Ex Libris 
 C. K. OGDEN I 
 
 THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 OF CALIFORNIA 
 
 LOS ANGELES
 
 A TREATISE 
 
 EMPLOYED IN 
 
 SURVEYING, LEVELLING, AND ASTEONOMY, 
 
 fyc. 8fc.
 
 A TREATISE 
 
 ON THK 
 
 PRINCIPAL 
 MATHEMATICAL INSTRUMENTS 
 
 EMPLOYED IN 
 
 SURVEYING, LEVELLING, AND ASTRONOMY 
 
 EXPLAINING THEIR 
 
 CONSTRUCTION, ADJUSTMENTS, AND USE 
 
 u:, an& Cables, 
 
 FREDERICK W. SIMMS, F.R.A.S., F.G.S., M.lNS.C.E., 
 
 CIVIL ENGINEER, 
 
 AUTHOR OP A TREATISE ON PRACTICAL TUNNELLING, &C. &C. 
 
 FIFTH EDITION. 
 
 LONDON : 
 MESSRS. TROUGHTON AND SIMMS, 138, FLEET STREET. 
 
 '1844,
 
 <ntfrrtf at Stationer*' $>all. 
 
 Tyler & Reed, Printers, Bolt-court, London.
 
 A 
 
 1844 
 PREFACE 
 
 TO THE FIRST EDITION. 
 
 THE want of a work containing a concise and popular descrip- 
 tion of the principal Instruments used in Practical Astronomy 
 and Surveying has long been felt, as the requisite information 
 with respect to such instruments can only be obtained by con- 
 sulting various expensive publications, which are not within tho 
 reach of many to whom such information is highly interesting 
 and important. 
 
 It was the original object of the writer of this little tract, to 
 place at the disposal of the young surveyor a description of the 
 instruments which are required in his profession, and such an 
 account of the method of examining and rectifying their adjust- 
 ments, as would enable him to obtain from them the most accu- 
 rate results ; but he found that without greatly increasing the 
 size of the book, he might materially add to its utility, by 
 including in his plan the most approved Astronomical Instru- 
 ments, that amateur astronomers as well as scientific travellers 
 might have at hand a manual of instructions, which would 
 enable them to use their instruments with the utmost advantage. 
 
 Usefulness being the author's chief object, he has not scrupled 
 to extract from the works of others whatever he found adapted 
 to his own purpose ; and to some kind literary and scientific 
 friends he is under obligations, for which, if he had obtained 
 their permission, he would be glad to thank them by name in 
 this place. 
 
 /; 2 
 
 : " !: "~/1 '"?' 
 *J
 
 V J PREFACE TO THE FIRST EDITION. 
 
 Of Surveying Instruments, those only have been described 
 which are applied in modern practice, no reference being made 
 to those which, having been superseded by better ones, may be 
 said to be out of use. 
 
 To the article on Levelling has been added a description of 
 Mr. TROUGHTON'S Improved Mountain Barometer, with an 
 easy and accurate method of computing differences of level from 
 barometrical observations. Table II. employed for this purpose, 
 has been carefully recomputed from Mr. BAILY'S Formula?. 
 The other Tables will, for their several purposes, be found con- 
 venient and useful. Tables I. and VIII. are new. 
 
 Much attention has been paid to the accuracy of the formulae 
 given for performing the various computations, and each has 
 been thrown into the form of a practical rule, that persons un- 
 acquainted with algebraic notation may be enabled notwith- 
 standing to make the requisite calculations. 
 
 With respect to such astronomical problems as appertain 
 chiefly to Navigation, and require extensive and special tables 
 for their convenient solution, it has been thought better to omit 
 all reference to them in this work, as in Mr. RIDDLE'S Treatise 
 on Navigation, Captain THOMPSON'S Lunar and Horary Tables, 
 and other similar works, all necessary information on the subject 
 may be readily obtained. 
 
 The Appendix relates chiefly to the protraction of the work 
 after a survey has been completed, and seems a suitable supple- 
 ment to the account of Surveying Instruments given in the pre- 
 ceding part of this treatise.
 
 TO THE SECOND EDITION. 
 
 IN preparing for the press a second edition of my " Treatise 
 on Mathematical Instruments," I have endeavoured to make 
 such additions and improvements as would render it still 
 more acceptable to the reader. To the account of Surveying 
 Instruments I have added a few remarks on the use of the 
 Land Chain ; and given some additional particulars, with an 
 engraving of Captain EVEREST'S Theodolite, which has hitherto 
 been extensively used in India, and is now frequently 
 employed in this country. To the Levelling Instruments 
 I have added a representation and description of Mr. 
 GRAVATT'S modification of the Spirit Level, and also of the 
 new Levelling Staves ; the article on Levelling has also 
 been remodelled, and I hope will be found by the young be- 
 ginner to contain some useful practical information. 
 
 Through the kind friendship of EDWARD RIDDLE, Esq., 
 I have been enabled to insert his latest improvements in the 
 practical solution of the problem for determining the Longitude 
 by the Moon and Moon-culminating Stars ; which will also be 
 contained in the third edition of his valuable Treatise on 
 Navigation, now in the press. Some additional examples and 
 formula have likewise been given in the account of the Portable 
 Transit Instrument, which it is hoped will not be without their 
 use.
 
 v jjj PREFACE TO THE SECOND EDITION. 
 
 To the Appendix is added an account of the various methods 
 of copying and reducing or enlarging Plans, &c. including a 
 description of the Pentagraph. Also an account of the method 
 of executing a Survey, for the purposes of a Railway or Turn- 
 pike Road. 
 
 Table VII. has been improved in its arrangement, and the 
 last four Tables have been added to this edition. 
 
 GREENWICH, F. W. S. 
 
 Feb. 17, 1836.
 
 CONTENTS. 
 
 SURVEYING INSTRUMENTS. 
 
 The Land Chain Page 1 
 
 The Surveying Cross and Optical Square 3 
 
 The Prismatic Compass ib 
 
 The Vernier 5 
 
 The Plane Table 9 
 
 Method of using the Plane Table . . . . 11 
 
 The Theodolite 14 
 
 Adjustments of the Theodolite 17 
 
 Captain Everest's Theodolite 20 
 
 Method of observing with the Theodolite 22 
 
 LEVELLING INSTRUMENTS. 
 
 The Y Spirit-level 26 
 
 Adjustments of the Y Level 27 
 
 Troughton's Improved Level 28 
 
 Adjustments of Troughton's Improved Level 29 
 
 The Method of determining Distances by a Micrometer Scale 32 
 
 Mr. Gravatt's Level ib. 
 
 Adjustments of Mr. Gravatt's Level 34 
 
 Levelling Staves 36 
 
 Troughton's Levelling Staves 37 
 
 The New Levelling Staves ib. 
 
 On Levelling 38 
 
 With the Spirit Level r .. 39 
 
 Form of Field Book 41 
 
 Method of Reducing Levels 42 
 
 The Theodolite 44 
 
 Description of the Mountain Barometer 46 
 
 Levelling with the Mountain Barometer ///.
 
 CONTENTS. 
 
 ASTRONOMICAL INSTRUMENTS. 
 
 Page 
 
 The Sextant 49 
 
 The Principle of its Construction ib. 
 
 Description of the most approved Sextant 50 
 
 Adjustments of the Sextant 52 
 
 Method of Observing with the Sextant 54 
 
 Troughton's Reflecting Circle 57 
 
 Directions for observing with the Reflecting Circle 58 
 
 The Box Sextant 61 
 
 Adjustments of the Box Sextant 62 
 
 The Artificial Horizon 63 
 
 Roof Horizon ib. 
 
 Plane Glass Horizon 64 
 
 Method of using the Artificial Horizon ib. 
 
 The Dip Sector 65 
 
 Method of observing with the Dip Sector 66 
 
 Tire Portable Transit Instrument 68 
 
 The Adjustments of the Transit Instrument 70 
 
 Methods of determining the Meridional Deviation of Transit 71 
 
 by a Circumpolar Star 72 
 
 by two Circumpolar Stars 73 
 
 by High and Low Star 76 
 
 Method of observing and registering Observations 79 
 
 Finding the Error of a Time-keeper 81 
 
 Correcting observed Transits for the Error of Collimation 84 
 
 ,, ,, for Error of Level 85 
 
 ,, for Meridional Deviation 86 
 
 Method of determining the Longitude by observed Transits of the Moon and 
 
 Moon-culminating Stars 88 
 
 Professor Bessel's Method of finding the Latitude with a Transit Instrument. . 91 
 
 The Altitude and Azimuth Instrument 1)2 
 
 The Adjustments of the Altitude and Azimuth Instrument 95 
 
 Description and Adjustments of the Reading Microscope 97 
 
 Use of the Altitude and Azimuth Instrument 99 
 
 To compute the Reduction to the Meridian 101 
 
 To determine the Latitude of a Place 102 
 
 Method of observing equal Altitudes and Azimuths 103 
 
 Determination of Time by equal Altitudes 104 
 
 the true Meridian by equal Azimuths 106 
 
 ,, ,, by a Circumpolar Star 108 
 
 ,, ,, by the Azimuth of a Celestial Object .... 10!) 
 Method of finding Differences of Latitude and Longitude by Trigonometrical 
 
 Measurement Ill 
 
 *Method of finding the Longitude by the Eclipses of Jupiter's Satellites 112
 
 CONTENTS. x [ 
 
 APPENDIX. 
 
 On Protracting and Plotting, Sfc. and the Instruments employed. 
 
 On the Protraction of Angles Page 114 
 
 Method of performing and plotting a Road Survey 115 
 
 Method of forming a Protractor on Paper 117 
 
 Description of the best Circular Protractor 120 
 
 Plotting Scale 122 
 
 Method of Copying, &c. Plans ib. 
 
 Description and Method of using the Pentagraph 123 
 
 Method of Surveying for a Railway or Turnpike Road 125 
 
 Marine Survey, and description of Station-Pointer, &c 126 
 
 On Plotting Scales 121) 
 
 TABLES. 
 
 I. To reduce the Apparent to the True Level. 
 II. For determining Altitudes with the Barometer. 
 
 III. For Converting Intervals of Sidereal into corresponding Intervals of 
 
 Mean Solar Time. 
 
 IV. For Converting Intervals of Mean Solar into corresponding Intervals 
 
 of Sidereal Time. 
 V. & VI. For computing the Longitude from the Observed Transits of the Moon 
 
 and Moon-culminating Stars. 
 
 VII. For Computing the Reduction to the Meridian. 
 
 VIII. The Length of a Second of Longitude and Latitude in Feet, for dif- 
 ferent Latitudes. 
 IX. Reduction in Links, &c. upon each Chain's Length, in Measuring on an 
 
 Inclined Plane, for various Angles of Elevation. 
 
 X. Rate of Inclination of Inclined Planes for various Angles of Inclination. 
 XI. Correction of the Moon's Meridian Passage. 
 XII. Effect of a Change of 1 Declination on the Moon's Semidiameter.
 
 A DESCRIPTION OF 
 THE PRINCIPAL INSTRUMENTS 
 
 EMPLOYED IN 
 
 SURVEYING, LEVELLING, AND ASTRONOMY, 
 
 WITH THEIR ADJUSTMENTS AND USE. 
 
 SURVEYING INSTRUMENTS. 
 
 THE LAND CHAIN. 
 
 GUNTER'S Chain is the one now commonly used in taking the 
 dimensions of land ; it is sixty-six feet, or four poles, in length, 
 and is divided into 100 links, each of which is joined to the next 
 by three rings ; the length of each link, including the connect- 
 ing rings, is 7,92 inches, and at the end of every tenth link is 
 attached a piece of brass (each of a different shape,) for more 
 readily counting the odd links. 
 
 " The English acre contains 4840 square yards, and Gunter's 
 chain is 22 yards in length, and the square chain, or 22 multi- 
 plied by 22, gives 484, exactly the tenth part of an acre ; and 
 ten square chains are equal to one acre ; consequently, as the 
 chain is divided into 100 links, every superficial chain contains 
 100 multiplied by 100, that is 10,000 square links ; and 10 
 superficial chains, or one acre, contains 100,000 square links. 
 
 " If therefore the content of a field, cast up in square links, 
 be divided by 100,000, or, (which is the same thing) if from the 
 content we cut off the last five figures, the remaining figures to- 
 wards the left hand give the content in acres, and consequently 
 the number of acres at first sight ; the remaining decimal frac- 
 tion, multiplied by 4, gives the roods, and the decimal part of 
 this last product, multiplied by 40, gives the poles or perches." 
 
 Short" distances, or off-sets from the chain-line, are usually 
 measured with a rod, called an off-set staff, the most convenient 
 length for which is 6 feet 7,2 inches, being equal to 10 links of 
 the chain, and it should be divided accordingly. 
 
 With the chain must be provided ten arrows, which may be 
 made of strong iron wire, about 12 or 15 inches long, pointed at 
 one end for piercing the ground, and turned up at the other, in 
 the form of a ring, to serve as a handle : their use is to fix in 
 the ground at each extremity of the chain whilst measuring, and 
 to point out the number of chains measured. 
 
 // B
 
 g SURVEYING INSTRUMENTS. 
 
 The operation of measuring with the chain requires at least 
 two persons, one to lead and the other to follow and direct, the 
 first or leader (taking the ring at one end of the chain upon two 
 fingers of his right hand with one arrow, and the remaining nine 
 in his left,) lays his end of the chain, by direction of the fol- 
 lower, in a straight line with the station to be measured to, and 
 there fixes an arrow, while the latter holds the other end of the 
 chain at the starting point ; the leader now proceeds onwards 
 until the follower comes to the arrow first laid down, to which 
 he places his end of the chain, and again directs the leader to 
 place a second arrow in line with the forward station : the leader 
 will now have an opportunity of checking the directions of the 
 follower at every succeeding chain's length, by observing if the 
 latter is also in a line with the back station, at the time he 
 directs him to place one of his arrows in the direction of the for- 
 ward station : they proceed in this manner till the whole line is 
 measured, or the leader has spent all his ten arrows, which, upon 
 counting, he will find in the possession of the follower, (unless 
 some error has been committed,) who must restore them to the 
 leader, and remark in the field-book that they had made one 
 change, or measured 1000 links; they then proceed onwards as 
 before, the leader taking all the ten arrows, until they are again 
 spent, when a second change must be made and entered in the 
 book, and if the line is measured out before a third change takes 
 place, the follower will have in his hand as many arrows as there 
 have been chains laid out upon the last measured part of the 
 distance ; which, together with the odd links and the former 
 two changes (or 2000), will make up the entire length of the 
 line. 
 
 For the purposes of plotting, &c. it will be necessary to re- 
 duce the measurement of the lines which alternately ascend and 
 descend to the correct horizontal measure, for it is evident that 
 the distance between two points, if measured over uneven 
 ground, will be greater than if measured perfectly straight in a 
 horizontal plane. Some surveyors attempt this correction as 
 they proceed, by holding the lower end of the chain above the 
 ground, as nearly horizontal as they can estimate, and if they 
 aim at considerable accuracy, will have a plumb, which they 
 allow to hang from the hand that holds the chain, over the arrow 
 or mark in the ground. In passing over very steep ground, they 
 frequently take half or even a quarter of a chain's length to 
 accomplish their measurements with, as the whole length would 
 be too great to be held horizontal, when the inclination is con- 
 siderable. But the most correct method is to take the vertical 
 angles along the undulations of the line after it has been mea- 
 sured, and compute the horizontal distances (by a rule in plain 
 trigonometry,) as the whole line is then supposed to be divided 
 into a number of right-angled triangles, the measured portion
 
 THE PRISMATIC COMPASS. 
 
 3 
 
 being the hypothenuse, and the horizontal line the base ; or it 
 may be more expeditiously accomplished by our Table IX. 
 which shows the quantity to be subtracted from each chain's 
 length for various angles of inclination of the ground, which 
 at once reduces the oblique or hypothenusal measure, to the 
 horizontal. 
 
 THE SURVEYING CROSS AND OPTICAL SQUARE. 
 
 The instrument formerly employed for laying out perpendi- 
 cular lines, was the cross-staff, of which there were various con- 
 structions ; but that in most general use consisted of four sights, 
 fixed at right angles upon a brass cross, and adapted to the top of 
 a staff; which being thrust into the ground with two of the sights 
 placed in any given direction, the other two pointed out the per- 
 pendicular required. But this instrument has been almost super- 
 seded by the optical square, which is much superior to it both for 
 convenience and expedition ; and it has also the advantage of 
 greater portability, not being larger than a shallow circular snuff- 
 box, which it resembles in shape. It is made of brass and con- 
 tains the two principal glasses of the sextant, viz., the index and 
 horizon glasses, fixed at an angle of 45 ; hence, while viewing 
 an object by direct vision, any other, forming a right angle with 
 it, at the place of the observer, will be referred by reflection, so 
 as to coincide with the object viewed. Thus a line may be laid 
 out perpendicular to a station-line, and from any point on it, by 
 simply standing with the instrument over the given point, and 
 looking through it along the line, having a person to go with a 
 mark or station-staff in the direction the perpendicular is required, 
 and signing to him by hand to move to the right or the left, 
 until his staff is seen by reflection to coincide with some object 
 on the line along which the observer is looking, when the place 
 of the staff will be in a perpendicular to the station-line at the 
 place of the observer. 
 
 If it be required to find on a line the place of a perpendicular 
 from a fixed object, as a house, &c. the observer himself must 
 move along the line until the image of the object appears, as 
 before, in the direction of the line, and the place where he then 
 stands, will be the spot where such perpendicular would fall. 
 
 THE PRISMATIC COMPASS. 
 
 The use of this little instrument is to measure horizontal angles 
 only, and from its portability is particularly adapted for military 
 surveying, or where but little more than a sketch map of the 
 country is required. It is also very useful in filling in the detail 
 of a map, where all the principal points have been correctly 
 fixed by means of the theodolite ; and for this purpose has been 
 extensively employed by the gentlemen engaged on the Ordnance 
 
 B 2
 
 4, SURVEYING INSTRUMENTS. 
 
 survey. It may likewise be used for determining approximately 
 the direction of the true meridian, the variation being determined 
 by comparing the observed azimuth of a celestial object, with 
 its true azimuth deduced from an observation made for the 
 purpose. 
 
 In the following figure, A represents the compass-box, and B 
 the card, which being attached to the magnetic needle, moves 
 as it moves, round the agate centre, a, on which it is suspended. 
 The circumference of the card is usually divided to 15' of a de- 
 gree, but it is doubtful whether an angle can be measured by it 
 even to that degree of accuracy : c is a prism, which the observer 
 looks through in observing with the instrument. The perpendi- 
 cular thread of the sight-vane, E, and the divisions on the card 
 appear together on looking through the prism, and the division 
 with which the thread coincides, when the needle is at rest, is 
 the magnetic azimuth of whatever object the thread may bisect. 
 The pi-ism is mounted with a hinge joint, D, by which it can be 
 turned over to the side of the compass-box, that being its posi- 
 tion when put into the case. The sight-vane has a fine thread 
 stretched along its opening, in the direction of its length, which 
 is brought to bisect any object, by turning the box round hori- 
 zontally ; the vane also turns upon a hinge joint, and can be laid 
 flat upon the box, for the convenience of carriage. F is a mirror, 
 made to slide on or off the sight-vane, E ; and it may be re- 
 versed at pleasure, that is, turned face downwards ; it can also 
 
 be inclined at any angle, by means of its joint, d; and it will 
 remain stationary on any part of the vane, by the friction of its 
 slides. Its use is to reflect the image of an object to the eye of 
 the observer when the object is much above or below the hori- 
 zontal plane. When the instrument is employed in observing 
 ths azimuth of the sun, a dark glass must be interposed; and
 
 THE VERNIER. 
 
 the coloured glasses represented at G, are intended for that pur- 
 pose ; the joint upon which they act, allowing them to be turned 
 down over the sloping side of the prism-box. 
 
 At e, is shown a spring, which being pressed by the finger at 
 the time of observation, and then released, checks the vibrations 
 of the card, and brings it more speedily to rest. A stop is like- 
 wise fixed at the other side of the box, by which the needle may 
 be thrown off its centre ; which should always be done when the 
 instrument is not in use, as the constant playing of the needle 
 would wear the point upon which it is balanced, and upon the 
 fineness of the point much of the accuracy of the instrument 
 depends. A cover is adapted to the box, and the whole is 
 packed in a leather case, which may be carried in the pocket 
 without inconvenience. 
 
 The method of using this instrument is very simple. First 
 raise the prism in its socket, b, until you obtain distinct vision 
 of the divisions on the card, and standing at the place where the 
 angles are to be taken, hold the instrument to the eye, and look- 
 ing through the slit, c, turn round till the thread in the sight- 
 vane bisects one of the objects whose azimuth, or angular dis- 
 tance from any other object, is required ; then, by touching the 
 spring, e, bring the needle to rest, and the division on the card 
 which coincides with the thread on the vane, will be the azimuth 
 or bearing of the object from the north or south points of the 
 magnetic meridian. Then turn to any other object and repeat 
 the operation ; the difference between the bearing of this object 
 and that of the former, will be the angular distance of the objects 
 in question. Suppose the former bearing to be 40 30' and the 
 latter 10 15', both east, or both west, from the north or south, 
 the angle will be 30 15'. The divisions are generally numbered 
 5, 10, 15, &c. round the circle to 360. A tripod stand similar 
 to those of the theodolite described at page 15, can be had with 
 the instrument, if required, on which to place it when observing, 
 instead of holding it in the hand. 
 
 THE VERNIER. 
 
 This is a contrivance for measuring parts of the space between 
 the equidistant divisions of a graduated scale. It is a scale 
 whose length is equal to a certain number of parts of that to be 
 subdivided, depending on the degree of minuteness to which the 
 subdivision is intended to be carried ; but it is divided into 
 parts, which in number are one more or one less than those of 
 the primary scale taken for the length of the vernier : in modern 
 practice the parts on the vernier are generally one more than 
 are contained in the same space on the primary scale. 
 
 If it is required to measure to hundredths of an inch, the 
 parts of a scale which is graduated to lOths, it may be done by 
 means of a scale whose length is nine tenths of an inch, and 
 divided into 10 equal parts ; or by one whose length is eleven
 
 (J SURVEYING INSTRUMENTS. 
 
 tenths of an inch, and divided into 10 equal parts ; for in either 
 case the difference between the divisions of the scale so made 
 and those on the primary scale is the hundredth of an inch. Such 
 a scale made to move along the edge of that to be subdivided 
 is called a vernier; and we shall explain how by its application, 
 either to straight lines or arcs of circles, the subdivisions of gra- 
 duated instruments are read off. For this purpose, let us take 
 as a general example the method of reading the sextant, as a 
 person acquainted with the graduations upon this instrument 
 will find no difficulty in becoming familiar with those on any 
 other. 
 
 It will be observed,* that some of the divisional lines on the 
 limb of the instrument are longer than others, and that they are 
 numbered at every fifth, thus, 0. 5. 10. 15, &c., the being the 
 starting point, or zero. The spaces between these lines represent 
 degrees ; and they are again subdivided by shorter lines, each 
 smaller space representing a certain number of minutes. For 
 instance, if the spaces are subdivided into four parts, then there 
 will be three short lines, each of which will indicate the termi- 
 nation of a space of 15 minutes; if there are six parts, there 
 will be five short lines, and each will be at the end of a space 
 of 10 minutes, reckoned from the commencement of the divi- 
 sions. Likewise it will be observed, that some of the divisions 
 on the vernier are longer than others : these indicate in the same 
 manner single minutes, and they are numbered from right to 
 left : the extreme right one is the zero, or commencement of 
 the index divisions, and it is marked or ; the shorter divi- 
 sions show fractions of minutes. If the spaces between each 
 minute (or long division) contain three lines, each space will be 
 15 seconds, and if five, 10 seconds ; the number of subdivisions 
 between the minutes of the vernier is usually, but not neces- 
 sarily, the same as between the degrees on the limb, so that if 
 the limb is divided into 20' the vernier is divided into 20"; if the 
 former is divided to 10' the latter is divided to 10", &c. 
 
 The limb of the instrument now before us is divided to 10', 
 and the vernier reads to 10", and by showing the manner of 
 reading it off, we shall explain sufficiently the method of reading 
 verniers in general. If the zero division of the vernier coincide 
 (or form a straight line) with any line on the limb, then that line 
 indicates the required angle ; thus, if it coincide with the line 
 marked 60, then sixty degrees is the angle; if with the next long 
 division, then 61 degrees will be the angle ; but if it coincide 
 with one of the shorter lines between 60 and 61, then the angle 
 will be 60 degrees and a certain number of minutes, according to 
 which of the short lines it coincides with. If it be the first, 
 (of the instrument before us) the angle will be 60 10', but if it 
 
 * The reader is supposed to have an instrument before him while perusing 
 these instructions.
 
 THE VERNIER. 
 
 coincide with the second, it will be 60 20', if with the third, 
 60 30', &c. But when it happens that the zero division of the 
 index does not coincide with any division upon the limb, but 
 stands between two of them, we must observe how many degrees 
 and minutes are denoted by the division it has last passed, and 
 look for a line on the vernier that does coincide with one on the 
 limb ; and the number of minutes and seconds from that line 
 to the zero of the index, added to the number read off upon 
 the limb, gives the angle required. Thus, supposing the index 
 to stand between 10' and 20' beyond 60, and the line on the 
 vernier denoting 6' 10" (which is the line next beyond the one 
 marked 6) coincides with any one on the limb, then this quan- 
 tity, added to 60 10', gives 60 16' 10", the angle required. 
 
 When the arc of excess on the limb of the sextant (the nature 
 of which will be explained hereafter) is required to be read off, 
 observe what quantity is passed to the right of zero by the zero 
 division of the vernier, and find the remaining minutes and 
 seconds to be added to it, by reading the vernier backwards ; 
 that is, consider the last numbered division to the left hand as 
 the zero : thus, suppose that (on our instrument) the index stood 
 beyond the third short division on the arc of excess, this would 
 be 30', and if the third long division from the last numbered 
 one on the left hand (marked 10,) coincided with a line on the 
 limb, this would denote 3' to be added to the former, making 
 33' for the reading on the arc of excess. 
 
 On the limbs of small theodolites, the spaces between the 
 degrees are generally divided into two parts, consequently the 
 short division represents 30', and the divisions on the vernier 
 are single minutes ; a smaller subdivision must be estimated by 
 the eye, which by a person accustomed to the instrument can be 
 done to 15". 
 
 The subdivision of a straight line, as the scale of a mountain 
 barometer, is likewise effected by a vernier, and is read off in the 
 following manner. The scale is divided into inches, which are 
 subdivided into 10 parts ; these tenths are again divided into 
 two, by a shorter division, which will be 5 hundredths of an inch. 
 The long divisions upon the vernier show each of them one 
 hundredth of an inch, and they are numbered at every fifth ; 
 these are again subdivided by shorter lines, representing thou- 
 sandths. Now, to read it off, observe where the zero division 
 of the vernier stands on the scale ; suppose a little above 
 30 inches and 4 tenths, and as it does not reach the short 
 line denoting 5 hundredths, observe what line on the vernier 
 coincides with one on the scale : if it is a long division, then it 
 is so many hundredths to be added, and if a short division, it 
 will be so many hundredths and thousandths to be added to 
 make up the measurement, and the readings are written deci- 
 mally thus, 30-435 inches. 
 
 In the subjoined figures, which are given for the purpose of
 
 8 
 
 SURVEYING INSTRUMENTS. 
 
 illustration, A B represents a portion of the graduated limb of 
 an instrument, and C D a portion of the vernier scale, the zero 
 point being at C. 
 
 Fiy. l. 
 B ;"> 45 
 
 2 
 Fig. 2. 
 
 1 
 
 1 1 
 
 1 1 
 
 1 1 
 
 o 
 
 1 1 
 
 2 
 1 
 
 
 1 1 
 
 6|0 
 
 1 1 1 1 1 I ! i 
 
 1 1 1 
 
 u 
 
 1 1 1 
 
 | 
 
 1 1 1 1 1 ! 
 
 k 
 
 20 
 
 Fig. 4. 
 
 1 1 
 
 1 1 
 
 2 
 1 1 
 
 
 
 I 1 
 
 1 
 
 1 1 
 
 1 1 
 
 1 
 
 1 1 
 
 31- 
 
 i 8 - 
 
 ! - 
 ' 6- 
 
 30- 
 
 -c?C 
 
 In the first figure the limb is divided to 15', 
 and these divisions are subdivided by the ver- 
 nier to 15". In the second figure, the limb is 
 divided to 10', and subdivided by the vernier 
 to 10". In the third, the limb is divided to 
 20', and subdivided by the vernier to 30"; and 
 in the fourth, the limb is divided to 20', and 
 subdivided by the vernier to 20". E, on each 
 figure, is placed where a division on the ver- 
 nier coincides with one on the limb. In the 
 first, the reading is 45 46' 30" ; in the second, 
 60 21' 20" ; in the third, 21 23' 30"; and in 
 the fourth, it is 17 2', and between 0' and 
 20", and as the 2' line is about as much in ad- 
 vance of the one on the limb near to it, as the 
 20" line is behind the one near to it, the read- 
 ing may be taken as 17 2' 10". The fifth 
 figure represents the scale of a barometer, 
 reading 30*435 inches, and is drawn much 
 larger than the reality, to render it more in- 
 telligible.
 
 THE PLANE TABLE. 9 
 
 THE PLANE TABLE. 
 
 Before the theodolite came into general use, the Plane-table 
 was extensively employed in the practice of surveying : it is 
 still sometimes, though seldom, used in surveying small plots of 
 ground, or (where great accuracy is not required) in forming a 
 sketch-map, or laying down the details of a country where the 
 relative situations of the principal conspicuous objects have been 
 previously fixed by triangulation. The expedition with which 
 such work may be performed, by a person who is expert in the 
 use of this instrument, is its chief recommendation. 
 
 The construction and size of the plane table has been varied 
 at different times, to suit both the convenience and intentions of 
 the surveyor ; but the annexed figure is a representation of that 
 which is now in most general use. It is a board, as A, about 
 sixteen inches square, having its upper edge rabbetted, to receive 
 a box-wood frame, B, which being accurately fitted, can be placed 
 on the board in any position, with either face upwards. This 
 frame is intended both to stretch and retain the drawing paper 
 upon the board, which it does by being simply pressed down into 
 its place upon the paper, which for this purpose must be cut a 
 little larger than the board. 
 
 One face of the frame is divided to 360 degrees, from a centre, 
 C, fixed in the middle of the board, and these are subdivided as 
 minutely as the size of the table will admit. The divisions are 
 frequently numbered each way, to show at sight both an angle 
 and its complement to 360. There is sometimes a second centre 
 piece, D, fixed on the table, at about a quarter of its width from 
 one of the sides, and at exactly half its length in the other di- 
 rection. From this centre, and on the other side of the frame, 
 there is graduated 180: each of these degrees is subdivided to 
 30 minutes, and numbered, 10, 20, 30, &c., both ways, to 180. 
 The object of these graduations is, to make the plane-table
 
 JO SURVEYING INSTRUMENTS. 
 
 supply the place of the theodolite, and an instrument formerly 
 in use called a semicircle. The reverse face of the frame is 
 usually divided into equal parts, as inches and tenths, for the 
 purpose of ruling parallel lines or squares, and for shifting the 
 paper, when the work requires more than one sheet. G is a 
 compass-box, let into one side of the table, with a dove-tail 
 joint, and fastened with a milled-headed screw, that it may be 
 applied or removed at pleasure. The compass, beside rendering 
 the plane-table capable of answering the purpose of a circumfer- 
 enter, is principally useful in setting the instrument up at a new 
 station parallel to any position that it may have had at a former 
 station, as well as a check upon the progress of the work. 
 
 The ruler or index, E, is made of brass, as long as the diag- 
 onal of the table, and about two inches broad; it has a sloping 
 edge, liked that of a Gunter's scale, which is called the fiducial 
 edge. A perpendicular sight- vane, F F, is fixed to each ex- 
 tremity of the index, and the eye looking through one of them, 
 the vertical thread in the other is made to bisect any required 
 distant object. Upon the flat surface of the index, there are 
 frequently engraved scales of various kinds, such as lines of equal 
 parts, with diagonal scales, a line of chords, &c. 
 
 To the under side of the table, a centre is attached with a 
 ball and socket, or parallel plate-screws like those of the theodo- 
 lite, by which it can be placed upon a staff-head ; and the table 
 may be set horizontal, by means of a circular spirit-level placed 
 upon it for that purpose. 
 
