A practical treatise on mining, land and railway surveying, engineering, &c

Hoskold, H[enry] D[avis] [from old catalog] · 1863 · Historical

A practical treatise on mining, land and railway surveying, engineering, &c by Hoskold, H[enry] D[avis] [from old catalog] (1863). Full text and reference in…

Public-domain full text preserved in the Mountain Man Mining Library. Original source: archive.org.

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Published By

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AN ANALYSIS OF / Fr'.XT DoMKSTIO AKCIHTEC- 'lUlJ?:; a!i .Will",.!; .V k, ;:'viji' the li.-t in (;rr.'t limit.tli;. iiMii lr:.\% !il iK'kt.mvniefits iHkJs on iIm *.Mit. by F. T. Do!lm:''. :if"!r:. t. ; -.l P.. JoMins i.f UK) 4io. il:Ui"s, o\- vniu 'n til*' I.-:- M- n;;t !. rt.r-juvrfj}, iu o Vois. hall i:iIrotvu t-.i

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Miners transit theodolite.

MANUFACTURED ONLY BY JOHN ARCHBUTT & SONS, "20, WESTMINSTER BRIDOE ROAaUMBETH.S.

BiblisharsAtchley&C° 106, Oreal Russell Bedford SguareLondon.

A Practical Treatise

Ok

lining, %uMia nrkjing, npemng, h.

THE EEEOES OF THE MAGNETIC NEEDLE; PEACTICiX GEOMETET AND TKIGONOMETETi

DESCRIPTION, USE, AND ADJUSTMENTS OF THE MINEKS' NEW" TRANSIT THEODOLITE;

ALSO, A NEW PLAN OF SETTING OUT UNDERGROUND RAILWAY CURVES; DNDEEGKOOND LEVELLING;

CONSTItDCTIOIT OP OF MIHSB, STQ.

|llii5lnittli bi lnuniiii |lists nit Mnliats. By H. D. HOSKOLD,

HtNIKO XKQIHEKK AND 8UBTET0B.

London: Atohley And Co.,

Enqineerinq And Architectural Pubushers,

106, Great Russell Street, Bedford Squabe.

[The right oj Trantlaiin U rttened.'\

f {[is Wmi, u rtsirdfsUs

TO THa

DEAN FOBElrMPN COMPANY,

Ik Grateful Acknowledgment Of Many

VAYoilBS BKCEIVED,

By The Author

Preface.

In the following pages I have endeavoured to place before the student of Mineral Surveyings a complete system of Practical Geometry as relating to the surface and underground surveying of mines. I have to the best of my ability, given a practical solution to the various problems which have occurred to me in my own practice, in the hope that this work may be the means of introducing a more accurate system of accomplishing what is so essentially important in all rniniTig operations — viz., correct plans and statistical records of all subterranean drivages and excavations.

I have given, throughout the work, the result of my own observations, and the demonstrations of mathematical truths, and in my own opinion, and what I have no doubt will have been found in the experience of others, a desideratum in useful tables at the end of the work.

Incorrect results from surveying are well known to every surveyor to be a cause of much anxiety and perplexity ; and as I have myself arrived at satis£actofy and accurate results from pursuing the methods which I now prominently bring forward in the present work, I presume to express a hope that this fiwjt of itself may be a sufficient recommendation of the methods to others.

H. D. Hoskold.

CiKDBBroBD, Dean Fobbst, June, 1863.

Table Of Contents.

VAAS tNTSODUCTION xiil

The Miners' Compass.

Description of the Miners* Compass — Tho probable reason why the drde came to be divided into 360° — Intended for minor survs aind rough work only — And its inadmissibility in titaniferous iron mines . I

Errors caused by the derangement of the magnetic needle — Electrical state of the atmosphere — Inaccurate readings of the needle— Nonparallelism, lui %

Influence of diurnal variation not to be neglected — TaUe of mean ranatioa for each hour in the dify — Table of mean variation for each month in the year ..., S

Great care necessary when taking a bearing by the needle— Table of annual variation from 1576 to 1861 — Fluctuations of the magnetic needle 4

The impossibility of an agreement between surveys executed at different times — Errors committed by the compass — Trigonometry will not compensate for inattention in observations— The amount of deviation from the truth — For a difference of SO' on the reading . 5

The amount of deflection of the magnetic needle deduced from experiment—

The difficulties and losses that would result ., 6

Superior instruments recommended — Thing done at ail should be done well — The possibility of the magnetic compass oonstrmcted so as not to be so much liable to local disturbing causes 7

Practical Geometry.

Introduction — To construct a scale of equal parts ... . 8

Comparison of angles-Angles and their measure— The division of tiie

drcle — Method df writing degrees, minutes, and seconds

To describe the angles with the compass and scale— <Hven ihe hypothenuse andtk sideof a right-angled triangle, to construct it and measure the dtker parts— CKven the two sides about a right-angled triangle, to construct it and measure the other parts 10

Given the hypotiienuse and one of Idie tfblique angles of a right-misled 'triangle, to construct it and measure the other parts — Given the side and an oblique*angle, to construct and measure the triangle — Given one side and two angles, to construct the triangle 11

The preceding problems applied to the sinking shafts, driving headings, drifts, cut-outs, galleries, undeiound tunnels, either horizontal, vertical, or in oblique pfaines ' . . . . . . 12

h 2

Vm TABLE OF CONTENTS.

Pjigb

The preceding problems also practically applied to the finding any distances . on the surface between two distant shafts, from one or two stations, having given the horizont-al angles, and angles of elevation or depression with one measured side, plotted diagrams illustrative of each

To find by observation, and by geometry, the difference of level under any conditions, by taking the horizontal and vertical angles, plotted diagram 18

Practical methods of measuring inaccessible distances and avoiding impediments in the running of lines 20

Practical Right-Angled Plane Trigonometry.

Introduction and trigonometrical definitions 24

Propositions and properties of triangles 25

New method of finding the side of any triangle by common arithmetical

proportion 26

Given the hypothennse and one of the acute angles of a right-angled triangle, to find the other parts 27

Calculations by sines, cosines, and tangents, and the preceding trigonometrical problems applied to sinking shafts, driving headings, cut-outs, tunnels, and in horizontal or oblique planes, plotted diagram ... 28

The preceding problems applied to finding any distances on the surface between two shafts from one or two stations, having given the horizontal angles and angles of elevation and depression with one measured side, also the difference of level under any condition 30

To find arithmetically the sine of the angle of direction, and length of a side to be driven from a fixed point below ground so as to strike at right angles to another fixed point, having given two measured sides . 32

Given the depth of a shaft with the distance from a fixed point on the surface, to find the length' and angle of elevation of an incline to be driven from the bottom of the shaft so as to strike the fixed point at surface . 33

AU the cases of right-angled trigonometry solved by logarithms independent

of the preceding rules 34

Nature And Use Of Logarithms.

Introductory remarks — Logarithmic series and systems ; , 37

Logarithmic indices — Best kind for use — Table of numbers and logarithms

explanatory of the method of supplying the indices 38

Method of finding the logarithm of numbers in tables ...'... 39

To perform multiplication, division, and the rule of three direct, by logarithms 41

Practical Oblique-Angled Trigonometry.

Proposition and rules 42

The side of oblique triangles found by problem the first of right-angled

trigonometry 43

Given two angles and a side of an oblique-angled triangle, to find the other

parts 44

Table Of Contents. Ix

Given two sides and the containing angle of an oblique-angled triangle, to

find the other parts 45

Given the three sides of an oblique-angled triangle to find the angles and

the difference of the segments of the base . 47

The preceding problems applied to the finding of any inaccessible distance

between two shafts or otherwise, or their distances from the stations . 48

Application to heights and distances 60

Proof by natural tangents 61

The Vernier Scale.

Description and general remarks 63

With explanatory diagrams 64

Subterraneous Or Mining Surveying.

Section I.

Introdaction — Remarks — Underground surreys are performed with less care than the surface, and the necessity for as good instruments in mines as on the surfEice 67

Description of the new miners' transit theodolite, with plate of the instrument — Remarks on its importance as an angular instrument — Its difference from an ordinary transit 67

Adjustment Of The Theodolite.

To detect and rectify parallax 61

To adjust the line of coUimation 61

New method to adjust do 62

To adjust the azimuthal axis 63

To adjust the vertical circle by a new method 63

To measure horizontal angles from the zero by the back observation and by

repetition, &c 64

To measure vertical angles 66

Traversing Underground.

Remarks on underground traversing — Necessity of tutoring assistants to

measure lines accurately 67

To peg out the lines to be measured — Proof of the correctness of angles . 67

Manipulating the theodolite and keeping the survey- book 68

Plotted diagrams from plane of latitude, and form of survey-book No. 1. 69 Rules for reducing any number of horizontal angles to plotting angles from

one station 70

Tables of the reduced angles 71

To construct the plan with a circular protractor divided as the theodolite— The plotting angles and form of survey-book — The plan may be constructed without the protractor, viz., by co-ordinates 71

Rules for reducing the observed angles to the meridian and latitude . . 72

X Table Of Contents.

Fa6B

Description of offi form No. 1 73

Multiplication q£ sines and cosines, with the office form necessary to receive

the calculated data 74

Explanation and use of the columns in office form No. 1 75

To take the survey of a circuitous heading with diagrams plotted from

planes of latitude and meridian 76

Survey-hpol No. 2, with explanation of its columns 77

Tahle of i:educed angles to the meridian-Parallel of the meridian — Latitude

and parallel of latitude 78

Explanation of office form No. 2, and to construct the plan from it . . . 79 Office form necessary to receive the reduced horizontal angles, and the sines

How to construct the.pUn, ,coitiivue4 , 81

The great advantage of office form No. 2 — Its utility in offering data, whereby

any point on the surface may be discovered corresponding to any below

ground' 82

Setting out such points 83

The new method of connecting underground workings with the surface

without the aid of the needle — Diagram of plan , . 84

Diagram explanatory, and calculation of the deviation of the theodolite from

meridian 85

To connect the sur&ce with underground working by chains and theodolite 87 To find a point on the surface corresponding to any one previously fixed

below ground 88

Sbotiok IJ.

Survey of a heading taken preparatory to sinking a shaft into the heading,

and driving a tunnel from one shaft to another 89

Survey-book No. 3 — Another office form required 90

To determine when an angle should be plus or minus, with explanatory

diagram 91

Office form No. 3, containing the reduced horizontal angles, distances, and

total distances from planes of the horizon, meridian, and latitude . . 92 To set out the shaft to be sunk into the heading below ground, with calculation ,' 93

To determine the levels distance, and direction of the tunnel to be drawn

from one side 94

Details 95

To drive the same tunnel from two ends at the same time 96

New method to set out underground curves from one side, with method to

calculate radius . . 97

Working diagram, with form of Not-e-book containing data for setting out

the curve 98

Explanation of Note-book, and to set out the curve 99

To set out the curve from two sides at the same lime, with working

diagram 100

To drive a tunnel on a curve and gradient from two points 101

Levelling underground with the spirit-level, with best kind of levelling

book 102

Table Of Contents, Xi

Best kind of levelling staves — General underground survey-book, with remarks on plans and section of collieries . . . . ' 103

Form of the general underground survey-book, connecting diQareQt vein of coal by drifts— -Difficulties to be encountered when no section if hand . . I . . I ] '. '. . . . 10

Points to be attended to in driving headings, so a? tp ayqid iimdatg |;)ie mine — Colliery mines are oftentimes badly conducted, and thiQgf be observed in order to be successful 105

To set out the tn;ie ix;ieridi by equal altitude? of fly celeste 10$

To se out the meridian by th azimuth of the , ith explatprj

diagram 107

To find the latitude, and to calculate the azimuth 110

Land Or Surface Surveying.

Section III.

Introduction Ill

Tq survey a road and enclosure with the chain 11

Cteneraji principles of ohain furvying 113

Extension of triangles : , ; |.14

Example in extensive chain surveying .; ' 115 3urveyiig loioilities for working plans — remarks on m9$

and plans for mines, their construction, direptipn for running line

Reducing the angles, and plotting the work 119

pescriptipn of scales for working plaxis, ajid colouring uid liUepiiig

Setting Out Raclways To "Mines.

Greneral remarks on choice of ground, and directions a railway should take

to avoid impediment and extra expense 122

Working pocket section of part of a railway, with direction for setting out

cutting and embankment, with calculation for curves 124

Table of gradients for underground railways or other roads 125

To set out a railway curve 126

Table of offsets for setting out railway curves . 127

To set out a compound and a inverse curve 128

To set out a gradient of any itio to a fixed height 129

Longitudinal And Transverse Sections.

Introduction and general detail 130

Field levelling book for section, with reduced data 131

General utility of longitudinal and transverse sections with their construction 132

Levelling with the miners' transit theodolite for section — The manner of

taking the section 133

The advantages of the theodolite over the level in precipitous places . . 135

Considerations of refractions where they affect observation 135

Table of curvature and refraction 136

Xu Table Of Contents.

Calculation Of Areas.

FAOl

Boles for finding the contents of any number of fields in acres, roods, and

perches 137

Computation of irregular spaces, and equalization of boundaries 138 Calculation of the area of a five-sided figure from observation taken from

one station 139

To prove the process logarithmically 140

Calculating the area by having given the me-asurement of the exterior boundary — Algebraical formula, process by the square root, and proof by logarithms . . 141

Heading to the New Set of Tables of Distances from Planes of Meridian

and Latitude, or Traverse Tables 143

Explanation to the Traverse Tables 146

To find the base and perpendicular to any angle and hypothenuse . . . 147

Example of the utility of the table 148

Proof of the problems in trigonometry by the tables 150

Tables of Distances from Planes of the Meridian and Latitude . . 151 — 196

Introduction to Tables of Natural Sines, Cosines, &c 197

To find the sine of degrees, minutes, and seconds 198

How to calculate the table of natural sines and cosines, with examples 199 — 201 To calculate natural tangents and secants, with the necessary algebraical,

geometrical, and arithmetical operations 202

Tables of Natural Sines, Secants, and Tangents 203 — 247

Tables of Batios of Inclined Planes, corresponding to different Angles 248, 249

Intkoduction,

By mark FRYAR, F.G.S/

BNOIHBXB 07 MIHIS, iOTD LXOTUBEB ON XUTINa IN THS QLA800W 80H0OL

Of Mines.

Progress in Science and Art has worked wonderftd revolutions in every industrial and commercial pursuit followed by the Christian nations of the world. Wherever we discover, by the aid of history or tradition, any phase of the of either mining, manufacture, or navigation, we become at once involuntarily impressed by a contrast with the present. Whilst, however, a condition of progress must be admitted as characterizing generally all the industrial arts, we must not close our eyes against the fact that there are very widely diflFering degrees of progress as respectively applicable to the various branches of industry ; and we may, perhaps, derive some useful lesson by investigating the causes which have operated against one branch advancing in economy and development in an equal degree with another.

The rapid strides which have been made during the past century in the construction and application of steam-engines, has been the means of vastly improving both productive and manufacturing art; and in common with other branches the art of mining has participated in the benefits arising from such -a source. Although, however, the mineral fields of Europe are unquestionably now being developed in a way which, for extent both in area and depth, has never been equalled in past times, it is nevertheless an undeniable &ct, that in some of the main departments of subterranean operations there is an evident stubborn conservatism of plans and modes of operation which time and intelligence have yet to overcome. By defective methods of working minerals —

Xiv Introduction.

coals especially — there are year by year hundreds of thousands of pounds' worth of property irrecoverably lost. Defective ventilation, bad carriages and carriage-roads for transporting the minerals, vicious of suppprtmg the roof and wUs of beds and lodes, and most injudicious division of labour, may at this day be met with to a surprising extent in many mining districts; and by neglecting to survey, or by surveying inaccurately, accidents to both life and property are by no means rare.

Great engineering difficulties are constantly occurring in the practice of Mining Engiuee; ; nd to overcome them safely, and with economy, requires no small share of tact and judgment. Underground work birrjed way from the " public eye,' and is seen only by the few whose duty it is to descend the shaft or ingoing eye.' Many mines are well conducted, and you may see at once in both arrangements and condition qf workings, that a kind of liberal economy is practised, and tat the superintending mind is one not only of intelligence, but also of order and care ; but in other cases, owing to want of perseverance under difficulties, and to the exercise of common judgment and prudence, as well as to thjQ &ct the results of such cannot be seen and understood by anyone habitually descending the mine; the whole raining operations are marked by si;ich a blundering, day-by-day, " makeshift" character, as to be both productive of accident and cause of unnecessary expenditure.

There is a class of colliery and other mine managers who have been trained fcom youth upwards in all the departments of mine engineering, or who have been educated liberally preparatory to entering upon an apprenticeship in naine mwagepient under some competent and conscientious engineer, and who are tljerefore fully alive to the importance of attending closely and carefully to the minor details of management. Men of this stamp and character of qualification ae to be found in charge of the most extensive and best-conducted collieries and na,etalliferous jnines of the United Kingdom ; but in many instances, through a mistaken notion of

Ijixbopuction. Xv

economy by proprietors men are employed responsible managers who are totally ignorant of the most elementary principles of mechanics the laws of gases the principle of th safety-lamp the geometry of figures formed by faults and heaves and the theory and accurate practice of surveying and levelling.

There cannot be directly any blame attached to eith employed or employers for the evil which we have thus indicated. Employers on the one hand are naturally anxious to obtain servants at the lowest possible remuneration ; and the employed on the pther are ready to enter upon the responsibility of situations in which they can obtain more wages for less arduous work than has been their lot in the capacity of common workmen. The speediest and most effective remedy that we can suggest is> that all responsible managers of mines be required by Grovernment to obtain Certificates of practical and theoretical competency for their situations before being allowed to enter upon them. Local Boards of Examiners could easily be established for this purpose ; and it would be no difficult task so to combine the practical in examinations with the educational and scholastic 83 to preclude the possibility of any mere theorist obtaining a certificate* We could adduce xnany instances of gross mismanagement in various departments of minings resulting in loss of life and prcrty ; but as it is our province in the present instanxse, to deal with the surveying of mines, .we must refer chiefly to this subject,

The want of accurate surveying of mines has often been forcibly shown by law-suits, by fatal accidents, and by blunders in mine engineering, which have been confined to the proprietors purse. Any attempt, therefore, which is made to introduce improvements into this department of mining work should be hailed with encouragement and patronage by all interested in mining operations.

The writer had the importunity of reading in manuscript the work to which this Introducticm is prefixed, ajid he can affirm that in this work the student of Mineral Surveying will find various

Xvi Introduction.

ingenious devices in surveying, and the principles of the improved systems of surveying, introduced to him in a manner not to be met with in any other published work.

Mr. Hoskold is a notable practical surveyor, and his book has evidently, as a consequence, received a stamp of character — straightforward and to the point — which is best suited to the studies of practical men. The instruction aflfbrded is likely to be the means of increasing accuracy in surveying mines, and of thereby saving both human life and valuable property.

It is well known to surveyors of mines, that the common miners* dial " is liable to lead to erroneous results, owing to various deviations of the needle firom parallel magnetic planes. It must, however, be admitted, on the other hand, that there are many cases of remarkable accuracy in surveying where the common dial or compass has been the only instrument used ; and for subterranean surveying, where boundaries, long drivages between shafts, or other important matters, are not involved, the dial is perhaps the begt instrument for the purpose, as it is certainly more handy and expeditive than more complicated, although more accurate instruments.

Correct manipulation and readings may perchance give accurate results with the dial, when properly constructed and in good working order; with a transit theodolite possessing these qualifications, and with accurate surveying, the results must be correct. It therefore follows that for all important surveys the magnetic needle should be dispensed with, and the angles made by the respective lines of survey with each other be measured, instead of measuring the angles which these lines make with the magnetic needle.

The methods of uniting underground surveys to surface ones will be easily understood, and I have no doubt appreciated. These methods have already been ably explained and illustrated by Mr. Beanlands in his paper read before the Newcastle-upon-Tyne Institute of Engineers, and by the late Mr. Herbert Mackworth

INTBODUCTION. XVll

of Bristol ; but lutlierto a separate transit instrument has been required for determining line in the horizontal surface plane which shall be at the same time -in the vertical plane of the surveying instrument placed at the bottom of a shaft, whilst Mr. Hoskold has provided an ingenious instrument which combines ihe purposes of the transit instrument with those of the theodolite and dial; and moreover, it mu$t be stated that there are points in Mr. Hoskold's pla of uniting the surveys which, so far as the writer is aware, have never yet been published.

The plan of setting-out curves for underground railways and other purposes is novel and useful, and will at once recommend itself to the intelligent mining engineer. It will be particularly useful for the long curves leading into level roads from inclined engine planes.

The plan of plotting surveys by means of co-ordinates from two vertical planes intersecting each other at right angles will be found to be correctly and clearly described, and the advantages it possesses, in comparison with the ordinary mode of plotting, should be duly considered by the practical surveyor.

The office form of recording the surveys deserves particular attention. It serves, indeed, more than all the purposes of a duplicate plan : as has been shown and illustrated by the author, it frimishes at all times a ready means of ascertaining any point in the underground survey coming vertically under any given point at the surface ; and it is at the same time a handbook record of all the particulars relating to sections of level and to all horizontal surveys of the mine, which plans and sections, in whole or in parts, can at any time be procured. Had there been during the last half century local mining record offices where such records of mines had been stored up by the Legislature, easily and readily accessible to mine owners, how much of human life and suffering would have been saved, and the irreparable loss of how much property would have been prevented !

The Traverse Tables are newly arranged, and cannot £ul to

be xst gft settice tb tforrieyort* The Tabled of Netulal Sines hare ell beeh tbwiaugMy revised, and will be at once apptiated as a valtLable additidii td the foimet of the book. Since 1811, when that useful book appeared, known as Fenwick's Bnbtertaiiean Snrteying, we have been favooid with such trotks as fudge's Miners' Guide>*' Williams' Geodesy," and Boards' Mifters* Manual,** all odntailaiiig useM information OA flie subject -of itiinifg surveying j but the esent work will be found to occupy a position from any of the works referred to, isind t6 be much better adapted than they are to the re(|uirements of mineral surveyors*

MagnC influences find their causes are not thoroughly and dearly understood ; it is Well knowii that the surveyor's hand, when in a heated condition, is quite capable of disturbing the iieedle simply by being placed aboV the needle whilst holding tihe (atidle or lap fefr the piorpose of reading a bearing. It is the hafid and not the lamp which disturbs the needle, the surveyor of o6U4*se beiiag careM to use lamp of copper or brass. Iron plates or rails teed for tramways, &c., frequently exhibit, in tailag degtie of intensity, diffeint magnetic poles ; North and Sout iBs heittg otten Iformed in variotis plaoes thouout the tlitk of a plate, to speak scientifically, the bar is broken up ia%6 pcd: to talk> thefreiot, of Ilidng le surveying dial befcWteii i9i6 traSwy ]ates sO tiist (me plate may neutralize the of iiie 'Ofrti, ii to igUot this dsrcianstance. Hie only iiiUHores, Hditeh aie l&ely to alfect the magnetic needle, are Magnetite and Ihnfi] (Mimetic OKide of iron and titanate of i!i*on). Specular 'ore— the Bed flsematite— is sometimes slightly inagtatetic, btit by holding a mass ctf it to the needle im) perceptible clffefct is Irou-fltofmes of the coal-measures are only logic e!r calcinatioli, but they are occasionally calcined by heat from trapdykes. An altered piece of black-band from the neighbourhood of a trap-dyke ill an Ayrshtpe coal-pit has bela Idndly analysed for by Mr. J. Napiex and is found

iWttODtbtlOK. xix

to cotitdti 4 pY petit. oi ihbgntie otidet it Itcts the mtug* netic needle powerfully.

Ill 1861 I made ft k& df etperiijanhi, ivith ihte di#&rent lifarvejriiig diane, in to tippet setai of Banktlead Colliety, hettf Glasgow, the 1r6sults of Wluch We gOhe fitr to '6oiivitice me of the and linsatisfectOiy dbtairtctiet of stoveys tith thl cOmmoti dial. The first ob&erratioii was made with an instrument placed in the seam at a distance of 4 feet froin the near inalin of the shaft. When the cage, in descending, approached the seam, the north end of the needle was repelled 10° ; in ascending, the cage attracted the north end of the needle 20°. At a distance of 8 feet from the shaft the descending cage repelled the needle 5° ; at 15 feet 1° ; and at 27 feet i°. Bearings taken with the three instruments, when the cage was stationary and at the surface, were as follows : —

With 1st instrument the bearing was S. 32° E.

With 2nd „ same line was S. 30 E.

With 3rd „ „ S. 29i E.

The same amount of difierence between the reading of No. 1 and No. 3 instruments was observed at the surface, a discrepancy which, of course, must have arisen from defective construction.

An ordinary cast-iron tram-plate, four feet long, when placed horizontally at a distance of 2 ft. 9 in. below the instrument, and 14 inches on one side of the centre line of the dials position, produced a difference of J® in the position of the needle ; two plates so placed made a difference of 1°. The same two plates standing on end at 14 inches on one side of the instrument repelled the needle 4° ; when placed on contrary end thay attracted the needle 15° ; placed at 6 feet the instrument they made a difference of i° ; when at 9 feet distance they produced no sensible effect. The same two plates placed horizontally two feet on one side of the instrument, end to- end, and similar poles being together, repelled the needle in the one case 2 °, and in the other attracted it 3°. When dissimilar poles were placed together they neu-

Xx Intboduction.

tralized each other but one of the plates had to be broken to get at the place of its pole.

The methods of surveying and systems of keeping records of such surveys by plans and other means should be of the best possible kinds where so much of life and wealth is at stake ; and it is to be hoped that the efforts made by Mr. Hoskold and others may lead to the introduction of improvements into this department of mining work.

The Miner'S Compass

Its Errors.

The miner's compass or dial, as it is generally called consists of a brass box containing a ring or divided circle, graduated into 360 divisions called degrees — (how all circles came to be so divided is a matter of spectQation; it might have arisen thus, because the radius of every circle is equal to the chord of 60®, or two-thirds of any right angle, which, repeated round the circumference, would exactly come in at six times, consequently 60° x or it is more probable that the idea first took its rise from the circumstance of the ancient year having been divided into 360 days, as the earth was supposed to describe its orbit in that time), — and numbered froln 0, or zero, round to 360. The bottom of the box also is divided into four quadrants, of 90° each, commencing &om a line passing through zero and 180°, or the north and south limb; which line is called the dial's meridian, and is numbered 10°, 20°, 30°, &c., to 90° each way, which division marks the east and west points of the compass. In the centre of the compass box there is fixed a pin finely pointed, and made to carry a steel bar accurately balanced and magnetized, called the magnetic needle. When in use it is allowed to play fi-eely inside the graduated circle until it stands at rest, and then points to its own north, which end is marked differently to the other for the sake of distinction. The box being thus far completed, is fitted with a glass cover to keep the needle fi-ee from the eflTects of the air from the exterior. There are also two projections of brass cast to the bottom plate of the compass box, to receive perpendicular sights. These sights are made to fold over the compass box by means of a pin joints and

B

2 THE miner's compass AND ITS ERRORS.

are each fastened to the north and south limb of the compass. From the description here given, it will be observed that this kind of instrument is of no very nice construction, and is only intended for rough work, where mere sketches are required only, and as such is well adapted for military and other sketching. It is much to be regretted that the dial still continues in almost general use at the present time among and in mining districts, where, the very nature of the instrument, it should be quite inadmissible, even if it had no other liability to derangement and inaccuracy than from the proximity to the iron rails in such mines when in use. However, the instrument is liable to so many kinds of derangements, that it should be entirely discarded as an instrument to be depended on for anything like accurate results. The magnetic needle of the above description of instrument coincides with its own magnetic meridian, which meridian is constantly variable, depending on the electrical state of the atmosphere, and is not the same in all places nor at all times of the year, and is called the magnetic variation. There is also a local attraction or variation, caused by metallic or mineral veins iiji hiUs, large masses of iron near at hand. Some kind of surface stones will also produce an eflFect on the lieedle.

Having occasion, some time since, to mark out a line through a very rugged inclosure with the miner's dial, and having proceeded some time with the measurement, we came to a large heap of some kind of stones, when, all at once, the needle began to play freely ; and after waiting some time for it to settle it was not at all disposed to do so — which circumstance was indicative of iron near at hand — and after a vain search none whatever could be found. Our attention was now directed to the heap of stones, and, after some examination, they were found to be sandstones, and strongly tinctured with iron. The line was now proceeded with, and after passing the spot a few yards the eflfiects entirely disappeared, proving at once the cause of the deflection of the needle.

The above-quoted variations in the miners dial are not the only drawbacks, among other causes of inaccuracy, the instrument is liable to. Angles cannot be read nearer the truth than from onequarter to one-half a degree ; the limb, or divided circle of the dial, rarely admits of a less subdivision ; also the horizonJ;al angles pointed out by the needle cannot be read accurately, because the needle is not always parallel to the plane of the instrument, but has one of its poles raised some distance from that plane. Again,

The Minebs Compass And Its Errors. 3

the diurnal variations of the needle* are too important to be neglected, as they oftentimes amount to three-quarters of a degree in twelve hours; and in the same localities variations of one degree and a half have been observed in thirty days, and in other places even more than 'this. To corroborate this, the following tablet is given jfrom the Philosophical Transactions, Vol. LI. : —

Hours.

Minutes.

Declination West.

Fahr. Ther. Deg.

Degrees.

Minutes.

e

The following table is also taken as the mean variation of

every montli in the year : —

January ... 0° 7' 8"

July .

. . . 0° 13' 18

February ... 8 58

August.

. . . 12 20

March . . . . 11 17

September ,

. . . 11 43

April . . . . 12 26

October

. . 10 86

May 13

November

June 13 14

December .

, . . 6 58

\fi

It will have been shown clearly by this time that the common compass is not at all applicable to long and extensive underground surveys, especially in mines containing iron ore, and such minerals as are undoubtedly possessed of kinetic properties. The dial, at best, should never be employed where niceties are required ; and when its assistance is required in a survey a bearing

Annates des Mines, tome ix., 1836. Williams's Practical Geodesy, t Phil, Trans,, vol. li. Taken by Mr. Canton.

B 2

4 THE miner's compass AND ITS ERRQRS.

of one line only should be taken, and even then with the greatest possible caution. All iron rails and other metallic substances should be removed to some distance, and the bearing of the same line taken at different points, the mean of which should be employed as the working bearing. We shall now insert a table of annual variations from the year 1576 to the present time, and then will foUow examples of work done by the compass : —

Year.

Declination.

11°

15' 0"

1683 . ,

1700 . ,

1717 . ,

, 10

1735 , ,

1740 . ,

, 17

Year.

Declination.

. 23°

36'

0"

at Lcudon.

M

ff

37 30 at Cinderford.

