .28517 3.5066 1.0399 .03834 .96166 5 56 . 7452 . 2548 .6427 . 8519 .5028 .0399 . 3842 . 6158 4 57 . 7480 . 2520 .6390 . 8580 .4989 .0400 . 3850 . 6150 3 58 . 7508 . 2492 .6353 . 8611 .4951 .0401 . 3858 . 6142 2 59 . 7536 . 2464 .6316 . 8643 .4912 .0402 . 3866 . 6134 1 60 . 7564 . 2436 .6279 . 8674 .4874 .0403 . 3874 . 6126 M. Cosine. Vrs. Bin. Secant. Cotang, Tang. Cosec'nt Vrs. COS. Sine. M. 105° 23 74° 338 NATURAL J<'U^'CT1UJN«. 16° Natural Trigonometrical Functions. 163° M. Sine. Vrs. COB. Cosec'nt Tang. Cotang, Secant. Vrs. sin. Cosine. M. .27564 .72436 3.6279 .28674 3,4874 1,0403 .o;k-4 .96126 60 1 . 7592 . 2408 .6243 . 8706 .4836 .0404 . 3,882 . 6118 59 2 . 7620 . 2380 .6206 . 8737 .4798 .0405 . 3890 . 6110 58 3 . 7648 . 2352 .6109 . 8769 .4760 .0406 . 3898 . 6102 57 4 . 7675 . 2324 .6133 . 8800- .4722 .0406 . 3906 . 6094 56 5 .27703 .72296 3.6096 .28832 3,4684 1.0407 .03914 .90086 55 6 . 7731 . 2268 .6060 . 8863 .46-16 .0108 . 3922 . 6078 54 7 . 7759 . 2240 .6024 . 8895 .4608 .0409 . 3930 . 6070 53 8 . 7787 . 2213 .5987 . 8926 .4.570 .0410 . 39.38 . 6062 52 9 . 7815 . 2185 .5951 . 8958 .4533 .0411 . 3946 . 6054 51 10 .27843 .72157 3.5915 .28990 3.4495 1.0412 .03954 .96045 60 11 . 7871 . 2129 .5879 . 9021 .4458 .0413 . 3962 . 6037 49 12 . 7899 . 2101 .5843 . 9053 .4420 .0413 . 3971 . 6029 48 13 . 7927 . 2073 .5807 . 9084 .4383 .0414 . 3979 . 6021 47 14 . 7955 . 2045 .5772 . 9116 .4346 Mlb . 3987 . 6013 46 15 .27983 .72017 3.5736 .29147 3,4308 1.0416 .03995 .96005 45 16 . 8011 . 1989 .5700 . 9179 .4271 .0417 . 4003 . 5997 44 17 . 8039 . 1961 .5665 . 9210 .4234 .0418 . 4011 . 5989 43 18 . 8067 . 1933 .5629 . 9242 .4197 .0419 . 4019 . 5980 42 19 . 8094 . 1905 .5594 . 9274 .4160 .0420 . 4028 . 5972 41 20 .28122 .71877 3.5559 .29305 3,4124 1.0420 .04036 .95964 40 21 . 8150 . 1849 .5523 . 9337 .4087 .0421 . 4014 . 5956 39 22 . 8178 . 1822 .5488 . 9368 .4050 .0422 . 4052 . 5948 38 23 . 8206 . 1794 .5453 . 9400 .4014 .0123 . 4060 . 5940 37 24 . 8234 . 1766 .5418 . 9432 .3977 .0124 . 4069 . 5931 36 25 .28262 .71738 3.5383 .29463 3.3941 1,0125 .04077 .95923 36 26 . 8290 . 1710 .5348 . 9495 .3904 .0426 . 4085 . 5915 34 27 . 8318 . 1682 .5313 . 9526 .3868 .0427 . 4093 . 5907 33 28 . 8346 . 1654 .5279 . 9558 .3832 .0428 . 4101 . 5898 32 29 . 8374 . 1626 .5244 . 9.590 .3795 .0428 . 4110 . 5890 31 30 .28401 .71608 8,5209 .29621 3.3759 1,0429 .04118 .9.5882 30 31 . 8429 . 1570 .5175 . 9653 .3723 ,0430 . 4126 . 6874 29 32 . 8457 . 1543 .5140 . 9685 .3687 ,0431 . 4131 . 5865 28 33 . 8485 . 1515 .5106 . 9716 .3651 ,0432 . 4143 . 5857 27 34 . 8513 . 1487 .5072 . 9748 .3616 ,0433 . 4151 . 5849 26 35 .28541 .71459 3,5037 .29780 3,3580 1,04.34 .04159 .9.5840 25 36 . 8569 . 1131 .5003 . 9811 ,3514 ,0135 . 4168 . 5832 24 37 . 8597 . 1403 .4969 . 9843 .3509 .0436 . 4176 . 5824 23 38 . 8624 . 1375 .4935 . 9875 .3473 .0437 . 4184 . 5816 22 39 . 8652 . 1347 .4fl01 . 9906 .3438 ,0438 . 4193 . 5807 21 40 .28680 .71320 8,4867 .29938 3.3402 1,0438 .04201 .95799 20 41 . 8708 . 1292 .4833 . 9970 .3367 ,0439 . 4209 . 5791 19 42 . 8736 . 1204 .4799 .30001 .3332 .0440 . 4218 . 5782 18 43 . 8764 . 1236 .4766 . 0033 .3296 .0441 . 4226 . 5774 17 44 . 8792 . 1208 .4732 . 0065 .3261 .0442 . 4234 . 5765 16 45 .28820 .71180 3.4698 .30096 3.3226 1,0443 .04243 .96757 15 46 . 8847 . 1152 .4665 . 0128 .3191 ,0144 . 4251 . 5749 14 47 . 8875 . 1125 .4632 . 0160 .3156 ,0445 . 4260 . 5740 13 48 . 8903 . 1097 .4598 . 0192 .8121 ,0446 . 4268 . 5732 12 49 . 8931 . 1069 .4565 . 0223 .3087 .0447 . 4276 . 5723 U 50 .28959 .71041 3.4532 .30255 3.3052 1.0448 .04285 .95715 10 51 . 8987 . 1013 .4498 . 0287 .3017 .0448 . 4293 . 5707 9 52 . 9014 . 0985 .4465 . 0319 .2983 .0449 . 4302 . 5698 8 53 . 9042 . 0958 .4432 . 0350 .2948 .0150 . 4310 . 5690 7 64 . 9070 . 0930 .4399 , 0382 .2914 .0451 . 4319 . 5681 6 55 .29098 .70902 3.4366 ,30414 3,2879 1,04.52 .04327 .95673 6 56 . 9126 . 0874 .4334 0446 ,2845 ,0453 . 4335 . 5664 4 57 . 9154 . 0846 .4301 . 0178 .2811 ,0454 . 4344 . 5656 3 58 . 9181 . 0818 .426S . 0509 ,2777 ,0455 . 43.52 . 5647 2 59 . 9209 . 0791 .42:-;o . 0541 .2712 .0456 . 4361 . 5639 I 60 . 9237 . 0763 .4203 . 0573 .2708 .0457 . 4369 . 5630 M. Cosine. Vrs. sin. Secant. Co tang. Tung, Cosec'nt Vrs. COS. Sine. M. 106° 73° Table 3. NATUEAL FUNCTIONS. 339 ,70 Natural Trigonometrical Functions. 162° M. Sine. Vra. COB. Cosec'nt Tang. Cotang. Secant. Vrs. sin. Cosine. M. .29237 .70763 3.4203 .30573 3.2708 1.0457 .04369 .95630 60 1 . 9265 . 0735 .4170 . 0605 .2674 .0468 . 4378 . 6622 59 2 . 9293 . 0707 .4138 . 0637 .2640 .0459 . 4386 . 5613 58 3 . 9321 . 0679 .4106 . 0668 .2607 .0460 . 4395 . 5606 67 4 . 9348 . 0651 .4073 . 0700 .2573 .0461 . 4404 . 5596 55 5 .29376 .70624 3.4041 .30732 3.2639 1.0461 .04412 .96688 55 6 . 9404 . 0596 .4009 . 0764 .2605 .0462 . 4421 . 6579 54 7 . 9432 . 0568 .3977 . 0796 .2472 .0463 . 4426 . 6671 63 8 . 9460 . 0540 .3945 . 0828 .2438 .0404 . 4438 . 5562 .52 9 . 9487 . 0512 .3913 . 0859 .2405 .0465 . 4446 . 5564 51 10 .29515 .70485 3.3881 .30891 3.2371 1.0466 .04455 .95545 50 11 . 9543 . 0457 .3849 . 0923 .2338 .0467 . 4463 . 5536 49 12 . 9571 . 0429 .3817 . 0955 .2305 .0468 . 4472 . 5628 48 13 . 9598 . 0401 .3785 . 0987 .2271 .0469 . 4481 . 5519 47 14 . 9626 . 0374 .3754 . 1019 .2238 .0470 . 4489 . 5511 46 15 .29654 .70346 3.3722 .31051 3.2205 1.0171 .04498 .95502 45 16 . 9682 . 0318 .3690 . 1083 .2172 .0472 . 4507 . 5493 44 17 . 9710 . 0290 .3669 . 1115 .2139 .0473 . 4615 . 5485 43 18 . 9737 . 0262 .3627 . 1146 .2106 .0474 . 4624 . 5476 42 19 . 9765 . 0235 .3596 . 1178 .2073 .0475 . 4532 . 5467 41 20 .29793 .70207 3.3565 .31210 3.2041 1.0476 .04541 .96469 40 21 . 9821 . 0179 .3534 . 1242 .2008 .0477 . 4550 . 5450 39 22 . 9848 . 0151 .3502 . 1274 .1975 .0478 . 4558 . 6441 38 23 . 9876 . 0124 .3471 . 1306 .1942 .0478 . 4567 . 5433 37 24 . 9904 . 0096 .3440 . 1338 .1910 .0479 . 4576 . 5424 36 25 .29932 .70068 3..S409 .31370 3.1877 1.0480 .04585 .96416 35 26 . 9959 . 0040 .3378 . 1402 .1845 .0481 . 4593 . 5407 34 27 . 9987 . 0013 .3347 . 1434 .1813 .0482 . 4602 . 5398 33 28 .30015 .69982 .3316 . 1466 .1780 .0483 . 4611 . 5389 32 29 . 0043 . 9967 .3286 . 1498 .1748 .0484 . 4619 . 5380 31 30 .30070 .69929 3.3265 .31530 3.1716 1.0486 .04628 .95372 30 31 . 0098 . 9902 .3224 . 1662 .1684 .0486 . 4637 . 6363 29 32 . 0126 . 9874 .3194 . 1594 .1652 .0487 . 4646 . 5354 28 33 . 0154 . 9846 .3163 . 1626 .1620 .0488 . 4654 . 6345 27 34 . 0181 . 9818 .3133 . 1658 .1588 .0489 . 4663 . 5337 26 35 .30209 .69791 3.3102 .31690 3.1556 1,0490 .04672 .95328 25 36 . 0237 . 9763 .3072 . 1722 .1524 .0491 . 4681 . 5319 24 37 . 0265 . 9735 .3042 . 1754 .1492 .0192 . 4690 . 5310 23 38 . 0292 . 9707 .3011 . 1786 .1460 .0493 . 4698 . 5301 22 39 . 0320 . 9680 .2981 . 1818 .1429 .0494 . 4707 . 5293 21 40 .30348 .69652 3.2951 .31850 3.1397 1.0496 .04716 .96284 20 41 . 0375 . 9624 .2921 . 1882 .1366 .0496 . 4725 . 5276 19 42 . 0403 . 9597 .2891 . 1914 .1334 .0497 . 4734 . 6266 18 43 . 0431 . 9569 .2861 . 1946 .1303 .0498 . 4743 .5257 17 44 . 0459 . 9541 .2831 . 1978 .1271 .0499 . 4751 . 6248 16 45 .30486 .69513 3.2801 .32010 3.1240 1.0500 .04760 .95239 15 46 . 0514 . 9486 .2772 . 2042 .1209 .0501 . 4769 . 5231 14 47 . 0542 . 9458 .2742 . 2074 .1177 .0502 . 4778 . 5222 13 48 . 0569 . 9430 .2712 . 2106 .1146 .0503 . 4787 . 5213 12 49 . 0597 . 9403 .2683 . 2138 .1115 .0604 . 4796 . 5204 11 50 .30625 .69375 3.2653 .32171 3.1084 1.0505 .04805 .95195 10 51 . 0653 . 9347 .2624 . 2203 .1053 .0506 . 4814 . 5186 9 52 . 0680 . 9320 .2594 . 2235 .1022 .0507 . 4823 . 5177 8 53 . 0708 . 9292 .2565 . 2267 .0991 .0508 . 4832 . 5168 7 54 . 0736 . 9264 .2535 . 2299 .0960 .0509 . 4840 . 6169 6 55 .30763 .69237 3.2506 .32331 3.0930 1.0510 .04849 .95150 5 56 . 0791 . 9209 .2477 . 2363 .0899 .0511 . 4868 . 5141 4 57 . 0819 . 9181 .2448 . 2395 .0868 .0612 . 4867 . 5132 3 58 . 0846 . 9154 .2419 . 2428 .0838 .0513 . 4876 . 5124 2 59 . 0874 . 9126 .2390 . 2460 .0807 .0514 . 4885 . 6115 1 60 . 0902 . 9098 .2361 . 2492 .0777 .0515 . 4894 . 5106 M. Cosine. Vrs. sin. Secant. Cotang. Tang. Cosec'nt Vrs. cos. Sine. iT 107° 72° 340 NATURAL FUNCTIONS. Table 3. 18 o Natural Trigonometrical Functions. 161° M. Sine. Vrs. COB. Cosec'nt Tanpr. Cotang. Secant. Vrs. sin Cosine. M. .30902 .69098 3.2361 .32492 3.0777 1.0515 .04894 .96106 60 1 . 0929 . 9071 .2332 . 2524 .0746 .0516 . 4903 . 5097 59 2 . 0957 . 9043 .2303 . 2656 .0716 .0517 . 4912" . 5088 58 3 . 0985 . 9015 .2274 . 2588 .0686 .0518 . 4921 . 5079 bl 4 . 1012 . 8988 .2245 . 2621 .0655 .0519 . 4930 . 5070 56 5 .31040 .68960 3.2216 .32653 3.0625 1.0520 .04939 .95061 55 6 . 1068 . 8932 .2188 . 2685 .0595 .0521 . 4948 . 5051 54 7 . 1095 . 8905 .2159 . 2717 .0565 .0622 . 4957 . 5042 53 8 . 1123 . 8877 .2131 . 2749 .0535 .0523 . 4966 . 5033 52 9 . 1160 . 8849 .2102 . 2782 .0505 .0524 . 4975 . 5024 51 10 .31178 .68822 3.2074 .32814 3.0475 1.0525 .04985 .95015 60 11 . 1206 . 8794 .2045 . 2846 .0445 .0626 . 4994 . 5006 49 12 . 1233 . 8766 .2017 . 2878 .0415 .0.527 . 5003 . 4997 48 13 . 1261 . 8739 .1989 . 2910 .0385 .0528 . 5012 . 4988 47 14 . 1289 . 8711 .1960 . 2943 .0356 .0529 . 5021 . 4979 46 15 .31316 .68684 3.1932 .32975 3.0326 1.0530 .05030 .94970 45 16 . 1344 . 8656 .1904 . 3007 .0296 .0531 . 5039 . 4961 44 17 . 1372 . 8628 .1876 . 3039 .0267 .0532 . 5048 . 4952 43 18 . 1399 . 8601 .1848 . 3072 .0237 .0533 . 5057 . 4942 42 19 . 1427 . 8573 .1820 . 3104 .0208 .0534 . 6066 . 4933 41 20 .31151 .68645 3.1792 .33136 3.0178 1.0535 .06076 .94924 40 21 . 1482 . 8518 .1764 . 3169 .0149 .0536 . 5085 . 4915 39 22 . 1510 . 8190 .1736 . 3201 .0120 .0537 . 5094 . 4906 38 23 . 1537 . 8163 .1708 . 3233 .0090 .0538 . 5103 . 4897 .37 24 . 1565 . 8435 .1681 . 3265 .0061 .0539 . 5112 . 4888 36 25 .31592 .68407 3.1653 .33298 3.0032 1.0540 .06121 .94878 35 26 . 1620 . 8380 .1625 . 3330 .0003 .0641 . 5131 . 4869 34 27 . 1648 . 8352 .1598 . 8362 2.9974 .0642 . 5140 . 4860 33 28 . 1675 . 8325 .1570 . 3395 .9945 .0543 . 6149 . 4851 32 29 . 1703 . 8297 .1543 . 3427 .9916 .0544 . 6158 . 4841 31 30 .31730 .68269 3.1615 .33459 2.9887 1.0645 .05168 .94832 30 31 . 17S8 . 8242 .1488 . 3492 .9858 .0546 . 5177 . 4823 29 32 . 1786 . 8214 .1461 . 3524 .9829 .0547 . 5186 . 4814 28 S3 . 1813 . 8187 .1433 . 3557 .9800 .0648 . 5195 . 4805 27 34 . 1841 . 8159 .1106 . 3589 .9772 .0549 . 5205 . 4795 26 35 .31868 -68132 3.1379 .33621 2.9743 1.0560 .05214 .94786 25 36 . 1896 . 8104 .1352 . 3654 .9714 .0561 . 5223 . 4777 24 37 . 1923 . 8076 .1325 . 3686 .9686 .0552 . 5232 . 4767 23 38 . 1951 . 8049 .1298 . 3718 .9657 .0553 . 5242 . 4758 22 39 . 1978 . 8U21 .1271 . 37.61 .9629 .0654 . 5251 . 4749 21 40 .32006 .67994 3.1244 .33783 2.9600 1.0555 .05260 .94740 20 41 . 2034 . 7966 .1217 . 3816 .9672 .0556 . 5270 . 4730 19 42 . 2061 . 7939 .1190 . 3848 .9544 .0557 . 5279 . 4721 18 43 . 2089 . 7911 .1163 . 3880 .9515 .0658 . 5288 . 4712 17 44 . 2116 . 7884 .1137 . 3913 .9487 .0559 . 5297 . 4702 16 45 .32144 .67866 3.1110 .33945 2.9459 1.0660 .05307 .94693 15 46 . 2171 . 7828 .1083 . 3978 .9431 .0561 . 5316 . 4684 14 47 . 2199 . 7801 .1057 . 4010 .9403 .0562 . 5326 . 4674 13 48 . 2226 . 7773 .1030 . 4043 .9375 .0563 . 5335 . 4665 12 49 . 2254 . 7746 .1004 . 4075 .9347 .0565 . 5344 . 4655 11 60 .32282 .07718 3.0977 .34108 2.9319 1.0566 .05354 .94646 10 51 . 2309 . 7691 .0951 . 4140 .9291 .0.567 . 6363 . 4637 9 52 . 2337 . 7663 .0925 . 4173 .9263 .0568 . 5373 . 4627 8 53 . 2364 . 7636 .0898 . 4205 .9235 .0569 . 5382 . 4618 7 64 . 2392 . 7008 .0872 . 4238 .9208 .0570 . 5391 . 4608 Q 55 .32419 .67581 3.0846 .34270 2.9180 1.0571 .05401 .94599 5 56 . 2447 . 7653 .0820 . 4303 .9162 .0672 . 5410 . 4590 4 57 . 2474 . 7526 .0793 . 4335 .9326 .0573 . 5420 . 4580 3 58 . 2502 . 7498 .0767 . 4368 .9097 .0574 . 5429 . 4571 69 2529 7471 .0741 . 4400 .9069 .0575 . 5439 4561 i 60 . 2557 . 7443 .0715 . 4433 .9042 .0576 ,6448 .' 4552 M. Cosine. Vrs. sin. Secant. Colang. Tang. ;Josec'nt Vrs. cos. Sine. Im. 108 o 7 1° Table 3. NATURAL FUNCTIONS. 341 19° Natural Trigonometrical Functions. 160° M. Sine. Vra. COS. Cosec'nt Tang. Cotang. Secant. Vrs. sin. Cosine. M. .32557 .67443 3.0715 .34433 2.9042 1.0576 .06448 .94552 60 1 . 2584 . 7416 .0690 . 4465 .9015 .0577 . 6458 . 4542 59 2 . 2612 . 7388 .0664 . 4498 .8987 .0578 . 6467 . 4533 68 3 . 2639 .7361 .0638 . 4530 .8960 .0579 . 5476 . 4523 57 4 . 2667 . 7383 .0612 . 4563 .8933 .0580 . 5486 . 4614 56 5 .32694 .67306 3.0586 .34595 2.8905 1.0581 .05495 .94604 55 6 . 2722 . 7278 .0561 . 4628 .8878 .0582 . 5505 . 4495 54 7 . 2749 . 7251 .0535 . 4661 .8851 .0584 . 5515 . 4485 63 8 . 2777 .7223 .0509 . 4693 .8824 .0585 . 5524 . 4476 52 9 . 2804 . 7196 .0484 . 4726 .8797 .0586 . 5534 . 4466 51 10 .32832 .67168 3.0458 .34758 2.8770 1.0587 .05643 .94467 60 11 . 2859 . 7141 .0433 . 4791 .8743 .0588 . 5553 . 4447 49 12 . 2887 . 7113 .0407 . 4824 .8716 .0589 . 5562 . 4438 48 13 . 2914 . 7086 .0382 . 4856 .8689 .0590 . 5572 . 4-128 47 14 . 2942 . 7058 .0357 . 4889 .8662 .0591 . 6581 . 4418 46 15 .32969 .67031 3.0331 .34921 2.8636 1.0392 .05591 .94409 45 16 . 2996 . 7003 .0306 . 4954 .8609 .0693 . 5601 . 4399 44 17 . 3024 . 6976 .0281 . 4987 .8582 .0594 . 6610 . 4390 43 18 . 3051 . 6948 .0256 . 5019 .8555 .0695 . 5620 . 4380 42 19 . 3079 . 6921 .0281 . 5052 .8529 .0596 . 5629 . 4370 41 20 .33106 .66894 3.0206 .35085 2.8502 1.0598 .06639 .94361 40 21 . 3134 . 6866 .0181 . 5117 .8476 .0599 . 6649 . 4351 39 22 . 3161 . 6839 .0156 . 5150 .8449 .0600 . 6658 . 4341 38 23 . 3189 . 6811 .0131 . 5183 .8423 .0601 . 5668 . 4332 37 24 . 3216 . 6784 .0106 . 5215 .8396 .0602 . 6678 . 4322 36 25 .33243 .66756 3.0081 .35248 2.8370 1.0603 .05687 .94313 35 26 . 3271 . 6729 .0066 . 5281 .8344 .0604 . 6697 . 4303 34 27 . 3298 . 6701 .0081 . 6314 .8318 .0605 . 6707 . 4293 33 28 . 3326 . 6674 .0007 . 5346 .8291 .0606 . .5716 . 4283 32 29 . 3353 . 6647 2.9982 . 5379 .8265 .0607 . 5726 . 4274 31 30 .33381 .66619 2.9957 .35412 2.8239 1.0608 .05736 .94264 30 31 . 3408 . 6592 .9933 . 5445 .8213 .0609 . 6745 . 4254 29 32 . 3435 . 6564 .9908 . 5477 .8187 .0611 . 5766 . 4245 28 33 . 3463 . 6537 .9884 . 5510 .8161 .0612 . 5765 . 4285 27 34 . 3490 . 6510 .9859 . 5543 .8136 .0613 . 5776 . 4225 26 35 .33518 .66482 2.9835 .35576 2.8109 1.0614 .05784 .94215 25 36 . 3545 . 6455 .9810 . 6608 .8083 .0615 . 6794 . 4206 24 37 . 3572 . 6427 .9786 . 5641 .8057 .0616 . 6804 . 4196 23 38 . 3600 . 6400 .9762 . 5674 .8032 .0617 . 5814 . 4188 22 39 . 3627 . 6373 .9738 . 5707 .8006 .0618 . 5823 . 4178 21 40 .33655 .66345 2.9713 .35739 2.7980 1.0619 .05833 .94167 20 41 . 3682 . 6318 .9689 . 5772 .7964 .0620 . 5843 . 4157 19 42 . 3709 . 6290 .9665 . 5805 .7929 .0622 . 6853 . 4147 18 43 . 3737 . 6263 .9641 . 6838 .7903 .0623 . 5863 . 4137 17 44 . 3764 . 6236 .9617 . 5871 .7878 .0624 . 6872 . 4127 16 45 .33792 .66208 2.9593 .35904 2.7852 1.0626 .06882 .94118 15 46 . 3819 . 6181 .9569 . 5936 .7827 .0626 . 5892 . 4108 14 47 . 3846 . 6153 .9545 . 5969 .7801 .0627 . 5902 . 4098 13 48 . 3874 . 6126 .9521 . 6002 .7776 .0628 . 5912 . 4088 12 49 . 3901 . 6099 .9497 . 6035 .7751 .0629 . 6922 . 4078 11 50 .33923 .66071 2.9474 .36068 2.7726 1.0630 .06932 .94068 10 51 . 3956 . 6044 .9450 . 6101 .7700 .0632 . 6941 . 4058 9 52 . 3983 . 6017 .9426 . 6134 .7675 .0633 . 5951 . 4049 8 53 . 4011 . 5989 .9402 . 6167 .7650 .0634 . 5961 . 4039 7 54 . 4038 . 5962 .9379 . 6199 .7625 .0636 . 6971 . 4029 6 55 .34065 .65935 2.9355 .36232 2.7600 1.0636 .05981 .94019 5 56 . 4093 . 5907 .9332 . 6265 .7574 .0637 . 5991 . 4009 4 57 . 4120 . 5880 .9308 . 6298 .7549 .0638 . 6001 . 3999 3 58 . 4147 . 5853 .9285 . 6331 .7524 . .0639 . 6011 . 3989 2 59 . 4175 . 5825 .9261 . 6364 .7500 .0641 . 6021 . 3979 1 60 . 4202 . 5798 .9238 . 6397 .7475 .0642 . 6031 . 3969 mT Cosine. Vrs. sin. Secant. Ootang. Tang. Cosec'nt Vrs. cos. Sine. M^ 109° 70° 342 NATURAL FUNCTIONS. Table 3. 20 3 Natural Trigonom etrical Functions. 159° M. Sine. Vrs. COS. Cosec'nt Tang. Cofcing. Secant. Vrs. gin Cosine. M. .34202 .65798 2.9238 .36397 2.7475 1.0642 .06031 .93969 60 1 . 4229 . 5771 .9215 . 6430 .7450 .0643 . 6041 . 3959 59 2 . 4257 . 5743 .9191 . 6463 .7426 .0644 . 6051 . 3949 58 3 . 4284 . 5716 .9168 . 6196 .7400 .0645 . 6061 . 3939 57 4 . 4311 . 5689 .9115 . 6529 .7376 .0646 . 6071 . 3929 56 5 .34339 .65661 2.9122 .36562 2.7351 1.0647 .06080 .93919 55 6 . 4366 . 5634 .9098 . 6595 .7326 .0643 . 6090 . 3909 54 7 . 4393 . 5607 .9075 . 6628 .7302 .0650 . 6100 . 3899 53 8 . 4421 . 5579 .9052 . 6661 .7277 .0651 . 6110 . 3889 52 9 . 4448 . 5552 .9029 . 6694 .7262 .0652 . 6121 . 3879 61 10 .34475 .65525 2.9006 .36727 2.7228 1.0653 .06131 .93869 50 11 . 4502 . 5497 .8983 . 6700 .7204 .0654 . 6141 . 3859 49 12 . 4530 . 5470 .8960 . 6793 .7179 .0655 . 6151 . 3849 48 13 . 4557 . 5443 .8937 . 6826 .7155 .0656 . 6161 . 3839 47 14 . 4584 . 5415 .8915 . 6859 .7130 .0658 . 6171 . 3829 46 15 .34612 .65388 2,8892 .36892 2.7106 1.0659 .06181 .93819 45 16 . 4639 . 5361 .8869 . 6925 .7082 .0660 . 6191 . 3809 44 17 . 4666 . 5334 .8846 . 6958 .7058 .0661 . 6201 . 3799 43 18 . 4693 . 5306 .8824 . 6991 .7033 .0662 . 6211 . 3789 42 19 . 4721 . 5279 .8801 . 7024 .7009 .0663 . 6221 . 3779 41 20 .34748 .05252 2.8778 .37057 2.6985 1.0664 .06231 .93769 40 21 . 4775 . 5226 .8756 . 7090 .6961 .0666 . 6241 . 3758 39 22 . 4803 . 5197 .8733 . 7123 .6937 .0667 . 6251 . 3748 38 23 . 4830 . 5170 .8711 . 7156 .6913 .0668 . 6262 . 3738 37 24 . 48.57 . 5143 .8688 7190 .6889 .0669 . 6272 . 3728 36 25 .34884 .65115 2.8666 .37223 2.6865 1.0670 .06282 .93718 35 26 . 4912 . 5088 .8644 . 7256 .6841 .0671 . 6292 . 3708 34 27 . 4939 . 5061 .8621 . 72X9 .6817 .0673 . 6302 . 3698 33 28 . 4966 . 5034 .8599 . 7322 .6794 .0674 . 6312 . 3687 32 29 . 4993 . 5006 .8577 . 7356 .6770 .0675 . 6323 . 3677 31 30 .35021 .64979 2.8554 .37388 2.6746 1.0676 .06333 .93667 30 31 . 5048 . 4952 .8532 . 7422 .6722 .0077 . 6343 . 3657 20 82 . 5075 . 4926 .8510 . 7155 .6699 .0678 . 6353 . 3647 28 33 . 5102 . 4897 .8488 . 7488 .6675 .0679 . 6363 . 3637 27 34 . 5130 . 4870 .8466 . 7521 .6652 .0681 . 6373 . 3626 26 35 .35157 .6-1843 2.8444 .37554 2.6628 1.0682 .06384 .93616 25 36 . 5184 . 4816 .8422 7587 .6604 .0683 . 6394 . 3606 24 37 . 5211 . 4789 .8400 7G21 .6581 .0681 . 6404 . 3596 23 38 . 5239 . 4761 .8378 . 7654 .6558 .0686 . 6414 . 3585 22 39 . 5266 . 4734 .8356 7687 .6534 .0686 . 6425 . 3575 21 40 .35293 .64707 2.8334 .37720 2.6511 1.0688 .06435 .93565 20 41 . 5320 . 4680 .8312 77M .6487 .0689 . 6445 . 3555 19 42 . 5347 . 4652 .8290 . 7787 .6464 .0690 . 6456 . 3544 18 43 . 5375 . 4625 .8269 . 7820 .6441 .0691 . 6466 . 3534 17 44 . 5402 . 4598 .8247 . 7853 .6418 .0692 . 6476 . 3524 16 45 .35429 .64571 2.8225 .37887 2.6394 1.0694 .06486 .93513 15 46 . 5456 . 4544 .8204 . 7920 .6371 .0695 . 6497 . 3503 14 47 . 5483 . 4516 .8182 . 7953 .6348 .0696 . 6507 . 3493 13 48 . 5511 . 4489 .8160 . 7986 .6325 .0697 . 6517 . 3482 12 49 . 5538 . 4462 .8139 . 8020 .6302 .0698 . 6528 . 3472 11 50 .35665 .64435 2.8117 .38053 2.6279 1.0699 .06638 .93462 10 51 . 5592 . 4408 .8096 . 8086 .6266 .0701 . 6548 . 3451 9 52 . 5619 . 4380 .8074 . 8120 .6233 .0702 . 6559 . 3441 8 53 . 5647 . 4353 .8053 . 8153 .6210 .0703 . 6569 . 3431 7 54 . 5674 . 4326 .8032 . 8186 .6187 .0704 . 6579 . 3420 6 55 .35701 .64299 2.8010 .38220 2.6164 1.0705 .06590 ■ .93410 5 56 . 5728 . 4272 .7989 . 