 In preparing the plane-table for use, the first thing to be 
 done is to cover it with drawing paper ; the usual method of 
 doing which is the same as that of covering a common drawing 
 board, by damping the under side of the paper, and laying it on 
 the board in an expanded state ; press the frame into its place, 
 so that the paper may be squeezed in between the frame and the 
 edge of the table ; and the paper shrinking as it dries, assumes a 
 flat surface for the work to be performed upon. There is one 
 great objection however to this mode of putting on the paper, as 
 when it has once been damped and strained, it is easily acted 
 upon by any change in the hygrometrical state of the atmosphere. 
 We therefore prefer putting the paper on dry, taking care to 
 keep it straight and smooth whilst pressing the frame into its 
 place ; but it must be acknowledged that this cannot be done so 
 nicely as when it is damped. We have been informed, that if 
 the under side of the paper be covered with the white of an 
 egg well beat up, it may be laid on the board with the greatest 
 nicety, and that when so prepared it is not easily affected by 
 atmospheric changes. 
 
 When the survey has been carried to the edge of the paper on 
 the table, and there is occasion to extend the operation further, 
 another sheet must be substituted ; but before removing the old
 
 THE PLANE TABLE. J J 
 
 one, a line should be drawn on it, through some particular sta- 
 tions or points of the survey that can be made common to both 
 sheets of paper ; then, by drawing a similar line upon the new 
 sheet, and transferring to this line the points or stations that are 
 upon the line in the former sheet, as well as the direction of the 
 last station lines, the survey may be renewed and continued in 
 the same manner, from sheet to sheet, till the whole is com- 
 pleted. In drawing the corresponding line upon the second 
 sheet, it is necessary to pay due regard to the general direction 
 of the future survey, that the line may be so drawn as to admit 
 the greatest possible quantity of work into each sheet of paper. 
 
 Such is the description of the plane-table as formerly, and as 
 now generally constructed ; but for our own use we could dis- 
 pense with the graduations on the box-wood frame altogether, 
 except perhaps those of equal parts, which are sometimes useful 
 when shifting the paper. Indeed, in our method of using the 
 instrument, a plain board made of well-seasoned but soft wood 
 (as pine or cedar) to admit readily of a fine pin or needle being 
 fixed in it, would, with the compass-box, answer every pur- 
 pose ; as we should prefer pasting, or gluing, a thick sheet of 
 drawing paper or fine pasteboard over the surface of the table, as 
 the errors caused by changes in the moisture of the air would 
 then be greatly diminished. A fair copy of the plan can be 
 afterwards made out at leisure, and if one board is not suffici- 
 ent to contain the whole of the survey, others similarly prepared, 
 and adapted to the same staff-head, may be provided, to con- 
 tinue the work. 
 
 Having explained the general construction of the instrument, 
 we shall show the manner of using it by means of an example. 
 
 In the annexed diagram, let the points marked ABC, &c. 
 be a few of an extensive series of stations, either fixed or tem- 
 porary, the relative situations of which are required to be laid 
 down upon the plan. Select two stations, as I and K, (consi- 
 derably distant from each other,) as the extremities of a base 
 line, from which the greatest number of objects are visible ; 
 then, if the scale to which the plan is to be drawn is fixed, 
 the distance, I K, must be accurately measured, and laid off 
 upon the board to the required scale ; otherwise a line may be 
 assumed to represent that distance ; and at some subsequent 
 part of the work the value of the scale thus assumed must be 
 determined, by measuring a line for that purpose, and com- 
 paring the measurement, with its length, as represented on the 
 plan. 
 
 Set up the instrument at one extremity of the base, suppose 
 at I, and fix a needle in the table at the point on the paper re- 
 presenting that station, and press the fiducial edge of the index 
 gently against the needle. Turn the table about until the me- 
 ridian line of the compass-card coincides with the direction of
 
 12 SURVEYING INSTRUMENTS. 
 
 the magnetic needle, and in that position chimp the table firm. 
 Then, always keeping the fiducial edge of the index against the 
 needle, direct the sights to the other station, K, and by the side 
 of the index draw a line upon the paper, to represent the base, 
 I K; when, if the scale is fixed, the exact length must be laid 
 off, otherwise the point K may be assumed at pleasure on the 
 line so drawn. 
 
 C, o 
 
 But it is sometimes necessary to draw the base line first, when 
 required, on some particular part of the board, so as to admit of 
 the insertion of a greater portion of the survey. When this is 
 the case, the index must be laid along the line thus drawn, and 
 the table moved till the further end of the base line is seen 
 through both the sights ; then fix the table in that position, and 
 observe what reading on the compass-card (or bearing) the needle 
 points to, for the purpose of checking the future operations, and 
 also for setting the table parallel to its first position, wherever it 
 may afterwards be set up. It should be observed, that in plac- 
 ing it over any station, that spot on the table representing such 
 station, and not the centre of the table, should be over the sta- 
 tion on the ground : it may be so placed by dropping a plumb- 
 line from the corresponding point on the under side of the table. 
 
 Having fixed the instrument and drawn the base line, move 
 the index round the point I, as a centre, direct the sights 
 to the station A, and keeping it there, draw the line I A along 
 the fiducial edge of the index. Then direct in the same manner
 
 THE PLANE TABLE. 13 
 
 to B, and draw the line I B ; and so proceed with whatever ob- 
 jects are visible from the station, drawing lines successively in the 
 direction of C D E, &c., taking care that the table remains steady 
 during the operation. 
 
 This done, remove the instrument to the station K, and placing 
 the edge of the index along the line I K, turn the table about 
 till the sights are directed to the station I, which if correctly 
 done, the compass-needle will point to the same bearing as it did 
 at the former station (in our example it was set to the meridian). 
 Now remove the needle from I, and fix it in the point K ; lay 
 the edge of the index against the needle, and direct the sights in 
 succession to the points ABC, &c., drawing lines from the point 
 K, in their several directions, and the intersection of these lines, 
 with those drawn from the point I, will be their respective situ- 
 ations on the plan. 
 
 To check the accuracy of the work, as well as for extending 
 the survey beyond the limits of vision at I and K, the table may 
 be set up at any one or more of the stations thus determined, as 
 at E : the needle being now fixed in the point E on the board, 
 and the edge of the index placed over E and I (or K,) the table 
 may be moved round till the station I is seen through both the 
 sights, and then clamped firm : the compass will now again, (if 
 all be correct) point? out its former bearing, and any lines drawn 
 from E, in the direction of A B C, &c. in succession, will pass 
 through the intersection of the former lines denoting the rela- 
 tive places of those objects on the board ; but should this not be 
 the case with all, or any of the lines, it is evident that some error 
 must exist, which can be detected only by setting the instrument 
 up and performing similar operations at other stations. 
 
 Having a number of objects laid down upon the plan, the situ- 
 ation of any particular spot, as the bend of a road, &c., may at 
 once be determined, by setting the instrument up at the place, 
 and turning the table about till the compass has the same bearing 
 as at any one of the stations. Clamp the table firm, and it will 
 now be parallel to its former position, if no local attraction pre- 
 vents the magnetic needle from assuming its natural position at 
 the different stations. Fix a needle in the point representing one 
 of the stations, and resting the edge of the index against it, 
 move the index till the station itself is seen through both the 
 sights, and then draw a line on that part of the paper where the 
 point is likely to fall. Remove the needle to another point or 
 station on the board, and resting the index against it, direct the 
 sights to the corresponding station on the ground, and draw a 
 line along the edge of the index : the point where this line inter- 
 sects the last, will be the situation on the paper of the place of 
 the observer. But, as a check upon the accuracy of the work, 
 a third or even a fourth line should be drawn in a similar manner
 
 14 SURVEYING INSTRUMENTS. 
 
 in the direction of other fixed points, and they ought also to in- 
 tersect in the same point. 
 
 In this manner the plane-table may be employed for filling in 
 the details of a map : setting it up at the most remarkable spots, 
 and sketching by the eye what is not necessary should be more 
 particularly determined, the paper will gradually become a repre- 
 sentation of the country to be surveyed. 
 
 THE THEODOLITE. 
 
 As an angular instrument, the theodolite has from time to time 
 received such improvements, that it may now be considered as 
 the most important one employed in surveying. Instruments of 
 this kind, of the best construction, may to a certain extent be 
 used as altitude and azimuth instruments ; and several astrono- 
 mical operations, such as those required for determining the time, 
 the latitude of place, &c., may be performed by them, and to a 
 degree of accuracy sufficient for most of the purposes that occur 
 in the ordinary practice of a surveyor. 
 
 There are various modes of constructing theodolites to suit the 
 convenience or the views of purchasers ; but we shall confine 
 ourselves to a description of one of the most perfect, as a person 
 acquainted with the details of its adjustments and use will find 
 no difficulty in comprehending those of others. 
 
 Description of the Theodolite. 
 
 This instrument (as represented in the next page) consists of 
 two circular plates, A and B, called the horizontal limb, the 
 upper or vernier plate, A, turning freely upon the lower, both 
 having a horizontal motion by means of the vertical axis, C. 
 This axis consists of two parts, external and internal, the former 
 secured to the graduated limb, B, and the latter to the vernier 
 plate, A. Their form is conical, nicely fitted and ground into 
 each other, having an easy and a very steady motion ; the external 
 centre also fits into a ball at D, and the parts are held together 
 by a screw at the lower end of the internal axis. 
 
 The diameter of the lower plate is greater than that of the 
 upper one, and its edge is chamfered off and covered with silver, 
 to receive the graduations : on opposite parts of the edge of the 
 upper plate, or 180 apart, a short space, a, is also chamfered, 
 forming with the edge of the lower plate a continued inclined 
 plane : these spaces are likewise covered with silver, and form 
 the verniers. The lower limb is usually graduated to thirty 
 minutes of a degree, and it is subdivided by the vernier to single 
 minutes, which being read off by the microscope, E, half, or even 
 quarter minutes, can easily be estimated. 
 
 The parallel plates, F and G, are held together by a ball and
 
 THE THEODOLITE. 
 
 socket at D, and are set firm and parallel to each other, by four 
 milled-headed screws, three of which, b b b, are shown in the 
 figure : these turn in sockets fixed to the lower plate, while their 
 heads press against the under side of the upper plate, and being 
 set in pairs, opposite each other, they act in contrary directions ; 
 the instrument by this means is set up level for observation. 
 
 Beneath the parallel plates is a female screw adapted to the 
 staif head, which is connected by brass joints to three mahogany 
 legs, so constructed that when shut up they form one round 
 staff, secured in that form for carriage by rings put on them ; 
 and when opened out, they make a very firm stand, be the 
 ground ever so uneven. 
 
 The lower horizontal limb can be fixed in any position, by 
 tightening the clamping screw, H, which causes the collar, c, to 
 embrace the axis, C, and prevent its moving ; but it being 
 requisite that it should be fixed in some precise position more
 
 Ifl SURVEYING INSTRUMENTS. 
 
 exactly than can be done by the hand alone, the whole instru- 
 ment, when thus clamped, can be moved any small quantity by 
 means of the slow-motion screw, I, which is attached to the 
 upper parallel plate. In like manner the upper or vernier plate 
 can be fixed to the lower, in any position, by a clamp, (in the 
 plate this clamp is concealed from view) which is also furnished 
 with a slow motion, the screw of which is generally called the 
 tangent-screw. The motion of this limb, and of the vertical arc, 
 hereafter to be described, is sometimes effected by a rack and 
 pinion ; but this is greatly inferior, where delicacy is required, 
 to the slow motion produced by the clamp and tangent-screw. 
 
 Upon the plane of the vernier plate, two spirit-levels, d d, 
 are placed at right angles to each other, with their proper ad- 
 justing screws : their use is to determine when the horizontal 
 limb is set level : a compass also is placed at J. 
 
 The frames K and L support the pivots of the horizontal axis 
 of the vertical arc (or semicircle) M, on which the telescope is 
 placed. The arm which bears the microscope N, for reading the 
 altitudes or depressions, measured by the semicircle, and denoted 
 by the vernier, e, has a motion of several degrees between the 
 bars of the frame, K, and can be moved before the face of the 
 vernier for reading it off. Another arm clamps the opposite end 
 of the horizontal axis by turning the screw, O, and has a tangent- 
 screw of slow motion at P, by which the vertical arc and tele- 
 scope are moved very small quantities up or down, to perfect the 
 contact when an observation is made. 
 
 One side of the vertical arc is inlaid with silver, and divided 
 to single minutes by the help of its vernier ; and the other side 
 shows the difference between the hypothenuse and base of a 
 right-angled triangle, or, the number of links to be deducted from 
 each chain's length, in measuring up or down an inclined plane, 
 to reduce it to the horizontal measure. The level, which is 
 shown under and parallel to the telescope, is attached to it at 
 one end by a joint, and at the other by a capstan-headed screw, 
 /, which being raised or lowered, will set the level parallel to the 
 optical axis of the telescope, or line of collimation ; the screw, 
 <7, at the opposite end, is to adjust it laterally, for true parallelism 
 in this respect. The telescope has two collars, or rings, of bell 
 metal, ground truly cylindrical, on which it rests in its supports, 
 h h, called Y's, from their resemblance to that letter ; and it is 
 confined in its place by the clips, i i, which may be opened by 
 removing the pins,^', for the purpose of reversing the telescope, 
 or allowing it a circular motion round its axis, during the ad- 
 justment. 
 
 In the focus of the eye-glass are placed three lines, formed of 
 spider's web, one horizontal, and two crossing it, so as to include 
 a small angle between them ; a method of fixing the wires, which 
 is better than having one perpendicular wire, because an object
 
 THE THEODOLITE. ] 7 
 
 at a distance can be made to bisect the said small angle with more 
 certainty than it can be bisected by a vertical wire. The screws 
 adjusting the cross wires are shown at m: there are four of 
 these screws, two of which are placed opposite each other, and 
 at right angles to the other two, so that by easing one, and tight- 
 ening the opposite one of each pair, the intersection of the cross 
 wires may be placed in adjustment. 
 
 The object-glass is thrust outwards by turning the milled head, 
 Q, on the side of the telescope, that being the means of adjust- 
 ing it to show an object distinctly. 
 
 A brass plummet and line are packed in the box with the 
 theodolite, to suspend from a hook under its centre, by which it 
 can be placed exactly over the station from whence the observa- 
 tions are to be taken : likewise, if required, two extra eye-pieces 
 for the telescope, to be used for astronomical observations : the 
 one inverts the object, and has a greater magnifying power, but 
 having fewer glasses possesses more light ; the other is a diagonal 
 eye-piece, which will be found extremely convenient when ob- 
 serving an object that has a considerable altitude ; the observer 
 avoiding the unpleasant and painful position he must assume in 
 order to look through the telescope when either of the other 
 eye-pieces is applied. A small cap containing a dark coloured 
 glass is made to apply to the eye-end of the telescope, to screen 
 the eye of the observer from the intensity of the sun's rays, 
 when that is the object under observation. A magnifying glass 
 mounted in a horn frame, a screw-driver, and a pin to turn the 
 capstan-screws for the adjustments, are also furnished with the 
 instrument. 
 
 The Adjustments. 
 
 The first adjustment is that of the line of collimation ; that 
 is, to make the intersection of the cross wires coincide with the 
 axis of the cylindrical rings on which the telescope turns : it is 
 known to be correct, when an eye looking through the telescope 
 observes their intersection continue on the same point of a dis- 
 tant object during an entire revolution of the telescope. The 
 usual method of making this adjustment is as follows: 
 
 First, make the centre of the horizontal wire coincide with 
 some well-defined part of a distant object ; then turn the tele- 
 scope half round in its Y's till the level lies above it, and observe 
 if the same point is again cut by the centre of the wire ; if not, 
 move the wire one half the quantity of deviation, by turning two 
 of the screws at m, (releasing one, before tightening the other,) 
 and correct the other half by elevating or depressing the tele- 
 scope ; now if the coincidence of the wire and object remain 
 perfect in both positions of the telescope, the line of collimation 
 in altitude or depression is correct, but if not, the operation 
 must be repeated carefully, until the adjustment is satisfactory.
 
 ]g SURVEYING INSTRUMENTS. 
 
 A similar proceeding will also put the vertical line correct, or 
 rather, the point of intersection, when there are two oblique 
 lines instead of a vertical one. 
 
 The second adjustment is that which puts the level attached 
 to the telescope parallel to the rectified line of collimation. The 
 clips, i i, being open, and the vertical arc clamped, bring the air- 
 bubble of the level to the centre of its glass tube, by turning 
 the tangent-screw, P ; which done, reverse the telescope in its 
 Y's, that is, turn it end for end, which, must be done carefully, 
 that it may not disturb the vertical arc, and if the bubble resume 
 its former situation in the middle of the tube, all is right; but 
 if it retires to one end, bring it back one half, by the screw/, 
 which elevates or depresses that end of the level, and the other 
 half by the tangent-screw, P : this process must be repeated 
 until the adjustment is perfect ; but to make it completely so, 
 the level should be adjusted laterally, that it may remain in the 
 middle of the tube when inclined a little on either side from its 
 usual position immediately under the telescope, which is effected 
 by giving the level such an inclination, and if necessary turning 
 the two lateral screws at g ; if making the latter adjustment 
 should derange the former, the whole operation must be carefully 
 repeated. 
 
 The third adjustment is that which makes the azimuthal axis, 
 or axis of the horizontal limb, truly vertical. 
 
 Set the instrument as nearly level as can be done by the eye, 
 fasten the centre of the lower horizontal limb by the staff-head 
 clamp, H, leaving the upper limb at liberty, but move it till the 
 telescope is over two of the parallel plate-screws ; then bring the 
 bubble of the level under the telescope, to the middle of the tube, 
 by the screw P ; now turn the upper limb half round, that is 
 180, from its former position ; then, if the bubble returns to the 
 middle, the limb is horizontal in that direction ; but if otherwise, 
 half the difference must be corrected by the parallel plate-screws 
 over which the telescope lies, and half, by elevating or depress- 
 ing the telescope, by turning the tangent-screw of the vertical 
 arc ; having done which, it only remains to turn the upper limb 
 forward or backward 90, that the telescope may lie over the 
 other two parallel plate-screws, and by their motion set it hori- 
 zontal. Having now levelled the limb-plates by means of the 
 telescope level, which is the most sensible upon the instrument, 
 the other air-bubbles fixed upon the vernier plate, may be 
 brought to the middle of their tubes, by merely giving motion 
 to the screws which fasten them in their places. 
 
 The vernier of the vertical arc may now be attended to ; it is 
 correct if it points to zero when all the foregoing adjustments 
 are perfect ; and any deviation in it is easily rectified, by releas- 
 ing the screws by which it is held, and tightening them again 
 after having made the adjustment : or, what is perhaps better,
 
 THE THEODOLITE. J9 
 
 note the quantity of deviation as an index error, and apply it, 
 plus or minus, to each vertical angle observed. This deviation is 
 best determined by repeating the observation of an altitude or 
 depression in the reversed positions, both of the telescope and 
 the vernier plate : the two readings will have equal and opposite 
 errors, one half of their difference being the index error. Such 
 a method of observing angles is decidedly the best, since the 
 mean of any equal number of observations taken with the tele- 
 scope reversed in its Y's, must be free from the effects of any 
 error that may exist in the adjustment of the vernier, or zero of 
 altitude. 
 
 The theodolite, as constructed in the manner we have described, 
 is not inconveniently heavy, as the diameter of the horizontal 
 limb seldom exceeds five inches ; but when the diameter is in- 
 creased, the other parts must be made proper tionably large and 
 strong, and the instrument becomes too weighty and cumber- 
 some to be easily carried from station to station. The object of 
 increasing the dimensions, is to enable the instrument to furnish 
 more accurate results, by applying a telescope of greater power, 
 and by a more minute subdivision of the graduated arcs. With 
 the increase of size, a small variation takes place in the construc- 
 tion, principally consisting in the addition of a second telescope, 
 and in the manner of attaching the supports, K and L, (page 1 5) 
 to the horizontal limb, to afford the means of adjusting the hori- 
 zontal axis, and of course, making the telescope and vertical arc 
 move in a vertical plane. In the smaller instruments this is done 
 by construction, but in the larger ones, the supports, K and L, 
 are attached to a stout frame, which also carries the compass- 
 box, instead of being fixed as represented in our figure, to the 
 upper horizontal plate. The frame is attached to the limb by 
 three capstan-headed screws, forming an equilateral triangle, 
 two of them lying parallel to the horizontal axis, and the third 
 in the direction of the telescope ; the adjustment is made by 
 means of these screws. To prove its accuracy, set up the theo- 
 dolite in such a situation that some conspicuous point of an 
 elevated building may be seen through the telescope, both 
 directly and by reflection, from a basin of water, or, what is 
 better, of oil or quicksilver. Let the instrument be very cor- 
 rectly levelled, and if, when a vertical motion is given to the 
 telescope, the cross-wires do not cut the object seen, both 
 directly and by reflection, it is a proof that the axis is not hori- 
 zontal ; and its correction is effected by giving motion to the 
 screws above spoken of, which are at right angles to the tele- 
 scope, or in the direction of the horizontal axis ; or a long plumb- 
 line may be suspended, and if the cross-wires of the telescope, 
 when it is elevated and depressed, pass exactly along the line, it 
 will be a proof of the horizontality of the axis. The third screw, 
 
 c 2
 
 20 SURVEYING INSTRUMENTS. 
 
 or that which is under the telescope, serves for adjusting the zero 
 of altitude, or vernier of the vertical arc. 
 
 A second telescope is sometimes attached to the instrument 
 beneath the horizontal limb ; it admits of being moved, both in 
 a vertical and horizontal plane, and has a tangent-screw attached 
 for slow motion : its use is to detect any accidental derangement 
 that may occur to the instrument whilst observing, which may 
 be done by it in the following manner. After levelling the 
 instrument, bisect some very remote object with the cross-wires 
 of this second telescope, and clamp it firm ; if the instrument 
 is steady, the bisection will remain permanent whilst any num- 
 ber of angles are measured, and by examining the bisection from 
 time to time, during the operation at the place where the instru- 
 ment is set up, any error arising from this cause may be detected 
 and rectified. 
 
 At the suggestion of Captain EVEREST, surveyor-general of 
 India, several small theodolites, differing considerably in con- 
 struction from that which we have been describing, have lately 
 been made by Messrs. TROUGHTON and SIMMS, for the great 
 Indian survey. In principle they are similar to the theodolites 
 of much larger dimensions, and consequently the whole of their 
 essential adjustments are made in the same manner. We shall 
 here give a description and engraving of this kind of instrument, 
 with the particulars of its adjustments, which must be under- 
 stood as equally applicable to the larger theodolites usually 
 employed in extensive trigonometrical operations. 
 
 The horizontal circle (or limb), A, of this instrument consists 
 of one plate only, which, as usual, is graduated at its circuin-
 
 THE THEODOLITE. 2 J 
 
 ference. The index is formed with four radiating bars, a, b, c, d, 
 having verniers at the extremities of three of them, for reading 
 the horizontal angles, and the fourth carries a clarnp to fasten 
 the index to the edge of the horizontal limb, and a tangent- 
 screw for slow motion. These are connected with the upper 
 works which carry the telescope, and turning upon the same 
 centre, show any angle through which the telescope has been 
 moved. The instrument has also the power of repeating the 
 measurement of an angle ; for the horizontal limb being firmly 
 fixed to a centre, movable within the tripod support, B, and 
 governed by a clamp and tangent-screw, C, can be moved with 
 the same delicacy, and secured with as much firmness, as the 
 index above it. Large theodolites, when required, have the power 
 of repeating given them, by means of a particular kind of stand, 
 called a repeating table. 
 
 The tripod support, which forms the stand of the instrument, 
 has a foot-screw at each extremity of the arms which form the 
 tripod ; the heads of the foot-screws are turned downwards, and 
 have a flange (or shoulder) upon them, so that when they rest 
 upon a triangular plate fixed upon the staff-head, another plate 
 locks over the flange, and being acted upon by a spring, retains 
 the whole instrument firmly upon the top of the staff, which is 
 similar to that of the theodolite represented at page 15. The 
 great advantage of the tripod stand is, that it can easily be 
 disengaged from the top of the staff, and placed upon a parapet 
 or other support, in situations where the staff cannot be used. 
 
 The telescope is mounted in the manner of a transit instru- 
 ment, that is, the horizontal axis, L, and the telescope, M, form 
 one piece, the axis crossing the telescope about its middle, and 
 terminating at each extremity in a cylindrical pivot. The pivots
 
 22 SURVEYING INSTRUMENTS. 
 
 rest upon low supports, (only one of them, D, being visible in the 
 figure), carried out from the centre, on each side, by a rial hori- 
 zontal bar, F, to which a spirit-level, G, is attached for adjusting 
 the axis to the horizontal plane. The vertical angles are read 
 off on two arcs of circles, H H, which have the horizontal axis 
 as their centre, and being attached to the telescope, move with 
 it in a vertical plane. An index, upon the same centre, carries 
 two verniers, I I, and it has a spirit-level, K, attached to it, by 
 which the index can be set in a horizontal position, so that 
 whatever position the telescope, and consequently the graduated 
 arcs, may have, when an observation is made, the mean of the 
 two readings will denote the elevation or depression of the object 
 observed, from the horizontal plane. 
 
 The following are the adjustments of this instrument : first, to 
 set the instrument level : to accomplish this, bring the spirit- 
 bubble, G, attached to the horizontal bar in a direction parallel 
 to two of the foot-screws, and by their motion cause the air- 
 bubble to assume a central position in the glass-tube ; then turn 
 the telescope, level, &c., half round, and if the bubble is not 
 central, correct half the deviation by raising or lowering one 
 end of the level itself, and the other half by the foot-screws, 
 which in this instrument perform an office similar to that of the 
 parallel plate-screws of the theodolite already described. Hav- 
 ing perfected this part, turn the telescope a quarter round, and 
 the level will be over the third foot-screw, which must be moved 
 to set the level correct, and this part of the adjustment will be 
 complete. 
 
 The line of collimation must be next attended to : direct the 
 telescope to some well-defined object, and make the vertical wire 
 bisect it ; then turn the axis end for end, an operation which of 
 course inverts the telescope, and if the object be not now bisected 
 by the vertical wire, correct half the deviation by the collimat- 
 ing screws at the eye-end of the telescope, and the other half by 
 giving motion in azimuth to the instrument, and this must be 
 repeated till the adjustment is satisfactorily accomplished. 
 
 Finally, for the zero of altitude. Take the altitude or depres- 
 sion of an object with the vertical sector in reversed positions ; 
 half the sum will be its true altitude, or depression, and to this, 
 let the verniers be set. Again carefully direct the telescope to 
 the object, making the bisection by the screws which retain the 
 index in a horizontal position, and finally correct the level by 
 the adjusting screws at one of its ends. 
 
 The Method of Observing with the Theodolite. 
 
 In describing the use of the theodolite, it is not our intention 
 to enter upon an account of the different ways in which it is ap- 
 plied to the purposes of land-surveying, since we do not profess
 
 THE THEODOLITE. 
 
 23 
 
 to write a treatise upon that subject, but in addition to what we 
 here insert, some further particulars will be found in the Ap- 
 pendix, where we purpose explaining the manner of surveying 
 roads, boundaries, &c., in connection with the method of using 
 a circular protractor. Confining ourselves therefore to the man- 
 ner of measuring angles by its assistance, we observe, that the 
 instrument being placed exactly over the station from whence 
 the angles are to be taken, by means of the plumb-line suspended 
 from its centre, it must be set level by the parallel plate-screws, 
 b b, &c., (page 15) bringing the telescope over each pair alter- 
 nately ; one must be unscrewed while its opposite one is screwed 
 up, until the two spirit-levels on the vernier plate steadily keep 
 their position in the middle of their tubes, while the instrument 
 is turned quite round upon its staff-head, when it will be ready 
 for commencing operations. (We are now supposing that the 
 adjustments before described have been carefully examined and 
 rectified, otherwise the observations will be good for nothing.) 
 First, clamp the lower horizontal limb firmly in any position, and 
 direct the telescope to one of the objects to be observed, moving 
 it till the cross-wires and object coincide ; then clamp the upper 
 limb, and by its tangent-screw make the intersection of the 
 wires nicely bisect the object ; now read off the two verniers, the 
 degrees, minutes, and seconds of (either) one, which call A,* and 
 the minutes and seconds only of the other, which call B, and 
 take the mean of the readings thus : 
 
 A =112 36' 30" 
 B = , 37 
 
 Mean =142 56 45 
 
 Next release the upper plate and move it round until the tele- 
 scope is directed to the second object (whose angular distance 
 from the first is required), and clamping it, make the cross-wires 
 bisect this object, as was done by the first; again read off the 
 two verniers, and the difference between their mean, and the 
 mean of the first reading, will be the angle required. 
 
 Some persons prefer making their first reading r= zero, by 
 clamping the upper to the lower plate of 360, and bisecting the 
 first object by the clamp and slow motion of the lower limb ; 
 then their second reading will be the absolute angle subtended 
 by the two objects : but as both verniers seldom read exactly 
 
 * It would be better to have the letters A, B, &c., engraved over the verniers, 
 making it a rule always to read the degrees from the one called A, which would 
 prevent confusion, and the possibility of a mistake when observing a number of 
 objects from one station. This is always done (by the makers) upon the verniers 
 of large instruments.
 
 24 SURVEYING INSTRUMENTS. 
 
 alike,* the mean of them should still be taken, unless one vernier 
 alone is used, which should never be the case ; therefore it 
 matters not at what part of the lower, the upper limb is clamped, 
 provided the angle is read off every time an object is bisected, 
 for the difference between any two readings will be the angle 
 subtended by the objects observed. 
 
 It would appear from the above statement, that it is not 
 necessary for the lower horizontal plate to have any motion at 
 all, which is certainly the case when angles are simply to be 
 measured ; but its use is important, as it gives us the means of 
 repeating the measure of any angle we may wish to determine 
 with great accuracy, it being evident that a mean of a number 
 of observations will give a more correct result than a single one. 
 To repeat an angle, therefore, after making the second bisection 
 as above directed, leave the upper plate clamped to the lower, 
 and release the clamp of the latter ; now move the whole instru- 
 ment (bodily) round towards the first object, till the cross-wires 
 are in contact with it ; then clamp the lower plate firm, and 
 make the bisection with the lower tangent-screw. Leaving it 
 thus, release the upper plate, and turn the telescope towards the 
 second object, and again bisect it by the clamp and slow motion 
 of the upper plate. This will complete one repetition, and if 
 read off, the difference between this and the first reading will be 
 double the real angle. It is, however, best to repeat an angle 
 four or five times ; then the difference between the first and last 
 readings (which are all that it is necessary to note) divided by the 
 number of repetitions will be the angle required. 
 
 The magnetic bearing of an object is taken, by simply reading 
 the angle pointed out by the compass-needle, when the object is 
 bisected ; but it may be obtained a little more accurately by 
 moving the upper plate (the lower one being clamped) till the 
 needle reads zero, at the same time reading off the horizontal 
 limb ; then turning the upper plate about, bisect the object and 
 read again : the difference between this reading and the former 
 will be the bearing required. 
 
 In taking angles of elevation or depression, it is scarcely 
 necessary to add, that the object must be bisected by the* hori- 
 zontal wire, or rather by the intersection of the wires, and that 
 after observing the angle with the telescope in its natural position, 
 it should be repeated with the telescope turned half round in its 
 Y's, that is, with the level uppermost; the mean of the two 
 measures will neutralize the effect of any error that may exist 
 in the line of collimation. 
 
 * The reason of their reading differently, arises from the errors of eccentricity, 
 or of graduation, and perhaps of both ; the object of having two readings is to 
 diminish the effect of these errors, which is more effectually done by three ver- 
 niers ; but this being inconvenient in small instruments, two only are applied.
 
 THE THEODOLITE. 05 
 
 The proof of the accuracy of a number of horizontal angles, 
 if they quite surround the station from whence they are taken, 
 is to add them altogether, and their sum, if correct, will be 360. 
 If they are taken at several stations, consider them as the internal 
 angles of a geometrical figure, and the lines connecting the 
 stations as the sides of such figure ; then, if the figure has three 
 sides, their sum will = 180, if four sides, = 360; if more than 
 four, multiply 90 by double the number of sides, and subtract 
 360 from the product ; the remainder will be the sum of the 
 internal angles. 
 
 The altitude and azimuth of a celestial object may likewise be 
 observed with the theodolite, the former being merely the eleva- 
 tion of the object taken upon the vertical arc, and the latter its 
 horizontal angular distance from the meridian. 
 
 The following particulars refer to the engraving at page 21. 
 
 The figure at page 21 shows the triangular plate, or base, upon 
 which the instrument is set when in use. A is the screw where- 
 by the plate is attached to the head of the tripod staff, or legs of 
 the instrument. D shows the edge of the lower or main plate, 
 that is so screwed on to the staff head. B, B, B, is the upper 
 plate, which slides on the surface of the lower one, D ; each 
 angle of the upper plate is perforated to admit of the passing of 
 the flanged heads of the footscrews of the tripod to cells made in 
 the lower plate for their reception, and when the instrument is 
 thus dropped into its place on the stand, it is there secured by 
 sliding the upper plate into the position shown in the above 
 figure, whereby the narrow part of the perforations are brought 
 over the heads or flanges of the foot-screws, and they are then 
 retained in their places. The two plates are thus kept in the 
 position now described, by the catch C, which acts with a spring, 
 and prevents their having any lateral motion. By the above 
 contrivance the theodolite is readily attached to the stand, or 
 vice versa.
 