Thus it will be seen that the magnetic variation has been decreasing eastward at about an average 0° 2' 9'' 87''' for a period of 81 years, or from 1576 to 1657, at which time the magnetic meridian coincided with the true meridian. Since that period the variation has been veering to the west, at about 0° 10' 16" per year, for a period of 146 year, or until the year 1803, at which

THE miner's compass AND ITS ERRORS. 5

date it appears to have attained its maximum variation. After this time the variation has decreased westwardly ; and is about 22° 37' 30'' at this time, April 3rd, 1861, at Cinderford. From the foregoing statement it is a positive established fact, that a survey made in any one year cannot possibly coincide with the same survey made a long time afterwards. That is, any survey performed with the magnetic needle of the miner's dial, and plotted either on the surface or on a drawing, cannot agree with a survey of the same place, and performed with the same instrument, in years afterwards. This occurs from the annual variation. Errors of a like nature happen from local diurnal variations, as also from non-parallelism of the needle; for an error of 30 on a line of 40 yards, if continued to one mile, produces an error of 15 yds. 1 ft. '9 in. ; and Herbert Mackworth, Esq., in his lecture to the Bristol Mining School, gives an example of this that was presented in the Standedge Canal tunnel, made some years since, when, in driving three-quarters of a mile directly from shaft to shaft the two levels missed each other by more than ten yards. Another example came under my own notice in a colliery in this neighbourhood, the proprietors of which wished to drive a slope or incline from a given point below to the surface : the bearing was said to be S. 64°, E. 33 yards ; instead of which, when the incline w&s driven to land, the bearing was found to be S. 27°, E. 106 yards, and being at a distance of more than 60 yards from the point at land it was expected to come out at.*

Others will teach a method of working out dialing by plane trigonometry, and after working the traverse forwards and backwards, and the resulting calculations approximating, take it for granted they have a mathematical demonstration of its accuracy. Nothing more absurd than to suppose that a trigonometrical calculation can possibly right what the compass or magnetic needle and their own inability have done wrong. To prove that metallic substances produce a great effect on the magnetic needle at a long distance, the following experiments were resorted to for determining the amount of deflection of the magnetic needle. A line of 1000 links was measured on part of an iron permanent railway, and the mean of several observations taken as the true

Several cases may- be adduced to prove the insufficiency and carelessness of those using the miner's compass, but from want of space are not noticed here.

6 THE miner's compass AND ITS ERRORS.

bearing per The line C D, E F, and G H, Pig. 3, Plate I., were then set out (with the theodolite) parallel to that of A B, and (in an open field free from all iron except the adjacent railway), and at a distance of 40 yards from each other, the bearing of the respective lines C D, E F, and G H, were then careftdly taken as before, when it was ascertained that the bear- . ing of C D, was N. 5° 35' E. differing from A B, 20' ; that of E F, N. 5° 56' E. differing from C D, 11'; and that of G H, N. 6° 0' E. differing from C D, 4'. The experiment was now deemed complete, inasmuch as the last bearing, G H, differing so little from E F, it was not necessary to measure off another parallel line, as probably its bearing would not have differed more than a minute or so the one last taken. Assuming the bearing of G H as the correct one, we shall readily arrive at a reliable conclusion, especially as the bearing of the last line was taken at a distance of 120 yards from the railway, and therefore comparatively free from its influence. The difference in the first and last bearing is 45' on a line of 1000 links; this would produce an error amounting to exactly 13 links ; and the error on a line of 4000 links would be equal to something more than 52 links, or 11 yds. 1 ft. 4 in. From my own experience in surveying in iron mines with the magnetic needle, I am of opinion that the effects produced from all causes taken together would be about equivalent to the above discrepancy. Any person will at once perceive that if an underground tunnel had been commenced from A, it would never have ended at B, the point intended, but would have terminated 13 links on one side for a distance of iOOO links, or 52 links on a line of 4000 ; and if the tunnel had been commenced from opposite sides, as at A, and B, it could not possibly have joined when the heading came opposite, but would have been 52 links asunder ; that is, if the line A B was set out from A, N. 5° 15' E., it would have been 26 links on one side of the centre line at 500 links, and also if the same bearing reversed wafe set out from B, S. 5° 15' W., it would also have been 26 links from the centre line at a distance of 500 links from B, and if any longer lines were required the difference would then accumulate in the same ratio. In concluding this chapter on the compass, I would just mention that I have made plans of iron mines from surveys taken with the greatest possible care with the magnetic needle, and am confident that at a great distance from the shaft, say two miles, the position of headings as pointed out on the surface coidd not be depended on from 5 to 10 yards ; and this is exactly the case

THE miner's compass AND ITS ERRORS. 7

with all maps and plans* (of iron mines at least) constructed from the entire use of this description of instrument. I would therefore recommend any student who intends perfecting himself in mining surveying, that it is much better to go to the price of an instrument that is not only the best but the cheapest in the end, even if it costs three times the price of those generally oflfered for sale, and not for work, for mining purposes.

The magnetic compass is perhaps more adapted for nse in lead and coal mines, where there is much less chance of derangement, than in iron mines ; and I think it not altogether improbable that at a future day the compass may be so constructed as not to be liable to derangements itom local disturbing causes.-

Practical Geometry.

Practical Geometry is intended to explain the method of constructing geometrical figures, and of describing lines according to any possible given conditions. There exist three kinds of magnitudes : lines, surfaces, and solids.

The following chapter refers to figures and lines described upon a plane surface, and contains only those problems and examples that are necessary to the subject in hand.

To Construct A Scale Of Equal Parts.

Take any opening of the compasses, and apply it to any line previously drawn as A B, Fig. 1, and repeated ten times from A to ; then with an opening of the compasses equal to A 0, mark ofi the other equal parts or divisions, 10, 20, 30, Sec. ; then each division in A will be units, the distance from to 10 will be 10 units, from to 20 wiU be 20 units, and so on to the end. Now if each division in A be taken as 10 units, then the divisions on the scale equal to A will be hundreds. And if

Fia. 1.

A be taken as a unit, then each division on A will be onetenth. Then when each division in A represents 10 feet, A B will be equal to 60 feet.

Angles And Their Measure.

What is meant by angles is the opening formed by two straight lines, meeting each other in a point called their vertices.

Practical Geometry.

Fig. 2.

Fig. 3.

As an example, take the angle ABC, Fig. 2, or the angle at B, formed by the lines B A and B C.

Now if we wish to ascertain whether or not the angle at C, Fig. 3, is equal to the angle at B, we put the lines forming the angle C upon the lines forming angle B, in such a way that the point C shall fall upon B, and the line C B shall cover the line B C, then if the line C A cover exactly B A, the angles are equal ; thus it appears that the angles do not depend on the length of the lines which form them, but they may be longer or shorter without altering the angle.

The line D C in the semicircle A G B, meeting the line A B, inclines to the right more than to the left ; the z D C A is evidently greater than the z D C B, q if the lines C G be drawn equidistant between A G B, thereby making the z G C B equal to the angle G C A, then they are called right angles, and the line C G is perpendicular to the line A B ; and the angle D C B is called acute because it is less than a right angle, and the angle D C A is called obtuse because it is more than a right angle.

The circumference of every circle is divided into 360 parts, each part is called a degree, and the angle formed by any two lines drawn from the points of division to the centre D, Fig. 5, of the circle, will be an angle of 1 degree. Take 30 of these divisions, it will be equal to the z S D B to 30° 0' 0'. The angle formed by the lines C D B will be a right angle, or 90°, and the semicircle A C B wiU contain 180°. Divide any one degree into 60 parts, ach part is called a minute, and one of these divisions divided agaiQ into 60 parts, each part is called a second. And we write it thus, 40° 20' 40'' : forty degrees, twenty minutes, forty seconds.

B

B

Practical Geometry.

To describe any angle on a drawing by the compasses and scale draw the line A C, Fig. 6, any length; take from a brass circle or line of chords (divided as above) the angle of 60°, set one foot of the compasses at A,* and describe the arc B C. ; take from the same circle the angle required, Fia. 6. say 45°, set the compasses at C,

and describe another arc at B; now join A B, and the lines A C and A B will contain the said angle.

Problem 1.

Given the hypothenuse and a side of a right angled triangle, to

construct it, and measure the other parts.

Example. — Given the hypothenuse A B,

Fig. 7=468 yards, and the side AC 268

yards, to find the other parts. Draw the line

A C=268 yards, and draw also the line C B

perpendicular to A C ; then from the point A

with the radius =468 yards, cut the line C B

in B, draw A B, and A C B is the triangle

required ; measure the perpendicular C B on

the scale, and it will be found to 380 yards,

the angle A to 55° and z B to 35°.

Fia. 7. ®

Problem 2.

Given the two sides about a right-angled triangle, to construct it, and measure all the other parts.

Example. — Given the base A C, Fig. 8,= 250 yards, and the perpendicular C B 490 yards. Draw the line A C 250 yards, and make the line C B perpendicular to A C, and =490 yards, draw also the line A B, and A C B is the required triangle ; measure A B on the scale and it will be found to 550 yards, and the angle A 61° 30', and the z B =to 28° 30'.

Horn centres may be obtained at sixpence each, which will prevent the compasses damaging or perforating the paper.

Fig. 8.

Practical Geombtrt,

Fia. 9.

Problem 3.

Given the hypothenuse, and one of the oblique angles of a right-angled triangle, to construct it, and measure the other parts.

Example, — Given the hypothenuse A B, Fig. 9=4'26 yards, and the angle at 51° 30 Draw any line A C, apply the protractor to the line A C, and mark off 51° 30', then draw A B=426 yards, let fall a perpendicular to C, and the triangle A C B is the one required; then B C will be found =334 yards, and the angle at A =51° 30', z at B=38° 30'. If the angle at B is given instead of angle A, then deduct the angle B from 90° and it will give angle A, and proceed as before.

Problem 4.

Given one of the sides of a right-angled triangle and an oblique angle, to construct and measure the triangle.

Example. — Given A C, Fig. 10 280 yards, and the angle at A 64° 20', to find the other parts. Draw the line A C=280 yards, and apply the protractor to A, mark off the angle 64° 20', then from the point C draw the perpendicidar C B, now draw A B through the angular point and A C B is the required triangle, A B and C B=648 yards and 580 yards. If angle B is given instead of angle A, proceed as before.

Problem 5.

Given one side and two angles of an oblique-angled triangle, to construct it, and measure the other parts.

Example. — Given the base A C, Fig. 11=350 yards, and the angle CAB, 34° 30', and A C B=29° 40'. Make the line A C B=350 yards, and with a protractor mark off the angles at A and C, 34° 30' and 29° 40' respectively; then produce the lines A B and C B until they join at B, then the triangle A B C is that required, and A B and B C measured on the scale =270 and 290

Fia. K).

Fig. 11.

Practical Geometry.

yards, and the angle at B 119° 50'. If the angles A and B, or C and B are given, the amount of both A and B, or C and B must be added together and deducted from 180, which will give the other angle C or A, as may be required.

Problem 6.

It is required to find the depth of a mine shaft B C Fig. 12, the distance from the croppings out of the underlie or A B, and

the depression of the said underlie or pitching with the horizon are given.

Example, — Given the distance A B 680 links, and the depression of A C or the angle C A B to 60° 30 Draw the horizontal line A B, and set oflF the angle 60° SO' of depression, or the line A C ; measure with the scale 680 links, from A to B, now let fall a line perpendicular from B, and the intersection of B C with A C will be the point required; measure the line B C on the scale, and it will give the depth required in links.

Problem 7.

The depth of a shaft is given, with the distance from the out-croppings, to find the length of heading required to meet the underlie.

Example. — Given the depth of a shaft, 200 yards, and its distance from the out-croppings, 840 yards, with the angle of depression of 69°. Draw the line A B, Fig. 13 840 yards, and protract the angle of depression to 69° (from A) ; let faU a line perpendicular from B, and set off the depth of the shaft, 200 yards. Draw also the heading C D

Fia. 12.

Fig. 13.

Practical Geometry.

Fig. 14

parallel with A B, and measure it on the scale which will give the distance required in yards In like manner the distance from D to A at land may be ascertained by applying the scale to the line D A.

' Problem 8.

It is required to find the depth of a shaft where a heading shall strike the underlie when its length is given, with the distance of the shaft from the out-cropping, and the depression of the pitching.

Example, — Given the distance A B, Fig. 14 to 300 yards j the depression 60°, and the length of heading 10 yds. Draw the horizontal line A B 300 yards, and from A set off the z of 60° for the depression of the underlie ; let fall a perpendicidar from the point B to any depth, apply a parallel ruler to the line A B and run the same downward until the heading of 10 yards will fall exactly between the shaft and underlie. The point where the heading touches the shaft will be the depth (which is at C). Now measure the shaft B C on the scale, and it will give the distance required.

Problem 9.

The distance of a shaft from the out-croppings and the depression of the underlie are given, to find the depth of the shaft at the point of intersection with the underlie, and also to find the length of a returned heading, when the shaft is continued below the point of intersection.

Example, — Given the distance of a shaft B, Fig. 15, from 400 yards, and the angle of depression =62° 30, with the shaft continued 60 yards below the point of intersection with the underlie. Draw the line A B to any length, and from the point A set off with the protractor the z 62° 30' of de-

Fia. 15.

Practical Geohbtbt.

presBioQ with the horizon ; let fall a perpendicular from S, and the point of intersection at E vill be the depth required ; now meaaure B £ on the scale, and it will give the distance. Continue the shaft yards below the point E to F, then apply the parallel ruler to the Une A B, and run it downwards to the point F; draw F H, and it will be the proposed returned heading, apply the scale to F H, and it will give the required distance.

Peoblem 10.

It is required to find the distance between two shafts, A and B, on opposite sides of a lake.

Example. — Plant the theodolite at C, Fig. 16, and measure the angle A C B, say to 94° 45', measure also the line C A and

C B, say 60 and 70 chains respectively. Draw the line C A, and mark ofi the angle at C with the protractor to 94° 45', apply the scale to C A and mark off 60 chains, also to the line, C B 70 ; join the line A B, and apply the same scale, which will give the distance required.

Pboblem U.

To find the distance between two mine shafts, A B, Fig.

Fio. 17. 17, inaccessible from each other by the intervention of the river S O.

Practical Geometry. 15

Example. — Set up a staff at B in line with A on the other side of the river and draw a line at right angles with it and set up a staff at D. Draw a line with D H at right angles with B D and set up a pole at H. Now measure off a given distance B, say B C, and set up a pole at C ; then direct the staff H to be brought in a direct line with C and A, and measm'e the distance B C, CD, and D H ; then because the angle B C A is to the z D C H and the z B is to the z D, being each right angles, /. C D : D H : : C B : B A ; let B C 400 feet, D C 200 feet, and D H 140 feet; then 200 : 140 : : 400 : 280 feet to B A, the distance required.

Problem 12.

Required the distance between two mine shafts, A and B, which could only be seen at the same time from one station, D.

Example, — Set up a theodolite at D and also a staff at C and E at a given distance. Now measure the angles A D B, ADC, and A C D ; the distance from to D being known,

Fig. 18.

the length of the line A D can be ascertained by. Problem 5 ; then measure the angles B D E and DEB and the length of the line D B wiU be found ; then the lines A D and B D being ascertained and also the angle A D B, the length of the line A B wiU be found by Problem 10.

Problem 13.

The depth of a mine shaft and the angle of elevation of another shaft, with the hypothenuse, are given, to find the depth of the shaft.

Practical Geometry.

Example. — Given the depth of the shaft A B, Fig. 19=120 yards, and the angle of elevation E A C=28° 30' 40 and the hypothenuse A C 1060 yards, to find the depth of the shaft C D. Draw the horizontal line B D, measure (with any scale) B upwards to A =120 yards,, apply the protractor to the horizontal

/I

,.'

Q

E

B

Fig. 19.

Une B D with its centre at B, mark ofiF the angle 28° 30' 40'', set a parallel ruler to the line B a, and run it upwards to the point A ; draw the line A C, then the zEAC= zDB; now measure oflF on the line A C 1060 yards, and let fall a perpendicular, D C, which measured on the same scale wiU give the depth of the shaft required.

Problem 14.

The depth of two shafts, with the angles of depression and the hypothenuse/ are given, to find their differences of level.

Example. — Given the depth of two shafts, A B and C D, Fig. 20=130 and 160 yards respectively, with the angles of depression T H A=24° 20' 40", z S H C 10° 34', and A H and H C =800 and 640 links, to find their differences of level. Draw the horizontal line T H S, and H with the protractor, mark off the angles of depression T H A=24° 20' 40", and S H C=10° 34' ; draw the lines H A and H C, making H A 800, and H C =640 links, then the points A and C will be the top of the shafts ; let fall the perpendiculars T A and S C, which measure

Practical Geometry.

on the scale, and their diflFerence will equal the difiFerence of elevation of the to shafts ; produce downward the lines A B and C D, and measure on them 130 and 160 yards respectively;

Fig. 20.

draw also the line B D parallel with T S, and the point of intersection of B D with A B wiU give the diflFerence of level of the bottoms of the shafts.

Problem 15.

Given the angles of elevation of two distant shafts, with the

measured hypothenuses, to find their diflFerences of level at the surface.

Example. — Given the angles of elevation H C A, Pig. 31,

Fia. 21.

=30° 40 and S C B=26° 10' 40", and the line A C and B 1100 and. 980,.. to find the -diflFerences of level. -Draw the. hon-

Practical Oeometbt.

zontal line H S, and mark oflP the observed angles from C ; H C A=30° and S C B=26° KT 40''; measure on the lines A C and C B the distances 1100 and 980 links then the difference of H A and S B, measured on the scale, will give the diflFerence of level of the two shafts, or set up a transit theodolite at B, and after careftdly levelling it, direct an assistant to move a staff (the same height as the instrument) along the surface from C towards A, then when the upper part of the staff is seen in . line with the cross hairs in the telescope at b, it will be on the same level as the instrument at B a ; now the difference of level between the point of intersection of the telescope with the staff and A should give the same difference of elevation as before.

Problem 16.

Given the angles of elevation and depression of the top of two shafts fi'om two distant stations, with the distance between the stations, to find their differences of level, and the distance of the shafts from each other.

Fio. 22.

Ewample. — Given the observed angles of deviation, taken with a transit instrument from A to B, Fig. 22, =25° 30' 20'',' from A to C=34° 40' 40", and from A to D=55° 45' 40". The line A D was measured and found to be 1250 links. The instrument was again planted at D, when the angles of depression from D to B, or i: S D 45' 20 and that of devafcioB. from D to C,

Practical Geometry. 19

or z S D C, 3° 32' 40''. Required the distances A B, A C, B D, and D C, and the difference of elevation &om B to C.

Construction. — Draw the horizontal line E P, and set the protractor to the point A, marked by the axis of the theodolite, mark off the angles of elevation, A to B or z F A B 25° SO' 20", A to C or z F A C=34° 40' 40", A to D or z E A D=55° 45' 40", and call them respectively Nos. 1, 2, and 3 ; from station No. 1 draw the lines A B, A C, and A D, making A D 1250 links. To prevent conftision, draw also the line D S parallel to A F, and from the point D, the axis of the instrument at the second station, mark off the angles of depression D to B or z S D B=20° 45' 20", and that of elevation DCorzSDC=3° 32' 40", and call them Nos. 1 and 2 ; from station No. 2 draw the lines D B and D C, and the point of intersection of D B with A B will be the top of the shaft No. 1. Draw also the line D C, and its intersection with the line A C in C will give the position of shaft No. 2; now apply the same scale (that the line A D was measured with) to the lines a B and F C, and their difference will equal the difference of elevation of the two shafts; also, F C will be the total elevation above the station A. In like manner, the scale, applied to the lines F C and A ft, will give the difference of level of station No. 2. Below the shaft No. 2 it is also evident that the scale applied to the lines A B, A C, D B, and D C, will give the distances required in links.

Problem 17.

Given the observed horizontal angles with two shafts from two distant stations, with the distance between the stations, to find the distance of the two shafts from each other and also from each station.

Example, — Given the observed horizontal angles taken with a transit theodolite z C A B=:108° 46' 40", the z D A 40' 20", z A B C=24° 10' 40", and z A B D=55° 27' 40", and the line A B=960 links, to find the distance C D, A C, A D, B C, and B D.

Construction. — Draw the horizontal line A B, Fig. 23, and make it the exact length of 960 links (with any convenient scale), apply the protractor to the line A B with its centre at A, and mark off the z C A B=108° 46' 40", and with the protractor in the same position mark off the z D A B=89° 40' 20" ; remove the instrument, and draw the lines A C and A D to any convenient

Practical Oeometbt.

length ; again set the protractor to the line A with its centre at B, maA off thez A B C=24° KX andz A B ; again remove the instrument and draw the line B C and B D ; the of intersection of B C with A C will give the posi-

Fig. 23.

tion of the first shafts the intersection of B D with A D wiU also give the position of the second shaft ; now apply the scale to the line C D, which will give the distance between the shafts A C, A D. Then B C and B D, measured on the same scale will give the distances of the two stations A and B from each shaft respectively.

PRACTICAL METHODS OF MEASURING INACCESSIBLB DIS- TANCES, AND AVOIDING IMPEDIMENTS IN THE RUNNING OF LINES.

Many cases of obstruction in the measurement of lines wiU occur in wooded districts ; and from the intervention of lakes buildings rivers, &c. &c. The following problems are intended to assist the surveyor in surmounting these difficulties ; and as the more simple cases may be effected by the chain only, those more complex will require a goodangular instrument at hand, together with the aid of plane trigonometry, which will enable any person to overcome these difficulties. The student should be very carefril to range his lines so that they may pass clear of all impediments, if possible. But after all his ingenuity has been expended, he wiU find it quite impossible to avoid them entirely.

Pbactical Geometry.

In measuring a Kne, A B, Fig. 24, an obstruction of a lake came in the direct Kne of measurement. To avoid which, set up staves at C D E, and at right angles to the points o o' and after having measured this new line past the obstruction and

-8

Fig. 24.

parallel to it, a return was made' by setting up staves 8 a" at right angles to £ C H, points in the new line, and at the same distance as before the original direction of the line is continued to the end.

To pass a similar object with the use of an angular instrument and the chain, as follows : —

In measuring the Une A B, Fig. 25, I come against an impenetrable inclosure ; to avoid the obstacle, I measured an angle A C D 120°, and after measuring along the line C D a sufficient distance to clear the inclosure at D, the theodolite was set

B

Fig. 25.

up at D, and an angle measured off equal to the supplement of 120°, or in this instance equal to 60° ; then meastire along the line D E exactly the same distance as C D, which will be a point in the same straight line A B; and as it is impossible to see the back station A, the direction of the forward station B will be found by setting up the transit at E, directing the telescope on D, make the instrument read exactly 120°, or equal the angle DEB; now a staff brought into the line marked by the cross hairs of the telescope will give the exact direction ; then, if it is possible to sight the station A from B, the station E should be in the same straight line, if the above process has been gone through correctly.

" Required aloi fr distance H O, iuacci'siM

Iicai. Geomktrt.

line A 11, 36 continued, the lUrcct nn'usnivnicii* witli the chain."*

Fio. 26.

Set up a staff at C, and make the line B C perpendicular to A B, then set out the line C D per[)cndicular to C O, and continue it until it joins A B in D : measure B D ; then as B D : B C : : B C : B O.

Given B D 400 and B C 600 feet. Then as 400 : 600 : : 600 : 900=B O.

To solve the above problem by the instrument, set up the theodolite at B, Fig. S7, and raise a perpendicular B D to any

convenient length, then remove the theodolite to the station D, and make the angle B D A=:to z B D C ; measure B C, then B C Trill equal B A,

It is required to find the distance B A, Fig. 28. Set up the

theodolite at B, and measure any angle, say about then

Williams's FractUat Oeodety.

Practical Geombtey. 23

Tralk along the line B D until yon find the instrameat reads half the angle, or 45° ; then measure the side B Dj and B D will be equal to B A, the distance required.

The following is another method of passing obstacles by equal triangles : — After having measured up to A, Fig. 29, the line A D

Fio. 29. was broken by the intervention of a sheet of water and nursery. Set up the theodolite at B, and continue the lines B A to C, and also B D to E, then measure the lines 6 A and B D, making fi A equal to B C, and B D equal to B £ ; then measure the line ; C E will be equal to A D, the distance required.

A great many other methods, more or less simple, may be contrived by the ingenious student to solve any of those questions. They all, more or less, are based on the principle of simUar and equal-sided triangles constructed on accessible planer, in conjunction with those on inaccessible planes.

Practical Plane Trigonometry

Right-Angled.

Trigonometry is one branch of the mathematics which treats of the relations existing between the sides and angles of triangles. The kind introduced here wiQ be as simple and practical as possible and sufficiently extensive for all the purposes of the undergroimd surveyor ; and as I have proposed to work out all the necessary calculations by one method, or by the table of natural sines, tangents, &c., any other methods of calculation will not be resorted to except where they can be more advantageously employed.

Definitions.

1. The complement of an arc is the diflference of that arc from a quadrant, or 90°, either in defect or excess.

2. The supplement of any arc is the difference of the arc from a semicircle, or 180°.

3. The complement of an angle is the difference from a right angle.

, 4. The supplement of an angle is the difference of the angle fim two right angles.

5. The sine of an arc is a straight line drawn from one extremity of the arc perpendicular to the radius drawn through the other extremity.

6. The tangent of an arc is the straight line drawn from the outer extremity of the radius, touching the circle in that point.

7. The secant of an arc is the straight line passing through the circumference of the circle, and from its centre, until it touches the tangent produced.

8. The sine, tangent, and secant of the complement of an arc are called the cosine, cotangent, cosecant of that arc.

Practical Plane Trigonometry.

Right-Angled Trigonometry.

Fig. 30.

Proposition, — If in a right-angled triangle we make the hypothenuse radius, then its side becomes the sine of the opposite angle, or the cosine of the adjacent angle. Then if the hypothennse A C, Pig. 30, is radius of the arc C D, B C will then be the sine of C D, and is also the sine of the z A. Also, if C is the centre of the arc" A E, described by the radius C A, and cutting the line C E in B, then A B is the sine of A E, is also the sine z C, that is, A B and B C are the sines of the opposite angles. The three angles of every triangle are to two right angles. Hence the oblique angles of a right-angled triangle are each other's complements.

Therefore, since the angles A and C are each other's complements, B C, the sine of A, is also the cosine of C, and A B, the sine of C, is the cosine of A, that is, A B and B C are the cosines of the adjacent angles. When one of the sides about the right angle of a right-angled triangle is made radius, the other side becomes the tangent of the opposite angle, and the hypothenuse the secant of the same, or the other side becomes the cotangent of the adjacent angle, and the hypothenuse the cosecant of the same. Let A B, Fig. 31, be a side of the right-angled triangle ABC, and made radius, then B C is a tangent of the angle A, and therefore also the cotangent of the angle C, which is the complement of A. Also, A C is the secant of the angle A, or cosecant of C.

Fig. 31.

Problem 1.

If we have given in any right-angled triangle one of the sides containing the right angle, and one of the acute angles, to find the other sides.

Before proceeding to a more general method of calculation, I shall here introduce a simple and accurate method of finding

Fm. 32.

26 Practical Planb Tbi60N01Cbtbt.

arithmetically the sides of any plane triangle having the base and one of the angles given.

Example.— Given the z A, Fig. 32, 32'', and the side A B=440 links, to find the other sides and angles.

Rule. — Square the angle at A, and add to the resolt of itself, which call the dividend, then add 100 to the angle at A=: 132, call this number the divisor, then add to the quotient the z at C=58®, and call this new number the hypothenuse, by supposition to A C. - Then say by proportion, as 58°, the z at C : to the base A B 440 links : : 68.34 the else hypothenuse to 518.34= the true hypothenuse.

(32)' or 32x32+=1365-1- 132=10.34 + 58=68.34 the else hypothenuse. Then say 58° : 440 : : 68.34 : 518.43 the true hypothenuse.

To find the Perpendicular.

Rule. — Square the z at A, and multiply it by 3. Then divide this number by 1000, and add to the quotient the constant number 57.3. Call the result the false perpendicular, then say by proportion, as the false perpendicular 60.0 is to the true hypothenuse, so is the angle at A to the true perpendicular.

+ 57.3=60.37 the false perpendicular.

Then 60.37 : : 518.43 : 32° : 274.8 the true perpendicular. The proof of the hypothenuse may be obtained from the 46th P roposition of Euclid, as follows : \/(AB)=*+(BC)'=AC, or (274.8)'= AC the hypothenuse.

We will now prove the truth of the perpendicular from the tangent of the angle at A.

Natural tangent z 32° .624869 Multiplied by the base A B 440

Perpendicular 274.942360

The result, as obtained by the above process, is of course more accurate than that found by the arithmetical process, but the

Practical Plane Trigonometry.

difference in the latter is so small that it may be relied on as a near approximation to the truth.

Problem 2.

Given the hypothenuse and one of the acute angles of a rightangled triangle, to find the other sides and angle.

Example, — In the rightangled triangle ABC, Fig. 33, given A C=800 links, and the angle at A =20° 30', to find the other parts.

Fig. 33.

To find the adjacent z C in all cases. 90-20° 30' z at A=69° 30' z at C.

Natural sine of the z at A =20° 30' .3502074 Which multiplied by the line A C 800

The perpendicular B C 280.1659200

To find the Base A B.

Natural cosine of the z A=20° Which multiply by the line A C

Therefore the base A B 749.3377600

Problem 3.

Given the depth of a shaft sunk

from a point in the out-croppings

with the angle of depression, to find

the length of a cross drift that shall

again strike the underlie.

Example, — The depth of a shaft, A C, Fig. 34, 80 yards, and the z B A C .40 30, to find the length B C.

Natural tangent of z B A C 40" 30' Multiplied by the depth AC. .

A

Fig. 34.

The cross drift B 68.326400

Practical Plans Trioonometbt.

Problem 4.

It is required to find the depth of a mine shafts B C, Fig. 35, that shall strike the underlie when its distance troxn. the outcroppings and the z of depression of the underlie are given.

Fig. 36.

Example, — Given the distance A B=700 yards, and the depression of A C=32° 15'.

Natural tangent of the z C A B 32" 15'= .630953 Which multiplied by the distance A B 700

Depth of shaft required 441 .667100 Or B C is about 441 yards.

Problem 5.

The distance of a mine shaft from the out-cropping, with its depth and the depression of the underlie, are given, to find the length of a cross cut, to be driven from the bottom of the shaft so as to strike the underUe.

Fm. 36.

Example, — Given the distance A B, Pig. 36, 420 yards, and the depth B C 140 yards, with the angle of depression 41° 15'.

Practical Plane Trigonometry.

Natural tangent of the z E A B=41° 15' Which multiply by the distance A B

Total distance f5rom B to E From which deduct from B to C

Therefore the distance of C E

.876976 420

368.329920 140

228.329920

To find the Angle at E.

90°- z at A=41° 15'= z at E=48° 15'. We have now the distance E C and the z at E, to find the cross cut C D.

Natural tangent of the z at E=48° 15' 1.120405 Which multiply by the distance C E 228

Therefore the length of the cross cut C D 255.452340

Problem 6.

Given the distance of a shaft from the out-cropping and the angle of depression, to find the depth of the shaft at its intersection

A B

Fig. 37.

with the underlie, also the length of two cross outs, one at a given distance above, and the other below the point of intersection.

Practical Plans Trigonombtry.

Example. — Given the distance A B, Fig. 37, =220 yards, and the z of depression =46'' 20', B C=60 yards, and E H 80 yards, to find B E, C D and H F.

To Jind BE.

Natural tangent of the z at A =46° 20' Which multiply by the distance A B

Depth of the shaft at the intersection Deduct the depth from B to C .