8263 .6142 .0707 . 6600 . 3400 4 57 . 5755 . 4245 .7968 . 8286 .6119 .0708 . 6611 . 3389 3 58 . 5782 4217 .7947 . 8320 .6096 .0709 . 6621 . 3379 2 59 . 5810 . 4190 .7925 . 8353 .6073 .0710 . 6631 . 3368 1 60 . 5837 . 4163 .7904 . 8386 .6051 .0711 . 6642 . 3358 M. Cosine. Vrs. sin. Secant. Cotang. Tang. ('osec'nt Vrs. COS. Sine. M. 110 o X no Table 3. NATURAL FUNCTIONS. 343 21° Natural Trigonometrical Functions. 158° mT Sine. Yre. C06. Cosec'nt Tang. Cotang. Secant. Vrs. bin. Cosine. M. .35837 .64163 2.7904 .38386 2.6051 1.0711 .06642 .93368 60 1 . 5864 . 4136 .7883 . 8420 .6028 .0713 . 6652 . 3348 59 2 . 5891 . 4109 .7862 . 8453 .6006 .0714 . 6663 . 3337 58 3 . 5918 . 4082 .7841 . 8486 .5983 .0715 . 6673 . 3327 57 4 . 5945 . 4055 .7820 . 8520 .5960 .0716 . 6684 . 3316 66 5 .35972 .64027 2.7799 .38553 2.5938 1.0717 .06694 .93306 55 6 . 6000 . 4000 .7778 . 8587 .5916 .0719 . 6705 . 3295 54 7 . 6027 . 3973 .7757 . 8620 .6893 .0720 . 6715 . 3285 53 8 . 6054 . 3946 .7736 . 8654 .5871 .0721 . 6726 . 3274 52 9 .6081 . 3919 .7715 . 8687 .5848 .0722 . 6736 . 3264 51 10 .36108 .63892 2,7694 .38720 2.6826 1.0723 .06747 .93253 50 11 . 6135 . 3865 .7674 . 8754 .5804 .0725 . 6757 . 3243 49 12 . 6162 . 3837 .7653 . 8787 .5781 .0726 . 6768 . 3232 48 13 . 6189 . 3810 .7632 . 8821 .5759 .0727 . 6778 . 3222 47 14 . 6217 . 3783 .7611 . 8854 .5737 .0728 . 6789 . 3211 46 15 .36244 .63756 2.7591 .38888 2.5715 1.0729 .06799 .93201 45 16 . 6271 . 3729 .7570 . 8921 .5693 .0731 . 6810 . 3190 44 17 . 6298 . 3702 .7550 . 8955 .6671 .0732 . 6820 . 3180 43 18 . 6325 . 3675 .7629 . 8988 .5640 .0733 . 6831 . 3169 42 19 . 6352 . 3648 .7609 . 9022 .5627 .0734 . 6841 . 3158 41 20 .36379 .63621 2.7488 .39055 2.5605 1.0736 .06862 .93148 40 21 . S406 . 3593 .7468 . 9089 .5583 .0737 . 6863 . 3137 30 22 . 6433 . 3566 .7447 . 9122 .5661 .0738 . 6873 . 3127 30 23 . 6460 . 3539 .7427 . 9156 .5639 .0739 . 6884 . 3116 37 24 . 6488 . 3512 .7406 . 9189 .5517 .0740 . 6894 . 3105 36 26 .36515 .63485 2.7386 .39223 2.6496 1.0742 .06905 .93095 36 26 . 6542 . 3458 .7366 . 9267 .6473 .0743 . 6916 . 3084 34 27 . 6569 . 3431 .7346 . 9290 .6451 .0744 . 6926 . 3074 33 28 . 6596 . 3404 .7325 . 9324 .5430 .0746 . 6937 . 3063 32 29 . 6623 . 3377 .7305 . 9357 .5408 .0747 . 6947 . 3052 31 30 .36660 .63360 2.7285 .39391 2.5386 1.0748 .06958 .93042 30 31 . 6677 . 3323 .7265 . 9425 .6366 .0749 . 6969 . 3031 29 32 . 6704 . 3296 .7216 . 9468 .6343 .0750 . 6979 . 3020 28 33 . 6731 . 3269 .7226 . 9492 .5322 .0751 . 6990 . 3010 27 34 . 6758 . 3242 .7205 . 9525 .5300 .0753 . 7001 . 2999 26 35 .36785 .63214 2.7185 .39559 2.5278 1.0754 .07012 .92988 25 36 . 6812 . 3187 .7165 . 9593 .5257 .0756 . 7022 . 2978 24 37 . 6839 . 3160 .7145 . 9626 .5236 .0756 . 7033 . 2967 23 38 . 6866 . 3133 .7126 . 9660 .5214 .0758 . 7044 . 2956 22 39 . 6893 . 3106 .7106 . 9694 .5193 .0769 . 7054 . 2946 21 40 .36921 .63079 2.7085 .39727 2.6171 1.0760 .07065 .92935 20 41 . 6948 . 3052 .7065 . 9761 .6150 .0761 . 7076 . 2924 19 42 . 6975 . 3025 .7046 . 9796 .5129 .0763 . 7087 . 2913 18 43 . 7002 . 2998 .7026 . 9828 .5108 .0764 . 7097 . 2902 17 44 . 7029 . 2971 .7006 . 9862 .5086 .0765 . 7108 . 2892 16 45 .37056 .62944 2.6986 .39896 2.5066 1.0766 .07119 .92881 15 46 . 7083 . 2917 .6967 . 9930 .5044 .0768 . 7130 . 2870 14 47 . 7110 . 2890 .6947 . 9963 .5023 .0769 . 7141 . 2859 13 48 . 7137 . 2863 .6927 . 9997 .5002 .0770 . 7151 . 2848 12 49 . 7164 . 2836 .6908 .40031 .4981 .0771 . 7162 . 2838 11 50 .37191 .62809 2.6888 .40065 2.4960 1.0773 .07173 .92827 10 51 . 7218 . 2782 .6869 . 0098 .4939 .0774 . 7184 . 2816 9 52 . 7215 . 2755 .6849 . 0132 .4918 .0775 . 7195 . 2805 8 53 .7272 . 2728 .6830 . 0166 .4897 .0776 . 7205 . 2794 7 54 . 7299 . 2701 .6810 . 0200 .4876 .0778 . 7216 . 2784 6 55 .37326 .62674 2.6791 .40233 2.4855 1.0779 .07227 .92773 5 56 . 7353 . 2647 .6772 . 0267 .4834 .0780 . 7238 . 2762 4 57 . 738C . 2620 .6762 . 0301 .4813 .0781 . 7249 . 2751 3 68 . 7407 . 2593 .6733 . 0336 .4792 .0783 . 7260 . 2740 2 59 . 7434 . 2566 .6714 . 0369 .4772 .0784 . 7271 . 2729 1 CO . 7461 . 2539 .6695 . 0403 .4761 .0785 . 7282 . 2718 M. Cosine. Vrs. Bin. Secant. Cotang. Tang. Cosec'nt Vrs. COS. Sine. M. 11° 68° 344 NATUEAL FUNCTIONS. Table 3. 22° Natural Trigonometrical Punctions. 1 57° M. Sine. Vra. coa. Cosec'nt Tang. Co tang. Secant. Vrs. sin Cosine. il. .G7461 .62639 2.6695 .40103 2.4761 1.0785 .07282 .92718 60 1 . 7488 . 2612 .6675 . 0136 .4730 .0787 . 7292 . 2707 69 2 . 7614 , 2485 .6656 . 0470 .4709 .0788 . 7303 . 2696 58 3 . 7641 . 2458 .6637 . 0504 .4689 .0789 . 7314 . 2686 57 4 . 7568 . 2431 .6618 . 0538 .4668 .0790 . 7325 . 2675 56 5 .37595 .62404 2.6599 .40572 2.4647 1.0792 .07336 .92664 55 6 . 7622 . 2377 .6680 . 0606 .4627 .0793 . 7347 . 2653 54 7 . 7649 . 2351 .6561 . 0640 .4606 .0794 . 7358 . 2642 63 8 . 7676 . 2324 .6542 . 0673 .4586 .0795 . 7369 . 2631 62 9 . 7703 . 2297 .6523 . 0707 .4566 .0797 . 7380 . 2620 51 10 .37730 .62270 2.6504 .40741 2.4515 1.0798 .07391 .92609 50 11 . 7757 . 2243 .6485 . 0775 .4525 .0799 . 7402 . 2598 49 12 . 7784 . 2216 .6466 . 0809 .4504 .0801 . 7413 . 2587 48 13 . 7811 . 2189 .6447 . 0843 .4484 .0802 . 7424 . 2676 47 14 . 7838 . 2162 .6428 . 0877 .4463 .0803 . 7436 . 2565 46 15 .37865 .62135 2.6410 .40911 2.4443 1.0804 .07446 .92554 45 16 . 7892 . 2108 .6391 . 0945 .4423 .0806 . 7457 . 2543 44 17 . 7919 . 2081 .6372 . 0979 .4403 .0807 . 7468 . 2532 43 18 . 7946 . 2054 .6353 . 1013 .4382 .0808 . 7479 . 2521 42 19 . 7972 . 2027 .6335 . 1047 .4362 .0810 . 7490 . 2610 41 20 .37999 .62000 2.6316 .41081 2.4342 1.0811 .07501 .92499 40 21 . 8026 . 1974 .6297 . 1116 .4322 .0812 . 7512 . 2488 39 22 . 8063 . 1947 .6279 . 1119 .4302 .0813 . 7523 . 2477 38 23 . 8080 . 1920 .6260 . 1183 .4282 .0815 . 7534 . 2466 37 24 . 8107 . 1893 .6242 . 1217 .4262 .0816 . 7546 . 2455 36 25 .38134 .61866 2.6223 .41251 2.4242 1.0817 .07556 .92443 35 26 . 8161 . 1839 .6206 . 1285 .4222 .0819 . 7567 . 2432 34 27 . 8188 . 1812 .6186 . 1319 .4202 .0820 . 7679 . 2421 33 28 . 8214 . 1786 .6168 . 1353 .4182 .0821 . 7690 . 2410 32 29 . 8241 . 1758 .6150 . 1387 .4162 .0823 . 7601 . 2399 31 30 .38268 .61732 2.6131 .41421 2.4142 1.0824 .07612 .92388 30 31 . 8295 . 1705 .6113 . 1465 .4122 .0825 . 7623 . 2377 29 32 . 8322 . 1678 .6095 . 1489 .4102 .0826 . 7634 . 2366 28 33 . 8349 . 1651 .6076 . 1524 .4083 .08'28 . 7645 . 2354 27 34 . 8376 . 1624 .6058 . 1558 .4063 .0829 7667 . 2343 26 35 .38403 .61597 2.6040 .41592 2.4043 1.0830 .07668 .92332 25 36 . 8429 . 1570 .6022 . 1626 .4023 .0832 . 7679 . 2321 24 37 . 8456 . 1514 .6003 . 1660 .4004 .0833 . 7690 . 2310 23 38 . 8483 . 1617 .6985 . 1694 .3984 .0834 . 7701 . 2299 22 39 . 8510 . 1490 .5967 . 1728 .3964 .0836 . 7712 . 2287 21 40 .38537 .61463 2.6919 .41762 2.3945 1.0837 .07724 .92276 20 41 . 8564 . 1436 .6931 . 1797 -.3925 .0838 . 7735 . 2265 19 42 . 8591 . 1409 .6913 . 1831 .3906 .0840 . 7746 . 2254 18 43 . 8617 . 1382 .6895 . 1865 .3886 .0841 . 7757 . 2242 17 44 . 8644 . 1366 .6877 . 1899 .3867 .0842 . 7769 . 2231 16 45 .38671 .61329 2.6859 .41933 2.3847 1.0814 .07780 .92220 15 46 . 8698 . 1302 .5841 . 1968 .3828 .0846 7791 . 2209 14 47 . 8725 . 1275 .5823 . 2002 .3808 .0816 . 7802 . 2197 13 48 . 8751 . 1248 .5805 . 2036 .3789 .0817 . 7814 . 2186 12 49 . 8778 . 1222 .5787 . 2070 .3770 .0849 . 7826 . 2175 11 50 .38805 .61195 2.5770 .42105 2.3760 1.0850 .07836 .92164 10 61 . 8832 . 1168 .5762 . 2139 .3731 .0861 . 7847 . 2152 9 52 . 8869 . 1141 .6734 . 2173 .3712 .0863 . 7869 . 2141 8 53 . 8886 . 1114 .6716 . 2207 .3692 .0854 . 7870 . 2130 7 54 . 8912 . 1088 .5699 . 2242 .3673 .0855 . 7881 . 2118 6 55 .38939 .61061 2.5681 .42276 2.3654 1.0857 .07893 .92107 5 66 . 8966 . 1034 .5663 . 2310 .3636 .0858 . 7904 . 2096 4 57 . 8993 . 1007 .5646 . 2344 .3616 .0859 . 7915 . 2084 3 68 . 9019 . 0980 .6628 . 2.379 .3597 .0861 . 7927 . 2073 2 59 . 9046 . 0954 .6610 . 2413 .8677 .0862 . 7938 . 2062 1 60 . 9073 . 0927 .6593 . 2447 .3558 .0864 . 7949 . 2050 M. Cosine. Vrs. sin. Secant. Co tang. Tang. Cosec'nt Vrs. COS. Sine. M. Table 3. NATURAL FUNCTIONS. 345 23° Natural Trigonometrical Functions. 156° M. Sine. Vr8. COS. Oosec'nt Tang. Cotang. Secant. Vrs. sin. Cosine. M. .39073 .60927 2.5593 .42447 2.3658 1.0864 .07949 .92050 60 1 . 9100 . 0900 .5575 . 2482 .3539 .0865 . 7961 . 2039 59 2 . 9126 . 0873 .5558 . 2616 .3520 .0866 . 7972 . 2028 58 3 . 9153 . 0846 .5540 . 2550 .3501 .0868 . 7984 . 2016 67 4 . 9180 . 0820 .5523 . 2581 .3482 .0869 . 7995 . 2005 56 5 .39207 .60793 2.5506 .42619 2.3463 1.0870 .08006 .91993 65 6 . 9234 . 0766 .5488 . 2654 .3445 .0872 . 8018 .1982 54 7 . 9260 . 0739 .5471 . 2688 .3426 .0873 . 8029 . 1971 53 8 . 9287 . 0713 .5453 . 2722 .3407 .0874 . 8041 . 1959 52 9 . 9314 . 0686 .5436 . 2757 .3388 .0876 . 8052 . 1948 51 10 .39341 .60659 2.5419 .42791 2.3369 1.0877 .08063 .91936 50 11 . 9367 . 0632 .5402 . 2826 .3350 .0878 . 8075 . 1925 49 12 . 9394 . 0606 .5384 . 2860 .3332 .0880 . 8086 . 1913 48 13 . 9421 . 0579 .5367 . 2894 .3313 .0881 . 8098 . 1902 47 14 . 9448 . 0552 .5350 . 2929 .3294 .0882 . 8109 . 1891 48 15 .39474 .60526 2.5333 .42963 2.3276 1.0884 .08121 .91879 45 ii; . 9501 . 0499 .5316 . 2998 ■• .3257 .0885 . 8132 . 1868 44 17 . 9528 . 0472 .5299 . 3032 .3238 .0886 . 8144 . 1856 43 18 . 9554 . 0445 .5281 . 3067 .3220 .0888 . 8155 . 1845 42 19 . 9581 . 0419 .5264 . 3101 .3201 .0889 . 8167 . 1833 41 20 .39608 .60392 2.5247 .43136 2.3183 1.0891 .08178 .91822 40 21 . 9635 . 0365 .5230 . 3170 .3164 .0892 . 8190 . 1810 39 22 . 9661 . 0339 .5213 . 3205 .3145 .0893 . 8201 . 1798 38 23 . 9688 . 0312 .5196 . 3239 .3127 .0895 . 8213 . 1787 37 24 . 9715 . 0285 .5179 . 3274 .3109 .0896 . 8224 . 1775 36 25 .39741 .60258 2.5163 .43308 2.3090 1.0897 .08236 .91764 35 26 . 9768 . 0232 .5146 . 3343 .3072 .0899 . 8248 . 1752 34 27 . 9795 . 0205 .5129 . 3377 .3053 .0900 . 8259 . 1741 33 28 . 9821 . 0178 .5112 . 3412 .3035 .0902 .8271 . 1729 32 29 . 9848 . 0152 .5095 . 3447 .3017 .0903 . 8282 . 1718 31 30 .39875 .60125 2.5078 .43481 2.2998 1.0904 .08294 .91706 80 31 . 9901 . 0098 .5062 . 3516 .2980 .0906 . 8306 . 1694 29 32 . 9928 . 0072 .5045 . 3550 .2962 .0907 . 8317 . 1683 28 33 . 99f)5 . 0045 .5028 . 3585 .2944 .0908 . 8329 . 1671 27 34 . 9981 . 0018 .6011 . 3620 .29-26 .0910 . 8340 . 1659 26 sr-, .40008 .59992 2.4995 .43654 2.2907 1.0911 .08352 .91648 25 36 . 0035 . 9965 .4978 . 3689 .2889 .0913 . 8364 . 1636 24 37 . 0061 . 9938 .4961 . 3723 .2871 .0914 . 8375 . 1625 23 38 . 0088 . 9912 .4945 . 3758 .2853 .0915 . 8387 . 1613 22 39 . 0115 . 9885 .4928 . 3793 .2835 .0917 . 8399 . 1601 21 40 .40141 .59858 2.4912 .43827 2.2817 1.0918 .08410 .91590 20 41 . 0168 . 9832 .4895 . 3862 .2799 .0920 . 8422 . 1578 19 42 . 0195 . 9805 .4879 . 3897 .2781 .0921 . 8434 . 1566 18 43 . 0221 . 9778 .4862 . 3932 .2763 .0922 . 8445 . 1554 17 44 . 0248 . 9752 .4846 . 3966 .2745 .0924 .8457 . 1643 16 45 .40275 .59725 2.4829 .44001 2.2727 1.0925 .08469 .91631 15 4G . 0301 . 9699 .4813 . 4036 •2709 .0927 . 8480 . 1519 14 47 . 0328 . 9672 .4797 . 4070 .2691 .0928 . 8492 . 1508 13 48 . 0354 . 9645 .4780 . 4105 .2673 .0929 . 8504 . 1496 12 49 . 0381 . 9619 .4764 . 4140 .2655 .0931 . 8516 . 1484 11 50 .40408 .59592 2.4748 .44176 2.2637 1.0932 .08527 .91472 10 51 . 0434 .9566 .4731 . 4209 .2619 .0934 . 8639 . 1461 9 52 . 0461 . 9539 .4715 . 4244 .2602 .0935 .8551 . 1449 8 53 . 0487 . 9512 .4699 . 4279 .2584 .0936 . 8563 . 1437 7 54 . 0514 . 9486 .4683 . 4314 .2566 .0938 . 8575 . 1425 6 55 .40541 .59459 2.4666 .44349 2.2548 1.0939 .08586 .91414 6 66 .0567 . 9433 .4660 . 4383 .2531 .0941 . 8598 . 1402 4 57 . 0594 . 9406 .4634 . 4418 .2513 .0942 . 8610 . 1390 3 58 . 0620 . 9379 .4618 4453 .2495 .0943 . 8622 .1378 2 59 . 0647 . 9353 .4602 . 4488 .2478 .0945 . 8634 . 1366 1 60 . 0674 . 9326 .4586 . 4523 .2460 .0946 . 8646 . 1354 M. Cosine, Vrs. sin. Secant. Cotang. Tang. Cosec'nt Vrs. COS. Sine. M. 346 NATURAL FUNCTIONS. Table 3. 24° Natural Trigonometrical Functions. iSS° M. Sine. Vrs. COS. Cosec'nt Tang. Cotang. Secant. Vra. sin. Cosine. M. .40674 .59326 2.4586 .44523 2.2460 1.0946 .08645 .91354 60 1 . 0700 . 9300 .4570 . 4558 .2443 .0948 . 8657 . 1343 59 2 . 0727 . 9273 .4554 . 4593 .2425 .0949 . 8669 . 1331 58 3 . 0753 . 9247 .4538 . 4627 .2408 .0951 . 8681 . 1319 57 4 . 0780 . 9220 .4622 . 4662 .2390 .0952 . 8693 . 1307 56 5 .40806 .59193 2.4506 .44697 2.2373 1.0953 .08705 .91295 55 6 . 0833 . 9167 .4490 . 4732 .2355 .0955 . 8716 . 1283 54 7 . 0860 . 9140 .4474 . 4767 .2338 .0956 . 8728 . 1271 53 8 . 0886 . 9114 .4458 . 4802 .2320 .0958 . 8740 . 1260 62 9 . 0913 . 9087 .4442 . 4837 .2303 .0959 . 8752 . 1248 51 10 .40939 .59061 2.4426 .44872 2.2286 1.0961 .08764 .91236 50 11 . 0966 . 9034 .4411 . 4907 .2268 .0962 . 8776 . 1224 49 12 . 0992 . 9008 .4395 . 4942 .2251 .0963 . 8788 . 1212 48 13 . 1019 . 8981 .4379 . 4977 .2234 .0965 . 8800 . 1200 47 14 . 1045 . 8955 .4363 . 5012 .2216 .0966 . 8812 . 1188 46 15 .41072 .58928 2.4347 .45047 2.2199 1.0968 .08824 .91176 45 16 . 1098 . 8901 .4332 . 5082 .2182 .0969 . 8836 . 1164 44 17 . 1125 . 8875 .4316 . 5117 .2165 .0971 . 8848 . 1152 43 18 . 1151 . 8848 .4300 . 5152 .2147 .0972 . 8860 . 1140 42 19 . 1178 . 8822 .4285 . 5187 .2130 .0973 . 8872 . 1128 41 20 .41204 .58795 2.4269 .45222 2.2113 1.0975 .08884 .91116 40 21 . 1231 . 8769 .4254 . 5257 .2096 .0976 . 8896 . 1104 39 22 . 1257 . 8742 .4238 . 5292 .2079 .0978 . 8908 . 1092 38 23 . 1284 . 8716 .4222 . 5327 .2062 .0979 . 8920 . 1080 37 24 . 1310 . 8689 .4207 . 5362 .2045 .0981 . 8932 . 1068 36 25 .41337 .58663 2.4191 .45397 2.2028 1.0982 .08944 .91056 36 26 . 1363 . 8636 .4176 . 5432 .2011 .0984 . 8956 . 1044 34 27 . 1390 . 8610 .4160 . 5467 .1994 .0985 . 8968 . 1032 33 28 . 1416 . 8584 .4145 . 5502 .1977 .0986 . 8980 . 1020 32 29 . 1443 . 8557 .4130 . 5537 .1960 .0988 . 8992 . 1008 31 30 .41469 .58531 2.4114 .45573 2.1943 1.0989 .09004 .90996 30 31 . 1496 . 8504 .4099 . 5608 .1926 .0991 . 9016 . 0984 29 32 . 1522 . 8478 .4083 . 6643 .1909 .0992 . 9028 . 0972 28 33 . 1549 . 8451 .4068 . 5678 .1892 .0994 . 9040 . 0960 27 34 . 1575 . 8425 .4053 . 5713 .1875 .0995 . 9052 . 0948 26 35 41602 .58398 2.4037 .45748 2.1S59 1.0997 .09064 .90936 25 36 . 1628 . 8372 .4022 . 5783 .1842 .0998 . 9076 . 0924 24 37 . 1654 . 8345 .4007 . 5819 .1825 .1000 . 9088 . 0911 23 38 . 1681 . 8319 .3992 . 5854 .1808 .1001 . 9101 . 0899 22 39 . 1707 . 8292 .3976 . 5889 .1792 .1003 . 9113 . 0887 21 40 .41734 .58266 2.3961 .45924 2.1775 1.1004 .09125 .90875 20 41 . 1760 . 8240 .3946 . 5960 .1758 .1005 . 9137 . 0863 19 42 . 1787 . 8213 .3931 . 5995 .1741 .1007 . 9149 . 0851 18 43 . 1813 . 8187 .3916 . 6030 .1725 .1008 . 9161 . 0839 17 44 . 1839 . 8160 .3901 . 6065 .1708 .1010 . 9173 . 0826 16 45 .41866 .58134 2.3886 .46101 2.1692 1.1011 .09186 .90814 15 46 . 1892 . 8108 .3871 . 6136 .1675 .1013 . 9198 . 0802 14 47 . 1919 . 8081 .3856 . 6171 .1658 .1014 . 9210 . 0790 13 48 . 1945 . 8055 .38'11 . 6205 .1642 .1016 . 9222 . 0778 12 49 . 1972 . 8028 .3826 . 6242 .1625 .1017 . 92.34 . 0765 11 50 .41998 ,58002 2.3811 .46277 2.1609 1.1019 .09247 .90753 10 51 . 2024 . 7975 .3796 . 6312 .1592 .1020 . 9259 . 0741 9 52 . 2051 . 7949 .3781 . 6348 .1576 .1022 . 9271 . 0729 8 53 . 2077 . 7923 .3766 . 6383 .1559 .1023 . 9283 . 0717 7 54 . 2103 . 7896 .3751 . 6418 .1543 .1025 . 9296 . 0704 6 55 .42130 .57870 2.3''36 .46454 2.1.527 1.1026 .09308 .90692 5 56 . 2156 . 7844 .3'21 . 6-189 .1510 .1028 . 9320 . 0680 4 57 . 2183 . 7817 .3706 . 6524 .1494 .1029 . 93.32 . 0668 3 58 . 2209 . 7791 .3691 . 6560 .1478 .1031 . 9.S45 . 0655 2 59 . 2235 . 7764 .3677 . 6595 .1461 .1032 . 9357 . 0643 1 60 . 2262 . 7738 .3662 . 6631 .1445 .1034 . 9369 . 0631 M. CoKine. Vrs. Bin. Secant. Cotang. Tang. Coeec'nt Vrs. COB. Sine. M. 114° 65° Table 3. NATUKAL FUNCTIONS. 347 25° Natural Trigonometrical Functions. J 54° M. Sine. Vrs. COS. Cosec'nt Tang. Cotang. Secant. Vrs. sin. Cosine. M. .42262 .57738 2.3662 .46631 2.1445 1.1034 .09369 .90631 60 1 . 2288 . 7712 .3647 . 6666 .1429 .1035 . 9381 .n618 59 2 . 2314 . 7685 .3632 . 6702 .1412 .1037 . 9394 . 0606 58 3 . 2341 . 7659 .3618 . 6737 .1396 .1038 . 9106 . 0594 67 4 .2367 . 7633 .3603 . 6772 .1380 .1040 . 9118 . 0681 56 5 .42394 .57606 2.3588 .46808 2.1364 1.1041 .09131 .90569 55 C . 2420 . 7580 .3574 . 6843 .1348 .1013 . 9413 . 0557 54 7 . 2446 . 7554 .3559 . 6879 .1331 .1044 . 9165 . 0611 53 8 . 2473 . 7527 .3544 . 6914 .1315 .1046 . 9468 . 0532 52 9 . 2499 . 7501 .3530 . 6950 .1299 .1047 . 9180 . 0520 51 10 .42525 .57475 2.3515 .46985 2.1283 1.1049 .09192 .90507 50 U . 2552 . 7418 .3601 . 7021 .1267 .1050 . 9605 . 0195 49 12 . 2578 . 7422 .3486 . 7056 .1251 .1062 . 9617, . 0183 48 13 . 2604 . 7396 .3472 . 7092 .1235 .1053 . 9530 . 0170 47 14 . 2630 . 7369 .3457 . 7127 .1219 .1056 . 9542 . 0458 46 15 .426.57 .57343 2.3443 .47163 2.1203 1.1056 .09551 .90115 45 16 . 2683 . 7317 .3428 . 7199 .1187 .1058 . 9567 . 0433 44 17 . 2709 . 7290 .3414 . 72.34 .1171 .1059 . 9579 . 0421 43 18 . 2736 . 7264 .3399 . 7270 .1155 .1061 . 9592 . 0408 42 19 . 2762 . 7238 .3385 . 7306 .1139 .1062 . 9604 . 0396 41 20 .42788 .57212 2.3371 .47341 2.1123 1.1061 .09617 .90383 40 21 . 2815 . 7185 .3356 . 7376 .1107 .1065 . 9629 . 0371 39 22 . 2841 . 7159 .3342 . 7412 .1092 .1067 . 9641 . 0358 3S 23 . 2867 . 7133 .3328 . 7448 .1076 .1068 . 9661 . 0316 37 24 . 2893 . 7106 .3313 . 7483 .1060 .1070 . 9666 . 0333 36 25 .42920 .57080 2.3299 .47519 2.1014 1.1072 .09679 .90321 35 26 . 2946 . 7054 .3285 . 7555 .1028 .1073 . 9691 . 0308 34 27 . 2972 . 7028 .8271 . 7590 .1013 .1075 . 9704 . 0296 33 28 . 2998 . 7001 .3256 . 7626 .0997 .1076 . 9716 . 0283 32 29 . 3025 . 6975 .3242 . 7662 .0981 .1078 . 9729 . 0271 31 30 .43051 .66949 2.3228 .47697 2.0966 1.1079 .09741 .90258 30 31 . 3077 . 6923 .3214 . 77.S3 .0950 .1081 . 9764 . 0216 29 32 . 3104 . 6896 .3200 . 7769 .0934 .1082 . 9766 . 0233 28 33 . 3130 . 6870 .3186 . 7805 .0918 .1081 . 9779 . 0221 27 34 . 3156 . 6844 .3172 . 7810 .0903 .1085 . 9792 . 0208 26 35 .43182 .56818 2.3158 .47876 2.0887 1.1087 .09804 .90196 25 36 . 3208 . 6791 .3143 . 7912 .0872 .1088 . 9817 . 0183 24 37 . 3235 . 6765 .3129 . 7948 .0856 .1090 . 9829 . 0171 23 38 . 3261 . 6739 .3115 . 7983 .0840 .1092 . 9842 . 0158 22 39 . 3287 . 6713 .3101 . 8019 .0825 .1093 . 9854 . 0115 21 40 .43313 .56685 2.3087 .18055 2.0809 1.1095 .09867 .90133 20 41 . 3340 . 6660 .3073 . 8091 .0794 .1096 . 9880 . 0120 19 42 . 3366 . 6634 .3069 . 8127 .0778 .1098 . 9892 . 0108 18 43 . 3392 . 6608 ..3046 . 8162 .0763 .1099 . 9905 . 0095 17 44 . 3418 . 6582 .3032 . 8198 .0747 .1101 . 9917 . 0082 16 45 .43444 .56555 2.3018 .48234 2.0732 1.1102 .09930 .90070 15 46 . 3471 . 6529 .3004 . 8270 .0717 .1101 . 9913 . 0057 14 47 . 3497 . 6503 .2990 . 8306 .0701 .1106 . 9956 . 0014 13 48 . 3523 . 6477 .2976 . 8342 .0686 .1107 . 9968 . 0032 12 49 . 8549 . 6451 .2962 . 8378 .0671 .1109 . 9981 . 0019 11 SO .43575 .56424 2.2949 .48414 2.0655 1.1110 .09993 .90006 10 61 . 3602 . 6398 .2936 . 8449 .0640 .1112 .10006 .89991 9 52 . 3628 . 0372 . .2921 . 8485 .0625 .1113 . 0019 . 9981 8 53 . 3654 . 6346 .2907 . 8521 .0609 .1115 . 0031 . 9968 7 54 . 3680 . 6320 .2894 . 8557 .0891 .1116 . 0044 . 9956 6 55 .43706 .56294 2.2880 .48593 2.0579 1.1118 .10057 .89943 5 56 . 3732 . 6267 .2866 . 8629 .0564 .1120 . 0070 . 9930 4 57 . 3759 . 6241 .2853 . 8665 .0518 .1121 . 0082 . 