 LEVELLING INSTRUMENTS. 
 
 THE Y SPIRIT-LEVEL. 
 
 The above figure represents this instrument ; it has an achro- 
 matic telescope, mounted in Y's like those of the theodolite, and 
 is furnished with a similar system of cross wires for determining 
 the axis of the tube, or line of collimation. By turning the 
 milled-headed screw, A, on the side of the telescope, the internal 
 tube, a, will be thrust outwards, which carrying the object-glass, 
 it is by this means adjusted to its focal distance, so as to show a 
 distant object distinctly. 
 
 The tube, c c, carrying the spirit-bubble, is fixed to the under 
 side of the telescope by a joint at one end and a capstan-headed 
 screw at the other, which sets it parallel to the optical axis of 
 the telescope ; at the opposite end is another screw, e, to make 
 it parallel in the direction sidewise. One of the Y's is supported 
 in a socket, and can be raised or lowered by the screw, B, to 
 make the telescope perpendicular to the vertical axis. Between 
 the two supports is a compass-box, C, (having a contrivance to 
 throw the magnetic needle off its centre when not in use) : it is 
 convenient for taking bearings, and is not necessarily connected 
 with the operations of levelling, but extends the use of the in- 
 strument, making it a circumferentor. The whole is mounted 
 on parallel plates and three legs, the same as the theodolite. 
 
 It is evident, from the nature of this instrument, that three 
 adjustments are necessary. First, to place the intersection of 
 the wires in the telescope, so that it shall coincide with the axis
 
 THE Y SPIRIT-LEVEL. 
 
 27 
 
 of the cylindrical rings on which the telescope turns ; secondly, 
 to render the level parallel to this axis ; and lastly, to set the 
 telescope perpendicular to the vertical axis, that the level may 
 preserve its position while the instrument is turned quite round 
 upon the staves. 
 
 To Adjust the Line of Collimation. 
 
 The eye-piece being drawn out, to see the wires distinctly, 
 direct the telescope to any distant object, and by the screw, A, 
 adjust to distinct vision ;* bring the intersection of the cross 
 wires to coincide with some well-defined part of the object, then 
 turn the telescope round on its axis as it lies in the Y's, and 
 observe whether the coincidence remains perfect during its revo- 
 lution : if it does, the adjustment is correct, if not, the wires 
 must be moved one-half the quantity of error, by turning the 
 little screws near the eye-end of the telescope, one of which 
 must be loosened before the opposite one is tightened, which, if 
 correctly done, will perfect this adjustment. 
 
 To set the Level parallel to the Line of Collimation. 
 
 Move the telescope till it lies in the direction of two of the 
 parallel plate-screws, (the clips which confine the telescope in 
 the Y's being laid open,) and by giving motion to the screws, 
 bring the air-bubble to the middle of the tube, shown by the 
 two scratches on the glass. Now reverse the telescope carefully 
 in its Y's, that is, turn it end for end ; and should the bubble not 
 return to the centre of the level as before, it shows that it is not 
 parallel to the optical axis, and requires correcting. The end 
 to which the bubble retires must be noticed, and the bubble 
 made to return one-half the distance by the parallel plate-screws, 
 and the other half by the capstan-headed screw at the end of 
 the level, when, if the halves have been correctly estimated, the 
 air-bubble will settle in the middle in both positions of the tele- 
 scope. This and the adjustment for the Collimation generally 
 require repeated trials before they are completed, on account of 
 the difficulty in estimating exactly half the quantity of deviation. 
 
 To set the Telescope perpendicular to the vertical Axis. 
 
 Place the telescope over two of the parallel plate-screws, and 
 move them (unscrewing one while screwing up the other) until 
 
 * The eye-piece must first be drawn out until the cross wires are perfectly well 
 defined, then the object-glass moved till distinctvision is obtained without parallax, 
 which will be the case if, on looking through the telescope at some distant object, 
 and moving the eye sidewise before the eye-glass, the object and the wires remain 
 steadily in contact ; but if the wires have any parallax, the object will appear 
 flitting to arid from them.
 
 gg LEVELLING INSTRUMENTS. 
 
 the air-bubble of the level settles in the middle of its tube ; 
 then turn the instrument half round upon the vertical axis, so 
 that the contrary ends of the telescope may be over the same 
 two screws, and if the bubble again settles in the middle all is 
 right in that position ; if not, half the error must be corrected 
 by turning the screw, B, and the other half by the two parallel 
 plate-screws over which the telescope is placed. Next turn the 
 telescope a quarter round, that it may lie over the other two 
 screws, and make it level by moving them, and the adjustment 
 will be complete. 
 
 Before making observations with this instrument, the adjust- 
 ments should be carefully examined and rectified, after which 
 the screw B should never be touched ; the parallel plate-screws 
 alone must be used for setting the instrument level at each sta- 
 tion, and this is done by placing the telescope over each pair 
 alternately, and moving them until the air-bubble settles in the 
 middle. This must be repeated till the telescope can be moved 
 quite round upon the staff-head, without any material change 
 taking place in the bubble. 
 
 A short tube, adapted to the object-end of the telescope, will 
 occasionally be found useful in protecting the glass from the in- 
 tensity of the sun's rays, and from damp in wet weather. 
 
 TROUGHTON'S IMPROVED LEVEL. 
 
 This modification of the instrument has a very decided advan- 
 tage over the Y level, inasmuch, as in its construction it is 
 more compact, and the adjustments when once made are less 
 liable to be deranged ; although, to a person unused to the in- 
 strument, they will at first appear more tedious to accomplish. 
 
 The telescope, A B, rests upon the horizontal bar, a b, which 
 turns upon the staff-head (similar to the one employed in the
 
 TROUGHTON'S IMPROVED LEVEL. gg 
 
 Y level and the theodolite). On the top of the telescope, and 
 partly imbedded within its tube, is the spirit-level, cd t over 
 which is supported the compass-box, C, by four small pillars ; 
 thus admitting the telescope to be placed so close to the hori- 
 zontal bar, ab, that it is much more firm than in the former 
 instrument. The bubble of the level is sufficiently long for its 
 ends to appear on both sides of the compass-box ; and it is 
 shown to be in the middle by its coinciding with scratches made 
 on the glass tube as usual. 
 
 The wire plate (or diaphragm) is generally furnished with 
 three threads, two of them vertical, between which the station- 
 staff may be seen ; and the third, by which the observation is 
 made, is placed horizontally. Sometimes a pearl micrometer- 
 scale is fixed perpendicularly on the diaphragm instead of wires. 
 This consists of a fine slip of pearl, with straight edges, one of 
 which is divided into a number of parts, generally hundred ths 
 or two-hundredths of an inch ; and it is so fixed, that the divided 
 edge intersects the line of collimation, the central division indi- 
 cating the point upon the staff where the observed level falls. 
 The scale itself may be employed in approximately determining 
 distances, as will be shown hereafter. It is also very useful in 
 roughly estimating equal distances from the instrument in any 
 direction. Thus, if a man in attendance holds up a staff at any 
 distance, and the observer, looking at it through the telescope, 
 notices how many divisions of the micrometer-scale the staff 
 appears to subtend, then, if the man moves in any other direc- 
 tion, retiring until the same staff appears to cover an equal 
 number of divisions, he will be at the same distance from the 
 instrument as before. We have seen a successful application 
 of a delicate wire micrometer to a levelling telescope precisely 
 similar to those applied to astronomical instruments, by means 
 of which distances can be determined with great precision, and 
 will fail only when the wind is too high to permit the instru- 
 ment or staff to remain steady. 
 
 The telescope is generally constructed to show objects in- 
 verted ; and as such a telescope requires fewer glasses than one 
 which shows objects erect, it has the advantage in point of bril- 
 liancy ; and when an observer is accustomed to it, the apparent 
 inversion will make no difference to him. A diagonal eye-piece, 
 however, generally accompanies the instrument, and by it ob- 
 jects can be seen in their natural position. A cap is adapted to 
 the object-end of the telescope, to screen the glass from the rays 
 of the sun, or from the rain : when the cap is used, it should be 
 drawn forwards as much as possible. 
 
 The requisite adjustments for this instrument are the same as 
 those of the Y level ; viz., that the line of collimation and the 
 level be parallel to each other, and that the telescope be exactly 
 perpendicular to the vertical axis ; or, in other words, that the
 
 30 LEVELLING INSTRUMENTS. 
 
 spirit-bubble preserve its position while it is turned round hori- 
 zontally on the staff-head. The adjustment of the level is effected 
 by correcting half the observed error by the capstan-screws, ef, 
 which attach the telescope to the horizontal bar, and the other 
 half by the parallel plate-screws : the capstan-screws, e,f, have 
 brass covers to defend them from injury or accidental disturb- 
 ance, but admit their adjustment when necessary. 
 
 The spirit-level itself has no adjustment, being firmly fixed in 
 its cell by the maker, and therefore the line of collimation must 
 be adjusted to it, by means of two screws, near the eye-end of 
 the telescope : the manner of doing this is as follows : Set up 
 the instrument on some tolerably level spot of ground, and after 
 levelling the telescope by the parallel plate-screws, direct it to a 
 staff held by an assistant at some distance (from ten to twenty 
 chains) ; direct him by signals to raise or depress the vane, until 
 its wire coincides with the horizontal wire of the telescope (or 
 central division of the micrometer scale ;) now measure the height 
 of the centre of the telescope above the ground, and also note 
 the height of the vane on the staff; let, for example, the former 
 be four feet and the latter six, their difference shows that the 
 ground over which the instrument stood is two feet higher than 
 where the staff is placed. Next make the instrument and staff 
 change places, and observe in the same manner as before, and if 
 it gives the same difference of level, the instrument is correct ; 
 if otherwise, take half the difference between the results, and 
 elevate or depress the vane that quantity, according as the last 
 observation gives a greater or less difference than the first. 
 Again, direct the telescope to the staff, and make the coinci- 
 dence of the horizontal wire and that on the vane perfect, by 
 turning the collimation-screws. 
 
 Suppose the instrument to be set up at A, and the staff at B. 
 C D will be the line of sight. A C the height of the instrument 
 = 4< feet, B D the height of the vane = 6 feet ; their difference 
 = 2 feet. On removing the instrument to B, and the staff to A, 
 c d will be the line of sight, giving for the difference of height 
 between B c and A d=2 feet, as before, if the adjustment is 
 correct ; but if it is incorrect, the direction of the line of sight 
 will be either above or below cd,as is shown by the dotted lines. 
 If above it, the difference will be greater than two feet, and the
 
 TROUGHTON'S IMPROVED LEVEL. gi 
 
 vane must be lowered half that quantity, and the collimation- 
 screws moved to correct the other half ; if below the line c d, the 
 difference will be less than two feet, and the vane must be raised 
 half that quantity, &c. 
 
 Another method of proving the adjustment of the line of 
 collimation is as follows : Let there be two staves held upright 
 at any convenient distance from each other ; call one staff A, 
 and the other B ; then place the instrument nearly in a line with 
 the staves, at about one or two chains' length beyond that called 
 A, and having set it level by the parallel plate-screws, read off 
 both the staves ; having done this, remove the instrument to 
 about the same distance from the staff B, set it level, and again 
 read the staves : now, if half the sum of the readings upon the 
 staff A, and also of those upon the staff B, be taken, they will 
 give two points upon the staves that are truly horizontal, by 
 which, or by any other points equidistant therefrom, as may best 
 suit the height at which the instrument is set up, so as to be seen 
 in the field of the telescope, the horizontal wire maybe adjusted, 
 that is, moved by its proper screws, so as to coincide with both 
 those points (or readings on the staves). 
 
 A third method of adjustment is by means of a sheet of water, 
 and when practicable, is both convenient and accurate : thus, at 
 the distance of a few chains, drive two stakes close to the water's 
 edge, so that their upper ends may be even with the surface of 
 the water ; let the level be set up over one of the stakes, and a 
 staff held perpendicular upon the other. Now, having measured 
 the height of the centre of the telescope above the stake over 
 which it is placed, it remains but to move the horizontal wire, 
 either up or down, till it points out exactly the same height on 
 the staff, if it does not already do so. 
 
 The adjustment of TROUGHTON'S level may also be effected by 
 employing as a collimator the telescope of a theodolite, or Y 
 level, in the following manner : First ascertain that the adjust- 
 ment of the collimating telescope is perfect ; then set both in- 
 struments up with their telescopes nearly at the same height, and 
 their object glasses opposed to each other, so that, upon placing 
 the eye to either instrument, you may be able to look through 
 both telescopes at once ; or, to speak more correctly, you must 
 see the image of the field of the further telescope with its cross- 
 wires distinctly. Both instruments must be carefully levelled, 
 and the telescopes adjusted to about the focus for distinct vision 
 of a remote object : this done, look through the telescope of 
 TROUGHTON'S level, and by the rack motion obtain distinct 
 vision of the cross-wires in the collimating instrument ; and if 
 the horizontal lines of them both exactly coincide, the adjust- 
 ment is perfect, if not, they must be made to do so by means of 
 the screws that act upon the wire plate. It should be remarked, 
 that the level of the instrument employed as the collimator
 
 32 LEVELLING INSTRUMENTS. 
 
 should be at least as sensible as that of the instrument under 
 adjustment, otherwise this method will be very uncertain. 
 
 It would be advisable, when the instrument is in perfect ad- 
 justment, to fix a level mark on some permanent spot, as a wall, 
 &c., to which the level may be from time to time referred, by 
 simply setting it up at a certain height from the ground, and 
 looking through the telescope at the mark ; any error in collima- 
 tion will be immediately detected, and may be corrected by the 
 collimation-screws only. 
 
 The Method of approximately determining Distances by the 
 Micrometer Scale. 
 
 First ascertain the value of the divisions on the scale, and 
 arrange them in a tabular form ; to do which, measure off one 
 chain's length from the object end of the telescope, and having 
 set up a staff there, observe how many divisions and tenths of a 
 division on the scale are occupied by the whole length of the 
 staff, or any part of it. Do the same when it is placed at 2, 3, 
 4, &c., chains, as far as 10, and place the results in a table. 
 
 Now to determine any distance, set up the same staff, or one 
 of equal length, at the distant spot ; observe how many divisions 
 and tenths on the scale its whole length subtends, and take from 
 your table the nearest number of divisions and parts, which make 
 the first term of an inverse proportion, the second term is the 
 number of chains corresponding thereto, the third the observed 
 divisions and parts, and the fourth will be your answer, viz., the 
 distance required. 
 
 In making the observations, great care is required in esti- 
 mating the number of divisions, &c., subtended on the scale by 
 the distant staff, as an error of half a division would occasion a 
 considerable error in the final result. 
 
 MR. GRAVATT'S LEVEL. 
 
 The folio wing engraving represents Mr. GRAVATT'S modification 
 of the spirit-level, whereby he obtains advantages, both optical 
 and mechanical ; the former by adapting an object-glass of large 
 aperture and short focal length to the telescope, for the purpose 
 of obtaining the light and power of a large instrument without the 
 inconvenience of its length ; and the latter, by various contri- 
 vances, described as follows : A A is the telescope, having a 
 diaphragm with cross-wires placed in the usual manner ; the 
 internal tube or slide which carries the eye-piece, &c., is nearly 
 equal in length to the external or telescope tube, which being 
 sprung at its aperture, (as shown in the cut,) secures to the slide 
 and the eye-piece a steady and parallel motion when adjusting
 
 GRAVATT'S LEVEL. gg 
 
 for distinct vision of a distant object by the milled-head P. The 
 spirit-level is represented at B, placed above the telescope, and 
 attached to two rings passing round it by the capstan-headed 
 screws C C, which are the means of adjusting the air-bubble of 
 the level for parallelism with the line of collimation. D repre- 
 sents a small level placed across the telescope at right angles to 
 the principal level C C ; it is very convenient in setting the 
 instrument up approximately level by means of the legs only, 
 which saves time, and also the wear of the parallel plate-screws. 
 (Practical men are aware of the uncertainty in judging by the 
 eye alone when the instrument is set nearly level, especially on 
 the side of sloping ground, and duly appreciate the application 
 of the cross-level, by which their valuable time and the wear of 
 their instruments are saved.) Having directed the sight to the 
 staif, and adjusted for distinct vision, the two levels at once show 
 which of the screws require touching, to perfect the level, before 
 noting the observation. 
 
 A mirror, mounted by a hinge-joint on a spring piece of brass, 
 is placed on the telescope, as represented at.E; its use is to 
 reflect the image of one end of the air-bubble (in the principal 
 level) to the eye, so that the observer (after having carefully 
 adjusted his level) can, at the same time that he is reading the 
 staff, see that the instrument retains its position, by noticing if 
 the reflected end of the air-bubble coincide with the proper 
 division of the small scale fixed on the bubble-tube:* this is 
 particularly useful in windy weather, or when levelling over soft 
 
 * The small scale spoken of in the text is not applied by the maker unless par- 
 ticularly ordered, hut in all cases the requisite marks or divisions are made on the 
 bubble-tube.
 
 $4 LEVELLING INSTRUMENTS. 
 
 or boggy ground, where the least movement of the observer 
 will materially alter the level of the instrument, as by keeping 
 both eyes open, with a little practice, the cross-wires, the bubble, 
 and the staff, can be all three seen at the same time, and by a 
 slight pressure of the hand upon one of the level-legs, any dis- 
 placement of the bubble may be corrected. The parallel plates 
 and screws, G G, are similar in every respect to those of the 
 former-described instruments. We may remark, that it is con- 
 venient to have one of the screws resting in a notch, or Y, fixed 
 on the lower plate exactly over one of the legs ; then, by giving 
 motion to that leg only, after the other two are fixed in the 
 ground, the instrument can be set up so nearly level, that a very 
 small motion of the parallel plate-screws will be required to 
 perfect it. 
 
 H H represent two capstan-screws, the same as in TROUGH- 
 TON'S level, and for a similar purpose, viz., to make the spirit- 
 bubble maintain a central position in its tube, while the instru- 
 ment is turned completely round on the staff- head. I, is the 
 compass, which contains either a floating card or graduated 
 silver ring, mounted on the needle, the divisions of which are 
 magnified by a lens, K, which slides in a socket, (not shown in 
 the figure,) affording the means of reading to 10 minutes of a 
 degree : the rapid vibrations of the card or needle are checked 
 and speedily brought to rest by a contrivance, in which a spiral 
 spring is moved by a milled-headed screw ; this acts upon the 
 needle independent of its centre, which is thus secured from its 
 liability in the ordinary construction to get blunted, whereby 
 the sensibility of the needle is destroyed : the same milled-head 
 will clamp the needle when not in use, and prevent the mis- 
 chievous consequences which would arise from suffering it 
 continually to play upon its centre. 
 
 The adjustments of this instrument may be examined and rec- 
 tified in the same manner as described for TROUGHTON'S level, 
 but much more correctly as described by Mr. GRAVATT himself, 
 which we have his permission for inserting. By his method the 
 instrument may be so adjusted, that any imperfections in the slide 
 or tube of the telescope, arising from their not being straight, 
 may not in the least cause the intersection of the cross-wires to 
 deviate from the optical axis of the telescope in its motion, 
 during adjustment for distinct vision. 
 
 To examine and correct the Collimation. 
 
 " On a tolerably level piece of ground drive in three stakes, at 
 intervals of about four or five chains, calling the first stake, a, 
 the second, b, and the third, c. 
 
 " Place the instrument half way between the stakes a and b, 
 and read the staff A, placed on the stake a, and also the staff
 
 GRAVATT'S LEVEL. 35 
 
 B, placed on the stake /; ; call the two readings, A' and B'; then, 
 although the instrument be out of adjustment, yet the points 
 i-ead off will be equidistant from the earth's centre, and conse- 
 quently level. 
 
 " Now remove the instrument to a point half way between b 
 and c. Again read off the staff B, and read also a staff placed 
 on the stake c, which call staff C (the one before, called A, being 
 removed into that situation). Now by adding the difference of 
 the readings on B (with its proper sign) to the reading on C, 
 we get three points, say A', B', and C', equidistant from the earth's 
 centre, or in the same true level. 
 
 " Place the instrument at any short distance, say half a chain 
 beyond A, and, using the bubble merely to see that you do not 
 disturb the instrument, read all three staffs, or, to speak more 
 correctly, get a reading from each of the stakes, a, b, c : call 
 these three readings, A" B" C". Now, if the stake b, be half 
 way between a and c, then ought C" C' (A" A') be equal 2 
 [B" B' (A" A')] ; but if not, alter the screws which adjust 
 the diaphragm, and consequently the horizontal spider-line, or 
 wire, until such be the case ; and then the instrument will be 
 adjusted for collimation. 
 
 " To adjust the spirit-bubble, without removing the instrument, 
 read the staff, A, say it reads A'", then adding (A'" A') with its 
 proper sign to B' we get a value, say B'". 
 
 " Adjust the instrument by means of the parallel plate-screws, 
 to read B"' on the staff B. 
 
 " Now, by the screws attached to the bubble-tube, bring the 
 bubble into the centre of its run. 
 
 " The instrument will now be in complete practical adjustment, 
 for level, curvature, and horizontal refraction, for any distance 
 not exceeding ten chains, the maximum error being only ToVo^ 
 of a foot." 
 
 EXAMPLE. 
 
 Tlje instrument being placed half way between two stakes, a 
 and 6," (at one chain from each,) the staff on a or A' read 6'/>3, and 
 staff on b or B' read 3*34, placing the instrument half way be- 
 tween the stakes b and c, (three chains from each) the staff on b 
 read 4'01, and the staff on c read 5'31. 
 
 Hence, taking state a as the datum, we have 
 
 Stake. Above Datum, 
 
 a or A' = 0-00 
 b or B' 3-19 
 
 c or C' 1-89 
 
 The instrument being now placed at d, (say five feet from a, 
 but the closer the better,) the staff on a or A" read4'01, on b or 
 B", 1 *03, and on c, or C", 3'07. Now had the instrument been 
 
 D 2
 
 36 LEVELLING INSTRUMENTS. 
 
 in complete adjustment (under which term curvature and refrac- 
 tion are included), when the reading on staff a was 4*0 1, the 
 readings on /; and c should have been respectively 0'82 and 2' 12. 
 
 The instrument therefore points upwards, the error at b being 
 0*2 1, and the error at c, 0*95: now, were the bubble only in 
 error, (as is supposed in all other methods of adjustment,) the 
 error at c ought to be four times as great as at b, but 4 X O21 
 0*84 only, there is an error, therefore, of 0*95 0'84=0'11 
 not due to the bubble. 
 
 For the purpose of correcting this error, (and be it remem- 
 bered contrary to former practice, for this purpose only,) we must 
 use the capstan-headed screws at the eye end of the telescope, 
 and neglecting the actual error of level, we are only to make the 
 error at b one-fourth that of c. 
 
 After a few trials, whilst the reading at a continued 4'01, the 
 reading on b became 0*75, and that on c, I '84. 
 
 Now 0-82 0-75 = 0-07, and 2' 12 1 -84 = 0-28. 
 
 And as 4 X 0-07 = 0'28, the telescope is now adjusted for 
 collimation. 
 
 All that remains to be done, is to raise the object end of the 
 telescope by means of the parallel plate-screws, until the staff at 
 c reads 2' 12, and then, by means of the nuts which adjust the 
 bubble-tube to bring the bubble into the centre of its run. 
 
 The operation of collimating, when once performed upon 
 levels on Mr. GRAVATT'S construction, will scarcely ever need 
 being repeated. 
 
 OF THE LEVELLING STAVES. 
 
 Two mahogany station-staves generally accompany the spirit- 
 level ; they consist of two parts, capable of being drawn out 
 when considerable length is required. They are divided into 
 feet and hundredths, or feet, inches, and tenths, and have a sliding 
 vane, with a wire placed across a square hole in the centre, as 
 shown in the annexed figure : this vane being raised or lowered 
 by the assistant, until the cross-wire corresponds with the 
 horizontal wire of the telescope, the height of the 
 wire in the vane, noted on the staff, is the height of 
 the apparent level above the ground at that place. 
 
 When both the staves are used, they should be 
 set up at equal distances on each side of the spirit- 
 level : the difference of the heights of their vanes 
 will be the absolute difference of level between the 
 two stations. But when one staff only is employed, 
 the difference between the height of the vane and 
 the height of the centre of the telescope of the 
 instrument, will be the apparent difference of level, 
 which, if the distance between the staff and instru-
 
 LEVELLING STAVES. 37 
 
 inent is great, requires to be corrected for the curvature of the 
 earth. The method of computing this correction will be pre- 
 sently shown. 
 
 TROUGHTON'S LEVELLING-STAVES. 
 
 These consist of three sliding rods of mahogany, each about 
 four feet long, and they are divided into feet, &c., as those which 
 have just been described. The sliding vane is circular, having at 
 the lower edge a square aperture, one side of which is bevelled ; 
 and a line on the bevelled side denotes 
 the reading of the staff. The face of 
 the vane is made of white holly, with 
 an inlaid lozenge of ebony, forming at 
 once a conspicuous object, and one easy 
 of bisection. A circular spirit-level is 
 attached to the top of the hindermost 
 rod, to guide the assistant in holding it 
 perpendicular. 
 
 In levelling, the vane must be moved 
 up or down, until the horizontal wire 
 of the telescope bisects the acute angles 
 of the lozenge, or in other words passes 
 through its horizontal extremities, as 
 shown in the figure. 
 
 The line on the bevelled edge at a (as before stated) denotes 
 the reading of the staff; therefore, a piece equal in length to 
 the distance, a b, is cut off from the bottom of the staff, or rather 
 the divisions commence at that number of inches above 0. 
 
 When the observation requires that the vane be raised to a 
 greater height than four feet, the object is effected by leaving it 
 at the summit of the rod in front, and then sliding this rod up 
 upon the one which is immediately behind it, this will carry the 
 vane up to eight feet ; and from that to twelve may be obtained 
 by similarly sliding the second upon the third rod. In the latter 
 steps, the reading is at the side of the staff, the index division 
 remaining stationary, and at four feet from the ground, a cir- 
 cumstance which affords greater facility in reading off. 
 
 M 
 
 THE NEW LEVELLING-STAVES. 
 
 Several years ago, WILLIAM GRAVATT, Esq., had constructed 
 for his own use a new kind of levelling-staff, which now appears 
 likely to come into general use. They have no vane to slide up 
 and down, but the face of each staff is made broad enough to 
 contain sufficiently large graduations and figures, for the observer 
 to read with certainty to the one-hundredth part of a foot, at 
 the distance of twelve chains or more, which is sufficient for
 
 gg LEVELLING INSTRUMENTS. 
 
 most practical purposes, thus securing greater certainty and 
 expedition in the work ; for it not unfrequently happened in 
 using the old staves, that when, by a succession of signals, the 
 staff-holder had nearly brought the wire of the vane to coincide 
 with that of the telescope, he would, in his attempt to perfect 
 it, remove the vane further from coincidence than at first ; and 
 we have been informed, that on one occasion the man held the 
 staff upside down, which introduced an error of several feet. 
 To obviate these difficulties, Mr. GRAVATT proposed, that the 
 observer should read the staff himself, which is now successfully 
 practised. 
 
 The newly-constructed staff consists of three parts, which 
 pack together for carriage in a neat manner, and when opened 
 out for use form a staff seventeen feet long, jointed together, 
 something after the manner of a fishing rod : the whole length 
 is divided into hundredths of a foot, alternately coloured black 
 and white, and occupying half the breadth of the staff; but for 
 distinctness, the lines denoting tenths of feet are continued the 
 whole breadth, every half foot or five tenths being distinguished 
 by a conspicuous black dot on each side. The whole contrivance 
 is very successful, and in some late levelling operations in which 
 we were engaged, we were able perfectly to read the staff, with 
 only a fourteen-inch level, at the distance of twelve chains. 
 
 ON LEVELLING. 
 
 " Levelling is the art of finding a line parallel to the horizon 
 at one or more stations, to determine the height or depth of one 
 place with respect to another. Two or more places are on a true 
 level, when they are equally distant from the centre of the earth. 
 Also, one place is higher than another, or above the level of it, 
 when it is further from the centre of the earth ; and a line equally 
 distant from that centre in all its parts, is called a line of true 
 level. Hence, because the earth is round, that line must be a 
 curve, and make a part of the earth's circumference, or at least 
 be parallel to it ; as the line I B C F G, which has all its points 
 equally distant from A, the centre of the earth; considering it 
 as a perfect sphere. 
 
 " But the line of sight, B, D, 
 E, &c., given by the operation of 
 levels, called the apparent line 
 of level, is a tangent, or a right 
 line perpendicular to the semi- 
 diameter of the earth at the point 
 of contact, B, rising always higher 
 above'the true line of level, the 
 further the distance is. Thus, C D is the height of the ap- 
 parent level above the true level, at the distance BC or BD:
 
 ON LEVELLING. vjf) 
 
 alsoFE is the excess of height at F; G H, that at G, &c. 
 The difference, it is evident, is always equal to the excess of the 
 secant of the arc of distance above the radius of the earth. 
 
 "Now the difference C D, between the true and apparent level 
 at any distance B C or B D, may be found thus : by a well- 
 known property of the circle, 2AC + CD:BD::BD: 
 C D. But because the diameter of the earth is so great with 
 respect to the line C D, at all distances to which an operation 
 of levelling commonly extends, 2 AC may be taken for 2 A C 
 -f- CD in this proportion without sensible error. The propor- 
 tion then will be2AC:BD::BD:CD; 
 
 , T^. BD 2 BC 2 
 
 whence ] o is= _ or - nearly; 
 
 that is, the difference between the true and apparent level, is 
 equal to the square of the distance between the places, divided 
 by the diameter of the earth ; and consequently it is always 
 proportional to the square of the distance." 
 
 Now the diameter of the earth being nearly 41,796,480 feet, 
 or 7916 miles ; if we first take B C equal 1 mile, then the 
 
 B C 2 1 
 
 excess - is - , of a mile, which is 8,004 inches, for the 
 
 <v A. \_/ 
 
 height of the apparent above the true level at the distance of 
 one mile. Otherwise, if to the arithmetical complement of the 
 logarithm of the diameter, or 2,3788603, we add double the 
 logarithm of the distance in feet, we shall obtain the logarithm 
 of the difference of the true and apparent level in decimals of 
 the same, to be subtracted from the height given by the instru- 
 ment to reduce it to the true level. In this manner the correc- 
 tions have been computed contained in Table I., which shows 
 the difference in decimals of a foot between the true and ap- 
 parent level, corresponding to any distance from 20 to 5000 feet. 
 
 The usual method of obtaining the difference of level between 
 any two places, is by a tangent, whose point of contact is exactly 
 in the middle of the level line : this method may be practised 
 without regarding the difference between the apparent and true 
 level ; for it is clear, that if from the same station, two points of 
 sight be observed equally distant from the eye of the observer, 
 they will be also equidistant from the centre of the earth. Thus, 
 let the instrument be placed at B, (see the last figure,) equally 
 distant from the station staves at C and I, the two points of 
 sight, D and J, marked upon them by the tangent, J D (or J H,) 
 will be level points, and the difference in height between C D 
 and I J, will show how much the one place is higher than the 
 other. 
 
 Suppose it were required to determine the difference of level 
 between the two places A and B. First set up your instrument 
 at any convenient distance from A in a line towards B, then hav- 
 ing a staff set up perpendicular at A, measure the distance, and
 
 40 LEVELLING INSTRUMENTS. 
 
 erect another staff beyond you, in the same line, and at the same 
 distance, as the first, that the instrument may be equally distant 
 from each staff; then direct the telescope towards the first staff, 
 
 and sign to a person holding it, to move the vane higher or lower, 
 until the wire placed across it coincides with the intersection of 
 the cross wires in the telescope ; he is then to note the height 
 marked by the wire on the staff, which suppose to be 4'69 feet. 
 Now turn the telescope about, and point it towards the second 
 staff, and direct that its vane be raised or lowered, as the former 
 was, until the cross wire is intersected by the wires of the tele- 
 scope ; it must then be likewise read off ; and suppose the read- 
 ing to be 9 '93 feet. 
 
 Having completed the first level, let the first staff take the 
 place of the second, and the second to be set up further on in 
 the required direction, as at B. Then, midway between them, 
 set up your level, and direct it to the first staff, and then to the 
 second, making the necessary observations, as before, when the 
 staves being read off, the operation is completed. Let us. sup- 
 pose the first to read 0*64 feet, and the second, 1 1 '88 feet ; the 
 work will then stand thus : 
 
 Reading of the first staff 
 
 (or back station) 
 
 Feet. 
 