The distance from E to C

The angle at E=90°-.46° 20'=43° 40'= z at E.

230.484980

170.484980

To Jind G J).

Natural tangent of the z at E=43° 40' .954508 Which multiplied by the distance E P 170

Therefore the cross cut C D 162.266360

Then to find H F we have given the distance E H, and the z at E.

The natural tangent of the z at E=43° 40' .954508 Which multiplied by the distance E H 80

Therefore the cross cut H F 76.360640

Problem 7.

It is required to find the diflFerence of level of two mine

Fig. 38.

shafts, having given the measured hypothenuses and angles of elevation.

Pbactical Plane Trigonometbt.

ExampU, — Given the angles of elevation taken with the transit from A, B, C, D and E, Fig. 38, =20° 20', 15° 40 24° 0', and 20°, and the hypothenuses A B, B C, C D, and D E,=to 200, 300, 340, and 600 yards, to find the diflference of level and the total length of the base F G.

To find the Perpendiculars.

Multiply the sine of the angles by the hypothenuses.

z at A =20° sine .3474812 x 200= 69.4962400 z at B 15 40 sine .2700403 x 300= 81.0120900 z at C=24 00 sine .4067366 x 340=138.2904440 zatD=20 00 sine .342020 x 600=205.2120000

Total diJBFerence of level 494.0007740

To find the Base G.

Multiply the cosine of the angles by the hypothenuses.

z A =20° cosine .9376869 x 200= 187.5373800 z B =15 40 cosine .9628490 x 300= 288.8547000 z C =24 00 cosine .9135455 x 340= 310.6054700 z D=20 00 cosine .9396926 x 600= 563.8155600

The total length of base F 1350.8131100

These calculations have resolved the above diagram into one large triangle, whose perpendicular G E is equal 494.0 and the base F Q=to 1350.8 yards.

Problem 8.

It is required to find the depth of a mine shaft at the same level with another shaft whose depth and angles of elevation with the hypothenuses are given.

Example. — Given the depth of the shaft A L, Fig. 39, 100 yards, and the observed ver-

tical angles=to angle at A 31° 21' and z at B 24° 10', with the hypothenuses A B and B C 920 and 840 yards, to find C D.

Multiply the sine of the Z of elevation by the hypothenuse

Practical Plane Trigonometry.

tofr the perpendicular and the cosine of the z of eleratii

the base.

z at A =31° 21'= sine .520246 x 920=478.644320 z at B=24 15 sine .4107189 x 840=345.0038760

Total difference of level C E Dh of the shaft A L .

Total depth of shaft C D

=823.6473080 + 100

=923.6473080

The base may be obtained from the cosines of the angles and measures of the hypothennses as preyionsly explained. The student should carefully construct these diagrams as explained in practical geometry from the data given in the example then if the result as obtained from the construction approximates with the calculation it will be a satisfactory conclusion ; otherwise it must be recalculated or constructed until they do agree.

Problem 9.

It is required to find the angle of direction and length of a heading to be driven a fixed point below ground so as to strike at right angles to another fixed pointy having given two measured distances.

Ride. — Divide the given side opposite the required angle by the hypothenuse, the quotient is the sine of the angle of direction.

Fig. 40.

Example, — Given two under-ground drifts A B and B C (Pig. 40) =420 and 160 yards, to find z of direction at A and the side AC.

22° 24'

To find the z at B] 3360 900-22 24'= z at h 4000 67° 36' ) 3780

Practical Plank Tbioonombtrt.

The length of the side A C may.now be found from Problem 3 from the cosine of the Z at or from Problem 8 from the tangent of the Z B.

To find the Side K G.

By Problem 2. Natural cosine of Z A 22 24'

Multiplied by A B

184909200

The side A C 388.3093200

By Problem 3. Natural tangent z B 67" 36'

Multiplied by the side B C

Therefore ' the side A nearly the same asl before i

160

.145570860

888.188960

Problem 10.

Given the depth of a shafts with its distance from a fixed point on the surface to find the length and angle of elevation of an incline to be driven from the bottom of the shaft so as to strike the fixed point at surface.

B

%.

Fig. 41.

v%

Rule. — Divide the side opposite the required angle by the other side ; the quotient will be the natural tangent of the angle.

Examph. — Given the depth of a shaft B C, Pig. 41, 260, and the distance A B 180 yards, to find the angle of elevation CAB and the length of the incline C A.

Pbactical Plane Tbigonomktby.

tangent M"" 4Si'

To find the z C A B 90°- z at C 34° 42'= z at A 55° 18'.

To find the Hypothenuse A C.

Natural secant z at A 55° 18' 1.7566063 Multiplied by A B 180

1405285040

.-. The incline or side A C 316.1891340

I have been induced to add the following examples calculated by logarithms, to suit the convenience of those who may hayQ occasion to deal in large numbers ; as in that case the multipHcatiou in the preceding method becomes very laborious.

The general rule in logarithmic calculations* is to add together the two last terms, deducting therefrom the first term for the

answer.

Problem 11.

Fig. 42.

Given the hypothenuse and one of the oblique angles, to find the other parts.

Example, — Given the hypothenuse A C, Pig. 42, 849 yards, and the z A =40° 16', to find the other parts.

z at C 90°- z A 40° 16' z C 49° 44'.

It is desirable that the student should read the chapter on Logarithms before proceeding to calculate with them.

Pbactical Plane Tbigonometbt.

3S

To find B C-

Logarithm radius . . . : Logarithm sine of the Z A 40° 16' Logarithm of A C 849 . . .

Logarithm B C 548.75 2.7393727

-10.0000000 +9.8104650 +2.9289077

TofindA.'Q.

Logarithm radius

Logarithm cosine of A 40° 16' .

Logarithm of A B 647.06 . . .

-10.0000000

+9.8825499

+2.92839S9

2.8109458

Given the base of a right-angled triangle and one of the oblique angles to find the other parts.

Example. — Given the base A B, Fig. 43 1620 chains and the z at A 53° to find the other parts.

To find "RQ.

Logarithm tangent Z A 53° 7' Logarithm A B 1620 . . .

Logarithm B C 2159.6 . . ,

Fia. 43.

-10.0000000

+10.1247266

+ 3.2096160

3.3342416

Pboblem 13.

Given the base and perpendicular of a right-angled triangle, to find the other parts.

Example.— dvr&n A B 5300 and B C 6700 yards, to find the other parts of the triangle.

D 2

36 Practical Plans Tbigonobibtbt.

To find the / at A.

Logarithm of A B 5300 3.7242759 Logarithm of B C 6700 . . . . + 3.8260748' Logarithm radius -f lO.OOOOOOQ

13.8260748 Deduct the first term 3.7242759

Logarithm tangent of z A 51° 39' lO'' 10.1017989

To find AG.

Logarithm radius =-10.0000000

Logarithm secant of z A 51° 39" 10'' + 10.2073837 Logarithm of A B 5300 . . . + 3.7242759

The side A C 8543.9 . . . . 3.9316596

Nature And Use Of Logarithms.

In tlie progress of tlie previous calculations no great difficulties have been encountered, but wben some of the problems require a much greater nimiber of figures to be employed, the computation becomes a work of much labour, and it is then of great importance to be enabled to perform trigonometrical calculations with greater facility; for which purpose. a series of artificial or decimal numbers has been invented and tabulated for use, called logarithms.

We may define logarithms to be thjB numerical exponent of ratios, or a series of numbers in arithmetical progression answering to another series of numbers in geometrical progression.

Numbers said to be in arithmetical progression are those that continually increase or decrease by the constant addition or subtraction of the same numbers.

Thus, 1, 2, 3, 4, 5, 6, 7, 8, 9 are in arithmetical progression; and numbers in geometrical progression axe those which are constantly multiplied by the same numbers. Thus, 1, 2, 4, 8, 16, 32, 64, 128, 256, &c., are in geometrical progression. If now we arrange the arithmetical series of progression (having previously prefixed a cypher to the commencement of the series) above the geometrical series, it will stand thus :

Indices or logarithms . 0, 1, 2, 3, 4, 5, 6, 7, 8, &c. Geometrical proportion 1, 2, 4, 8, 16, 32, 64, 128, 256, &c.

Indices . . . 0, 1, 2, 3, 4, 5, &c.

Progression . . 1, 10, 100 1000, 10,000, "100,000, &c.

It will now be evident that the same indices serve equally for any geometrical series, and that th6re may be an endless variety of systems of logarithms to the same common number by only changing the second term, 2, 3, or 10, &c., of the geometrical series of whole numbers. It now plainly appears that*if any two indices are added together, their sum will be the index of that number which is equivalent to the produce of the two terms in the geometrical progression to which the indices belong.

38 Logarithms.

" the indices 4 -1-3 =7, and the corresponding terms 16 x 8, to those indices, produce 128, which correspond to 7. It will also appear that if any one index be subtracted from another, the difference will be the index of that number that is equal to the quotient of the two corresponding terms. Thus, the index 7 — the index 3=4, and the terms corresponding to these indices are 128 and 8, whose quotient is 16, the number corresponding to the index 4, and if the logarithm of any number be multiplied by the index of its power, the produce will be equal to the logarithm of that power. Thus the index or log. of 4 in the above series is =2, which multiply by 3=6, which is the logarithm 64 or the cube of 4.

" And if the logarithm of any number be divided by the index of its root, the quotient will be equal to the logarithm of that root ; the index or logarithm of 64 (in the first series) is 6, if divided by 2=3, which is the logarithm of 8 or the square root of 64.'*

The logarithms most convenient for use are such as are adapted to the gmetrical series, increasing in a tenfold proportion, as in the last or preceding series, and are those which are generaUy found in common tables.

In which system of Icarithms, the index or logarithm of 1 is 0, that of 10 is 1, of 100 is 2, of 1000 is 3, of 10,000 is 4, &c., whence it is evident that the logarithm of any intermediate number between 1 and 10 and 100 will be some fractional part, and so on for any number whatever.

And as we always know the value of the index of a logarithm by the nimiber of integrals in the natural number, the index to the logarithms is not prefixed to the tables, but must be supplied by the calculator himself.

To explain the method of supplying the indices, the following numbers and logarithms are given. Thus :

JN'umbers. Logarithms.

1806 3.2567177

180.6 2.2667177

18.06 1.2567177

1.806 0.2567177

J806 1.2567177

.01806 2.2567177

.001806 3.2567177

Moore's Nautical Aitronomy,

Logarithms. 8&

The index of tlie logarithm of a number is one less than the number of integral figures contained in the natural number. That is, if the number contains 4 intel'al figures, the index is 3 ; if it contains 3, the index is 2, and so on. Sometimes the number has no integral figures, then the index of its logarithm is negative, and is 1 more than the number of cyphers immediately after the decimal place. Thus in the table we find the number 1806, its logarithm =.25 67177; then, because there are four places in the natural number 1806, I prefix the index 3 before the decimal part of the logarithm ; it will then stand thus : 3.2567177, showing that the index is 1 less than the number of integrals in the said number. ' In like manner take the same number, first prefixing a decimal point before it, thus .1806= its logarithm .2567177. Then, because there ai*e no integrals in the number, but all decimals, the index will be one more than the number of cyphers prefixed to the number, consequently the index to the nimiber

.1806 will be negative or 1 with the logarithm following; —

thus, number .1806, its log. 1.2567177, and so on of others. Supposing the student has procured a set of tables, he should proceed to find the logarithm of any numbers, and to supply the indices. Find the logarithm of the number 1686, the decimal part of the logarithm as found in the tables is =.2268576, and as the number contains four integral places, we must annex the index 3 as a whole number. Then the logarithm will stand thus : 3.2268576. Find the logarithms .of the following numbers : —

8464 4689.25

.3148 -000214

It is required to find the number to the following logarithm : 3.3181886. At first sight we know that the number must contain four places, and to find it run down the columns until the first part of the logarithm 318 is found, and to the left and opposite is the number 2080, and as the number is not complete, we look into the adjoining columns for the remainder of the logarithm, or 1886, which number will complete the logarithm, and will stand as above, 3.3181886. The number answering to the last part of the logarithm is .6, which annex to the number previously found ; it will then stand thus, 2080.6, the number required.

To find the produce of any two numbers by the logarithms as follows :

Rule. — Add the logarithms of the given numbers together,

40 Logabithms.

and find the number corresponding to their sum, which nuniher is the produce required.

Example 1. — What is the produce of 4864 multiplied by 26 ?

Logarithm of 4864 =3.6869986

„ „ 26 1.4149733

„ „ the produce 12646.4 51019669

Thus it appears that multiplication is performed by addition. The utility of logarithms is thus sufficiently obvious when large numbers have to be dealt with.

Example 2. — Find the continual produce of 384. .462 and

Logarithm of 384 . . ... . =2.5843312

„ .462 ! =1.6646420

„ 89.8 . . . . . . 1.9532763

„ . „ the produce, 15930.1234=4.2022495

Division byi Logarithms. .

Rule. — Subtract the logarithm of the divisor from the logarithm of the dividend, the remainder is the logarithm of the quotient.

Example 1.— Divide 6854.3 by 422.

Logarithm of 6854.3 =4.8359631 „ ,,422 =2.6253125

9f

„ the quotient 162.4141=2.2106506

Example 2.— Divide .462 by .023.

Logarithm of .462 . . . . 1.6646420 „ „ .023 . . , . 2.3617278

„ „ the quotient 20.086= + 1.3029142

7%e Utile of Three direct by Logarithms.

Rule, — Add the second and third term together, and from that sum deduct the first term ; the remainder is the logarithm of the fourth term.

Logabjthms. 41

Example. — Find a fourth proportional to the nmnbers 69 800, and 164.

Logarithm of 800 =2.9030900

„ ,,164 2.2148438

Sum of logarithms of 2nd and 3rd term =5.1179338 Logarithm of 69, 1st term 1.8388491

„ „ 4th term, 1901.44 . , 3.2790847

The student will now be prepared to work out any problems in trigonometry by the logarithms, and witli the assistance of the logarithm, sines, tangents, &c., which will all be found in Chambers Mathematical Tables, a work at once so portable and cheap that all persons making these kind of calculations should be provided with a copy.

.#

Practical Plane Trigonometry:

Oblique-Angled.

In any oblique-angled triangle when two angles and a side opposite to either of them are given, all the other parts can be found.

The sides of all oblique triangles are in proportion to the sines of the opposite angles.

Rule. — To find a side, begin with the angle opposite the given side ; that is, the sine of the angle opposite the given side is to the sine of the angle opposite the required side as the given side is to the required side. And when two angles of any oblique triangle are given, the third may be found by subtracting their sums from two right angles.

Before proceeding to work out other trigonometrical quantities by the logarithms, I shall insert in this place a method of finding the sides of oblique triangles by Problem 1 of rightangled trigonometry.

Problem 14.

Given two angles and a side of an oblique-angled triangle, to find the other parts.

Example, — Given the angles DAB and A D B=40® 12' and 10 30', to find the other parts. (Fig. 44.)

To find the Angles

A B S, S B D, and D B C. 90- /. at A, 40 12'=to 49° 48' Z at B. And 90°- z S D B, 10° 30' 79° 30' z at D B S. Also to find the z D B C, 180°- z A B S+ z D B S 180°-49° 48' -79° 30' =50° 42' z D B C.

To find the Side 1& S.

Rule. — Divide 48', the fractional parts of a degree, in the angle A B S by 6=.8, enter down the degrees in the same angle

Practical Plane Tbigonometbt.

as a whole number, witli the decimal .8 appended : 49.8. Square this number, or 49.8, by 49.8=2480.04, add onethird of this number to itself =3306.72, also add to the first

number, or 49.8, 100=149.8. Divide =22.07; to this

' ' 149.8

quotient add the angle at A, 4/OP 12' =62° 20, which call the

hypothenuse.

D

Fia. 44.

Then say, as 62.20, the false hypothenuse, is to the base A B 884, so is 40.2, the angle at A, to the side B S =248.18. The operation is given in as follows : —

49.8 Then

as 62.20 : 384 : : 40.2

S

Fbactical Plans Tbigonometby.

To find the Side BD.

Rule. — Divide SCX, the odd minutes in the angle at D, by 6 =.5, enter down 10° in the angle at D as a whole number, with the decimal .5 appended as before =10.5. Square this number, or 10.5x10.5 110.25, multiply this number by 8 330.75, which divide by 1000, the quotient, .3, ..to which add the constant number 57.3=57.6; then say, as the angle at D 10.5 is to the side B S, so is the number 57.6 to the side B D.

( .3 57.3= the constant number.

57!6

Then say, as 10.5 : 248.2 : : 57.6 : 1361.05 D.

Problem 15.

Given two angles and a side of an oblique-angled triangle, to find the other parts.

Example. — In the triangle ABC, Fig. 45, there are given z A 34° 20' and z B=40° 40', with the side B C=800 Unks.

Fig. 45.

To find the Angle C. 180f-A+B=180°-34° 20'+40° 40'=180°-75°=lb5

To find the Side A

The logarithmic sine of A 34° 20'

The logarithm sine of B 40° 40' . The logarithm of B C 800 . .

The two last terms added . . Deduct the first term

Fourth term the side A 924.32 2.9658250

- 9.7612842

+ 9.8140192 + 2.9030900

12.7171092 9.7512842

Fbacticai. Flanb Tbigonometbt.

To find the Side A B.

sme A 34 30' . . .

-9.7512843

+9.9849438

„ B C800.

+2.9030900

The two last terms added

12.8880338

The first term deducted

9.7512842

The fourth term or side A B=1370.

8.1367496

In any oblique-angled triangle, when two sides and an angle opposite either of them are given, to find the other parts.

Rule. — To find an angle, commence with the side opposite the given angle; that is,. the side opposite the given angle is to the side opposite to the required angle as the sine of the given angle is to the sine of the required angle.

Problem 16.

Given two sides of an oblique*-angIed triangle and the containing anglcj to find the other ,pai:t.

Fio. 46.

Example. — Given the side A C and B C, Fig. 46, 640 and 470 yards, with the containing angle A C B equal to 79° 40.

To find the Angles A and B.

The side A C is greater than the side B C, and the angles opposite the former exceed the zA opposite the latter. Also the z A-|-B=180— z at C.

Practical Flanb Trioonohrtbt.

9f

Logarithm A C + B 0=640+470=1110 . -3.0453230

of A C-B 0=640-470=170 +2.2304489

tangentof i (A + B)=5010'. + 10.0787534

The two last terms added together . . . 12.3092023

Deduct the first term 3.0453230

Logarithm tangent of i (B+A)=10° 24' 9.2638793 To which add the angle i (B + A) 50 10

Hence the angle at B =60° 34'

ThenzO + /B 79° 40 + 60° 140° 14', and 180° 140° 14'= to 39° 46', the angle at A.

To find the Side A B.

Logarithm sine of z B 60° 34' .

z 79° 40' of A 640 yards .

The two last terms added together The first term deducted

The side A B =722.93 . .

9f

-9.9399823

+9.9928984

+2.8061800

12.7990784

9.9399823

2.8590961

Problem 17.

Given the three sides of an oblique-angled triangle, to find the angles and the difference of the segments of the base.

Fio. 47.

RtUe. — Make the longest side the base, and let fall a perpendicular npon it £rom the -opposite angle. Then, as the base or snm of the segments made hj the perpendicular is to the sum of the other two sides, so is the difference of those sides to the difference of the segment of the base, and half the difference

Practical Plane Thigonometby. 47

added to the half base will give the greater segment or that nearest the greatest side, and subtract it from it, will give the less. Then in each of the right-angled triangles formed by the perpendiculars, two sides are known, consequently all the other parts can be found.

Ea?ample.— Given AC, B C, and A D, Fig, 47, 667, 446, and 800 yards.

To Jind the Difference of the Segments.

A D and D B segment of the base. The logarithm of the base A B, 800 . -2.9030900

The side A C=667-hB 0=446=1113 =+3.0464952 „ AC=667-B C=446= 221 =+2.3443923

The two last terms added . . . . 5.3908875 Deduct the first term 2.9030900

Logarithm difference between segments) 4877075 of the base =307.46 ) -i

Then the half base= — =400=half base, and 400 + 153.86,

or half difference of the segment to 553.86= to A D, and 400—

153.73 246.14 to D B.. A D =553.86 + 246.14 DB 800.00

total base.

To Jind the Angle at C in Triangle A C B. Logarithm of the side A C 667 . . =-2.8241258

„ Ab=553.86 +2.7434000

radius zD . . . . 10.0000000

The two last terms added . . . 12.7434000

The first term deducted . v . . . 2.8241258

The logarithmic sine Z C 56° . . 9.9192742 90°- / C 56° / at A33° 51'.

Fractical Flake Trigonombtbt.

To find the Angle Gin the Triangle D G B. Logarithm of the side B C =446 -2.6493349

Bd=246.14 +2.3911822

radius

The two last terms added The first term deducted .

Logarithm sine of z 0=33'' 29"

10.0000000

12.3913315

2.6493349

9.7418473

90°- / C 33° / B 56° 31'.

To find the Angle A C B.

Add together the angle ACD + the/DCB 66° 6' + 83° SO' =89° 36', the angle required.

Pboblem 18.

To find the distance between two mine shafts on the opposite sides of a river from the observer.

Observation, — Measure the base line A B, Fig. 48, and set

Fio. 48.

up the transit instrument at the stations A and B. Measure the angles (B A D), (C A D), (A B D) and (D B C), then the

Practical Plane Trig0K0Mbtr7.

z C 180° - (A -f B), and in the triangle B A D, the Z D 180°- (A -hB).

Example. — Required the distance between the two pits D and C in the above figure froin the following data : the angles ABC =105° 20'; ABD, 60°40'; BAD, 96°4y; B A C, 45° 40 20', and the base A B 2400 yards. In the triangle A B C z C=180° "-(Ah-B)=28° 59' 40", and in the triangle BAD, D=180°- (A-hB)=22°30'40 -

To find A C in the Triangle ABC.

Logarithm sine of z C=28° 59' 40' -9.6853452

of AB2400 . . . sine of zB=74°40'

The two last terms added together The first term deducted . .

The side A C=4777

+ 3.3802112 +9.9842589

13.3644701 9.6853452

3.6791249

To find A D in the Triangle A D B.

Logarithm sinezD=22° 3040" . -9.5831445

The two last terms added The first term deducted

The side A D =5463.56 .

+3.3802112

+9.9404091

13.3206203

9.5831445

3.7374758

To find the Angles D and C in the Triangles ADC and B C D. (D + C) is=to 180-A=180°-51° 9'=128° 51'.

Logarithm of A C+ A 10292.05 -4.0124998

„ „ A C-A D=738.04 . +2.8680564

The tangent of J B + C=64° 25' 30" 10.3198802

13.1879366 The first term deducted . . . . 4.0124998

Audi (D + C) . . . 64 25 30

Hence the z D . . . 72 56 30

90°- z D 72° 56' 30" z at C 17 3' 80".

B

50 FBACnCAL FLANK TSIGONOHBTBT.

To find D C in triangle ADC. Logarithm 8me/D=72° 56' 80" . -9.9804415

„ „ of/AsSPg'. . +9.8914208

„ of the side AD =5515.14 +3.7415566

The two last terms added , . . 13.6329774 The first term deducted . . . 9.9804415

Therefore the side D C=4493 . . 3.6529359

.% The distance between the two mine shafts is 4493 that required.

When the figure in Problem 13 is employed to find the height of an object as D the same rule may be applied as that for the side D S, or B D. In this case the angle BCD must be the one to be squared/ and the quotient thus becomes the first term in the proportion ; the side B D is the second term, and the angle at D C B the third term. The fourth term will be the height of the object C D.

If in the figure Problem 13, it is required to find the distance A D, the angles B S A and BSD may be found by setting up an instrument at the station B, and observing the angles ; then the angles at A and D will be the complement of those found by observation. The side B S must be found according to the rule given for that purpose.

The side A D may now be foiind either by the natural tangent or by logarithms, as follows :

Example. — In figure Problem 13, given the side S B, and the z A B S and SB D=49° 48' and 79° 30 to find the side AD.

The natural tangent of the z A B S 49° 48'= 1,183340 Multiplied by the side S B 248

The side AS =293.468320

FBACTICAL FLABB iSKKfUOiinit. 51

To find the Side S D.

Natural tangent z S B D=79*' 80' . 595517 Multiplied by the side B S . . . 248

21S82068

The side S D =1838.088216

As 293.468320

Therefore side A D =1631.556536

To prove the preceding process by logarithms as follows : Logarithm radius 10.0000000

„ tangent 49 48' . . . 10.0731096

„ of the side S B 248 . . 2.3944517

The side A 298.46 2.4675618

Logarithm radius =10,0000000

„ tangent z 79 30' . . . =10.7320831

The logarithm of the side S B=248 . 2.8944517

The side S D =1838,08= 3.1264848

„ As 293.46

„ Ad =1681.54

The preceding trigonometrical problems are the principal ones employed in general surveying in order to determine the height and distance of any distant objects and for mining purposes they are sufficiently extensive. Cases may occur in practice requiring some modification ; but from what has been previously shown it will be evident that in every triangle three parts must be known, and at least one side must , be given in order to arrive at a conclusion either by calculation or construction.

B 2

52 Pbactical Plans Tbigonohetbt.

When it is required to find the height of an object, as from C D, or the distance A D, in Problem 13, the base A B should be taken more in proportion to the distance A D, so as to form well-conditioned triangles. The angle D in the triangle A B D is therefore an ill-conditioned one, arising from the non-proportion existing between the base A B and the distance AD. In all cases the angle at D in the triangle should never be less than about 25 if possible ; the results as obtained will then be more to be depended on, being free from errors, which are liable when small angles are employed.

The Vernier Scale As Applied To Surveying Instruments.

The vernier is a most beautiful modern contrivance for measuring small parts of space contained between the equidistant divisions of a graduated scale.* Such a scale ip made equal to a certain number of parts of the one to be subdivided depending on the degree of minuteness to which the subdivisions are intended to be carried, but it is divided into parts which in number are one more or one less than those on the primary scale taken for the length of the vernier in modem instruments ; the parts on the vernier are generally one more than are contained in the same space on the primary scale. The spaces between the degrees on the limb of the miners transit theodolite are divided into two parts; the short divisions represent, and are equal to 30', and the divisions on the vernier are each equal to 1' j and on the six-inch transit theodolite the spaces between the degrees are divided into three parts representing 20' each, and by the help of the vernier reads to 20''/

The subjoining diagrams are given to illustrate the method of reading off the limbs of the above-mentioned instruments.

Fig. 49 represents part Df the limb of the miners transit, but the divisions are made much larger than the reality for the sake of distinctness. The divisions on the limb are 30' each ; they are subdivided by the vernier to 1', the reading is 2° 36' ; and as a general rule, when reading this instrument, the observer takes an account of the degrees and parts of a degree passed over by the index of the vernier on the primary arc (or lower plate of the instrument), then for the fractional parts he must look along the vernier until he finds one of its divisions coincide with a division on the primary arc, then this part added to that previously read off wiU be the angle. Take figure 49 as an example; on examining

Troughton and Simms on Mathematical Instruments,

Thb Vbrnieb.

the vernier I find its arrow has passed firom the zero or 860 of the instmment to the left something more than and as the vernier arrow is in advance of the half degree the reading is not the correct angle, but I find that the 6th division the arrow towards the left coincides with a division on the primary arc, therefore it will be 6' to be added to the angle previously found ; the reading will now be 2° -f 6' 2® 86', thie true angle pointed out. And if the vernier precedes, or does not come up to the half degree on the primary scale, the minutes must then be taken from the vernier altogether, without reference to the primary arc, except for the coincidence of one of its divisions, that is, the fractional part being under 30' cannot be read from the primary scale. If the student has an instrument of the above description before him, he will observe that in reading angles, when the vernier arrow has passed any number of whole degrees, its first division may be opposite a division on the lower

Fio. 40.

plate ; in that case the additional part to be added would be 1' ; by turning the tangent screw the vernier may be made to pass successively 1' to 29', which is its greatest reading under half a degree on the primary arc. If then it is made to move 1' more the reading will be confined to the lower limb of the transit or primary arc, and also in reading from the half degree to the left ; the greatest reading of the vernier under a whole degree will also be 29' + SC, the half degree on the lower plate 59', and by moving it exactly 1' more we shall make up another whole degree, &c.

Figure 50 represents part of the limb of the six-inch transit theodolite, the divisions of which are drawn much larger than the reality for the purpose of rendering it more intelligible, and are each equal to 20', and are subdivided by the vernier into 20". A general rule for reading with the vernier may be expressed as follows : — First observe the number of parts on the primary scale

The tebttier. 85

(or lover plate of the transit) that is equal in length to the same number plus one on the vernier; this last number will he the denominator of a fraction, the numerator of which will he equal to unity, which expresses the subdivision of the parts on the lower plate of the transit.

Take the figure number 50 as an example. , The lower plate is divided into 360°, each degree being subdivided by shorter lines into 20', and for the length of the vernier a space equal in length to 59 of these subdivisions is taken and divided in 60 parts ; therefore each of these parts on the vernier ia smaller than those on the lower plate by ot 20', or equal to 20". In reading the same rule applies as that for the other instrument spoken of before, only in this instrument there are three quantities instead

3X< Primary Are or Circle.

of two, viz., degrees, minutes, and seconds ; the degrees and parts of a degree arc first read off from the primary circle, or lower plate of the theodolite, and the remaining minutes and seconds (if any) from the vernier. Taking the above figure as an example, I find the vernier arrow has passed from the zero on the lower plate to 3°, and something more, which cannot be estimated on the primary circle. By running the eye along the vernier scale towards the left, I find that there are two minutes and something more to be added to that previously read oflF for the true angle, and as we said before that every short line between the large ones is twenty seconds each, the second subdivision coincides with a division on the primary circle; the quantity therefore to be added b 2' 40' + 3° 0' 3° 2' 40", the angle

56 Thb Yebnibr.

pointed out by the vernier. The student wiQ eacperience a little more difficulty in reading with this kind of vernier than he would with the one previously explained but after a little examination and practice it may be as easily read as the one for single minutes. If the student should be lucky enough to have an instrument of the above description before him he will do well to make the vernier pass through a whole degree, noting at the same time the different readings ; thus the first division on the vernier coincides with one on the primary arc, then the angle will be 20"; the third also coincides, the angle will be equal to 1'; the 14th division coincides, the angle will be equal to 4' 20' ; the 31st coincides, the angle is equal W 20" ; the 59tli coincides equal to 19' 59", the 60th coincides equal 20. The vernier will now have passed through one of the divisions on the primary arc, and so on through a whole degree.

Subterraneous Or Mining

Surveying.

Section I.

Underground surveying has often been considered of less importance than that of the surface, where, from the circumstance of its aflfording greater facility than is oflfered underground, it is presumed that no approach to accuracy can be obtained except when the best instruments are applied ; and if this is established, how does it come to pass that underground purveys may be performed with less attention and with instruments of an inferior description ? Most of our eminent civil engineers of the present day bring to their aid (in setting out working tunnels and shafts for railways) the transit instrument of best construction. In mines it is very rard' to find employed even the ordinary theodolite ; in fact, I know of but few instances where it is used in this district, and then only occasionally. I have been at much trouble and expense in having produced an instrument that shall be alike suitable for surface and underground surveys, and have no hesitation in recommending it as the best yet produced for the latter purpose, for which it was principally constructed.