9918 3 58 . 3^85 . 6215 .2839 . 8701 .0633 .1123 . 0095 . 9905 2 59 . 3811 . 6189 .2825 . 8737 .0518 .1124 . 0108 . 9892 1 60 . 3837 . 6163 .2812 . 8773 .0503 .1126 . 0121 . 9879 mT Cosine. Vrs. sin. Secant. Cotang. Tang. Cosec'nt Vrs. coo. Sine. M. 348 NATURAL FUNCTIONS. Table 3. 26<: Natural Trigonometrical Functions. 153° M. Sine. Vre. CO.S. Cosec'nt Tang. Co tang. Secant. Vrs. Bin. Cosine. M. .4S837 .56163 2.2812 .48773 2.0503 1.1126 .10121 .89879 60 1 . 3863 . 6137 .2798 . 8809 .0488 .1127 . 0133 . 9867 59 2 . 3aS9 . 6111 .2784 . 8845 .0473 .1129 . 0146 . 9854 58 8 . 3915 . 0084 .2771 . 8881 .0458 .1131 . 0159 . 9841 57 4 . 3942 . 0058 .2757 . 8917 .0443 .1132 . 0172 . 9828 56 5 .43968 .56032 2.2744 .48953 2.0427 1.1134 .10184 .89815 55 6 . 3994 . 6006 .2730 . 8989 .0412 .1135 . 0197 . 9803 54 7 . 4020 . 5980 .2717 . 9025 .0397 .1137 . 0210 . 9790 53 8 . 4046 . 5954 .2703 . 9062 .0382 .1139 . 0223 . 9777 52 9 . 4072 . 5928 .2690 . 9098 .0367 .1140 . 0236 . 9764 51 10 .44098 .55902 2.2676 .49134 2.0352 1.1142 .10248 .89751 50 11 . . 4124 . 5875 .2663 . 9170 .0338 .1143 . 0261 . 9739 49 12 . 4150 . 5849 .2650 . 9206 .0323 .1145 . 0274 . 9726 48 IS . 4177 . 5823 .2636 . 9242 .0308 .1147 . 0287 . 9713 47 14 . 4203 . 5797 .2623 . 9278 .0293 .1148 . 0300 . 9700 46 15 .44229 .55771 2.2610 .49314 2.0278 1.1150 .10313 .89687 45 16 . 4255 . 5745 .2.596 . 9351 .0263 .1151 . 0326 . 9674 44 17 . 4281 . 5719 .2583 . 9387 .0248 .1153 . 0338 . 9661 43 18 . 4307 . 5693 .2570 . 9423 .0233 .1155 . 0351 . 9619 42 19 . 4333 . 5667 .2556 . 9459 .0219 .1156 . 0364 . 9636 41 20 .14359 .55641 2.2543 .49495 2.0204 1.1158 .10377 .89623 40 21 . 4385 . 5615 .2530 . 9.532 .0189 .1159 . 0390 . 9610 39 22 . 4411 . 5S89 .2517 . 9668 .0174 .1161 . 0403 . 9697 38 23 . 4437 . 5562 .2503 . 9604 .0159 .1163 . 0416 . 9684 37 24 . 4463 . 5536 .2490 . 9640 .0145 .1164 . 0429 . 9571 36 25 .44489 .55510 2.2477 .49077 2.0130 1.1166 .10442 .89658 35 26 . 4516 . 5484 .2464 . 9713 .0115 .1167 . 0455 . 9515 34 27 . 4542 . 5458 .2451 . 9749 .0101 .1169 . 0468 . 9532 33 28 . 4568 . 5432 .2438 . 9785 .0086 .1171 . 0481 . 9519 32 29 . 4594 . 5406 .2425 . 9822 .0071 .1172 . 0493 . 9.506 31 30 .44620 .55380 2.2411 .49858 2.0058 1.1174 .10606 .89493 30 31 . 4646 . 5354 .2398 . 9894 .0042 .1176 . 0619 . 9480 29 32 . 4672 . 5328 .2385 . 9931 .0028 .1177 . 0532 . 9467 28 33 . 4698 . 6302 .2372 . 9967 .0013 .1179 . 0545 . 9454 27 34 . 4724 . 5276 2359 .50003 1.9998 .1180 . 0558 . 9441. 26 35 .44750 .65250 2^2346 .50040 1.9984 1.1182 .10571 .89428 25 36 . 4776 . 5224 .2333 . 0076 .9969 .1184 . 0584 . 9415 24 37 . 4802 . 5198 .2320 . 0113 .9955 .1185 . 0598 . 9402 23 38 . 4828 . 6172 .2307 . 0149 .9940 .1187 . 0611 . 9389 22 39 . 4854 5146 .2294 . 0185 .9926 .1189 . 0624 . 9376 21 40 .44880 .55120 2.2282 .50222 1.9912 1.1190 .10637 .89363 20 41 . 4906 . 6094 .2269 . 0258 .9897 .1192 . 0650 . 9350 19 42 . 4932 . 5068 .2256 . 0295 .9883 .1193 . 0663 . 9337 18 43. . 4958 . 5042 .2243 . 0331 .9868 .1195 . 0676 . 9324 17 44 . 4984 . 5016 .2230 . 0368 .9854 .1197 . 0689 . 9311 16 45 .45010 .54990 2.2217 .50404 1.9840 1.1198 .10702 .89298 15 46 5036 . 4964 .2204 . 0441 .9825 .1200 . 0715 . 9285 14 47 . 5062 . 4938 .2192 . 0477 .9811 .1202 . 0728 . 9272 13 48 5088 4912 .2179 . 0514 .9797 .1203 . 0741 . 9258 12 49 . 5114 . 4886 .2166 . 0550 .9782 .1206 . 0754 . 9215 11 50 .45140 .54860 2.2153 .50587 1.9768 1.1207 .10768 .89232 10 51 . 5166 . 4834 .2141 . 0623 .9754 .1208 . 0781 . 9219 9 52 . 5191 . 4808 .2128 . 0660 .9739 .1210 . 0794 . 9206 8 53 . 5217 . 4782 .2115 . 0696 .9725 .1212' . 0807 . 9193 7 54 . ,5243 . 4756 .2103 . 0733 .9711 .1213 . 0820 . 9180 6 65 .45269 .54730 2.2090 .50769 1.9697 1.1215 .10833 .89166 5 56 . 6295 . 4705 .2077 . 0806 .9683 .1217 . 0846 . 9153 4 57 . 5321 . 4679 .2065 . 0843 .9668 .1218 . 0860 . 9140 3 58 5347 . 4653 .2052 . 0879 .9654 .12-20 . 0873 . 9127 2 59 . 5373 . 4627 .2039 . 0916 .9640 12,22 . 0886 . 9114 1 60 . 5399 . 4601 .2027 . 0952 .9626 .1223 . 0899 . 9101 M. Cosine. Vrs. sin. Secant. Co tang. Tang. Cosec'nt VrB. cos. Sine. M. 116° 63° Table 3. NATURAL FUNCTIONS. 349 27° Natural Trigonometrical Functions. 152" M. Sine. Vre. COB. Cosec'nt Taug. Co tang. Secant. Vi-a. sin. Cosine. M. .45399 .54601 2.2027 .50952 1.9626 1 1223 .10899 .89101 60 1 . 5425 . 4575 .2014 . 0989 .9612 .1225 . 0912 . 9087 59 2 . 5451 . 4549 .2002 . 1026 .9598 .1226 . 0926 . 9074 58 3 . 5477 . 4523 .1989 . 1062 .9684 .1228 . 0939 . 9061 57 4 . 5503 . 4497 .1977 . 1099 .9570 .1230 . 0952 . 9048 56 5 .45528 .54-171 2.1964 .51136 1.9656 1.1231 .10965 .89034 55 6 . 6554 . 4145 .1962 . 1172 .9542 .1233 . 0979 . 9021 54 7 . 5580 . 4420 .1939 . 1209 .9528 .1235 . 0992 . 9008 53 8 . 5606 . 4394 .1927 . 1246 .9514 .1237 . 1005 . 8995 52 9 . 5632 . 4368 .1914 . 1283 .9500 .1238 . 1018 . 8981 51 10 .45658 .51342 2.1902 .51319 1.9486 1.1240 .11032 .88968 50 11 . 6684 . 4316 .1889 . 1356 .9472 .1242 . 1045 . 8955 49 12 . 5710 . 4290 .1877 . 1393 .9458 .1243 . 1058 . 8942 48 13 . 5736 . 4264 .1865 . 1430 .9444 .1245 . 1072 . 8928 47 14 . 5761 . 4238 .1852 . 1466 .9430 .1247 . 1085 . 8915 46 15 .45787 .54213 2.1840 .51603 1.9416 1.1248 .11098 .88902 45 16 . 6813 . 4187 .1828 . 1540 .9402 .1250 . 1112 . 8888 44 17 . 5839 . 4161 .1815 . 1677 .9388 .1252 . 1125 . 8875 43 18 . 6865 . 4135 .1803 . 1614 .9375 .1253 . 1138 . 8862 42 19 . 6891 . 4109 .1791 . 1651 .9361 .1255 . 1152 . 8848 41 20 .45917 .54083 2.1778 .51687 1.9347 1.1267 .11165 .88835 40 21 . 5942 . 4057 .1766 . 1724 .9333 .1258 . 1178 , 8822 39 22 . 5968 . 4032 .1754 . 1761 .9319 .1260 . 1192 . 8808 38 23 . 5994 . 4006 .1742 . 1798 .9306 .1202 . 1205 . 8795 37 24 . 6020 . 3980 .1730 . 1835 .9292 .1264 . 1218 . 8781 36 25 .46046 .53954 2.1717 .51872 1.9278 1.1265 .11232 .88768 35 26 . 6072 . 3928 .1705 . 1909 .9264 .1267 . 1245 . 8765 34 27 . 6097 . 3902 .1693 . 1946 .9251 .1269 . 1259 . 8741 33 28 . 6123 . 3877 .1681 . 1983 .9237 .1270 . 1272 . 8728 32 29 . 6149 . 3851 .1669 . 2020 .9223 .1272 . 1285 . 8714 31 30 .46175 .53825 2.1657 .52057 1.9210 1.1274 .11299 .88701 30 31 . 6201 . 3799 .1645 . 2094 .9196 .1275 . 1312 . 8688 29 32 . 6226 . 3773 .1633 . 2131 .9182 .1277 . 1326 . 8674 28 33 . 6252 . 3748 .1620 . 2168 .9169 .1279 . 1339 . 8661 27 34 . 6278 . 3722 .1608 . 2205 .9155 .1281 . 1353 . 8647 26 35 .46304 .53696 2.1596 .52242 1.9142 1.1282 .11366 .88634 25 36 . 6330 . 3670 .1584 . 2279 .9128 .1284 . 1380 . 8620 24 37 . 6355 . 3645 .1572 . 2316 .9115 .1286 . 1393 . 8607 23 38 . 6381 . 3619 .1560 . 2353 .9101 .1287 . 1407 8593 22 39 . 6407 . 3593 .1548 . 2390 .9088 .1289 . 1420 . 8580 21 40 .46433 .53567 2.1536 .52427 1.9074 1.1291 .11434 .88666 20 41 . 6458 . 3541 .15'25 . 2464 .9061 .1293 . 1417 . 8563 19 42 . 6484 . 3516 .1513 . 2501 .9047 .1294 . 1461 . 8539 18 43 . 6510 . 3490 .1501 . 2638 .9034 .1296 . 1474 . 8526 17 44 . 6536 . 3464 .1489 . 2675 .9020 .1298 . 1488 . 8512 16 45 .46561 .53438 2.1477 .52612 1.9007 1.1299 .11501 .88499 15 46 . 6587 . 3413 .1465 . 2660 .8993 .1301 . 1515 . 8485 14 47 . 6613 . 3387 .1453 . 2687 .8980 .1303 . 1528 . 8472 13 48 . 6639 . 3361 .1441 . 2724 .8967 .1305 1642 . 8458 12 49 . 6664 . 3336 .1430 . 2761 .8953 .1306 . 1555 . 8444 11 60 .46690 .53310 2.1418 .52798 1.8940 1.1308 .11569 .88431 10 51 . 6716 . 3284 .1406 . 2836 .8927 .1310 . 1583 . 8417 9 52 . 6741 . 3258 .1394 . 2873 .8913 .1312 . 1596 . 8404 8 63 . 6767 . 3233 .1382 . 2910 .8900 .1313 . 1610 . 8390 7 64 . 6793 . 3207 .1371 . 2947 .8887 .1315 . 1623 . 8376 6 55 .46819 .53181 2.1359 .52984 1.8873 1.1317 .11637 .88363 5 66 . 6844 . 3156 .1347 . 3022 .8860 .1319 . 1651 . 8349 4 67 . 6870 . 3130 .1335 . 3059 .8847 (1320 . 1664 . 8336 3 68 . 6896 . 3104 .1324 . 3096 .8834 .1322 . 1678 . 8322 2 59 . 6921 . 3078 .1312 . 3134 .8820 .1324 . 1691 . 8308 1 60 . 69J7 . 3053 .1300 . 3171 .8807 .1326 . 1705 . 8295 M^ Cosine. Vrs. sin. Secant. Cotang. Tang. Cosec'nt Yrs. cos. Sine. M. Ii7° 62° S50 NATDEAL FUNCTIONS. Table 3. 28<^ Natural Trigonometrical Functions. JS1° M. Sine. Vrs. COS. Cosec'ut Tang. Cotang. Secant. Vrs. sin. Cosine, M. .46947 .53053 2.1300 .53171 1.8807 1.1326 .11705 ,88295 60 1 . 6973 . 3027 .1289 . 3208 .8794 .1327 . 1719 . 8281 59 2 . 6998 . 3001 .1277 . 3245 .8781 .1329 . 1732 . 8267 58 3 . 7024 . 2976 .1266 . 3283 .8768 .1331 . 1746 . 8254 57 4 . 7050 . 2950 .1264 . 3320 .8754 .1333 . 1760 . 8240 56 5 .47075 .52924 2.1242 .63358 1.8741 1.1334 .11774 .88226 55 6 . 7101 . 2899 .1231 . 3395 .8728 .1336 . 1787 . 8213 54 7 . 7127 . 2873 .1219 . 3432 .8715 .1338 . 1801 . 8199 5S 8 . 7152 . 2847 .1208 . 3470 .8702 .1340 . 1815 . 8185 52 9 . 7178 . 2822 .1196 . 3507 .8089 .1341 . 1828 . 8171 51 10 .47204 .52796 2.1185 .53545 1.8676 1.1343 .11842 .88158 50 11 . 7229 . 2770 .1173 . 3582 .8603 .1345 . 1856 . 8144 49 12 . 7255 . 2745 .1102 . 3619 .8650 .1347 . 1870 . 8130 48 13 . 7281 . 2719 .1150 . 3657 .8637 .1349 . 1883 . 8117 47 14 . 7306 . 2694 .1139 . 3694 .8624 .1350 . 1897 . 8103 46 15 .47332 .62668 2.1127 .53732 1.8611 1.1362 .11911 ■88089 45 16 . 7367 . 2642 .1116 . 3769 .8598 .1364 . 1925 . 8075 44 17 . 7383 . 2617 .1104 . 3807 .8585 .1356 . 1938 . 8061 43 18 . 7409 . 2591 .1093 . 3844 .8572 .1357 . 1952 . 8048 42 19 . 7434 . 2565 .1082 . 3882 .8569 .1359 . 1966 . 8034 41 20 .47460 .52540 2.1070 .53919 1.8546 1.1361 .11980 .88020 40 21 . 7486 . 2514 .1069 . 3957 .8533 .1363 . 1994 . 8006 39 22 . 7511 . 2489 .1048 . 3996 .8520 .1365 . 2007 . 7992 38 23 . 7537 . 2463 .1036 . 4032 .8507 .1366 . 2021 . 7979 37 24 . 7562 . 2437 .1025 . 4070 .8495 .1368 . 2035 . 7965 36 25 .47588 .52412 2.1014 .51107 1.8482 1.1370 .12049 .87951 35 26 . 7613 . 2386 .1002 . 4145 .8469 .1372 . 2063 . 7937 34 27 . 7639 . 2361 .0991 4183 .8456 .1373 . 2077 . 7923 33 28 . 7665 . 2335 .0980 . 4220 .8443 .1375 . 2090 . 7909 32 29 . 7690 . 2310 .0969 . 4268 .8430 .1377 . 2104 . 7895 31 30 .47716 .52284 2.0957 .54296 1.8418 1.1379 .12118 .87882 30 31 . 7741 . 2258 .0946 . 4333 .8405 .1381 . 2132 . 7868 29 32 . 7767 . 2233 .0935 . 4371 .8392 .1382 . 2146 . 7854 28 33 . 7792 . 2207 .0924 . 4409 .8379 .1384 . 2160 . 7840 27 34 . 7818 . 2182 .0912 . .4446 .8367 .1386 ■ . 2174 . 7826 26 36 .47844 .52156 2.0901 .54484 1.8354 1.1388 .12188 .87812 25 36 . 7869 . 2131 .0890 . 4522 .8341 .1390 . 2202 . 7798 24 37 . 7895 . 2105 .0879 . 4659 .8329 .1391 . 2216 . 7784 23 38 . 7920 . 2080 .0868 . 4597 .8316 .1393 . 2229 . 7770 22 39 . 7946 . 2054 .0867 . 4635 .8303 .1395 . 2243 . 7756 21 40 .47971 .52029 2.0846 .54673 1.8291 1.1397 .12257 .87742 20 41 . 7997 . 2003 .0835 . 4711 .8278 ,1399 . 2271 . 7728 19 42 . 8022 . 1978 .0824 . 4748 .8265 .1401 . 2285 . 7715 18 43 . 8048 . 1952 .0812 . 4786 .8253 .1402 . 2299 . 7701 17 44 . 8073 . 1927 .0801 . 4824 .8240 .1404 . 2313 . 7687 16 45 .48099 .51901 2.0790 .54862 1.8227 1.1406 .12327 .87673 16 46 . 8124 . 1876 .0779 . 4900 .8215 .1408 . 2341 . 7659 14 47 . 8160 . 1850 .0768 . 4937 .8202 .1410 . 2355 . 7645 13 48 . 8175 . 1825 .0757 . 4975 .8190 .1411 . 2369 . 7631 12 49 . 8201 . 1799 .0746 . 5013 .8177 .1413 . 2383 . 7617 11 50 .48226 .61774 2.0736 .55051 1.8165 1.1415 .12397 .87603 10 61 . 8252 . 1748 .0726 6089 .8152 .1417 . 2411 . 7688 9 52 . 8277 . 1723 .0714 . 6127 .8140 .1419 . 2425 . 7574 8' 63 . 8303 . 1697 .0703 . 6165 .8127 .1421 . 2439 . 7560 7 64 . 8328 . 1672 .0692 . 5203 .8115 .1422 . 2453 . 7546 6 85 .48354 .51646 2.0681 .55241 1.8102 1,1424 .12468 .87532 5 66 . 8379 . 1621 .0670 . 5279 .8090 .1426 . 2482 . 7518 4 67 . 8405 . 1695 .0659 . 5317 .8078 .1428 . 2496 . 7504 3 58 . 8430 . 1670 .0648 . 5355 .8065 .1430 . 2510 . 7490 2 59 . 8455 . 1644 .0637 . 6393 .8063 ,1432 . 2524 . 7476 1 60 . 8481 . 1519 .0627 . 6431 .8040 .1433 . 2538 . 7462 M. Cosine. Vrs, sin. Secant. Ootang. Tang. Gosec'nt Vrs, cos. Sine. jE 118° 6J° Table 3. NATURAL FUNCTIONS. 351 29° Natural Trigonom etrical Functions. . J 50° M. Sine. Vrs. COS. Cosfc'nt Tang. Cotang. Secant. Vrs. sin. Cosine. M. .48481 .51519 2.0627 .55431 1.8040 1.1433 .12538 .87462 60 1 . 8506 . 1493 .0616 . 5469 .8028 .1435 . 2552 . 7448 59 2 . 8532 . 1468 .0605 . 5507 .8016 .1437 . 2566 . 7434 58 3 . 8557 . 1443' .0594 . 5545 .8003 .1439 . 2580 . 7420 57 4 . 8583 . 1417 .0683 . 5583 .7991 .1441 . 2594 . 7406 66 5 .48608 .51392 2.0573 .55621 1.7979 1.1443 .12609 .87391 55 6 . 8533 . 13G6 .0562 . 6659 .7966 .1445 . 2623 . 7377 64 7 . 8659 . 1341 .0551 . 5697 .7964 .1446 . 2637 . 7363 53 8 . 8684 . 1316 .0540 . 5735 .7942 .1448 . 2661 . 7349 52 9 . 8710 . 1290 .0530 . 5774 .7930 .14,50 . 2665 . 7335 51 10 .48735 .51265 2.0519 .55812 1.7917 1.1452 .12679 .87320 50 11 . 8760 . 1239 .0508 . 6850 .7905 .1454 . 2694 . 7,306 49 12 . 8786 . 1214 .0498 . 5888 .7893 .1456 . 2708 . 7292 48 13 . 8811 . 1189 .0487 . 5926 .7881 .1458 . 2722 . 7278 47 14 . 8837 . 1163 .0476 . 6964 .7868 - .1459 . 27,36 . 7264 46 15 .48862 .51138 2.0466 .56003 1.7866 1.1461 .127.50 .87250 45 16 . 8887 . 1112 .0465 . 6041 .7844 .1463 . 2765 . 7235 44 17 . 8913 . 1087 .0444 . 6079 .7832 .1465 . 2779 . 7221 43 18 . 8938 . 1062 .0434 . 6117 .7820 .1467 . 2793 . 7207 42 19 . 8964 . 1036 .0423 . 6156 .7808 .1469 . 2807 . 7193 41 20 .48989 .51011 2.0413 .66194 1.7795 1.1471 .12821 .87178 40 21 . 9014 . 0986 .0402 . 6232 .7783 .1473 . 2836 . 7164 39 22 . 9040 . 0960 .0.392 . 6270 .7771 .1474 . 2850 . 7150 38 23 . 9065 . 0935 .0381 . 6309 .7759 .1476 . 2864 . 7136 37 24 . 9090 . 0910 .0370 . 6347 .7747 .1478 . 2879 . 7121 36 25 .49116 .60884 2.0360 .66385 1.7735 1.1480 .12893 .87107 35 26 . 9141 . 0859 .0349 . 6424 .7723 .1482 . 2907 . 7093 34 27 . 9166 . 0834 .0339 . 6462 .7711 .1484 . 2921 . 7078 33 28 . 9192 . 0808 .0329 . 6500 .7699 .1486 . 2936 . 7064 32 29 . 9217 . 0783 .0318 . 6539 .7687 .1488 . 2950 . 7050 31 30 .49242 .60758 2.0308 .56577" 1.7675 1.1489 .12964 .87035 30 31 . 9268 . 0732 .0297 . 6616 .7663 .1491 . 2979 . 7021 29 32 . 9293 . 0707 .0287 . 6654 .7651 .1493 . 2993 . 7007 28 33 . 9318 . 0682 .0276 . 6692 .7639 .1495 . 3007 . 6992 27 34 . 9343 . 0656 .0266 . 6731 .7627 .1497 . 3022 . 6978 26 35 .49369 .50631 2.0256 .66769 1.7615 1.1499 .13036 .86964 25 36 . 9394 . 0606 .0245 . 6808 .7603 .1501 . 3050 . 6949 24 37 . 9419 . 0580 .0235 . 6846 .7591 .1503 . 3065 . 6935 23 38 . 9445 . 0565 .0224 . 6886 .7579 .1505 . 3079 . 6921 22 39 . 9470 . 0530 .0214 . 6923 .7567 .1607 . 3094 . 6906 21 40 .49495 .50505 2.0204 .66962 1.7565 1.1608 .13108 .86892 20 41 . 9521 . 0479 .0194 . 7000 .7544 .1610 . 3122 . 6877 19 42 . 9M6 . 0454 .0183 . 7039 .7532 .1512 . 3137 . 6863 18 43 . 9571 . 0429 .0173 . 7077 .7520 .1614 . 3151 . 6849 17 44 . 9596 . 0404 .0163 . 7116 .7608 .1516 . 3166 . 6834 16 45 .49622 .50378 2.0152 .67165 1.7496 1.1518 .13180 .86820 15 46 . 9647 . 0363 .0142 . 7193 .7484 .1520 . 3194 . 6805 14 47 . 9672 . 0328 .0132 . 7232 .7473 .1522 . 3209 . 6791 13 48 . 9697 . 0303 .0122 . 7270 .7461 .1524 . 3223 . 6776 12 49 . 9723 . 0277 .0111 . 7309 .7449 ■ .1526 . 3238 . 6762 11 50 .49748 .60252 2.0101 .67.348 1.7437 1.1528 .13252 .86748 10 51 . 9773 . 0227 .0091 . 7386 .7426 .1530 . 3267 . 6733 9 52 . 9798 . 0202 .0081 . 7425 .7414 .1531 . 3281 . 6719 8 53 . 9823 . 0176 .0071 . 7464 .7402 .1633 . 3296 . 6704 7 54 . 9849 . 0151 .0061 . 7602 .7390 .1535 . 3310 . 6690 6 55 .49874 .50126 2.00.50 .57541 1.7379 1.1637 .13325 .86675 5 56 . 9899 . 0101 .0040 . 7580 .7367 .1539 . 3339 . 6661 4 57 . 9924 . 0076 .0030 . 7619 .7365 .1541 . 3354 . 6646 3 58 . 9950 . 0050 .0020 . 7657 .7344 .1543 . 3368 . 6632 2 69 . 9975 . 0025 .0010 . 7696 .7332 .1645 . 3383 . 6617 1 60 .50000 . 0000 .0000 . 7735 .7320 .1547 . 3397 . 6602 mT Cosine. Vrs. sin. Secant. Cotang. Tang. Cosec'nt Vrs. COS. Sine. M. U9° 60° 352 NATURAL FUNCTIONS. Table 3. 30 3 Natural Trigonometrical Functions. 149° M. Sine. Vrs. COS. Cosec'nt Tang. Co tang. Secant. Vrs. sin. Cosine. M. .50000 .50000 2.0000 .57735 1.7320 1.1547 .13397 .86602 60 1 . 0025 .49975 1.9990 . 7774 .7309 .1549 . 3412 . 6588 69 2 . 0050 . 9950 .9980 . 7813 .7297 .1551 . 3426 . 6573 58 S . 0075 . 9924 .9970 . 7851 .7286 .1553 . 3441 . 6559 67 4 . 0101 . 9899 .9960 . 7890 .7274 .1565 . 3456 . 6544 56 5 .50126 .49874 1.9950 .57929 1.7262 1.1567 .13470 .86630 55 6 . 0151 . 9849 .9940 . 7968 .7261 .1559 . 3485 . 6516 54 7 . 0176 . 9824 .9930 . 8007 .7239 .1561 . 3499 . 6500 53 8 . 0201 . 9799 .9920 . 8046 .7228 .1562 . 3514 . 6486 62 9 . 0226 . 9773 .9910 . 8085 .7216 .1564 . 3529 . 6171 61 10 .60252 .49748 1.9900 .58123 1.7205 1.1666 .13543 .86457 .50 11 . 0277 . 9723 .9890 . 8162 .7193 .1568 . 3568 . 6442 49 12 . 0302 . 9698 .9880 . 8201 .7182 .1570 . 3572 . 6427 48 13 . 0327 . 9673 .9870 . 8240 .7170 .1672 . 3587 . 6413 47 14 . 0352 . 9648 .9860 . 8279 .7169 .1574 . 3602 . 6398 46 15 .50377 .49623 1.9850 .58318 1.7147 1.1576 .13616 .86383 45 16 . 0402 . 9597 .9840 . 8357 .7136 .1578 . 3631 . 6369 44 17 . 0428 . 9572 .9830 . 8396 .7124 .1680 . 3646 . 6354 43 18 . 0453 . 9547 .9820 8435 .7113 .1582 . 3660 . 6339 42 19 . 0478 . 9522 .9811 . 8474 .7101 .1584 . 3675 . 6325 41 20 .50503 .49497 1.9801 .58513 1.7090 1.1.586 .13690 .86310 40 21 . 0528 . 9472 .9791 . 8552 .7079 .1588 . 3704 . 6295 39 22 . 0553 . 9447 .9781 . 8591 .7067 .1590 . 3719 . 6281 38 23 . 0578 . 9422 .9771 . 8630 .7056 .1592 . 3734 . 6266 37 24 . 0603 . 9397 .9761 . 8670 .7044 .1594 . 3749 . 6251 36 25 .50628 .49371 1.9752 .58709 1.7033 1.1596 .13763 .86237 35 26 . 0653 . 9346 .9742 . 8748 .7022 .1698 . 3778 . 6222 34 27 . 0679 . 9321 .9732 . 8787 .7010 .1600 . 3793 . 6207 33 28 . 0704 . 9296 .9722 . 8826 .6999 .1602 . 3807 . 6192 32 29 . 0729 . 9271 .9713 . 8865 .6988 .1604 . 3822 . 6178 31 30 .50754 .49246 1.9703 .58904 1.6977 1.1606 .13837 .86163 30 31 . 0779 . 9221 .9693 . 8944 .6965 .1608 . 3852 . 6148 29 32 . 0804 . 9196 .9683 . 8983 .6954 .1610 . 8867 . 6133 28 33 . 0829 . 9171 .9674 . 9022 .6943 .1612 . 3881 . 6118 27 34 . OS.'M . 9146 .9664 . 9061 .6931 .1614 . 3896 . 6104 26 35 .50879 .49121 1.9654 .59100 1.6920 1.1616 .13911 .86089 25 36 . 0904 . 9096 .9645 . 9140 .6909 .1618 . 3926 . 6074 24 37 . 0929 . 9071 .9635 . 9179 .6898 .1620 . 3941 . 6059 23 38 . 0954 . 9040 .9625 . 9218 .6887 .1622 . 3955 . 6044 22 39 . 0979 . 9021 .9616 . 9258 .6875 .1624 . 3970 . 6030 21 40 .6a0O4 .48996 1.9006 .59297 1.6864 1.1626 .13985 .86015 20 41 . 1029 . 8971 .9596 . 93.36 .6853 .1628 . 4000 . 6000 19 42 . 1054 . 8946 .9587 . 9376 .6842 .1630 . 4015 . 5985 18 43 . 1079 . 8921 .9577 . 9415 .6831 .1632 . 4030 . 5970 17 44 . 1104 . 8896 .9568 . 9454 .6820 .1634 . 4044 . 5965 16 45 .51129 .48871 1.9558 .59494 1.6808 1.1636 .14059 .85941 16 46 . 1154 . 8846 .9549 . 9533 .6797 .1638 . 4074 . 5926 14 47 . 1179 . 8821 .9539 . 9572 .6786 .1640 . 4089 . 5911 13 48 . 1204 . 8796 .9530 . 9612 .6775 .1642 . 4104 . 5896 12 49 . 1229 . 8771 .9520 . 9651 .6764 .1644 . 4119 . 6881 11 60 .51254 .48746 1.9510 .59691 1.6753 1.1646 .14134 .85866 10 51 . 1279 . 8721 .9501 . 9730 .6742 .1648 . 4149 . 5851 9 52 . 1304 . 8696 .9491 . 9770 .6731 .1650 . 4164 . 5836 8 53 1329 . 8671 .9482 . 9809 .6720 .1652 . 4178 . 5821 7 54 . 1354 . 8646 .9473 . 9849 .6709 .1654 . 4193 . 5806 6 55 .51379 .48621 1.9463 .59888 1.6698 1.1656 .14208 .85791 5 56 . 1404 . 8596 .9454 . 9928 .6687 .1658 . 4223 . 5777 4 57 . 1429 . 8571 .9444 . 9967 .6676 .1660 4238 . 5762 3 58 . 1454 . 8546 .9435 .60007 .6665 .1662 . 4253 . 5747 2 69 . 