 First reading . . . 4,69 
 Second reading . . 0,64 
 
 Sum . . . 5,33 
 
 Reading of the second staff 
 
 (or forward station) 
 
 Feet. 
 
 First reading . . . 9,93 
 Second reading . . 11,88 
 
 Sum . . 21,81 
 
 The difference of these sums shows that the ground at B is 
 16'48 feet, or 16 feet 5'76 inches lower than at A. 
 
 By continuing the above process, the operation of levelling 
 may be carried on for many miles, the relative height of every 
 station being determined. Also, if the height from the ground 
 of the centre of the levelling telescope be taken at each place 
 at which it is set up, the relative height of that spot will also be 
 determined. 
 
 In common levelling operations, it is not usual to be particular 
 about placing the instrument in the centre between the staves, 
 but to observe each way, as far as the inclination of the ground 
 will admit. It is also found most eligible to employ staves, such 
 as those described at page 37, in preference to those with sliding 
 vanes, as the observer himself can note and register the various 
 readings, the assistant having nothing more to do than to hold 
 the staff upright. This is a far preferable mode of procedure,
 
 ON LEVELLING. 
 
 41 
 
 for it not unfrequently happens, that sufficiently intelligent per- 
 sons cannot be procured in obscure country places to hold and 
 read off the staff. Besides the greater facility and less liability 
 to error, the observer, with scarcely any loss of time, can make 
 his calculations as he proceeds, so that following each level, the 
 book contains the absolute difference of level of any station, 
 above or below a horizontal line drawn through a point assumed 
 as the standard level or point of comparison. This saves a deal 
 of after trouble in the office, as every requisite is prepared in 
 the field for laying down the section. 
 
 The following example shows the form of a levelling book; 
 in which the first staff is called the back station, and the second 
 staff the forward station ; when the first forward station will 
 become the second back station, the second forward, the third 
 back, &c. 
 
 No. of 
 Sta- 
 tion. 
 
 Back Station. 
 
 Forward Station. 
 
 Reduced 
 Levels. 
 
 Bearing. 
 
 Distance. 
 
 Staff. 
 
 Staff. 
 
 Distance. 
 
 Bearing. 
 
 1 
 
 o 
 300-20 
 
 140 
 
 Ft. 
 2-15 
 
 Ft. 
 
 14-97 
 
 358 
 
 120-10 
 
 100-00 
 
 14-97 
 
 85-03 
 2-15 
 
 2 
 
 300-40 
 
 89 
 
 50 
 
 15-14 
 
 420 
 
 120-12 
 
 87-18 
 15-14 
 
 72-04 
 0-50 
 
 3 
 
 300-15 
 
 106 
 
 0-54 
 
 14-12 
 
 275 
 
 120-0 
 
 72-54 
 14-12 
 
 58-42 
 0-54 
 
 4 
 
 300-10 
 
 109 
 
 0-83 
 
 15-31 
 
 337 
 
 120-0 
 
 58-96 
 15-31 
 
 43-65 
 0-83 
 
 5 
 
 300-0 
 
 128 
 
 1-49 
 
 12-15 
 
 609 
 
 120-0 
 
 44-48 
 12-15 
 
 32-33 
 1-49 
 
 6 
 
 300-0 
 
 592 
 
 5-96 
 
 10-50 
 
 Bottom 
 
 of River 
 
 33-82 
 10-50 
 
 23-32 
 5-96 
 
 
 
 
 10-50 
 
 3-78 
 
 215 
 
 120-0 
 
 29-28 
 3-78 
 
 25-50 
 10-50 
 
 7 
 
 300-30 
 
 221 
 
 8-84 
 
 0-90 
 
 128 
 
 119-40 
 
 36-00 
 0-90 
 
 35-10 
 
 8-84 
 
 
 
 
 
 
 
 
 43-94
 
 42 LEVELLING INSTRUMENTS. 
 
 The contents of each column may be known by the various 
 headings. At the commencement of the operation, the first 
 back station is assumed to be 100 feet above the horizontal line, 
 which is done to avoid the introduction of plus and minus signs 
 in the calculation : this being placed at the top of the column to 
 contain the reduced levels, the reading of the forward station 
 must be subtracted from it, and to the remainder must be added 
 the reading of the back station, which completes the first level, 
 the result being the height of the forward station above the 
 assumed horizontal line ; thus, in the first level, from 100 sub- 
 tract 14-97, and it leaves 85'03, to which add 2' 15, and the 
 result is 87*18, which is the height of the second station, assum- 
 ing the first to have been 100 feet above the horizontal line : the 
 operation for each of the succeeding levels is precisely similar. 
 The difference of level between any two points in the section 
 may be obtained by simply taking the difference of the heights : 
 thus to obtain the difference between the first and fourth level, 
 from 87*18 subtract 44*48 :=42*70 the difference required, and 
 the difference between 100 feet (the assumed height of the start- 
 ing point) and any other level in the book will be the difference 
 of altitude between those points : thus, in our example, to find 
 how much the bed of the river is below the point of commence- 
 ment, subtract 29*28 from 100, and the remainder, 70*72, is the 
 quantity required. 
 
 As a check upon the accuracy of the computation, it is neces- 
 sary to add up the contents of the two columns on each page 
 containing the back and fore observations, and subtracting the 
 less from the greater, the difference shows the whole amount of 
 the rise or fall of the ground (as in the example given at page 
 40) ; and if both computations have been correctly made, it will 
 be identical with the difference as shown in the column of re- 
 duced levels : thus in our example, page 41, the sum of the 
 back sights is 30*81, and the sum of the fore sights is 86'87 ; 
 their difference is 56*06, which, taken from 100, (because the 
 first station was assumed to be that height) leaves 43*94, the 
 same as given by the reduced levels. 
 
 There is also another mode of reducing levels, by having two 
 columns, one headed " rise," and the other " fall," in one of 
 which the difference between the back and fore sight must be 
 entered, according as the ground is rising or falling ; and then, 
 by the continual adding or subtracting of these quantities, (in 
 a separate column,) the reduced levels are obtained, without 
 assuming the first station to be 100, or any other number of feet 
 high. 
 
 In the practice of levelling, it is usual to leave at convenient 
 intervals, what are called bench-marks : these mostly consist of 
 permanent objects, such as gate-posts, stumps of trees, &c., on 
 which it is usual to cut a distinguishing mark, that it may be
 
 ON LEVELLING. 
 
 known hereafter. Their use is chiefly for future reference, in 
 the event of its being necessary, either to check the levels by 
 repetition, t,o change the direction of the line of levels from any 
 point, or to take up and continue the levels at the commence- 
 ment of a day's work, a bench-mark having been left at the close 
 of the day preceding : in the latter case, it is more common to 
 leave a peg driven into the ground to renew the work at. When 
 the staff is placed on a bench-mark, the bed of a river, or on any 
 object out of the direct line of levels, the same method of entry 
 and computation must be adopted, as shown in our example 
 where the bed of the river was taken, no bearing or distance 
 was noticed, but the name of the object entered instead ; 
 the computation is precisely the same as before, but when 
 the next forward station was about to be observed, the read- 
 ing taken to the bed of the river, viz., 10'50, was entered 
 in the column as a back observation, against which, the for- 
 ward reading with its bearing and distance was placed ; but 
 as both observations were taken from the same spot, they are 
 considered as belonging to the sixth station, as also would any 
 number of intermediate levels ; the seventh station being that 
 which has the next back observation taken in the actual line of 
 levelling, this distinction is sufficiently conspicuous to prevent 
 the draftsman plotting the wrong levels in the section, as com- 
 mon bench-marks are not usually noticed in it. In making a 
 section, it is of importance to take the level of all considerable 
 hollows in the ground, the bed of any river, as near the centre 
 as can be obtained, and also of public and private roads. 
 
 It is not common to apply the correction for the curvature of 
 the earth, except where extreme accuracy is required ; but by 
 way of illustrating the use of Table I. we have annexed the 
 following example. 
 
 
 
 Dist. of 
 
 
 
 
 Dist. of 
 
 
 
 No. of 
 
 
 Instru- 
 
 Correct. 
 
 Height 
 
 
 Instru- 
 
 
 
 back 
 
 Back 
 
 ment 
 
 for 
 
 of 
 
 Forward 
 
 ment 
 
 Correct. 
 
 Remarks. 
 
 Sta- 
 
 Station. 
 
 from 
 
 Curvat. 
 
 Instru- 
 
 Station. 
 
 from 
 
 for 
 
 
 tion. 
 
 
 Station. 
 
 
 ment. 
 
 
 Station. 
 
 Curvat. 
 
 
 
 Ft. In. Dec. 
 
 Feet. 
 
 In. Dec. 
 
 Ft. In. 
 
 Ft. In. D. 
 
 Feet. 
 
 In. Dec. 
 
 
 1 
 
 3 1,7 
 
 1200 
 
 0,413 
 
 4 4 
 
 11 2,6 
 
 800 
 
 0,184 
 
 
 2 
 
 6 1,6 
 
 480 
 
 0,066 
 
 4 6 
 
 8 1,7 
 
 960 
 
 0,264 
 
 
 3 
 
 1 7,3 
 
 1479 
 
 0,629 
 
 4 3 
 
 6 2,4 
 
 1220 
 
 0,427 
 
 
 4 
 
 2 4,8 
 
 984 
 
 0,276 
 
 4 7 
 
 10 8,3 
 
 2160 
 
 1,339 
 
 
 5 
 
 4 8,3 
 
 764 
 
 0,166 
 
 4 
 
 9 3,8 
 
 1190 
 
 0,406 
 
 
 6 
 
 10,2 
 
 280 
 
 0,022 
 
 4 1 
 
 11 7,3 
 
 340 
 
 0,033 
 
 
 7 
 
 7 8,7 
 
 1640 
 
 0,772 
 
 4 5 
 
 8 2 ; 1 
 
 3100 
 
 2,759 
 
 
 8 
 
 2 5,4 
 
 660 
 
 0,125 
 
 4 4 
 
 4 3,4 
 
 1700 
 
 0,829 
 
 
 Sum 
 
 29 0,0 
 
 7487 
 
 2,469 
 
 
 69 7,6 
 
 11470 
 
 6,241 
 
 
 Cor. 2,47 6,24 
 
 28 9,53 69 1,36
 
 44 LEVELLING INSTRUMENTS. 
 
 Sum of forward stations corrected for curva- Ft - In - 
 
 ture = 69 1,36 
 
 Sum of back stations = 28 9,53 
 
 Difference of level, between extreme stations = 40 3,83 
 
 Sum of distances from the instrument to the Feet - 
 
 back stations 7487 
 
 Sum of distance to forward stations 11470 
 
 Whole distance levelled = 1 8957 
 
 In this example, the difference of level being taken in feet and 
 inches, the corrections from Table I. have been multiplied by 
 12, in order to render them decimals of inches, they being, as 
 contained in the Table, decimals of feet ; consequently, if the 
 levels had been taken in feet and decimals of feet, the correc- 
 tions would have been applied at once as taken from the Table. 
 
 LEVELLING WITH THE THEODOLITE. 
 
 The use of the theodolite as a levelling instrument consists in 
 taking a series of angles of elevation and depression along the 
 line, the section of which is required. This must be done at 
 every point where the inclination of the line changes, and the 
 distance measured between the instrument and the station-staff. 
 This distance, it will be evident, is the hypothenuse of a right- 
 angled triangle ; the perpendicular of which is the difference of 
 level. To insure accuracy, the angles should be observed both 
 forwards and backwards, by making the instrument and staff 
 change places ; and a mean of the two measures should be taken 
 as the correct angle. The instrument should be set up (as nearly 
 as possible) at a constant height from the ground, and the staff 
 used for the observations should have a fixed vane, or conspicuous 
 mark on it, at exactly the same height from the ground as the 
 centre of the telescope, which mark must be bisected by the 
 cross-wires in observing. Great care should be taken that the 
 adjustments of the instrument are correct, more particularly thaj 
 of the line of collimation, and the level attached to the telescope. 
 
 With the measured distance, and the observed angle, the dif- 
 ference of level may be computed, by adding to the logarithm 
 of the measured distance, the log. sine of the vertical angle, and 
 their sum, rejecting 10 from the index, will be the log. of the 
 difference of level, (in feet or links, as the distance was measured 
 in.*) Having a series of elevations and depressions, the final 
 
 * When the distance has been measured in links of Gunter's chain, and the 
 difference of level is required in feet, it may be obtained by adding to the above 
 logarithm, the constant log. 9-8195439, when the sum, rejecting 10 from the index, 
 will be the log. of the difference of level in feet.
 
 ON LEVELLING. AK 
 
 difference of level between the extreme, or any two stations, 
 may be found by simply taking the difference of the sums of the 
 intervening elevations and depressions. 
 
 A theodolite of larger dimensions than those we have described 
 at page 15, &c., and capable of measuring vertical angles with 
 great accuracy, may be advantageously applied to take the levels 
 of a continually rising line of section ; thus, suppose the the- 
 
 odolite set up at A, and the telescope elevated so that the line 
 of sight, A B, may coincide with the vane on a staff exactly at 
 the same height from the ground as the instrument ; suppose 
 the staff placed at B, the angle of elevation being carefully noted, 
 the instrument must remain perfectly steady whilst the observer 
 is watching an assistant passing along the line with a staff, which 
 he successively holds up at every change of inclination, as at 
 C D E, &c., the staff-man raising or lowering the vane until the 
 observer perceives the cross wires of the telescope (or line of 
 sight, A B,) to coincide with that on the vane ; the height on 
 the staff is then read off and noted, which gives the depression 
 of that spot of ground below the line A B, which being done 
 along the whole distance, and a mark made on the ground at 
 each spot, that the distances may likewise be measured, unless 
 determined by a micrometer, as explained at page 29, the undu- 
 lations of the surface below the line A B is determined ; and 
 the inclination of the line of sight being likewise obtained with 
 the theodolite, the requisite data for drawing the section is ob- 
 tained. This method has been successfully practised by JOHN 
 MACNEILL, Esq., the engineer of the London and Holyhead 
 roads, with an instrument purposely constructed by Messrs. 
 TROUGHTON and SIMMS, which is not exactly a theodolite, but 
 rather a large spirit-level, capable of measuring vertical angles 
 with great precision, and has a delicate wire micrometer attached 
 to the eye-end of the telescope, by which the distances of the 
 staff from the instrument are accurately determined. 
 
 After having obtained the difference of level from station to 
 station, either with a spirit-level or theodolite, the rates of incli- 
 nation of the surface may be found by dividing the distance by 
 the difference of height ; thus, if the distance be 760 feet, and
 
 45 LEVELLING INSTRUMENTS. 
 
 the height 38 feet, 760 divided by 38 gives 20 ; showing the rate 
 of inclination to be 1 in 20. 
 
 The rate of inclination may likewise be found by observing 
 the angle of elevation or depression; and Table X., at the end 
 of the volume, shows the corresponding inclination. 
 
 LEVELLING WITH THE MOUNTAIN BAROMETER. 
 
 The employment of the barometer for the determination of 
 heights has caused it to become an interesting instrument to the 
 philosopher and the traveller ; and many attempts have been made 
 to improve it, and render it portable, that it may be conveyed from 
 place to place, without much inconvenience or risk. The following 
 figure represents the portable barometer as constructed by Mr. 
 TROUGHTON. In the brass box, A, which covers the cistern of mer- 
 cury, near the bottom of the tube, are two slits, made horizontally, 
 precisely similar and opposite to each other, the plane of the upper 
 edges of which represent the beginning of the scale of inches, or 
 zero of the barometer. The screw B, at the bottom, performs a 
 double office; first, it is the means of adjusting the surface of the 
 mercury in the glass cistern to zero, by just shut- 
 ting out the light from passing between it and 
 the upper edges of the above-named slits ; and, 
 secondly, by screwing it up, it forces the quick- 
 silver upwards, and by filling every part of the 
 tube, renders the instrument portable. 
 
 The divided scale on the upper part, is sub- 
 divided, by the help of a vernier, to the five hun- 
 dredth part of an inch. The screw C, at the top, 
 moves a sliding piece, on which the vernier scale 
 is divided, the zero of which is at the lower end of 
 the piece. In taking the height of the mercury, 
 this sliding piece is brought down and set nearly 
 by the hand, and the contact of the zero of the 
 vernier with the top of the mercurial column is 
 then perfected by the screw C, which moves the 
 vernier the small quantity that may be required 
 just to exclude the light from passing between 
 the lower edges of the sliding piece, and the 
 spherical surface of the mercury. 
 
 The barometer is attached to the stand by a ring, in which it 
 turns round with a smooth and steady motion, for the purpose of 
 placing it in the best light for reading off, &c. ; and the tripod 
 stand, when closed, forms a safe and convenient packing case for 
 the instrument. 
 
 A thermometer is always attached to the lower part of the ba- 
 rometer, to indicate its temperature ; while another, detached
 
 LEVELLING WITH THE MOUNTAIN BAROMETER. 4,7 
 
 from the instrument, is employed at the same time, to show the 
 temperature of the surrounding air. 
 
 The barometrical method of determining differences of level, 
 is founded upon the principle that the strata of air decrease in 
 density, in a geometrical proportion, when the elevations above 
 the surface of the earth increase in an arithmetical one. There- 
 fore, from the known relation between the densities and the 
 elevations, we can discover the elevations by observations made 
 on the densities by means of the barometer. 
 
 Observe at the same time the height of the mercurial columns 
 at both the stations, whose difference of elevation is required, 
 and also the temperature of the instrument by the thermometer 
 attached thereto ; and that of the surrounding air by another, 
 called the detached thermometer.* 
 
 The computations for deducing the difference of height from 
 these observations, is rendered very easy by means of Table II. 
 which is computed by the formula given by Mr. Baily, in his 
 volume of Astronomical Tables and Formulae, and is similar to 
 Table XXXVI. in the same volume, but more extended. 
 
 The following is the method of using the Table. 
 
 Find in the column headed " S" the sum of the degrees read 
 on the detached thermometers at the two stations, and take out 
 the corresponding number from the adjoining column, headed 
 "A" ; next in the column D, find the difference of the degrees 
 read on the attached thermometers, and take out the opposite 
 number in the column B ; lastly, from the column C, take out 
 the number opposite the latitude of the place of observation, 
 found in the column L ; then, 
 
 When the upper thermometer reads less than the lower one. 
 
 To the number called B, add the log. of the height of the ba- 
 rometer at the upper station, and subtract their sum from the 
 log. of the height of the barometer at the lower station, and call 
 the remainder R ; then take out the log of R and add it to the 
 numbers A and C, and the sum, rejecting the tens from the 
 index, will be the log. of the difference of the altitudes of the 
 two stations in feet. 
 
 When the upper thermometer reads more than the lower one. 
 
 To the log. of the height of the barometer at the lower station 
 add the number called B, and from their sum subtract the log. 
 of the height of the barometer at the upper station, and call the 
 remainder R ; then take out the log of R, and add it to the 
 numbers A and C, and the sum, rejecting the tens from the 
 index, will be the log. of the difference of the altitudes of the 
 two stations in feet. 
 
 * The mean result of several observations should be taken as that to be used 
 for computation.
 
 48 LEVELLING INSTRUMENTS. 
 
 EXAMPLE. 
 
 The following observations were made in the transit-room of 
 the Royal Observatory, and at the base of the statue of 
 George II. in Greenwich Hospital, latitude 51 28' to deter- 
 mine the difference of altitude. 
 
 Upper Station. Lower Station. 
 
 Detached thermometer . . . 71 5 . . . 71 5 
 Attached ditto ... 70 ... 70 
 Barometer, mean of 5 obs. . 29 870 inches 30 014 in. 
 A =4-81719 
 
 B = 0-00000 
 
 C = 9-99976 
 
 log. of bar. upper station, 1-47524 
 
 1-47524 
 log. of bar. lower station, 1-47732 
 
 7-31806 log R = 0-00208 
 
 Sum 2-13501 log. of 136-46 feet, the diff. of altitude. 
 
 The difference of altitude, as obtained by levelling with the 
 spirit-level, (Phil. Trans. 1831, Part I.) = 135'57 feet, differing 
 only 0*89 feet from that obtained above. The observations 
 should be made simultaneously at both stations, but to do this, 
 two observers and two barometers are required. When there is 
 only one observer, he should, after making his first observations, 
 lose no time in hastening to his second station, to make his 
 observations there ; which, if done quickly, and the atmosphere is 
 undergoing no change at the time, will answer nearly as well as 
 if simultaneous observations were made by a barometer at each 
 station.
 
 ASTRONOMICAL INSTRUMENTS. 
 
 THE SEXTANT. 
 
 It was our intention, before describing tbe Sextant, to devote 
 some space to an account of HADLEY'S Quadrant ; but as the 
 construction of both instruments is essentially the same, we 
 shall confine ourselves to a description of the sextant and its 
 uses only, as comprehending the other instrument, and per- 
 forming with greater correctness all the operations to which the 
 quadrant can be applied. The principle of its construction may 
 be understood from the following demonstration. 
 
 Let ABC represent a sextant, having # ^ 
 an index, A G, (to which is attached a 
 mirror at A,) movable about A as a cen- \ | 
 
 tre, and denoting the angle it has moved \ \ 
 
 through, on the arc, BC; also let the !* .'.'j^V 
 
 half-silvered (or horizon) glass, a b, be 
 fixed parallel to A C ; now a ray of light, 
 S,A, from a celestial object, S, imping- 
 ing against the mirror, A, is reflected off 
 at an equal angle, and striking the half- 
 silvered glass at D, is again reflected 
 to E, where the eye likewise receives 
 through the transparent part of that glass 
 a direct ray from the horizon. Then the altitude, S A H, is 
 equal to double the angle C A G, measured upon the limb, B C, 
 of the instrument. 
 
 For the reflected angle, BAG (or DAF) = the incident an- 
 gle, S A I, and the reflected angle, b D E the incident a D A 
 = DAE = DEA, because a b is parallel to A C. Now, H A I 
 DFA = (FAE + FEA,) and DAE being equal to DEA, 
 it follows that HAI = (DAE + FAE.) From HAI and 
 (DAE + FAE) take the equal angles, SAI and DAF, and 
 there remains S A H = 2 F A E, or 2 G A C ; or, in other 
 words, the angle of elevation, S A H, is equal to double the 
 angle of inclination of the two mirrors, D G A, being equal to 
 GAC. 
 
 Hence the arc on the limb, B C, although only the sixth part 
 of a circle, is divided as if it were 120, on account of its double 
 being required as the measure of CAB, and it is generally ex- 
 tended to 140.
 
 50 ASTRONOMICAL INSTRUMENTS. 
 
 The annexed figure represents a sextant of TROUGHTON'S con- 
 struction, having a double frame, A A, connected by pillars, a a, 
 &c., thus uniting strength with lightness. The arc. B C, is ge- 
 nerally graduated to 10' of a degree, commencing near the end, 
 C, and it is numbered towards B. The divisions are also con- 
 tinued on the other side of zero, towards C, forming what is 
 called the arc of excess, which is useful in determining the index 
 error of the instrument, as will be explained hereafter. The 
 limb is subdivided by the vernier, E, into 10", the half of which 
 (or 5") can be easily estimated : this small quantity is easily dis- 
 tinguishable by the aid of microscope, H, and its reflector, b, 
 which are connected by an arm with the index, I E, at the point c, 
 round which it turns as a centre, affording the means of examin- 
 ing the whole vernier, the connecting arm being long enough to 
 allow the microscope to pass over the whole length of it. 
 
 K 
 
 To the index is attached a clamp to fasten it to the limb, and 
 a tangent screw, J, (in the plate, the clamp is concealed from 
 view,) by which the index may be moved any small quantity, 
 after it is clamped, to render the contact of the objects observed 
 more perfect than can be done by moving it with the hand alone. 
 The upper end, I, terminates in a circle, across which is fixed the 
 silvered index-glass, F, over the centre of motion, and perpen- 
 dicular to the plane of the instrument. To the frame at G is 
 attached a second glass, called the horizon-glass, the lower half 
 of which only is silvered : this must likewise be perpendicular to
 
 THE SEXTANT. 
 
 51 
 
 the plane of the instrument, and in such a position that its plane 
 shall be parallel to the plane of the index-glass, F, when the 
 vernier is set to (or zero) on the limb, B C. A deviation from 
 this position constitutes the index error before spoken of. 
 
 The telescope is carried by a ring, L, attached to a stem, e, 
 called the up-and-down piece, which can be raised or lowered 
 by turning the milled screw, M : its use is to place the telescope 
 so that the field of view may be bisected by the line on the hori- 
 zon-glass that separates the silvered from the unsilvered part. 
 This is important, as it renders the object seen by reflection, and 
 that by direct vision equally bright ;* two telescopes and a plain 
 tube, all adapted to the ring, L, are packed with the sextant, one 
 showing the objects erect, and the other inverting them ; the last 
 has a greater magnifying power, showing the contact of the 
 images much better. The adjustment for distinct vision is ob- 
 tained by sliding the tube at the eye-end of the telescope in the 
 inside of the other ; this also is the means of adapting the focus 
 to suit different eyes. In the inverting telescope are placed two 
 wires, parallel to each other, and in th^e middle of the space be- 
 tween them the observations are to be made, the wires being first 
 brought parallel to the plane of the sextant, which may be judged 
 of with sufficient exactness by the eye. When observing with 
 this telescope, it must be borne in mind, that the instrument 
 must be moved in a contrary direction to that which the object 
 appears to take, in order to keep it in the field of view. 
 
 Four dark glasses, of different depths of shade and colour, are 
 placed at K, between the index and horizon glasses; also three 
 more at N, any one or more of which can be turned down to 
 moderate the intensity of the light, before reaching the eye, when 
 a very luminous object (as the sun) is observed. The same pur- 
 pose is effected by fixing a dark glass to the eye-end of the tele- 
 scope : one or more dark glasses for this purpose generally accom- 
 pany the instrument. They, however, are chiefly used when the 
 sun's altitude is observed with an artificial horizon, or for ascer- 
 taining the index error, as employing the shades attached to the 
 instrument for such purposes, would involve in the result, any 
 error which they might possess. The handle, which is shown at 
 O, is fixed at the back of the instrument. The hole in the mid- 
 dle is for fixing it to a stand, which is useful when an observer 
 is desirous of great steadiness. 
 
 * This is not the case when one object is much brighter than the other, as the 
 sun and moon ; in taking the distance between which, the screw, M, should be 
 moved more than above stated, until they are both nearly of the same brightness, 
 as an observation can be made better when this is the case than when otherwise.
 
 ASTRONOMICAL INSTRUMENTS, 
 
 Of the Adjustments. 
 
 The requisite adjustments are the following : the index and 
 horizon glasses must be perpendicular to the plane of the instru- 
 ment, and their planes parallel to each other when the index di- 
 vision of the vernier is at on the arc, and the optical axis of 
 the telescope must be parallel to the plane of the instrument. 
 We shall speak separately of each of these adjustments. 
 
 To examine the Adjustment of the Index-glass. 
 
 Move the index forward to about the middle of the limb ; then, 
 holding the instrument horizontally with the divided limb from 
 the observer, and the index-glass to the eye, look obliquely down 
 the glass, so as to see the circular arc, by direct view and by re- 
 flection, in the glass at the same time ; and if they appear as one 
 continued arc of a circle, the index-glass is in adjustment. If 
 it requires correcting, the arc will appear broken where the re- 
 flected and direct parts of the limb meet. This in a well-made 
 instrument is seldom the case, unless the sextant has been ex- 
 posed to rough treatment. As the glass is in the first instance 
 set right by the maker, and firmly fixed in its place, its position 
 is not liable to alter, therefore no direct means are supplied for 
 its adjustment. 
 
 To examine the horizon-glass, and set it perpendicular to the 
 Plane of the Sextant. 
 
 The position of this glass is known to be right, when by a 
 sweep with the index, the reflected image of any object passes 
 exactly over or covers its image as seen directly ; and any error 
 is easily rectified by turning the small screw, i, at the lower end 
 of the frame of the glass. 
 
 To examine the Parallelism of the Planes of the two Glasses, 
 when the Index is set to Zero. 
 
 This is easily ascertained ; for, after setting the zero on the 
 index to zero on the limb, if you direct your view to some object, 
 the sun for instance, you will see that the two images (one seen 
 by direct vision through the unsilvered part of the horizon-glass, 
 and the other reflected from the silvered part) coincide or appear 
 as one, if the glasses are correctly parallel to each other ; but if 
 the two images do not coincide, the quantity of their deviation 
 constitutes what is called the index error. The effect of this 
 error on an angle measured by the instrument is exactly equal
 
 THE SEXTANT. AJ3 
 
 to the error itself: therefore, in modern instruments, there are 
 seldom any means applied for its correction, it being considered 
 preferable to determine its amount previous to observing, or 
 immediately after, and apply it with its proper sign to each ob- 
 servation. The amount of the index error may be found in the 
 following manner : clamp the index at about 30 minutes to the 
 left of zero, and looking towards the sun, the two images will 
 appear either nearly in contact or overlapping each other; then 
 perfect the contact, by moving the tangent-screw, and call the 
 minutes and seconds denoted by the vernier, the reading on the 
 arc. Next place the index about the same quantity to the right 
 of zero, or on the arc of excess, and make the contact of the two 
 images perfect as before, and call the minutes and seconds on 
 the arc of excess* the reading off the arc ; and half the differ- 
 ence of these numbers is the index error ; additive when the 
 reading on the arc of excess is greater than that on the limb, 
 and subtractive when the contrary is the case. 
 
 EXAMPLE. 
 
 Reading on the arc . . . 31 56 
 off the arc 31 22 
 
 Difference 34 
 
 Index error . . 017 
 
 In this case the reading on the arc being greater than that on 
 the arc of excess, the index error, =17 seconds, must be sub- 
 tracted from all observations taken with the instrument, until it 
 be found, by a similar process, that the index error has altered. 
 One observation on each side of zero is seldom considered enough 
 to give the index error with sufficient exactness for particular 
 purposes : it is usual to take several measures each way; "and 
 half the difference of their means will give a result more to be 
 depended on than one deduced from a single observation only 
 on each side of zero." A proof of the correctness of observations 
 for index error is obtained by adding the above numbers together, 
 and taking one-fourth of their sum, which should be equal to the 
 sun's sernidiameter, as given in the Nautical Almanac. When 
 the sun's altitude is low, not exceeding 20 or 30, his horizontal 
 instead of his perpendicular diameter should be measured, (if the 
 observer intends to compare with, the Nautical Almanac, other- 
 wise there is no necessity ;) because the refraction at such an 
 altitude affects the lower border (or limb) more than the upper, 
 
 * When reading off the arc of excess, the vernier must be read backwards, or 
 froui its contrary end, as explained at page 7.
 
 54 ASTRONOMICAL INSTRUMENTS. 
 
 so as to make his perpendicular diameter appear less than his 
 horizontal one, which is that given in the Nautical Almanac : in 
 this case the sextant must be held horizontally. 
 
 To make the Line of Collimation of the Telescope parallel to the 
 Plane of the Sextant. 
 
 This is known to be correct, when the sun and moon, having 
 a distance of 90 or more, are brought into contact just at the 
 wire of the telescope which is nearest the plane of the sextant, 
 fixing the index, and altering the position of the instrument to 
 make the objects appear on the other wire ; if the contact still 
 remains perfect, the axis of the telescope is in proper adjust- 
 ment ; if not, it must be altered by moving the two screws which 
 fasten, to the up-and-down piece, the collar into which the tele- 
 scope screws. This adjustment is not very liable to be deranged. 
 
 Having now gone through the principle and construction of 
 the sextant, it remains to give some instructions as to the man- 
 ner of using it. 
 
 It is evident that the plane of the instrument must be held in 
 the plane of the two objects, the angular distance of which is 
 required : in a vertical plane, therefore, when altitudes are mea- 
 sured ; in a horizontal or oblique plane, when horizontal or ob- 
 lique angles are to be taken. As this adjustment of the plane of 
 the instrument is rather difficult and troublesome to the beginner, 
 he need not be surprised nor discouraged, although his first at- 
 tempts may not answer his expectations. The sextant must be 
 held in the right hand, and as slack as is consistent with its 
 safety, for in grasping it too hard the hand is apt to be rendered 
 unsteady. 
 
 When the altitude of an object, 'the sun, for instance, is to be 
 observed, the observer, having the sea horizon before him, must 
 turn down one or more of the dark glasses, or shades, according 
 to the brilliancy of the object ; and directing his sight to that 
 part of the horizon immediately beneath the sun, and holding 
 the instrument vertically, he must with the left hand lightly 
 slide the index forward, until the image of the sun, reflected 
 from the index glass, appears in contact with the horizon, seen 
 through the unsilvered part of the horizon glass. Then clamp 
 it firm, and gently turn the tangent-screw, to make the contact 
 of the upper or lower limb of the sun and the horizon perfect, 
 when it will appear a tangent to his circular disc.* If an artificial 
 
 * If the observer knows his latitude approximately, he may find the meridional 
 altitude nearly, to which he may previously set his instrument ; when he will not 
 only find his object more easily, but have only a small quantity to move the index 
 to perfect the observation. 
 