DESCRIPTION OF THE NEW MINER's TRANSIT THEODOLITE.

The miner's transit theodolite, as an angular instrument, may be considered as the most important one applicable to underground surveys. The above description of instrument is only made by Messrs. John Archbutt and Sons, mathematical instrument makers, 20, Bridge-road, London, to whom I had given directions as to its construction. The instrument diflfers from an ordinary transit theodolite in the arrangement of the upper plate, which is made to project over the lower plate in the places

68 Mining Surveying,

where tlie upright supports are attached to it in order to allow of greater space on the plate than is usual to receive a large compass. AVhcn the limb of the instrument is made five inches in diameter, it will carry a 4J-inch compass differing half an inch only from the size of its own limb ; whereas in the ordinarily constructed theodolite the same sized limb admits of a compass 2; inches only, which is quite useless except for ornament. Another advantage in the new over the old construction is, that the compass of the former is quite free and open, and can be read quite round the circle, but in the latter it is so darkened up with the appendages of the upper plate, as to render it impossible to be read at all. Another and more valuable addition to the transit is the application of a particular diagonal eye-piece, which when screwed into the telescope will allow it to be. pointed directly vertical or in the zenith, and can then in that position be as easUy read as when horizontal.

This superiority over all other instruments hitherto applied to mining purposes will be appreciated by those who find it a frequent and necessary operation to connect underground with surface surveys, which can be easily performed by this instrument without the assistance of the compass. The instrument will also be found superior in traversing, as its construction allows the telescope to revolve, and consequently back and fore observations may be taken in the same manner as by the common dial, the readings in all cases to be taken from the limb. When it is required to take a bearing with this instrument per compass, the vernier arrow should be first brought to zero of the lower limb, and the whole body turned round until the needle points to its own zero ; then after releasing the upper plate, and bisecting the object with the cross hairs of the telescope, the bearing will be obtained to thirty seconds, from the theodolite limb instead of from the compass. I shall now explain the use and adjustment of the different parts, with a sketch of the instrument.

This instrument, as represented on the subjoining page, is a mechanical contrivance of a number of parts, the most important consisting in two horizontal circular plates, A and B, called the lower and upper limb. The upper or vernier plate. A, is made to move upon the lower, both having a free horizontal motion communicated to them by the vertical axis C. This axis is made in two parts, external and internal; the former is made fast to the graduated circle B, and the latter to the vernier circle A. They are

opno7 -wnb pJPJP*e ,S IPssrrjf imiio '901 oO npPf

; 'Hi3anti avoa looiaa M3iSNiNj.S3Moz'SkOS unaHoyw NHor xa ahno aaanxovjnijw

SIHOIS NlVld 3dO0S3T31 Ay V1N3 W31ddnS HUM 3inoa03Hl J.1S_NLVHI SXHNrW

Mining Surveying. 59

made conical in form and are nicely fitted and ground into each other, which gives an easy and steady motion ; the external centre also fits into a ball at H, and the parts are held together by means of a screw at the bottom of the internal axis.

The edge of the diameter of the lower plate is chamfered off and covered with silver to receive the graduations; the upper plate is made to project over the lower one, where the upright supports are attached (and also projects downwards like a conical concentric ring to protect the graduations on lower plate,) to it as at I. This arrangement gives a greater space between the telescope uprights, to admit a larger compass than is usual. On opposite sides of this plate, or 180 apart, a ishort space is also chamfered and covered with silver, which forms the vernier. The lower limb is graduated to 30' of a degree, and subdivided to minutes and seconds, as explained when speaking of the vernier.

There is also a second or supplementary telescope attached to the main centre of the theodolite and under the lower plate, and has a free horizontal motion qidte round the centre ; it has also a vertical motion given to it of about 20® of elevation or depression. The principal use of this second telescope being to determine whether the lower plate has in the least moved fi*om the place it was set when the first observation was made ; and we may at all times find out if any displacement has taken place, while measuring any number of angles with the upper work and telescope.

The plan of using the second telescope is, after the theodolite is set up and levelled, and the upper telescope set on the back station — to point it to either the back station or other object, and clamp it fast ; then measure the required angle with the upper telescope; now, by looking t&rough the lower telescope, we can see if any deviation has taken place, to affect the angles as observed jfrom the upper plate.

The parallel plates D E are held together by a ball and socket joint at H, and are set parallel to each other by four conjugate or milled-headed screws, of which three only arie seen in the diagram as a a!' ; they turn in sockets bored and bossed into the upper parallel plate ; their heads press against the upper side of the lower plate; they are set in pairs opposite each other, and of course act in contrary directions, and by this means the instrument is brought level for use.

Underneath the parallel plates there is a screw called the female screw, joined to the staff head, and connected by brass joints to three legs forming the tripod stand of the instrument.

60 MINING 8UBy£TING.

There is a clamp fixed at o, round the exterior axis C, and by tightening its screw S, the lower horizontal plate can be fixed in any position and the bisection of any object with the cross hairs of the telescope is made by turning the tangent screw G. In the same manner the upper, plate can be fixed to the lower in any position by its clamp and tangent screws b and c.

Upon the ends of the projections of the vernier plates outside the upright supports a spirit level d is fixed and at right angles to it and screwed to the back upright is another spirit level, as at €j with their proper adjusting screws ; their use is to bring the instrument to a level position.

There is also a large 4-inch compass fixed on top of the upper plate at K, and being open and tree firom adjacent appendages can be read quite round the circle.

The upright supports L L carry the pivots of the horizontal transit axis of the vertical circle P.* The telescope Q is screwed into the axis, and forms one piece, which revolves vertically altogether with the vertical circle. The vertical circle has three verniers at h h' h"y reading in opposite directions, connected by a brass bar and made to move on the transit axis as a centre. To this bar is fixed a spirit level, m, to bring the vernier to a level position; there are two screws not seen in the diagram, for bringing the axis bubble or level to a level position, and consequently the verniers also ; the circle has also its clamp and tangent screw, as shown at o and p. The vertical circle is inlaid with silver, and divided down to SCK' by the help of its verniers, which are read through the microscopes at 10, 10'. The other side shows the difference between the hypothenuse and base of a right-angled triangle up to about 50° or 60°. There are placed in the focus of the eye-glass two lines, one horizontal and the other perpendicular, by which the distant object is nicely bisected ; the screws adjusting the diaphragm or wire plate are shown at i V There is also a focus screw for pushing out the object glass of the telescope, to show an object distinctly. There are also four eye-pieces, an astronomical inverted and also an erect eye-piece, for long distances ; there is also a diagonal, and an extra one to be screwed into the diagonal one at D, or set for

♦ The theodolite is now made with two telescopes, the supplementary one being placed under the lower plate, and has a free motion round the main centre, its principal arc being a check on all angles measured with the instrument.

Mining Surveying.

sighting directly vertical, or from the bottom to the top of a shaft. There are also fixed to the top of the telescope at 20' 20'' a pair of plain hinged jointed sights, similar to those on the common dial ; their use is to enable persons to take short sights where the telescope would not command. The whole are packed in a mahogany case quite free from injury.

Adjustments Of The Miner'S Transit Theodolite.

Draw out the tube of the eye-piece until the cross hairs appear well defined, and to be free from the effects of parallax* that may arise from optical displacement of the cross hairs when the eye is placed in a lateral position. If when the telescope of the instrument is pointed to any distant object its image remains fixed when the eye is moved to the right or left, or in a lateral position out of the optical axis of the telescope, no parallax exists j on the contrary, if the image of the distant object does not remain steady, the tube must be drawn out, more or less, until the required steadiness of the image take place.

To adjmt the Line of CoUimation.

Set up the theodolite at B, Fig. 51, as a station, and after careftdly levelling the instrument, set the index of the upper plate to zero on the lower limb ; direct the telescope to a distant object A ; fix the cross hairs on the object with the body tangent; now release the clamp of the upper plate, and bisect the object C with the tangent of that plate; read off the angle, say 60° 82' 20", leaving the upper part of the instrument clamped ,at the last reading; revolve the telescope vertically in the direction of D; turn the whole body of the instrument round in the direction of C ; clamp and bisect the object C with the body tangent ; release the upper plate, and turn it round in the direction of A ; and again clamp and bisect with the upper tangent. If

Parallax is a term in general use to denote the difference between the true and apparent place of any objects.

d

the index of the yemier points exactly to sero on the lower plate the line of eollimation is correct but if it does not half the difference is the amount of error in arc.

The line of eollimation may also be adjusted as follows : place the instrument at last figure, and level it; damp all fiist; direct the cross hairs by means of the body tangent-screw, to an object D; revolve the telescope vertically with great care, so as not to disturb the instrument in position ; fix the cross hairs on the object C, then release the lower clamp, and turn the body of the instrument quite round ; clamp it again, and make the cross hairs bisect D ; then again revolve the telescope vertically, and if the cross hairs strike C as before, it is in adjustment, but if not, fix another object, a, in the place where it does strike; now, by altering the screw placed in the wire-plate, the cross hairs must be made to strike half way between C and a ; a few more trials will generally be needed before the adjustment is perfect.

Another method is sometimes employed to perfect this adjust ment. The instrument is set up as usual at B, and the telescope directed to C, the cross hairs of which are made to bisect the object nicely. The telescope, circle, axis, &c.> must now be carefully lifted out of its bearings, or Y, Y, and replaced with the axis reversed ; revolve vertically, and point the telescope again to the object C; and if it be still bisected, the adjustment for the line of eollimation is correct. If, on the contrary, it is not bisected, move the wire one-half the error by timing the screws, holding the wire-plate (near the eye end of the telescope), and the other half by the body tangent-screw, which wiQ complete the adjustment if half the difference has been correctly estimated. If after a second trial, it is not exactly in adjustment, the same operation must be gone through again until it is quite perfect. Or the adjustment may be performed as follows : measure some horizontal angle, say, 128® 89' 20'', revolve the telescope vertically, let go the lower screws, and measure the same angle again in this reversed position ; then if the angle is the same, the adjustment is correct ; but if one angle is larger than the other, take a mean,

39' 20" + 127® 42' 40" thus : =128® 11' 30". The vernier

should now be set to this angle without moving the instrument the least in position, and the vertical hair moved by giving motion to the diaphragm screws until it cuts through the object. Then

Minikg Surveying. 63

if the angle is measured in reversed position it will be the same and consequently, the line of collimation correct.

To adjust the Azimuthal Axis.

Bring the bubble of the axis level to the centre of its run by the screw attached to the arm ; then release the upper plate, and revolve it half round ; then if the bubble remain in the centre it is correct in this direction; but if otherwise, half the diflTerence must be corrected by the parallel plate-screws, over which the telescope lies, and half by elevating or depressing the axis level by the milled head adjusting screw attached to the bar carrying the verniers and level ; after having done which, turn the upper plate backward 90°, and the telescope wiU be over the other pair of parallel screws, and by their motion set it horizontal, after having levelled the instrument with the axis level. The other bubbles fixed on the vernier plate and frame, may be brought to their runs by giving motion to the screws which fasten them in their places.

To adjust the Vertical Circle.

After the preceding operation has been gone through, level the

bubble to vertical circle with the conjugate screws, and point the

telescope to some weU-defined distant object— the more distant

the better — and make the horizontal wire cut the object ; read off

the vertical circle, say, vernier A =24° 26', and vernier B=24i° 27',

and =24° 26' 80" mean angle. Reverse the in-

2

strument: attend to the axis level; revive the telescope vertically, and again point the telescope to the same distant object ; read off the angle from vertical circle as before, say, vernier

28° 40' +28° 41'

28° 40', and B=28° 41'; then =26° 32' 30".

24° 26'- 30" +26° 32' 30" ,

Then =25° 29' 30". Set the verniers

to this mean angle, taking care all through the operation that

the axis bubble is perfect, and move the vertical screws in the

wire plate until' the horizontal wire cuts nicely through the

distant object etaployed for the purpose ; then if the operation

has been accurately gone through, the vertical angle will be found

to be the same in both positions. Sometimes, however, a few

64 Mining Suryetinq.

trials will be required to perfect this important adjustment for if it is neglected for a long time, any angles taken with the circle cannot possibly give accurate results.

TO MEASURE HORIZONTAL ANGLES WITH THE MINER's

Transit Theodolite.

The adjustments, as previously explained, having beau carefully tested and rectified, the instrument is set over a driven into the ground, to form the station, &om whence the angles are to be taken by means of a plummet suspended to a hook screwed into the lower side of the table plate of the staflF head. The instrument is then set level by the parallel plate screws by bringing the telescope and vertical circle clamped to zero over each pair in succession.

There are several methods of measuring angles with this instrument ; the first of which is to clamp the upper plate at zero and the telescope with the body tangent-screw, to the left-hand station ; release the upper plate, and turn it round in the direction of the right-hand station ; clamp and bisect with the slowmotion screw attached to the upper plate : the vernier will now point out the angle in degrees and minutes ; the minutes and seconds, if any, should also be read from the opposite vernier, the average of the two to be used as the mean correct angle. A second method of observing angles is to clamp the lower horizontal plate in any position ; direct the telescope to one of the stations ; clamp the upper plate, and turn the tangent screw attached until the bisection of the hairs and object takes place ; read off the .two verniers which are marked A and B, entering down the degrees, minutes, and seconds iBrom vernier A, and the minutes and seconds from vernier B, taking the mean of the two readings respectively.

Then release the upper plate, and turn it round until the telescope is pointed to the second station : clamp the upper plate, and make the bisection with its tangent screw as before ; again read off the verniers, taking their mean ; the difference between this and first mean reading will be the angle required : —

Vernier A. Vernier B. Mean angle.

The first reading =164° 42' 40'' . 43' 6" . 164° 42' 50"

The second reading =246 82 20 .32 40 . 246 32 30 Correct angle =diff 81 49 40

The reason of reading off the two verniers placed opposite to

Mining Surveying. 65

each other is to counteract any error that may arise from inaccurate centering and graduating the plates which is found in practice to be ahnost an impossibility.

Another, and more perfect method of observing angles (when great accuracy is required), is to set up the instrument and measure the angles as in the first method, noting down the reading ; then release the body clamp screw, and turn the whole instrument round in the direction of the first station, leaving the upper plate clamped at the last reading ; make the bisection with the body tangent, and release the upper plate ; turn it round in the direction of the second station; make the bisection with the upper tangent, and read off as before, entering down the reading each time ; repeat the operation as often as thought necessary, always remembering to set the last reading on the fin station ; then either of the readings divided by the number of repetitions that produce it wiU be the correct angle free from eccentricity, &c. : —

Second reading= 143° 39' 0"-r2 7r 49' 30'' Third „ =215 28 20 -f 3 71 49 27 Fourth „ =287 18 -t-4=71 49 30 Fifth „ =359 7 30 -r5 71 49 30

The above example shows that either of the readings would be suiBSciently accurate, and is only given to explam the method; others, more simple, would be found convenient in practice.

Another method, somewhat similar to the above, is in general use amongst the best surveyors. The instrument is set up at the commencement of the traverse, and the first angle measured by setting the zero on the back, or first station ; and at every successive station the proceeding forward readings are set on the back station ; the reading of the horizontal limb for the forward angles shows the direction of each station with reference to the first line on which the telescope of the instrument was set, and in that case becomes the first meridian ; the traverse may therefore be calculated by trigonometry, the registered angle and measured distance become one of the angles and sides of a right-angled triangle.

The last method I shall mention is that which, fi*om its simplicity, should be employed in mining operations : it can only, however, be practised with an instrument carrying a revolving telescope, and. is quite new as far as I know : clamp the upper plate to the lower, and bring the vernier arrow to zero ; release

F

66 Miniko Surveyiko.

the body of the transit and torn it round; bisect the back station ninth the telescope wires by the body tangent screw then revolve the telescope vertically, and release the upper plate ; turn it round until the telescope is in the direction of the forward station ; clamp and bisect with the upper clamp and tangent screw ; then if the forward line inclines to the right of the back line, the vernier will have moved forward in the direction of the graduations, but if to the left, it will have moved backwards or in a contrary direction to the graduations.

In the former case the angles, as read off, become minus (or to be deducted), and in the latter plus (or to be added). When the successive angles are reduced to plotting angles firom one station, the angles in the former case require no change previous to their being employed in the reduction ; but in the latter the angles, as read off, must in all cases be deducted 360 ; the difference is that to be employed in the reduction.

To Measure Vertical Angles.

Vertical angles are those that are measured in a vertical plane, taken from an imaginary line passing through the transit axis carrying the telescope and circle, and parallel to the senidble horizon, and are said to be positive or negative as they are above or below this line. Bring the bubble of the spirit level attached to the verniers of the vertical circle to the centre of its run, by giving motion to the milled-head screws working in the vernier arm bar; or if the levels on the instrument are all parallel (which they should be) they may be all levelled at the same time by the parallel plate screws ; then direct the telescope to the object whose altitude or depression is required, making the bisection with the tangent screw to vertical circle, the angle may then be obtained from either of the verniers.

Traversing Underground.

The term traverse is used and applied to imdergroimd surveys to denote a continuous line in a zigzag direction ; and it is tlie business of the surveyor to determine the amount of inclination of the lines composing the traverse with reference to one another, and also to the first line continued as a common meridian to the whole ; and if he intends carrying out his surveys with all possible accuracy he must tutor his assistants to their work to do this. He should instruct them to measure a surface line (whose exact length he knows) backwards and forwards as many times as he should think necessary, or until they bring the measure three or four times the same ; and also enforce upon them the necessity of the greatest attention to the accurate measurement of the lines composing a survey; or if he is not sufficiently confident in the ability of his chainmen, he may proceed to go through all the operations himself separately.

To Peg Out The Lines, Etc,

Suspend a fine copper wire or chain, with a plumb-bob attached, down the centre of the shaft A, Fig. 52, or, in absence of these, make a countersink hole in the cast-iron stricker-plate in the centre of the shaft ; then place a lamp directly over this, and advancing along the line to the next turn in the road, drive into the ground ftn iron peg as at B, the first station ; proceed along the line, and at every bend drive an iron peg to mark the successive stations ; it will also be a good precaution to mark the sides of the rock with white paint, to point out the stations more readily. The surveyor may now return to the commencement, and after setting the instrument exactly over the centre of the station B, level it, and clamp the upper plate to zero of the lower plate, and release the body damp; turn the whole instrument round in the direction of the lamp A, in the centre of the shaft, and bisect it, with the cross hairs of the upper telescope, by the body tangent; release the upper plate, and turn it round till the telescope bears on the second or forward station C (or the flame of the lamp placed over the iron peg previously driven) ; clamp and make the bisec-

F 2

68 Mining Subybting.

tion as before with the upper tangent screw ; read off the angle, say 140 If now the same operation is gone through of bringing the plates to zero, and sighting to the forward station C as a back station, and also on A as a forward station, the second reading plus the first should be exactly 360°, say the second reading, 219° 40' + 140° 20=360°.* The instrument may now be removed to the second station C, and again set up and levelled, thus bringing the plates again to zero, and on the lamp brought forward to the first station B ; the upper plate is then released, and directed to the lamp placed over the third station D ; the bisection is now made, and the angle read off as before. The instrument is then taken forward to the fourth station E, and the same operation gone through, the theodolite and lamps changing places alternately from station to station, until the whole is completed.

The surveyor may then return and proceed to measure the distances between the stations ; the lamps are again to be placed on the forward stations in turns, 3nd the surveyor, taking the hind part of the chain, directing the forechain man in line, and towards the forward station, entering down from time to time the measure of the different lengths (as also any particulars or remarks he may find necessary) until the whole are measured. He must be prepared with a survey-book ruled into as many columns as will be required, and headed — 1st, No. of draffcs or hypothenuses ; 2nd, angles of inclination ; 3rd, measured distance in links ; 4th, horizontal angles; 5th, remarks, &c. The surveyor being now provided with proper tutored assistants, and a book of the form stated above, he will commence a survey of the heading ABODE, &c., to L. From a chain suspended down the shaft at A set up the theodolite at B, and after levelling and clamping the plates at zero, bisect the chain with the cross hairs of the telescope ; release the upper plate, and direct the telescope to the forward station C, entering down or zero in the column headed horizontal angles, and the length of the line A B in its proper column opposite it ; now read off the angle pointed out on the instrument, and enter it down in the column appointed for the horizontal angles. His assistants will then measure the line B C, which he will enter down in its own column, also any remarks

If the surveyor is certain of his first reading, and has brought the second or lower telescope into use as a cheek, there may be no occasion to measure the supplementary angle*

MDnNG SUBTBTINO.

69'

that may occur at the time ; remove the instrument to the station C, level, clamp at zero, and bisect the back station B ; again release the upper plate, and point the wires to the station D, read the angle off the limb, and enter it down in its column, while the chainmen are measuring the line C D ; they will now

FloiUdfrom the Flam ofLatUude.

call out the measurement, which must be entered down as before. It is to be remembered that, after reading off the horizontal angles, the vertical ones are also to be read off, if any, from the Tertical circle, and entered down as elevation or depression accordingly. Proceeding as above, we arrive at the conclusion of the survey, which will stand in the survey-book as follows : —

SURVEY BOOK. No. 1.

Aogleaoflndinotioo.

Anglei.

Bemwlui.

n

o-o-

0" 0'

OORiro

/ Opposite No. 1, croBscut

140° 20'

f Opposite to a feult, I going up.

K

y

/ Ojipoaite No. 2, crosscut.

Ii)

Minihg Stjrvetino.

If it is required to draw a plan of the trarene tram the preceding snrvey-book it may be accomplished with the assistance of a circular protractor and chain-scale the former to be divided into minntes, as on the theodolite. The horizontal angles are however to be reduced to angles firom the first line; as a meridian to the whole previous to their being plotted.

To Beduce Horizontal Angles To Plottinq Angles

From One Station.

Bule 1st. If the first line in any survey is assumed as a meridian, and going southward, and the first observed angle from it is less than 180% it inclines to the east, but if more than 180 to the west of the first line ; in the former the angle as observed is to be called No. 1, and in the latter, the difference between it and 180° is to be called No. 1 plotting angle ; but if the first line, assumed as a meridian, is going northward, and the observed angles are less than 180°, it inclines to the west, and if more to the east of the first line, the angle in the former, and its difference firom 180° in the latter, to be called No. 1 plotting angle.

Bule 2nd. To No. 1 plotting angle add the next observed horizontal angle; if the sum exceeds 180°, take that amount fi*om it, the remainder will be No. 2 plotting angle ; to No. 2 add the next horizontal angle, and proceed as before for the plotting angles. Sometimes, however, the sum will exceed 860°, and in that case a whole circle, or 360°, must be deducted it for the plotting angle, taking the horizontal angles in surveybook No. 1 as an example. The first angle is 140° 20, and tp the eastward of the first meridian ; and the reduction to angles firom one station are explained as follows : —

140° 20' 0" No. 1 Plotting Angle.

No. -No. No.

)1

97'*

40' 0"

No. 4 Plotting Angle.

No. 5

No. 6

No. 7

Mining Surveying.

5r 0' 0" No. 7 Plotting Angle.

N"o. 9

tf

80° 0' 0' No. 9 Plotting Angle.

84 20 =No.l0

n

The reduced angles should now be collected (from amongst the preceding work) into a tabular form ready to be plotted on the drawings, as below —

Pi

otto

g Angles.

No. 1

140° 20'

0''

,, 2

., 3

„ 4

:i=

Plotting Angles.

No. 6 66° 20

O''

„ 7 51

„ 8 63 20

„ 9 80

„ 10 84 20

To Construct The Plan.

Draw a line through any convenient part of the paper intended to receive the plan and make a fine pencil mark on it ; apply the centre of the protractor to this mark and the notches marking the zero and 180® on the instrument on the same line, (to prove if the protractor has been properly set, bring the vernier to zero, and unfold the arms of the protractor, carrying small fine needle points at each end ; if these points strike exactly in the line it is properly set) ; the angles No, 1, 2, 3, 4, &c., to 10, may now be marked ofl', taking care to number them in the drawing in succession as they were taken from the table ; remove the protractor, and apply a chain scale; mark off 400 links on the back, or zero line, A B, Fig. 52 ; set a rolling parallel ruler to angle No. 1, and the centre it was marked off from ; run it to statioli B, draw the line in this direction, and lay off with the same scale 700 links ; bring up also angle No. 2 to the station C, and draw also another line in this direction ; mark off on it the distance 300 links; apply the parallel ruler to each angle in succession, bringing them to the last station arrived at, remembering to make every line its proper length before bringing up another angle ; proceed in this manner until you arrive at the last station L ; lines may now be drawn parallel to the centre

72 Mdidio Sukvjciibo.

ones, which will represent the sides ct the heading; and the phin will be completed. The plan mar also be constmcted from data calculated from the tlemcLt griven in the snircT-book without the u:se of the protractor. The detail of which method I now intend to explain ; and the first step is that of reducing the ohsened horixontal angles contained in sunrer-book No. 1 to angles finom the first hue A B as a meridian, or their oomplementSy £h:>m the east line B N.

Rule 1st. If the angles are leas than 180 deduct it from 180°, and call the difference plus +; and if more than 180° deduct 180° from it, and call the difference minus — .

Rule 2nd. Add together the angles marked plus + and deduct those marked minus the remainder will be the angles to be employed, if the horixontal angle were taken from the east line or plane of latitude, or their complements;, if taken from the assumed meridian.

Taking again the obserred angles in survey-book No. 1 as an example, the reduction will be as follows : —

No. 1 (K - 140° 20' 39° 40 +

2 180 - 160 10 =19 50 +

3 180 - 170 40 9 20 +

4 180 - 166 30 =13 30 +

5 180 - 172 20 7 40 +

6 180 - 156 20 =23 40 +

7 180 - 164 40 =15 20 +

8 192 20 - 180 12 20 -

9 196 40 - 180 16 40 - 10 184 20 - 180 4 20 -

The angles marked + and — may now be collected and disposed in the following manner : —

Complements to the first Series*

19 50 + 15 20 +

2 59 30 7 39 30 30

9 20 + 12 20-

3 68 50 „ 8 26 40 21 10

13 30 + 16 40-

4 82 20 „ 9 10 7 40

7 40 + 4 20 -

5 90 ,,10= 5 40 '000

The angles No, 1, 2, 8, &c., to 10, are those to be employed

k

Mining Surveying. 73

in the calculation and are separated into two series the first as far as No. 5, which last is in an east direction; and it would be much easier in the plotting to assume this as a new zero line. The data fourd by calculation from the first series may then be called a positive, and that 'from the second a negative quantity. The sines of the complements of angles of the former, nmltipHed by the measured distances, will give the perpendicular, and the cosine the base ; in the latter, the sines of the angles, multiplied by the measured distance in links, will give* the perpendicular, and the cosines of the angles, multiplied by the measured distance, the base.

To find the difierent quantities it will be necessary to enter the particulars in a ruled form similar to the Table on the next page.

The form is ruled into eight columns, the first contains the number of drafts, or hypothenuses, the second and third the reduced angles of direction with reference to the plane of the meridian ; the fourth, the complements of the angles of direction, or the angles of direction with reference to the plane of latitude ; the fifth contains the sines of the complements of angles of direction x by the measured distance, and producing the perpendiculars ; the sixth, the cosines of the complements of the angles of direction x by the measured distance, or hypothenuse, producing the bases ; the seventh, the distance in links of each station fim the plane of latitude ; and the eighth, the total bases or distance of each station from the plane of the meridian. The calculations having been completed in the preceding manner, are entered as explained in the columns seven and eight, which wiQ contain the projections of perpendiculars and bases requisite for constructing the plan ; to do which, draw the line A B, Fig. 52, passing through the centre of the shaft and first station, and mark off on it 400 links ; continue this line to an indefinite length, and call it the assumed meridian ; draw also another line B N through the first station, and at right angles to the former, this line wiU then be the east, or plane of latitude ; apply any chain scale, say a three-chain scale, to the line B N, with its zero at the first station B, and mark off the different bases from column eight as 416.82, 705.30, &c., to the last 2973.98. Now apply the parallel ruler to the meridian A B, and run it to the several pencil marks left on the line B N, and draw perpendiculars ; then with the same scale mark off on these perpendiculars the several distances from column seven, as 528.82, 691.09, to the last

HININQ 8UBTRTZHQ. OFFICE FOBIC, No. 1.

AnglM of direction

irith reftmic* to plum

fll

Ijlneiafcoiiiplc otdiitic,n=

nitaneeof each UUdn

DMunrf rithnfoma

+

+

+

+

+

+

39° 40'

fiO'20'

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152.26 +

258.48 +

£38.8997000

449.8240700

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.S036S84

Soi. 09 144,43 +

373.01 +

lS-2.n61&-ZV{)

25S.4Ss7S00

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373.0136000

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420.00 +

6;04Soo

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40.6933600

348.8713500

S

4 2U. 00000

RTPotbenoux

228,57 +

+

+

+

268.08 +

0° 0'

23' 40'

0- 0'

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200307500

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84.3680300

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267G.4G 298.68 +

S

134.8397600

268.0397800

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4S5.02

19fi96l£60

38.2025820

216.6577100

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£9.0220300

298.5338600

Mining Surveying. 75

495.02 ; draw ink lines from point to point in the perpendiculars thus formed, and they will represent a plan of the traverse required. The student wiQ perceive that I have employed the signs + plus, and— *minus9 in a different sense to what they have sometimes been made to represent ; thus, in the reduction of the angles by. the second method the first angle =39° 40' is made positive, as also the quantity obtained from it by calculation ; the second reduced angle =19° 30' is also a positive angle, and therefore to be added to the preceding one, 39° 40' + 19° 30=59° 10'. These angles, as reduced, are entered in their respective columns in Office Form No. 1, the sines and cosines corresponding to the angles are also entered in the columns five and six, and the calculation made as we proceed. Thus the perpendicular from the first angle is =538.83 and base 446.82; that for the second angle 152.26 and 255.48 ; and as they are all plus, and to be added together, they are entered in columns seven and eight; thus 538.83 + 152.26=691.09, and 446.82 + 258.48=705.30, and are the lengths of the lines required to determine the stations from the plane of latitude and the meridian. In the second series, commencing at No. 6, the signs + changed to — sho that all quantities calculated from the angles thus entered are to be deducted the total lengths of the last perpendiculars in the first series; thus the angle 23° 40'— produces 100.35— for its perpendicular, and 882.21-100.35 781.86, the length of the next perpendicular determining the station H from the line B N, the same process to be continued until a change in the signs takes place. Thus it appears that the signs + plus and — minus mean that the data found by calculation from the angles so marked are to be added to or subtracted from the last plotted perpendicular, so as to determine the next station, when the plottings are made from the first meridian or plane of latitude.

To Survey, Calculate, And Construct The Plan Of A

Circuitous Underground Heading.