1479 . 8521 .9425 . 0046 .6654 .1664 . 4268 . 5732 1 60 . 1504 . 8496 .9416 . 0086 .6643 .1666 . 4283 . 5717 M, Cosine. Vrs. sin. Secant. Co tang. Tang. Cosec'nt Vrs. COS. Sine. M. 120° 59° Table 3. NATURAL FUNCTIONS. 353 31° Natural Trigonometrical Functions. 148° M. Sine. Vrs. COS. CoBoc'nt Tang. Co tang. Secant. Vrs. sin. Cosine. M. .51504 .48496 1.9416 .60086 1.6643 1.1666 .14283 .85717 60 1 . 1529 . 8471 .9407 . 0126 .6632 .1668 . 4298 . 5702 59 2 .1554 . 8446 .9397 . 0165 .6621 .1670 . 4313 . 5687 58 3 . 1578 . 8421 .9388 .0205 .6610 .1672 . 4328 . 5672 57 4 . 1603 . 8396 .9378 . 0244 .6599 .1674 . 4343 . 5657 56 5 .51628 .48371 1.9369 .60284 1.6588 1.1676 .14358 .85642 55 6 . 1653 . 8347 .9360 . 0324 .6577 .1678 . 4373 . 5627 54 7 . 1678 . 8322 .9350 . 0363 .6566 .1681 . 4388 . 5612 53 8 . 1703 . 8297 .9311 . 0403 .6555 .1683 . 4403 . 6597 52 9 . 1728 . 8272 .9332 . 0443 .6544 .1685 . 4418 . 5582 51 10 .51753 .48247 1.9322 .60483 1.6534 1.1687 .14433 .85566 50 11 . 1778 . 8222 .9313 . 0522 .6523 .1689 . 4418 . 5551 49 12 . 1803 . 8197 .9304 . 0562 .6512 .1691 . 4163 . 5536 48 13 . 1827 . 8172 .9295 . 0602 .6501 .1693 . 4479 . 5521 47 14 . 1852 . 8147 .9285 . 0642 .6490 .1695 . 4494 . 5606 46 15 .51877 .48123 1.9276 .60681 1.6479 1.1697 .14509 .85491 45 16 . 1902 . 8098 .9267 . 0721 .6469 .1699 . 4524 . s-ne 44 17 . 1927 . 8073 .9258 . 0761 .6468 .1701 . 4539 . 5461 43 18 . 1962 . 8048 .9248 . 0801 .6447 .1703 . 4554 . 5446 42 19 . 1977 . 8023 .9239 . 0841 .6436 .1705 . 4569 . 5431 41 20 .52002 .47998 1.9230 .00881 1.6425 1.1707 .14581 .86416 40 21 . 2026 . 7973 .9221 . 0920 .6415 .1709 . 4599 . 6400 39 22 . 2051 •. 7949 .9212 . 0960 .6404 .1712 . 4615 . 5385 38 23 . 2076 . 7924 .9203 . 1000 .6393 .1714 . 4630 '. 5370 37 24 . 2101 . 7899 .9193 . 1040 .6383 .1716 . 4645 . 5355 36 25 .52126 .47874 1.9184 .61080 - 1.6372 1.1718 .14660 .85340 35 26 . 2151 . 7849 .9175 . 1120 .6361 .1720 . 4675 . 5325 34 27 . 2175 . 7824 .9166 . 1160 .6350 .1722 . 4690 . 5309 33 28 . 2200 . 7800 .9157 . 1200 .6340 .1724 . 4706 . 5294 32 29 . 2225 . 7775 .9148 . 1240 .6329 .1726 . 4721 . 5279 31 30 .52250 .47760 1.9139 .61280 1.6318 1.1728 .14736 .85264 30 31 . 2275 . 7725 .9130 . 1320 .6308 .1730 . 4751 . 5249 29 32 . 2299 . 7700 .9121 . 1360 .6297 .1732 . 4766 . 5234 28 33 . 2324 . 7676 .9112 . 1400 .6286 .1734 . 4782 . 5218 27 34 . 2349 . 7651 .9102 . 1440 .6276 .1737 . 4797 . 6203 26 35 .52374 .47626 1.9093 .61480 1.6265 1.17.39 .14812 .86188 25 36 . 2398 . 7601 .9084 . 1520 .6255 .1741 . 4827 . 6173 '24 37 . 2423 . 7577 .9075 1560 .6244 .1743 . 4842 . 5157 23 38 . 2448 . 7552 .9066 . 1601 .6233 .1745 . 4858 . 5142 22 39 . 2473 . 7527 .9057 . 1611 .6223 .1747 . 4873 . 5127 21 40 .52498 .47502 1.9048 .61681 1.6212 1.1749 .14888 .85112 20 41 . 2522 . 7477 .9039 . 1721 .6202 .1751 . 4904 . 6096 19 42 . 2547 . 7453 .9030 . 1761 .6191 .1753 . 4919 . 6081 18 43 . 2572 . 7428 .9021 . 1801 .6181 .1756 . 4934 . 6066 17 44 . 2597 . 7403 .9013 . 1842 .6170 .1758 . 4949 . 6050 16 45 .52621 .47379 1.9004 ■ .61882 1.6160 1.1760 .14965 .85035 15 46 . 2616 . 7354 .8995 . 1922 .6149 .1762 . 4980 . 6020 14 47 . 2671 . 7329 .8986 . 1962 .6139 .1764 . 4995 . 5004 13 48 . 2695 . 7304 .8977 . 2004 .6128 .1766 . 5011 . 4989 12 49 . 2720 . 7280 .8968 . 2043 .6118 .1768 . 5026 . 4974 11 50 .52745 .47255 1.8959 .62083 1.6107 1.1770 .15041 .84959 10 61 . 2770 . 7230 .8950 . 2123 .6097 .1772 . 5067 . 4943 9 52 . 2794 . 7205 .8941 . 2164 .6086 .1775 . 6072 . 4928 8 53 . 2819 . 7181 .8932 . 2204 .6076 .1777 . 5087 . 4912 7 54 . 2844 . 7156 .8924 . 2244 .6066 .1779 . 5103 . 4897 6 55 .52868 .47131 1.8915 ■ .62285 1.6055 1.1781 .15118 .84882 5 56 . 2893 . 7107 .8906 . 2325 .6045 .1783 . 6133 . 4806 4 57 . 2918 . 7082 .8897 . 2366 .6034 .1785 . 5149 . 4851 3 58 . 2942 . 7057 .8888 . 2406 .6024 .1787 . 5164 . 4836 2 59 . 2967 . 7033 .8879 . 2416 .6014 .1790 . 5180 . 4820 1 60 . 2992 . 701'S .8871 . 2487 .6003 .1792 5195 . 4805 mT Cosine. Vrs, Bin, Secant. Co tang. Tang. Coeec'nt Vrs. cos. Sine. M. 121° 58° 354 NATURAL FUNCTIONS. Table 3. 32<: Natural Trigonometrical Functions. 147° M. Sine. Vrs. COS. Cosec'nt Tang. Co tang. Secant. Vrs. sin. Cosine. M. .52992 .47008 1.8871 .62487 1.6003 1.1792 .15195 .84805 60 1 . 3016 . 6983 .8862 . 2627 .5993 .1794 . 5211 4789 59 2 . 3041 . 6959 .8853 . 2568 .5983 .1796 . 5226 . 4774 58 3 . 3066 . 6934 .8844 . 2608 .5972 .1798 . 5241 . 4758 57 4 . 3090 . 6909 .8836 . 2649 .5962 .1800 . 5267 . 4743 56 5 .53115 .46885 1.8827 .62689 1.5952 1.1802 .15272 .84728 '55 6 . 3140 . 6860 .8818 . 2730 .5941 .1805 . 5288 . 4712 54 7 . 3164 . 6835 .8809 . 2770 .5931 .1807 . 5303 . 4697 53 8 . 3189 . 6811 .8801 . 2811 .5921 .1809 . 5319 . 4681 62 9 . 3214 . 6786 .8792 . 2851 .5910 .1811 . 5334 . 4666 51 10 .53238 .46762 1.8783 .62892 1.5900 1.1813 .15350 .84650 50 11 . 3263 . 6737 .8775 . 2933 .5890 .1815 . 5365 . 4635 49 12 . 3288 . 6712 .8766 . 2973 .5880 .1818 . 5381 . 4619 48 13 . 3312 . 6688 .8757 . 3014 .5869 .1820 . 5396 . 4604 47 14 . 3337 . 6663 .8749 . 3056 .5869 .1822 . 5412 . 4588 46 15 .53361 .46638 1.8740 .63095 1.5849 1.1824 .15427 .84673 45 16 . 3386 . 6614 .8731 . 3136 .5839 .1826 . 5443 . 4557 44 17 . 3111 . 6589 .8723 . 3177 .5829 .1828 . 5458 . 4542 43 18 . 3435 . 6565 .8714 . 3217 .5818 .1831 . 5474 . 4526 42 19 . 3460 . 6540 .8706 . 3258 .5808 .1833 . 5489 . 4511 41 20 .53484 .46516 1.8697 .63299 1.5798 1.1835 .16505 .84495 40 21 . 3509 . 6191 .8688 . 3339 .5788 .1837 . 5520 . 4479 39 22 . 3533 . 6466 .8680 . 3380 .5778 .1839 . 5536 . 4464 38 23 . 3558 . 6442 .8671 . 3121 .5768 .1841 . 5582 . 4448 37 24 . 3583 . 6417 .8663 . 3462 .5757 .1844 . 5567 . 4433 36 25 .53607 .40393 1.8654 .63603 1.5747 1.1846 .15583 .84417 35 26 . 3632 . 6368 .8646 . 8643 .5737 .1848 . 6698 4402 ,34 27 . 3656 . 6344 .8637 . 3684 .5727 .1850 . 5614 4386 33 28 . 3681 . 6319 .8629 . 3625 .5717 .1862 . 5630 . 4370 32 29 . 3705 . 6294 .8620 . 8666 .6707 .1865 . 5645 . 4355 31 30 .53730 .46270 1.8611 .63707 1.5697 1.1857 .15661 .84339 30 31 . 3754 . 6245 .8603 . 3748 .5687 .1859 . 5676 . 4323 29 32 . 3779 . 6221 .8595 . 3789 .5677 .1861 . 5692 . 4308 28 33 . 3803 . 6196 .8586 . 3830 .5667 .1863 . 5708 . 4292 27 34 . 3828 . 6172 .8578 . 3871 .6657 .1866 . 5723 . 4276 26 35 .53852 .46147 1.8569 .63912 1.5646 1.1868 .15739 .84261 25 36 . 3877 . 6123 .8561 . 3953 .5636 .1870 . 5755 . 4245 24 37 . 3901 . 6098 .8552 . 3994 .5626 .1872 . 5770 . 4229 23 38 . 3926 . 6074 .8544 . 4035 .6616 .1874 . 5786 . 4214 22 39 . 3950 . 6049 .8535 . 4076 .6606 .1877 . 5802 . 4198 21 40 .53975 .46025 1.8527 .64117 1.5596 1.1879 .15817 .84182 20 41 . 3999 . 6000 .8519 . 4168 .5586 .1881 . 5833 . 4167 19 42 . 4024 . 5976 .8510 . 4199 .6577 .1883 . 5849 . 4151 18 43 . 4048 . 5951 .8502 . 4240 .5567 .1886 . 5865 . 4135 17 44 . 4073 . 5927 .8493 . 4281 .5557 .1888 . 5880 . 4120 16 45 .54097 .45902 1.8485 .64322 1.5547 1.1890 .16896 .84104 15 46 . 4122 . 5878 .8477 . 4363 .5537 .1892 . 5912 . 4088 14 47 . 4146 . 5854 .8468 . 4404 .5527 .1894 . 5927 . 4072 13 48 . 4171 . 5829 .8460 . 4446 .5517 .1897 . 5943 . 4057 12 49 . 4195 . 5805 .8452 . 4487 .5607 .1899 . 5959 . 4041 11 50 .54220 .45780 1.8443 .64528 1.5497 1.1901 .15975 .84025 10 51 . 4244 . 5756 .8435 . 4569 .5487 .1903 . 5991 . 4009 y 52 . 4268 . 5731 .8427 4610 .5477 .1906 . 6006 . 3993 8 53 . 4293 . 5707 .8418 . 4052 .6467 .1908 . 6022 . 3978 7 54 . 4317 . 5682 .8410 . 4693 .5458 .1910 . 6038 . 3962 6 55 .54342 .45658 1.8402 .64734 1.5448 1.1912 .16064 .83946 5 66 . 4366 . 5634 .8394 . 4775 .5438 .1915 . 6070 . 3930 4 57 . 4391 . 5609 .8385 . 4817 .5428 .1917 . 6085 . 3914 3 58 . 4415 . 5585 .8377 . 4868 .5418 .1919 . 6101 . 3899 2 59 . 4439 . 5560 .8369 . 4899 .5408 .1921 . 6117 . 3883 1 60 . 4464 . 5536 .8361 . 4941 .6399 .1922 . 6133 . 3867 H. Cosine. Vre. sin. Secant. Cotang. Tang. Cosec'nt Vrs. COB. Sine. M. 122° 57° Table 3. NATURAL FUNCTIONS. 355 33° Natural Trigonometrical Functions. 146° M. Sine. Vrs. COS. Oosec'nt Tang. Cotang. Secant. Vrs. Bin. Cosine. M. .54464 .45536 1.8361 .64941 1.5399 1.1924 .16133 .83867 60 1 . 4488 . 5512 .8352 . 4982 .5389 .1926 . 6149 . 3851 59 2 . 4513 . 5487 .8344 . 5023 .5379 .1928 . 6165 . 3835 58 3 . 4537 . 5463 .8336 . 6065 .5369 .1930 . 6180 . 3819 57 4 . 4561 . 5438 .8328 . 5106 .5359 .1933 . 6196 . 3804 56 5 .54586 .45414 1.8320 .65148 1.5350 1.1935 .16212 .83788 55 6 . 4610 . 5390 .8311 . 5189 .5340 .1937 . 6228 . 3772 54 7 . 4634 . 5365 .8303 . 5231 .5330 .1939 . 6244 . 3756 53 8 . 4659 . 5341 .8295 . 5272 .5320 .1942 . 6260 . 3740 52 9 . 4683 . 5317 .8287 . 5314 .5311 .1944 . 6276 . 3724 51 10 .54708 .45292 1.8279 .65355 1.5301 1.1946 .16292 .83708 50 11 . 4732 . 5268 .8271 . 5397 .5291 .1948 . 6308 . 3692 49 12 . 4756 . 5244 .8263 . 5438 .5282 .1951 . 6323 . 8676 48 13 . 4781 . 5219 .8255 . 5480 .5272 .1953 . 6339 . 3660 47 14 . 4805 . 5195 .8246 . 5521 .5262 .1955 . 6355 . 3644 48 15 .54829 .45171 1.8238 .65563 1.5262 1.1958 .16371 .83629 45 10 . 4854 . 5146 .8230 . 5604 .5243 .1960 . 6387 . 3613 44 17 . 4878 . 5122 .8222 . 5646 .5233 .1962 . 6403 . 3597 43 18 . 4902 . 5098 .8214 . 5688 .5223 .1964 . 6419 . 3581 42 19 . 4926 . 5073 .8206 . 5729 .5214 .1967 . 6435 . 3565 41 20 .54951 .45049 1.8198 .65771 1.6204 1.1969 .16451 .83549 40 21 . 4975 . 5025 .8190 . 5813 .5195 .1971 . 6467 . 3533 39 22 . 4999 . 5000 .8182 . 5864 .6185 .1974 . 6483 . 3617 38 23 . 5024 . 4976 .8174 . 5896 .5175 .1976 . 6499 . 3601 37 24 . 5048 . 4952 .8166 . 5938 .5166 .1978 . 6515 . 3485 36 25 .55072 .44928 1.8158 .65980 1.6156 1.1980 .16531 .83469 35 26 . 5097 . 4903 .8150 . 6021 .5147 .1983 . 6547 . 3453 34 27 . 5121 . 4879 .8142 . 6063 .6137 .1985 . 6563 . 3437 33 28 . 5145 . 4855 .8134 . 6105 .6127 .1987 . 6679 . 3421 32 29 . 5169 . 4830 .8126 . 6147 .6118 .1990 . 6595 . 3405 31 30 .55194 .44806 1.8118 .66188 1.5108 1.1992 .16611 .83388 30 31 . 5218 . 4782 .8110 . 6230 .6099 .1994 . 6627 . 3372 29 32 . 5242 . 4758 .8102 . 6272 .6089 .1997 . 6643 . 3356 28 33 . 5266 . 4733 .8094 . 6314 .6080 .1999 . 6660 . 3340 27 34 . 5291 . 4709 .8086 . 6356 .5070 .2001 . 6676 . 3324 26 35 .55315 .44685 1.8078 .66398 1.5061 1.2004 .16692 .83308 25 3G . 5339 . 4661 .8070 . 6440 .5051 .2006 . 6708 . 3292 24 37 . 5363 . 4637 .8062 . 6482 .5042 .2008 . 6724 . 3276 23 38 . 5388 . 4612 .8054 . 6524 .5032 .2010 . 6740 . 3260 22 39 . 5112 . 4588 .8047 . 6666 .6023 .2013 . 6756 . 3244 21 40 ..55436 .44564 1.8039 .66608 ■ 1.5013 1.2015 .16772 .83228 20 41 . 5460 . 4540 .8031 . 6650 .5004 .2017 . 6788 . 3211 19 42 . 5484 . 4515 .8023 . 6692 .4994 .2020 . 6804 . 3195 18 43 . 5509 . 4491 .8015 . 6734 .4985 .2022 . 6821 . 3179 17 44 . 5533 . 4467 .8007 . 6776 .4975 .2024 . 6837 . 3163 16 45 .55557 .44443 1.7999 .66818 1.4966 1.2027 .16853 .83147 15 46 . 5581 . 4419 .7992 . 6860 .4957 .2029 . 6869 . 3131 14 47 . 5605 . 4395 .7984 . 6902 .4947 .2031 . 6885 . 3115 13 48 . 5629 . 4370 .7976 . 6944 .4938 .2034 . 6901 . 3098 12 49 . 5654 . 4346 .7968 . 6986 .4928 . 6918 . 3082 11 50 .55678 .44322 1.7960 .67028 1.4919 1.2039 .16934 .83066 10 51 . 5702 . 4298 .7953 . 7071 .4910 !2041 . 6950 . 3050 9 52 . 5726 . 4274 .7945 . 7113 .4900 .2043 . 6966 . 3034 8 53 . 5750 . 4250 .7937 . 7155 .4891 .2046 . 6982 . 3017 7 54 . 5774 . 4225 .7929 . 7197 .4881 .2048 . 6999 . 3001 6 55 .55799 .44201 1.7921 .67239 1.4872 1.2050 .17015 .82985 5 56 . 5823 . 4177 .7914 . 7282 .4863 .2053 . 7031 . 2969 4 57 . 5847 . 4153 .7906 . 7324 .4853 .2055 . 7047 . 2962 3 58 . 5871 . 4129 .7898 . 7366 .4844 .2057 . 7064 . 2936 2 59 . 5895 . 4105 .7891 . 7408 .4835 .2060 - 7080 . 2920 1 60 . 5919 . 4081 .7883 . 7451 .4826 .2062 . 7096 . 2904 M. Cosine. Vrs. sin. Secant. Cotang. Tang. Cosec'nt Vrs. cos. Sine. M. 356 NATUnAL FUNCTIONS. Table 3. 34° Natural Trigonometrical Functions. 145° M. Sine. Vrs. C08. Cosec'nt Tang. Cotang. Secant. Vre. sin. Cosine. M. .55919 .44081 1.7883 .67451 1.4826 1.2062 .17096 .82904 60 1 . 5943 . 4057 .7875 . 7493 .4816 .2064 . 7112 . 2887 59 2 . 5967 . 4032 .7867 . 7535 .4807 .2067 . 7129 . 2871 58 3 . 5992 . 4008 .7860 . 7578 .4798 .2069 . 7145 . 2855 57 4 . 6016 . 3984 .7852 . 7620 .4788 .2072 . 7161 . 2839 56 5 .56040 .43960 1.7844 .67663 1.4779 1.2074 .17178 ,82822 55 6 . 6064 . 3936 .7837 . 7705 .4770 .2076 . 7194 . 2806 54 7 . 6088 . 3912 .7829 . 7747 .4761 .2079 . 7210 . 2790 53 8 . 6112 . 8888 .7821 . 7790 .4751 .2081 . 7227 . 2773 52 9 . 6136 . 3864 .7814 . 7832 .4742 .2083 . 7243 . 2757 51 10 .56160 .43840 1.7806 .67875 1.4733 1.2086 .17259 .82741 50 11 . 6184 . 3816 .7798 . 7917 .4724 .2088 . 7276 . 2724 49 12 . 6208 . 3792 .7791 . 7960 .4714 .2091 . 7292 . 2708 48 13 . 6232 . 3768 .7783 . 8002 .4705 .2093 . 7308 . 2692 47 14 . 6256 . 3743 .7776 . 8045 .4696 .2095 . 7325 . 2675 46 15 .56280 .43719 1.7768 .6S087 1.4687 1.2098 .17341 .82659 45 16 . 6304 . 3695 .7760 . 8130 .4678 .2100 . 7357 . 2643 44 17 . 6323 . 3671 .7753 . 8173 .4669 .2103 . 7374 . 2626 43 18 . 6353 . 3647 .7745 . 8215 .4659 .2105 . 7390 . 2610 42 19 . 6377 . 3623 .7738 . 8258 .4650 .2107 . 7406 . 2593 41 20 .56101 .43599 1.7730 .68301 1.4641 1.2110 .17423 .82,577 40 21 . 6425 . 3575 .7723 . 8343 .4632 .2112 . 7439 . 2561 39 22 . 6449 . 3551 .7715 . 8386 .4623 .2115 . 7456 . 2544 38 23 . 6473 . 3527 .7708 . 8429 .4614 .2117 . 7472 . 2528 37 24 . 6497 . 3503 .7700 . 8471 .4605 .2119 . 7489 . 2611 36 25 .56521 .43479 1.7093 .68514 1.4,595 1.2122 .17505 .82495 35 26 . 6545 . 3455 .7685 . 8.557 .4586 .2124 . 7521 . 2478 34 27 . 6569 . S4S1 .7678 . 8600 .4577 .2127 . 7538 . 2462 33 28 . 6593 . 3407 .7670 . 8642 .4568 .2129 . 7554 . 2445 32 29 . 6617 . 3383 .7663 . 8685 .4559 .2132 . 7571 . 2429 31 30 .56641 .43359 1.7655 .68728 1.4550 1.2134 .17587 .82413 30 31 . 6664 . 3335 .7648 . 8771 .4541 .2136 . 7604 . 2396 29 32 . 6688 . 3311 .7640 . 8814 .4532 .2139 . 7620 . 2380 28 33 . 6712 . 3287 .7633 . 8857 .4523 .2141 . 7637 . 2363 27 34 . 6736 . 3263 .7625 . 8899 .4514 .2144 . 7653 . 2.347 26 35 .56760 .43239 1.7618 .68942 1.4505 1.2146 .17670 .82330 25 36 . 6784 . 3216 .7610 . 8985 .4496 .2149 . 7686 . 2314 24 37 . 6808 . 3192 .7603 . 9028 .4487 .2151 . 7703 . 2297 23 38 . 6832 . 3168 .7596 . 9071 .4478 .2153 . 7719 . 2280 22 39 . 6856 . 3144 .7588 . 9114 .4469 .2156 . 7736 . 2264 21 40 .56880 .43120 1.7581 .69157 1.4460 1.2158 .17752 .82247 20 41 . 6904 . 3096 .7573 . 9200 .4451 .2161 . 7769 . 22,31 19 42 . 6928 . 3072 .7566 . 9243 .4442 .2163 . 7786 . 2214 18 43 . 6952 . 3048 .7559 . 9286 .4433 .2166 . 7802 . 2198 17 44 . 6976 . 3024 .7551 . 9329 .4424 .2168 . 7819 . 2181 36 45 .57000 .43000 1.7514 .69372 1.4415 1.2171 .17835 .82165 15 46 . 7023 . 2976 .7537 . 9415 .4406 .2173 . 7852 . 2148 14 47 . 7047 . 2952 .7529 . 9459 .4397 .2175 7868 . 2131 13 48 . 7071 . 2929 .7522 . 9502 .4388 .2178 . 7885 . 2115 12 49 . 7095 . 2905 .7514 . 9.545 .4379 .2180 . 7902 . 2098 11 50 .57119 .4'2881 1.7507 .69588 1.4370 1.2183 .17918 .82082 10 61 . 7113 . 2857 .7500 . 9631 .4361 .2185 . 7935 . 2066 9 52 . 7167 . 2833 .7493 . 9674 .4352 .2188 . 7951 . 2048 8 63 . 7191 . 2809 .7485 . 9718 .4343 .2190 . 7968 . 2032 7 54 . 7214 . 2785 .7478 . 9761 .4335 .2193 . 7985 . 2015 6 55 .57238 .42761 1.7471 .69804 1.4326 1.2195 .18001 .81998 5 56 . 7262 . 2738 .7463 . 9847 .4317 .2198 . 8018 . 1982 4 57 . 7286 . 2714 .7456 . 9891 .4308 .2200 . 8035 . 1965 3 68 . 7310 . 2690 .7449 . 99.S4 .4299 .2203 . 8051 . 1948 2 59 . 7334 . 2666 .7412 . 9977 .4290 .2205 . 8068 . 1932 1 60 . 7358 . 2642 .7434 .70021 .4281 .2208 . 8085 . 1915 M. Cosine. ViB. sin. Secant. Co tang. Tang. CoBec'nt Vrs. COB. Sine. M. J 24° 55° Table 3. NATURAL FUNCTIONS. 357 35' Natural Trigonometrical Functions. 144° M. Sine. Vre. COS. CoBec'nt Tang. Cotang. Secant. Yrs. Bin. Cosine. M. .57358 .42642 1.7434 .70021 1.4281 1.2208 .18085 .81915 60 1 . 7:«1 . 2618 .7427 . 0064 .4273 .2210 . 8101 . 1898 59 2 . 7405 . 2595 .7420 . 0107 .4264 .2213 . 8118 . 1882 58 3 . 7429 . 2571 .7413 . 0151 .4255 .2215 . 8135 . 1865 57 4 . 7453 . 2547 .7405 . 0194 .4246 .2218 . 8151 . 1848 56 5 .57477 .42523 1.7398 .70238 1.4237 1.2220 .18168 .81832 55 6 . 7500 . 2499 .7391 . 0281 .4228 .2223 . 8185 . 1815 54 7 . 7524 . 2476 .7384 . 0325 .4220 .2225 . 8202 . 1798 53 8 . 7548 . 2452 .7377 . 0368 .4211 22?« . 8218 . 1781 52 9 . 7572 . 2428 .7369 . 0412 .4202 .2230 . 8235 . 1765 51 10 .57596 .42404 1.7362 .70455 1.4193 1.2233 .18252 .81748 50 11 . 7619 . 2380 .7355 . 0499 .4185 .2235 . 8269 . 1731 49 12 . 7643 . 2357 .7348 . 0542 .4176 .2238 . 8285 . 1714 43 13 . 7667 . 2333 .7341 . 0586 .4167 .2240 . 8302 . 1698 47 14 . 7691 . 2309 .7334 . 0629 .4158 .2243 . 8319 . 1681 46 15 .57714 .42285 1.7327 .70673 1.4150 1.2245 .18336 .81664 45 16 . 7738 . 2262 .7319 . 0717 .4141 .2248 . 8353 . 1647 44 17 . 7762 . 2238 .7312 . 0760 .4132 .2250 . 8369 . 1630 43 18 . 7786 . 2214 .7305 . 0804 .4123 .2253 . 8386 . 1614 42 19 . 7809 . 2190 .7298 . 0848 .4115 .2255 . 8403 . 1597 41 20 .57833 .42167 1.7291 .70891 1.4106 1.2258 .18420 .81580 40 21 . 7857 . 2143 .7284 . 0935 .4097 .2260 . 8437 . 1563 39 22 . 7881 . 2119 .7277 . 0979 .4089 .2263 . 8453 . 1546 38 23 .7904 . 2096 .7270 . 1022 .4080 .2265 . 8470 . 1530 37 24 . 7928 . 2072 .7263 . 1066 .4071 .2268 . 8487 . 1513 36 25 .57952 .42048 1.7256 .71110 1.4063 1.2270 .18504 .81496 35 26 . 7975 . 2024 .7249 . 1154 .4054 .2273 . 8521 . 1479 34 27 . 7999 . 2001 .7242 . 1198 .4045 .2276 . 8538 . 1462 33 28 . 8023 . 1977 .7234 . 1241 .4037 .2278 . 8555 . 1445 32 29 . 8047 . 1953 .7227 . 1285 .4028 .2281 . 8571 . 1428 31 30 .58070 .41930 1.7220 .71329 1.4019 1.2283 .18588 .81411 30 31 . 8094 . 1906 .7213 . 1373 .4011 .2286 . 8605 . 1395 29 32 . 8118 . 1882 .7206 . 1417 .4002 2?88 . 8622 . 1378 28 33 . 8141 . 1859 .7199 . 1461 .3994 .2291 . 8639 . 1361 27 84 . 8165 . 1835 .7192 . 1505 .3985 .2293 . 8656 . 1344 20 35 .58189 .41811 1.7185 .71549 1.3976 1.2296 .18673 .81327 25 36 , 8212 . 1788 .7178 . 1593 .3968 .2298 . 8690 . 1310 24 37 . 8236 . 1764 .7171 . 1637 .3959 .2301 . 8707 . 1293 23 38 . 8259 . 1740 .7164 . 1681 .3951 .2304 .8724 . 1276 22 39 . 8283 . 1717 .7157 . 1725 .3942 .2306 . 8741 . 1259 21 40 .58307 .41693 1.7151 .71769 1.3933 1.2309 .18758 .81242 20 41 . 8330 . 1669 .7144 . 1813 .3925 .2311 . 8775 . 1225 19 42 . 8354 . 1646 .7137 . 1857 .3916 .2314 . 8792 . 1208 18 43 . 8378 . 1622 .7130 . 1901 .3908 .2316 . 8809 . 1191 17 44 . 8401 . 1599 .7123 . 1945 .3899 .2319 . 8826 . 1174 16 45 .58425 .41575 1.7116 .71990 1.3891 1.2322 .18843 .81157 15 46 . 8448 . 1551 .7109 . 2034 .3882 .2324 . 8860 . 1140 14 47 . 8472 . 1528 .7102 . 2078 .3874 .2327 . 8877 . 1123 13 48 . 8496 . 1504 .7095 . 2122 .3865 .2329 . 8894 . 1106 12 49 . 8519 . 1481 .7088 . 2166 .3857 .2332 . 8911 . 1089 11 60 .58543 .41457 1.7081 .72211 1.3848 1.2335 .18928 .81072 10 51 . 8566 . 1433 .7075 . 2'255 .3840 .2337 . 8945 . 1055 9 62 . 8990 . 1410 .7068 . 2299 .3831 .2340 . 8962 . 1038 8 53 . 8614 . 1386 .7061 . 2344 .3823 .2342 . 8979 . 1021 7 54' . 8637 . 1363 .7054 . 2388 .3814 .2345 . 8996 . 1004 6 55 .58661 .41339 1.7047 .72432 1.3806 1.2348 .19013 .80987 5 56 . 8684 . 1316 .7040 . 2477 .3797 .2350 . 9030 . 0970 4 57 . 8708 . 1292 .7033 . 2521 .3789 .2353 . 9047 . 0953 3 58 . 8731 . 1268 .7027 . 2565 .3781 .2355 . 9064 . 0936 2 59 . 8755 . 1245 .7020 . 2610 .3772 .2358 . 9081 . 0919 1 60 . 8778 . 1221 .7013 . 