 Take from the Nautical Almanac the declination of the object, and if it be of the 
 same name with the latitude, add it to the co-latitude; if of a different name, 
 subtract it : the sum or difference will be the meridian altitude.
 
 THE SEXTANT. 55 
 
 horizon is employed, the two images of the sun must be brought 
 into contact with each other ; but this will be explained when 
 speaking of that instrument. To the angle read off apply the 
 index error, and then add or subtract the sun's semidiameter, as 
 given in the Nautical Almanac, according as the lower or upper 
 limb is observed, to obtain the apparent altitude of the sun's 
 centre. Before we can use this observation for determining the 
 time, the latitude, &c., it must be further corrected for refrac- 
 tion and parallax, to obtain the true altitude, subtracting the 
 former and adding the latter ; and when the sea horizon is em- 
 ployed, a quantity must also be subtracted for the dip, which is 
 unnecessary when the altitude is taken by means of an artificial 
 horizon. 
 
 Tables for obtaining the above corrections may be found in 
 Mr. BAILY'S Astronomical Tables, &c., in the Requisite Tables, 
 or in any modern work on navigation. 
 
 EXAMPLE. 
 
 o / a 
 
 Obs. alt. of the sun's lower limb =61 13 5 
 
 Index error . = 17 
 
 Apparent altitude =61 12 48,0 
 
 * (Sun's semidiameter = + 15 46,9 
 
 ( ,, parallax = + 4,0 
 
 Refraction 34,4 ^ 61 28 38,9 
 
 Dip of the horizon, for an \ . A > = 4 37,4 
 elevation of 18 feet.../ ~ d ' U j 
 
 True altitude of the sun's centre = 61 24 1,5 
 
 If the observer is ignorant of the precise moment of the ob- 
 ject's being on the meridian, he should, by a slow and gradual 
 motion of the tangent-screw, keep the observed limb in contact 
 with the horizon as long as it continues to rise ; and immediately 
 on the altitudes appearing to diminish, cease from observing, and 
 the angle then read on the instrument will be the meridian 
 altitude. 
 
 After what has been advanced, little need be said about ob- 
 serving lunar distances, whether of the moon and the sun, or the 
 moon and a fixed star or planet, except that the instrument must 
 be held in the plane of the two objects, and it is generally pre- 
 ferable to direct the telescope to the fainter object, particularly 
 if a star, as it can be more easily kept in view when seen directly 
 
 * An observation of a star requires no correction for either parallax or semi- 
 diameter.
 
 tyQ ASTRONOMICAL 1NSTUUMENTS. 
 
 than it can when seen by reflection. If the brighter object is to 
 the left, the sextant must be held with the face downwards. 
 
 The enlightened limb of the moon is always to be brought into 
 contact with the sun or star, even though the moon's image is 
 made to pass beyond the sun or star before the desired contact 
 can be obtained. 
 
 Perhaps the best method of taking a lunar distance is, not to 
 attempt to make the contact perfect by the tangent-screw, but 
 when the nearest limbs are observed, make the objects overlap 
 each other a little when they are receding, or leave a small 
 space between them when they are approaching, and wait till the 
 contact is perfect, and the reverse, when the furthest limbs are 
 observed. 
 
 The altitudes of the two objects should be observed at the same 
 instant as the distance, and the time noted by a chronometer, or 
 watch : this would require several observers ; but one person may 
 take them all, by having recourse to the following method : 
 " First, observe the altitude of the sun or star ; secondly, the 
 altitude of the moon ; then any number of distances ; next the 
 altitude of the moon, and lastly the altitude of the sun or star, 
 noting the times of each by a watch. Now add together the 
 distances and times when they were observed, and take the mean 
 of each ; and in order to reduce the altitudes to the mean time, 
 making the following proportion : As the difference of times be- 
 tween the observations is to the difference of their altitudes, so is 
 the difference between the time that the first altitude was taken 
 and the mean of the times at which the distances were observed, 
 to a fourth number: which, added to or subtracted from the first 
 altitude, according as it is increasing or decreasing, will give the 
 altitude reduced to the mean time." 
 
 The angular distances of terrestrial objects are measured by 
 the sextant in the same manner as those of celestial ones ; but if 
 the objects are not in the same horizontal plane, a reflecting 
 instrument will not give their horizontal angular distance. But 
 this may be obtained nearly by measuring their angular distances 
 from an object in or near the horizon, which subtends a great 
 angle with both, and the sum, or the difference of the angles so 
 measured, will be nearly the required horizontal angle. 
 
 Of the sextant, it has been said, that it is in itself a portable 
 observatory; and it is doubtless one of the most generally useful 
 instruments that has ever been contrived, being capable of fur- 
 nishing data to a considerable degree of accuracy for the solution 
 of a numerous class of the most useful astronomical problems ; 
 affording the means of determining the time, the latitude and 
 longitude of a place, &c., for which, and many other purposes, 
 it is invaluable to the land surveyor as well as the navigator.
 
 TROUGHTON'S KKl-'LliCTlNG CIRCLE. 
 
 TROUGHTON'S REFLECTING CIRCLE. 
 
 The above figure represents this instrument, which in principle 
 and use is the same as the sextant. It has three vernier readings, 
 ABC, moving round the same centre as the index-glass, E, 
 which is upon the opposite face of the instrument. One of the 
 verniers, B, carries the clamp and tangent-screw. D, represents 
 the microscope for reading the verniers ; it is similar to the one 
 used in reading the sextant, and is adapted to each index-bar, 
 by slipping it on a pin placed for that purpose, as shown in the 
 figure. The horizon-glass is shown at F. The barrel, G, 
 contains the screws for giving the up-and-down motion to the 
 telescope ; it is put in action by turning the milled-head under 
 the barrel. H is the telescope, adapted to the instrument in a 
 manner similar to that of the sextant. I and J are two handles 
 fixed parallel to the plane of the circle, and a third handle, K, 
 is screwed on at right angles to that plane, and can be transferred 
 to the opposite face of the instrument by screwing it into the 
 handle, 1; the use of this extra handle is for convenience in 
 reading and in holding the instrument, when observing angles 
 that are nearly horizontal ; it can be shifted, according as the 
 face of the instrument is held upwards or downwards. The 
 requisite dark glasses are attached to the frame-work of the 
 circle, to be used in the same manner and for the same purposes 
 as those of the sextant. With respect to the adjustments and 
 application of this instrument, we cannot do better than use the
 
 58 ASTRONOMICAL INSTRUMENTS. 
 
 words of the inventor, Mr. TROUGHTON, contained in a paper 
 which he calls 
 
 " Directions for observing with Troughtons Reflecting Circle. 
 
 " Prepare the instrument for observation by screwing the tele- 
 scope into its place, adjusting the drawer to focus, and the wires 
 parallel to the plane, exactly as you do with a sextant : also set 
 the index forwards to the rough distance of the sun and moon, 
 or moon and star ; and holding the circle by the short handle, 
 direct the telescope to the fainter object, and make the contact 
 in the usual way. Now read off the degree, minute, and second, 
 by 'that branch of the index to whicb the tangent-screw is at- 
 tached ; also, the minute and second shown by the other two 
 branches ; these give the distance taken on three different sex- 
 tants ; but as yet, it is only to be considered as half an observa- 
 tion : what remains to be done, is to complete the whole circle, 
 by measuring that angle on the other three sextants. Therefore 
 set the index backwards nearly to the same distance, and reverse 
 the plane of the instrument, by holding it by the opposite handle, 
 and make the contact as above, and read off as before what is 
 shown on the three several branches of the index. The mean 
 of all six is the true apparent distance, corresponding to the 
 mean of the two times at which the observations were made. 
 
 " When the objects are seen very distinctly, so that no doubt 
 whatever remains about the contact in both sights being perfect, 
 the above may safely be relied on as a complete set ; but if, from 
 the haziness of the air, too much motion, or any other cause, the 
 observations have been rendered doubtful, it will be advisable to 
 make more : and if, at such times, so many readings should be 
 deemed troublesome, six observations, and six readings may be 
 conducted in the manner following : Take three successive sights 
 forwards, exactly as is done with a sextant ; only take care to read 
 them off on different branches of the index : also make three 
 observations backwards, using the same caution : a mean of these 
 will be the distance required. When the number of sights taken 
 forwards and backwards are unequal, a mean between the means 
 of these taken backwards and those taken forwards will be the 
 true angle. 
 
 " It need hardly be mentioned, that the shades, or dark 
 glasses, apply like those of a sextant, for making the objects 
 nearly of the same brightness ; but it must be insisted on, that 
 the telescope should, on every occasion, be raised or lowered, by 
 its proper screw, for making them perfectly so. 
 
 " The foregoing instructions for taking distances, apply equally 
 for taking altitudes by the sea or artificial horizon, they being 
 no more than distances taken in a vertical plane. Meridian alti- 
 tudes cannot, however, be taken both backwards and forwards
 
 TROUGHTON'S REFLECTING CIRCLE. 59 
 
 the same day, because there is not time : all therefore that can 
 be done, is, to observe the altitude one way, and use the index 
 error ; but even here, you have a mean of that altitude, and this 
 error, taken on three different sextants. Both at sea and land, 
 where the observer is stationary, the meridian altitude should be 
 observed forwards one day, and backwards the next, and so on 
 alternately from day to day ; the mean of latitudes, deduced 
 severally from such observations, will be the true latitude ; but 
 in these there should be no application of index error, for that 
 being constant, the result would in some measure be vitiated 
 thereby. 
 
 " When both the reflected and direct images require to be 
 darkened, as is the case when the sun's diameter is measured 
 and when his altitude is taken with an artificial horizon, the 
 attached dark glasses ought not to be used: instead of them, 
 those which apply to the eye-end of the telescope will answer 
 much better : the former having their errors magnified by the 
 power of the telescope, will, in proportion to this power, and 
 those errors, be less distinct than the latter. 
 
 " In taking distances, when the position does not vary from 
 the vertical above thirty or forty degrees, the handles which are 
 attached to the circle are generally most conveniently used ; but 
 in those which incline more to the horizontal, that handle which 
 screws into a cock on one side, and into the crooked handle on 
 the other, will be found more applicable. 
 
 " When the crooked handle happens to be in the way of read- 
 ing one of the branches of the index, it must be removed, for 
 the time, by taking out the finger-screw, which fastens it to the 
 body of the circle. 
 
 " If it should happen that two of the readings agree with each 
 other very well, and the third differs from them, the discordant 
 one must not on any account be omitted, but a fair mean must 
 always be taken. 
 
 " It should be stated, that when the angle is about thirty de- 
 grees, neither the distance of the sun and moon, nor an altitude 
 of the sun, with the sea horizon, can be taken backwards ; 
 because the dark glasses at that angle prevent the reflected rays 
 of light from falling on the index-glass ; whence it becomes 
 necessary, when the angle to be taken is quite unknown, to ob- 
 serve forwards first, where the whole range is without interrup- 
 tion ; whereas in that backwards, you will lose sight of the 
 reflected image about that angle. But in such distances, where 
 the sun is out of the question, and when his altitude is taken 
 with an artificial horizon, (the shade being applied to the end of 
 the telescope,) that angle may be measured nearly as well as any 
 other ; for the rays incident on the index-glass will pass through 
 the transparent half of the horizon-glass, without much diminu- 
 tion of their brightness.
 
 (JO ASTRONOMICAL INSTRUMENTS. 
 
 " The advantages of this instrument, when compared with 
 the sextant, are chiefly these : the observations for finding the 
 index error are rendered useless, all knowledge of that being put 
 out of the question, by observing both forwards and backwards. 
 By the same means the errors of the dark glasses are also cor- 
 rected ; for, if they increase the angle one way, they must dimi- 
 nish it the other way by the same quantity. This also perfectly 
 corrects the errors of the horizon-glass, and those of the index- 
 glass very nearly. But what is still of more consequence, the 
 error of the centre is perfectly corrected by reading the three 
 branches of the index ; while this property combined with that 
 of observing both ways, probably reduces the errors of dividing 
 to one-sixth part of their simple value. Moreover, angles may 
 be measured as far as one hundred and fifty degrees, consequently 
 the sun's double altitude may be observed when his distance 
 from the zenith is not less than fifteen degrees ; at which altitude, 
 the head of the observer begins to intercept the rays of light 
 incident on the artificial horizon ; and, of course, if a greater 
 angle could be measured, it would be of no use in this respect. 
 
 " This instrument, in common with the sextant, requires three 
 adjustments. First, the index-glass perpendicular to the plane 
 of the circle. This being done by the maker, and not liable to 
 alter, has no direct means applied to the purpose ; it is known 
 to be right, when, by looking into the index-glass, you see that 
 part of the limb which is next you, reflected in contact with the 
 opposite side of the limb, as one continued arc of a circle : on 
 the contrary, when the arc appears broken, where the reflected 
 and direct parts of the limb meet, it is a proof that it wants to 
 be rectified. The second is, to make the horizon-glass perpendi- 
 cular. This is performed by a capstan-screw, at the lower end 
 of the frame of that glass ; and is known to be right, when, by 
 a sweep of the index, the reflected image of any object will pass 
 exactly over, or cover the image of that object seen directly. 
 The third adjustment is, for making the line of collimation 
 parallel to the plane of the circle. This is performed by two 
 small screws, which also fasten the collar into which the telescope 
 screws to the upright stem on which it is mounted ; this is known 
 to be right, when the sun and moon, having a distance of one 
 hundred and thirty degrees, or more, their limbs are brought in 
 contact, just at the outside of that wire which is next to the 
 circle ; and then, examining if it be the same, just at the outside 
 of the other wire: its being so is the proof of adjustment. 
 
 " Should these hints about the adjustments set any over-handy 
 gentleman on tormenting his instrument, it will not be what was 
 intended by them ; they were added, that, in case of accident, 
 those who are so unfortunate, might be enabled thereby to put 
 their own instrument in order."
 
 TIIK BOX SEXTANT. 
 
 TPIE BOX SEXTANT. 
 
 This useful little instrument, which is represented in the above 
 figure, might, perhaps with more propriety, have been classed as 
 a surveying instrument, it being chiefly used in that business. 
 The principle of its construction and adjustments is precisely 
 the same as the sextant before described ; a minute description, 
 therefore, would be little more than a recapitulation of what has 
 already been advanced. A is the index, which instead of being 
 moved along the divided limb, ef, by the hand, has a motion given 
 to it by a rack and pinion, concealed within the box, and turned 
 by the milled head B, which acts as the tangent-screw does to 
 the index of the large sextant. The glasses (shown at C and 
 D) are within the box, by which they are protected from injury, 
 and their adjustments, when once perfected, kept secure ; so 
 much so, that it would require considerable violence to derange 
 them. The horizon-glass, D, alone has a contrivance for adjust- 
 ment at a and d, both to set it perpendicular to the plane of the 
 instrument, and to correct or reduce the index error, which, in 
 this instrument, had better be kept correct, as it is not so likely 
 to get out of order as in the large sextant, which, as we have be- 
 fore observed, seldom admits of its index error being rectified. 
 The key, c, is formed to fit both squares at a and d, to make 
 the adjustments, and it is generally tapt into some spare place 
 in the instrument, as at c, that it may be always safe and at 
 hand. 
 
 It is supplied with a telescope, E, which screws into a shoul- 
 der-piece, F, and can be attached to the box by the screw, G : 
 this can be applied or not, at the pleasure of the observer, as 
 there is a contrivance at H to enable him to observe without 
 the telescope, if he prefers plain sights. Two dark glasses are 
 placed within the box, and there is also one adapted to the 
 eye-end of the telescope. 
 
 The angle is read off by the help of the glass, I, which being 
 mounted with a joint, can be moved- over the vernier on any part
 
 (52 ASTllONOMICAL INSTRUMENTS. 
 
 of the limb. The instrument is divided to 30 minutes of a 
 degree, and by the vernier is subdivided to single minutes, one- 
 half of which, or 30 seconds, can be obtained by estimation. 
 
 The divided limb is numbered both to the right and left, 
 commencing at to 120, and backwards from 120 to 180, 
 and beyond to 230 ; the latter row of figures are furthest from 
 the divisions, and belong to the supplementary angles; their 
 zero division of the vernier is at the end, contrary to that of the 
 angles, reading from to 120. 
 
 Beneath the index-glass is fixed a similar one, in such a 
 manner as always to reflect the image of an object to the eye 
 when applied to a hole in the side of the box near the division 
 120, at the constant angle of 90; whence the observer must 
 direct his sight towards the right hand, and at right angles, to 
 the real place of the object. When the index is set to 180, its 
 glass will also reflect an opposite image to the eye at right 
 angles to the left hand, (the two glasses then being exactly 
 across each other ;) consequently an eye looking through the 
 hole near the division 120 will (if the adjustments be perfect) 
 perceive objects 180 apart to coincide, at right angles to a 
 line connecting them. Thus a point can be found in line 
 between two stations : the observer, with the instrument set as 
 above, having placed himself as nearly in the line as he can 
 guess, must apply his eye to the hole near 120, and looking at 
 right angles to his station line, step backwards or forwards, 
 until he perceives the two distant objects to coincide, when the 
 spot he stands on will be a point in the line joining the objects : 
 to verify this, he should then turn himself half round, and look- 
 ing in the opposite direction, see if the two objects still coincide, 
 which they will do, if the adjustments of the instrument are cor- 
 rect. If they do not appear in junction, move as before, until 
 you find the spot where they do : then, half way between the two 
 spots so found, will be the true point on the line required. 
 
 ** The adjustment of this part, as well as the method of ob- 
 serving supplemental angles with it, is performed thus : Choose 
 two objects in the horizon, the further apart the better, but not 
 nearer than 140 ; turn your face at right angles to the right- 
 hand object, so as to get sight of its image in the fixed glass; 
 then, by moving the index, bring the image of the other object, 
 seen in the index-glass, exactly to coincide with it on the line of 
 separation of the two glasses : read off the angle, turn yourself 
 half round, and take in like manner the angle which the same 
 objects make the other way. It is evident that the sum of the 
 two angles should be 360, and also, that if they exceed that 
 quantity, half the excess must be subtracted, and if they fall 
 short of it, half the defect must be added, to obtain the true 
 angle. It is, perhaps, better to allow for the errors than to ad- 
 just them ; but the latter may be done by applying the key, c, 
 to a square underneath the box."
 
 THE ARTIFICIAL HORIZON. 
 
 The lid of the box is contrived to screw on the bottom, (as is 
 shown in the plate,) where it makes a convenient handle for hold- 
 ing the instrument. 
 
 Since writing the above, we have been shown by Mr. Macneill, 
 an excellent contrivance of his, for taking altitudes or depres- 
 sions with the box-sextant, which consists of two small spirit- 
 levels fixed at the back of the horizon-glass, at right angles to 
 each other, so that standing before the object, you look perpen- 
 dicularly down through the plane-sight, and moving the index 
 bring the image of the object to appear with the levels, which 
 must have their air-bubbles in the centre of their tubes. The 
 reading of the instrument will then show the supplement of the 
 zenith distance, and its complement to 90 will be the angle re- 
 quired ; elevated if more than 90, and depressed if less than 90. 
 
 THE ARTIFICIAL HORIZON. 
 
 When the altitude of a celestial object is to be taken at sea, 
 the observer has the natural (or sea) horizon, as a line of depar- 
 ture ; but on shore, he is obliged to have recourse to an artificial 
 one, to which his observation may be referred : this consists of a 
 reflecting plane parallel to the natural horizon, on which the 
 rays of the sun or other object falling, are reflected back to an 
 eye placed in a proper position to receive them ; the angle be- 
 tween the real object and its reflected image, being then mea- 
 sured with the sextant, is double the altitude of the object above 
 the horizontal plane. 
 
 Various natural as well as artificial reflecting surfaces have 
 been made by mechanical arrangements, to afford the means of 
 obtaining double angles : such as pouring water, oil, treacle, or 
 other fluid substances into a shallow vessel ; and to prevent the 
 wind giving a tremulous motion to its surface, a piece of thin 
 gauze, talc, or plate-glass, whose surfaces are perfectly plane and 
 parallel, may be placed over it, when used for observation. But 
 the most accurate kind of artificial horizon is that in which fluid 
 quicksilver forms the reflecting surface, the containing vessel be- 
 ing placed on a solid basis, and protected from the influence of 
 the wind. The adjoining figure represents an instrument of this 
 kind. The mercury is contained in an oblong 
 wooden trough, placed under the roof A, in 
 which are fixed two plates of glass whose sur- 
 faces are plane and parallel to each other. This 
 roof effectually screens the surface of the metal 
 from being agitated by the wind, and when it has its position re- 
 versed at a second observation, any error occasioned by undue 
 refraction at either plate of glass will be corrected. 
 
 Another and more portable contrivance for an artificial hori- 
 zon, is represented in the following figure, which consists of a
 
 (JJ. ASTRONOMICAL INSTHUMKNTS. 
 
 circuit! plate of black glass about two inches 
 diameter, mounted on a brass stand, half an 
 inch deep, with three foot-screws, a b c, to 
 set the plane horizontal; the horizontality 
 being determined thus by the aid of a short 
 spirit-level, d, having under the tube a face 
 ground plane on which it lies in contact with the 
 reflecting surface ; place the level on the glass 
 in a direction parallel to the line joining two of the three foot- 
 screws, as a and b, then move one of these screws till the bubble 
 remains in the middle of the tube in both the reversed positions 
 of the level, and the plate will be horizontal in that direction ; 
 then place the level at right angles to its former position, and 
 turn the third foot-screw back or forwards till the bubble again 
 settles in the middle of its tube, the former levelling remaining 
 undisturbed, and the plane will then be horizontal. This instru- 
 ment, from its portability, is extremely convenient for travellers, 
 as when packed in its case it can be carried in the pocket with- 
 out being any incumbrance. 
 
 When an artificial horizon is used, the observer must place 
 himself at such a distance that he may see the reflected object as 
 well as the real one ; then having the sextant properly adjusted, 
 the upper or lower limb of the sun's image (supposing that the 
 object) reflected from the index-glass, must he brought into con- 
 tact with the opposite limb of the image reflected from the arti- 
 ficial horizon, observing that when the inverting telescope is 
 used, the upper limb will appear as the lower, and vice versa;* 
 the angle shown on the instrument, when corrected for the index 
 error, will be double the altitude of the sun's limb above the 
 horizontal plane ; to the half of which, if the semidiameter, re- 
 fraction, and parallax be applied, the result will be the true alti- 
 tude of the centre. 
 
 EXAMPLE. 
 
 Observed angle 122 25 50,00 
 
 Index error 17,05 
 
 2} 122 25 32,95 
 
 App. alt ~61 12 46,47 
 
 Semidiameter -f- 15 46,91 
 
 Parallax + 4,00 
 
 61 28 37,38 
 Refraction 34,40 
 
 True alt. of sun's centre . . 61 28 2,98 
 
 * When the contact is formed at the lower limb, the images will separate shortly 
 after the contact has been made, if the altitude be increasing ; but if the altitude 
 be decreasing, they will begin to overlap ; but when the contact is formed at the 
 upper limb, the reverse takes place. An observer, if in doubt as to which limb he 
 has been observing, should watch the object for a short time after he has made 
 the observation.
 
 THE DIP-SECTOR. 
 
 When the late Professor VINCE was engaged in making ob- 
 servations upon extraordinary refraction at Ramsgate, Mr. 
 TROUGHTON contrived and constructed for his use an instrument 
 which he called a Refraction-Sector. About five years after- 
 wards, when preparations were making for the first of the late 
 North Polar Expeditions, Mr. TROUGHTON was applied to by 
 the late Dr. WOOLL ASTON, to make him an instrument on the 
 principle of the back observation with the quadrant, to send with 
 the expedition, to measure the dip of the horizon ; but upon Mr. 
 TROUGHTON'S producing his Refraction-Sector, which was as 
 well adapted to Dr. WOOLL ASTON 's purpose as that for which it 
 was devised, the Doctor immediately ordered one to be made for 
 him, and named it a Dip-Sector ; proposing at the same time an 
 improvement in the construction of the handle, which, on his 
 suggestion, was made to turn on a centre, to be placed in any 
 position, for convenience in use, or packing in its case ; that 
 made for Mr. VINCE having two fixed handles, at right angles 
 to the face of the instrument. 
 
 The preceding figure represents this instrument: A is the 
 sector, B the index, with its clamp and tangent-screw, exactly 
 similar to that of the sextant : the index-glass, C, and the hori- 
 zon-glass, D, are fixed at right angles to the plane of the instru- 
 ment. The telescope, E F, is fitted into a collar, having an 
 up-and-down motion given to it by turning the screw H ; the 
 
 F
 
 g(j ASTRONOMICAL INSTRUMENTS. 
 
 two images of the horizon can thus be made to appear of the 
 shades most favourable for observation. G represents the eye- 
 piece fixed at right angles to the telescope, and a diagonal mirror 
 is placed in the telescope at F, to change the direction of the rays 
 of light, from E F, to F G, in which the observer looks. 
 
 The handle, I, turns upon a centre, and is held firmly in any 
 position by tightening the clamp-screw, J. In use it is fixed 
 perpendicular to the length of the instrument, and when wanted 
 it can be turned half round, and fixed in a similar position on the 
 other side, a position in which it is required to be when the in- 
 strument is reversed for the second observation; it is turned 
 under and parallel to the instrument when packed in its case. 
 
 The dip of the horizon, which varies with the height of the 
 observer above the surface of the earth, may always be computed 
 when the height is known ; but as a correction of altitudes ob- 
 served from the horizon of the sea, it is combined with the effects 
 of refraction upon the apparent place of the horizon, which ap- 
 pears elevated above its true place ; and as the effects of refrac- 
 tion are extremely variable, the dip obtained by computation is 
 necessarily very uncertain. Tables containing the dip for vari- 
 ous altitudes, allowing for the mean effect of refraction, are to 
 be found in all collections of nautical tables. But these tabular 
 dips are at times found to differ so considerably from the truth, 
 especially in tropical climates, that some experienced navigators 
 have lately been induced to measure the actual dip by means of 
 the sector, when any important determination has depended on 
 the observation of altitudes. 
 
 In the above diagram, A a represents a portion of the 
 earth's surface, and O the place of an observer ; H O H will be 
 his true horizon, O A and O a his visible horizon ; these rays 
 being tangents to the earth's surface at A and a ; the angle, 
 H O A, or H O a, is the dip of the horizon, which it is the 
 business of the dip-sector to measure. But the arcs to be
 
 THE DIP-SECTOR. 
 
 67 
 
 measured by this instrument, for the purpose of obtaining the 
 dip, are A Z a and A N a, the former of which is 180 + double 
 the dip, and the latter 180 double the dip, therefore the fourth 
 part of the difference is the measure of the dip. But as the in- 
 strument is constructed, only double the dip affected by index 
 error is read from it, and the index error is made so great that 
 the readings are both on the same side of zero, therefore the 
 fourth part of the difference of the readings is the dip angle. 
 
 In observing, the face of the instrument must be held in a 
 vertical plane, and lengthwise, in a line with the opposite parts 
 of the horizon whose dip is required ; the eye-tube, G F, (page 
 65) will then be horizontal, and the observer will be looking at 
 right angles to those points of the horizon which he wishes to ob- 
 serve. Suppose the instrument to be held as represented in our 
 engraving, with the index uppermost, the observer will be look- 
 ing in the direction, G F, when by giving motion to the index B, 
 its glass C will receive a ray from the visible horizon on the left 
 hand, and reflect it to the silvered part of the horizon-glass D, 
 and from thence to the telescope ; at the same time the whole 
 instrument being moved vertically round the hand as a centre, 
 a ray from the opposite part of the horizon to the right hand will 
 pass through the plain or upper part of the horizon-glass, and 
 both rays moving together will pass down the telescope E to F, 
 where by the diagonal mirror they will be reflected, at right 
 angles, to the eye of the observer at G. The index must now 
 be clamped, and by giving motion to the tangent-screw, the two 
 images of the horizon must be made exactly to coincide with 
 each other, and appear as one. To determine when the coincidence 
 is perfect, a slight motion of the instrument will cause the two 
 images to cross each other, by which a judgment may be formed 
 of the accuracy of the observation. This being satisfactorily 
 done, the angle may now be read off, which is the measure of 
 H O A + H O a, or double the dip of the horizon, subject to 
 the index error of the instrument. This must be considered but 
 half an observation, and to obtain the correct result, a second 
 observation must be taken with the instrument held in an in- 
 verted position, the index being now undermost. This is done 
 by releasing the clamp-screw, J, and turning the handle half 
 round, observe in the same manner as before : but when the 
 brightness of the two opposite parts of the horizon differ con- 
 siderably, the observer, to avoid the necessity of altering the 
 shades of the two images, (regulated by the up-and-down motion 
 of the telescope,) should reverse his own position as well as that 
 of his instrument, that is, turn himself exactly half round, for 
 then the telescope will be directed to the same part of the hori- 
 zon as before, and he will make the second observation under 
 precisely the same circumstances as he did the first, which, as 
 well as the due adjustment of the shades, is essential to good ob- 
 
 F 2
 
 Q8 ASTRONOMICAL INSTRUMENTS. 
 
 serving. The reading of the second observation will also give 
 double the dip, affected by the same index error as before ; and 
 as both readings are on the same side of zero, one fourth of their 
 difference will give the true result. Several observations should 
 be taken in each position of the instrument, and the mean taken 
 as the final result. 
 
 "In using this instrument at sea for the first time, considerable 
 difficulty arises from the constant change in the plane of the in- 
 strument, from the perpendicular position in which it is abso- 
 lutely necessary that it should be held, in order to obtain a cor- 
 rect observation. What at first appears to be a defect, however, 
 is a real advantage, namely, that whenever it is held in the least 
 degree out of the vertical plane, the two horizons (that seen 
 direct and the reflected one) cross each other, and it is only when 
 the plane is vertical that the horizons can appear parallel." 
 
 THE PORTABLE TRANSIT-INSTRUMENT.
 
 THE PORTABLE TRANSIT-INSTRUMENT. QQ 
 
 The Transit is a meridional instrument employed, in conjunc- 
 tion with a clock or chronometer, for observing the passage of 
 celestial objects across the meridian, either for obtaining correct 
 time, or determining their difference of right ascension ; the 
 latter of which, in the case of the moon and certain stars near 
 her path, that differ but little from her in-right ascension, affords 
 the best means of determining the difference of longitude between 
 any two places where corresponding observations may have been 
 made. Such being more especially the use of the portable transit 
 instrument, it forms a valuable accession to the apparatus of the 
 scientific traveller, who remaining a short time at any station, 
 is enabled thereby to adjust his time-keepers, both with ease and 
 accuracy, and to obtain the best data for finding his longitude. 
 It also may be employed very successfully in determining the 
 latitude.* 
 
 The preceding figure represents this instrument as constructed 
 by Mr. TROUGHTON, when the telescope does not exceed twenty 
 inches, or two feet focal length. The telescope-tube, A A, is in 
 two parts, and connected together by a sphere, B, which also 
 receives the larger ends of two cones, C C, placed at right angles 
 to the direction of the telescope, and forming the horizontal 
 axis. This axis terminates in two cylindrical pivots, which rest 
 in Y's fixed at the upper end of the vertical standards, D D. 
 One of the Y's possesses a small motion in azimuth, communi- 
 cated by turning the screw, a ; in these Y's the telescope turns 
 upon its pivots. But, that it may move in a vertical circle, the 
 pivots must be precisely on a level with each other, otherwise 
 the telescope will revolve in a plane oblique (instead of perpen- 
 dicular) to the horizon. The levelling of the axis, as it is called, 
 is therefore one of the most important adjustments of the instru- 
 ment, and is effected by the aid of a spirit-level, E, which is 
 made for this purpose to stride across the telescope, and rest on 
 the two pivots. 
 
 The standards, D D, are fixed by screws upon a brass circle, 
 F, which rests on three screws, bed, forming the feet of the 
 instrument, by the motion of which the operation of levelling is 
 performed. The two oblique braces, G G, are for the purpose 
 of steadying the supports, it being essential for the telescope to 
 have not only a free but a steady motion. On the extremity of 
 one of the pivots, which extends beyond its Y, is fixed a circle, 
 H, which turns with the axis while the double vernier, e e, re- 
 mains stationary in a horizontal position, and shows the altitude 
 to which the telescope is elevated. The verniers are set hori- 
 zontal by means of a spirit-level, /, which is attached to them, 
 
 * The transit-instrument is also now much employed by the most eminent civil 
 engineers, in setting out the lines of direction, and the working shafts, in tunnel- 
 ling ; which it is capable of doing with the greatest precision.
 