The surtey of the heading, commencing from the shaft Fig. 53, and continued to the stations No. 1, 2, 3, 4, &;c., to 27 is taken exactly in the same manner as described when taking that of Fig. 52. The instrument is set up at the station and its plates dkunped to zero; the cross hair is then made to

MIHI90 SUBYXmOo

biflect a lamp placed in centre of the abaft at A; thenpper plate is then released, and the telescope turned round towards the forward lamp placed oyer the station 2, the croas hairs of whidi are made to Insect it ; the angle is then read off and entered, and the instrument carried forward to the station 2 ; the same opera-

tion is then gone through at each successive station, until we arrive at the last one, No. 27 ; the survey will then close on the lamp in centre of the shaft, and placed in the same relative position it had at the commencement of the survey. The entries in the survey-book having been made according to that of No. 1, will have the following tabular form : —

Mining Surveying.

SUEVEY BOOK, No. 2.

No. of Hypotbenuses.

Angles of Inclination.

Measnred Distance in Links.

Observed

Horixontal

Angles.

Bemarks.

Elevation.

Depression.

Zero. 32'

r Zero of instrament on I lamp in centre of shaft.

.

I .

.

.

. .

. .

.

1

1 .

( Turn suddenly to the I right.

U

Turn to the right.

a

1 .

.

1

.

2r

a

Turn to the right. ( Return to the shaft ( commenced at.

There are in this survey 28 measured distances and 27 observed angles being always one less than the number of sides composing it. The survey was taken on an horizontal plane consequently the measured distances require no reduction to that plane. The student will observe that when the direction of the hues in a survey takes a sudden change either right or left, it must be remarked in the proper column, and a line drawn through the column of angles, and underneath the last angle in the old direction, as at 207° 10', turn to the right, &c. The present survey being a circuitous one, is divided into four series — the

MINUra 8UBVBTIHa,

first and third are in the general direction of the meridian and the second and fourth in that of the planes of latitude. The rules for reducing the observed horizontal angles to those planes are the same as employed for Fig. 52 ; the student is therefore referred to that part of the work where the process is fiilly explained. The following are the angles as reduced to the preceding planes from survey-book No. 2 : —

No.

24:

Angles as reduced to Planes of Meridiam

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7 52 +

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8 25 + . . . . . 10 5 + . 20 35 + . 26 15 + . 29 45 + . 11 25 + . 12 15 + . 22 55 + . 4 35 + . 21 55 -

Aisles as redqeed to Planes of Ijatitade.

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It will be observed that when the reduction is made to the meridian and an angle in that reduction amounts to more than 45, it should then be referred to the plane of latitude as at No. 11, the angle 67'' 25'+ ; this angle, deducted from 90% will give the angle from the plane of latitude. Again, when arriving at No. 16, the angle amounts to 98® 25', and 90® deducted from it, wiU give 8® 25' -f-, the angle of direction with reference to the plane of the meridian, and so on at each turn for the other series. The preceding reduced angles must now be entered in

MININa SURYBTINQ. 79

their proper columns in an oflSce form similar to that of No. 1, with the natural sines and cosines corresponding, as also the measured lengths which are to be multiplied by the sines and cdsines opposite the respective angles. After having performed the multiplication we are then ready for another form, wherein are to be collected, classed, and entered all the data from the form No. 1. This second form is then a general sheet of reference and companion to the plan afterwards to be constructed from it. This Office Form, No. 2, is ruled into nine double and three single colimins. The 1st contains the nimiber of hypothenuses ; the 2nd, the measured lengths; 3rd and 4th, the angles of inclination ; the 5th, the observed horizontal angles ; the 6th and 7th, the angles of direction, with reference to the first line, continued as a common meridian ; the 8th and 9th, the angles of direction with reference to the plane and parallel of planes of latitude; the 10th and 11th, the distance £rom planes of the horizon; 12th and 13th, the distance from the planes of the meridian, in which the quantities become perpendicular and base alternately in the succeeding series; the 14th and the 15th, the distance planes and parallels of planes of latitude, in which the quantities become base and perpendicular alternately in the succeeding series ; the 16th, 17th, 18th, 19th, 20th, and 21st, the total distance from the planes of the horizon, meridian, and latitude. The columns 18 and 20 are deduced from those of 12,13, 14, and 15, by constantly adding together the bases and perpendiculars in the separate series, and entering their totals in the former.

To Construct The Plan From Office Form No. H.

Draw a line on a sheet of paper representing the meridian A B, Fig. 53, and mark ofTfrom a chain-scale from A to station 1 460 links through this station, and at right angles to the meridian A B ; draw the line C D, representing the first plane of latitude ; apply the chamfered edge of the same scale to the line A B, with its zero division coinciding exactly with the first station 1 ; mark off on the paper this scale the bases or distances in column 20, as far as No. 11, as 185.20, 835.03, 564.66, &c., to 2623.49, which will finish the bases in the first series ; take the offset scale belonging to the chain-scale, and slide it along the edge of the latter fixed to the line A B, and at every pencil point on the paper representing the successive bases ; mark off the perpendiculars, or distances, taken from colimin 18, as far as No. 11, as

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MINING SUBYEYINa. 81

147.27, 211.84, &c., to 83.76, which wiU finish the perpendiculars in the first series ; apply the parallel ruler to the line C D, and run it to the successive points, &om which draw dotted perpendicular lines from the plane of the meridian; join the extreme ends of these perpendiculars by drawing lines from end to end, the junction of which with the perpendiculars will represent the stations, and the lines the measured distances in this part of the traverse. Set the parallel again to the line C D, and run it up to the last station in the first series ; draw the line E F through this station, which call the parallel of latitude ; apply the same scale to the line E P, with its zero to the station No. 11 ; mark off from it the bases in column 18, second series, as 277.14, &c., to 866.25, which will end the bases in this direction ; slide the offset scale again along the edge of the chain-scale, and mark off perpendiculars from column 20, second series, as 114.78, &c., to 281.26, which will end the perpendiculars for this series, the ends of which, joined as before, will give the stations and direction of the lines composing this part of the traverse.

The meridian A B must now be brought up and drawn through the station 16 as G H, which call the parallel of the meridian. The scale must be applied to this line also, and the bases marked off from it out of column 20, third series, as 603.41, &c., to 3173.36, the end of the bases in this direction. The offset scale is again to be made pass along the edge of the previous one, and the perpendiculars marked off from it in column No. 18, third series, as 89.24, &c., to 642.39, the last; the ends joined as before will complete this part of the traverse also. Set the parallel to the first plane of latitude C D ; run it up and draw the line 0' through the station No. 26 ; apply the scale to this line with its zero division coinciding with the station; the bases and perpendiculars are then to be marked off from the chain and offset scale out of column 18 and 20, fourth series, the extreme ends of the perpendiculars joined by lines as before will complete the traverse. If the survey has been correctly taken, and the calculations accurately made, the last line in the traverse should strike exactly through the centre of the shaft at A, the point of commencement, otherwise one part or the other is at fisiult.

This method of plotting a traverse is vastly superior to the many others employed, as by it the several stations are plotted independent of each other ; consequently an error committed in one part of the work could not possibly affect that immediately succeeding it; whereas, by the old method <rf

82 Mininq Surveying.

plotting with protractors the slightest error committed in one of the angles is certain to be communicated throughout all the succeeding parts of the work. A great advantage is also to be derived from the plans so constructed from OflSce Form No. 2, as we can at any time by inspection find the distances and positions of all the most important points in the survey which can with the greatest facility be referred to the sur£EU when occasion requires.

Example. — Suppose we wish to sink a shaft from the surfisice to the station 16 at G. By inspecting the OflSce Form No. 2 we are in possession of aU the requisite data for calculating the direction and distance from station 1, or, in other terms, what we wish to find is only the hypothenuse, and one of the angles of a right-angled triangle. Thus, taking the total base from column 20, first series =2623.49, the distance between the parallel of latitude C D and E E, Fig. 54, to this distance add from G to the parallel E P, the last perpendicular in second series column 20=281.26, and 2623.49 + 281.26= 2904.75 =total length of base, or the distance from station 1 to a. Then, to find the perpendicular, take from column 18, second series, the number 866.25, deducting therefrom the last perpendicular in the first series, or 83.76, and 866.25-83.76=782.49, total length of perpendicular. The hypothenuse and angle may now be found by Problem 18th in Bight-angled Trigonometry, as follows : —

To find the Angle of Direction. Logarithm of the base 1 a =2904.75 . 3.4621088

perpendicular a G =782.49= 2.8934788 „ radius =10.0000000

12.8984788 Deduct the first term 3.4621080

The logarithm tangent, / =15° 6' 37'' 9.4318708

To find the HypotheniLse.

Logarithm radius 10.0000000

„ secant =15® 6' 37''. . . =10.0155695

„ of the base =2904.75 . . 3.4621088

Hypothenuse =3000.38 3.4776783

Mining Subveting. 83

Therefore the angle of direction (of the point at surf&ce, corresponding to the station No. 16) &om station 1, with reference to the first meridian A B, is=to 15° 6' 37'', and the distance 3000.38 links, which, if accurately set out, will find the point to the greatest degree of exactness ; the chances for error are also diminished 30 times less than it would have been in case the surveyor traced all the angles, and measured the distances until he arrived at the point intended. In the former we have one angle and one line, but in the latter 15 angles and 15 lines to be measured. The surveyor has also an opportunity to correct the position as determined, by setting out the angle and distance in the first case ; for if with a good instrument he ranges out the line A B, or base from station 1, until he arrive at a point determined by the distance 290 4?. 75, and from this point set out the line a G at a perfect right angle to a 1, and also measure 782.49 links in this direction, the termination of which line should reach to and strike through the point previously determined at G. If the case proves to be otherwise (as it will frequently happen in hilly or woody districts, or from inattention to a careful measurement of the line), the three sides of the triangle must be again set out and tested with as much care as the circumstances of the case admit of, as the surveyor should rest satisfied with nothing short of perfect coincidence of the termination of the lines at this point. If any other point in the traverse was decided on, as at station 9 or 22, the bases and perpendiculars to these points are also given ; consequently, the same logarithmic process will find the angle of direction and lengths.

THE NEW METHOD OF COimECTING UNDERGROUND WORKINGS

With The Surface.

The following. Fig. 54, represents a vertical section of the shaft A (plotted traverse. Fig. 52). The first line, A to station 1, corresponds to that of A B in the section, and is required to be produced in the same direction on the surface. To perform this, we plant the transit theodolite in the centre of the shaft at A ; it is then careftdly levelled, and the hairs of the telescope made to bisect a oiall bright light placed on the peg at B ; the clamp to the vertical circle is then released, and the telescope pointed up the shaft in the same vertical plane, and in the direction of the dotted line a a ; the flame of a lamp or white peg is brought exactly into this line of sight. The surveyor should then direct the person

G 2

IdNIRG SUBYKTIHG.

attending to it to drive in a peg, or make any permanent mark exactly noder the lamp as at a' ; the lamp shonld again be placed over it, and tested with the telescope, which if iband to be correct, may be revolved a little until it strike the other side of the shaft in the direction of the dotted line b b', and the same operation gone through as befiire, we shall then have two points at Borface in the same vertical plane as the line A B below groond. The surveyor may now ascend the shaft (after having carefully

Fio. 64.

measured the line A B) to complete the operation, which will be done by stretching a strong line through the centres of the pegs or marks b' a", which continued, and the measure of the line A B, measured off from A to c, would evidently give a point directly over the centre of the peg previously driven below at B. A strong iron peg should be driven into the ground at c, and its centre determined by stretching the same line through the marks or points a b; a small bole should then be made in its centre for

MININa SUBVBYINa. 85

fdtare reference. The underground workings are all connected to this line which in its turn becomes connected to the surface and the accuracy of any distant point referred to or from this line with respect to those below, depends on the correctness with which it is set out.

If we wish to produce a line (previously marked out on the surfisu between two shafts) in the same relative direction below ground, so as to form a heading or tunnel from one shaft to another, it may be performed in the following manner : — Set up the transit theodolite at the bottom and in the centre of the shaft, as near as can be estimated by the eye, as at (a) Fig. 55, and after the instrument has been accurately levelled, and the zero of the upper made to coincide with the zero of the lower limb, sight up the shaft and make the cross hairs of the telescope bisect a mark

Fig. 55.

i,* Figure 55, in the line A B at surface, revolve the telescope

vertically a little, until the vertical wire strike the opposite side

of the shaft at (c) Figure 55. Measure the angle c b e {or what

c a e c a e is the same thing, the angle for the /. l c be), and

the radius of the shaft, or the distance a e. We may then easily find by calculation the exact deviation of the centre of the instrument from the meridian, or the vertical plane of the line A B at surface. Let the angular deviation the/ c a e=3° 10', and the radius of the shaft =60 inches ; then the sine /. c a e x

If any difficulty should be experienced in not being able to find ordinary marks, small lamps may be used instead, and these will be seen better at night than in the day-time.

86 Mining Subvbying.

radius or the distance a e=to the deyiation in inches. the

Lca =1'' 85'=sine iJ7639 x 60=16.5 inches.

The instrument may now be removed from a along the line B and towards a distance of 16.5 inches ; it will then occupy its true position, or be in the same vertical plane as the line A B at surface.

The theodolite may now be properly adjusted and levelled, and the telescope pointed up the shaft, when, if the preceding operation has been accurately performed, the vertical wire will exactly thread or pass through the marks previously fixed at e and i.

It is to be observed, that ordinary marks fixed in the sides of the shaft, and in the line A B at surface, wiQ not be distinctly seen, if the shaft is used as an occasional smoke chimney, and in that case small bright lights ought to be used instead.

It sometimes happens that shafts are to be found so badly arranged, that the surveyor will not be able to sight up them with the theodolite, in consequence of the water that is emitted from the sides of the shaft being allowed to fall down spontaneously. In such a case he may resort to the following plan for accomplishing his object. Procure two small chains, made from iron of about 1% in., and sufficiently long to reach the bottom of the shaft. Attach them to iron bolts with screws at each end, which are to be inserted into, and screwed to a stout piece of timber, sufficiently strong to endure the weight of the chains suspended to them. This piece of timber may then be mounted on a frame erected for that purpose, and by signals from a man at the bottom of the pit the ends of the chain support (E and F) Fig. 56, may be moved until a line passing through the two chains, a and J, is found to bisect the peg A in the heading below ground. The surveyor should then measure from c to A, and ascend the shaft, when again sighting out a line passing through the chains at surface, and setting off the distance from c to A, the point B will be arrived at, which should be perpendicular to that of A below ground. After the points A and B have been fixed by this means, a theodolite should be set up over each peg separately ; then after correctly adjusting the instrument, set the cross hairs on the chain

This rule assumes the angle to be that of a right-angled triangle instead of an isosceles triangle, which it is. The rule depending on the former will give residts sufficiently correct for small angles and short distances, and may be performed with great facility.

Mining Subvbying.

e, and clamp it fast ; remove the chain c aside a little, and look through the telescope, then if the cross hair bisect the centre of the chain d, the point A is accurately fixed. The same operation must be performed on the surface at B to test the accuracy of that point also. It is needless to say that permanent marks should be left at A and c and B and ff for fiiture reference.

This method is much superior to that with copper wires, as the weight of the chains is sufficient of themselves to prevent any oscillation that may be occasioned by a strong current of air.

Fm. 56.

TO FIND A POINT ON THE SURFACE CORRESPONDINa TO ANY ONE PREVIOUSLY FIXED ON BELOW GROUND.

The finding a point at surface corresponding to one below ground, is an operation which has often baffled the best attempts of some of the surveyors of the old school, who are so wedded to the method of plotting on the surface the bearings and lengths taken below, that it is no wonder they so seldom find the point proposed ; indeed, it would be a much greater wonder if they found even a distant approximation to the truth by means of the miners circumferentor ; to perform it accurately requires much

88 MINIKO SURYETIKa

tact and care when the best instnnnents are employed. The surveyor or student is referred to the plotted diagram Pig. 53, firom which we will take station 16 at G as a point for an example. The length of the line firom 1 to G, with the angle of direction, has already been calculated at 3000.38 links and 15° 6' 37"; to set it out the miner's transit theodolite is placed over the station 1 (in the line AB previously connected from below); and after carefully levelling it and bisecting a distant station staff at B, the upper plate is released and the vernier made to read the angle of direction 15® 6' 37'', the cross hairs wiU then point to the station at G, Fig. 53. If this station cannot be seen from station 1 the line must be sighted out with the telescope, and station staves and pegs driven in at different distances, to mark the line so sighted out. The calculated distance 3000.38 links must be measured in this direction, the termination of which will indicate the position of a point directly above the one proposed below ground. A good caution against error will be to run out the base and perpendicular belonging to the hypothenuse just set out, and if the termination of the last line or perpendicular coincide with the position as indicated by that of the hypothenuse, the surveyor may rest satisfied his work is correct.

Mining Surveying. 89

SURVEY OF A HEADING, TAKEN PREPARATORY TO SINKING A SHAFT INTO THE HEADING, AND DRIVING A TUNNEL FROM ONE SHAFT TO ANOTHER.

Section Ii.

The survey of tlie heading fipom the shaft at C (Fig. 1, Plate 1) to the point where the shaft is intended to be sunk down to C, is taken in a somewhat different way to that abeady taught in the preceding part of Section I. The surveyor plants his instrument at station and levels it as usual ; the cross hairs of the telescope are brought in contact with a lamp placed in centre of the shaft at c, the upper plate is then released and the forward object sighted ; through the telescope the horizontal and vertical angles are then read off, and entered in Survey Book No. 3 ; the instrument is then removed to the next station 2, when, after proper adjustment, the body of the instrument is released &om its clamp screw, and turned round in the direction of a lamp placed over the back station 1,* the vernier remaining fixed at the previous forward reading, which becomes the back reading when the hairs of the telescope are made to bisect the lamp at station 1. The upper plate is then released from its clamp and turned in the direction of the next forward station 3 ; the horizontal and vertical angles are then read off and entered as before. The instrument is again removed, and carried forward to the station 3 ; the same operation is again gone through, always remembering to leave the upper plate clamped to the last forward reading, which becomes alternately the back reading. It will be obvious to the student that the 360° and 180° of the instrument are constantly

It is of the greatest importance in traversing by this method that the supplementary or lower telescope is brought to bisect the back station before the upper telescope is released from it, and thus checking the angle by looking through the lower one to determine if any of the upper or lower works have shifted during the interval.

Mining Subvbyinq*

the meridian and consequently parallel to the first line the telescope was set on and the angles as read .off are the horizontal angles with respect to this line, and not to the lines composing the traverse, as in the preceding method of referring the zero to the back line continually.

The survey book necessary for the entries of the observations taken in this survey is ruled into nine columns, the first four of which are common to all ; the fifth and sixth contain the back and fore horizontal reading of the limb ; and the seventh and eighth the horizontal angles with reference to the meridian ; the ninth remarks, &c.

SUBVEY BOOK, No. 3.

H

Angles of Inclination.

Elevation.

Depression.

Horizontal Beadings.

Back.

Fore.

185" 30'

Horizontal Angles with Meridian.

+

5' 30'

0°25'

Remarks.

Zero on a lamp in centre of shaft.

Finish at a point in the heading where the shaft must strike firom the surface.

It is now evident that we require an oflBice form similar to that of Nos. 1 and 2, Section 1, in which to enter the measured distances from column four, which is to be multiplied by the sines and cosines belonging to the separate angles of elevation and depression in columns two and three ; the former producing the difierence of level or the distance from the planes of the horizon, and the latter the reduced horizontal measure ; which last is also to be multiplied by the sines and cosines of the angles of direction with the plane of the meridian, producing the distance from the planes of the meridian in the former, and the distance from the planes of latitude in the latter.

To Determine When An Angle Is Plus Ob Minus.

Rule, — In any traverse whose general direction inclines to the right of the first line in the traverse, and the first angle is more than 180®, its difierence is called plus, the whole succeeding angles less than 180°, minus ; also when the instrument is reversed (as it will be at every alternate station), and the vernier pass over

Mining Suhveyino.

or indicate the line to be to the right of 860®, it is called plus, and if less than 360°, its diflTerence to be called mints. If, on the contrary, the traverse inclines to the left of the first line, and the first angle is less than 180®, its difference must be taken and called plus ; the whole succeeding angles more than 180®, their difference to be called minus. These rules will be better understood by reference to Fig. 57, where it will be seen in the traverse A C that all lines diverging from the meridian D E are called plus, being in the third quadrant, and all those inclining towards it minus, being actually in or belonging to the second quadrant. Again, aU lines in the traverse A B, inclining towards the meridian A E, are called minus, being in or belonging to the third quadrant ; and all those diverging from it plus, being in or belonging to the second quadrant.

The data, as calculated in the preparatory office form, should now be collected under their proper signs in the Office Form No. 3. This office form is analogous to that of No. 2, and thefore will reqtiire no explanation of its columns. The plan may be accurately constructed from the columns nine and eleven, with the chain and offset scales. The surveyor should guard against the possibility of any of the lines in the traverse crossing the meridian. This may be avoided by drawing a datum line or meridian, A B, parallel to the first one at a 6 (Fig. 1, Plate 1), and at a fixed distance from it, as a A, say 1000 links. This number must be

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Mining Surveying. 93

entered in column nine, containing the perpendiculars, each perpendicular or distance from the plane of the meridian must be added to this number, which will express the distance of each station from the datum or assumed meridian. The object of the present survey being to arrive at a conclusion whereby to determine the position of a shaft to be sunk down to the last point arrived at in the survey below ground as C, and also to connect another shaft D by means of a tunnel from C, I shall now hasten to this important part of the work, thinking the student has received suf&cient insight into the preceding methods of conducting a subterraneous survey as to be enabled to undertake any work of a similar nature.

TO SINK A SHAFT SO THAT IT SHALL STRIKE A FIXED POINT IN A HEADING BELOW GROUND.

By referring to Office Form No. 3, we have given the base or distance a b (Fig. 1, Plate 1) 6267.97 links, and the perpendicular C 624.84 links being the base and perpendicular of a right-angled triangle, we have therefore to find the hypothenuse and angle b a C -, this method has already been explained in Section I., but for the sake of example the calculations will be fully gone into.

For the Angle of Direction.

Logarithm of the base a 6267.97= 3.7971270

ofperpendicularC= 624.84= 2.7957688 Logarithmic radius 10.0000000

12.7957688 Deduct the first term 3.7971270

Logarithmic tangent =5° 41' 34'' . . 8.9986418

For the Line a C.

Logarithmic radius 10.0000000

=5° 41' 34'' 1 10.0022102

Logarithm of the base a 6267.97 3.7971270 The hypothenuse a 6299.95 . . 3.7993372

Therefore the distance from a to 6299.95 links, and the angle

94 Mining Surveying.

of direction 4 a C 6° 41' 34''. Now if the transit instrument is set up at tlie station and this angle run off from the line a b, and 6299.95 links measured in this direction will give the position of the intended shaft. It is presumed that the line c a has been connected to the surface as previously explained and as a great deal depends on the accuracy of the point C thus found, it is requisite that the surveyor should run out the base a 6 to the measure indicated in the Office Form No. 8 ; and when arrived there set up the theodolite and run off the perpendicular to its measure also, he will then be able to judge about the correctness of the position.

TO DETERMINE THE LEVELS, DISTANCE, AND DIRECTION OP THE TUNNEL TO BE DRIVEN FROM D TO C.

The difference of level between the tops of the two shafts D and C (Fig. 2, Plate 1) is 30 feet, the depth of D E 450 feet, and that of C F 320 feet ; we have therefore to determine at what distance from the surface at D down the shaft towards E, or from E upwards towards D, to commence the tunnel, so as to strike the point F in the heading (Fig. 2) ; therefore D G=to C F-C H, or 320 feet-30=290 feet=to D G, and E G=to D E + C H - C F, or 450 feet + 30 - 320 feet 1 60. The distance therefore from the surface down the shaft to G and on a level with F=290 feet, or from the bottom at E to the same point G 160 feet. The distance C D (Fig. 1 and corresponding to C F, Fig. 2) may be found by direct measurement, if no obstacle come in the way ; on the contrary, it will be best to continue the line C b to d, which being measured, and the lines set out at right angles, wiU be two sides of a triangle to find D C, this may be performed by the same logarithmic rule applied for ascertaining the length of C a ; the distance between D and C, found as above, is 2234.8 links, and the angle D C rf=15° 50' 23''. The surveyor will now plant his instrument at C, and measure off the previous calculated angle D C rf=15° 50' 23"; this line continued will strike through the shaft at D. Two permanent marks should now be fixed in the sides of the shaft directly in this line, so sighted out. The shaft at C may now be commenced, and also a door or platform fiixed in the shaft D E at G, or 290 feet from the top of the shaft. If the surveyor chooses to take the compass bearing of the line C D he may do so, and in the absence of all iron set the headers to work, driving the tunnel on this bearing

Mining Surveying. 95

from G ; it must, however be totally abandoned after the tunnel is driven a few yards, to make room for carrying on other operations. To set out this line D C accurately from G in the shaft D E (Fig. 2), so as to strike the point F, the surveyor should plant the miners transit theodolite on the stage at G,* and move it about the centre of the shaft after calculating its deviation, as previously explained, until he finds the vertical hairs of the telescope strike exactly through the two marks previously fixed in the sides of the shaft D, and in the line C D (care being taken that the instrument is careftdly levelled at the same time), he may then bring down or revolve the telescope to a horizontal position ; the same vertical wire will now point troxn. G to F (Fig. 2, Plate 1), and consequently give the correct direction for the tunnel to be driven, being in the same vertical plane as the line C D at surface. Two lines, with plumb bobs appended to their ends, must be fastened to the roof of that part of the tunnel already driven, which are placed directly in the line of sight and coinciding with the cross hairs of the telescope ; these lines should be examined from time to time, and others suspended in the same line of sight as the work proceeds, the workmen will then have two points or more given by which to drive the tunnel straight ; it is, however, very objectionable to depend entirely on the skill of any workmen to continue the tunnel without frequent examination, and indeed the surveyor or mining engineer should not rest satisfied with a single operation in setting out this line, and if needs be it shoxdd be traced again on the surface, and the old marks in the sides of the shaft verified. The operation of bisecting these marks, thus corrected from G with the instrument, is again to be performed, and the telescope brought down to a horizontal position ; if the wires bisect the lines previously suspended it will be much ease and satisfaction to the operator. The direction may also be corrected by the following means : produce the line erected perpendicular to C b ai d through the shaft, and leave marks in the sides of the shaft in this line as before. Plant the instrument again at G, and after repeated trials of levelling and sighting up the shaft, the cross hairs wiU be found to strike through these two marks ; also bring the upper plate of the in-

If the engineer has a connterbalaneing stand for his instrument, he may set the theodolite at surface near the edge of the shaft and in the line D C, when it may be brought into such a position that the telescope will sight down the shaft ; the direction of the line C D may then be produced below ground.

96 Mining Subyeying.

strument to the zero of the lower, clamp them together, and again bisect the two marks up the shaft ; bring the telescope down, and release the upper plate ; make the vernier point to the angle or measure between the lines D C and D rf, or the complement of the previous calculated angle B C rf 74° 9' 27'' ; then the vertical hair wUl bisect the plumb-line before suspended, if the work is correct ; if it does not agree, the operations must all again be gone through, from first to last, until they do agree.

To Drive The Same Tunnel From Two Opposite Points

At The Same Time.

If the shafts D E and C F (Pig. 2, Plate 1), are both sunk to their depths before commencing to drive the tunnel (CD, Pig. 1) between them, and it is requisite that the tunnel should be driven from these two points at the same time, it may be set out from the shafts as before explained. I have given a explanation how the tunnel may be set out from G in the shaft D E, consequently a similar operation from the bottom of the shaft at P would bring the vertical hair of the telescope to bear on or point to G, and the tunnel so driven from the two points must meet at about I. If the surveyor desires an easy check on the direction of the tunnel as set out from P by the preceding method, he may correctly do so as follows : by referring to the Survey Book No. 3, he wUl find that the last horizontal angle. No. 8, was 357° 18', and consequently 2° 45'— from the meridian; if to this angle 2° 45' we add 90 + 15° 50' 23" 108° 35' 23" or the angle of direction of the tunnel from the line passing through C and the last station, or No. 8 Station in the Survey Book ; if the theodolite is planted in the centre of the shaft represented by P in the section, and C in the plan, and the telescope is brought to bisect the last station (No. 8 below groimd), and the vernier made to read the angle 108° 35' 23", the cross hairs should then point to or bisect the plimib-line before fixed to the roof of the tunnel. It should also be remembered that the tunnel must be driven perfectly level from both sides, or it cannot be expected to meet on the same horizontal plane.

Setting Out Underground Curves.

Curves, i.e,, pieces of circles, are sometimes employed in mines to join headings or tunnels that have been driven in two directions and converging to a point or apex. There are many methods of

MINING SURVETINa. 97

raising Unes on the surface; but from the want of su£Scient space below ground are not applicable. The following, though different, is equally reliable and well adapted to mining purposes.

To set out a Curve from One Point.

It is proposed to drive a tunnel on a curve from the point B in the heading to join that of A in the heading (Fig. 58).

It is also presumed that the surveyor has made a preparatory survey of the two headings leading from two separate shafts to the points B and A, and also to have connected the shafts in the manner before described. The survey should be reduced to a form similar to that of Office Form No. 3, and plotted to a large scale (say two inches to a chain) with a beam compass, reading to the fraction of a unit. The lines E B and F A are produced until they meet at C, and one of the sides, as B C, measured with the beam compass, we shall then have one side with the angle A C B, to find the radius necessary to draw the curve tangentially to the points A and B. The angle A C B may be calculated by the 16th Problem of Oblique-angled Trigonometry, first taking the measure of the lines A C, A B, and C B, or it may be measured direct with a circular protractor. The angle between the lines C A and C B=105 1', and the measure of the line A C 605 links, to find the radius.

To find the Radius. As the cosine of J z A C 9.8994667

Is to the sine of z A C B 105® V, -.9 9349099

complement 59' J

So is the logarithm of the length A C)

The two last terms added 12.6882013

The first term deducted 9.8994667

Therefore the radius A D 629 links . . =2.7887846

Take this distance, 629 links, from a scale with the compass, and set one of its points at A, strike the arc 11'; set the point also at B, and strike another arc 2 2', as at D. The point of the compass set in the intersection of these arcs, and with a sweep of the other point will describe the curve tangentially at the points A and B. The curve may then be divided into any number of equal parts

H

MIKIlfO SUBYETIKG*

which in this case are six. Right-angled triangles should now be formed upon these divisions as hypothenuses, by drawing lines parallel to B C through the points a, b, c, d, e, &5C., to A, and also the perpendiculars a o', 6 6', c c', &c. ; we shall then be enabled

Fig. 58.

to calculate the angles* by which to determine the direction of the lines abb &c., to e A ; measure the lines B a' abbd d e and as bases ; also a of, b b'y c c', c df e/ apd e as perpendiculars and enter them in a tabular form as follows :—

Form Of Note Book Contaniig Pata Fob Setting Out The Curve,

Cq

Afagles of Diiction.

6° 20' 30'' 18 '26 30 43 bS 21

Oompleinents

of Aoglei of

Direction.

Beduced

Angles for

setting on tthe

curve.

20' 30"

Remarks.

The curve to be set from one points

Or these angles may be referred to a spare space on the drawing, and measured at once with the protractor.