2654 .3764 .2361 . 9098 . 0902 jr. CosiDe. Vrs. sin. Secant. Cotang. Tang. Cosec'nt Vrs. COS. Sine. M. 125° 358 NATURAL FUNCTIONS. Table 3. 36° Natural Trigonometrical Functions. 143° M. Sine. Vrs. coe. Coscc'nt Tang. Cotang. Secant. Vrs. sin. Cosine. M. .58778 .41221 1.7013 .72654 1.3764 1.2361 .19098 .80902 60 1 . 8802 . 1198 .7006 . 2699 .3755 .2363 . 9115 . 0885 59 2 . 8825 . 1174 .6999 . 2743 .3747 .2366 . 9132 . 0867 58 3 . 8849 . 1151 .6993 . 2788 .3738 .2368 . 9150 . 0850 57 4 . 8873 . 1127 .6986 . 2832 .3730 .2371 . 9167 . 0833 56 5 .58896 .41104 1.6979 .72877 1.3722 1.2374 .19184 .80816 55 6 . 8920 . 1080 .6972 . 2921 .3713 .2376 . 9201 . 0799 54 7 . 8943 . 1057 .6965 . 2966 .3705 .2379 . 9218 . 0782 53 8 .8967 . 1033 .6959 . 3010 .3697 .2382 . 9235 . 0765 52 9 . 8990 . 1010 .6952 . 3055 .3688 .2384 . 9252 . 0747 51 10 .59014 .40986 1.6945 .73100 1.3680 1.2387 .19270 .80730 50 H . 9037 . 0963 .6938 . 3144 .3672 .2389 . 9287 . 0713 49 12 . 9060 . 09.39 .6932 . 3189 .3663 .2392 . 9304 . 0696 48 13 . 9084 . 0916 .6925 . 8234 .3655 .2395 . 9321 . 0679 47 14 . 9107 . 0892 .6918 . 8278 .3647 .2397 . 9338 . 0662 46 15 .59131 .40869 1.6912 .73323 1.3638 1.2400 .19365 .80644 45 16 . 9164 . 0845 .6905 . 3368 .3630 .2403 . 9373 . 0627 44 17 . 9178 . 0822 .6898 . 3412 .3622 .2405 . 9390 . 0610 43 18 . 9201 . 0799 .6891 . 3457 .3613 .2408 . 9407 . 0593 42 19 . 9225 . 0775 .6885 . 3502 .3005 .2411 . 9424 . 0576 41 20 .69248 .40752 1.6878 .73547 1.3597 1.2413 .19442 .80558 40 21 . 9272 . 0728 .6871 . 3592 .3588 .2416 . 9459 . 0641 39 22 . 9295 . 0705 .6865 . 3637 .3580 .2419 . 9476 . 0524 38 23 . 9318 . 0681 .6858 . 3681 .3672 .2421 . 9493 . 0507 37 24 . 9342 . 0658 .6851 . 3726 .3564 .2424 . 9511 . 0489 36 25 .59365 .40635 1.6845 .73771 1.3655 1.2127 .19528 .80472 35 26 . 9389 . 0611 .6838 . 3816 .3547 .2429 . 9545 . 0455 34 27 . 9412 . 0588 .6831 . 3861 .3539 .2432 . 9562 . 0437 33 28 . 9435 . 0564 .6825 . 3906 .3531 .2435 . 9580 . 0420 32 29 . 9459 . 0541 .6818 . 3951 .3522 .2437 . 9597 . 0403 31 30 .59482 .40518 1.6812 .73996 1.3514 1.2440 .19614 .80386 30 31 . 9506 . 0494 .6805 . 4041 .3506 .2443 . 9632 . 0368 29 32 . 9529 . 0471 .6798 . 4086 .3498 .2445 . 9649 . 0351 28 33 . 9562 . 0447 .6792 . 4131 .3489 .2448 . 9666 . 0334 27 34 . 9576 . 0424 .6785 . 4176 .3481 .2451 . 9683 . 0316 26 35 .59599 .40401 1.6779 .74221 1.3473 1.2453 .19701 .80299 25 36 . 9622 . 0377 .6772 . 4266 .3465 .2456 . 9718 . 0282 24 37 . 9646 . 0354 .6766 . 4312 .3457 .2459 . 9736 . 0264 23 38 . 9669 . 0331 .6759 . 4357 .3449 .2461 . 9753 . 0247 22 39 . 9692 . 0307 .6752 . 4402 .3440 .2464 . 9770 . 0230 21 40 .59716 .40284 1.6746 .74447 1.3432 1.2467 .19788 .80212 20 41 . 9739 . 0261 .6739 . 4492 .3424 .2470 . 9805 . 0195 19 42 . 9762 .0237 .6733 . 4538 .3416 .2472 . 9822 . 0177 18 43 . 9786 . 0214 .6726 . 4583 .3408 .2475 . 9840 . 0160 17 44 . 9R09 . 0191 .6720 . 4628 .3400 .2478 . 9867 . 0143 16 45 .59832 .40167 1.6713 .74673 1.3392 1.2480 .19875 .80125 15 46 . 9856 . 0144 .6707 . 4719 .3383 .2483 . 9892 . 0108 14 47 . 9879 . 0121 .6700 . 4764 .3375 .2486 . 9909 . 0090 13 48 . 9902 . 0098 .6694 . 4809 .3367 .2488 . 9927 . 0073 12 49 . 9926 . 0074 .6687 . 4855 .3359 .2491 . 9944 . 0056 11 60 .59949 .40051 1.6681 .74900 1.3351 1.2494 .19962 .80038 10 51 . 9972 . 0028 .6674 . 4946 .3343 .2497 . 9979 . 0021 9 52 . 9995 . 0004 .6668 . 4991 .3335 .2499 . 9997 . 0003 8 53 .60019 .39981 .6661 . 5037 .3327 .2502 .20014 .79986 7 54 . 0042 . 9958 .6655 . 5082 .3319 .2505 . 0031 . 9968 6 55 .60065 .39935 1.6648 .75128 1.3311 1.2508 .20049 .79951 5 56 . 0088 . 9911 .6642 . 5173 .3303 .2510 . 0066 . 9933 4 57 . 0112 . 9888 .6636 . 5219 .3294 .2513 . 0084 . 9916 3 68 . 0135 . 9866 .6629 . 5264 .3286 .2516 . 0101 . 9898 2 59 . 0158 . 9842 .6623 . 6310 .3278 .2519 . 0119 . 9881 1 60 . 0181 . 9818 .6616 . 6355 .3270 .2521 . 0136 . 9863 M. Cosine. Vrs. sin. Secant. Cotang. Tang. Cosec'nt Vrs. COS. Sine. M, 126° 53° Table 3. NATURAL FUNCTIONS. 359 37= Natural Trigonometrical Functions. 142° mT Sine. Vra. COS. Cosec'nt Tang. Cotang. Secant. Vrs. sin. Cosine. M. .60181 .39818 1.6616 .75355 1.3270 1.2521 .20136 .79863 60 1 . 0205 . 9795 .6610 . 5401 .3262 .2524 . 0154 . 9846 59 2 . 0228 . 9772 .6603 . 5447 .3254 .2527 . 0171 . 9828 58 3 .0251 . 9749 .6597 . 5492 .3246 .2530 . 0189 . 9811 57 4 . 0274 . 9726 .6591 . 5538 .3238 .2532 . 0206 . 9793 56 5 .60298 .39702 1.6584 .75584 1.3230 1.2535 .20224 .79776 55 6 .0320 . 9679 .6578 . 5629 .3222 .2538 . 0242 . 9758 54 7 . 0344 . 9656 .6572 . 5675 .3214 .2541 . 0259 . 9741 53 8 . 0367 . 9633 .6565 . 5721 .3206 .2543 . 0277 . 9723 52 9 .0390 . 9610 .6559 . 5767 .3198 .2546 . 0294 . 9706 51 10 .60413 .39586 1.6552 .75812 1.3190 1.2549 .20312 .79688 50 11 . 0437 . 9563 .6546 . 5858 .3182 .2552 . 0329 . 9670 49 12 . 0460 . 9540 .6540 . 5904 .3174 .2554 . 0347 . 9653 48 13 . 0483 . 9517 .6533 . 5950 .3166 .2557 . 0365 . 9635 47 14 . 0506 . 9494 .6527 . 5996 .3159 .2560 . 0382 . 9618 46 15 .60529 .39471 1.6521 .76042 1.3151 1.2563 .20400 .79600 45 16 . 0552 . 9447 .6514 . 6088 .3143 .2565 . 0417 . 9582 44 17 . 0576 . 9424 .6508 . 6134 .3135 .2568 . 0435 . 9565 43 18 . 0599 . 9401 .6502 . 6179 .3127 .2571 . 0453 . 9547 42 19 . 0622 . 9378 .6496 . 6225 .3119 .2574 . 0470 . 9530 41 20 .60645 .39355 1.6489 .70271 1.3111 1.2577 .20488 .79512 40 21 . 0668 . 9332 .6183 . 6317 .3103 .2579 . 0505 . 9494 39 22 . 0691 . 9309 .6477 . 6364 .3095 .2582 . 0523 . 9477 38 23 . 0714 . 9285 .6470 . 6410 .3087 .2585 . 0541 . 9459 37 24 . 0737 . 9262 .6464 . 6156 .3079 .2588 . 0558 . 9441 36 25 .60761 .39239 1.6458 .76502 1.3071 1.2591 .20576 .79424 35 26 . 0784 . 9216 .6152 . 6548 .3064 .2593 . 0594 . 9406 34 27 . 0807 . 9193 .6445 . 6594 .3056 .2596 . 0611 . 9388 33 28 . 0830 . 9170 .6439 . 6640 .3048 .2599 . 0629 . 9371 32 29 . 0853 . 9147 .6433 . 6686 .3040 .2602 . 0647 . 9353 31 30 .60876 .39124 1.6427 .76733 1.3032 1.2605 .20665 .79335 30 31 . 0899 . 9101 .6420 . 6779 .3024 .2607 . 0682 . 9318 29 32 . 0922 . 9078 .6414 . 6825 .3016 .2610 . 0700 . 9300 28 33 . 0945 . 9055 .6408 . 6871 .3009 .2613 . 0718 . 9282 27 34 . 0963 . 9031 .6402 . 6918 .3001 .2616 . 0735 . 9264 26 35 .60991 .39008 1.6396 .76964 1.2993 1.2619 .20753 .79247 25 36 . 1014 . 8985 .6389 . 7010 .2985 .2622 . 0771 . 9229 24 37 . 1037 . 8962 .6383 . 7057 .2977 .2624 . 0789 . 9211 23 38 . 1061 . 8939 .6377 . 7103 .2970 .2627 . 0806 . 9193 22 39 . 1084 . 8916 .6371 . 7149 .2962 .2630 . 0824 . 9176 21 40 .61107 .38893 1.6365 .77196 1.2954 1.2633 .20842 .79158 20 41 . 1130 . 8870 .6359 . 7242 .2946 .2636 . 0860 . 9140 19 42 . 1153 . 8847 .6352 . 7289 .2938 .2639 . 0878 . 9122 18 43 . 1176 .8824 .6346 . 7335 .2931 .2641 . 0895 . 9104 17 44 . 1199 . 8801 .6340 . 7382 .2923 .2644 . 0913 . 9087 16 45 .61222 .38778 1.6334 .77428 1.2915 1.2647 .20931 .79069 15 46 . 1245 . 8755 .6328 . 7475 .2907 .2650 . 0949 . 9051 14 47 . 1268 . 8732 .6322 .7521 .2900 .2653 . 0967 . 9033 13 48 . 1290 . 8709 .6316 . 7568 .2892 .2656 . 0984 . 9015 12 49 . 1314 . 8686 .6309 . 7614 .2884 .2659 . 1002 . 8998 11 50 .61337 .38663 1.6303 .77661 1.2876 1.2661 .21020 .78980 10 51 . 1360 . 8640 .6297 . 7708 .2869 .2664 . 1038 . 8962 9 52 . 1383 . 8617 .6291 . 7754 .2861 .2667 . 1056 . 8944 8 53 . 1405 . 8594 .6285 . 7801 .2853 .2670 . 1074 . 8926 7 54 . 1428 . 8571 .6279 . 7848 .2845 .2673 . 1091 . 8908 6 55 .61451 .38548 1.6273 .77895 1.2838 1.2676 .21109 .78890 5 56 . 1474 .8525 .6267 . 7941 .2830 .2679 . 1127 . 8873 4 57 . 1497 . 8503 .6261 . 7988 .2822 .2681 . 1145 . 8855 3 58 . 1520 . 8480 .6255 . 8035 .2815 .2684 . 1163 . 8837 2 59 . 1543 . 8457 .6249 . 8082 .2807 .2687 . 1181 . 8819 1 60 . 1566 . 8434 .6243 . 8128 .2799 .2690 . 1199 . 8801 M. Cosine. Vrs. flin. Secant. Cotang. TanK. Cosec'nt Vrs. COS. Sine. M. 360 NATURAL FUXCTTONS. Table 3. 38' Natural Trigonometrical Functions. 141° M. Sine. Vrs. COS. Cosec'nt Tang. Cotang. Secant. Vrs. ein. Cosine. M. .61566 .38434 1.6243 .78128 1.2799 1.2690 .21199 .78801 60 1 . 1589 . 8411 .6237 . 8175 .2792 .2693 . 1217 . 8783 59 2 . 1612 . 8388 .6231 . 8222 .2784 .2696 . 1235 . 8765 58 3 . 1635 . 8365 .6224 . 8269 .2776 .2699 . 1253 . 8747 67 4 . 1658 . 8342 .6218 .8316 .2769 .2702 . 1271 . 8729 56 5 .61681 .38319 1.6212 .78363 1.2761 1.2705 .21288 .78711 55 6 . 1703 . 8296 .6206 . 8410 .2753 .2707 . 1306 . 8693 54 7 . 1726 . 8273 .6200 . 8457 .2746 .2710 . 1324 . 8675 63 8 . 1749 . 8251 .6194 . 8504 .2738 .2713 . 1342 . 8657 52 9 . 1772 . 8228 .6188 . 8561 .2730 .2716 . 1360 . 8640 61 10 .61795 .38205 1.6182 .78598 1.2723 1.2719 .21378 .78622 50 11 . 1818 . 8182 .6176 . 8645 .2715 .2722 . 1396 . 8604 49 12 . 1841 . 8159 .6170 . 8692 .2708 .2726 . 1414 . 8586 48 13 . 1864 . 8136 .6164 . 8739 .2700 .2728 . 1432 . 8568 47 H . 1886 . 8113 .6159 . 8786 .2692 .2731 . 1450 . 8550 46 15 .61909 .38091 1.6153 .78834 1.2685 1.2734 .21468 .78532 45 16 . 1932 . 8068 .6147 . 8881 .2677 .2737 . 1486 . 8514 44 17 . 1955 . 8045 .6141 . 8928 .2670 .2739 . 1504 . 8496 43 18 . 1978 . 8022 .6135 . 8975 .2662 .2742 . 1622 . 8478 42 19 . 2001 . 7999 .6129 . 9022 .2655 .2745 . 1540 . 8460 41 20 .62023 .37976 1.6123 .79070 1.2647 1.2748 .21558 .78441 40 21 . 2046 . 7954 .6117 . 9117 .2639 .2751 . 1576 . 8423 39 22 . 2069 . 7931 .6111 . 9164 .2632 .2754 . 1694 . 8405 38 23 . 2092 . 7908 .6105 . 9212 .2624 .2757 . 1612 . 8387 37 24 . 2115 . 7885 .6099 . 9259 .2617 .2760 . 1631 . 8369 86 25 .62137 .37862 1.6093 .79306 1.2609 1.2763 .21649 .78351 35 26 . 2160 . 7840 .6087 . 9354 .2602 .2766 . 1667 . 8333 34 27 . 2183 . 7817 .6081 . 9401 .2594 .2769 . 1685 . 8315 33 28 . 2206 . 7794 .6077 . 9449 .2587 .2772 . 1703 . 8297 32 29 . 2229 . 7771 .6070 . 9496 .2579 .2776 . 1721 . 8279 31 30 .62251 .37748 1.6064 .79543 1.2572 1.2778 .21739 .78261 30 31 . 2274 . 7726 .6058 . 9591 .2564 .2781 . 1767 . 8243 29 32 . 2297 . 7703 .6052 . 9639 .2557 .2784 . 1775 . 8224 28 33 . 2320 . 7680 .6046 . 9686 .2549 .2787 . 1793 . 8206 27 34 . 2312 . 7657 .6040 . 9734 .2542 .2790 . 1812 . 8188 26 35 .62365 .37635 1.6034 .79781 1.2534 1.2793 .21830 .78170 25 36 . 2388 . 7612 .0029 . 9829 .2527 .2795 . 1848 . 8152 24 37 . 2411 . 7589 .6023 . 9876 .2519 .2798 . 1866 . 8134 23 38 . 2433 . 7566 .6017 . 9924 .2612 .2801 . 1884 . 8116 22 39 . 2456 . 7544 .6011 . 9972 .2604 .2804 . 1902 . 8097 21 40 .62479 .37521 1.6005 .80020 1.2497 1.2807 .21921 .78079 20 41 . 2501 . 7498 .6000 . 0067 .2489 .2810 . 1939 . 8061 19 42 . 2524 . 7476 .5994 . 0115 .2482 .2813 . 1967 . 8043 18 43 . 2547 . 7453 .5988 . 0163 .2475 .2816 . 1975 . 8025 17 44 . 2570 . 7430 .5982 . 0211 .2467 .2819 . 1993 . 8007 16 45 .62592 .37408 1.5976 .80268 1.2460 1.2822 .22011 .77988 15 46 . 2615 . 7385 .5971 . 0306 .2462 .2825 . 2030 . 7970 14 47 . 2638 . 7362 .5965 . 0354 .2445 .2828 . 2048 . 7952 13 48 . 2660 . 7340 .5969 . 0402 .2437 .2831 . 2066 . 7934 12 49 . 2683 . 7317 .5953 . 0460 .2430 .2834 . 2084 . 7915 11 50 .62708 .37294 1.5947 .80498 1.2423 1.2837 .22103 .77897 10 61 . 2728 . 7272 .5942 . 0546 .2415 .2840 . 2121 . 7879 9 52 . 2751 . 7249 .6936 . 0594 .2408 .2843 . 2139 . 7861 8 53 . 2774 . 7226 .6930 . 0642 .2400 .2846 . 2157 . 7842 7 54 . 2796 .7204 .6924 .0690 .2393 .2849 . 2176 . 7824 6 55 .62819 .37181 1.5919 .80738 1.2386 1.2862 .22194 .77806 5 56 . 2841 . 7158 .6913 . 0786 .2378 .2865 . 2212 . 7788 4 57 . 2864 . 7136 .6907 . 0834 .2371 .2858 . 2230 . 7769 3 58 . 2887 . 7113 .6901 . 0882 .2364 .2861 . 2249 . 7751 2 59 . 2909 . 7090 .5896 . 0930 .2356 .2864 . 2267 . 7733 1 60 . 2932 . 7068 .6890 . 0978 .2349 .2867 . 2285 . 7715 M. Coeine. Vrs. sin. Secant. Cotang. Tang. Cosec*nt Vrs. COS. Sine. M. 128° Si" Table 3. NATURAL FUNCTIONS. 361 39° Natural Trigonometrical Functions. 140° M. Sine. Vrs. COB. Cosec'nt Tang. Co tang. Secant. Vrs. Bin. Cosine. M. .62932 .37068 1.5890 .80978 1.2349 1.2867 .22285 .77715 60 1 . 2955 . 7045 .5884 . 1026 .2342 .2871 . 2304 . 7696 59 2 . 2977 . 7023 .5879 . 1076 .2334 .2874 . 2322 . 7678 68 3 . 3000 . 700O .5873 . 1123 .2327 .2877 . 2340 . 7660 57 4 . 3022 . 6977 .5867 . 1171 .2320 .2880 . 2359 . 7641 56 5 .63045 .36955 1.5862 .81219 1.2312 1.2883 .22377 .77623 55 C . 3067 . 6932 .5856 . 1268 .2305 .2886 . 2395 . 7605 64 7 . 3090 . 6910 .5850 . 1316 .2297 .2889 . 2414 . 7586 53 8 .3113 . 6887 .5845 . 1364 .2290 .2892 . 2432 . 7568 52 9 . 3135 .6865 .5839 ; 1413 .2283 .2895 . 2450 . 7549 51 10 .63158 .36512 1.5833 .81461 1 9716 1.2898 .22469 .77531 50 11 . 3180 . 6820 .5828 . 1509 .2268 .2901 . 2487 . 7513 49 12 . 3203 . 6797 .5822 . 1558 .2261 .2904 . 2505 . 7494 48 13 . 3225 . 6774 .5816 . 1606 .2254 .2907 . 2524 . 7476 47 14 . 3248 . 6752 .6811 . 1655 .2247 .2910 . 2542 . 7458 46 15 .63270 .36729 1.5805 .81703 1.2239 1.2913 .22561 .77439 45 16 . 3293 . 6707 .6799 . 1752 .2232 .2916 . 2579 . 7421 44 17 . 3315 . 6684 .5794 . 1800 .2225 .2919 . 2597 . 7402 43 18 . 3338 . 6662 .5788 . 1849 2218 .2922 . 2616 . 7384 42 19 . 3360 . 6639 .5783 . 1898 .2210 .2926 . 2634 . 7366 41 20 .63383 .36617 1.5777 .81946 1.2203 1.2929 .22653 .77347 40 21 . 3405 . 6594 .5771 . 1995 .2196 .2932 . 2671 . 7329 39 22 . 3428 . 6572 .5766 . 2043 .2189 .2935 . 2690 . 7310 38 23 . 3450 . 6549 .5760 . 2092 .2181 .2938 . 2708 . 7292 37 24 . 3473 . 6527 .5755 . 2141 .2174 .2941 . 2727 . 7273 36 25 .63495 .36504 1.5749 .82190 1.2167 1.2944 .22745 .77265 36 26 . 3518' . 6482 .5743 . 2238 .2160 .2947 . 2763 . 7236 34 27 . 3540 . 6469 .6738 . 2287 .2152 .2950 . 2782 . 7218 33 28 . 3563 . 6487 .5732 . 2336 .2145 .2953 . 2800 . 7199 32 29 . 3585 . 6415 .5727 . 2385 .2138 .2956 . 2819 . 7181 31 30 .63608 .36392 1.5721 .82434 1.2131 1.2960 .22837 .77162 30 31 . 3630 . 6370 .6716 . 2482 .2124 .2963 . 2856 . 7144 29 32 . 3653 . 6347 .5710 . 2531 .2117 .2966 . 2874 . 7125 28 33 . 3675 . 6325 .5705 . 2580 .2109 .2969 . 2893 . 7107 27 34 . 3697 . 6302 .5699 . 2629 .2102 .2972 . 2912 . 7088 26 35 .63720 .36280 1.5694 .82678 1.2096 1.2975 .22930 .77070 25 36 . 3742 . 6258 .5688 .2727 .2088 .2978 . 2949 . 7051 24 37 . 3765 . 6235 .5683 . 2776 .2081 .2981 . 2967 . 7033 23 38 . 3787 . 6213 .5677 . 2825 .2074 .2985 . 2986 . 7014 22 39 . 3810 . 6190 .6672 . 2874 .2066 .2988 . 3004 . 6996 21 40 .63832 .36168 1.5666 .82923 1.2059 1.2991 .23023 .76977 20 41 . 3854 . 6146 .6661 . 2972 .2052 .2994 . 3041 . 6958 19 42 . 3877 . 6123 .5655 . 3022 .2045 .2997 . 3060 . 6940 18 43 . 3899 . 6101 .5650 . 3071 .2038 .3000 . 3079 . 6921 17 44 . 3921 . 6078 .5644 . 3120 .2031 .3003 . 3097 . 6903 16 45 .63944 .36056 1.6639 .83169 1.2024 1.3006 .23116 .76884 15 46 . 3966 . 6034 .5633 . 3218 .2016 .3010 . 3134 . 6865 14 47 . 3989 . 6011 .5628 . 3267 .2009 .3013 . 3153 . 6847 13 48 . 4011 . 5989 .5622 . 3317 .2002 .3016 . 3172 . 6828 12 49 . 4033 . 5967 .6617 . 3366 .1995 .3019 . 3190 . 6810 11 50 .64056 .35944 1.6611 .83415 1.1988 1.3022 .23209 .76791 10 51 . 4078 . 5922 .5606 . 3465 .1981 .3026 . 3227 . 6772 9 52 . 4100 . 5900 .5600 . 3514 .1974 .3029 . 3246 . 6754 8 53 . 4123 . 5877 .5595 . 3663 .1967 .3032 . 3265 . 6735 7 54 . 4145 . 5855 .5590 . 3613 .1960 .3035 . 3283 . 6716 6 55 .64167 .35833 1.5584 .83662 1.1953 1.3038 .23302 .76698 5 56 . 4189 . 6810 .6579 . 3712 .1946 .3041 . 3321 . 6679 4 57 . 4212 . 5788 .6573 . 3761 .1939 .3044 . 3339 . 6660 3 58 . 4234 . 6766 .5568 . 3811 .1932 .3048 . 3358 . 6642 2 69 . 4256 . 6743 .5663 . 3860 .1924 .3051 . 3377 . 6623 1 60 . 4279 . 5721 .5557 . 3910 .1917 .3054 . 3395 . 6604 M. Cosine. Vrs. Bin. Secant. Cotang. Tang. CoBec'nt Vrs. cos. Sine. M. 129° 50° 362 NATUEAL FUNCTIONS. Table 3. 40° Natural Trigonometrical Functions. 139° M. Sine. Vrs. COS. Cosec'nt Tang. Cotang. Secant. Yrs. sin. Cosine. M. .64279 .35721 1.5557 .83910 1.1917 1.3054 .23395 .76604 60 1 . 4301 . 5699 .5552 . 3959 .1910 .3057 . 34] <1 . 6686 59 2 . 4323 . 5677 .5546 . 4009 .1903 .3060 . 3433 . 6567 68 3 . 4345 . 5654 .5541 . 4059 .1896 .3064 . 3462 . 6548 57 4 . 4368 . 5632 .5536 . 4108 .1889 .3067 . 3470 . 6530 56 5 .64390 .35610 1.5530 .84158 1.1882 1.3070 .23489 .76611 55 . 4412 . 5588 .5525 . 4208 .1875 .3073 . 3508 . 6492 54 7 . 4435 . 5565 .5520 . 4267 .1868 .3076 . 3627 . 6473 53 8 . 4457 . 5543 .5514 . 4307 .1861 .3080 . 3545 . 6455 52 9 . 4479 . 5521 .5509 . 4357 .1854 .3083 . 3564 . 6436 ,51 10 .64601 .35499 1.6503 .81407 1.1847 1.3086 .23583 .76417 50 11 . 4523 . 5476 .5498 . 4457 .1840 .3089 . 3602 . 6398 49 12 . 4516 . 5-164 .5493 . 4506 .1833 .3092 . 3620 . 6380 48 13 . 4568 . 5432 .5187 . 4556 .1826 .3096 . 8639 . 6361 47 U . 4590 . 5410 .5482 . 4606 .1819 .3099 . 3658 . 6342 46 15 .61612 .36388 1.5477 .84656 1.1812 1.3102 .23677 .76323 45 16 . 4635 . 5365 .5471 . 4706 .1805 .3105 . 3695 . 6304 44 17 . 4657 . 5343 .5466 4756 .1798 .3109 . 3714 . 6286 43 18 . 4679 . 6321 .5161 . 4806 .1791 .3112 . 3733 . 6267 42 19 . 4701 . 5299 .5456 . 4856 .1785 .3115 . 3752 . 6248 41 20 .64723 .35277 1.5450 .84906 1.1778 1.3118 .23771 .76229 40 21 . 4745 . 6254 .6445 . 4956 .1771 .3121 . 3790 . 6210 39 22 . 4768 . 5232 .5440 . 5006 .1764 .3125 . 3808 . 6191 38 23 . 4790 . 5210 .5134 . 5056 .1757 .3128 . 3827 . 6173 37 24 . 4812 . 5188 .5429 . 5107 .1750 .3131 . 3846 . 6154 36 26 .64834 .35166 1.5424 .85157 1.1743 1.3134 .23865 .76135 35 26 . 4856 . 5144 .5419 . 5207 .1736 .3138 . 3884 . 6116 34 27 . 4878 . 5121 .5413 . 5257 .1729 .3141 . 3903 . 6097 33 28 . 4900 . 6099 .5408 . 5307 .1722 .3144 . 3922 . 6078 32 29 . 4923 . 6077 .5403 . 6358 .1715 .3148 . 3940 . 6059 31 30 .64945 .35055 1.5398 .85408 1.1708 1.3151 .23959 .76041 30 31 . 4967 . 5033 .6392 . 5458 .1702 .3154 . 3978 . 6022 29 32 . 4989 . 5011 .6387 . 6509 .1695 .3157 . 3997 . 6003 28 33 . 5011 . 4989 .6382 . 6559 .1688 .3161 . 4016 . 5984 27 34 . 6033 . 4967 ,5377 . 6609 .1681 .3164 . 4035 . 5965 26 35 .65055 .34945 1.5371 .85660 1.1674 1.3167 .24054 .76946 25 36 . 6077 . 4922 .5366 . 6710 .1667 .3170 . 4073 . 5927 24 37 . 5099 . 4900 .5361 . 6761 .1660 .3174 . 4092 . 5908 23 38 . 5121 . 4878 .5356 . 6811 .1653 .3177 . 4111 . 5889 22 39 . 5144 . 4856 .5351 . 5862 .1647 .3180 . 4130 . 5870 21 40 .65166 .34834 1.6345 .85912 1.1640 1.3184 .24149 .75851 20 41 . 5188 . 4812 .6340 . 5963 .1633 .3187 . 4168 . 5832 19 42 . 5210 . 4790 .6335 . 6013 .1626 .3190 . 4186 . 5813 18 43 . 5232 . 4768 .5330 . 6064 .1619 .3193 . 4205 . 5794 17 44 . 5254 . 4746 .5325 . 6115 .1612 .3197 . 4224 . 5775 16 45 .66276 .34724 1.5319 .86165 .1.1605 1.3200 .24243 .75766 15 46 . 5298 . 4702 .5314 . 6216 .1599 .3203 . 4262 . 5737 14 47 . 5320 . 4680 .5309 . 6267 .1592 .3207 . 4281 . 5718 13 48 . 6342 ,. 4658 .5304 . 6318 .1685 .3210 . 4300 . 5699 12 49 . 5364 . 4636 .5299 . 6368 .1578 .3213 . 4319 . 5680 11 50 .65386 .34614 1.5294 .86419 1.1571 1.3217 .24338 .75661 10 51 . 6408 . 4592 .5289 . 6470 .1565 .3220 . 4357 . 5642 9 52 . 5430 . 4570 .5'283 . 6521 .1558 .3223 . 4376 . 5623 8 63 . 5452 . 4548 .5278 . 6672 .1551 .3227 . 4396 . 5604 7 54 . 5474 . 4526 .5273 . 6623 .1544 .3230 . 4415 . 5585 6 55 .65496 .34504 1.5268 .86674 1.1637 1.3233 .24434 .75566 5 '56 . 6518 . 4482 .5263 . 6725 .1531 .3237 . 4453 . 5547 4 57 . 5640 . 4460 .5258 . 6775 .1524 .3240 . 4472 . 5528 3 58 . 5662 . 4438 .5253 . 6826 .1517 .3243 . 4491 . 6509 2 59 . 5584 . 4416 .6248 . 6878 .1510 .3247 . 4510 . 5490 1 60 . 5606 . 4394 .6242 . 6929 .1504 .3260 . 4529 . 5471 M. Cosine. Vrs. sin. Secant. Co tang. Tang. Cosec'nt Vrs. COB. Sine. M. 130° 49° Table 3. NATURAL FUNCTIONS. 