 7() ASTRONOMICAL INSTRUMENTS. 
 
 and they are fixed in their position by an arm of brass, g, clamped 
 to the supports by a screw at h : the whole of this apparatus is 
 movable with the telescope, and when the axis is reversed, can 
 be attached in the same manner to the opposite standard. 
 
 Near the eye-end, and in the principal focus of the telescope, 
 is placed the diaphragm, or wire-plate, which in the theodolite 
 or levelling telescope need only carry two cross wires, but in this 
 instrument it has five vertical and two horizontal wires. The 
 centre vertical wire ought to be fixed in the optical axis of the 
 telescope, and perpendicular with respect to the pivots of the 
 axis. It will be evident upon consideration, that these wires 
 are rendered visible in the daytime by the rays of light passing 
 down the telescope to the eye ; but at night, except when a very 
 luminous object, as the moon, is observed, they cannot be seen. 
 Their illumination is therefore effected by piercing one of the 
 pivots, and admitting the light of a lamp fixed on the top of one 
 of the standards, as shown at I ; which light is directed to the 
 wires by a reflector placed diagonally in the sphere B ; the re- 
 flector having a large hole in its centre, does not interfere with 
 the rays passing down the telescope from the object, and thus 
 the observer sees distinctly both the wires and the object at the 
 same time : when however the object is very faint, (as a small 
 star,) the light from the lamp would overpower its feeble rays : 
 to remedy this inconvenience, the lamp is so constructed, that 
 by turning a screw at its back, or inclining the opening of the 
 lantern, more or less light may be admitted to the telescope, to 
 suit the circumstances of the case. 
 
 The telescope is furnished with a diagonal eye-piece, by which 
 stars near the zenith may be observed without inconvenience. 
 
 Of the Adjustments. 
 
 Upon setting the instrument up, it should be so placed that 
 the telescope, when turned down to the horizon, should point 
 north and south as near as can possibly be ascertained. This of 
 course can be but approximate, as the correct determination of 
 the meridian can only be obtained by observation, after the other 
 adjustments are completed. 
 
 The first adjustment is that of the line of collimation. Direct 
 the telescope to some small, distant, well-defined object, (the 
 more distant the better,) and bisect it with the middle of the 
 central vertical wire; then lift the telescope very carefully out 
 of its angular bearings, or Y's, and replace it with the axis re- 
 versed ; point the telescope again to the same object, and if it 
 be still bisected, the collimation adjustment is correct ; if not, 
 move the wires one half the error, by turning the small screws 
 which hold the diaphragm near the eye- end of the telescope,
 
 THE PORTABLE TRANSIT-INSTRUMENT. 7J 
 
 and the adjustment will be accomplished ; but, as half the devi- 
 ation may not be correctly estimated in moving the wires, it 
 becomes necessary to verify the adjustment by moving the tele- 
 scope the other half, which is done by turning the screw a; this 
 gives the small azimuthal motion to the Y before spoken of, and 
 consequently to the pivot of the axis which it carries. Having 
 thus again bisected the object, reverse the axis as before, and if 
 half the error was correctly estimated, the object will be bisected 
 upon the telescope being directed to it ; if not quite correct, the 
 operation of reversing and correcting half the error, in the same 
 manner, must be gone through again, until, by successive approxi- 
 mations, the object is found to be bisected in both positions of 
 the axis; the adjustment will then be perfect. The collimation 
 adjustment may likewise be examined from time to time, by 
 observing the transit of Polaris, or any other close circumpolar 
 star, over the first three wires, which gives the intervals in time 
 from the first to the second, and from the first to the third wire ; 
 and then reversing the axis, observe the same intervals in a 
 reverse order, as the wires which were the three first, in the 
 former position, will now be the three last : if the intervals in 
 the first observations are exactly the same as the intervals in 
 the second, the collimation adjustment is correct; but should 
 the corresponding intervals differ, such difference points out the 
 existence of an error, which must be removed, as before de- 
 scribed, one half by the collimating screws, and the other half 
 by the azimuthal motion of the instrument. 
 
 It is desirable that the central, or middle wire (as it is usually 
 termed) should be truly vertical ; as we should then have the 
 power of observing the transit of a star on any part of it, as 
 well as the centre. It may be ascertained whether it is so, by 
 elevating and depressing the telescope : when directed to a dis- 
 tant object, if it is bisected by every part of the wire, the wire 
 is vertical ; if otherwise, it should be adjusted, by turning the 
 inner tube carrying the wireplate, until the above test of its 
 verticality be obtained, or else care must be taken that the observ- 
 ations are made near the centre only ; the other vertical wires 
 are placed by the maker equidistant from each other and parallel 
 to the middle one, therefore, when the middle one is adjusted, 
 the others are so too ; he also places the two transverse wires at 
 right angles to the vertical middle wire. These adjustments are 
 always performed by the maker, and but little liable to derange- 
 ment. When, however, they happen to get out of order, and 
 the observer wishes to correct them, it is done by loosening the 
 screws which hold the eye-end of the telescope in its place, and 
 turning the end round a small quantity by the hand until the error 
 is removed. But this operation requires very delicate handling, 
 as it is liable to remove the wires from the focus of the object- 
 glass.
 
 72 ASTRONOMICAL INSTRUMENTS. 
 
 The axis on which the telescope turns must next be set hori- 
 zontal : to do this, apply the level to the pivots, bring the air- 
 bubble to the centre of the glass-tube, by turning the foot-screw, 
 b, which raises or lowers that end of the axis, and consequently 
 the level resting upon it; then reverse the level by turning it end 
 for end, and if the air-bubble still remains central, the axis will 
 be horizontal, but if not, half the deviation must be corrected 
 by the foot-screw, b, and the other half by turning the small 
 screw, i, at one end of the level, which raises or lowers the glass- 
 tube (containing the air-bubble) with respect to its supports, 
 which rest upon the pivots. This, like most other adjustments, 
 frequently requires several repetitions before it is accomplished, 
 on account of the difficulty of estimating exactly half the error. 
 
 Having set the axis on which the telescope turns, parallel to 
 the horizon, and proved the correct position of the central wire 
 or line of collimation, making it describe a great circle perpen- 
 dicular to that axis, it remains finally to make it move in that 
 vertical circle which is the meridian. 
 
 We have supposed the instrument to be nearly in the meridian, 
 the next step is to determine the amount of its deviation, and 
 then by successive approximations to bring it exactly into that 
 plane : one of the methods of accomplishing this, is to observe 
 the time of both the upper and lower transits of Polaris, or any 
 other close circumpolar star, and as the middle wire of the instru- 
 ment, when exactly in the meridian, bisects the circle which the 
 star apparently describes, round the polar point, in 24 sidereal 
 hours, the time elapsed, during its traversing either the eastern 
 or western semicircle, will be equal to 12 sidereal hours; but 
 should the interval be greater or less, it is clear that the instrument 
 deviates from the meridian. If the eastern interval is greater 
 than the western, the plane in which the instrument moves from 
 the zenith to the north of the horizon, is westward of the true 
 meridian, and vice versa, if the western interval is greatest. 
 Having the difference of the interval from 12 hours, the quantity 
 of deviation measured on the horizon may be computed by the 
 following formula, the latitude of the place, and the polar distance 
 of the star, being both supposed to be known, at least approxi- 
 mately. 
 
 Deviation =log. -n + log. sec. L + log. tan. TT 20: 
 
 in which expression A = the difference of the intervals from 12 h 
 
 (reduced to seconds) 
 7T = the polar distance of the star 
 L = the latitude of the place. 
 
 This formula, in words, gives the following practical rule : 
 Add together the log. of half the difference of the intervals from 
 12 hours in seconds, the log. secant of the latitude, and the log. 
 tangent of the polar distance of the star : the sum, rejecting 20
 
 THE PORTABLE TRANSIT-INSTRUMENT. 73 
 
 from the index, will be the log. factor of deviation, which may 
 be converted into arc by multiplying it by 15. 
 
 The correction of this error may be effected by turning the 
 screw, a, if the angular value of one revolution be known, unless 
 the instrument possesses an azimuth circle, by which the tele- 
 scope may be set exactly that quantity from its present position. 
 
 But if the quantity of motion to be given to the adjusting- 
 screw, a, is not a matter of certainty, the observer, after ascer- 
 taining the difference of the intervals, must make the adjustment 
 which he considers sufficient, and again proceed to verify it by 
 observation, until, by continued approximation, he succeeds in 
 fixing his instrument correctly in the meridian. 
 
 The above method of determining the instrumental deviation, 
 is wholly independent of the tabulated place of the circumpolar 
 star, but it assumes some knowledge of the rate of the time- 
 keeper, and the perfect stability of the instrument for twelve 
 hours ; a condition which is rarely to be obtained, except in a 
 regular observatory. The method is still further limited in 
 practice, by the uncertainties of the weather, and the want 
 of stars sufficiently bright to be observed in the daytime, (Polaris 
 being the only star in the northern hemisphere fit for the pur- 
 pose, and there is no similar star in the southern.) There are, 
 however, two methods almost as good as the preceding, which 
 depend on the tabulated places of the stars only. These will 
 now be explained. 
 
 Take two well-known circumpolar stars, the nearer the pole 
 the better, differing about twelve hours in right ascension, and 
 observe one above and the other below the pole. Now it is 
 evident, that any deviation of the instrument from the meridian 
 will produce contrary effects upon the observed times of transit, 
 exactly as in the upper and lower culmination of the same star. 
 Hence, the time which elapses between the two observations 
 will differ from the time which should elapse according to the 
 catalogue, by the sum of the effects of the deviation upon the 
 two stars. Compute what effect a deviation of 15" will produce 
 on the interval, then the difference between the observed interval 
 and computed interval, divided by the quantity thus computed, 
 will be the factor of deviation to be used for correcting transits 
 observed the same night ; or, if the deviation itself be required 
 for altering the position of the instrument, multiply this factor 
 by 15, the result will be the deviation to the east or west of the 
 north in seconds of space. 
 
 The effect produced on the interval by a deviation of 15", is 
 to be computed as follows : let TT be the polar distance of the 
 upper star, TT' that of the star sub-polo, A the co-latitude of the 
 place : then the effect in time of a deviation of 1 5" is, for 
 
 sin. (X TT) , ,, , sin. (A. -f- TT'), 
 
 the upper star .-- -J- and for the star sub-polo 
 
 sin. TT sin. TT
 
 74 ASTRONOMICAL INSTRUMENTS. 
 
 acting contrary ways upon the time of transit of each star re- 
 spectively, and hence affecting the interval by their sum, or by 
 
 -. ; -, -. Hence the factor for instrumental devia- 
 
 sin. TT sin. TT 
 
 tion rz ' '- - X the difference between the ob- 
 
 sin. A. sin. (TT + TT ) 
 
 served and computed intervals. When -n =. TT', or the same star 
 is observed at the upper and lower culmination, this factor be- 
 comes - X the difference. 
 
 2 sin. A 
 
 Practical Rule. To the log. of the difference in seconds be- 
 tween the tabulated and observed interval, add the log. sines of the 
 polar distances of the two stars, the log. secant of the latitude, 
 and the log. co-secant of the sum of the two polar distances, 
 reject 40 from the index, and the result will be the log. factor of 
 the deviation, (to be used according to the formula, page 86, in 
 correcting the transits of all stars observed the same night.) 
 And, as before observed, when it is intended to correct the posi- 
 tion of the instrument, this quantity, multiplied by 15, will give 
 the deviation from the meridian in space to the east or west of 
 the north. 
 
 In determining the direction of the deviation, it must be re- 
 collected, that when the deviation is to the east, the star above 
 pole passes too early, and that below pole too late, and therefore, 
 if the upper star precedes, the interval is increased, but if the 
 lower precedes, then vice versa. When the deviation is to the 
 west, the star above pole passes too late ; while the star below 
 pole passes too early. Hence, if the former precedes, the inter- 
 val is diminished, and vice versa. 
 
 EXAMPLE. 
 
 At page 79 we have inserted an extract from the Greenwich 
 
 Observations for 1834, page 10 : we shall take for an example 
 the stars Cephei 51 Hev. and b Urs. Min. S. P. 
 
 March 18th, 1834. 
 
 Observation. Naut. Aim. 
 
 H. M. S. H. M. S. 
 
 Cephei 51 Hev. . . . = 6 20 59,00 . 6 20 19,61 
 
 Urs. Min. S. P. . . = 6 26 32,50 . 6 25 51,24 
 
 Interval observed . = 533,50 . 531,63 
 
 tabulated . = 5 31,63 
 
 Diff. of Intervals . = 1,87 Jug. 0*27184
 
 THE PORTABLE TRANSIT-INSTRUMENT. 
 
 Diff. of Intervals 
 
 75 
 
 (Brought forward) 
 1,87 log. 0-27184 
 
 TT the P. D. of Cephei 
 a 8 Urs. Min. 
 
 Latitude 
 
 (TT + TT') sum of the) 
 Polar distances f 
 
 = 2 43 50 
 
 = 3 25 4 
 
 51 28 39 
 
 sine 
 sine 
 secant 
 
 8-67796 
 8-77536 
 0-20564 
 
 = 6 8 54 co-sec. 0-97020 
 
 Multiply by 
 
 0,080 = 
 15 
 
 8-90100 
 
 Deviation from the meridian in space 1,200 
 
 In determining the direction of the above deviation, we must 
 observe according to our precepts, that the upper star precedes, 
 and the observed interval is greater than the tabulated interval, 
 therefore the deviation is to the east of north. 
 
 This method may now be practised very conveniently, as the 
 apparent places of 8 Ursas Minoris and Cephei 51 Hev. are given 
 in the Nautical Almanac. In like manner, Polaris may be com- 
 bined, though less advantageously, with the stars of the Great 
 Bear. 
 
 Again, Polaris, or any close circumpolar star, the place of 
 which is accurately known, may be combined with any star dis- 
 tant from the pole. The simplest mode of considering this is, 
 that the star which is distant from the pole, gives the time or 
 error of the time-keeper ; and again, if Polaris gives the same 
 error, that the instrument must be in the meridian,* the for- 
 mula for computation is the same as in the next following 
 method, commonly called that of high and low stars, but is 
 much more accurate. 
 
 The last method we shall speak of for correcting the position 
 of the instrument, is by observing the transit of any two stars 
 differing from each other considerably in declination, (at least 
 40) and but little in right ascension. The nearer the right 
 
 * Persons desirous of avoiding computation, and who do not -want the greatest 
 possible accuracy, may proceed conveniently thus : Get the error of the time- 
 keeper from stars as near the zenith as may be, levelling with the utmost care 
 before each observation, and reversing the instrument once during the series. By 
 taking a mean of the whole, an excellent error of the time-keeper will be found, 
 unaffected by errors of deviation or collimation, and, if the levelling has been 
 performed with all care, of inclination too. With this error, find what time, by 
 the time-keeper, Polaris, S Ursae Minoris, or Cephei 51 Hev., should transit, and 
 adjust the azimuthal screws accordingly. If the observer has made out, as he 
 always ought to do, the time between each wire, and the middle wire, as well as 
 the value of the revolutions of his adjusting-screw, he may compute the time for 
 each wire, and examine his success at each, as the star passes through the field 
 of the telescope. It is necessary to add, that the level should always be examined 
 after touching the azimuth-screws.
 
 7(3 ASTRONOMICAL INSTRUMENTS. 
 
 ascensions of the stars are to each other the better, as this pre- 
 vents the possibility of any error arising from a change in the 
 rate of the time-keeper affecting the observations. And as the 
 apparent places of one hundred principal stars are now given in 
 the Nautical Almanac for every tenth day, it will be better to 
 select a pair from thence, which will save the trouble of com- 
 puting their apparent right ascensions ; and, as many suitable 
 pairs are contained therein, it will seldom happen, but that the 
 passage of some of them will occur at a convenient time for 
 observation. 
 
 The times of the transits of the two stars being observed 
 (without regard to the error of the time-keeper), the deviation of 
 the instrument from the plane of the meridian may be thus de- 
 termined : Take the difference between the observed passages of 
 the two stars, and also the difference of their computed right as- 
 censions (calling the differences + when the lower star precedes 
 the higher, and vice versa) ; and if these differences be exactly 
 equal, the instrument will be correctly in the plane of the meri- 
 dian ; if they are not equal, their difference, that is to say, the 
 difference of the observed times of transit, minus the difference 
 of the computed right ascensions, will point out a deviation from 
 that plane, to the eastward of the south when it is -{-; and west 
 when it is . As an example, let us take the following : 
 
 Observed Time. Apparent A.R. 
 
 H. M. s. H. M. s. 
 
 Higher star 5 46 51,91 ... 5 46 53,50 
 
 Lower 6 37 25,66 ... 6 37 33,66 
 
 Difference = 50 33,75 ... 50 40,16 
 
 Subtractdiff.of A.R.= 50 40,16 
 
 + 6,41 -=. the difference of 
 
 time minus the difference of right ascension, which being + 
 shows that the instrument deviates to the eastward of the south 
 point of the horizon. It is evident that a high star will be less 
 affected by deviation, than one in any other situation, and that 
 a star between the pole and zenith will be differently affected 
 from a star south of the zenith, it being observed sooner than it 
 ought when the latter is observed later, and vice versa. 
 
 The deviation in azimuth may now be computed from the fol- 
 lowing formula : 
 
 Deviation in azimuth = D. sin. TT sin.Tj-' co-sec. (-7r7 ?r')sec. L. 
 
 In which D represents the difference of times minus the dif- 
 ference of right ascensions ; TT and TT' the polar distances of the
 
 THE PORTABLE TRANSIT-INSTRUMENT. 77 
 
 higher and lower stars, and L, as before, the latitude of the 
 place of observation. 
 
 This formula, in words, gives the following rule : To the log. 
 of the difference of times minus the difference of right ascen- 
 sions, add the log. sin. of the polar distance of the higher star, 
 the log. sin. of the polar distance of the lower star, the log. co- 
 secant of the difference or sum of the polar distances of both the 
 stars, (the difference when they are both above the pole, and the 
 sum when one is above and the other below the pole,) and the 
 log. secant of the latitude : the sum will be the log. of the azi- 
 muthal deviation, which multiplied by 15 will be the deviation in 
 arc. 
 
 As a complete example, let us take a high and low star, from 
 the same day's work at Greenwich that we took our former ex- 
 ample from, and see how nearly alike the deviation comes out by 
 the two calculations. 
 
 t 
 March 18th, 1834. (See page 79.) 
 
 Observed Time. A PP a ^ ent f A -&- from 
 Naut. Aim. 
 
 H. JI. 8. H. M. S. 
 
 Higher star, Cephei 51 Hev. 6 20 59,00 ... 6 20 19,61 
 Lower Sirius 6 38 30,88 ... 6 37 49,76 
 
 Difference = 17 31,88 . . 17 30,15 
 
 Subtract diff. of A. R. ..= 17 30,15 
 
 1,73 = the difference of 
 
 time minus the difference of right ascension, which being 
 shows that the deviation is to the west of south, agreeing with 
 our former determination, (page 75) viz., east of north. Let us 
 now compute the deviation in azimuth by our last formula, and 
 see how nearly they agree. 
 
 Difference of intervals l s ,73 . . log. . . 0'23805 
 
 TT the P.D. of Cephei 2 43' 50" . . sine . . 8-67796 
 v' f Sirius 106 29 50 .. sine .. 9-98174 
 
 (TT' TT) the diff. of Polar distances . . co-sec. 0'01266 
 L = the latitude =51 28' 39" sec. , . 0-20564 
 
 0,131 =9-11605 
 
 Multiply by 15 
 
 The azimuthal deviation in arc, \ , ORt , 
 
 west of south / 
 
 " The time employed in making these observations is supposed 
 to be sidereal time, therefore, if a clock or watch be used which 
 marks mean solar time, the interval between the observations 
 must be corrected accordingly/' This correction is made by
 
 78 ASTRONOMICAL INSTRUMENTS. 
 
 adding to the difference of the observed times, the acceleration 
 of the fixed stars for that interval, (Table IV.,) which will con- 
 vert that portion of mean into an equivalent portion of sidereal 
 time ; so that by means of this correction it will be indifferent 
 whether the clock shows sidereal or mean time. 
 
 " If, before or after the passage of the stars, the telescope be 
 pointed to the horizon and compared with some object there, a 
 meridian mark may be set up, which may be corrected from time 
 to time by subsequent observations on various stars similarly 
 situated, and when once correctly fixed, it will serve to verify 
 both the meridional position of the instrument, and the adjust- 
 ment of the collimation." 
 
 Having, by means of the previous adjustments, made the line 
 of collimation describe a great circle passing through the zenith 
 of the place, and the north and south points of the horizon, the 
 instrument will be in a fit state for making observations. We 
 have said that the telescope contains five vertical and two hori- 
 zontal wires, placed a short distance from each other ; these last 
 are intended to guide the observer in bringing the object to pass 
 across the middle of the field, by moving the telescope until it 
 appears between them : the centre vertical is the meridional 
 wire, and the instant of a star's passing it will be the time of 
 such star's being on the meridian ; but as, in noting the time, 
 it will not often happen that an exact second will be shown by 
 the clock, when the star is bisected by the wire, but it will pass 
 the wire in the interval between two successive seconds, the 
 observer must, therefore, whilst watching the star, listen to the 
 beats of his clock, and count the seconds as they elapse ; he will 
 then be able to notice the space passed over by the star in every 
 second, and consequently its distance from the wire at the second 
 before it arrives at, and the next second after it has passed it, 
 and with a little p'ractice he will be able to estimate the fraction 
 of a second at which the star was on the wire, to be added to the 
 previous second : thus, suppose the observer counted 4, 5, 6, 7, 
 8 seconds, whilst watching the passage of a star, which passed 
 the wire between the 7th and 8th, at which times it appeared 
 equally distant on each side of it, the time of the transit would 
 then be 7 8 ,5 ; but if it appeared more distant on one side than 
 the other, it would be 7 8 ,3, or 7 S ,7, &c., according to its ap- 
 parent relative distance from the wire. 
 
 This kind of observation must be made at each of the five 
 wires, and a mean of the whole taken, which will represent the 
 time of the star's passage over the mean or meridional wire. The 
 utility of having five wires instead of the central one only will 
 be readily understood, from the consideration that a mean result 
 of several observations is deserving of more confidence than a 
 single one; since the chances are, that an error which may have 
 been made at one wire will be compensated by an opposite error
 
 THE PORTABLE TRANSIT-INSTRUMENT. 79 
 
 at another ; thus destroying each other's effect, the mean result 
 will come out very nearly the same as the observation at the 
 middle wire, if they are made with any tolerable degree of ac- 
 curacy, and if the intervals of the wires are uniform. 
 
 The annexed Table is an ex- 
 ample of the Greenwich mode 
 of registering observations made 
 with a transit-instrument. 
 
 The heading at the top of the 
 columns sufficiently explains the 
 nature of their contents. The 
 error of the clock from sidereal 
 time is obtained, by taking the 
 difference between the mean of 
 the wires, and the apparent 
 right ascension of the object as 
 given in the Nautical Almanac ; 
 and the daily rate is the differ- 
 ence of such errors, divided by 
 the number of days elapsed be- 
 tween the observations. In ob- 
 serving the sun, the times of 
 passing of both the first and 
 second limb over the wires are 
 observed and set down as dis- 
 tinct observations, the mean of 
 which gives the time of the pas- 
 sage of the centre across the 
 meridian, as is shown in the an- 
 nexed example. The wires of 
 the instrument are generally 
 placed by the maker at such a 
 distance from each other, that 
 the first limb of the sun shall 
 have passed all of them before 
 the second limb arrives at the 
 first, and the observer can thus 
 take the observations without 
 hurry or confusion. 
 
 One limb only of the moon 
 can be observed, except when 
 her transit happens to be within 
 an hour or two of her opposi- 
 tion ; and in observing the larger 
 planets, the first and second limb 
 may be observed alternately over 
 the five wires ; that is to say, the 
 first limb over three wires, viz., 
 
 
 
 ttjco 
 
 1 
 
 
 ^O ^ -J r ' ^ 
 
 fl 
 
 i 
 
 2 S "~ ^ ""* 
 
 J 
 
 ? 
 
 V5 "^ c -^ "~* ^ 
 
 c 
 
 l 
 
 ^ w o ^ '^ ^ 
 
 
 
 ^0/^g 
 
 O 4. 
 
 < 
 
 5 | 
 
 . . ** CO . .CO 
 
 
 
 H o o 
 
 
 
 . .00 . .0 
 
 ,! 
 
 i 
 
 , . -00 -0 
 
 
 
 
 <M rH CO 
 
 a 
 
 C 
 
 e< l c T. "! 
 
 
 C 
 
 C 
 
 w 
 
 j 
 
 
 
 S . .00 . .0 
 
 Of 
 
 i 
 i- 
 
 
 CO O CO O O O 00 
 
 t^ C5 CO CO O O 00 
 
 ^ 
 
 
 02 rH <N i-l in CO CO 
 
 C 
 
 
 o <N -H m o co oo 
 
 e 
 
 I 
 
 
 S in n in c^ (N co 
 
 jj 
 
 
 aSSS 100500 
 
 
 . 
 
 riS5-5 -S 
 
 
 
 ^< in 'in 
 
 
 ,; 
 
 ^ l-H^CO >^ FH 
 03 r-To" e>f ' G5 
 
 to 
 
 1-1 
 
 co Tj< co ' m 
 
 H 
 
 
 
 
 0} 
 
 
 TJ 
 
 _t 
 
 >. as to o m i>. 
 
 a 
 M 
 
 ^ 
 
 18 p-i w ' m co co 
 
 V 
 
 'C 
 
 . . 
 
 1 
 
 B 
 
 S m m ' IN N co 
 
 a 
 
 s 
 
 
 S 
 
 
 
 3 
 
 . 
 
 * o m oo 
 
 11 
 
 
 OJTfico'*O^' "i-t 
 
 
 
 n -"i" i i 
 
 
 1 J 
 
 CO << ' i^t * " i 
 
 8 
 
 
 oo 
 eo 
 
 
 
 
 ft 
 
 
 "S
 
 ASTRONOMICAL INSTRUMENTS. 
 
 the first, third, and last ; and the second limb over the second 
 and fourth ; which being reduced in the same manner as the 
 observation of the sun, will give the meridional passage of the 
 centre. When an observation at one or more of the wires has 
 been lost, it is impossible to take the mean in the same way as 
 in a perfect observation. If the centre wire is the one that is 
 deficient, the mean of the other four may be taken as the time 
 of the meridional passage, or the mean of any two equally dis- 
 tant on each side of the centre, (supposing the interval of the 
 wires to be equal ;) but when any of the side wires are lost, and 
 indeed under any circumstance of deficiency in the observation, 
 the most correct method of proceeding is as follows : By a con- 
 siderable number of careful observations over all the wires, the 
 equatorial interval between each side wire and the centre one is 
 to be deduced and set down for future use. Then, when part of 
 the wires only are observed, each wire is to be reduced to the 
 mean, by adding or subtracting, as the case may be, to the time 
 of observation, the equatorial interval between that wire and the 
 centre wire, multiplied by the secant of the declination of the 
 star, as in the following rule. 
 
 To the log. of the equatorial interval (from the wire at which 
 the observation was made to the centre) add the log. secant of 
 the star's declination (or co-sec, of its polar distance,) the sum, 
 rejecting ten from the index, will be the log. of the inteival from 
 the wire at which the transit was taken, to the centre wire, 
 which being added to observations made at the first or second 
 wire, or subtracted from those made at the fourth or fifth, will 
 give the time of the star's passing the meridional wire. 
 
 The equatorial intervals of the wires may readily be com- 
 puted by the following rule, from observations made upon any 
 star whose declination is known. To the log. of the interval 
 occupied by the star in passing from any wire to the centre wire, 
 add the log. cosine of the star's declination (or sine of its polar 
 distance ;) the sum, rejecting ten from the index, will be the log. 
 of the equatorial interval, which being determined for each wire, 
 from observations of a number of stars having different declina- 
 tions, the mean will be a very correct result. The equatorial 
 intervals of the wires of the transit at the Royal Observatory, 
 were found to be, 
 
 s. 
 
 From the first wire to the third = 36,647 
 
 second = 18,305 
 
 fourth = 18,309 
 
 fifth = 36,605 
 
 The middle wire at Greenwich coincides with the mean of the 
 wires, the intervals being very nearly equal, but when this is not 
 the case, the observer must correct the mean of the wires for the
 
 THE PORTABLE TRANSIT-INSTRUMENT. J 
 
 difference from the centre wire, to obtain a correct mean ; the 
 correction to be applied to the mean of the wires may be com- 
 puted as follows : divide the difference between the sum of the 
 first two and sum of the last two equatorial intervals by 5, and 
 to the log. of the quotient add the log. co-secant of the polar 
 distance of the star ; the sum will be the log. of the correction 
 required, plus if the sum of the two first intervals is greater than 
 the second, otherwise minus. Such inequality in the intervals 
 should never be allowed to remain, unless circumstances pre- 
 vented their rectification. 
 
 In regular observatories, the transit-instrument is employed, 
 not only for the determination of time, but in forming catalogues 
 of the right ascensions of the fixed stars, and other important 
 operations in astronomy ; purposes for which instruments of a 
 superior class, and fixed in their respective places, are required. 
 But, from the small size and low optical power of the portable 
 transit-instrument, it can be applied with good effect only to the 
 determination of time, and of the longitude by observations of 
 the moon and moon-culminating stars. The Nautical Almanac 
 contains the true apparent right ascension of the sun, and of one 
 hundred of the principal fixed stars ; that is, the sidereal time 
 when each of them, respectively, is on the meridian, or on the 
 centre wire of a properly adjusted transit-instrument; and if 
 the instant when a star so passes the central wire, be noted 
 by a clock correctly adjusted to sidereal time, the time shown 
 by the clock will be the right ascension of the star as given in 
 the Almanac. The difference therefore between the time shown 
 by a clock, and such right ascension, will be the error of the 
 clock from sidereal time + (or too fast) when the clock time is 
 greater than the right ascension, and (or too slow) wh.-ii it is 
 less. Thus, on March 18th, 1834, (page 79,) 
 
 II. M. S. 
 
 The observed passage of Capella by clock ..55 6,60 
 Right ascension by Naut. Aim 54 25,29 
 
 Clock error + 41,31 
 
 In the same manner the error of the clock is deduced from an 
 observed transit of the sun's centre, the time of which, as before 
 shown, is derived from a mean of the observations of the first 
 and second limbs; but when, from intervening clouds or other 
 circumstances, one limb only can be observed, the passage of the 
 centre may be found, by adding or subtracting the sidereal 
 time of the suns semidiameter passing the meridian, as given in 
 the Nautical Almanac, according as the first or second limb may 
 be observed. 
 
 If the clock error be determined in this manner from a num- 
 ber of observations each day, the mean of the whole will pro-
 
 ga ASTRONOMICAL INSTRUMENTS. 
 
 bably be a very accurate determination of the error for the mean 
 of the times at which the observations were made. In like 
 manner the mean daily rate may be found by taking the differ- 
 ence between the errors as determined by the same object from 
 day to day ; and if more than one day has elapsed between the 
 observations, dividing the change in the error by the number of 
 days elapsed ; the rate, when the clock is too fast, will be + (or 
 gaining) when the second error is greater than the first, and 
 (or losing) when the second error is the least ; and vice versa, 
 when the clock is too slow. 
 
 When a clock or chronometer, showing mean solar time, is 
 employed, its error from such time may be found, by computing 
 the mean time of the passage of the object over the meridian of 
 the place, and the difference between such mean time, and the 
 observed time of the object's meridian passage, will, as before, 
 be the error of the clock from mean time. 
 
 The following is the method of computing the mean solar time 
 of the transit of a star across the meridian. 
 
 From the right ascension of the star, subtract the sidereal 
 time at mean noon for the given day, taken from the Nautical 
 Almanac, (adding 24 hours to the former when the latter exceeds 
 it) the remainder is the sidereal interval after noon of that day. 
 From this, subtract the acceleration of sidereal upon mean time, 
 and the result is the required mean solar time of the passage. 
 As an example, suppose it were required to find the mean time 
 of the passage of Capella on March 18th, 1834: 
 
 H. M. s. 
 
 Right ascension of Capella (+ 24 hours) 29 4 25,29 
 
 Sidereal time at mean noon 23 42 15,64* 
 
 Sidereal interval, past noon = 5 22 9,65 
 
 Acceleration of sidereal on mean time\ r ^ _Q 
 
 for the interval / 
 
 Mean time of passage = 5 21 16,87 
 
 The acceleration of sidereal on mean time is to be taken from 
 Table III. ; thus, in the above example : 
 
 M. S. 
 
 Acceleration for 5 hours 49,118 
 
 22 minutes .... 3,604 
 
 ,, 9 seconds .... 0,025 
 
 65 hundredths . . . 0,003 
 
 For the whole interval 52,7<SO 
 
 * The sidereal time as given in the Nautical Almanac, is for mean noon at 
 Greenwich, and therefore must be corrected for any other meridian, as directed in 
 the explanation of the articles, given at the end of the Almanac.
 