Mining Subybtino. 99

The Note Book contains eight colnmnB : the first, the number of stations; the second the length of the chords required to set out the curve; thirds the bases; and the fourth perpendiculars of the right-angled triangle formed upon the chord as an hypothenuse; the fifth and sixths the calculated angles of the triangles formed with the chord lines; the seventh the reduced angles required in setting out the curve; the eighth remarks &c. The angles in column five are calculated by adding the logarithm of the perpendicular to radius, deducting the logarithm of the base. The remainder is the logarithm tangent of the angle. The last two calculated angles 33 50' and 21® 39' 10'' are those of dee' and e A in the triangles consequently as we are setting out the curve one side, or &om B, their complements are to be employed instead. Then the z A fl i' — the zaBa', z c b cb aby zcb c, ze d e - Lde d, andzAe-Zerfe, =18® 26' 30"-6® 20' 30" z 28® 29' 46" - 18® 26' 30" z 43® 53' 21" - 28® 29' 46" z 56® 39' 10"-43® 58' 21", and z 68® 20' 50"-56® 39' 10"=to 6° 20' 30", 12® 6', 10® 3' 16", 15® 20' 35", 12® 40' 49", and 11® 41' 40", reduced angles for setting out the curve.

To set out the Curves.

Plant the transit theodolite in the heading at station B, and

after adjusting it, sight back along the heading to a mark at E ;

release the upper plate and make the vernier read the first reduced

angle 6® 20' 30", the cross hairs of the telescope will then point

towards the station a ; revolve the telescope vertically, and point

in the direction from B to 1 ; at this point cause a plumb-bob to

le suspended ; also suspend a second line over the station B, the

workman will then have two points by which to drive the length

B tf . After the line B a is partly driven, its direction should then

tested by setting up tiiie instrument at B, and measure the

angle 6® 20' 30". It is a very important matter to get this first

line correct, as the curve is hinged at B, and a small error com*

mitted on it would accumulate throughout the succeeding chords,

rendering it impossible lor the curve to join the point A, When

the point a is attained, or 142 links from B, the theodolite may be

set up at this station also ; and after being levelled, thie cross

hairs are made to bisect the point B ; the telescope is then re*

volvedf id the vernier made to read the angle 12 6' ; the cross

hairs will then point to the next station V in the curve* Hie

H 2

lao

MINING . SURVBTINa.

telescope may then be revolved and directed towards a point at 2, at whicli point suspend another plumb-bob line ; a second line is also to be suspended over the station V ; these lines will then be a guide for driving from a to as before. This operation must be repeated at every station, as at b. c, d, and e, which last must strike tih the poSt A previously defined upon and tangentially 1;o it.

To set out the same Curve from Two Points.

Divide the curve into any number, of equal parts as before, draw a line through these points parallel to B C and A C (Fig. 59)

-vfhich call the base draw also perpendiculars to these lines, we then have three right-angled triangles formed upon B C and A C, and upon the chord divisions as hypothenuses, as a B a', hah\ cb &, and a A a", b a and c b These angles may be calculated by the same rule as that of Fig. 58 ; but in this case the angle? calctdated for the three first triangles from the line B C, will be equal to those formed upon the line A C. These angl, therefore, set off B and A alternately as previously explained, will find the point c in the middle of the curve, the operation being exactly the same as described when setting out

Mining Sukveying. 101

To Drivb A Tunnel On A Curve And Gradient

From Two Points.

It is required to drive an underground tunnel from two points A and B, Fig. 59, on a curve and gradient. The position of these points has been previously determined by survey, and the horizontal distance between them found after plotting to a large scale equal to 56 feet. The difference of level between the points A and B=50 feet ; the length of the incline wiU then be found by the 13th Problem of Right-angled Plane Trigonometry, viz., by adding together the logarithm of the total rise, or 50 feet to radius, deducting the logarithm of the horizontal distance between A and B, the remainder equals the logarithm tangent of the angle of depression 5° 17' from A, or angle of elevation from B ; secondly, add together the logarithm secant of the angle of depression to radius, deducting the logarithm of the distance 562 feet, the remainder equals the logarithm of the incline 564.40 ; divide this number by six, we have 94.06 feet for the length of each tjhord required in setting out the curve. This distance 94.06 may be taken between the compasses and run along the curve from A towards B, which will form hypothenuses to two series of rightangled triangles formed upon it, whose bases are parallel to the lines AC and B C ; the bases and perpendiculars of which triangles are to be employed in calculating the angles of direction of these chord lines, and determining points in the curve. These angles of direction may also be calculated by the forementioned problem in trigonometry, which are to be treated exactly in the same manner as those for setting out the horizontal curve (Pig. 58). The ratio of the incline may be found by dividing the length of the incline between the points A and B by the total rise, thus — 564.40 feet -i- 50 feet 11.288, or about 1 in 11. The rise or fall per chain may be obtained by dividing the total rise by the number of chains in the incline 50.000 feet -r by 8.55 chains =5. 84; also to find the fall per yard, divide the number of feet in the yard by the ratio, thus— 3.000 feet -f 11.288 .265 ; therefore when the chord A a has been set out as previously explained, every yard forward must show a difference in level of .265, or for every chain 5.84 feet, and the termination of the chord line at a must be at 8.29 feet below A; on the contrary, when the chord B & is set out, the point b must be elevated 8.29 above B, in order to pre serve the proper gradient ; and every forward station determined by the angles of direction and the length of the chord 94.40, must show this difference in elevation or depression from B or A. The.

Miking Subyeying.

gradient may also be set out by planting tbe transit tbeodoUte at A and B successively, and causing a staflF to move along that part of the line A a and B b already driven the same height as the instrument, the vernier of the vertical circle pointing to the previous calculated angle of deprion or elevation 5° 17' ; a strong peg must then be driven into the ground until the cross hairs bisect the horizontal line on the vane of. the staff; a similar process must be gone through at B, but the instrument will show an angle of elevation of instead as before of depression. The best method, however would be to mount a spirit level on a short stand and using a graduated staff, make each chaia's length ward show the previous mentioned difference of level in elevation or depression.

Levelling Underground With The Spirit Level.

Levelling underground may be performed to a considerable degree of correctness by the miners transit theodolite, and with the aid of Problem 7 of Bht-angled Plane Trigonometry ; but when great niceties are required, it would be better to employ the spirit level.* The method of using it is as foUows. Set up the level (similar to the theodolite) between the two stations whose difference of level is sought, then after bringing the bubble to the centre of its run, sight back to a graduated staff, and take the readings say 3.42 feet; turn the telescope round to another staff placed at some distance along the road, read off again, say 3.96, the difference of these readings will then show the difference of level ; the former to be called the back sight and the latter the fore sight. The surveyor may proceed to level any length of road in a similar manner.

UNDEEGROTJlD LEVELLING BOOK.

Rifle.

Back Sight.

Fore Sight.

Fall

Reduced Height.

r Distance in Links.

Remarks.

T 'e'

B M bottom and centre of shaft.

2Q

The 14-inch dumpy level is in most general use, and is the best for surface or underground operations.

MIKING SUUVETTSOf. 108

If now we add up the columns of the back and fore Bights the difference will equal the difference of level between the first and last places arrived at or the difference of the sums of the falls and rises will give the same figures.

The best kind of levelling staves for underground use are those painted and graduated to feet tenths and hundredths of a foot and about from to 6 feet in length. When reading these graduations through the telescope a person attending to the staff must bring the flame of two lamps to bear on its face &om each side in such a manner as to illuminate it and not obstruct the line of sight of the observer. The surveyor will then be able to read off the divisions with greater certainty. If the cross hairs of the telescope cannot be seen distinctly the observer must hold a faint light to the object-glass of the telescope.*

In general, when surveys are taken requiring ofisets to determine the width of roads, and for taking off all the workings, the Survey Book will most convenient when begun at the bottom, and written upwards as the work proceeds. The following sketch, Pig. 60, is offered as a good form wherein to record the elements taken in a survey. The horizontal angles are written in the book in the order they were taken, with an horizontal line drawn under them ; the angles of elevation or depression are written opposite, but in a direction at right-angles to the former; they are also characterized by having a line drawn under them with the letters E or D, elevation or depression ; the chainage is taken and entered as in the sketch ; remarks or names are written on the survey sketch as they occur without any regard to position.

When the mining surveyor or engineer is called upon to inspect and survey existing works or collieries (of which he has had no previous knowledge), for the purpose of driving or setting a tunnel or drift, to extend and connect different veins of coal one above another by that means, it is absolutely necessary that the mining-agent, or manager in charge of the works, should be able to furnish a sectional drawing or record of all particulars relative to the different strata and veins of coal through which the shaft has been sunk, as their thickness, depth &om surface, or height from Jbottom of the shaft; and, above all things, the angle of depression as obtained to the greatest nicety ;

Those persons who hare constant occasion for levelling underground sbotdd have their level constructed to carry a small lantern, which illuminates the cross hairs through a perforation in the side of the telescope.

Mssma subyxtinq;

there -will then be no difSculty whatever, urith ordinary care, to start from any convenient selected spot in any of the nndeiround heading, iriUi a catout or drift that ahall have a certain length and gradient by which to arrive at uwther rein of coal at a remote point.

A GENEBAX UNDESQBOUND SUJtVET BOOK. To heading fue

Of B, oolliery for wording plan, Commeiuw tarred at the aliaft No. 1, Janaarj 20th, 1860.

It ifl of fignent occnirence in coal-mining that long driiitB are required to connect veins of coal at a higher altitude ; and it aa often happens tiiat it is necessary that such drifts should terminate in a point which shall be perpendicular to another selected or fixed mark at surface. This mark may be the boundary line defining other adjacent property — a common known drainage-gutter, an old dam or dike of water, and the like, pent up below — all of which the surveyor should avoid coming in contact with. In the absence of such a section or register as before mentioned, the surveyor will most certainly be at a loss in performing these operatious accurately, for should the pitch or depression of the vein of coal be a emaJl angle, and that depression represented to be different to what it actually is, thre will be no end of anxiety.

MINING SUBVBYINa. 105

find the result will most probably be that the drifts when set out will terminate perhaps ten or twenty yards short of distance or further off than was at first intended ; and it may be such a case would be the immediate cause of inundating the mine or producing some other dreadful catastrophe. By these remarks I do not mean to presume that every coUiery is not provided with the necessary means to enable a man to perform work accurately. I know to the contrary. The present race of colliery managers and agents taken by the majority are too well up to their own interest, and that of the proprietors, to allow any chance slip of bringing science to bear on their undertakings, which may account for the general success of such imdertakings. At the same time, I am aware that many works are sadly neglected, and that it has been so heedlessly conducted, that if you were to inquire of them the amount of depression or thickness of a certain measure out of sight, you may get for answer, " I don't know ;" or " We never troubled about it.'' Such a plan would no doubt involve a man in an engineering difficulty of which he may not easily get free.

For the assistance of those young men, or students, who may one day be connected with mines, I have constructed a sectional diagram {Plate 2), which if followed would, in my opinion, prevent perplexities and uncertainties in after years. The diagram is so plain as to require little explanation. The first column is devoted to a description, with the names of all strata as passed through in course of sinking the shaft ; the second shows a section of the strata on one side of the shaft, and may be coloured or otherwise, at the option of the constructor; three, four, and five, the thickness of each stratum ; six, seven, and eight, the total depths from surface; nine contains the angle of depression; ten, the number of gallons of water produced from each stratum per minute. (This is an important item to notice while sinking the shaft, as it is the usual practice to put in tubbing to keep back these springs, thereby decreasing the size and power of the pumping apparatus ; but should it be thought necessary to work any of the upper veins from the same shaft by drifts, while the under ones are in progress, ample provisions should be made in the pumping apparatus in case the water so tubbed back in the shaft may again be let into the work by such drifts.) Eleven contains remarks as to the nature and solidity of the strata ; and twelve, the cost per yard in course of sinking. This wiU also prove of service in proportioning the friture cost of driving headings, &c.

106 laNjasTG subybting.

tlurongli the same kinds of strata in a horizontal or oblique direction. Other columns may be added at the discretion of the person making the section such as the nature and quantities of all noxious gases found in particular places all heaves or dislocations of strata, and every other particular that will be likely to prove of servi,; in afterTears. If the student is at aU versi in geology and chemistry he may have special coltmins devoted to the geological description of the strata, and the period to which each belongs, all fossils, flora, &c., as likewise the chemical constituent parts contained in each separate rock or stratum.

To Set Out The True Meridian By Equal Altitudes

Of Any Celestial Object,

The direction of the true meridian may be found by setting up the miners transit theodolite over the station or peg from which the meridian is intended to be set out, as at A, Fig. 61 ; and after carefully levelling the instrument, bring the zero of the upper to coincide with the zero of the lower limb or plate ; clamp them fast; direct the telescope to any celestial object (as a planet or fixed star), when it is about two hours distant from the meridian,* and make the cross hairs cut through it by turning the body tangent, and tangent to vertical circle. The vertical circle must now be left at this reading, releasing the upper plate, and waiting imtil the star descends in the western horizon ; attention must be given to the instrument at intervals, so as not to allow the star to descend below the line of sight before completing the second observation. When the star appears in the field of the telescope, it must be followed by moving the tangent screw of the upper plate (without moving the instrument the least in position), by which the vertical hair of the telescope should be made to cut the star continually until it descends and appears to thread the horizontal wire, at which time the vertical hair must be made to coincide with the centre of the object ; the observation will then be complete. We may now take the reading of the horizontal limb, half of which angle will point out the meridian— say the reading equals 37° 18' 20" -f 2 18° 39' 10'' ; if, therefore, the vernier is set to this angle, the cross hairs

This refers, of course, to observAtions made at night ; and the student is referred to Hannay's Astronomical JSphemeris, price sixpence, for the elements of the planets and fixed stars, by the aid of which they may be known, and their distance from the meridian approximately determined.

MINING SUBYETIKa.

will point in the direction of the true meridian ; a strong peg should then be driven into the ground in the line of sights and on opposite sides from the observer and small nails driven into these pegs marking the precise direction of the meridian for future reference. The meridian may also be set out by equal altitudes of the sun but in this case there is a correction to be made for the sun's declination during the interval, amounting to several minutes. To calculate the amount of this correction, the time of each observation must be noted.

Fig. 61.

Practical Rule, — To the log. secant of the latitude, add the log. of half the change of declination during the interval, and the log. cosecant of half the interval of time between the observations turned into space ; the total sum — 20 will be the log. of the correction in seconds of space.

To Set Out The Meridian By The Azimuth Of The Sun.

The azimuth of the sun is the distance of its centre from the north point or meridian at the instant of observation. But as it cannot be calculated without a knowledge of the latitude of the place of observation, I presume it will not be out of place to introduce here all the necessary elements, in order to enable the student to perform his calculations without reference to other works.t

To Find The Latitude.

Set up the transit instrument at station A, Fig. 61, and after adjusting and putting a coloured glass on the eye-end of the telescope, direct it to the sun just before it arrives at the point D ;

Frederick William Simma on Mathematical Instruments. t Those who desire to enter more fully into astronomical calculations maj consult Part the Second of Chambers' Practical Mathematics, Woodhouse's Astronomy, Herschers Astronomy, or Button's Mathematical Dictionary,

108 MINING StJBVEYIKG.

take the horizontal wire coincide with the lower limb, and follow it by turning the tangent screw to the body and vertical circle, until by continually reading off the vertical circle, it appears to remain stationary, and at the same altitude ; the reading on the vertical circle will then point out the sun's observed meridian altitude say 48° 9'; then adding the semidiameter=16' 68'', wiQ give the altitude of the sun's centre =48° 24' 8", deducting 50" for refraction* =48° 24' 8", and adding 6" for paraUax=48° 24' 14''= the true altitude of the sun's centre, and 90°-48° 24' 14"=41° 35' 46"= the zenith distance. If the place is situated northward, and the sun's declination also to the north, add the declination corrected for the longitude to the zenith distance, the sum wiU express the latitude of the place of observation ; thus the sun's declination =10° 13' 10'', adding 8' for difference in longitude west 10° 13' 18" +41° 35' 46", the zenith distance =61° 49' 4"= true latitude ; but an average of several observations gives 61° 49' 27" 35"', for the true latitude of Cinderford. The polar distance of any celestial object may be found by deducting its declination from 90°, when the declination and latitude are north ; but when of different names add the declination to 90, the difference or sum in the first or second case will be the polar distance. The colatitude of a place is the latitude of the same place deducted from 90°.

To Calculate The Azimuth.

Observation. — Set up the instrument at A, Fig. 61, and as directed for finding the meridian altitude ; direct the telescope to the sun after it has passed the meridian about 2 or 2i hours, as at C, and make the horizontal wire touch the lower limb, and the vertical wire the eastern limb ;t read off the altitude from vertical circle, and clear it from refraction, parallax, &c.=32° 40' 5"; this deducted from 90' =47° 19' 56", the zenith distance, the polar distance is =78° 42' 7", and the colatitude 38° 10' 32"; add together the polar distance, colatitude, and zenith distance, and divide them by two; call the remainder S; from the remainder, or S, deduct the polar distance, or P ; call the remainder S — P ; then

Tables of mean refraction parallax, and with others equally necessary, will be found in Chambers' Mathematical Tables, price 3s. 6d,

t It is to be understood that when the observation is not made to the eentre of the sun, its semidiameter must be added or deducted from the computed azimuth according to which limb of the sun was observed.

Mining Surveying. 109

add together the log. sine of S + the log. sine S— P+log. cosecant of C + the log. cosecant of Z ; half these logarithms will =log. cosine of half the azimuth &om the north.

Example.

Polar distance 78° 42' 7'' Colatitude . 38 10 33 Zenith distance 47 19 55

s p

-p

82 11 78 42

s-

3

Logarithm cosine of 8 82*11' 17''= 9.9959458

„ „ S-P= 3 29 10 8.7836048

„ cosecant of C 38 10 32 .2090459

„ „ Z 47 19 55 .1336466

2 )19.1222431 Xogarithm cosine J 68° 39' 9.5611215

\Z' Ji / Sun's semidiameter from the Q lggO north. Trueaamuth* . 137 33 50

If now the upper plate is released and the vernier made to Tead this angle, or 137° 83' the cross hairs will strike through the peg at E previously fixed in the meridian, if the operation has been accurately performed. The meridian may also be set out by equal altitudes of a circumpolar star ; but this involves several hours before the observation can be made complete : it is therefore preferable to perform it by a single observation, thus. Set up the instrument at A as before ; direct the telescope to the star (Alioth) in the constellation (Ursa Major), and follow it until it has attained its greatest eastern or western elongation, as at a or i ; leave the instrument clamped, and make the calculation by the following rule : from the log. cosine of the starts declination increase the index by ten, deduct the log. cosine of the latitude, and the remainder will be log. sine of the azimuth, or the star's distance from the meridian the arc c d. Then if the zero of the instrument was set on the star at its greatest

If the western ]imb of the sun was observed, add the semidiameter ; and if the eastern limb deduct it from the azimuth.

110 Mining Sueyeyino.

elongation, or at a =860°,— the star's distance will give the angle on the limb ; the vernier must be set in order for the telescope to point due north, or in the true meridian. When speaking of observations taken at night, if the instrument used has no lantern (which it should have), by which the cross hairs may be illuminated through the transit axis, the observer must hold a faint light to the object-glass in order to see the cross hairs distinctly.

To Determine The Difference Between The True And

Magnetic Meridian.

Set up the miner's transit theodolite at the peg A, and set the zero of the upper ind lower plate to coincide; turn the whole instrument round until the compass-needle points to its own north, or zero, which it should be made to do with the body tangent screw ; release the upper plate, and bisect the peg D or E with the cross hairs ; the angle may then be obtained from the horizontal limb ; the minutes from the opposite vernier should also be read off, and an average of the two used ; the angle as pointed out on the horizontal limb will be the difference between the true and magnetic meridian.

Land Or Surface Surveying In Connexion With Underground.

To Survey With The Chain Only.

Chain surveying has not much pretension to accuracy and is much more limited in its capabilities when not accompanied with an angular instrument being in all cases confined to one figure — viz., a triangle ; it will, however, serve for temporary purposes when the miner has no angular instrument at hand, and when his surface operations are not extensive. To complete a survey with any degree of precision, the student should range a line through the middle of that part of the surface he intends surveying, calling it the base line ; he may then form a triangle Upon it, enclosing aU he possibly can of the estate in it. The whole detail of fences may then be run in by forming other triangles upon any of the sides of the original ones, in a continuous order, until complete ; or he lay perform it by enclosing the whole extent of surface in one large triangle, as before, and then running the base or tie-line from the apex of the triangle through the greatest extent of groimd to be surveyed. In measuring lines it must be remembered that if the ground is irregular or hilly, it must be reduced to horizontal measure. The student wiQ find no difficulty in performing this after reading the chapter on Trigonometry in this work. It may, however, be done in the field by holding one end of the chain some distance from the ground until it appears to be horizontal, suspending a plumb-bob on the end of it, which will point out the place for the chain pin. When measuring lines in any given direction, the position of all fences, houses, brooks, &c., within a chain of- the line, must be determined by o&settings taken to them, and entered down in the field-book as right and left offiets ; they are generally taken with a deal rod, ten links in length* The link lines in the diagrams represent the boundary fences, and the dotted lines the measured lines. ...

Surface Surveying

The survey of a road A b, &c., to B, Fig. 62, may be taken by setting up a pole at A, and ranging the line A and B, which is carefiilly measured, taking offsets from it to all the bends in the road, as A a, b b, &c. to e as left, and /, &c. to B, as right offsets. A survey of the woodland. Fig. 63, may be taken by ranging the lines A B, B C, and A D, with poles ; continue the line

A-i

Fig. 62.

D A to a, say A a=100 links; continue B A to A the same distance as A a ; measure a A, determining the angle DAB; measure A D and A B, taking offsets to all the irregularities in the boundary ; continue the line A B to e 100 links, and also C B to d the same distance ; measure e d, which will determine the angle ABC; also the lines B C and C D may now be measured.

Fig. 63.

taking o&ets to the boundary as before ; the survey will then be complete. The method of plotting the preceding survey will be obvious to the student ; he may draw the line A B at pleasure, making it the exact measure ; set off from A to A, and B to <r, each 100 links ; take with the compasses, £rom the same scale, the distance a b; set one point at b, and describe an arc of a circle ; describe an arc firom A, with 100 links, as radius, the intersection

In Connexion With Underground.

of these two arcs will be a point in the line A D ; draw the line A D through the points a A, and produce it to D ; set off its measure which will determine the position D ; the same operation at B will also determine that of C. It is now evident that all is right if the line C D will come in : but if it is found too short or too long an error has been committed in some part of the work. To survey Fig. 64, it will be first necessary to run a line through its greatest length at A B, which will form a base for the other lines, and divide it into two triangles ; the lines A B, A B C, B D, and D A, being all measured, the plan may be easily constructed by drawing the line A B its exact length ; then describing two arcs with the measured distances B C and A their intersection will determine the position of C; that of D may be found in a similar manner. If the surveyor desires to

Fie. 64

extend his survey, he may do so by taking one of the side lines D A or B C as a new base, as in Fig. 65, where B C becomes a base to the triangle C B F. EC becomes also a base to the triangle C E F ; or if the point F can be seen from B, run the line B F as a base to the whole.

Before commencing a survey of the boundary and interior fences of Fig. 67, it would be advisable for the student to examine the ground, in order to run the necessary lines (forming the triangle which incloses the greater part of the estate) in the most advantageous direction. Having determined on this, we will commence running the lines. Plant a strong flagstaff at B, Fig. 66, this point being chained on from A, and as you proceed 'with the measurement of this line, as with all other principal lines, leave marks, or false stations, before or after crossing all

114 Subface Subvbyino

fences, brookB roads &c., entering down in tlie fieldbook the total distance firom the commencement to these points leaving marks at d and e, as most probably they vill be required in ninninff other necessary lines at a more advanced stase of the mrv we next plan7a staff at C, chaining on it Z, leaving fiedse stations at a and f, as before ; erect another pole at A, the starting pointy chaining on it also leaving false stations at b and g; we shall then have enclosed a greater part of the six fields oom posing the estate in one large triangle ABC. The next process will be to run a line A D through the centre of the property ; this line should alao be accurately measured leaving false stations at J and k ; enter the total lengths to these points in the field-book ;

Fig. 65.

the line D E must then be ranged and measured through the &lse station previously fixed at ff; from E measure to A, ranging the line E A through the false station h, previously fixed. We now return to the false station b on the line C A, ranging it through the stations left at k and d, measuring as we proceed, until arriving at d ; the line should be continued on to c ; we then commence again at ranging and measuring a line through c to G; and trora G we range another line through the false station e on A B, continuing it to a ; we must now conmience at B, ranging and measuring a line to and fi*om D to C, and finally from h, on the line E through the fidse station J, to the boundary ; the survey wiU then be complete. It will hardly be necessary to say that whoi plotting the preceding survey the

IN CONNSZIOir WITH UNDEBOBOUND.

flame order must be observed in plotting as in measuring the lines in the field ; and that when the triangle ABC and the centre line A D are accurately laid down, and pencil marks left representing the false stations in the field, the other lines drawn through these marks cannot but come in for length and direction. In a similar manner the student may survey any number of fidd. by extending the triangolation ; that I oder triangle, formed upon the side of one of those previously employed, as A or C D, &c., for a new series. The student should be careful to test the accuracy of his chain by some standard every day he commences his work, as by continual use some of the links get curved, or shortened, and others longer, by stretching, and of course finally give inaccurate results.

Fia.66.

SXmVBTING MINERAL LOCALITIES FOB WOBKINQ FLANSi ETC.

Before commencing the plant at any mine, a general survey of that part of the mineral locality which is intended to be opened tip should be made, and a correct plan constructed firom it. The map so constructed should contain (in addition to all that is usual to be found in maps) the boundaries of the mineral property, also

116 Surface Survbying

the position and extent of old cropping workings, where the mineral vein or load crops out, likewise that of any existing faults or dislocations, its direction, fee, the nature of the adjoining strata, &c, &c., with any other geological or mineralogical information that is likely to be useful, and may tend to assist the mining agent or engineer in setting out the winding shaft in such a position that it shall embrace as great a depth, and as long a range of level course, as may be conveniently obtained, having due regard to the proximity to existing railways, which is generally a ruling condition. If the surveyor has no reference to existing old plans of the locality about to be surveyed, he should examine the ground, and make a pencil-sketch plan of it, by which he will at once perceive the best method to be employed in pegging it out. In performing this he must exercise some judgment and care in running his lines, so that they may pass clear of all obstacles, embracing, at the same time, the greatest amount of work. He will find that, by having due regard to this point, he will save himself a vast amount of valuable time when completing the survey ; much, however, depends on the facilities offered by the locality, as in some places greater difficulties occur than in others. If the property is open and free from impediments, the survey may generally be accomplished by enclosing it in two or more large triangles, which wUl be sufficient for obtaining the exterior boundary ; the interior details may be obtained by forming other smaller triangles on the sides of the large or primary ones. If, on the contrary, the mineral property is covered with thick woodlands, such as enclosures, it will then be a difficult matter to run straight lines in any given direction without obstructions ; and it would be much better to make a circuitous survey or traverse of it, going by any open roads in the general direction. The boundary marks having been previously found, may be connected to the circuitous survey (if not too far distant) by oflfeetting, otherwise a point in the traverse must be selected, and a new series of angles taken or traversed in the direction, and until arriving at the separate boundary marks indicating the limits of the mineral property. It will be needless to say that when the survey is plotted, lines drawn these points will give the figure and extent of the gale or mineral property in question.

If we take Plate 3 as an example, it will be seen that the greater portion of the surface to be surveyed was enclosed by the triangle ABC and A B D, by first careftdly measuring the

In Connexion With Underground. 117

angles B A B A and C B ranging and measuring the lines A B, B C, C A, A D, and B D. The interior details were then taken by ranging and measuring other lines from known else stations previously fixed in the lines forming the primary triangle, as explained in chain surveying. Fig. 66 ; all the boundarios, natural or artificial objects, were thus readily and correctly obtained. The student will now readily understand how the other positions of the surface shown on the plan may be obtained by ranging other lines frm the angular points D, B, and C, thus enclosing it by forming another triangle upon AC, CD, and B D, as bases. The student will do well to copy the working plan contained in Plate 3, when he may supply the requisite lines for taking off the other portion of the surfiEice.

Best Kind Op Field Survey Book.

A great difference of opinion exists among practical surveyors as to the best mode or form a survey or field-book should be kept in ; one party prefers and urges that it should be kept in a tabular form j others are advocates for a hand sketch while a third party informs us that a mixture of the two would be the best to be adopted. In some cases, the first method would possibly be employed to advantage, but when the survey is extensive, and contains many intricacies, the hand sketch is most undoubtedly to be preferred, the former requiring to be plotted immediately, whereas the latter may remain aside any length of time, and then be plotted as readily as when taken. However, it is a matter of indifference which mode is preferred, so long as the student abides "by his choice, as it is a very bad practice indeed to sketch in one portion of a survey, and tabulate another ; such a practice would confuse the field-book, and, of course, render it worse than useless. That mode of keeping a field-book to which I have always Tesorted, has been a mixture of the tabular and hand sketch, being, in my opinion, the most concise, and requiring the least space in the field-book.

The Field Book, Plate 4, by which part of the working plan, opiate 3, was constructed, was taken in this manner. It will be seen by reference to the working plan, and compared to the field notes, that the lines, boundaries, &c., are sketched in the direction in which they run, and the distances written at right angles to the direction in which they were measured. Thus we measure

118 6Urfacb Subybtikg

from A to 2480 links fhe first Mse station which is distinguished

from other numbers thus, F 2480* S, sketching every object that

occurs in its proper direction ; we then measure from F S 2480

to F S 2920 on the line B C ; return to F S 1020 we measure to F S 1800 on the line C A ; we now return and continue the base line from 2480 to the next false station or F S 3760; measuring from 3760 to F S 2190 on the line B we obtain the fences in this direction. Thus bj continuing the process we get all the detail of fences and other objects from the base line to the right. The extreme boundaries may be better obtained when measuring the side of the triangle on the line B C. It is now a general rule and of the greatest advantage to begin the entries in the field-book at the bottom of the page writing upwards ; the reason is obvious, as it places the survey-book in the same relative position as the station lines with respect to the surveyor, who advances towards the distant station. The method of keeping the survey-book has been sufficiently explained by this time to enable the student to enter down any survey in a similar manner.

Bbducing The Angles And Plotting The Work.

When a survey is taken by enclosing the surfEtce in one or more large triangles, the angles formed by the intersection of the sides of the triangles with each other require no reduction, and may therefore be plotted at once ; but when the survey cannot be taken in this way, but by a circuitous route, it then becomes necessary to reduce the series of angles taken to a common meridian, to be protracted from one line by the theodolite protractor. The method of the reduction is clearly explained by example in section 1 of Subterraneous Surveying, under the head of. To Reduce Horizontal Angles to Plotting Angles from one Station ; or it may be more accurately performed by preparing a form similar to Office Form No. 1, Section 1, using the natural sines and rules appended for that purpose. The working plan may be constructed by first assuming a line on the drawing as a base line A B, making it the exact length ascertained by measurement ; the points C and D may then be accurately foimd either by protracting the observed horizontal angles form by the lines and the base A B, or by intersection from their measure of length.