363 4«° Natural Trigonometrical Functions. 138° mT Sine. Vra. COB. C!osec'nt Tang. Cotang. Secant, Yrs. sin. Cosine, M. .65606 .34394 1.5242 .86929 1.1504 1,3250 .21529 .75471 60 1 . 5628 . 4372 .6237 . 6980 .1497 .3253 .4648 . 5462 59 2 . 5650 . 4350 .5232 .7031 .1490 .3257 . 4567 . 5133 58 3 .6672 . 4328 .5227 . 7082 .1483 .3260 . 4586 . 5414 57 A . 5694 . 4306 .6222 . 7133 .1477 .3263 . 4605 . 5394 56 C .65716 .34284 1.5217 .87184 1.1470 1.3267 .21624 .75375 65 6 . 5737 . 4262 .5212 . 7235 .1463 .3270 .4644 . 6356 54 7 . 5759 . 4210 .5207 . 7287 .1456 .3271 . 4663 . 5337 53 8 . 5781 .4219 .5202 . 7338 .1450 .3277 . 4682 . 5318 52 9 . 5803 . 4197 .5197 . 7389 .1443 ,3280 . 4701 . 5299 51 10 .65825 .31175 1.6192 .87441 1,1436 1.3284 .21720 ,75280 50 11 . 5847 . 4153 .5187 . 7192 .1430 .3287 . 1739 . 5261 49 12 . 5869 . 4131 .5182 . 7513 .1423 .3290 . 4758 . 5241 48 13 . 5891 . 4109 .5177 . 7595 .1416 .3294 . 4778 . 5222 47 14 . 5913 . 4087 .5171 . 7616 .1409 .3297 . 4797 . 5203 46 15 .65934 .31065 1.5166 .87698 1.1103 1.3301 .24816 .76184 45 16 . 5956 . 4043 .5161 . 7719 .1396 .3304 .4835 . 5165 44 17 . 5978 . 4022 .5156 . 7801 .1389 .3307 . 4861 . 5146 43 18 .6000 . 4000 .5151 . 7852 .1383 .3311 . 4873 . 5125 42 19 . 6022 . 3978 .5146 . 7904 .1376 .3314 . 1893 . 5107 41 20 .66044 .33956 1.5141 .87955 1.1369 1.3318 .21912 ,7.5088 40 21 . 6066 . 3931 .6136 . 8007 .1363 .3321 . 4931 . 5069 39 22 . 6087 . 3912 .5131 . 8058 .1356 .3324 . 1950 . 5049 38 23 . 6109 . 3891 .5126 . 8110 .1319 .3328 . 4970 . 5030 37 24 . 6131 . 3869 .5121 . 8162 .1313 .3331 . 1989 . 6011 36 25 .66153 .33847 1.5116 .88213 1.1336 1.3335 .25008 .74992 35 26 . 6175 . 3825 .5111 . 8265 .1329 .3338 , 5027 . 4973 34 27 . 6197 . 3803 .5106 . 8317 .1323 .3342 . 5017 . 4953 33 28 . 6218 . 3781 .5101 . 8369 .1316 .3345 . 5066 . 4934 32 29 . 6240 . 3760 .5096 . 8121 .1309 .3318 . 5085 . 4915 31 30 .66262 .33738 1.8092 .88172 1.1303 1.3352 .25104 .74896 30 31 . 6284 . 3716 .6087 . 8521 .1296 .3355 . 5124 . 4876 29 32 . 6305 . 3694 .5082 . 8576 .1290 .3359 . 5143 . 4857 28 33 . 6327 . 3673 .5077 . 8628 .1283 .3362 , 5162 . 4838 27 34 . 6349 . 3651 .5072 . 8680 ,1276 .3366 . 5181 . 1818 26 35 .66371 .33629 1.5067 .88732 1.1270 1,3369 ,25201 .74799 25 36 . 6393 . 3607 .6062 . 8781 .1263 ,3372 . 5220 . 4780 24 37 . 6414 . 3586 .5057 . 8836 .1257 .3376 . 5239 . 4760 23 38 . 6436 . 3564 .5052 . 8888 .1250 .3379 . 5259 . 4741 22 39 . 6158 . 3542 ..5017 . 8940 ,1243 .3383 . 5278 . 4722 21 40 .66479 .33520 1.5012 .88992 1.1237 1.3386 .25297 .74702 20 41 . 6501 . 3499 .5037 . 9041 .1230 .3390 . 5317 . 4683 19 42 . 6523 . 3477 .5032 . 9097 .1224 .3393 . 5336 . 4664 18 43 . 6545 . 3455 .5027 . 9119 .1217 .3397 . 5355 . 4644 17 44 . 6566 . 3433 .5022 . 9201 .1211 .3400 . 5375 . 4626 16 45 .66588 .33412 1.5018 .89253 1.1204 1.3404 .26394 .74606 15 46 . 6610 . 3390 .5013 . 9306 .1197 .3407 . 6414 . 4586 14 47 . 6631 . 3368 .5008 . 9358 .1191 .3411 . 5433 . 1567 13 48 . 6653 .3347 .6003 . 9110 .1184 .3414 . 5462 . 4518 12 49 . 6675 . 3325 .1998 . 9163 .1178 .3418 . 6472 . 1528 11 50 .66697 .33303 1.1993 .89515 1.1171 1.3421 .25491 .71509 10 51 . 6718 . 3282 .4988 . 9567 .1165 ,3126 . 5510 . 4489 9 52 . 6740 . 3260 .4983 . 9620 .1158 .3428 . 5530 . 4170 8 63 . 6762 . 3238 .4979 . 9672 .1152 .3432 . 5619 . 4450 7 54 . 6783 . 3217 .4971 . 9725 .U45 .3435 . 5569 . 4131 6 55 .66805 .33195 1.4969 .89777 1.1139 1.3439 .25588 .74412 5 56 . 6826 . 3173 .4964 . 9830 .1132 .3442 . 5608 . 4392 4 57 . 6848 . 3152 .4959 . 9882 .1126 .3446 . 5627 . 4373 3 58 . 6870 . 3130 .4954 . 9935 ,1119 .3449 . 5617 . 4353 2 69 . 6891 . 3108 .4949 . 9988 .1113 .3153 . 5666 . 4334 1 60 . 6913 . 3087 .4945 .90040 .1106 .3156 . 5685 . 4314 M. Cosine. Vrs. sin. Secant, Co tang. Tang. Cosec'nt Vrs. COB, Sine, M. 131° 48° 364 NATURAL FUNCTIONS. Table 3. 42° Natural Trigonometrical Functions. 137° M^ Sine. Vrs. COS. Cosec'nt Tang. Cotang. Secant. Yrs. Bin. Cosine. M. .66913 .33087 1.4945 .90040 1.1106 1.3456 .26685 .74314 00 1 . 6936 . 3065 .4940 . 0093 .1100 .3460 . 5705 . 4295 59 2 . 6956 . 3044 .4935 . 0146 .1093 .3463 . 5724 . 4275 58 3 . 6978- . 3022 .4930 . 0198 .1086 .3467 . 5744 . 4256 .67 4 . 6999 . 3000 .4925 . 0251 .1080 .3470 . 5763 . 4236 56 5 .67021 .32979 1.4921 .90304 1.1074 1.3474 .25783 .74217 55 6 7043 . 2957 .4916 . 0357 .1067 .3477 . 5802 . 4197 54 7 . 7064 . 2936 .4911 . 0410 .1061 .3481 . 5822 . 4178 .53 8 . 7086 . 2914 .4906 . 0463 .1054 .3485 . 5841 . 4168 52 9 . 7107 . 2893 .4901 . 0515 .1048 .3488 . 5861 . 4139 51 10 .07129 .32871 1.4897 .90568 1.1041 1.3492 .25880 .74119 60 11 . 7150 . 2849 .4892 . 0621 .1035 .3495 . 5900 . 4100 49 12 . 7172 . 2828 .4887 . 0674 .1028 .3499 . 5919 . 4080 48 13 . 7194 . 2806 .4882 . 0727 .1022 .3502 . 5939 . 4061 47 14 . 7215 . 2785 .4877 . 0780 .1015 .3506 . 5959 . 4041 46 15 .07237 .32763 1.4873 .90834 1.1009 1.3509 .25978 .74022 45 10 . 7258 . 2742 .4868 . 0887- .1003 .3513 . 5998 . 4002 44 17 . 7280 . 2720 .4863 . 0940 .0996 .3517 . 6017 . 3983 43 la . 7301 . 2699 .4868 . 0993 .0990 .3520 . 6037 . 3963 42 19 . 7323 . 2677 .4864 . 1046 .0983 .3524 . 6056 . 3943 41 20 .67344 .32656 1.4849 .91099 1.0977 1.3527 .26076 .73924 40 21 . 7366 . 2634 .4844 . 1153 .0971 .3531 . 6096 . 3904 39 22 . 7387 . 2613 .4839 . 1206 .0964 .3534 . 6115 . 3885 38 23 . 7409 . 2591 .4835 . 1259 .0953 .3538 . 6135 . 3865 37 24 . 7430 . 2570 .4830 . 1312 .0951 .3542 . 6154 . 3845 36 25 .67452 .32548 1.4825 .91366 1.0945 1.3545 .26174 .73826 35 26 . 7473 . 2527 .4821 . 1419 .0939 .3549 . 0194 . 3806 34 27 . 7495 . 2505 .4816 . 1473 .0932 .3552 . 6213 . 3787 33 28 . 7516 . 2484 .4811 . 1526 .0926 .3556 . 6233 . 3767 32 29 .7537 . 2462 .4806 . 1580 .0919 .3560 . 6253 . 3747 31 30 .67559 .32441 1.4802 .91633 1.0913 1.3563 .26272 .73728 30 31 . 7580 . 2419 .4797 . 1687 .0907 .3567 . 6292 . 3708 29 32 . 7602 . 2398 .4792 . 1740 .0900 .3571 . 6311 . 3688 28 33 . 7623 . 2377 .4788 . 1794 .0894 .3574 . 6331 . 3669 27 34 . 7645 . 2355 .4783 . 1847 .0888 .3578 . 6351 . 3649 26 35 .67666 .32334 1.4778 .91901 1.0881 1.3581 .26371 .73629 25 36 . 7688 . 2312 .4774 . 1955 .0875 .3585 . 6390 . 3610 24 37 . 7709 . 2291 .4769 . 2008 .0868 .3589 . 6410 . 3590 23 38 . 7730 . 2269 .4764 . 2062 .0862 .3592 . 6430 . 3570 22 39 . 7752 . 22J8 .4760 . 2116 .0856 .3596 . 6449 . 3551 21 40 .07773 .32227 1.4755 .92170 1.0849 1.3600 .26169 .73631 20 41 . 7794 . 2205 .4750 . 2223 .0843 .3603 . 6489 . 3511 19 J2 . 7816 . 2184 .4746 . 2277 .0837 .3607 . 6508 . 3491 18 43 . 7837 . 2163 .4741 . 2331 .0830 .3611 . 6528 . 3472 17 44 . 7859 . 2141 .4736 . 2385 .0824 .3614 . 6548 . 3452 16 45 .67880 .32120 1.4732 .92439 1.0818 1.3618 .26568 .73432 15 40 . 7901 . 2098 .4727 . 2493 .0812 .3622 . 6587 . 3412 14 47 . 7923 . 2077 .4723 . 2547 .0805 .3625 . 6607 . 3393 13 48 . 7944 . 2056 .4718 . 2601 .0799 .3629 . 6627 . 3373 12 49 . 7965 . 2034 .4713 . 2655 .0793 .3633 . 6647 . 3353 11 60 .67987 .32013 1.4709 .92709 1.0786 1.3636 .26066 .73333 10 51 . 8008 . 1992 .4704 . 2703 .0780 .3640 . 6686 . 3314 9 52 . 8029 . 1970 .4699 . 2817 .0774 .3644 . 6700 . 3294 8 53 . 8051 . 1949 .4695 . 2871 .0767 .3647 . 6726 . 3274 7 54 . 8072 . 1928 .4690 . 2926 .0761 .3651 . 6746 . 3254 6 65 .68093 .31907 1.4686 .92980 1.0755 1.3655 .26765 .73234 5 66 . 8115 . 1885 .4681 . 3034 .0749 .3658 . 6785 . 3215 4 57 . 8136 . 1864 .4676 . 3088 .0742 .3662 . 6805 . 3195 3 68 . 8157 . 1843 .4672 . 3143 .0736 .3666 . 6825 . 3175 2 r.9 . 8178 . 1821 .4667 3197 .0730 .3669 . 6845 . 3155 1 60 . 8200 . 1800 .4663 . 3251 .0724 .3673 . 6865 . 3135 M. Conine. VrB. ein. Secant. Cotang. Tang. Coeec'nt IVrB. COB. Sine. M. 132° 470 Table 3. NATURAL FUNCTIONS. 365 43° Natural Trigonometrical Functions. 136° mT Sinn. Vrs. COS. Cosec'nt Tang. Co tang. Secant. Yrs. sin. Cosine. M. .68200 .31800 1,4663 .93251 1.0724 1.3673 .26865 .73135 60 1 . 8221 . 1779 .4658 . 3306 .0717 .3677 . 6884 . 3115 59 2 . 8242 . 1758 .4654 . 3360 .0711 .3681 .6904 . 3096 58 3 . 8264 . 1736 .4649 .3415 .0705 .3684 . 6924 . 3076 67 4 . 8285 . 1715 .4614 . 3469 .0699 .3688 . 6944 . 3056 56 5 .68306 .31694 1.4610 .93524 1.0692 1.3692 .26964 .73036 55 6 .8327 . 1673 .4635 . 3578 .0686 .3695 . 6984 . 3016 54 7 . 8349 . 1651 .4631 . 3633 .0680 .3699 . 7004 . 2996 53 8 . 8370 . 1630 .4626 . 3687 .0674 .3703 . 7023 . 2976 62 9 . 8391 . 1609 .4622 . 3742 .0667 .3707 . 7043 . 2966 51 10 .68412 .31588 1.4617 .93797 1.0661 1.3710 .27063 .72937 50 11 . 8433 . 1566 .4613 . 3851 .0665 .3714 . 7083 . 2917 49 12 . 8455 . 1545 .4608 . 3906 .0649 .3718 . 7103 . 2897 48 13 . 8476 . 1524 .4604 . 3961 .0643 .3722 . 7123 . 2877 47 14 . 8497 . 1503 .4599 . 4016 .0636 .3725 . 7143 . 2857 46 15 .68518 .31482 1.4595 .94071 1.0630 1.3729 .27163 .72837' 45 16 . 8539 . 1460 .4590 . 4125 .0624 .3733 . 7183 . 2817 44 17 . 8561 . 1439 .4586 . 4180 .0618 .3737 . 7203 . 2797 43 18 . 8582 . 1418 .4581 . 4235 .0612 .3740 . 7223 . 2777 42 19 . 8603 . 1397 .4577 . 4290 .0605 .3744 . 7243 . 2757 41 20 .68624 .31376 1.4572 .94345 1.0599 1.3748 .27263 .72737 40 21 . 8645 . 13.55 .4568 . 4400 .0593 .3752 . 7283 . 2717 39 22 . 8666 . 1333 .4563 . 4455 .0587 .3756 . 7302 . 2697 38 23 . 8688 .1312 .4559 . 4510 .0581 .3759 . 7322 . 2677 37 24 . 8709 . 1291 .4554 . 4565 .0575 .3763 . 7342 . 2657 36 26 .68730 .31270 1.4550 .94620 1.0568 1.3767 .27862 .72637 35 26 . 8751 . 1249 .4545 . 4675 .0562 .3771 . 7382 . 2617 34 27 . 8772 . 1228 .4541 . 4731 .0556 .3774 . 7402 . 2597 33 28 . 8793 . 1207 .4536 . 4786 .0550 .3778 . 7422 . 2577 32 29 . 8814 . 1186 .4532 . 4841 .0644 .3782 . 7442 . 2557 31 80 .68835 .31164 1.4527 .94896 1.0538 1.3786 .27462 .72537 80 31 . 8856 . 1143 .4523 . 4952 .0532 .3790 . 7482 . 2517 29 32 . 8878 . 1122 .4518 . 5007 .0525 .3794 . 7503 . 2497 28 33 . 8899 . 1101 .4514 . 5062 .0519 .3797 . 7523 . 2477 27 34 . 8920 . 1080 .4510 . 5118 .0613 .3801 . 7643 . 2457 26 35 .68941 .31059 1.4505 .95173 1.0507 1.3805 .27563 .72437 25 36 . 8962 . 1038 .4501 . 5229 .0501 .3809 . 7583 . 2417 24 37 . 8983 . 1017 .4496 . 5284 .0496 .3813 . 7603 . 2397 23 38 . 9004 . 0996 .4492 . 5340 .0489 .3816 . 7623 . 2377 22 39 . 9025 . 0975 .4487 . 5395 .0483 .3820 . 7C.43 . 2357 21 40 .69046 .30954 1.4483 .95451 1.0476 1.3824 .27663 .72337 20 11 . 9067 . 0933 .4479 . 5506 .0470 .3828 . 7683 . 2317 19 42 . 9088 . 0912 .4474 . 5562 .0464 .3832 . 7703 . 2297 18 43 . 9109 .0891 .4470 . 5618 .0458 .3836 . 7723 . 2277 17 44 . 9130 . 0870 .4465 . 5673 .0452 .3839 . 7743 . . 2266 16 45 .69151 .30849 1.4461 .95729 1.0446 1.3843 .27764 .72236 15 46 . 9172 . 0828 .4457 . 5785 .0440 .3847 . 7784 . 2216 14 47 . 9193 . 0807 .4452 . 5841 .0434 .3851 . 7804 . 2196 13 48 . 9214 . 0786 .4448 . 5896 .0428 .3855 . 7824 . 2176 12 49 . 9235 . 0765 .4443 . 5952 .0422 .3859 . 7844 . 2156 11 50 .69256 .30744 1.4439 .96008 1.0416 1.3863 .27864 .72136 10 51 . 9277 . 0723 .4435 . 6064 .0410 .3867 . 7884 . 2115 9 52 . 9298 . 0702 .4430 . 6120 .0404 .3870 . 7904 . 2095 8 53 . 9319 . 0681 .4426 . 6176 .0397 .3874 . 7925 . 2075 7 54 . 9340 . 0660 .4422 . 6232 .0391 .3878 , 7945 . 2055 6 55 .69361 .30639 1.4417 .96288 1.0385 1.3882 .27965 .72035 5 56 . 9382 . 0618 .4413 . 6344 .0379 .3886 . 7985 . 2015 4 57 . 9403 . 0597 .4408 . 6400 .0373 .3890 . 8005 . 1994 3 58 . 9424 . 0576 .4404 . 6456 .0367 .3894 . 8026 . 1974 2 59 . 9445 . 0555 .4400 . 6513 .0361 .3898 . 8046 . 1954 1 60 . 9466 . 0534 .4395 . a569 .0355 .3902 . 8066 . 1934 M. Cosine. Vrs. sin. Secant. Gotang. Tang. Cosec'nt Vrs. cos. Sine. , M. 366 NATURAL FUNCTIONS. Table 3. 44 D Natural Trigonometrical Functions. J 35° M. Sino. Vrg. COS. Cosec'nt Tang. Cotang. Secant. Vrs. sin. Cosine. M. .59466 .30534 1.4395 .96569 1.0355 1.3902 .28066 .719.34 60 1 . 9487 . 0513 .4391 . 6625 .0349 .3905 . 8086 . 1914 59 2 . 9508 . 0492 .4387 . 6681 .0343 .3909 . 8106 . 1893 58 3 . 9528 . 0471 .4382 . 6738 .0337 .3913 . 8127 . 1873 57 4 . 9549 . 0450 .4378 . 6794 .0331 .3917 . 8147 . 1853 56 5 .69570 .30430 1.4374 .96850 1.0325 1.3921 .28167 .71833 55 6 . 9591 . 0409 .4370 . 6907 .0319 .3925 . 8187 . 1813 54 7 . 9612 . 0388 .4365 . 6963 .0313 .3929 . 8208 . 1792 53 8 . 9633 . 0367 .4361 . 7020 .0307 .3933 . 8228 . 1772 52 9 . 9654 . 0346 .4357 . 7076 .0301 .3937 . 8248 . 1752 51 10 .69675 .30325 1.4352 .97133 1.0295 1.3941 .28268 .71732 50 11 . 9696 . 0304 .4348 . 7189 .0289 .3945 . 8289 . 1711 49 12 . 9716 . 0283 .4344 . 7216 .0283 .3949 . 8309 . 1691 48 13 . 9737 . 0263 .4339 . 7302 .0277 .3953 . 8329 . 1671 47 14 . 9758 . 0242 .4335 . 7359 .0271 .3957 . 8349 . 1650 46 15 .69779 .30221 1.4331 .97416 1.0265 1.3960 .28370 .71630 45 16 . 9800 . 0200 .4327 . 7472 .0259 .3964 . 8390 . 1610 14 17 . 9821 . 0179 .4322 . 7529 .0253 .3968 . 8410 . 1589 43 18 . 9841 . 0158 .4318 . 7586 .0247 .3972 . 8431 . 1669 42 19 . 9862 . 0138 .4314 . 7643 .0241 .3976 . 8451 . 1549 41 20 .69883 .30117 1.4310 .97699 1.0235 1.3980 .28471 .71529 40 21 . 9904 . 0096 .4305 . 7756 .0229 .3984 . 8492 . 1.508 39 22 . 9925 . 0075 .4301 . 7813 .0223 .3988 . 8512 . 1488 38 23 . 9945 . 0054 .4297 . 7870 .0218 .3992 . 8532 . 1468 37 24 . 9966 . 0034 .4292 . 7927 .0212 .3996 . 8553 . 1447 36 25 .69987 .30013 1.4288 .97984 1.0206 1.4000 .28573 .71427 35 26 .70008 .29992 .4284 . 8041 .0200 .4004 . 8593 . 1406 34 27 . 0029 . 9971 .4280 . 8098 .0194 .4008 . 8614 . 1386 33 28 . 0049 . 9950 .4276 . 8155 .0188 .4012 . 8634 . 1366 32 29 . 0070 . 9930 .4271 . 8212 .0182 .4016 . 8654 . 1345 31 30 .70091 .29909 1.4267 .98270 1.0176 1.4020 .28675 .71325 30 31 . 0112 . 9888 .4283 . 8327 .0170 .4024 . 8695 . 1305 29 32 . 0132 . 9867 .4259 . 8384 .0164 .4028 . 8716 . 1284 28 33 . 0153 . 9847 .42.54 . 8441 .0158 .4032 8736 . 1264 27 34 . 0174 . 9826 .4250 . 8499 .0152 .4036 . 8756 . 1243 26 35 .70194 .29805 1.4246 .98556 1.0146 1.4040 .28777 .71223 25 30 . 0215 . 9785 .4242 . 8613 .0141 .4044 . 8797 . 1203 24 37 . 0236 . 9764 .4238 . 8671 .0135 .4048 . 8818 . 1182 23 38 . 0257 . 9743 .4233 . 8728 .0129 .4052 . 8838 . 1162 22 39 . 0277 . 9722 .4229 . 8786 .0123 .4056 . 8859 . 1141 21 40 .70298 .29702 1.4225 .98843 1.0117 1.4060 .28879 .71121 20 41 . 0319 . 9681 .4221 . 8901 .0111 .4065 . 8899 . 1100 19 42 . 0339 . 9660 .4217 . 8958 .0105 .4069 . 8920 . 1080 18 43 . 0360 . 9610 .4212 . 9016 .0099 .4073 . 8940 . 1059 17 44 . 0381 . 9619 .4208 . 9073 .0093 .4077 . 8961 . 1039 16 45 .70401 .29598 1.4204 .99131 1.0088 1.4081 .2S9S1 .71018 15 46 . 0422 . 9578 .4200 . 9189 .0082 .4085 . 9002 . 0998 14 47 . 0443 . 9557 .4196 . 9246 .0076 .4089 . 9022 . 0977 13 48 0463 . 9536 .4192 . 9304 .0070 .4093 . 9043 . 0957 12 49 . 0484 . 9516 .4188 . 9362 .0064 .4097 . 9063 . 0936 11 50 .70505 .29495 1.4183 .99420 1.0058 1.4101 .29084 .70916 10 51 . 0525 . 9475 .4179 . 9478 .0052 .4105 . 9104 . 0895 9 62 . 0546 . 9454 .4175 . 9536 .0047 .4109 . 9125 . 0875 g 53 . 0566 . 9133 .4171 . 9593 .0041 .4113 . 9145 . 0854 7 54 . 0587 . 9413 .4167 . 9651 .0035 .4117 . 9166 . 0834 6 55 .70608 .29392 1.4163 .99709 1.0029 1.4122 .29186 .70813 5 66 . 0628 . 9372 .4159 . 9767 .0023 .4126 . 9207 . 0793 4 57 . 0649 . 9351 .4154 . 9826 .0017 .4130 . 9228 . 0772 3 58 . 0669 . 9330 .4150 . 9884 .0012 .4134 . 9248 . 07,52 2 59 . 0690 . 9310 .4146 . 9942 .0006 .4138 . 9269 . 0731 1 60 . 0711 . 9289 .4112 1.0000 .0000 .4142 . 9289 . 0711 M. Cosine. Vrs. sin. Secant. Col^ang. Tang. Cosec'nt Vrs. cos. Sine. M. 134° 45° Table 4. SQUARES, CUBES AND KOOTS. 367 Squares, Cubes, Square Roots, Cube Roots, Circumferences AND Circular Areas ov Nos. from t to 520 No. Square Cube Sq. Root Cube Root Circle Circum. Area I I I 1. 0000 1 .0000 3.142 0.7854 2 4 8 1. 4142 1.2599 6.283 3.1416 3 9 27 I.7321 1.4422 9-425 7.0686 4 16 64 2 .0000 1-5874 12.566 12.5664 5 25 125 2.2361 1. 7100 15-708 19.6350 6 36 216 2.4495 I.8171 18.850 28.2743 7 49 343 2.6458 1.9129 21.991 38-4845 8 64 512 2.8284 2 .0000 25-133 50.2655 9 81 729 3.0000 2.0801 28.274 63.6173 10 100 1000 3.1623 2.1544 31.416 78.5398 II 121 1331 3.3166 2.2240 34-558 95-033 12 144 1728 3.4641 2.2894 37.699 113.097 13 169 2197 3.6056 2-3513 40.841 132.732 14 196 2744 3-7417 2.41OI 43.982 153-938 IS 225 3375 3-8730 2.4662 47.124 176-715 16 256 4096 4.0000 2.5198 50.265 201.062 17 289 4913 4-I23I 2-5713 53-407 226.980 18 324 5832 4.2426 2.6207 56-549 254-469 19 361 6859 4-3589 2.6684 59.690 283.529 20 400 8000 4.4721 2.7144 62.832 314-159 21 441 9261 4.5826 2.7589 65-973 346.361 22 484 10648 4.6904 2.8020 69.115 380.133 23 529 12167 4-7958 2.8439 72.257 415.476 24 576 13824 4.8990 2.8845 75-398 452.389 25 625 15625 5 .0000 2.9240 78.540 490.874 26 676 17576 5-0990 2.9625 81.681 530.929 27 729 19683 5.1962 3 .0000 84.823 572.555 28 784 21952 5-2915 3.0366 87-965 615.752 29 841 24389 5-3852 3-0723 91.106 660.520 3° 900 27000 5-4772 3.1072 94.248 706.858 31 g6i 29791 5-5678 3-1414 90.389 754.768 32 1024 32768 5-6569 3.1748 100.531 804.248 33 1089 35937 5-7446 3-2075 103.673 855.299 34 1156 39304 5-8310 3-2396 106.814 907.920 35 1225 42875 5-9161 3.271I 109.956 962.113 36 1296 46656 6.0000 3-3019 113.097 1017.88 37 1369 50653 6.0828 3-3322 116.239 1075.21 38 1444 54872 6.1644 3.3620 I19.381 1134.ll 39 1521 S9319 6.2450 3-3912 122.522 1194.59 40 1600 64000 6.3246 3.4200 125.660 1256.64 368 SQUARES, CUBES AND BOOTS. Table 4. Squares, Cubes, Squaee Roots, Cube Roots, Circumperences AND Circular Areas of Nos. from i to 520 No. Square Cube Sq. Root Cube Root Circle Circum. Area 41 1681 68921 6.4031 3-4482 128.81 1320.25 42 1764 74088 6.4807 3.4760 131-95 1385-44 43 1849 795°7 6-5574 3-5034 135-09 1452.20 44 1936 85184 6.6332 3-5303 138-23 1520.53 45 2025 91125 6.7082 3-5569 141-37 1590-43 46 2116 97336 6.7823 3-5830 144-51 1661.90 47 2209 103823 6-8557 3.6088 147-65 1734-94 48 2304 I 10592 6.9282 3-6342 150.80 1809.56 49 2401 1 1 7649 7.0000 3-6593 153-94 1885.74 50 2500 125000 7.07II 3.6840 157.08 1963.50 SI 2601 I3265I 7-1414 3-7084 160.22 2042.82 52 2704 140608 7.2III 3-7325 163-36 2123.72 53 2809 148877 7.2801 3-7563 166.50 2206.18 54 2916 157464 7-3485 3-7798 169.65 2290.22 55 3°25 166375 7.4162 3.8030 172.79 2375-83 56 3136 I756I6 7-4833 3-8259 175-93 2463.01 57 3249 I85I93 7-5498 3-8485 179.07 2551.76 58 3364 I95II2 7.6158 3.8709 182.21 2642.08 59 3481 205379 7.68II 3-8930 185-35 2733-97 60 3600 216000 7.7460 3-9149 188.50 2827.43 61 3721 226981 7.8102 3-9365 191.64 2922.47 62 3844 238328 7-8740 3-9579 194.78 3019.07 63 3969 250047 7-9373 3-9791 197.92 3117-25 64 4096 262144 8.0000 4.0000 201.06 3216.99 65 4225 274625 8.0623 4.0207 204.20 3318.31 66 4356 287496 8.1240 4.0412 207.35 3421.19 67 4489 300763 8.1854 4-0615 210.49 3525-65 68 4624 314432 8.2462 4.0817 213-63 3631.68 69 4761 328509 8.3066 4.1016 216.77 3739-28 70 4900 343000 8.3666 4.1213 219.91 3848-45 71 5041 3579II 8.4261 4.1408 223.05 3959-19 72 5184 373248 8.4853 4-1602 226.19 4071.50 73 5329 389017 8.5440 4-1793 229.34 4185.39 74 5476 405224 8.6023 4.1983 232.48 4300.84 75 5625 421875 8.6603 4.2172 235.62 4417.86 76 5776 438976 8.7178 4-2358 238.76 4536-46 77 5929 456533 8-7750 4-2543 241.90 4656.63 78 6084 474552 8.8318 4.2727 245-04 4778-36 79 6241 493039 8.8882 4.2908 248.19 4901.67 80 6400 512000 8.9443 4-3089 251-33 5026.55 Table 4. SQUAEES, CUBES AND ROOTS. 