 THE PORTABLE TRANSIT-INSTRUMENT, gg 
 
 Table III. will not answer for performing the reverse opera- 
 tion, viz., converting a portion of mean solar time into a corre- 
 sponding portion of sidereal time ; Table IV. must be employed 
 for this purpose, adding to the given portion of mean time the 
 quantity taken from the table corresponding thereto, and the 
 sum will be an equivalent portion of sidereal time. As an ex- 
 ample we will take the above case of Capella. 
 
 H. M. s. 
 
 Mean time 5 21 16,87 
 
 Table IV + 58,78 
 
 Sidereal interval 5 22 9,65 
 
 Sidereal time at mean noon 23 42 15,64 
 
 29 4 25,29 
 
 Sidereal time of the star's passage, or\ - A ~ OQ 
 
 .1, f - ' ff Tf sCt) t /Ci' 
 
 its right ascension J 
 
 The method of taking out the correction from Table IV. is 
 exactly similar to that given in the above example for Table III. 
 
 To find the error of a clock or chronometer intended to show 
 mean time from an observed transit of the sun, nothing more is 
 necessary than to apply the equation of time to 24 hours, and 
 the difference between the result and the time of the sun's 
 transit, as shown by the chronometer, is the error of the chro- 
 nometer for mean time; -}- when the chronometer time is the 
 greater, and when it is the less. 
 
 From the description which has been given of the .method of 
 bringing a transit-instrument into a state of perfect adjustment, 
 it might be inferred that it is essential it should be strictly so, to 
 obtain accurate results from the use of it. It is certainly desir- 
 able that the adjustments should be examined and rectified as 
 often as possible, as doing so ultimately saves the labour of 
 computing the corrections to be applied to each observation, on 
 account of the errors in the position of the instrument. But 
 in some established observatories, where large instruments are 
 employed, it is not attempted to put them in perfect adjustment, 
 but the amount of the various derangements is ascertained from 
 time to time, and the observations corrected accordingly. The 
 adoption of this method, with so small an instrument as the one 
 which we have been describing, where the adjustments are easily 
 examined and corrected, will give indeed more accurate results, 
 but, on account of the greater trouble, is not perhaps to be 
 generally recommended ; we shall, nevertheless, introduce in 
 this place, an account of the method of computing these cor- 
 
 G 2
 
 gj, ASTRONOMICAL INSTRUMENTS. 
 
 rections, that persons possessing transit-instruments may adopt 
 which method they think proper. 
 
 The first correction is for the deviation of the line of collima- 
 tion : the amount of the error may be determined by a micro- 
 meter attached to the eye-end of the telescope, by which, when 
 the telescope is directed towards any distant object, the angular 
 distance of that object from the central wire is measured in revo- 
 lutions and parts of the micrometer-screw. The instrument is 
 then reversed, and the distance of the same object from the cen- 
 tral wire again measured, when half the difference of the measures 
 is the error in collimation ; and the angular value of a revolution 
 of the screw being known, the corresponding value of the error 
 is likewise known. The correction on account of this error to 
 be applied to the time of each observation may be computed 
 from the following formula : 
 
 c 
 
 Correction = co-sec. IT 
 15 
 
 c = the error of collimation + if the deviation is toward the east. 
 TT = (as before) the polar distance of the star. 
 
 Hence we have in words this rule : To the log. of the devia- 
 tion in collimation, add the log. co-secant of the polar distance 
 of the star, and the arithmetical complement of the log. of 15: 
 the sum will be the log. of the correction in time required. 
 
 The next correction to be considered, is that arising from a 
 want of horizontality in the axis. The spirit-level, which we 
 described as striding across the instrument and resting on the 
 pivots, determines the amount of the inclination of the axis, and 
 also, as we have seen, enables the observer to correct it. Above 
 the glass tube, and parallel to its length, is placed a fine gradu- 
 ated scale, the reading of which points out the number of seconds 
 in arc that the pivots deviate from the true level, shown by the 
 air-bubble receding from the centre towards that pivot which is 
 the highest ; but as it is necessary, when correcting for the ad- 
 justment, to remove half the error, by giving motion to the 
 little screw on the level itself, so, for the same reason, in finding 
 the measurement of the error, it is necessary to reverse the level 
 on the axis, and read the scale at each extremity of the air- 
 bubble in both its positions ; that is, with the same end of the 
 level on both the east and west pivots alternately, and the dif- 
 ference of the sums of the two readings divided by the number 
 of readings will be the amount of deviation. This may be 
 illustrated by the following example, in which the divisions on 
 the scale represent seconds.* 
 
 * The value of the divisions of the scale may be had from the maker.
 
 THE PORTABLE TRANSIT-INSTRUMENT. 
 
 Readings of the Scale. 
 East End. West End. 
 
 85 
 
 .... 69,6 
 
 109,0 .... 69,8 
 
 108,8 .... 69,9 
 
 Level Reversed. 
 
 69.0 .... 109,0 
 68,6 .... 108,9 
 
 69.1 .... 109,0 
 
 533,5 Sums 536,2 
 533,5 
 
 Divide by the number of observations, 12) 2,7. 
 
 | difference = 0,23 = the amount 
 
 of deviation in arc, showing that the west end of the axis is 
 higher by that quantity than the east end, since the sum of the 
 western readings is greater than the sum of the eastern. This 
 quantity, divided by 15, will give the proper factor for inclin- 
 ation. It is more convenient that the scale should be divided 
 into units, each of which is 15". 
 
 Having in this manner determined the inclination of the axis 
 by the level, the correction to be applied to the time of observ- 
 ation of any star made during the existence of that error, may 
 be computed from the following formula : 
 
 Correction = b cos. (TT A) co sec. TT 
 
 b = the factor for the inclination of the axis + if the west 
 end be too high. 
 
 TT = the polar distance of the star. 
 A = the co-latitude of the place. 
 
 This formula in words gives the following practical rule. To 
 the log. of the factor for inclination of the axis, add the log. 
 co-secant of the polar distance, and the log. co-sine of the dif- 
 ference between the polar distance and the co-latitude : the sum 
 20 will be the log. of the correction in time required. 
 
 We have already explained the manner of ascertaining the 
 azimuthal deviation of the instrument from the plane of the 
 meridian, page 72, &c. The correction to be added algebraically 
 to the observed time of transit of any star whilst the instrument 
 so deviates, may be computed from the following formula : 
 
 Correction = a sin. (TT A) co-sec. TT
 
 gg ASTRONOMICAL INSTRUMENTS. 
 
 in which a = the factor for azimuthal deviation, -f when the 
 instrument deviates to the eastward of the south meridian. 
 
 TT = the polar distance of the star. 
 A = the co-latitude of the place. 
 
 This formula in words gives the following : 
 
 Rule. To the log. of the factor for azimuthal deviation, add 
 the co-secant of the polar distance, and the sine of the difference 
 between the polar distance and co-latitude : the sum will be the 
 log. of the correction required. 
 
 As an example, let us take the star e Bootis. (Pearson's 
 Astron. vol. ii. p. 344.) 
 
 H. M. S. 
 
 Observed time of transit . = 14 35 4, 86 
 
 Error of collimation . . 12" or = -f- 0, 80 
 
 Inclination of the axis . . . = 1 , 75 
 
 ^Deviation of Instrument in azimuth . = 4,737 
 
 The errors are in units, each of which = 15 
 
 o / 
 
 Polar distance . . = 61 12 
 
 Co-latitude . . . = 08 27 
 
 The correction for the Collimation. 
 
 Deviation = + 0,80 log. . . . = -.- fHHMMJ 
 Polar dist. 62 12' co-secant . =H <>05;j:.'<; 
 
 Correction = + O s ,904 . log. = + <>-05G35 
 
 The correction for the Level. 
 
 Deviation =1,75 log. .- . . = 0-2i,".01 
 Polar dist. 62 12' co-secant . . . = + 0'053~'6 
 Polar dist. minus co-lat. 23 45' cos. . = + 9*96157 
 
 Correction = 1 8 ,811 . log. 0-25787 
 
 The Correction in Azimuth. 
 
 Deviation = 4,737 log = 0*67550 
 
 Polar dist. 62 12' co-secant . . . . = + 0-05326 
 Polar dist. minus co-lat. = 23 45' sine = + 9-60503 
 
 Correction = 2 s , 157 .. log. = 0-33379 
 
 Now apply the sum of these corrections to the observed tiim 
 of the star's transit, and the actual time of transit will be obtained
 
 THE PORTABLE TRANSIT-INSTRUMENT. $7 
 
 as correctly as if the instrument had been in a state of perfect 
 adjustment when the observation was made. 
 
 u. M. s. 
 
 Observed time of transit =1435 4,860 
 
 Correction for the collirnation . . = + 0,904 
 
 level = 1,811 
 
 in azimuth = 2,157 
 
 Corrected observation = 14 35 1,796 
 
 Computed right ascension ... = 14 37 28,910 
 
 Clock slow on sidereal time . = 2 27,114 
 
 Besides the determination of time, the portable transit-instru- 
 ment may be successfully employed in determining the longitude. 
 The Nautical Almanac contains, for each lunation, a list of the 
 right ascensions and declinations of the moon-culminating stars, 
 whose meridional transits being observed, together with that of 
 the moon, at any two places, the differences of right ascension 
 thus obtained between the moon's illuminated limb and each of 
 those stars, form the data required for computation. " If the 
 moon had no motion, the difference of her right ascension from 
 that of a star would be the same at all meridians, but in the in- 
 terval of her transit over two different meridians, her right as- 
 cension varies, and the difference between the two compared dif- 
 ferences exhibits the amount of this variation, which, added to 
 the difference of meridians, shows the angle through which the 
 westerly meridian must revolve before it comes up with the 
 moon ; hence, knowing the rate of her increase in right ascen- 
 sion, the difference of longitude is easily obtained." 
 
 The necessity of having recourse to actual observation of the 
 same stars at the two places, in order to obtain the longitude, 
 may soon be dispensed with, since their apparent right ascen- 
 sions are given in the Nautical Almanac. At present, however, 
 and until the places of the moon-culminating stars are perfectly 
 well known, corresponding observations are required for the ac- 
 curate determination of differences of longitude. 
 
 The difference of longitude between the stations, is supposed 
 to be approximately known, or may be got near enough for an 
 approximation, by dividing the difference between the observed 
 and computed right ascension of the moon's bright limb by the 
 hourly motion given in the Nautical Almanac. 
 
 The formula for computation, with the necessary explanation, 
 may be found in the Memoirs of the Royal Astronomical So- 
 ciety, vol. ii. p. 1, &c. Availing myself of the kind permission 
 of Mr. RIDDLE, I am enabled to insert his method of perform- 
 ing the computation, together with Table XXXIII. of his valu- 
 able treatise on Navigation.
 
 ASTRONOMICAL INSTRUMENTS. 
 
 PRACTICAL RULE. 
 
 To the estimated longitude in time, add the correction from 
 Table XL, and apply the sum to the time of the moon's passing 
 the meridian of Greenwich, as given in the Nautical Almanac, 
 adding if the longitude is west, or subtracting it if east ; and the 
 sum or the remainder will be the approximate Greenwich date 
 for the moon's passing the given meridian. 
 
 Find the moon's right ascension, both for this time and the 
 time of her passing the meridian of Greenwich, and divide the 
 difference of her right ascensions by the hours, &c., in the dif- 
 ference of these times, and the quotient will be the mean hourly 
 change of the moon's right ascension in the interval, which is 
 the argument of Table V. 
 
 Take also the declinations roughly for the same two times. 
 
 With the mean of these declinations, and the change of the 
 moon's semidiameter, take the correction from Table VI. and 
 apply it to the interval between the transits of the star and the 
 moon's bright limb, as observed at or computed for the more 
 westerly meridian. 
 
 Again, with the mean of the declinations take the corrections 
 from Table XII., and multiplying it by the degrees in the moon's 
 change of declination, apply the product as a second correction 
 to the western interval. 
 
 The following formula will show the signs with which these 
 corrections are to be applied. 
 
 Sign of First Correction. 
 
 Moon's 
 semidiam. 
 increasing 
 
 Limb Cor- 
 Obsd. rect. 
 
 ?7W 
 
 star 3 E + 
 
 following ) W + 
 
 star 3 E 
 
 Moon, 
 'preceding 
 
 Moon. 
 
 Limb Cor- 
 Obsd. rect. 
 
 it , fprecedn 
 
 Moon s 
 
 I star 
 
 semidiam. < ,. ,, -. w 
 
 i . | following 7 W 
 
 decreasing I 5- E> 
 
 star \ J + 
 
 r") W + 
 
 }E 
 
 star 
 
 Sign of Second Correction. 
 
 Moon. 
 
 Limb Cor- 
 Obsd. rect. 
 
 Moon's (Preening* W + 
 j r ,. star 3 E 
 
 declination < /. n . ~\ \\j 
 
 \ following ) VV 
 increasing I > ^ 
 
 ' V star y E + 
 
 Moon. 
 
 Moon's ( 
 i T 4 - 
 declination < c . ,- 
 
 1 following f W + 
 decreasing I ^ 
 
 ' V. star 3 E 
 
 Limb Cor- 
 Obsd. rect. 
 
 W 
 
 J E 
 
 The change of semidiameter here spoken of is that taken 
 from the Ephemeris, without augmentation for altitude. 
 
 The interval at the more westerly meridian being thus cor- 
 rected, call the seconds of the differences of the intervals. A ;
 
 THJi PORTABLE TRANSIT-INSTRUMENT. 
 
 or, if more than one star has been observed, call the seconds in 
 the mean of the differences of the corresponding intervals A. 
 
 If either of the intervals be in mean time, add to it its 860th 
 part diminished by the 70th part of itself, and the sum will be 
 the corresponding interval in sidereal time. And if both are in 
 mean time, reduce their difference to sidereal time by the same 
 rule. Table IV. may also be used for this purpose. 
 
 If the moon precede the star at the easterly, and follow it at 
 the westerly meridian, the sum of the intervals instead of the 
 difference will be A. 
 
 Then add the logarithm of the seconds in A, the difference of 
 the sidereal intervals to the logarithm from Table V., and the sum 
 will be the logarithm of the difference of longitude in seconds of 
 time. 
 
 Note. The parts for hundredths in Table V. are found in the 
 column of ' parts' opposite the corresponding tenths. Thus, for 
 I m 42 s ,57, the log. for I m 42 s ,5, is 1*534256, and the part for seven 
 hundredths is 304 whence the log. is 1 "533952. Striking off 
 the figures on the right, in the column of ' parts,' the remaining 
 figures on the left are parts for thousandths. 
 
 EXAMPLE. 
 
 December 8th, 1834. 
 
 Star, &c. 
 
 Clock Transit, 
 
 observed 
 
 Rate of 
 
 Clock Transit, 
 observed 
 
 Rate of 
 
 
 at Greenwich. 
 
 Clock. 
 
 at Cambridge. 
 
 Clock. 
 
 
 H. M. S. 
 
 s. 
 
 H. M. S. 
 
 s. 
 
 96 Aquarii .. 
 n Piscium . . 
 
 23 10 58,18 
 23 39 35,32 
 
 0,68 
 
 23 10 14,40 
 23 38 51,72 
 
 2,56 
 
 ]) 's 1st Limb 
 s Piscium . . 
 n Ceti .... 
 
 23 47 29,86 
 23 57 1,18 
 21 45,08 
 
 
 23 46 45,52 
 23 56 17,53 
 21 1,32 
 
 
 First find the mean intervals between the passage of the stars 
 and the moon at both places, thus : 
 
 Greenwich Intervals. 
 
 M. s. 
 
 36 31,68 
 
 6 54,54 
 
 9 31,32 
 
 34 15,22 
 
 Cambridge Intervals. 
 
 M. 
 
 36 
 7 
 9 
 
 34 
 
 Intervals corrected for Rate. 
 
 36 31,70 36 
 
 7 54,54 7 
 
 9 31,32 9 
 
 34 15,23 34 
 
 s. 
 
 31,12 
 53,80 
 32,01 
 15,80 
 
 31,18 
 53,81 
 32,03 
 15,85
 
 90 ASTRONOMICAL JNSTHUML.MS. 
 
 Diff. Intervals. 
 52 
 73 
 71 
 62 
 
 Mean '65 
 
 On December 8th, 1834, the moon passed the meridian of 
 Greenwich at G h 40 m , the declination being then about 7, and 
 it would be about one thousandth of a degree different at Cam- 
 bridge; and the lOOOdth part of '134< (the number corresponding 
 to 7 of declination in Table XII.) is too small a quantity to be 
 worth attention. This also is the case with the effect of the 
 change in the moon's semidiameter, the change being not more 
 than a thousandth of a second of space ; and the effect of that 
 small change on the time of the moon's transit being clearly 
 beyond the reach of notice in ordinary observers. The Nautical 
 Almanac gives the following : 
 
 M. s. 
 
 Hourly change of 3> 's R. A, from 5 hours to 6 . 1 50, 19 
 
 6 7 . 1 50,03 
 
 7 8 . 1 49, 87 
 Hence, at 6 h 4>0 m the hourly rate of change would be about 
 
 l m 50 S ,08. 
 
 M. S. 
 
 1 50,08 Table V 1-502334 
 
 ,65 log =9-812913 
 
 Longitude of Cambridge in time 21 s , 3 1'3 15247 
 
 The longitude of the Cambridge Observatory has been deter- 
 mined by Professor AIRY to be 23 s , 5. The reader may perhaps 
 be surprised, that the above result differs 2 s , 8 from it ; but it 
 may be remarked, that by this method of finding longitude, it 
 is absolutely necessary, that a great number of results be taken 
 as a satisfactory determination. This arises mostly from the 
 errors made in observing the transit of the moon's limb, which, 
 it is well known to practical men, is a very difficult observation 
 to make correctly ; and a very small error in the observation 
 makes a considerable one in the final result: supposing the 
 transits of the stars to have been observed perfectly correct, 
 yet, if an error of only two tenths of a second be made in that 
 of the moon's limb at either Observatory, the longitude deduced 
 from such observations, would be incorrect to the amount of 6 
 seconds in time, at a mean rate of the moon's motion. When 
 both limbs of the moon can be observed at both Observatories,
 
 THE PORTABLE TRANSIT-INSTRUMENT. 91 
 
 which can only be the case when she is near the full at the time 
 of transit, a better result can be obtained. 
 
 There is a mode of finding the latitude by the transit-instru- 
 ment, pointed out by Professor BESSEL, and used with great 
 success in the Russian survey, which we will now explain in some 
 detail, as the method is not so commonly known or practised in 
 this country as it deserves to be. 
 
 Place the transit-instrument with its supports north and south, 
 so that the telescope when pointed to the horizon looks due east 
 and west. Observe the passage of a well-known star over the 
 middle wire when the telescope is pointing east, and again, ob- 
 serve the passage of the same star over the middle wire when the 
 telescope is pointing west, noting the time carefully. The star 
 should be near the zenith, (such a star as yDraconis, for instance, 
 in the latitude of London, and for a degree or two to the north- 
 wards,) as the observations take less time, and are therefore more 
 independent of the timekeeper employed ; the method is also 
 move accurate when the star is near the zenith than when other- 
 wise. 
 
 In the accompanying figure, P is the 
 pole, Z the zenith, E Z W the prime 
 vertical passing through the east and 
 west points, the dotted line S s the path 
 of the star ; all seen as projected on the 
 horizon from a point above Z. Then 
 in the right-angled spherical triangle, 
 P Z S, P S is the north polar distance 
 of the star, P Z the co-latitude, and the angle Z P S, half the 
 time elapsed from S to s, therefore, tan. P Z = tan. P S X * cos. 
 ZPS. 
 
 Let A" = half the interval in time reduced to arc between the 
 
 two transits of the star over the prime vertical, (a 
 
 circle which passes through the zenith, and east and 
 
 west points of the horizon.) 
 
 Tt = the N. P. D. of the star (taken from the Nautical 
 
 Almanac.) 
 
 A the co-latitude of the place, 
 then tan. \ = tan. TT cos. A" 
 
 or in words to the log. tangent of the star's N.P.D. add the 
 log. co-sine of half the time elapsed, and the sum 10 will be 
 the log. tangent of the co-latitude required. 
 
 It is essential to the accuracy of this method, that the instru- 
 ment should be well adjusted, or the errors known and allowed 
 for. The error caused in the latitude thus determined, by the 
 want of adjustment of level or collimation, will exactly equal the 
 error of the level and collimation. If the observation be repeated
 
 (J<J ASTRONOMICAL INSTRUMENTS. 
 
 on various nights, the telescope should be reversed. With these 
 precautions, and a level of the best kind, the latitude may be ob- 
 tained within a second or two, if the place of the star is suffi- 
 ciently well known ; and differences of latitude, whether the star 
 be known or not. 
 
 To find the time when the star will come to the proper position 
 for observation, viz., the prime vertical ; first ascertain when the 
 star will be on the meridian, by the method explained at page 
 82, then, by the following formula compute the time that would 
 elapse during the passage of the star from the prime vertical to 
 the meridian, or (referring to the above diagram) the angle, 
 S P Z, which time, subtracted from that of the meridian passage, 
 will give the time of the star's being on the prime vertical, or in 
 the required position for making the first observation. 
 
 Formula, cos. S P Z (or A") = tan. P.Z, cot. P S. 
 
 Practical Rule. To the log. tangent of the assumed co-lati- 
 tude add the log. co-tangent of the star's polar distance, the sum 
 will be the log. co-sine of half the elapsed time in arc, which 
 divided by 15 will give the time required. 
 
 THE ALTITUDE AND AZIMUTH INSTRUMENT.
 
 THE ALTITUDE AND AZIMUTH INSTRUMENT. C)g 
 
 To the centre of the tripod, A A, is fixed the vertical axis of 
 the instrument, of a length equal to about the radius of the 
 circle ; it is concealed from view by the exterior cone, B. On 
 the lower part of the axis, and in close contact with the tripod, 
 is centred the azimuth circle, C, which admits of a horizontal 
 circular motion of about three degrees, for the purpose of bring- 
 ing its zero exactly in the meridian ; this is effected by a slow- 
 moving screw, the milled head of which is shown at D. This 
 motion should, however, be omitted in instruments destined for 
 exact work, as the bringing the zero into the meridian is not re- 
 quisite, either in astronomy or surveying ; it is in fact purchas- 
 ing a convenience too dearly, by introducing a source of error 
 not always trivial. Above the azimuth circle, and concentric 
 with it, is placed a strong circular plate, E, which carries the 
 whole of the upper works, and also a pointer, to show the de- 
 gree and nearest five minutes to be read off on the azimuth 
 circle ; the remaining minutes and seconds being obtained by 
 means of the two reading microscopes, F and G : this plate, by 
 means of the conical part B, (which is carefully fitted to the axis) 
 rests on the axis, and moves concentrically with it. The conical 
 pillars H H, support the horizontal or transit axis, I, which be- 
 ing longer than the distance between the centres of the pillars, 
 the projecting pieces, c c, fixed to their top, are required to carry 
 out the Y's, a a, to the proper distance, for the reception of the 
 pivots of the axis ; the Y's are capable of being raised or low- 
 ered in their sockets by means of the milled-headed screws, b b, 
 for a purpose hereafter to be explained. The weight of the axis, 
 with the load it carries, is prevented from pressing too heavily on 
 its bearings, by two friction rollers on which it rests, one of which 
 is shown at e. A spiral spring, fixed in the body of each pillar, 
 presses the rollers upwards, with a force nearly a counterpoise to 
 the superincumbent weight ; the rollers on receiving the axis 
 yield to the pressure, and allow the pivots to find their proper 
 bearings in the Y's, relieving them, however, from a great por- 
 tion of the weight. 
 
 The telescope, K, is connected with the horizontal axis, in a 
 manner similar to that of the portable transit-instrument. Upon 
 the axis as a centre, is fixed the double circle J J, each circle 
 being close against the telescope, and on each side of it. The 
 circles are fastened together by small brass pillars ; by this circle 
 the vertical angles are measured, and the graduations are cut on 
 a narrow ring of silver, inlaid on one of the sides, which is 
 usually termed the face of the instrument : a distinction essential 
 in making observations. The clamp for fixing, and the tangent- 
 screw for giving a slow motion to the vertical circle, are placed 
 beneath it, between the pillars, H H, and attached to them, as 
 shown at L. A similar contrivance for the azimuth circle is re- 
 presented at M. The reading microscopes for the vertical circle,
 
 ASTRONOMICAL INSTRUMENTS. 
 
 are carried by two arms bent upwards near their extremities, 
 and attached towards the top of one of the pillars. The project- 
 ing arms are shown at N, and the microscopes above at O. 
 
 A diaphragm, or pierced plate, is fixed in the principal focus 
 of the telescope, on which are stretched five vertical and five 
 horizontal wires : the intersection of the two centre ones, denot- 
 ing the optical axis of the telescope, is the point with which a 
 terrestrial object is bisected, when observing angles for geodetical 
 purposes. The vertical wires are used for the same purpose as 
 those in the transit telescope, and the horizontal ones for taking 
 altitudes of celestial objects. A micrometer having a movable 
 wire is sometimes attached to the eye-end of the telescope, but 
 it is not generally applied to instruments of portable dimensions. 
 The illumination of the wires at night is by a lamp, supported 
 near the top of one of the pillars, as at d, and placed opposite 
 the end of one of the pivots of the axis, which being perforated, 
 admits the rays of light to the centre of the telescope-tube, 
 where falling on a diagonal reflector, they are reflected to the 
 eye, and illuminate the field of view : the whole of this contriv- 
 ance is precisely similar to that described as belonging to the 
 transit-instrument. 
 
 The vertical circle is usually divided into four quadrants, each 
 numbered, 1, 2, 3, &c., up to 90, and following one another 
 in the same order of succession ; consequently, in one position 
 of the instrument, altitudes are read off, and with the face of the 
 instrument reversed, zenith distances ; and an observation is not 
 to be considered complete, till the object has been observed in 
 both positions. The sum of the two readings will always be 90, 
 if there be no error in the adjustments, in the circle itself, or in 
 the observations. 
 
 It is necessary that the microscopes, O O, and the centre of 
 the circle, should occupy the line of its horizontal diameter; to 
 effect which, the up-and-down motion (before spoken of) by 
 means of the screws, b b, is given to the Y's to raise or lower 
 them, until this adjustment is accomplished. A spirit-level, P, 
 is suspended from the arms which carry the microscopes : this 
 shows when the vertical axis is set perpendicular to the horizon. 
 A scale, usually showing seconds, is placed along the glass-tube 
 of the level, which exhibits the amount, if any, of the inclination 
 of the vertical axis. This should be noticed repeatedly whilst 
 making a series of observations, to ascertain if any change has 
 taken place in the position of the instrument after its adjust- 
 ments have been completed. One of the points of suspension 
 of the level is movable, up or down, by means of the screw, /, 
 for the purpose of adjusting the bubble. A stri ding-level, simi- 
 lar to the one employed for the transit-instrument, and used for 
 a like purpose, rests upon the pivots of the axis. It must be 
 carefully passed between the radial bars of the vertical circle to
 
 THE ALTITUDE AND AZIMUTH INSTRUMENT. 95 
 
 set it up in its place, and must be removed as soon as the opera- 
 tion of levelling the horizontal axis is performed. The whole 
 instrument stands upon three foot-screws, placed at the extremi- 
 ties of the three branches which form the tripod,* and brass cups 
 are placed under the spherical ends of the foot-screws. A stone 
 pedestal, set perfectly steady, is the best support for this as well 
 as the portable transit-instrument. 
 
 Of the Adjustments. 
 
 The first adjustment to be attended to, after setting the instru- 
 ment up in the place where the observations are to be made, is 
 to set the azimuthal or vertical axis truly perpendicular to the 
 horizon : the method of doing this is to turn the instrument 
 about, until the spirit-level, P, is lengthwise in the direction of 
 two of the foot-screws, when by their motion the spirit-bubble 
 must be brought to occupy the middle of the glass-tube, which 
 will be shown by the divisions on the scale attached to the level. 
 Having done this, turn the instrument half round in azimuth, 
 and if the axis is truly perpendicular, the bubble will again 
 settle in the middle of the tube ; but if not, the amount of de- 
 viation will show double the quantity by which the axis deviates 
 from the vertical in the direction of the level ; this error must 
 be corrected, one half by means of the two foot-screws (in ques- 
 tion,) and the other half by raising or lowering the spirit-level 
 itself, which is done by the screw represented at /. The above 
 process of reversion and levelling should be repeated, to ascertain 
 if the adjustment has been correctly performed ; for, as we before 
 observed, when speaking of the transit-instrument, adjustments 
 of every kind can be made perfect only by successive trials and 
 approximations. 
 
 Next turn the instrument round in azimuth a quarter of a 
 circle, so that the level, P, shall be at right angles to its former 
 position ; it will then be over the third foot-screw, which may 
 be turned until the air-bubble is again central, if not already so, 
 and this adjustment will be completed : if delicately performed, 
 
 * The foot-screws are sometimes made in the following ingenious manner, as 
 described by Mr. TROUGHTON, in the Memoirs of the Astronomical Society, 
 vol. i. p. 37. "Each of the three screws is double, that is, a screw within a 
 screw : the exterior one, as usual, has its female in the end of the tripod, and the 
 female of the interior screw is within the exterior ; the interior one is longer than 
 the other, its flat end rests on a small cup on the top of the support, and its milled 
 head is a little above the other. Now by this arrangement we gain three distinct 
 motions : for by turning both screws together, an effect is produced equal to 
 the natural range of the exterior screw ; by turning the interior one alone the 
 effect produced is what is due to this screw ; and by turning the exterior one alone 
 (which may be done, because the friction of the interior screw in the cup is greater 
 than that which exists between the two screws,) an effect is produced equal to the 
 difference of the ranges of the two screws. Thus, were the exterior one to have 
 30 turns in an inch, and the interior 40, the effect last described will be exactly 
 equal to what would be produced by a simple screw of 120 threads in an inch."
 
 9(5 ASTRONOMICAL INSTRUMENTS. 
 
 the air-bubble will steadily remain in the middle of the level 
 during an entire revolution of the instrument in azimuth. These 
 adjustments should be first performed approximately, for if the 
 third foot-screw is much out of the level, it will be impossible 
 to get the other two right. The vertical axis is now adjusted. 
 
 The next adjustment is to set the vertical circle at such a 
 height that its two reading microscopes shall be directed to two 
 opposite points in its horizontal diameter, which is done by 
 raising or lowering the Y's which carry the horizontal axis. 
 
 The next adjustment is the levelling of the horizontal axis by 
 means of the striding-level, the whole of which operation is in 
 all respects the same as that described for levelling the transit 
 axis, to which therefore the reader is referred. After perform- 
 ing this, the preceding adjustment must be examined, as it will 
 probably be deranged. Indeed it is better first to set the axis 
 horizontal, and then, by equally raising or depressing the two 
 ends, to bring the miscroscopes into a diameter, and finally level 
 again. 
 
 The adjustment for the Hue of collimation requires not only 
 that the middle vertical wire shall describe a great circle, but 
 that the middle horizontal wire shall have a definite position with 
 respect to the divisions of the limb. It is usual to rectify the 
 position of one of these at a time, taking the middle vertical wire 
 first.* The error of this wire is ascertained and corrected, pre- 
 cisely in the same manner as that of the transit-instrument ; 
 with this difference, that, instead of taking the axis out of its 
 bearings and turning it end for end, the whole instrument is 
 turned half round in azimuth, which is an equivalent operation. 
 The middle horizontal wire may be adjusted in the following 
 manner: " Point the telescope to a very distant object, bisect it 
 by the middle horizontal wire (near the intersection of the wires,) 
 and read off by the microscopes the apparent zenith distance ; 
 now reverse the instrument in azimuth, and turning the telescope 
 again upon the same object, bisect it as before, and again read 
 off the angle which they show. One of these angles will be an 
 altitude, and the other a zenith distance ;" and, if there is no 
 error, the sum of the two readings will be 90, and half of what 
 it differs from 90 will be the error of collimation, which may be 
 either applied to correct any observation made during its exist- 
 ence, or removed in the following manner. One of the readings 
 being the zenith distance, and the other the altitude of the ob- 
 ject, reduce the zenith distance to an altitude, or vice versa, and 
 take the mean ; it is evident that " the mean of the two will 
 be the true zenith distance or altitude respectively ; and while 
 the telescope bisects the object, the microscopes must be ad- 
 
 * We speak of the middle -wire only, as the side wires are supposed to be fixed 
 parallel to it by tb.2 maker, and cannot be adjusted by the observer.
 