In Connexion With Undergbound. 119

When the primary triangles are thus laid down, their measure of length should be tested by the rules of oblique-angled trigonometry, assuming the base and angles to have been correctly taken. IS the process agree with the length of the lines ascertained by measurement, it is a satisfactory conclusion. The scale which the plan is intended to be drawn to should be applied to the base line A B, Plate 3, with its zero division coinciding with the commencement of the base or apex of the triangle, and all boundaries marked off from it, with the oflt scale, with all false stations. The same operation to be applied to the other lines B C, C A, A D, and D B ; the secondary lines should now be run in from the false stations on one line to those on another, exactly in the same order they were taken and occur in the field-book. The scale may be applied to these lines also, and the boundary fences and other objects marked off with the o£t scale, until they are all run in. If other false stations occur on any of the secondary lines, as they frequently win, lines fonniBg a third series ma/be run from Lm in the required direction, and treated with the scale as the other lines. The points left in the boundaries from the offset scale having been penciUed in, may now be permanently inked in with the drawing pen, when the plan wiQ be complete. The next thing to be attended to, before the lines of construction are rubbed out, is the boundary stone, or marks known as Gale or Royalty marks ; these should have their proper places assigned to them on the plan, as Nos. 1, 2, 3, 4, 5, and 6. Lines drawn No. 1 to 6, 1 to 2, 2 to 3, 3 to 4, 4 to 5, and 5 to 6, define the boundaries, figure, and extent of the QbIq Bioyalty, or mineral property under consideration.

K there are any underground workings to be connected to the Burfstce plan thus constructed, they may be connected by first sighting out or producing the line o A in direction of the cutout on the surface, according to the explanation given in Figs. 54 and 56, section 1 of Mining Surveying. The angle this line makes with the true meridian previously set out, may be measured and transferred to the plan, from which all the underground cutouts, headings, and other workings may at once be plotted down.

The underground excavations should be plotted from the planes of the meridian or latitude, as explained in section No. 1 of Subterraneous Surveying; the mine agent will then be enabled to rely on the accuracy of his surfieice plan, and the working so connected to it.

120 SURFACE SUBYETINa

Description Of Scales For Working Plans.

Working surface and underground plans ought, for the sake of clearness, to be plotted to as large a scale as possible, if the subterranean workings are of small extent, or if the. plan is required merely for driving a heading from one place to another, a scale of 1 chain to an inch would be convenient, or a scale of 2 inches to a chain would be still better, as in the latter case the plotting could be made to a single link ; but if the subterranean workings are extensive, then the plan would best suit drawn to a scale of 2 or 3 chains to an inch. In all cases the same scale should be used for the surface as for underground ;

scale 4ij or even 6 chains to an inch.

Colouring And Lettering Plans.

The mining agent or engineer, on whom generally devolves the task of constructing maps and plans for mining purposes, cannot, from the limited time he has to devote to it, be expected to produce beautifrdly drawn, finished plans ; neither, indeed, is it at all necessary or required, since it would be much more beneficial to devote the time required for such productions to other branches of his calling requiring immediate attention.* Colouring is optional, and if the service — that is, fields, woods, brooks, &c. &c., are to be coloured, it is much better to have the underground headings and other workings drawn in ink only ; on the contrary, if the underground excavations are to be coloured, then the surface would be much better drawn in ink only. Houses are generally coloured red with carmine ; brooks, rivers, or ponds of water, are coloured blue ; enclosures, pleasure fields, &c., green ; and all other fields, of whatever colour the draughtsman thinks most suitable, having a care to produce as much contrast as possible ; underground works are generally coloured of a dull red colour.f The plan should have the name of the mine, with

Plate 8 is given as a good specimen of conventional signs, and shows how different ground may be represented and coloured.

t Those who desire to practise colouring, mapping, lettering, &c., are referred to B. P. Wilme's Civil Engineer's Handhoohfor Mapping, where are to be seen some beautiful specimens of finished maps, plans, &c.

k

In Connexion With Undebgbound. 121

those of the owners, also those of the surveyors, written in plain lettering in a spare corner of the plan, and the date at which the plan was constructed, &c. The names of enclosures, fields, fee, should be written and disposed in parallel lines west to east ; those of railways, brooks, and other objects, may be written in the direction in which such objects occur. The scale by which the plan is constructed should be accurately drawn on the plan ; it will then lengthen and shorten, by change of temperature, in the same proportion as the plan itself; consequently, it is oftentimes more to be depended on than the original scale by which the plan was constructed.

Setting Out Railways To Mines,

Railways to mines may easily be set out when the natural features of the ground present no great irregularities; on the contrary the least expensive most convenient and best line of railway, between the starting-point and the mine is most satisfactorily determined by levelling.

The engineer may best determine on the route to be taken if at long distance, from a correct map, showing the direction of watercourses, which is a good guide as to the inclination of the country; and as watercourses, such as rivers, brooks, &c., present great uniformity in their inclination, they materially assist the eye in judging the inclination and direction the railway should take. Levelling operations are then commenced along two or more separate directions, and trial sections made representing the valley lines along which Jthe levelling was conducted; the engineer then selects the line represented by that section which has the slowest gradient, the least curves, and fewest engineering difficulties of construction. If the map by which the direction of the trial section was determined, was purposely constructed, and to be depended on, the straight and curved positions of the railway are laid down on the map, and are then to be transferred to the ground, and marked out roughly with strong pegs; a levelling operation is then carefully conducted along the entire line, and an accurate section made from it, by which may be determined the amoimt of cutting and embankment, height of roads, &c. fee. ; after which the entire line, with its curves, is to be accurately set out and marked on the ground ready for execution.

The general section having been prepared, and the gradients, if any, calculated by the rule given in section 2, on Subterraneous Surveying, the pocket section may then be deduced from it. The following example, Plate 6, wiU illustrate the method of setting out any part of a railway from the pocket section ; take Plate 5 for a cutting to be worked both sides.

At peg 27 chains we are to have a formation height of 114.50 feet above the datum line A D, and the height of sur&ce

Setting Out Railways To Mines. 123

is 110.25 ; set up the leyelling instrument at any convenient spot and direct a man to hold the levelling staff on the peg 114.50 feet ; read off suppose it to be =6.20 feet; then 114.50—110.25 4.25 feet height of embankment and 6.20—4.25 1.95; and if the staff is held up until the engineer can read through the level telescope 1.95 feet it is certain that the lower part of the levelling staff will be at 4.25 feet above the surface line or at 114.50 feet above datum and consequently at formation height. The engineer may then direct a staff to be carried along the centre line and in direction of the intended point of cutting until he reads through the levelling telescope 1.95 feet ; he will then have found a point along the centre line on a level with the proposed formation height at peg 110.25 feet. The gradient however is 1 in 132 or .5 per chain ; the intended formation, therefore, at this point, must be lower than 110.25 by 1 in 132, depending on the distance from the peg at 110.25 ; say this distance to 50 links, or half a chain, our formation height will then

be=to 110.25 — - =110.00, or .25 lower; now, by adding .25 to

1.95, we shall have 2.20; then if the staff is brought nearer, until the reading 2.20 is found, it will give a point on the surface, and at an inclination of 1 in 132 for the cutting ; a strong peg should now be driven in until reading off the staff we have 2.20, and another at 110.25, until we have 1.95 ; we shall then have two pegs at the required formation height 50 links asunder, and any excavators may find other points in the incline as the excavation proceeds by a process called boning, until other correcter levels are set out for his guidance. Boning is carried on with boning rods similar in formation to T squares, as follows : — Suppose a b, Fig. 67, are two pegs driven into the ground to a certain depth, and to any given inclination, if on each of these pegs there are placed boning rods of the same height, and held perfectly perpendicular, it is evident that the tops or heads of these rods will be parallel to the incline A B ; and if any person carries a third rod along the intended slope, the top of it will be in line with the tops of the other two, if the incline is correct ; on the contrary, if it is too high or too low, it must be depressed or raised until a Une agrees with the tops of the boning rods. This method is, of course, only a coarse approximation, being intended only for temporary guidance to the excavator. We will now return to Plate 5, and also to the other end of the intended catting. At formation 101.00, the surface height is =95 .30, and

124 Setting Out Bail Ways To Mines.

101.00—95,30=5.70 for embankment. Set up the level at a convenient point, and the staff on the peg at 101.00; read off, suppose =10.10 and 10.10 — 5.70=4.40; if the staff is raised up until this depth is read off, the foot of it will be again a formation height ; now direct the man with the staff to move along the centre line until this 4.40 is again read off, we shall then have a point level with the intended height of embankment. The gradient being uniform, allowance must be made for it as before, thus 4.40—0.5=3.90; if now the staff is moved along the centre line until from the level we read off 3.90, we shall find the new point for formation at the proper inclination. This example has been devised* purposely for the benefit of those who may have occasion to set out branch railways to works, which generally, however, are not of any great length.

Railway curves are of frequent occurrence, and, in fact, no very long length of line can be formed without them ; they, however, cannot be laid out accurately except by system, and must start from or join into straight pieces of railway at a tangent, that is, the straight portions should be tangential .to the curves at the point of junction. A few gauge rods shod with iron shoes, two or more plumb-bobs, a measuring rule and pegs, are requisite in setting out curves. "When the radius of the curve is given, it may be set out after calculating the tangential and deflection distance by the following rule : — When the chord is 100 link c d

in inches — — : — -— r— 792 being the number

radius of the curve in chains

of inches in a chain, or 100 links, if the curve is 20 chains radius,

i.e. A D=20 chains, then c rf= — =39.6 inches, and the tan-

gential distance a =19.8 inches. To obtain the first

point in the curve, measure from A to the point of contact, 100 links

Those who desire to enter more in detail of railway eonstmction may consult Brees's Bailway Practice,

SBTTINa OUT RAILWAYS TO MINES.

Table Of Gradients For Underground Railways Or

Other Roads.

Of a Foot. Of an Inch.

or

.15 per

Chain

n

2,25

9)

6M

6,15

1 in.

or 1 foot per Mile.

„ 9

„ 10

„ H

12

„ 13

„ 15

„ 16

„ 17

„ 19

„ 20

21

22

„ 23

24

„ 25

„ 26

27

„ 28

„ 29

„ 30

„ 31

32

„ 33

„ 34

„ 35

„ 36

37

„ 38

„ 39

„ 40

„ 41

„ 42

„ 43

,, 44

„ 45

„ 46

„ 47

126 Setting Out Railways To Minei.

in tlie same direction as the line O A, Fig. 68 ; set off the perpendicular &om a, or the tangential distance a i=19.8 inches then will b form a point in the curve ; set a pole at and range a line through the point b to any lengthy measure from i to c 100 Unks and from c set off perpendicular to the line b c the deflection distance c d 39.6 inches. Then will dhe a point in the curve ; the other points in the curve, as s, h, J, I, n, &c., may be found exactly in the same manner ; and in order to pass &om the curve A n to a tangent at B, use the tangential distance 19.8 instead of the deflection distance 39.6. K it is required to find the radius without construction, it may be found thus : — Set up a theodolite at c, and measure the z A C B, and one of the sides, as A C,

then as the cosine of half Z A C B is to the sine of Z A C

so is the log. of the side A C to the log. of the radius A D. K

we wish to set out the same curve by deflection angles, instead

of deflection distances, the angles may be calculated by the fol*

lowing rule, thus : — Divide half the given chord by the radius of

the curve in feet, the quotient will equal the natural sine of half

the deflection angle. Thus : =50 =:33 feet, and

2 ' 1320

.025, the sine of the angle 1° 25' and 1° 25' x 2 2° 50', the

deflection angle. The deflection distance may also be found by

multiplying double the natural sine by the chord in feet, calling

the first figure feet, and the remainder inches and parts of an

inch. Thus : .025 x 2 =.05 x 66=3300 or 39 inches, nearly the

8BTTINQ OUT RAILWAYS TO BflNBS.

Table Of Offsets Foe Settig Out Curves.

Badina

in Chains

Oflbetsin

feet,

Badius

in Chains

Oflbetsin

feet.

and Feet.

inches,

, and

parts.

and Feet.

inches, and

parts.

Chains.

Feet.

Feet, inohos, &c.

Chains.

Feet.

Feet, inches, &o.

132

2706

198

2772

2838

330

2904

396

2970

462

3036

3102

594

3168

660

3234

726

3300

792

3432

858

3564

924

3696

990

3828

1056

3960

1122

4224

1188

4488

1254

4620

1320

4884

1386

5148

1452

5280

1518

1584

5808

1650

5940

1716

6204

1782

6468

1848

6600

1914

6930

1980

7260

2046

7920

2112

8580

2178

9240

2244

2310

10564

2376

11220

2442

11880

2508

12540

2574

13200

2640

14520

Setting Out Railways To Mines.

same as before. Set up the theodolite at A, Kg. 68, and sight back along the straight part of the railway at O, revolve the telescope vertically in the direction A a, set the vernier to half the deflection angle 2° 50' =1° 25', which is the tangential angle, the cross hairs will then strike a point in the curve at a distance of 100 links from A or at A ; remove the instrument to b, and sight back to A, revolve again vertically, and produce the line A i in the direction towards c, set the vernier to the deflection angle =2° 50', and the cross hairs will again strike a point in the curve at a distance of 100 links from i, or at d. The same operation must be carried on at all the stations until arriving at n, when we must return to the point of contact with the tangent at B by the tangential angle, or 1° 25'. If it is required to pass from one curve to another of a different radius, set up the theodolite at C,

Fig. 69, sight back to the end of the 100 link chord at set the vernier of the instrument to the tangential angle A C a of the curve n a Cy the telescope wires will then form the line A C B, the common tangent at the juncture of the two curves C, lay off the tangential angle calculated for the other curve b m C, making the chords C ft=to 100 links. The curve just laid out would be what is generally termed a compound curve, and is of frequent occurrence in railway practice. If we wish to pass from one curve to another in an opposite direction, termed a reverse curve,

it may be performed by setting up the theodolite at Fig. 70, sighting back to the end of the 100 link chord at b, laying off the tangential angle b a e of the curve b d a, then from the tangent line e a set off the tangential angle o a n the curve a s n, making the chord a n=100 links. In deep cutting, it is of the greatest importance to well examine the soil through which the railway is to pass, in order to give the proper and necessary slopes. Experiment proves that the following materials wiU stand better at one angle than another, thus : — Fine dry sand at the angle 35° 30', loose dry shingles

Fio. 69.

Fig. 70.

SBTnNG OUT BAILWATS TO BONES. 129

at solid damp earth common gravel about 85 wet gravel at 36®, loose gravel at 37® 80. The slope most generally given to cuttings and embankments of the above description is 1 foot in 1, or 1 to 1, but this will very much depend on the nature of the soil, and therefore must be left to the judgment of the engineer. The contents of cutting, &c., may be found by direct computation, or more easily from tables published on purpose. A good table of this description is published by Bashford. Those who may wish more minute particulars in connexion with railway curves, &c., may consult those works published for that purpose.

In some cases where the height or total rise and ratio of incline are made a fixed quantity, it will be necessary to find the length of base or the horizontal distance between the termination and commencement of the incline, in order to give it the proposed ratio before commencing operations.

The following rule will be found a very useful and practical one in. such a case. Suppose it is found necessary to form an incline to a fixed height or total rises to B C, Fig. 71, of 50.50 feet, and to a ratio of 1 in 14, required the distance from the representing the total rise 50.50 feet at C to a point of commencement somewhere along the surface line between C and D.

Fm. 71.

Let a total rise and a? ss to required base, and 6 to the

a b 50.50 , 50.50 1 in.

fixed ratio. Then — =sft, therefore and ="77"

X X or 14

.-. 4? X 1 50.50 X 14 and x 707.0 feet for the length of the

incline A B.

Longitudinal And Transverse

Sections.

Longitudinal and transverse sections are of the greatest importance to the mining agent in ascertaining the depth from the surface line to any point selected in the mine ; it also presents to the eye all the vertical underground workings occurring in that part of the inine across which the section is taken.

The irregularities of the surface line may be obtained by a levelling operation commencing from any point selected on the plan and terminating at a similar one. The levelling book may be kept in the same manner — with an additional column for the reduced levels — as that in section 2 of Mine Surveying.

The levels may then be reduced, and entered in a column for that purpose, and the depth at the point of commencement used as a datum line. The section may then be plotted in the same maimer as a railway section. The underground vertical workings having been previously determined may be plotted — if they are above the datum previously fixed upon — from the same datum line as the surface. Deep or land headings having elevation or de-. pression must have their proper degree of inclination given to them ; all vertical workings will then occupy their true position with respect to those headings and to the surface line. If the line assumed for the datum is at the deepest point in the mine, any number of land headings may be plotted on the same section, although they would not occupy their true position with respect to the surface line — except the surface line was perfectly level — because they would be at a distance from each other transversely, and therefore parallels, yet as regards their vertical distance or height from datum, and from each other upwards, would be perfectly true. Parts of the Field Levelling Book are given, by the aid of which that part of the section from Deans shaft to the southern land boundary line were plotted ; the datum line was taken from the bottom of Dean's shaft, 360 feet below the surface line. This number of 360 feet, therefore, is entered at the head

LOKGirUDINAL AND TRANSYEBSB SECTIONS.

Field Levelling Book Fob Section.

Bcmaiki.

+

—

S.Io'20'E.

lliis lereDiDg operation waa peiv formed for a longitudinal Motion ror an iron mioA.

+ 8.60

+ 8.90

+ 4.70

+5.40

+6.60

+ 8.10

+ 4.50

+ 0.20

+ 0.80

+ 4.20

K 2

132 Longitudinal And Tbansvebsb Sections.

of column 8, and tlie numbers marked + and — in columns 5 and which are the difference of the back and fore sights added to or subtracted from this number, in order to determine the distance from datum to the surface at these points.

The section may be constructed by first drawing a line A B, representing the datum line ; the horizontal distances &om column 7 are then marked off on the Une, and perpendiculars raised from tte points thus made. The distances represented by the reduced numbers, column 8, as 360.00, 353.40, 342.95, &c., to 334.14 are then marked off on the perpendiculars previously raised. A line is then drawn the point so marked off on the first perpendicular, through all the succeeding ones, which will represent the surface line. The underground cut-outs, heading, working, &c., are then laid on the section according to their elevation from the datum line A B ; as the land heading at 220 feet above datum, and at an inclination of 1 in 600, the second land heading at 130 feet, and at an inclination of 1 in 300, kc. &c. In finishing the section, the same remarks apply as for underground working laid down on the plan, that is, if the section is uncoloured the workings would be more easily distinguished when coloured, and vice versA.

The transverse section, Plate 7, is generally employed to show the depth at which all vertical shafts intersect the lodes, or underlie the position of all cross drifts or levels driven at right angl to the principal headings, the intersection of one lode with another, Sdc. &c., and may also be made to represent a geological section of the mine across which it was taken, by colouring the different strata their natural colour, &c. The transverse section may be constructed exactly in the same manner as described for the longitudinal section.

Levelling With The Transit

Theodolite.

In levelling operations wlien the ground rises very abruptly, the " miner's transit theodolite" may be employed to more advantage than the " spirit level/ If the latter were used for the purpose over a line whose altitude would probably amount to &om to 50°, much time would be lost, to say nothing of the labour that would be entailed for accomplishing the object. The following method, therefore, will render the operation less tedious, and will be found sufficiently accurate for any every-day purpose.

The proposed line of section should be ranged, and pegs driven

at all the more prominent changes of the ground ; the instrument is then set up at one end of the line, and the horizontal wire of the theodolite made to intersect. A levelling staff at the same height or distance as the axis of the theodolite telescope was set up above the ground. The angle of elevation or depression is then read jfrom both verniers of the vertical circle, the mean of which are entered in the field-book.* The levelling staff is then taken along the entire length of line between the observer and the second station, and fixed at each peg previously driven. The surveyor then reads from the staff the different distances from each successive peg to the visual line passing through the axis of the telescope as it was fixed on the staff at the commencement. These distances are then entered in the field-book, and will re* quire no reduction, as they are the actual distances required to form the section.

When the line is too long to enable the observer to read the ordinary staff, it may be exchanged for a description of staff with

For conyenience, the staff should be longer than those generally employed for levelling purposes — say, about 18 feet would be a good length.

LBVELLmG WITH THE TRANSIT THEODOLITE.

a slidiiig vane and when it is takei along the line the assistant must slide this vane up or down until the intersection of the telescope hairs bisects it. The measures are then noted down by the assistant, and given to the surveyor at the completion of each separate line. Take diagram Fig. 72 for example, in which A represents the position of the theodolite at the first station, and B that of the staff held at the second station. The irregularities of the ground are represented by pegs driven in as at b, c,d,e, &c., upon which the staff is successively held, while the surveyor makes the several readings or heights, dj& a a/ b Vy c &, &c. The theodolite is now removed &om A to B, and before commencing a new line it would be well to take the reciprocal angle of elevation C, which would be a check on the accuracy of the angle as observed from A. The instrument is now turned round

m the direction of the section to be taken to E, and the same operation of clamping the theodolite in position, and noting the readings of the staff when carried along the line between B and E as before.

The section may be plotted by first drawing a horizontal line, &om which the vertical angles are marked off, and a line drawn through the angular marks representing the liiie of sight observed through the theodolite telescope. The measured distances are then marked off an this line from C to B, and perpendiculars to the line C F let fall through these points determined by measurement. The several distances from the field-book are then to be set off from the line C B downward, which will represent the different points in the irregularities of the ground, and by drawing a fine line through these points the surface line will be ob-

Levellikg With The Transit Thkodolitb. 185

tained. Tie total rise as determined by adding the columns in the field-book may be corrected by means of the vertical angles and measured hypothenuse, and by the rules already explained for that purpose in Trigonometry.

This mode of levelling has some objections to its general adoption except for very precipitous places because the surveyor has to depend on his assistant for noting the distances on the staff when the length of line is so great as to prevent the surveyor from taking the reading himself through his theodolite telescope. There is, however, this advantage — that the surveyor is not at all times obliged to take such long lines ; and for my own part I do not see why the section may not be accurately taken, especially with an instrument so finely graduated as the vertical circle of the '' miner's transit theodolite,'' which reads to 20'' of a dree. The observed vertical angles and measured distances also provide an effectual check on the heights, as determined by reading the levelling staff. And what practical man is there who would not employ the method, when by the ordinary process of levelling he had arrived at some dangerous and precipitous ravines, mountainous country, and the like, to cross which would consume much time, owing to the great difficulty in planting the levelling instrument in places from which the staff could easily be read, and from which the staff would not be removed probably more than 20 feet, and consequently require an endless number of readings in a short horizontal distance ? On the contrary, the theodolite would most likely take the whole line of section at one sight as before described.

It seldom happens, when levelling with the theodolite, that the line is sufficiently long to require an allowance for curvature and refraction, and on shorter lines than half a mile it would be useless to trouble about it, as there are other disturbing causes that would amount to more than that due to curvature and refraction.

For example, the correction for curvature and refraction, taken together, amounts to —

Curvature. Eetraction. For J mile .1668 - .0238 .1430 „ 1 „ .6670 - .0953 .5717

Allowances for curvature and refraction are seldom made in levelling operations for short distances; but should occasion require it, the following table will be found useful ; —

186 Levblmng With The Transit Theodolite.

Table Of Cuevatube Akd Eefeaction.

Curvature in

Befraction in

Refraction and

Maei.

Tarda.

Feet

Decimals of

Decimals of

Curvature in

Feet.

Feet.

Decimals of Feet.

n

H

H

1.167

H

H

H

H

k

Calculations Of Areas.

The contents of a field, or any nnmber of fields, in acres, roods, and perches, may be found by the following rules : — If the fields are square or rectangular, their contents may be ascertained by multiplying the length taken in links by the breadth in links, the result will express the area in square links, which, divided by 100.000, or what is the same thing, cut o£f five figures from the right, the figures to the left of the decimal point will express the number of acres ; the decimal part of the number, multiplied by 4, cutting off the same number of figures from the right, will be roods ; the remainder, multiplied by 40, if any, cutting off five figures from the right, will represent perches. IS the fields are triangular,* the area may be obtained in square links by mul tiplying the base or longest side of the triangle by half the perpendicular ; the result wiU express the contents.

K we wish to compute the area of Fig. 73, it may be performed

Fio. 73.

as follows : Run a line from A to L, taking of its right and left to all the points in the irregular boundary A B C D E F, kc.,

Any fields or nnmber of fields may be reduced mto triangles by drawing pencil lines in different directions on the map or plan representing fields, and treated accordingly for the area.

Calculations Of Areas,

to L, entering down the length of A L, and also that of the perpendiculars; the areas of the spaces contained in the interior of the boundary A B C D E F, and those on the exterior of the boundary F G H I S L, may be found thus : —

Interior.

60 + 80

80 + 60

60 + 40

30x100-000= 3000 50x150-100= 2500 70x250-150= 7000 50x300-250= 2500 20x400-300 2000

Area of A B D E F 17000

Exterior.

50 + 90

90 + 80

80 + 50

25x535-400= 3375 70x650-530= 8010 85x760-650= 9350 65x900-760= 9100 25x990-900= 2256

Area of F G H I S L 32091

Measuring from A to o, erecting the perpendicular o O, and also to s, erecting another perpendicular shj the calculation will then be performed thus : —

AN . . =±:1560 + o =210600

And adding the area ofABCDEFas above . . . 17000

Total area ofABCDELNOA 427280

Deduct the area exterior to the boundary EFGHISL= 32091

Total area in square links 3.95189

Therefore the area of the field. Fig. 73, is equal to 3 acres, 3 roods, 32.3 perches.

If we wish to reduce the field into two large triangles, by running a line so as to equalize the irregular boundary, and con-

Oalculatioks Of Arsa8. 139

sequently cutting off from the field as much as we take in it may be performed thus : —

Take the area of A B C D E P . . =17000 And theareaofFGHISL . . =32085

Difference of the areas =15085

Twice the difference of areas . . . =30170

Therefore =30 J links.

Thus by dividing twice the difference of the areas by the total length of the line A L=990 gives 30 links ; and if this 30 links is set off perpendicular from the point as L it will give a point which if a line is again set out from wiU exactly equalize the proposed irregular boundary and reduce the field into two large triangles by the lines A a and A which may be treated as previously explained for triangles.

The surveyor wishing to discover the area of the figure ABODE from one station plants his theodolite at S, Fig. 74, and measuring the angles A S B S C S D S and

Fig. 74.

sides S A, S B, S C, S D, and S E, are then measured, and found to be =800, 790, 575, 920, and 940 links respectively.

Putting =area of each triangle separately, the contents

140 Calculations Of Aheas.

of the whole figure may be found by any of the following

rules: —

sine z A S B=a?.

Or logarithmically thus —

Log. A S xlog. B S xlog. sine A S B — 10=a?.

AS,BS 800x790

X sine A S x .941666=297566.456

B S, C S 790x575

x „ z B S X .900065 =204427.263

CS, D S 575x920

— X „ z C S X .941666=249070.657

2

DS, ES 920x940

— x „ z D S X .928485 =401476.914

ES,AS 940x800

' X „ zESA= X. 998598=375472.848

A=15.28014.138

1.12056.552

4.82262.080

Therefore the contents of the figure A B C D E=to 15 acres 1 rood and 4.82 perches.

To prove the First Triangle A S B logariihmicaUy.

Logarithm 400 + 2.6020600

„ 790 2.8976271

Log. sine z 70 20' . . . =4- 9.9738971

297566.5482= 5.4735842

This last operation gives the area .09 of a square link in excess of the first operation of the natural sines.

Given the three sides of a triangle H B C=1200, 1600, and 2000 links, to find the area.

Calculations Of Areas. 141

Putting A=area a b and c=tliree several sides of the triangle and =lialf sum of the three sides we have : —

Or logarithmically — Logarithm A=i {log. *+log. a) 4- log. log. c).}

1200+1600+2000= =2400

2400-1200=1200

2400-1600= 800

2400-2000= 400

2400 +1200+ 800+ 400=921600000000 Unks,

And 'y921600000000=960000.=area.

By Logarithms.

The logarithm of 2400 . . . 3.3802112

„ „ 1200 . . . 3.0791812

„ „ 800 ... 2.9030900

„ „ 400 .. . 2.6020600

11.9645424

Arc of triangle =960000 .. . . 5.9822712

k

A New Set Of Tables Of Distances

Planes Of Meridian And Latitude,

Ob

Traverse Tables,

Calculated To Every Two Minutes In The Quadrant,

▲Nd

By Differences To Twenty Seconds,

Fob Ant Length Of Lines

irp TO

Five Hundred Thousand.

Explanation To The Traverse

Tables.

Trayersb Tables are those wherein are arranged in a tabular form, the two sides of all right-angled triangles to a given hypothennse and angle. Such a table properly arranged and calculated to a minute division of the circle suitable to improved modem instruments, would be invaluable to practical men ; and when we take into consideration the great amount of time and labour expended in the necessary direct calculations required in business, such a table (by the aid of which equal results may be obtained by inspection) must be considered a desideratum. Such a table I have not been able to procure — and, indeed, believe such a one does not exist in print — those that I have procured are badly arranged, and very inaccurate and incomplete, arising partly from the circumstance of their not having been calculated to a less subdivision than from 30' to 15' of a degree, which renders them comparatively useless for practical purposes when instruments are employed divided to single minutes, and oftentimes to 20'' ; and it is still more surprising that such tables, published nearly forty years since, should again be reprinted at the present date without the necessary alterations, which would render them more valuable when adapted to finely graduated instruments, for it is certain that a table calculated to every 1' or 2', or even to 20", could be used, and made available for an inst]:ument whose graduations were not less than from 30' to 15' ; but a table calculated for the latter division only could not be used (without much trouble) for observations taken with instruments graduated to the former division ; and to use the words of a very distinguished author — "They are necessarily very imperfect, because, to be complete, they should be very voluminous/ Under these circumstances, and for other reasons, I have therefore been induced to compute a new set of Traverse Tables, calculated to every two minutes of a degree and by differences io twenty

146 Explanation To Thr Traverse Tables.

seconds, and by applying the decimal system of notation for any length of lines under five hundred thousand. The tables thus arranged are concise, and not very voluminous ; and since they are calculated so fine, and to so great an extent, are consequently much more valuable. The proof sheets of these tables were carefully compared with the MS., corrected, and read over three separate times by different persons each time. A new proof was then taken from the press, compared with a duplicate of the preceding ones, and afterwards every number composing the table recalculated over again. They were then returned to the press for alteration, and finally stereotyped. I have, therefore, no hesitation in presenting to those of my subscribers and others .(who may not find sufficient time or inclination to work from the natural sines or logarithms as taught in the body of the preceding work) a table that will not only supply nearly all that is required in practice, but will be found to be as correct and reliable as any tables of this class can possibly be made. Indeed, after all the labour and care bestowed, I cannot suppose that a single error has crept in ; but aft;er the last proof of each page was read, and aU is said and done, it is quite possible that some accident might happen to the type while in press, and before stereotyping, thus introducing errors over which the author could have no possible control. However, I am not apprehensive on that account, as the work was printed by one of the best and most careftd houses* in London.

Nevertheless, if any of my readers should happen to discover any error either in the tables or body of the work, I should feel obliged by their communications, in order that the same may be put right in ftiture editions.