369 Sqtjases, Cubes, Square Roots, Cube Roots, Circui iferences AND Circular Areas of Nos. from i to 520 No. Square Cube Sq. Root Cube Root Circle Circum. Area 81 ■ 6561 S31441 9.0000 4.3267 254-47 5153-00 82 6724 SS1368 9-0554 4-3445 257-61 5281.02 83 6889 571787 9.II04 4.3621 260.7s 5410.61 84 ■7056 592704 9.1652 4-3795 263.89 5541-77 8S 7225 614125 9.219s 4.3968 267.04 5674-50 86 7396 636056 9.2736 4.4140 270.18 5808.80 87 7569 658503 9-3274 4.4310 273-32 5944-68 88 7744 681472 9.3808 4.4480 276.46 6082.12 89 7921 704969 9-4340 4.4647 279.60 6221.14 90 8100 729000 9.4868 4.4814 282.74 6361.73 91 8281 7S3S7I 9-5394 4.4979 285.88 6503.88 92 8464 778688 9-5917 4-5144 289.03 6647.61 93 8649 804357 9-6437 4-5307 292.17 6792.91 94 8836 830584 9-6954 4.5468 295-31 6939.78 9S 9025 857375 9.7468 4.5629 298.45 7088.22 96 9216 884736 9.7980 4-5789 301-59 7238.23 97 9409 912673 9.8489 4-5947 3°4-73 7389-81 98 9604 941192 9-8995 4.6104 307.88 7542.96 99 9801 970299 9.9499 4.6261 311.02 7697-69 100 lOOOO I 000000 10.0000 4.6416 314.16 7853-98 lOI I020I 1030301 10.0499 4.6570 317-30 8011.85 102 10404 1061208 10.0995 4.6723 320.44 8171.28 103 10609 1092727 10.1489 4-6875 323-58 8332.29 104 I0816 I I 24864 10.1980 4-7027 326.73 8494.87 loS 1 1025 1157625 10.2470 4-7177 329.87 8659.01 106 11236 1191016 10.2956 4.7326 333-01 8824.73 107 1 1449 1225043 10.3441 4-7475 336-15 8992.02 108 1 1664 1259712 10.3923 4.7622 339-29 9160.88 109 I1881 1295029 10.4403 4.7769 342-43 9331-32 no I2IOO 1331000 10.4881 4.7914 345-58 9503-32 III I232I 136763J 10.5357 4.8059 348.72 9676.89 112 12544 1404928 10.5830 4.8203 351-86 9852.03 "3 12769 1442897 10.6301 4.8346 355-00 10028.7 ti4 12996 1481544 10.6771 4.8488 358-14 10207.0 "S 13225 152087s 10.7238 4.8629 361.28 10386.9 116 13456 1560896 10.7703 4-8770 364.42 10568.3 117 13689 1601613 10.8167 4.8910 367-57 10751.3 118 13924 1643032 10.8628 4.9049 370.71 10935.9 119 I4161 1685159 10.9087 4.9187 373-85 11122.0 120 14400 1728000 10.9545 4-9324 376-99 11309.7 370 SQUARES, CUBES AND ROOTS. Table 4. Squares, Cubes, Squaee Roots, Cube Roots, Circumferences AND Circular Areas of Nos. from i to 520 No. Square Cube Sq. Root Cube Root Circle Circum. Area 121 14641 1771561 1 1 .0000 4.9461 380.13 I1499.0 122 14884 1815848 11.0454 4-9597 383-27 11689.9 123 15129 1860867 1 1 .0905 4-9732 386.42 11882.3 124 15376 1906624 11-1355 4.9866 389-56 12076.3 I2S 1562s 1953125 11.1803 5.0000 392.70 12271.8 126 15876 2000376 11.2250 5-0133 395-84 12469.0 127 16129 2048383 11.2694 ■ 5-0265 398.98 12667.7 128 16384 2097152 11-3137 S-°397 402.12 12868.0 129 16641 2146689 11-3578 5.0528 405.27 13069.8 130 16900 2197000 11.4018 5.0658 408.41 13273.2 131 17161 2248091 1 1 4455 5.0788 411-55 13478.2 132 17424 2299968 11.4891 5-0916 414.69 13684.8 133 17689 2352637 11.5326 5-1045 417-83 13892.9 134 17956 2406104 11-5758 5.1172 420.97 14102.6 135 18225 2460375 11.6190 5.1299 424.12 14313-9 136 18496 2515456 11.6619 5.1426 427.26 14526.7 137 18769 2571353 11.7047 5-1551 430.40 14741-1 138 19044 2628072 11-7473 5.1676 433-54 I49S7-1 T-39 19321 2685619 11.7898 5.1801 436.68 15174-7 140 19600 2744000 11.8322 S-1925 439.82 15393-8 141 19881 2803221 11.8743 5.2048 442.96 15614-S 142 20164 2863288 11.9164 5.2171 446.11 15836.8 143 20449 2924207 11-9583 5-2293 449-25 16060.6 144 20736 2985984 12.0000 5-2415 452-39 16286.0 I4S 21025 3048625 12.0416 5-2536 455-53 16513.0 146 21316 3112136 1 2 .0830 5-2656 458.67 16741.5 147 21609 3176523 12.1244 5.2776 461.81 16971.7 148 21904 3241792 12-1655 5.2896 464.96 17203.4 149 22201 3307949 12.2066 5-3015 468.10 17436.6 150 22500 3375000 12.2474 5-3133 471-24 17671.5 151 22801 3442951 12.2882 5-3251 474-38 17907.9 IS2 23104 3511808 12.3288 5-3368 477-52 18145.8 153 23409 3581577 12.3693 5-3485 480.66 18385.4 154 23716 3652264 12.4097 5-3601 483.81 18626.5 15s 24025 372387s 12.4499 S-3717 486.95 18869.2 156 24336 3796416 12.4900 5-3832 490.09 19113.4 157 24649 3869893 12.5300 5-3947 493-23 19359-3 158 24964 3944312 12.5698 5.4061 496-37 19606.7 159 25281 4019679 1 2 .6095 5-4175 499-51 19855-7 160 25600 4096000 12.6491 5.4288 502.65 20106.2 Table 4. SQUARES, CUBES AND ROOTS. 371 Squares, Cubes, Square Roots, Cube Roots, Circumferences AND Circular Areas of Nos. from i to 520 ClRPTTi' No. Square Cube Sq. Root Cube Root Circum . Area 161 25921 4173281 12.6886 5.4401 505.80 20358.3 162 26244 4251528 12.7279 5-4514 508.94 20612.0 163 26569 4330747 12.7671 5.4626 512.08 20867.2 164 26896 4410944 12.8062 5-4737 515-22 21124.I i6s 27225 4492125 12.8452 5.4848 518.36 21382.5 166 27556 4574296 12.8841 5-4959 521-50 21642.4 167 27889 4657463 12.9228 5-5069 524-65 21904.0 168 28224 4741632 12.9615 5-S178 527-79 22167. 1 169 28561 4826809 13.0000 5.5288 530-93 22431.8 170 28900 4913000 13.0384 5-5397 534-07 22698.0 171 29241 50002 I I 13.0767 5-5505 537-21 22965.8 172 29584 5088448 13.1149 S-5613 540.35 23235-2 173 29929 5177717 13-1529 5-5721 543-50 23506.2 174 30276 5268024 13.1909 S-5828 546.64 23778.7 175 30625 5359375 13.2288 5-5934 549-78 24052.8 176 30976 5451776 13.2665 5.6041 552-92 24328.5 177 31329 5545233 13-3041 5-6147 556.06 24605.7 178 31684 5639752 13-3417 5-6252 559-20 24884.6 179 32041 5735339 13-3791 S-6357 562.35 25164.9 180 32400 5832000 13.4164 5.6462 565-49 25446.9 181 32761 5929741 13-4536 5-6567 568.63 25730-4 182 33124 6028568 13.4907 5.6671 571-77 26015.5 183 33489 6128487 13-5277 5-6774 574-91 26302.2 184 33856 6229504 13-5647 5-6877 578-05 26590.4 i8s 34225 6331625 13.6015 5.6980 581.19 26880.3 186 34596 6434856 13.6382 5-7083 584-34 27171.6 187 34969 6539203 13.6748 5-7185 587-48 27464.6 188 35344 6644672 13-7113 5.7287 590.62 27759.1 189 35721 6751269 13-7477 5-7388 593-76 28055.2 190 36100 6859000 15.7840 5-7489 596.90 28352.9 191 36481 6967871 13.8203 5-7S90 600.04 28652.1 192 36864 7077888 13.8564 5.7690 603.19 28952.9 193 37249 7189057 13.8924 5-7790 606.33 29255-3 194 37636 7301384 13.9284 5.7890 609.47 29559-2 19s 38025 7414875 13.9642 5-7989 612.61 29864.8 196 38416 7529536 14.0000 5.8088 615-75 30171.9 197 38809 7645373 14-0357 5.8186 618.89 30480.5 198 39204 7762392 14.0712 5.828s 622.04 30790.7 199 39601 7880599 14.1067 5-8383 625.18 3 1 102 .6 200 40000 8000000 14.1421 5.8480 628.32 31415-9 S72 SQUAEES, CUBES AND ROOTS. Table 4. Squaees, Cubes, Square Roots, Cube Roots, Circumferences AND Circular Areas of Nos. from i to 520 Circle No. Square Cube Sq. Root Cube Root Circum. Area 201 40401 8120601 14.1774 5-8578 631.46 31730.9 202 40804 8242408 14.2127 5 -8675 634.60 32047.4 203 41209 8365427 14.2478 5-8771 637-74 32365.5 204 41616 8489664 14.2829 5.8868 640.89 32685.1 205 42025 8615125 14.3178 5. 8964 644-03 33006.4 206 42436 8741816 14.3527 5-9°S9 647-17 33329-2 207 42849 8869743 14.3875 S-915S 650.31 33653-5 208 43264 8998912 14.4222 5-9250 653-45 33979-S 209 43681 9129329 14.4568 5-9345 656-59 34307-0 210 44100 9261000 14.4914 5-9439 659-73 34636.1 211 44521 9393931 14.5258 5-9533 662.88 34966.7 212 44944 9528128 14.5602 5.9627 666.02 35298.9 213 45369 9663597 14-5945 5-9721 669.16 35632.7 214 45796 9800344 14.6287 5.9814 672.30 35968.1 2IS 46225 9938375 14.6629 5 -9907 675-44 36305.0 216 46656 10077696 14.6969 6.0000 678.58 36643.5 217 47089 10218313 14.7309 6.0092 681.73 36983.6 218 47524 10360232 14.7648 6.0185 684.87 37325-3 219 47961 10503459 14.7986 6.0277 688.01 37668.5 220 48400 10648000 14.8324 6.0368 691.IS 38013-3 221 48841 10793861 14.8661 6.0459 694.29 38359-6 222 49284 10941048 14.8997 6.0350 697-43 38707-6 223 49729 I1089567 14.9332 6.0641 700-58 39°57-i 224 50176 I 1239424 14.9666 6.0732 703.72 39408.1 225 50625 I 1390625 15.0000 6.0822 706.86 39760.8 226 51076 I1543176 i5-°333 6.0912 710.00 40115.0 227 51529 I 1697083 15.0665 6.1002 713-14 40470.8 228 51984 I1852352 15.0997 6.1091 716.28 40828.1 229 52441 12008989 15-1327 6.1180 719.42 41187.1 230 52900 1 2167000 15.1658 6.1269 722.57 41547.6 231 53361 12326391 15.1987 6.1358 725-71 41909.6 232 53824 12487168 15-2315 6.1446 728.85 42273.3 233 54289 12649337 15.2643 6.1534 731-99 42638.5 234 54756 12812904 15.2971 6.1622 735-13 43005.3 23s 55225 12977875 15-3297 6.1710 738-27 43373-6 236 55696 13144256 15-3623 6.1797 741.42 43743-5 237 56169 13312053 15-3948 6.1885 744.56 44115.0 238 56644 13481272 15.4272 6.1972 747-70 44488.1 239 57121 13651919 15-4596 6.2058 750.84 44862.7 240 57600 13824000 15.4919 6.2145 753-98 45238.9 Table 4. SQUARES, CUBES AND KOOTS. 373 Squares, Cubes, Square Roots, Cube Roots, Circumferences AND Circular Areas of Nos. from 110520 Square Cube Sq. Root Cube Root Circle No. Circum. Area 241 58081 13997521 15-5242 6.2231 757-12 45616.7 242 58564 14172488 15-5563 6.2317 760.27 45996.1 243 S9°49 14348907 15-5885 6.2403 763-41 46377.0 244 S9S36 14526784 15.6205 6.2488 766.55 46759-5 24S 60025 14706125 15-6525 6-2573 769.69 47143-5 246 60516 14886936 15.6844 6.2658 772.83 47529.2 247 61009 15069223 15.7162 6.2743 775-97 47916.4 248 61504 15252992 15.7480 6.2828 779.12 48305.1 249 62001 15438249 15-7797 6.2912 782.26 48695.5 250 62500 15625000 15,8114 6.2996 785-40 49087.4 251 63001 15813251 15.8430 6.3080 788.54 49480.9 252 63504 16003008 15-8745 6.3164 791.68 49875-9 253 64009 16194277 15.9060 6.3247 794.82 50272.6 2S4 64516 16387064 15-9374 6-3330 797-96 50670.7 2SS 65025 16581375 15.9687 6.3413 801.11 51070.5 256 65536 16777216 16.0000 6.3496 804.25 51471-9 2S7 66049 16974593 16.0312 6-3579 807.39 51874.8 258 66564 17173512 16.0624 6.3661 810.53 52279.2 2S9 67081 17373979 16.0935 6-3743 813.67 52685.3 260 67600 17576000 16.1245 6-3825 816.81 53092-9 261 68121 I 7779581 16.1555 6.3907 819.96 53502.1 262 68644 17984728 16.1864 6.3988 823.10 53912.9 263 69169 18191447 16.2173 6.4070 826.24 54325-2 264 69696 18399744 16.2481 6.4151 829.38 54739-1 265 70225 18609625 16.2788 6.4232 832.52 55154.6 266 70756 18821096 16.3095 6.4312 835-66 SS57I-6 267 71289 19034163 16.3401 6-4393 838.81 5599°-3 268 71824 19248832 16.3707 6.4473 841.95 56410.4 269 72361 19465109 16.4012 6-4553 845.09 56832.2 270 72900 19683000 16.4317 6.4633 848.23 57255-5 271 73441 19902511 16.4621 6.4713 851-37 57680.4 272 ^3984 20123648 16.4924 6.4792 854-51 58106.9 273 74529 20346417 16.5227 6.4872 857-66 58534-9 274 75076 20570824 16.5529 6.4951 860.80 58964.6 27s 75625 20796875 16.5831 6.5030 863.94 59395-7 276 76176 21024576 16.6132 6.5108 867.08 59828.5 277 76729 21253933 16.6433 6.5187 870.22 60262.8 278 77284 21484952 16.6733 6.5265 873-36 60698.7 279 77841 21717639 16.7033 6.5343 876.50 61136.2 280 78400 21952000 16.7332 6.5421 879.65 61575-2 374 SQUARES, CUBES AND EOOTS. Table 4. Squares, Cubes, Square Roots, Cube Roots, Circumferences AND Circular Areas of Nos. from i to 520 No. Square Cube Sq. Root Cube Root Circle Circum. Area 281 78961 22188041 16.7631 6.5499 882.79 62015.8 282 79524 22425768 16.7929 6-5577 885.93 62458.0 283 80089 22665187 16.8226 6.5654 889.07 62901.8 284 80656 22906304 16.8523 6.5731 892.21 63347.1 285 81225 23149125 16.8819 6.5808 895-35 63794.0 286 81796 23393656 16.9I15 6.5885 898.50 64242.4 287 82369 23639903 16.9411 6.5962 901.64 64692.5 288 82944 23887872 16.9706 6.6039 904.78 65144.I 289 83521 24137569 17.0000 6.6II5 907.92 65597-2 290 84100 24389000 17.0294 6.6I9I 911.06 66052.0 291 84681 24642 I 71 17.0587 6.6267 914.20 66508.3 292 85264 24897088 17.08S0 6.6343 917.3s 66966.2 293 85849 25153757 17.I172 6.6419 920.49 67425.6 294 86436 25412184 17.1464 6.6494 923.63 67886.7 •^95 87025 25672375 17.1756 6.6569 926.77 68349.3 296 87616 25934336 17.2047 6.6644 929.91 68813.5 297 88209 26198073 17.2337 6.6719 933 -05 69279.2 298 88804 26463592 17.2627 6.6794 936.19 69746.5 299 89401 26730899 17.2916 6.6869 939.34 70215.4 300 90000 2 7000000 17.3205 6.6943 942.48 70685.8 301 90601 27270901 17.3494 6.7018 945.62 71157.9 302 91204 27543608 17.3781 6.7092 948.76 71631.5 5°3 91809 27818127 17.4069 6.7166 951.90 72106.6 3°4 92416 28094464 174356 6.7240 955. °4 72583.4 305 93°25 28372625 17.4642 6.73-I3 958.19 73061.7 306 93636 28652616 17.4929 6.7387 961.33 73541.S 3°7 94249 28934443 17.5214 6.7460 964.47 74023.0 308 94864 29218112 17.5499 6-7533 967.61 74506.0 3°9 95481 29503629 17-5784 6.7606 970-75 74990.6 310 96100 29791000 17.6068 6.7679 973-89 75476.8 3" 96721 30080231 17.6352 6.7752 977.04 75964.5 312 97344 30371328 17.6635 6.7824 980.18 76453.8 Z^3 97969 30664297 17.6918 6.7897 983-32 76944.7 314 98596 30959144 17.7200 6.7969 986.46 77437-1 315 99225 31255875 17.7482 6.8041 989 .60 77931.1 316 99856 31554496 17.7764 6.8113 992.74 78426.7 317 100489 31855013 17.8045 6.8185 995.88 78923.9 318 101124 32157432 17.8326 6.8256 999.03 79422.6 319 101761 32461759 17.8606 6.8328 1002.20 79922.9 320 102400 32768000 17.8885 6.8399 1005.30 . - - --^^ ■- — 80424.8 Table 4. SQUARES, CUBES AND EOOTS. 375 Squares, Cubes, Square Roots, Cube Roots, Circumferences AND Circular Areas of Nos. from i to 520 No; Square Cube Sq. Root Cube Root Circle Circum. Area 321 I03041 33076161 17.9165 6.8470 1008.5 80928.2 322 103684 33386248 17.9444 6.8541 1011.6 81433.2 323 104329 33698267 17.9722 6.8612 1014.7 81939.8 324 104976 34012224 18.0000 6.8683 1017.9 82448.0 32s 105625 34328125 18.0278 6.8753 102 1. 82957.7 326 106276 34645976 18.0555 6.8824 1024.2 83469-0 327 106929 34965783 18.0831 6.8894 1027.3 83981.8 328 107584 35287552 18.1108 6.8964 1030.4 84496.3 329 108241 35611289 18.1384 6.9034 1033.6 85012.3 330 108900 35937000 18.1659 6.9104 1036.7 85529-9 331 109561 36264691 18.1934 6.9174 1039.9 86049.0 332 110224 36594368 18.2209 6.9244 1043.0 86569.7 333 1 10889 36926037 18.2483 6-9313 1046.2 87092.0 334 111556 37259704 18.2757 6.9382 1049.3 87615.9 335 II2225 37595375 18.3030 6.9451 1052.4 88141.3 336 I12896 37933056 .18.3303 6.9521 IOS5-6 88668.3 337 113569 38272753 18.3576 6.9589 1058.7 89196.9 338 I 14244 38614472 18.3848 6.9658 1061.9 89727.0 339 I 1492 I 38958219 18.4120 6.9727 1065.0 90258.7 340 I 15600 39304000 18.4391 6-9795 1068.1 90792.0 341 I16281 39651821 18.4662 6.9864 1071.3 91326.9 342 I16964 40001688 18.4932 6.9932 1074.4 91863.3 343 1 1 7649 40353607 18.5203 7.0000 1077.6 92401.3 344 I 18336 40707584 18.5472 7.0068 1080.7 92940.9 345 I 19025 41063625 18.5742 7.0136 1083.8 93482.0 346 119716 41421736 18.6011 7.0203 1087.0 94024.7 347 120409 41781923 18.6279 7.0271 1090.1 94569-0 121104 42144192 18.6548 7-0338 i°93-3 95114.9 349 121801 42508549 18.6815 7.0406 1096.4 95662.3 350 122500 42875000 18.7083 7-0473 1099.6 96211.3 351 123201 43243551 18.735° 7-0540 1102.7 96761.8 352 123904 43614208 18.7617 7.0607 1105.8 97314.0 353 124609 43986977 18.7883 7.0674 1109.0 97867.7 354 125316 44361864 18.8149 7.0740 1112. I 98423-0 355 126025 44738875 18.8414 7.0807 "15-3 98979-8 356 126736 45118016 18.8680 7.0873 1118.4 99538.2 357 127449 45499293 18.8944 7.0940 1121.5 100098 358 128164 45882712 18.9209 7.1006 1124.7' 100660 359 128881 46268279 18.9473 7.1072 1127.8 101223 360 129600 46656000 18.9737 7.1138 1131.0 101788 376 SQUARES, CUBES AND ROOTS. Table 4. Squaees, Cubes, Square Roots, Cube Roots, Circumferences AND Circular Areas of Nos. from i to 520 ClEf^I-T=; No. Square Cube Sq. Root Cube Root Circum. Area 361 130321 47045881 19.0000 7.1204 I134.I 102354 362 131044 47437928 19.0263 7.1269 II37-3 102922 363 131769 47832147 19.0526 7-1335 1 140.4 103491 364 132496 48228544 19.0788 7.1400 1 143 -5 104062 36s 133225 48627125 19.1050 7.1466 1146.7 104635 366 133956 49027896 I9.I3II 7-1531 1 149.8 105209 367 134689 49430863 19.1572 7-1596 "53-0 105785 368 135424 49836032 19-1833 7.1661 1156.1 106362 369 136161 50243409 19.2094 7.1726 1159.2 106941 370 136900 50653000 19-2354 7.1791 1162.4 107521 371 I 3 7641 51064811 19.2614 7-1855 1165.5 108103 372 138384 51478848 19.2873 7.1920 1168.7 108687 373 139 1 29 51895117 19.3132 7.1984 1171.8 109272 374 139876 52313624 19-3391 7.2048 1175.0 109858 375 140625 52734375 19.3649 7.2112 1 178.1 110447 376 I4I376 53157376 19.3907 7.2177 1181.2 II1036 377 I42I29 53582633 19.4165 7.2240 1184.4 111628 378 142884 54010152 19.4422 7.2304 1187.S 112221 379 I4364I 54439939 19.4679 7.2368 1190.7 I1281S 380 144400 54872000 19.4936 7-2432 1193.8 113411 3^' I45I6I 55306341 19.5192 7-2495 1196.9 114009 382 145924 55742968 19.5448 7-2558 1200.1 114608 383 146689 56181887 19.5704 7.2622 1203.2 115209 384 147456 56623104 19-5959 7.2685 1206.4 115812 38s 148225 57066625 19.6214 7.2748 1209.5 I16416 386 148996 57512456 19.6469 7.2811 1212.7 I17021 3f7 149769 57960603 19.6723 7.2874 1215.8 117628 388 150544 58411072 19.6977 7.2936 1218.9 118237 389 I5I32I 58863869 19.7231 7-2999 1222.1 118847 390 I52IOO 59319000 19.7484 7.3061 12*25.2 I 19459 391 I5288I 59776471 19-7737 7-3124 1228.4 120072 392 153664 60236288 19.7990 7.3186 1231-5 120687 393 154449 60698457 19.8242 7-3248 1234.6 121304 394 155236 611 62984 19.8494 7-331° 1237.8 121922 39S 156025 61629875 ig.8746 7-3372 1240.9 122542 396 I568I6 62099x36 19.8997 7-3434 1244.1 123163 397 157609 62570773 19.9249 7-3496 1247.2 123786 398 158404 63044792 19.9499 7-3558 1250.4 124410 399 I5920I 63521199 19.9750 7-3619 1253-5 125036 400 ■ 160000 64000000 20.0000 7.3684 1256.6 125664 Table 4. SQUARES, CUBES AND ROOTS. 377 Squares, Cubes, Square Roots, Cube Roots. Circumferences, AND Circular Areas or Nos. from i to 52° No. Square Cube Sq. Root Cube Root Circle Circum. Area 401 160801 64481201 20.0250 7-3742 1259.8 126293 402 161604 64964808 20.0499 7-3803 1262.9 126923 403 162409 65450827 20.0749 7.3864 1266.I 127556 404 163216 65939264 20.0998 7-3925 1269.2 128190 40s 164025 66430125 20.1246 7.3986 1272.3 128825 406 164836 66923416 20.1494 7.4047 1275-S 129462 407 165649 67419143 20.1742 7.4108 1278.6 130100 408 166464 67917312 20.1990 7.4169 1281.8 130741 409 167281 68417929 20.2237 7.4229 1284.9 131382 410 168100 68921000 20.2485 7.4290 1288.1 132025 411 168921 69426531 20.2731 7-4350 1291.2 132670 412 169744 69934528 20.2978 7.4410 1294.3 133317 413 170569 70444997 20.3224 7-4470 1297-5 133965 414 171396 70957944 20.3470 7-4530 1300.6 134614 41S 172225 71473375 20.3715 7-459° 1303-8 135265 416 173056 71991296 20.3961 7.4650 1306.9 135918 417 173889 72511713 20.4206 7.4710 1310.0 136572 418 174724 73034632 20.4450 7.4770 1313-2 137228 419 175561 73560059 20.4695 7-4829 1316.3 137885 420 176400 74088000 20.4939 7.4889 1319-5 138544 421 177241 746 I 846 I 20.5183 7.4948 1322.6 139205 422 178084 75 I 5 1448 20.5426 7.5007 1325-8 139867 423 178929 75686967 20.5670 7.5067 1328.9 140531 424 179776 76225024 20.5913 7.5126 1332.0 141 196 42s 180625 76765625 20.6155 7-5185 1335-2 141863 426 181476 77308776 20.6398 7-5244 1338-3 142531 427 182329 77854483 20.6640 7-5302 1341-5 143201 428 183184 78402752 20.6882 7-5361 1344.6 143872 429 184041 78953589 20.7123 7.5420 1347-7 144545 430 184900 79507000 20.7364 7-5478 1350-9 145220 431 185761 80062991 20.7605 7-5537 1354-0 145896 432 186624 80621568 20.7846 7-5595 1357-2 146574 433 187489 81182737 20.8087 7-5654 1360.3 147254 434 188356 81746504 20.8327 7-5712 1363-5 147934 435 189225 82312875 20.8567 7-5770 1366.6 1486x7 436 190096 82881856 20.8806 7.5828 1369-7 149301 437 190969 83453453 20.9045 7.5886 1372.9 149987 438 19 1844 84027672 20.9284 7-5944 1376.0 150674 439 192721 84604519 20.9523 7.6001 1379-2 151363 440 193600 85184000 20.9762 7.6059 1382.3 152053 378 SQUARES, CUBES AND ROOTS. Table 4. Squares, Cubes, Square Roots, Cube Roots, Circumferences AND Circular Areas of Nos. from i to 520 No. Square Cube Sq. Root Cube Root Circle Circum. Area 441 194481 85766121 2 1 .0000 7.6117 1385.4 152745 442 195364 86350888 21.0238 7.6174 1388.6 153439 443 196249 86938307 21.0476 7.6232 1391-7 154134 444 197136 87528384 21.0713 7.6289 1394.9 15483° 44S 198025 88121125 21.0950 7.6346 1398.0 155528 446 198916 88716536 2I.I187 7.6403 1401.2 156228 447 199809 89314623 21.1424 7.6460 1404.3 156930 448 200704 89915392 21.1660 7.6517 1407.4 157633 449 201601 90518849 21.1896 7.6574 1410.6 158337 45° 202500 91125000 21.2132 7.6631 1413-7 159043 451 203401 91733851 21.2368 7.6688 1416.9 159751 452 204304 92345408 21.2603 7.6744 1420.0 160460 453 205209 92959677 21.2838 7.6801 1423.I 161171 454 206116 93576664 21.3073 7.6857 1426.3 161883 455 207025 94196375 21.3307 7.6914 1429.4 162597 «56 207936 94818816 21.3542 7.6970 1432.6 163313 457 208849 95443993 21.3776 7.7026 1435-7 164030 458 209764 96071912 21.4009 7.7082 1438-9 164748 459 210681 96702579 21.4243 7.7138 1442.0 165468 460 211600 97336000 21.4476 7.7194 1445-1 166190 461 212521 97972181 21.4709 7.7250 1448.3 166914 462 213444 98611128 21.4942 7.7306 145 1. 