The table is in five double or pairs of colunms, and are base and perpendicular the commencement alternately, and are carried to five decimal places for the first five degrees, and to six places all through the table afterwards. The hypothenuse which in all cases is the measured distance, wiU be found at the tops and bottoms of the pages as 1, 2, 3, &c., to 5 ; or 10 20 80, &c., to 60; or 100, 200, to 500; or 1000, 2000, 3000, 8cc., to 5000 ; or 10,000, 20,000, to 50,000 ; or 100,000, 200,000, &c., to 500,000 ; and the angle* in degrees and minutes in the first and last columns of every page, the first reading downwards, as 2', 4', &c., to CO', and the last 0"" SO', 58', &c., to 2' upwards. As an illustration of the use of the tables, let us proceed to find the basQ and perpendictdar to an angle of 4i2', and hypotheiiuse

Sxflanation To Thb Travbrsb Tables.

Ul

measured distance of 10000 links. Find the angle the head of the first column the minutes may be found by running the eye downwards until we find 42', we then refer to the head of the first double column, and under 1 for measured distance, pr 10,000 links. Take out of the column called distance plan of latitude and opposite 42', the number 0.990983,* and that called distance from planes, of meridian, and opposite .42', the number 0.133986, which will express the length of the base, and perpendicular for the proposed hypothenuse and angle ; but these numbers thus found are decimals, we wish, therefore, to know the exact length of the lines expressed by them m integers and parts of an integer. The student will observe as a rule that the decimal nimibers in each column are calculated to the common integral units 1, 2, 8, 4, and 5, and when each integral unit is made to' represent 10, 20, 30> 40, and 50, to find the corresponding imits and parts in their respective columns the decimal point must be placed one figure to the right, and when each integral unit is made to represent 100, 1000, 10,000, or 100,000, the decimal point must be removed two, three, four, and five places to the right, the figures to the left of the decimal point so supplied will be integers, or whole numbers, and those to the right, if any, parts of an integer or decimals. Thus the base and perpendicular previously found is =9909.83, and 1345.62.

Tojind the Base and Perpendicular to an Angle 58'

and Hypothenuse 3424 Links.

The angle 7° 58' maj be found in the table as previously explained, but the measured distance, or hypothenuse, must be taken out at two or more times, thus : —

Hypothennae.

Base.

Perpendicular.

. - 2971.044 . .

, . 415.79100

. 396.139 . ,

. . 55.43880

. 19.069 . ,

, . 2.77194

. 3.961 . ,

. . 0.55438

. 3390.213 . ,

. . 474.55612

Therefore the base and perpendicular =3390.21 Sand 474.55612 links respectively.

The degrees are represented by figures larger and blacker than those for the minutes.

t2

Explanation To The Traverse Tables.

Take another case when the angle hypothenuse =94243 links :—

Hjpoiheniue.

Base.

Perpendicular.

50000 . .

. =49783.500 . .

. 4647.450

40000 . .

. =39826.800- . .

. 3717.960

. 3982.680 . .

. 371.796

. 199.134 . .

. 39.826 . .

. 2.987 . .

94243 . .

. =93834.927 . .

. 8759.790

The required base and perpendicular is therefore =93834.927 and 8759.790 links.

To find the Base and Perpendicular to an Angle of T 31' 30" and Hypothenuse 1000 Links.

Rule. — Take out of the table the base and perpendicular corresponding to the next lesser and greater angle than the one proposed the difference of the bases, divided by 2, and deducted &om the number, expressing the base, to the larger angle, will be required base ; and the difference of the perpendicular divided by 2, and added to the number, expressing the perpendicular for the lesser angle, will be the required perpendicular.

Next lesser angle =7° 30' Next larger angle

Difference

Example.

Base. Perpendicular.

991.444 138.526

991.368 131.103

0.076

difference 0.577

One minute =:half difference =0.038 half difference =0.288 Half minute quarter distance =0.019 quarter difference =0.144

Base.

Next larger angle =7® 30 0"= 991.444 Half difference 10 =-000.038

31 991.406 Quarter difference =-000.019

7 31 80

Perpendicolar.

+000.288

+ 000.144

BXFLiLNATIOlfr TO THE TRAVERSE, TABLES. 149

Therefore the required base and perpendicular is =991.387 and 130.958.

Many of the problems proposed in the body of the work may be solved by the tables; we will, therefore, prove a few of them : —

Problem 7th in Plane Trigonometry

Hypotheniue. Base. Perpendicular.

z 20"" 20' and distance 200 . 187.5372 . 69.4962

zl5 40 „ „ 300 . 288.8547 . 81.0120

_(300 . 274.0635 . 122.0208

/ 24 „ '' " C 40 . 36.5418 . 16.2694

_C300 . 281.9076 . 102.6060

F G, Pig. 38 =1350.8184 . 494.0104

Data as found trigonometrically 1350.8131 494.0077

Difference in calculation .0003 and .0024

Thus the base and perpendicular differs only .0003 and .0024 &om the operation at page 31, showing the great accuracy and utility of the tables in performing by addition what we should otherwise be obliged to do by the tedious process of multiplication.

Problem in Plane Trigonometry.

z 40° 16' and distance =849, to find A B and B C.

z 40"" 16']

Hypothenuse. Base. Perpendicular.

500 .. . 381.522 . . . 323.173

300 .. . 228.913 . . . 193.si03

40 . . . 30.521 . . . 25.853

9 . . . 6.867 . . . 5.817

A C=849 A B=647.823 and B 0=548.746

By logarithmic process =647.820 „ „ =584.750

Difference in calculations .003 .004

15Q Explanation To The Traverse Tables:

Problem 15th, Oblique Trigonometry" by the Tables.

{See Fig. 46.)

Hypothe- Larger Segment nuse. of Base.

300 . =230.5965

Z 39°46 300 . =230.5965

[ 40 . 30.7462

640 =491.9392

Hypothe- Larger Segment nuse. of Base.

(400 . =196.5640

z60°34'] 50 . 24.5706

( 20 . 9.8282

470 =230.9628

Then 491.9392 + 230.9628 722.9020 A B Data by logarithms . . 722.9300

Diflference .0280

The calculation by the taCbles comes up to the logarithmic conjputation, a;id differs but .080 of a , unit from it. The length of the Ime let fall from the apex of the triangle at C perpendicular to the base coidd have been found at the same time by taking out the quantities* opposite the bases. All the calculated data in OflSce Forms No. 1 and 2, may readily be found by merely inspecting the table ; but it must be remembered that the work and angles must be properly stated before any reference to the table is made — that is, the observed reduced angles are all to be referred to the plane of the meridian or latitude, according as they are found with respect to these lines. Thus in the table for measured distance 330 and angle 28 6', we find 262.5549 and 145.1373 for the distance from planes latitude and meridian, or the base and perpendicular ; but for 4', found at the bottom of the page, the same numbers become meridian and latitude ; that is, for the degrees at the top of the page each pair of columns read base and perpendicular downwards ; but for the degrees at the bottom the same columns read perpendicular and base upwards.

tllA VERSE

Tables.

gownal 1

eo| poi fHusa 1

g

IS5"s

§

1

& 1'

t s

Joku-W

dotsoo

f -B

§ 8"

d d 0' d d

d d d d d

1%

1 1"

1 S

S'

8'

Hutnia

-K,Qn.ia

s

B s

HUnu JO

e %

Irt

B' E!

dotio

d d d d d

=43

P

Ck.Ow

WUIl'HI

11}

Is

"1

0.0.01 0.0

nuiqiia

Miipibk

jDBuiid . )r..s 1

dcidood

d d d d d

Im

OOr-t-,-

OBOOf-

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Traverse Tables.

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TRAVEItSB TABLES.

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Introduction

To Thb

Tables Of Natural Sines, Secants, Tangents, Etc.

Ths method of computing with sines cosines, &c. is due to the invention of the Arabians, who were the first to improve on the ancient practice of calculating trigonometrical quantities by means of chords, in which state it was received by them from the Greeks. The science of trigonometry came to us from the Arabians, and has, since that period, received important alterations and improvements from many eminent European mathematicians, among whom may be mentioned '' Purbech,'' and his friend 'Begiomontanus,'' whose tables are little inferior to those we possess in. these times.

Lines drawn in and about a circle whose radius is=unity have been taken by these mathematicians as a standard measurement for triangles, and there may be any number of such triangles formed upon the radius, depending on the minuteness to which the circumference of the circle is conceived to be divided to. The length of a series of these lines, called or denominated sines, cosines, secants, tangents, &c., have at different times been determined to every minute of arc of the circle, arranged and tabulated ready for use. Such a table I have prepared to six decimal places (for the present work), and in their compilation inuph care has been bestowed in order to render the table as accurate as possible ; for which purpose they have not only been compared with Dr. Button's best edition, but with those of others equally reliable. The number answering to any degree or minute may be taken out of the table by inspection at sights by fiildingtiie'

198 Introduction To The Tables Of

number of degrees at the top or bottom of each page, and the single minutes down every first or up every last column.

If the sine, cosine, &c., answering to degrees, minutes, and seconds are wanted, they may be found in the following manner by differences : —

Suppose the sine of 10° 2V 20'' is required :—

Natural sine. . . 1021' (K'= .179661 Next less natural sine 10 20 .179375

J) 1 J) .000286 difference for 1' "O 20 -h .000095 „ ,, 20''

Therefore .179661 + .000095 .179756 / 10° 21' 20"

Required the Cosine of 10° 2V 20"-

Natural cosine of . '. 10° 20' 0"= .983781

next less 10 21 .983729

Jj 33.

i) 1 j).oooo5a

20 - .00001 7 difference for 20" TkCTefa* .988729 - .000017 .988712 /. 10° 21' 20".

To find the Sine of 40" 10' 10".

Next higbor natural smeO" 11' 0"= .646236 Nearest given „ „ 40 10 .645018

i) 1 i ) .000223 diff. for 1' Of 10 .000037 .. .. 10

M

Therefore .645013 +.000037=. 645050 s=5 / 40° 10' 10".

The proportional part found in this manner for any number of odd seconds must be added for sines, secants, and tangents, but subtracted for cosines, cosecants, and cotangents.

The tables of natural sines may be computed in the following manner: —

1. K we take A B C as a quadrant, or fourth part of the circle A B D E, and call the radius C B unity or=to 1, and conceive it to be divided into an indefinite number of decimal parts as 1000000, &c.; and if we take the chord B F=to the radius the arc subtended by it will be=:to 60°, because in every circle the radius is equal to the chord of 60. Now if we draw the

Hue Or ¥, tbe tine of tlie arc at right angles to C and also the cosine F H at right angles to C bisect B F in then will IS be the sine, and I N the cosine of the arc of I; bisect the chord of 80 or B I in O, then M O will beto the allie and T O the oosine of the arc of ISasB (X Now if we. continue the process until we have made 12 bisections, we should determine the arc of 0° (T 62'' W' 2/'" the cosine of which would uppioximate so closely to the radius C B, that it may be

taken, without sensible error, as of equal value, and the numerical measured value of this arc may be obtained by calculation as

follows : —

Euclid's corollary to the 15 Prop, of his 4 Book, teaches that the chord line B F, Fig. 75, is the side of a hexagon, inscribed in a circle, being the subteme of 60, and consequently, as before stated,=soothe radius C B. Now, if we take half the radius, or fi L, it wiU be equal to the sine of 80=sl S, which, consequently, must be =5000000. The sine of 80, having been thus fotmC

200 Introduction To The Tablbs Op

its cosine may also be deduced from it by the 47 Prop, of the 1 Book of Euclid ; for in the right-angled triangle I C S the hypothenuse 10 100000 and the perpendicular I S .500000 to find the base C S=:the cosine I N.

/. n/C S X C I-I S X I S=C N .8660254= I N, consequently the sine of and its cosine .8660254.

2nd. The perpendicular I S in the triangle I S B is

.5000000/ and the base CB-CS=SB .1339745, to find the

hypothenuse B I.

, .51763809

Therefore /I Sxl S + S BxS B=B

.25881904, or O T, its equal the sine of 15°, the cosine belonging

may now easily be found from it, for in the triangle OCT the

hypothenuse and perpendicular are given =100000 and .25881904

respectively, to find the base T C, or the cosine O M.

Therefore n/O CxO C-0 TxO T=T .96592582, consequently the sine of 15° is .25881904, and cosine =.96592582.

Now, if we were to continue the same process until we had attaitied the 12th bisection, we should arrive at the sine .000556634=to the arc 0° 0' 5' 44'' S'"' 4&"''\ This arc only differs from that of 1 minute by QP Q' T' m'" \W"", and as small arcs are nearly proportional to their corresponding sines, the measure of of arc may be deduced from the small

To find the Sine of V.

Ride.— As the arc 52'' 44'" 3"" 45'"" is to the arc of 1', so is the sine — previously found — to the sine of an arc of 1'.

Ea:ampl€.—As 52" 44'" 3"" 45'"" : 1' : : .00025566 : .00029088, which is therefore=to the sine of 1', and the cosine answering to this is nearly equal to radius.

To find the Sine of 2'.

Example.— Ab radius (1) : 2 : : .00029088 : .0005817 — ,0005817.

To find the Sine of 3'.

Example.— As radius (1) : 2 ; : .0005817 2 .0011635 and .0011635 -.0002908=.0008726-the sine of three minutes. By the same kind of operation the sines may be foimd to 30° or 60° iHe tables computed to te end by dditioja only.

Katubal 8Inb8, Secants, Tangents, Etc. 201

The tangents, cotangents, secants, and cosecants may also be found from the sines and cosines as follows. When referring to Fig. 75, the roman large letters represent lines, and the small ones — as a, b, and e — the arc of any circle to radius 1.

(Sine a)* + (cosecant a)' 1 . (Secant a)* — (tangent a)' 1 .

(Cosecant a)* — (cotangent a)*=l.

For in Fig. 75, in the right-angled triangle P A F H C, and K B, we have : —

F + H C? F C? (sine a)* + (cosecant a)*=l. C P-AP AC?/. (secant a) — (tangent a)=l. O — B C .-. (cosecant a)* —(cotangent a)*=l.

sine a cosine a

Again, for — : ; cotangent — ;; ; tangent

cosme a sme a

cotangent a.

For .3689702 tangent of 20

And ' JZ74i74i774i cotangent of 20

.8420201

For, by similar triangles in the same figure we have : —

A P P H sine a

tangent a

Ac Hc

Bk Hc

A B C B

tangent a

cosine a

cosine a

sine a

C B B K cotangent a

100000000

For .8639702 tangent of 2QP.

.27474774

Also to prove the secants and cosecants as follows : — Secant a — : ; cosme a

cosme a sme a

202 INTRODUCTION TO TABLES OF NATUBAL SINES, IfcTC.

C P

For we hare

C A

C F 1

— .. secant a — C H sine a

Ck

Op 1

C B

F H cosine a

For

1.0641778 secant of 20°.

. , 1000000

And

2.9238044 cosecant of 20°.

New tables may be thus constructed or the accuracy of those already calculated proved, from the commencement to the end of the quadrant.

TABLES 07 KATtJRAL SnTBS, SECANTS, ETC.

Ooeinea.

0'

O.OtMWOO

l.OOOOOO

8437,7488

1437.7487

0.0Uo583

1,000000

1718.8736

1718.8732

Es

0.00O873

1,000000

1146 9167

I14G.9153

869,48689

869.43630

G8

fi

687.64980

887.54887

o!99eB9

i.oooooa

573.95809

0.0O1T46

672.96721

0.00203S

491.10702

491.10800

B

0.0023-27

429.71873

429.71767

e

881.97230

881.79099

843.77518

843.77871

Go

0.003S0O

312.62297

313.62137

Is

388,47943

286.17773

Is

0,099903

364.44289

0. 003782

261.44080

246.S5402

215.55198

Is

229.18Ssb

229.18168

314.Ssb95

214.86783

0.004Ms

0.Bw93B

203.22123

202.31875

0.S90038

l.OOOOU

190.98880

190-98419

Is

0.00553T

180.93496

180.93220

So

O.0Os818

i.ooooir

171.88831

171.88640

0.0O6I0B

1S3.7032S

183.70019

S3

O.0O640O

0.(t99980

158.26223

0. 008400

158.36908

149.46837

149.48602

M

143,24061

113.28713

as

0.O0T17S

187.51108

137.50745

H

0.Oot568

182,32229

132.21861

n

J.0Oo031

127.32620

127.32134

3S

H

122.77803

0.00811 S

122.77398

se

1.0O0O38

118.51440

118.64018

so

114.69301

114.6S366

0. 0090 17

0,999960

110.89658

110.89205

So

sa

107.43114

107.12648

Si

0.00B6M

0.999B54

104.17674

101.17094

0.00B8M

101.11185

101.10690

So

H

98.233033

98.217943

0.01O4T9

95.494711

95.489476

It

92.913869

92.903487

S8

90.468883

90.463338

S8. 14 9244

88.143672

O.01163S

0.S9998S

85.946809

85.039791

S9.S49470

83.813507

4S

0.0122It

81.853150

81.847041

0.01 2S08

0.999U22

79.949484

79.943130

It

D.999B1S

78.132748

78.128343

4S

0,013090

76.396664

0.01 3091

78.890009

74.735858

71,729186

4T

73.146827

78.138991

71.823862

71.616070

70.160474

70.163346

Go

68.757360

88.760087

67.409272

67.401854

6G.1 13036

66.106473

84.866716

61.858008

E4

83.684595

83.668741

e

Gs

82.607153

62.499164

M

0.016E89

61.891050

81.882905

St

60.814110

60.306820

0.0Ib871

50.274308

69.286872

68.289765

68.261174

1.0001E2

57.298088

67.289962

Coriae.u

Sine*.

Tablbs Ot Natubal Sikbs, 8Ecakts, Etc.

r

"07999848

~i7oob'i52'

O.0174S2

67.298088

67.289962

60'

0.01 T743

0.99Ii843

£6.3.':9462

0,017746

66 850580

s

G.'i. 450534

66.441 51T

&3

64.670461

0,018328

61.561300

63.717896

63.708687

s

63.891564

62.882109

0. 01 91 97

1.0001S4

62.090272

62.080673

61.312902

61.303167

Ss

S

60.668396

0,019783

G0.61S606

S

49.825763

0,020071

49.816726

0.020Ss1

49.114062

0.020S66

19.103881

48.422*11

18.41208*

la

47.749974

17.789501

0.98B776

47.095901

17.085313

u

46.469626

0,021529

16.448862

46.840200

45.829361.

Is

45.237105

45.226141

44.89796

0. 023402

*4.6S8598

Ib

44-077458

11.066113

O.022078

43.619612

0,022984

13,608122

42.976713

12.081077

1,000278

42.446245

12.133*64

41.927717

41.815700

41.433660

41.410588

0.Bb9702

40.929630

10,917412

40.448201

a 024731

40,436837

39.877969

89,966460

39.51S549

39.606895

2S

39.069571

38.068771

S3

38.630683

38.617738

38.201660

38.188169

3D

Si

37.781849

37.768613

37.371273

37.367892

36.969528

86.956001

0.O27S40

Ss.6T6333

3S.563869

Sfi

0.0376Si

36.19U14

36.177696

36.814517

86,800653

0. 02821 2

1.00D39S

35.445391

0.02832*

36,431383

1.O0O4O7

3E.803S00

36.069546

84.720615

0,028806

84.716116

to

84.382319

84.387771

tl

1.O0O432

84.041994

34.027303

1.O0O441

33.708346

Ss.89S60B

33.381878

S3.368194

83.060300

33.016173

Is

82.746537

32.730264

32.436713

32.121296

0. 0996 16

0,031136

32.118099

31.836225

31.820516

4e

0,031702

31.6J424S

0,031717

31.528393

31.257677

31.241577

Bl

80.97607*

30.969928

80,699698

30.683307

G3

30.428077

30.411680

30.181201

30,144619

29.899026

29,882299

1.Ou067O

38.841373

29.624499

0.03402T

0,999421

39.38812*

39.371108

0,034S18

29,139160

29,123006

0,999401

28,894398

38.877080

28.653708

38.638263

Cosine..

Snm.

Coeocuiis.

88°

Tables Op Natceal Sines, Secants, Btc.

2°

Coseeanle.

Tmgenta.

0'

28.863708

28.636263

0,036190

23.416997

28,399397

1. 000630

28.184168

28.166422

27.966126

27.937233

27.729777

0.036D36

27.711740

27.608036

27.4S9363

2T.2S9811

27.271486

0.03G034

27.076030

27.056567

0,0372S5

1.000B94

26.863603

20-844981

26.655466

26.636690

0.037B07

26.450610

26.131600

So

36.248691

26.329638

2B.04S937

26.031730

2.'>.86416B

26.834823

26.861324

26.641832

26.471337

26.451700

26.284144

26.264301

26.099686

26.079767

Is

24.917900

21.897820

24.738731

21.718512

21.662123

21.641768

ai

21.388020

21.367609

21.216370

24.195714

21.047121

24.026320

23.S80221

23.869277

2S

0,999111

23.716630

23.694637

0,990098

23.663291

0.04 2196

23.632062

0.99903S

33.393161

23.371.777

23.286196

23.213686

33.079361

0,043370

23.057877

ai

So

23.926686

22.903786

22.77385T

22.761892

1.0011978

22.624126

22.002016

22.470352

23.454096

22.330492

22.308097

22.186528

22.163980

32.0M403

22.021710

i.oeioi4

21.904090

21.881261

0.04 604 4

21.766660

21.743666

0.9Bs931

31.828769

0.01684

21.606630

0.0466:&

0.99S917

1.001 081

31.493676

31.470101

0.04081 S

21.360373

21.330861

0.99Sb90

21.228616

21.204819

1. 001126

31.098378

21.071081

30.969821

20.945906

0.B03848

1,001153

30.843830

20.818828

20.717308

20.663220

U

20.693409

20.689116

4S

o.oisaso

O.998S0S

30.170926

20.440180

0.04B141

30.349893

20.326308

20.380384

20.206663

0.04972!

20.112076

20.087199

S3

0,060012

19.996241

19.970319

19.879768

19.864961

19.765804

19.710291

0.99S706

19.662764

19.027296

19.641187

18.618684

0.05146.

19.430882

19.406133

S

. 0.998600

19.321816

19.295932

16.213970

19.187930

19.107322

19.081137

Cosisra.

Cosecaats.

Secanli.

Tangents.

87°

Tables Of Hatubal Sines, 8Ecakt8, Etc._

Slaet 1 CBLneiL

Secanto.

Cosecant*.

Tangenla.

0.017*52

67.298683

57.289962

ev

5.So8162

66.360580

0. 01 8034

9B9837

65.150531

56.111517

0.01 8325

64.570461

61.561300

1.O00171

53.717896

63.708537

0.018So7

52.881564

52.882108

o.oieiOT

1,000184

62.090372

62.080673

51.312803

51.303157

50.558396

60.518503

19.825762

19.815726

19.111062

49.103881

0.020Be3

18.422411

48.112084

47.749974

47.739501

17.096861

47.085313

1. 000233

16.168625

16.448862

15.810260

45.829351,

8Bs76S

45.237185

45.226111

44.68795

11.633596

44.077458

14,066113

88873S

13.519612

13.608123

42.975713

12.964077

Si

42.445245

42.433161

0.03861

41.827717

41.915780

11.422660

0,021118

11.110583

40.929630

10.917412

10.418201

40.135837

O.026014

89.977969

39.966160

89.618519

38.605886

88.068571

38.056771

2a

0.02688S

1,000335

38.630683

38.617738

0,026177

1,000318

33.201660

38.188168

37.781819

87.768613

87.371273

87.357882

S3

36.969528

36.956001

1,000371

88.676332

36.663659

1,000382

36.191111

86.177596

se

35.811617

36,800653

0.O2S212

1,000398

86.115391

35.131282

35.803800

0.D235I5

35.068516

3a

31.730516

31.715115

999S77

31,382316

34.367771

S4.041994

31.027303

la

83,708316

33.693509

Ib

0.029B67

33.381976

O.02Bb71

33.366194

33.060300

33,045173

Is

Is

0,030539

82.715537

32.730261

1.0i,'O478

82.136718

32.421286

82.133663

82.118099

1.0001S4

81.836225

81.820516

Ib

89S487

31.541216

81.528392

Bo

31,257577

31.211577

]. 000523

30.976071

30.850928

£2

30.699598

30.633307

Bs

0.032S61

30.428077

30.111580

T

30.161201

30,141619

0.03311S

29.899026

0.03S165

29.882299

29.611373

29.621199

29.388124

29.371108

E8

29.139168

29.122005

38.891398

23.877089

28.653708

28.636253

Tangent..

8?

TABtES OF KATDSAL SINES, SBCANTS, BTC.

2"

Secanta.

Tungents.

0.0319oa

9B33Bi

28.653708

0.034B21

28.636253

80'

B9B3S1

28.116997

99B370

28.1S416B

28! 166422

6S

99B360

27.956125

37,937233

27.729777

0.0360S6

27.711740

27.608035

27.489853

0. 038614

B99328

27.2H9814

27.271186

27.075030

27.066S67

26,863603

26.844984

99B298

26.655166

26, 636690

0. 037807

26.460610

0.03783*

26.131600

0.03S0B7

26.248691

26,229888

0.0383B8

26.D1S937

26.031738

4B

26.851169

26.834823

It

Bb9210

25.66132*

0.0389B9

25.841832

9B922B

26.171337

26.451700

B9B218

26.234111

26,261361

B99206

26.099686

26.079757

Bb9191

21.917900

0.01018*

21.897826

Is

1.0O0S13

24.738731

3*.718612

0.01(]713

21.562123

0.0107*7

2*. 54 1758

24.388020

21.367508

24.216370

21.19571*

S8

21.0*7121

21.026320

S7

23.880321

23.859277

S6

23.716680

23.691637

S6

9990B8

23.653291

0.0121B6

23.633053

23.393161

23.371.777

23.285196

23.213666

23.079361

0.0*3370

23.067077

22.925686

22.903766

0,043910

22.773857

22.761892

22.624126

0. 014 214

22.602016

22. 176352

23.161096

at

998Bu7

22.330*92

O.044827

22.308097

23.186628

22.103980

22.011103

0,046410

22.021710

21.901090

0.01S701

21.881251

0.0J5944

21.766560

0.0169B3

21.742589

998B31

21.628759

21.605630

1.00108*

21.493676

21.470101

21.360273

21.Ss6861

l.OOlllI

21.228515

0.04 71 58

21.204948

B9S&76

1.00U25

21.098376

21.071661

20.909824

20.846966

B93848

20.842830

30.818828

Bb8831

20.717868

20.683220

9B8S20

20.693109

20.569116

4B

20.170936

20.1*6486

0.04B141

20.319893

30.325308

20.230284

20.305553

20.112076

20.087199

6'i

19.9952*1

19.970219

0. 050303

998T34

19.879758

16.854951

19.766604

18.710261

sa

19.652764

O.0.W950

16.637288

19.641187

0.0512*1

19.61868*

Ib. 430882

19-*05133

19.321818

18.285922

Ib. 213970

18.187930

19.107322

18.081137

Cosines.

Bines.

eecanti.

Tingenla.

ar

206 TABLES OF NATURAL SINKS, SECANTS, ETa

Sines.

Coflines.

Tangents.

Cotangents.

19.107328

19.081137

60'

19.101854

18.975523

18.897545

18.871068

18.794377

18.767754

18.692830

18.665562

18.591887

18.564473

18.491530

18.464471

18.892742

18.365537

18.295005

18.267654

18.198303

18.170807

18.102619

18.074977

18.007937

17.980150

17.914243

17.886310

17.821520

17.798442

17.729753

17.701529

17.638928

17.610559

17.549030

17.520516

17.460046

17.431385

17.371960

17.343155

17.284761

17.255809

17.198434

17.169387

1,001712.

17.112966

17.088724

17.028846

16.998957

16.944559

16.915025

1.861594

16.831915

16.779439

16.749614

16.698082

16.668112

16.617512

16.587396

16.537717

16.607456

16.458686

16.428279

16.880408

16.849855

16.802878

16.272174

16.226069

16.195225

16.149987

16.118998

16.074617

16.043482

15.999948

15.968667

15.925971

15.894545

15.852676

15.821105

15.780050

15.748387

15.708096

0,063791

15.676283

15.636798

15.604784

15.566135

15.588981

15.496114

15.468814

15.426721

15.394276

15.357949

15.325358

15.289789

15.257052

15.222231

15.189349

15.155270

15.122242

15.088896

15.055723

15.023103

14.989784

14.957882

14.924417

14.893226

14.859616

14.829128

14.795372

14.765580

14.781679

14.702576

14.668529

14.640109

14.605916

14.578172

14.548833

14.516756

14.482273

14.455859

14.421280

14.395471

14.860696

14.335587

14.800666

86

Oosines.

Sines.

Cofiecants.

Secants.

Cotangents.

Tangents.

XABLB8 or HATDBAL SINES, SBCAITTS, ITO.

4"

Coubm.

BMUrt>.

CoBBianta. T

0.O6Bts7

14.336687

11.300688

oo-

14.378200

14.241134

0.9fi7S23

14.217304

14.182092

14.168Sb4

14.123588

0.070B17

14.100963

11.065169

14.043604

14.007866

13.886514

13.950719

13.929985

13.894046

13.873913

13.837827

1S.918291

13.782060

13.763116

13.726Tbs

0,997838

13.708879

18.871868

0.0782S8

13.661077

13.617408

0.9972Bs

18,600206

18.563391

0,997273

13,516768

13.609799

0.S97250

1,002757

13.493731

13.4,'.6625

0.0743B9

1,002779

13.441118

13.103867

13.888914

13.351518

1,002823

13.337118

18.299571

Is

13.385719

13.218031

so

0.07Ss69

13.234717

18.196883

ai

13.184106

18.146127

13.133882

18.096767

18.08040*

13.045769

1,002868

13.034576

12.998160

13,985488

12.946924

3S

12.936765

12.896058

0.98H985

1,003024

12.888410

13.849,W7

S3

sa

0.9M963

I.003O48

12.840416

13.801117

12.782779

07S409

13.753634

So

13.745495

13.706206

Si

1.00311 B

12.608500

12.869125

1.00H138

12.651971

13.812390

0.9116849

12,606724

12.565997

O.079H19

12.659816

12.519942

3fi

1,003208

18.614240

12.171221

12.468995

12.128881

0,996766

12.424078

12,383788

1.0U327B

I2.37l'481

12.3S9028

1,003308

12.336210

12.294609

12.291263

12.260606

1,008860

12.247808

12.206716

0.081B39

12.201274

0Ss21S

12.163238

12.161248

12.120063

0,896690

12.118622

12.077192

Is

0. 082808

12.076098

12.031622

la

12.038970

0833S7

11.992319

4T

0.Os3388

11.992137

11.960370

4S

0,996493

11.950596

11.808682

1,003544

11.909340

11.867283

11.868370

11.828167

11.827683

11.786338

0,084887

1.0O3838

11.784274

11.741779

11.747111

11,701600

£4

0,906346

11.707282

11.864495

11.667693

11.624761

0,99Hs96

11.628372

11.585291

11.546093

11.650623

11.607161

0,996220

11.611590

11.468171

11.473713

11.430062

85°"

Crmiaen.

Sines.

Caaeeaat*.

SceaiiUi. (k

TABUU OF KATUBAL SINESf SKCANTS, KTC:

&u

CMdna. a

Cotugents.