4 167639 463 214369 99252847 21.5174 7.7362 1454.6 168365 464 215296 99897344 21.5407 7.7418 1457.7 169093 465 216225 100544625 21.5639 7.7473 1460.8 169823 466 217156 loi 194696 21.5870 7.7^29 1464.0 170554 467 218089 101847563 21.6102 7.7584 1467.1 171287 468 219024 102503232 21.6333 7.7639 1470.3 172021 469 219961 103161709 21.6564 7.769s 1473.4 172757 470 220900 103823000 21.6795 7.775° 1476-S 173494 471 221841 104487111 21.7025 7.7805 1479-7 174234 472 222784 I 05 I 5 4048 21.7256 7.7860 1482.8 174974 473 223729 105823817 21.7486 7.7915 1486.0 175716 474 224676 106496424 21.7715 7.7970 1489.1 176460 475 225625 107171875 21-7945 7.8025 1492-3 177205 476 226576 107850176 21.8174 7.8079 1495.4 177952 477 227529 108531333 21.8403 7.8134 1498.5 178701 478 228484 109215352 21.8632 7.8188 1501.7 I 7945 I 479 229441 109902239 21.8861 7.8243 1504.8 180203 480 230400 110592000 2 1 .9089 7.8297 1508.0 180956 Table 4. SQUARES, CUBES AND BOOTS. 379 Squares, Cubes, Square Roots, Cube Roots, Circumferences AND Circular Areas of Nos. from i to 520 Square Cube Sq. Root Cube Root Circle Circum. Area 231361 232324 233289 234256 235225 236196 237169 238144 239121 240100 241081 242064 243049 244036 245025 246016 247009 248004 249001 II1284641 II1980168 II 2678587 113379904 I14084125 114791256 iiS5°i3°3 116214272 116930169 I I 7649000 118370771 119095488 119823157 120553784 12128737s 122023936 122763473 123505992 124251499 250000 125000000 251001 252004 253009 254016 255025 256036 257049 258064 259081 260100 261121 262144 263169 264196 265225 266256 267289 268324 269361 270400 125751501 1 26506008 127263527 I 28024064 128787625 129554216 130323843 131096512 131872229 132651000 133432831 134217728 135005697 135796744 136590875 137388096 138188413 138991832 139798359 140608000 21.9317 21.9545 21.9773 22.0000 22.0227 - 22.0454 22.0681 22.0907 22.1133 22.1359 22.1585 22.1811 22.2036 22.2261 22.2486 22.2711 22.2935 22.3159 22.3383 22.3607 22.3830 22.4054 22.4277 22.4499 22.4722 22.4944 22.5167 22.5389 22.5610 22.5832 22.6053 22.6274 22.649s 22.6716 22.6936 22.7156 22.7376 22.7596 22.7816 22.8035 7-8352 7.8406 7.8460 7-8514 7-8568 7.8622 7.8676 7-8730 7-8784 7-8837 7.8891 7-8944 7.8998 7-9051 7-9105 7-9158 7.9211 7.9264 7-9317 7.9370 7-9423 7.9476 7-9528 7-9581 7-9634 7.9686 7-9739 7.9791 7-9843 7.9896 7.9948 8.0000 8.0052 8.0104 8.0156 8.0208 8.0260 8.03 II 8.0363 8.0415 1511.1 1514-3 1517-4 1520.5 1523-7 1526.8 1530.0 1533-1 1536-2 1539-4 1542.S 1545-7 1548.8 1551-9 1555-1 1558.2 1561.4 1564-5 1567-7 1570.8 1573-9- 1577-1 1580.2 1583-4 1586-S 1589-7 1592.8 IS95-9 IS99-I 1602.2 1605.4 1608.S 1611.6 1614.8 1617.9 1621.1 1624.2 1627.3 1630.5 1633.6 181711 182467 183225 183984 184745 185508 186272 187038 187805 188574 189345 190117 190890 191665 192442 193221 194000 194782 195565 196350 197136 197923 198713 199504 200296 201090 201886 202683 203482 204282 205084 205887 206692 207499 208307 2091 1 7 209928 210741 211556 212372 380 TEIGONOMETEIC FUNCTIONS. Table s. Trigonometric Functions and the Solution of Triangles In the accompanying figure the trig- onometric functions of the angle A between the lines B A and A C are as follows; sin ^ = ^ C cos A = A C tan ^ = EF cot A =G II sec A = A E cosec A = A II ex-sec A = B E In the right-angled triangle ABC let a equal the side B C opposite the angle A; let b equal the side A C opposite the angle B; let c equal A B, the side opposite the angle C. Let C = 90° The following formulae apply to right-angled triangles: Angles. A + B + C = 180° A + B = 90° A = 90° B = 90° sin ^ = ■ — c - B — A cos A = — ■ c tan^ = -p b A rea ab area 2 Sides, a =c sin ^4 = J tan A a = V{c + b) (c-b) b = ccos. A = - tan A b = V {c + a) (c - a) _ a _ b sin A cos A <;= y/ a? + l^ Oblique Triangles. Note. Where an angle is more than 90° its sine, cosine, and tangent are equal to that of the angle (180° — the angle in question); that is, if the sine of 1 20° is desired take the sine of (180° - 120°) = 60°. Table 5. TEIGONOMETMC TABLES. 381 Given Desired Formulse A,B, a A, a, b C,b c, K C = i8o—{A + B); b =- ■ sin B sin A sin(A+B);K= sin A a? sin B sin C 2 sin yl B,C sin B= 5^ 6; C = i8o°- (A + B) c = —. — - sin C sm A Two solutions are possible with B' as an acute angle and B as an obtuse angle C, a, b i(A+B) HA-B) A B c K 1{A+B) =9o''-§C tan|(^-B) = ''"* a + b A=\{A-frB) + \{A -B) B = \{A+B)-\{A-B) sin \U+B) C = {a — b) -. j— r-j ^r- sm f (4 — 5) K = \ ah sSnC tan \{A+B) a, b, c In the following formula s = \ {fl + b +c) sin hB -*/ (^ - «) (^ - ^) . . „ 2^/ s{s-a) [s-b) {s- -c) '■^^ ac K K = V s(s -a) (s - b) is - c) EXPLANATION OF TABLES. TABLE I. LOGARITHMS OE NUMBERS.— The log- arithm of any number to any base is the index of the power to which the base must be raised to equal the number. The logarithms given in Table I are Briggs or Common Logarithms in which the base is 10. Then 100 ^ 102, and the logarithm of 100 = 2. Also 200 = lOa-soios, and the logarithm of 200 = 2.30103. The integer of a logarithm is called the characteristic, and is one less than the number of integers in the number. The decimal part of the log- arithm is called the mantissa and is given in Table I. The mantissae of the logarithms in Table I are given to five places ; while the numbers are given to four significant figures. Where there are more than four significant figures in the number, the table of proportional parts may be used. The star opposite certain logarithms shows that the two figures at the left are to be taken from the line below. The logarithm of 1 is 0, and the logarithm of any number less than unity will be negative. It is much more convenient to use positive mantissae, and logarithms of numbers less than unity are written as cologarithms or modified logarithms in which the negative logarithm is sub- tracted from a positive integer as 10, 20, etc., 100, 200, etc. ; and the cologarithm or modified logarithm is written as a positive logarithm with the integer shown as subtracted from the logarithm. For example the logarithm of 0.2 =: logarithm of % = log. 1 — log. 5 = 0.00000 — 0.69893 = — 0.69893. The cologarithm or modified logarithm will be equal to the logarithm subtracted from 10 and is written 9.30103 — 10. The logarithm of .00625 = log. %oo = ^°S- 5 — log. 800 = 0.69897 — 3.90309=— 2.20412, or as a colog- arithm or modified logarithm r= 7.79588 — 10. The mantis- sae of the cologarithms of numbers less than unity are given in Table I. The following rules shovild be kept in mind in using the table of logarithms. 382 EXPLANATION OF TABLES. 383 1. The logarithin of a product is the sum of the loga- rithms of the factors. 2. The logarithm, of a quotient is the difference of the logarithms of the dividend and divisor. 3. The logarithm of a power of a number is equal to the logarithm of the number multiplied by the index of the power. 4. The logarithm of a root of a number is equal to the logarithm of the number divided by the index of the root. 5. The logarithm of a fraction is equal to the logaritlim of the numerator minus the logarithm of the denominator. 6. In dividing modified logarithms add a number to the positive and negative characteristics so that the resulting logarithm will have. — 10 following the logarithm. For example if 8.36748 — 10 is to be divided by 3, the logarithm should be written 39.36748 — 30 ; and dividing by 3 we have 9.45583 — 10. Eeverse the operation when multiplying modified loga- rithms. 7. The characteristic of the logarithm of an integer is always one less than the number of digits in the integral part of the number. 8. The characteristic of the cologarithm of a number less than unity (a decimal) is equal to 10 minus the number of the place to the right of the decimal point occupied by the first significant figure. TABLE II. LOGAEITHMIC FTTNCTIOITS OF ANGLES. — To avoid the use of negative characteristics the loga- rithms of the functions of angles are written as cologa- rithms, 10 being added to the characteristic of each loga- rithm. In adding the logarithms of the functions of angles the correct number of tens should be subtracted from the result. For angles from 0° to 45° and from 135° to 180° the headings at the tops of the columns are to be used ; while from 45° to 90° and from 90° to 135° the headings at the bottoms of the columns are to be used ; the minutes being read from the top down on the left of the page, and from the boittom up on the right of the page. In using the logarithmic functions of angles in connec- tion with logarithms of numbers it should be remembered that the logarithmic functions of angles are cologarithms and that 10 should be subtracted from each logarithmic function. 384 EXPLANATION OF TABLES. TABLE III. NATUBAL FUNCTIONS OF ANGLES.— For angles from 0° to 45° and from 135° to 180° the head- ings at the tops of the columns are to be used ; while from 45° to 90° and from 90° to 135° the headings at the bottoms of the columns are to be used ; the minutes being read from the top down on the left of the page and from the bottom up on the right of the page. INDEX. Page. Acres, Reduction to 33 Angles, Errors of 214, 216 Measurement of . . 19, 52, 100 (See chain, transit.) Areas 32, 34, 36, 54, 112 Axeman, Duties of ..23, 24, 190 Azimuth 100, 124, 150 Base line (see chain). Borrow pit (see level). Bubble vial 62 Calculations 12,85,224 Chain and Tape, The 13 Linear measuring instru- ments 14 Problems (see Contents) . . 23 I'nits of Measure 13 Use of chain and tape.. 16, 24 Angles 19, 30 Areas 32, 34, 36 Base line 40, 153, 155 Chaining. .16, 26, 28, 30, 40, 188 on a slope 17, 28 Comparison of chains ... 43 Curve, Locating 38 Errors of chaining . .40, 215 Location of objects .... 20 of points 19 Passing obstacle 36 Parallels 18, 38 Perpendiculars ...17, 32, 38 Standard of length . . 26, 40, 153 Standardizing chain . . 26, 40, 153 Surveys, Tie line 21, 33 Tape constants 43, 153 making standard .... 43 Taping (see chaining). Chaining (see chain). Chainman, Duties of 16, 188 Collimation, Line of .... 58, 103 (see telescope, level, etc.). Compass, The 45 Adjustments and tests . . 50, 56 Declination (variation) . .46, 51 Page. Local attraction 48 Problems (see Contents) . . 51 Types of 45 Use of 49, 51 Adjustment 50, 56 Angles 52, 101, 112, 185 Area 54 Comparison of compasses 56 Declination of needle. .46, 51 Traverse with compass. . 52 Variation (see declination). Computing, Methods of 223 Accuracy, Consistent .... 223 Arithmetical calculations. . 225 Addition 226 Checks 225 Division, Contracted.... 231 Divisor near unity .... 232 Multiples of 10 227 Multiplication 227 Contracted 230 Cross 228 Square root by subtrac- tion 233 Contracted 232 of small number 233 Computing machines 234 Logarithmic calculations . . 224 Reckoning tables 234 Contour leveling (see level). Cross-hairs 62, 93, 94 (see telescope). Cross-sectioning (see level, also see Railroad Survey- ing). Curve (see chain ; also see vertical). Declination (see compass). DifEerential leveling (see level). Dumpy level (see level).... 57 Eccentricity 48, 105, 133 Errors of Surveying 211 (see angles, chain, etc.). Probable error 211 Tests of precision 215 385 386 INDEX. Page. Angular errors 216 Leveling 222 Linear errors 215 Traverse surveys 216 Field notes (see instructions). Flagman, Duties of ..22, 24, 191 Forms for notes (see level notes ; transit notes ; prob- lems, etc.). Freehand Lettering 237 Freehand titles 246 Grade lines (see level). " Shooting in " 69, 101 Instructions, General 1 Field equipment, Care of . . 2 Adjusting screws 5 Axes and hatchets .... 5 Carrying instruments . . 3 Chains and tapes 5 Clamps 4 Exposure of instrument. 3 Eyepiece 4 Flag poles 5 Foot screws 4 Lenses 5 Leveling rods 5 Magnetic needle 5 Plumb bob, 5 Setting up 8 Stakes 6 Sunshade 3 Tangent screws 4 Tripod 2, 5 Field notes 6 Book, Field note 6 Character of notes .... 7 Criticism of notes 11 Cross referencing 7 Erasures 10 Form of notes 7 Indexing 7 Interpretation 6 Lettering 7 Numerical data 10 Office copies 10 Original 6 Pencil 7 Recording field notes ... 7 Scope 6 Sketches, Field note ... 7 Title page 7 Field work 1 Accuracy, Consistent ... 1 Correctness, Habitual . . 1 Page. Decorum, Field practice. 2 Duties, Alternation of. . 2 Instructions, Familiarity with 1 Instruments, Inferior . . 2 Speed 1 Office Work 12 Calculations 12 Drafting 12 Drafting room decorum. 12 equipment 12 Land Surveying 161 Functions of a surveyor. . 161 Metes and bounds, Surveys 172 Rectangular system, U. S. 163 Resurvey rules 161 Problems (see Contents).. . 173 City block, Resurvey of. . 179 Corner, Investigation of. 173 Perpetuation of 174 Metes and bounds 170 Partition of land 180 Quarter section corner . . 175 Section corner 176 Section, Resurvey of . . . . 176 Townsite, Design and survey for 180 Lettering (see Freehand Let- tering) Leveler, Duties of 65, 68, 70, 191 Level note forms. .67,79, 80, 82, 86, 87, 94, 192, 196, 203 Leveling rods 64, 95 Making rod 64, 95, 142 Types of 63, 96 Leveling (see level ; leveling). Level, The 57 Adjustments 72, 75 Practical hints 70 Problems (see Contents).. 76 Running lines 70 Telescope (see telescope). Types of 57 Use of 65, 76 Adjustment of dumpy level 75, 93 of wye level 72, 92 Bubble vial, Delicacy of 64, 90 Calculation of quantities 85 Comparison of levels . . 95 Contour leveling 69, 88 Contour map. Use of . . . 89 Cross-hairs, Stretching 62, 93 INDEX. 387 Page. Cross-sectioning 69, 86 Differential leTeling . .66, 77 Error of setting target. . 94 Errors of leveling. . .59, 72, 77, 222 Grade line. Establishing 83 Levels for street paving.... 86 Level vial (see bubble). Profile leveling ..68, 81, 82 Reciprocal leveling . . . 68, 89 Setting slope stakes .... 85 Sketching dumpy level.. 93 wye level 92 Staking out borrow pit. . 86 Tests ofi dumpy level. .75, 92 telescope (see telescope). wye level 72, 91 Vertical curve 83 Line of coUimation (see colli- mation. Line of). Line shafting, Survey of. . . . 116 Local attraction (see compass). Location survey 183, 201 Meridian (see transit). North, True (see transit). Note book, Field 7 Notes, Field (see instructions ; problems ; chain ; compass ; level; transit, etc.). Pace, Length of 24 Pacing, Distances by 24 Parallels (see chain). Perpendiculars (see chain; transit). Plane table 142 Problems 152 Preliminary survey ....183, 197 Profile leveling (see level). Race track survey 117 Railroad Surveying 183 Bridge and masonry party. 207 Cross-sectioning party .... 202 Land-line party 207 Level party 191 Office work 197 Estimates, Approximate. 200 of quantities and costs 201 Map, Location 201 Preliminary 197 Office copies 201 Profile, Location 201 Preliminary 200 Records, Right of way . . . 201 Report of reconnaissance 197 Page. Topography party 194 Topographer 194 Assistant topographer . . 195 Topography rodmau .... 196 Tapemau 197 Transit party 182 Chief of party 182 Transitman 182 Head chainman 188 Rear chainman 189 Stakeman 189 Axeman 190 Front flagman 191 Rear flagman 101 Problems (see Contents) . . 208 Adjustments, Review of. 208 Field equipment, Use of. 210 Curve practice, Field . . . 210 Curve problems, Office . . 210 Range pole practice 24 Ranging in lines (flagman).. 21 Reciprocal leveling (see level). Reconnaissance 183, 197 Rectangular surveys 163, 175, 176 Referencing out points 110 Resurveys (see land survey- ing). Rodman, Duties of ..66, 71, 193 Sextant 146, 152 Signals 22 Simpson's rule 36 Slope stakes (see level). Stadia 139, 148, 150, 157 Standard (see chain). Stakeman, Duties of. .22, 24, 189 Stakes and stake driving. ... 23 Survey tsee chain ; compass ; level ; transit ; stadia ; etc. ; topographic ; land ; rail- road ; reconnaissance ; pre- liminary ; location ; rectan- gular ; line shafting; race track; tie line, etc.). Tape (see chain). Tapeman (see chainman). Telescope, The . -. 58 Chromatic aberration ..58, 91, 132 CoUimation, Line of ..58, 103 Cross-hairs 62, 93, 94 Definition 61, 91, 132 Eye-piece 4, 60, 106 Field, Angular width.. 61, 91, 132 388 INDEX. Page. Illumination 61, 91, 132 Magnifying power.. 61, 92, 132 Objective 61, 92, 132 Parallax 4, 61, 71, 102 Spherical aberration . . . 60, 92, 132 Tests of 60, 02, 132 Tie line survey (see chain). Topographer, Duties of .... 194 Topographic Surveying 137 City topographic surveying 138, 180 Hydrography 139 Problems (see Contents).. 148 Base line measurement. . 155 tape coefficients 153 Plane table and stadia . 159 by intersection 151 by radiation 152 by traversing 152 three-point problem . . . 152 Sextant, Angles with... 152 Sketching topography . . 156 Stadia, Azimuth traverse with 150 constant. Fixed hairs. 148 reduction table 150 Topographic survey . . . 159 Transit and stadia sur- vey 157 Triangulation system . . 156 Transitman, Duties of 99, 184 Transit, The 97 Adjustment (see problems) 102 Problems (see Contents) . . 106 Telescope (see telescope). Types of 97 TJse of 99, 106 Adjustment 102, 133 Angles by repetition 101, 118 deflection 101, 112 Horizontal 100,106 Vertical 100, 114, 138 Page. Angular errors 101, 216, 217 Area, Transit traverse. . 112 Azimuth 101, 150 Comparison of telescopes 132 of transits 135 Cross-hairs, Stretching 62, 93 Deflection survey ..100, 112 Double sighting 106 Eccentricity 105, 132 Error of setting pole . 134 Height of tower 114 Interpolation of point. . 107 Intersection of lines .... 108 Leveling with transit 77, 101 Line shafting survey. . . . 116 Meridian, Determination of 119 direct observation .... 131 Polaris at any time . . 121 Polaris at elongation. . 119 solar attachment .... 127 Passing obstacle 110 Prolongation of line.. 99, 106 Referencing out a point. 109 Race track survey 117 Sketching transit 133 Staking out building. . . . 114 Tests of transit . . 102, 133 Traverse survey . . . .101, 112, 150 Triangulation across river llO Transit note forms. 107, 109, 111, 113, 115, 118, 120, 124, 130, 134, 149, 187 Traverse surveys. Errors of 54, 112, 218 Triangulation. 110, 156, 159, 215 Variation of magnetic decli- nation (see compass) .... 47 Vernier 49, 99, 1C5 Vertical curve 84 Wye level (see level) 57