 THE ALTITUDE AND AZIMUTH INSTRUMENT. 97 
 
 justed by their proper screws, so as to show that mean. This 
 process may be repeated for obtaining a greater degree of accu- 
 racy ; but its final determination should be deduced from observ- 
 ations upon many heavenly bodies, and the minute error that 
 may remain unadjusted had better be allowed for." This and 
 the preceding operation may be more conveniently performed 
 by a collimating telescope. 
 
 The adjustment for setting the cross-wires truly vertical, is 
 the same as that described as belonging to the portable transit ; 
 the position of the horizontal wires will then depend on the 
 maker, or the horizontal wire may be put right by making it 
 thread an equatorial star at its transit, when the vertical wires 
 will depend upon the maker. 
 
 In conclusion it may be observed, that during a series of ob- 
 servations, if the instrument should be detected to be a small 
 quantity out of level, (having previously gone through the prin- 
 cipal adjustments,) it may generally be restored by means of the 
 foot-screws only, when they require but a slight touch to eifect 
 it: this is more especially essential when the level of the hori- 
 zontal axis is the one deranged, as correcting it by moving the 
 Y's would derange the adjustment of the vertical circle with 
 regard to its reading microscopes, the construction and adjust- 
 ments of which it will next be necessary to describe. 
 
 The error of the vertical axis is to be detected by the hanging 
 level, and can very readily be allowed for in computing the ob- 
 servation : as a general rule, when great accuracy is required, it 
 is easier and safer to adjust by computation than by mechanical 
 contrivances. 
 
 THE READING MICROSCOPE. 
 
 The divisions on the graduated circle indicate spaces of five 
 minutes each, which are read off along with the degree, by means 
 of an index-pointer. The remaining minutes and seconds, if 
 any, are determined by the reading microscope, as was stated 
 when describing the construction of the circle ; it now remains
 
 t)g ASTRONOMICAL 1XSTKUMKNTS. 
 
 to explain the principal parts of the micrometer, the method of 
 adjusting it, and its application to practice. A A, fig. 1, repre- 
 sents the microscope, passing through a collar or support, B, 
 where it is firmly held by the milled nuts, g g, acting on a screw 
 cut on the tube of the microscope. These nuts also serve for 
 placing the instrument at the proper distance, for distinct vision, 
 from the divisions it is employed to read. In the body of the 
 microscope, at a, the common focus of the object and eye-glasses, 
 are placed two wires, crossing each other diagonally, and they 
 are made to traverse the field of view either up or down, by 
 turning the micrometer-screw, 6, working in the box, c c . Fig. 2 
 shows the field of view, with the magnified divisions on the in- 
 strument, as seen through the microscope. The shaded part 
 represents the diaphragm, with the cross-wires, the angle made 
 by which, may, by turning the micrometer-screw, b, be bisected 
 by any line on the circle in the field of view, as is shown in the 
 figure. On the left hand of the diaphragm appears the comb or 
 scale of minutes, each of the teeth representing one minute. 
 Movable with the wires along the comb is a small index or 
 pointer, e, which in the figure is represented at zero, the centre 
 of the scale, as is shown by its bisecting the small hole at the 
 back of the comb. Now one revolution of the screw, b, moves 
 the wires and the pointer over one tooth of the comb, that is, 
 over a space equal to one minute ; and part of a revolution moves 
 them but a fraction of a minute. To determine this fractional 
 quantity, a large cylindrical head, e e, is attached to the screw, 
 having its edge divided into 60 equal parts, representing seconds, 
 the index being fixed opposite the eye of the observer at /. In 
 reading off an angle by this instrument, observe first the degree 
 and nearest five minutes shown by the pointer on the graduated 
 circle ; then apply to the microscope, and by turning the screw, 
 b, in the order of the numbers upon the head e e, make the 
 nearest division nicely bisect the acute angles formed by the cross- 
 ing of the wires ; the number of teeth the pointer, e, has passed 
 over from zero, to produce such bisection, will be the number of 
 minutes to be added to the degree, &c. read off from the circle ; 
 and lastly, the odd seconds and tenths to be added, are to be 
 taken from the divided head, e e, as shown by the index,/. 
 
 The adjustments of the microscope, consist in making the 
 cross-wires in its focus, and the divisions on the circle, both 
 appear at the same time distinct, and free from parallax ; and 
 also making five revolutions of the screw, exactly measure a 
 five-minute space on the graduated circle. For the former of 
 these adjustments, draw out the eye-piece, d, until distinct vision 
 of the wires is obtained, and observe if the divisions of the in- 
 strument are also well defined, arid whether any motion of the. 
 eye causes the least apparent displacement or parallax of the 
 wires with respect to the graduations. If such a dancing motion
 
 THE ALTITUDE AND AZIMUTH INSTRUMENT. 99 
 
 be found, the microscope must be moved to or from the circle, 
 by turning the nuts, g g, unscrewing one and screwing the other, 
 until the wires and graduations both appear distinct, and no 
 parallax can be detected. 
 
 Next, to examine and adjust the run (as it is termed) of the 
 screw. If the run has been carefully adjusted by the maker, and 
 no alteration made in the body of the microscope, the image of 
 the space between two of the divisions will be exactly equiva- 
 lent to five revolutions of the screw, when the wires and divi- 
 sions are both seen distinctly. Let us, however, suppose that 
 the length of the microscope has been deranged, and that 
 the run is too great ; for example, that the space of 5' on the 
 limb is equal to 5' W by the micrometer, or that the image is 
 too large. Now the magnitude of the image formed by the 
 object-glass of the microscope, depends entirely on the distance 
 of the object-glass from the limb, and is diminished (in the 
 ordinary construction of the microscope) by increasing the dis- 
 tance between the limb and the object-glass, and vice versa. In 
 the case supposed, the image is too large, therefore the object- 
 glass h must be removed further from the limb. Let this be 
 done by turning the screw at h in or towards B. The image 
 now will not be formed at a, as it ought to be, but nearer to B, 
 and distinct vision must be gained by bringing the whole body 
 of the microscope nearer to the limb. In this way, by two or 
 three attempts cautiously conducted, we shall make five revolu- 
 tions of the cross-wires correspond exactly with the image of the 
 space between two divisions ; and for greater accuracy the 5' 
 should be read on each side zero, or 10' on the limb made equal 
 to 10 revolutions of the micrometer. 
 
 The screw, c, gives motion to the comb or scale of minutes ; 
 and the micrometer-head being adjustable by friction, can be 
 made to read either zero, or any required second, when the 
 cross-wires bisect any particular division, by holding fast the 
 milled-head, b, and at the same time turning the divided head, 
 e e, round until its zero, or whatever division you require, coin- 
 cides with the index,/: this, it will readily be perceived, is the 
 means of accomplishing the adjustment spoken of at page 96. 
 
 Use of the Altitude and Azimuth Instrument. 
 
 This is the most generally useful of all instruments for mea- 
 suring angles, being applicable to geoditical as well as astro- 
 nomical purposes. In the hands of the surveyor it becomes a 
 theodolite of rather large dimensions, measuring with great 
 accuracy both vertical and horizontal angles. It does not possess 
 the power of repetition ; but the effect of any error of division 
 on the azimuthal circle may be reduced or destroyed, by measur- 
 
 H 2
 
 10Q ASTRONOMICAL INSTRUMENTS. 
 
 ing the same angle upon different parts of the arc ; thus, After 
 each observation, turn the whole instrument a small quantity on 
 its stand, and adjusting it, again measure the required angle. 
 A fresh set of divisions is thus brought into use at every observ- 
 ation, and the same operation being repeated many times, where 
 great accuracy is required, the mean result may be considered 
 as free from any error that may exist in the graduation. A 
 repeating stand has, of late years, been frequently added to this 
 instrument, and is a most powerful and convenient appendage, 
 when great accuracy is required in the measurement of azimuthal 
 angles. The two opposite micrometers being read off at each 
 observation, will always remove the effect of any error in the 
 centring. The vertical angles should, in all cases, be taken 
 twice, reversing the instrument before taking the second observ- 
 ation, when (as hefore observed) one of the readings will be an 
 altitude, and the other a zenith distance ; the sum of the two 
 readings, therefore, if the observation be made with accuracy, 
 and no error exists in the adjustments or the instrument, will 
 be exactly 90 ; and whatever the sum differs from this quantity 
 is double the error of the instrument in altitude, and half this 
 double error is the correction to be applied + or to either of 
 the separate observations, to obtain the true altitude or zenith 
 distance, + when the sum of the two readings is less than 90, 
 and when greater. 
 
 In applying the instrument to astronomical purposes, it was 
 formerly the custom to clamp it in the direction of the meridian, 
 and after taking an observation, or series of observations, with 
 the face of the instrument one way, to wait till the next night, 
 or till opportunity permitted, and then take a corresponding 
 series of observations of the same objects, with the face of the 
 instrument in a reversed position. But this method being at- 
 tended both with uncertainty and inconvenience, it is now usual 
 to complete at once the set of observations, by taking the alti- 
 tudes in both positions of the instrument as soon as possible 
 after each other. When the meridian altitude is required, 
 several observations may be taken, a short time both before and 
 after the meridional passage ; with the face of the instrument 
 in one direction, and with it reversed, noting the time at each 
 observation ; and if we have the exact time of the object's 
 transit, its hour angle in time, or its distance from the meridian 
 at the moment of each observation, may be deduced. This, 
 with the latitude of the place (approximately known) and the 
 declination of the object, affords data for computing a quantity 
 called the reduction to the meridian, which added to the mean of 
 the observed altitudes, when the object is above the pole, and 
 subtracted when the object is below the pole, will give the 
 meridional altitude of the object, and vice versa for zenith dis- 
 tances. The nearer the observations are taken to the meridian,
 
 THE ALTITUDE AND AZIMUTH INSTRUMENT. 
 
 101 
 
 the less will the results depend upon an accurate noting or know- 
 ledge of the time. 
 
 To compute the Reduction to the Meridian. 
 
 Practical Rule. Take from Table VII. the natural versed 
 sines of the hour-angles, or times of each observation from the 
 time of transit separately, and take their mean ; then to the log. 
 of this mean, add the log. co-sine of the assumed latitude, the 
 co-sine of the declination, the co-secant of the meridian zenith 
 distance, and the constant log. 9.31443; the sum, rejecting the 
 tens from the index, will be the log. of the reduction in seconds 
 of space. 
 
 The meridional zenith distance employed in the computation 
 need only be approximate ; if the latitude of the place and the 
 declination of the body be nearly known, the meridional zenith 
 distance will be equal to the difference between the latitude and 
 the declination, when both are north, or both south, but equal 
 to their sum, when one is north and the other south : and the 
 meridian zenith distance of an object below the pole, is equal 
 to the difference between 180, and the sum of the latitude and 
 declination. 
 
 As an example, we shall take that given in WOODHOUSE'S 
 Astronomy, vol. i. page 4^#, of the star Arcturus, as observed 
 at the Dublin Observatory. 
 
 
 Face 
 of 
 Inst. 
 
 Observed Alt. 
 
 Hour Angle 
 in Time. 
 
 Versed 
 Sine. 
 
 East of meridian -j 
 
 E. 
 
 E. 
 
 56 40 5,2 
 42 22,9 
 
 m s. 
 10 35 3 
 6 35 3 
 
 1067 
 0413 
 
 West of meridian -j 
 
 W. 
 W. 
 
 45 10,0 
 43 23,1 
 
 2 47 7 
 
 7 48 7 
 
 0074 
 0580 
 
 Means 56 42 45,3 
 
 Reduction . . . + 1 52.4 
 
 533,5 
 
 log. = 2-7271 
 
 Refraction 
 
 56 44 37,7 
 37,8 
 
 Latitude 53 23 
 
 Declination ...20 7 
 Mer. Z. D. = 33 16 
 
 cos. 
 
 COS. 
 COS. 
 
 Meridian Alt 56 43 59,9 
 
 Constant log. 
 
 9-7756 
 9-9727 
 0-2608 
 9-3144 
 
 Reduction .. 1 52,4 = 112,4 log. 2.0506 
 
 If the star be supposed known, the meridian altitude thus 
 determined may be employed in correcting an assumed latitude ; 
 or, if the latitude be known, the star's declination may be 
 obtained. 
 
 The latitude of a place is its distance from the equator, north 
 or south, and it is equal to the elevation of the celestial pole
 
 ASTRONOMICAL INSTRUMENTS. 
 
 above the horizon, or to an arc of the meridian contained between 
 the zenith and the celestial equator ; which arc can readily be 
 determined, by observing the greatest or meridional altitude 
 of a celestial object whose declination at the time is known ; for 
 when the declination is greater than the zenith distance, both 
 being of the same denomination, (either both north or both south) 
 the latitude will be equal the declination, minus the zenith 
 distance. When the declination and zenith distance are of con- 
 trary denominations, then the declination plus the zenith distance 
 will be the latitude. And lastly, when the zenith distance is 
 greater than the declination, then the zenith distance minus the 
 declination will be the latitude. And always of the same 
 denomination as the greater of the two. 
 
 Another method of determining the latitude, is by observing 
 the meridional zenith distance of a circumpolar star, both at its 
 upper and lower culmination ; then, computing the refraction for 
 each observation, the co-latitude will be equal to half the sum of 
 the two zenith distances added to half the sum of the two re- 
 fractions. The latitude thus obtained does not depend on a pre- 
 vious knowledge of the declination of the object observed. 
 
 The method of determining the latitude by an observation of 
 the altitude of the pole-star at any time of the day, together 
 with the necessary tables, is given in the Nautical Almanac, (as 
 newly arranged.) 
 
 A very successful and useful application of this instrument, is 
 the determination of time and the direction of the true meridian, 
 by equal altitudes and azimuths ; the method of conducting a 
 series of observations of this kind has been so clearly explained 
 by the late Mr. WOLL ASTON, in his Fasciculus Astronomicus, 
 that we shall at once transcribe it nearly in the author's own 
 words. 
 
 " In the morning, two or three or more hours before noon, let 
 him (the observer) point the telescope toward the sun, and a little 
 above it, and clamping the vertical circle, let him follow the sun 
 till its upper limb just touches the first horizontal wire. Then, 
 noting down the exact second of time, as shown by his chrono- . 
 meter, when that happened, let him follow the sun again till its 
 upper limb just arrives at the second horizontal wire. After 
 setting that down as before, let him prepare for the third or 
 central wire ; by now clamping the instrument in azimuth like- 
 wise, and holding its adjusting screw between his finger and 
 thumb, let him bring the preceding limb of the sun just to touch 
 the third or central perpendicular wire, at the same instant that 
 the upper limb just touches the third or central horizontal one. 
 Noting that instant, and setting it down, let him now read off 
 the azimuth marked on the azimuth circle, and set it down under 
 the other, and then prepare for making the preceding limb to 
 touch the fourth perpendicular wire, at the same instant that the
 
 THE ALTITUDE AND AZIMUTH INSTRUMENT. J(jCJ 
 
 upper limb arrives at the fourth horizontal one ; setting that time 
 down again, and reading off the azimuth again, and setting it 
 down, let him do the same by the fifth wire at each way, and re- 
 cord them as before. He will now find the lower limb of the 
 sun, and its second or following limb, ready for observing in the 
 same way, at the first, second, and third wire, making each per- 
 pendicular wire a tangent to the sun's last limb, at the instant 
 that its lower limb just leaves the correspondent horizontal 
 wire ; and setting down the time, and after reading off the azi- 
 muth, setting that down too under the other. After these, the 
 instrument may be released in azimuth, and the lower limb alone 
 be observed, as it quits the fourth and fifth horizontal wires 
 respectively. 
 
 " As soon as the sun has thus passed all his wires, he should 
 read off at both the microscopes the zenith distance and altitude 
 at which he had clamped the vertical circle ; and if he has a 
 barometer and thermometer, he should set down their station at 
 the same time ; for though he probably will have no occasion to 
 regard the precise altitude at which he made these observations, 
 yet if anything should deprive him of the correspondent ones, 
 he may wish to have it in his power to deduce his time or his 
 azimuth from them, and the reading off the microscopes after all 
 is over is attended with very little trouble. 
 
 " These things will appear at first hurrying, and till a person 
 becomes a little accustomed to it they certainly will be so. But 
 after a little practice there will be found time enough to go 
 through the whole with ease ; for the vertical circle remains 
 clamped the. whole time, and all the six azimuths lie much within 
 the limits of their adjusting screw. 
 
 " The easiest method of keeping so many observations from 
 confusion, is to have a slate, or sheet of paper, ready ruled into 
 five columns, to correspond with the five wires in the telescope 
 as they occur in succession, in which to write down the observa- 
 tion belonging to each wire, whether that be time or azimuth ; 
 for if any cloud or accident should deprive him of any one or 
 more of his observations, he will then at once see afterwards 
 which of them is missing, when he comes to compare the two 
 sets together. 
 
 " Leaving the instrument clamped for altitude, and covered 
 entirely from the sun's rays, he must wait till it is at the same 
 distance from noon in the evening to resume his task. For that, 
 he must hold himself ready against the time comes ; and previous 
 to it, he will do well to re-examine the adjustment of his instru- 
 ment, to be certain that no change has happened in the stand or 
 the central cone, so as to throw its axis out of a perpendicular. 
 Let him then observe the same method in this second set of ob- 
 servations as he did in those of the forenoon, considering those 
 wires as first, at which the sun's limbs touch first, and setting
 
 104 ASTRONOMICAL INSTRUMENTS. 
 
 down the times of their appulse to each respective horizontal 
 wire, and bringing the preceding or subsequent limb to the cor- 
 responding perpendicular one, and reading off the azimuths just 
 as he did before. When all are passed, he may release all the 
 clamps, and replacing his shade, leave the instrument till he has 
 reduced his observations of corresponding altitudes : if he has 
 observed them all, he will have obtained ten pair, and of azi- 
 muths six pair, which he must now select from each other, and 
 properly class them, by taking the last in the morning, in con- 
 junction with the first in the evening, and so on, till each observ- 
 ation is paired with its opposite corresponding one." 
 
 The time of the meridional passage of the sun's centre, as indi- 
 cated by the time-keeper employed, will be very nearly equal to 
 half the sum of the times at which each pair of the observations 
 were made, and would be exactly so if the declination did not 
 change during the interval elapsed ; (similar observations being 
 made upon any star, the result will show the exact sidereal time 
 of its transit.) The correction to be applied to the time of the 
 sun's transit or apparent noon deduced as above, on account of 
 the change of declination, may be computed by the following 
 formula : * 
 
 Make - - _ = A 
 1440 sin. T 
 
 2 
 T 
 
 1440 tan. T 
 
 B 
 
 correction = + A . 5 . tan. L + B . 8 . tan. D 
 iu which L = the latitude of the place (minus when south) 
 
 8 = the double daily variation in the sun's declination 
 
 (deduced from the noon of the preceding day, to 
 
 the noon of the following day ; minus when the 
 
 sun is receding from the north) 
 
 D = the declination, at the time of noon, on the given 
 
 day (minus when south) 
 T = the interval of time between the observations ex- 
 
 pressed in hours. 
 
 Note. B is to be considered plus, when the interval of time is 
 less than 12 hours, otherwise minus. 
 
 Practical Rule. To the constant log. 3-1584 add the log. 
 sine of half the interval of time between the observations reduced 
 
 * Tables of equation of equal altitudes are contained in Mr. BAIJ,Y'S volume 
 of Astronomical Tables and Formula, and in Schumacher's Hiilfstafeln. The log. 
 of double the sun's daily variation in declination, is given in the Berlin Ephemeris 
 as log. p, in the page relating to true noon.
 
 THE ALTITUDE AND AZIMUTH INSTRUMENT. JQ/J 
 
 to space, and subtract their sum from the log. of the whole in- 
 terval, expressed in hours and decimals ; call the remainder A, 
 always minus. 
 
 To the constant log. 3' 1584, add the log. tangent of the above 
 half interval, and subtract their sum from the log. of the whole 
 interval as before, and call the remainder ^,plus, when the inter- 
 val is less than twelve hours. 
 
 To A, add the log. of double the daily variation of the sun's 
 declination, expressed in seconds, (minus when the sun is receding 
 from the north,) and the log. tangent of the latitude, (minus 
 when south,) the natural number corresponding to the sum to 
 be considered as seconds of time, &c., plus or minus as it may 
 result. 
 
 To B, add the log. of double the daily variation of the sun's 
 declination, as before, and the log. tangent of the sun's declina- 
 tion, minus when south, for noon of the given day, the natural 
 number corresponding to the sum, must be taken as seconds of 
 time with its proper sign. The algebraic sum of these two 
 quantities will be the correction required, and must be added 
 to, or subtracted from, the half sum of the times of observation, 
 according as it is plus or minus, to obtain the correct apparent 
 time. 
 
 EXAMPLE. 
 
 (From Mr. BAILY'S Volume of Astronomical Tables, 8fc., p. 227.) 
 
 On July 25, 1823, in N. Lat. 54 20' at 8 h 59 m 4" A.M., and 
 at 3 h O m 40 s P.M. the sun had equal altitudes. Required the 
 equation, or correction to be applied to the mean of those times, 
 in order to find the time of noon. The interval of time is 
 6 h l m 36 s , which converted into arc = 90 24', and by the Nau- 
 tical Almanac the declination of the sun, at noon on that day, 
 was -f 19 48' 29", and its double daily variation equal to 
 - 25' 29" = 1529". The operation will therefore stand 
 thus : 
 Constant log. . = 3*1584 3-1584 
 
 = 45 12' sin. = 9-8510 . . tangent = 0-0030 
 
 Sums . . . = 3-0094 . . '. . . = 3-1614 
 
 T = 6-0266 log. = 0-7801 0-7801 
 
 A = (Differences) = 7-7707 . . . . B = + 7-6187 
 
 8 = 1 529" log. 3-1844 log. 3-1844 
 
 L = +54 20' tan. +0-1441 D = 19 48' tan. + 9'5563 
 
 + 12 S ,57 = + 1-0992 . . 2 8 ,29 = 0-3594 
 correction = + 12 8 ,57 2",29 = + 10 S ,28
 
 10(5 ASTRONOMICAL INSTRUMENTS. 
 
 This value being added to the mean of the times of the observed 
 altitudes, or \ (20 h 59 m 4 s + 27 h O ra 40 s ) = 23 h 5<J m 52 s , will 
 give O h O m ^",28 for the time at apparent noon, to which, if the 
 equation of time be applied, the result will be the time of mean 
 noon. 
 
 The equal azimuths may similarly be employed for finding the 
 direction of the true meridian. They must be opposed to each 
 other in pairs, just in the same manner as corresponding alti- 
 tudes ; the first in the morning to the last in the evening, and 
 so of the rest. Then deducting the one from the other, and 
 applying half the difference between the two to the smallest 
 number in each pair, it will give a number of degrees, minutes, 
 and seconds, in which, if all the observations were perfect, the 
 whole six pair would coincide ; and if they do not, the fair mean 
 deduced from among them will approach nearly to the truth, i. e., 
 the error of 180 on the azimuth circle from the true meridian. 
 
 To that mean point, deduced from these observations, the 
 instrument must now be turned, and fixed there till the proper 
 correction can be applied to it. Upon the telescope being turned 
 down to the horizon each way, it may be observed what distinct 
 object there may be, either to the north or south, that coincides 
 with one of the perpendicular wires ; or if no such object^should 
 occur, a mark may be placed each way, or either way, to which 
 the instrument may be kept, till the correction can be investi- 
 gated, which is requisite, on account of the change of the sun's 
 declination during the interval between the morning and evening 
 observations ; for any alteration in his declination will affect the 
 azimuth deduced in this way, as it does the hour. >/Phis correc- 
 tion is greatest about the time of the equinoxes, as the change 
 in the sun's declination is then the most rapid : it may be com- 
 puted from the following formula ; but when deduced from a 
 star, no such correction is requisite. 
 
 (T' _ T) 
 Correction = | (D' D) sec. Lat. cosec.- - ' 
 
 In which expression, (D' D) = the change in the sun's de- 
 clination during the interval between the observations, and 
 (T T) = the interval itself. 
 
 Practical Rule. To the log. of half the change of declination, 
 add the log. secant of the latitude, and the log. co-secant of half 
 the interval of time converted into space : the sum 20 will be 
 the log. of the correction in seconds of space. 
 
 When the sun is advancing towards the elevated pole, the 
 middle point, or meridian, as found by equal altitudes, will be too 
 much to the west of the true meridian, by the amount of this 
 correction, and vice versa, when he is receding from the elevated 
 pole ; therefore, the telescope being shifted in azimuth by the 
 quantity thus computed, will be correctly in the meridian.
 
 THE ALTITUDE AND AZIMUTH INSTRUMENT. 1Q7 
 
 EXAMPLE. 
 
 On February 28, 1834. When the sun had equal altitudes, 
 the azimuth circle read 130 10' 15" and 32 36' 15", therefore 
 the middle point or reading of the approximate meridian was 
 81 23' 15". The interval of time between the observations was 
 5 hours, the half of which converted into arc = 37 30' 0". The 
 sun's hourly change of declination = 56",77j therefore the change 
 for half the interval = 141 ",92 (approaching the north pole.) 
 The latitude of the place 51 28' 39", required the correction to 
 be applied to the middle point to obtain the direction of the 
 true meridian. 
 
 i(D' D)= 141",92 log. = 2-1520436 
 
 Lat. = 5128'39" sec. = 0'2056388 
 
 ) = 3730',0 co-sec. = 0-2155529 
 
 Correction = 374",31 log. = 2*5732353 
 
 = 6'14",S1 , 
 
 Reading of the middle point = 81 23 15 
 
 Correction 6 14,31 
 
 Reading of the instrument when set ~1 . _ o i 17 o P9 
 to the true meridian J 
 
 Another, and an easy method of finding a meridian line, where 
 dependence can be placed upon the time shown by a chronome- 
 ter (or watch) is to compute the time of the meridional passage 
 of a star near the pole, either above or below the pole, and point- 
 ing the telescope of the instrument to the star, bisect it at the 
 exact moment ; when, if the adjustments of the instrument are 
 perfect, the telescope will be very nearly in the plane of the 
 meridian. 
 
 A third method, which admits of great accuracy, when instru- 
 ments of large dimensions are employed, consists in bisecting a 
 circumpolar star when at its greatest eastern and western elong- 
 ations ; a line bisecting the horizontal interval, contained be- 
 tween the two positions of the telescope, will be the direction of 
 the meridian ; this interval being measured on the azimuthal 
 circle, and the telescope moved through half that interval, from 
 either its eastern or western position, will place it in the meridian. 
 But it will often be inconvenient to wait till the star attains its 
 second greatest elongation ; and as one of the observations must 
 be made in the day-time, (except at particular seasons of the 
 year,) a star will not be visible through telescopes of small size. 
 To make a single observation available for the purpose, the azi-
 
 ASTRONOMICAL INSTRUMENTS. 
 
 muth (east or west) of a star, when at its greatest elongation, as 
 well as the time of its attaining such position, must be computed 
 (which may be done by the annexed rules), when the observer 
 must first bisect the star, and follow it in its slow motion, until 
 he is satisfied that it is stationary ; or, what is perhaps better, 
 if he is certain of his time, bisect it at the exact moment. The 
 azimuth circle must now be read off, and the position of some 
 fixed object, with respect to the azimuth of the star, should be 
 determined ; a lamp may at the time be placed at some distance 
 for reference, and its azimuth being thus obtained, other objects 
 may be referred to it at leisure. 
 
 To compute the azimuth of a circumpolar star, when at its 
 greatest elongation. 
 
 Rule. From the log. sine of the polar distance, subtract the 
 log. co-sine of the latitude : the remainder will be the log. 
 sine of the azimuth required. 
 
 To compute the time (before or after its meridional passage) of 
 a circumpolar star attaining its greatest elongation, either 
 east or west. 
 
 Rule. Add together the log. tangent of the polar distance, 
 and the log. tangent of the latitude: their sum, rejecting ten 
 from the index, will be the log. co-sine of the hour-angle (in- 
 space) ; which, divided by fifteen, will be the sidereal time a star 
 attains its greatest elongation before or after it passes the meri- 
 dian at its upper culmination ; therefore, having the time of the 
 meridional passage (computed, as explained at page 82} the time 
 of its greatest elongation will be known. 
 
 The star a Ursse Minoris, commonly known as the pole-star, 
 is well situated for determining the direction of the meridian by 
 the above method : its apparent motion when near its greatest 
 elongation is so small, that it appears stationary at that point for 
 a considerable time, affording us an opportunity of observing it 
 both by direct vision, and also by reflection ; an advantage parti- 
 cularly great, as we need not depend upon the spirit- bubble in 
 levelling the instrument, for the observations expose the slightest 
 deviation, and enables us to correct its position ; thus : 
 
 Suppose the pole-star, by previous calculation, is ascertained 
 to be at its greatest elongation at a certain time ; having set up 
 the instrument approximately level, place an artificial horizon 
 in a proper position to observe the star by reflection : then direct 
 the telescope to the star, and having bisected it with the inter- 
 section of the cross-wires, clamp the horizontal circle ; now de- 
 press the telescope till you see the reflected image of the star in 
 the artificial horizon, which if the instrument is perfectly level, 
 will also appear bisected ; if it does not, you must immediately 
 correct half the deviation by the foot-screws of the instrument, 
 which will set the instrument perfectly level. Bisect the reflected 
 image of the star by giving motion to the horizontal circle ; then
 
 THE ALTITUDE AND AZIMUTH INSTRUMENT. JQ^ 
 
 carefully elevate the telescope to the star itself, which will also 
 be bisected if the estimation of half the amount of deviation 
 has been correctly made, and if not, it will be a nearer approxi- 
 mation, which must be perfected by a similar process, that is, 
 by removing half the error with the foot-screws, and the other 
 half by the horizontal circle. 
 
 Having now set the instrument, so that, upon elevating and 
 depressing the telescope, both the direct and reflected images of 
 the star appear bisected, a satisfactory observation will have 
 been made. This being done at both the eastern and western 
 elongations, and the readings of the azimuth circle noted, the 
 middle point between the two readings will lie in the plane of 
 the true meridian. Or, as before observed, one observation may 
 be made available for the same purpose, by likewise observing a 
 fixed object, as a lamp or church tower, and computing, by the 
 foregoing rule, the azimuth of the star at that time, for the hori- 
 zontal angle between the star and the fixed object, plus or minus 
 the computed azimuth of the star (plus when the object is on 
 the same side of the meridian, and further from it than the 
 star ; and minus, when it is nearer the meridian than the star, 
 or when they are on opposite sides of the meridian) will give 
 the azimuth of the fixed object from the north, from which the 
 direction of the meridian may be found at any time. 
 
 It is only stars whose polar distance is less than the co-latitude 
 of the place of observation, that can be used in the two latter 
 methods of determining the direction of the meridian. 
 
 The last method which we shall advert to, and which is mostly 
 applied to objects south of the zenith, consists in computing the 
 azimuth of a celestial object, from an observation of its altitude, 
 the latitude being known ; at the same time observing the hori- 
 zontal angle contained between it and any fixed object; for the 
 difference or sum of the azimuth of the celestial body, and this 
 observed horizontal angle, will be the angular distance of the 
 fixed object from the meridian : the sum when the fixed object 
 is on the same side of, and further from, the meridian than the 
 celestial object, otherwise, the difference. 
 
 Formula for computing the azimuth of a celestial object from 
 its observed altitude, &c. 
 
 s . s 
 
 sin. z. sin. - A 
 
 Tang, i azimuth = 
 
 In which _= half the sum of the polar distance, the co-latitude 
 2 
 
 and the zenith distance, ir and A. = the polar distance and co- 
 latitude Z = the zenith distance of the object.
 
 ASTRONOMICAL INSTRUMENTS. 
 
 Practical Rule. Add together the polar distance, the co- 
 latitude, and the zenith distance, and call their sum S. To the 
 log. sine of half S minus the zenith distance, add the log. sine 
 of half S minus the co-latitude, and increasing the index by 20, 
 call the sum of the two logs. A. 
 
 To the log. sine of half S add the log. sine of half S minus the 
 polar distance, and call the sum of the two logs. B. 
 
 From A subtract B, and divide the remainder by 2 ; the quo- 
 tient will be the log. tangent of half the object's azimuth, which 
 doubled will be the whole azimuth, or horizontal angular distance 
 from the south meridian. 
 
 EXAMPLE. 
 
 On February 20, 1834, in latitude 51 28' 39". The zenith 
 distance of a Geminorum (east of the meridian) corrected for re- 
 fraction 56 20' 10", the azimuth circle reading 125 18' 24": 
 after which the clamps of the instrument were released, and a 
 fixed terrestrial object bisected, also to the east, but nearer the 
 meridian than the star, the azimuth circle now read 83 15' 20", 
 consequently the horizontal angle between the star and the 
 object = 42 3' 4" required the azimuth of the object from the 
 meridian. 
 
 77 (from the N. A.) = 57 45 16 
 
 A = 38 31 21 
 
 z = 56 20 10 
 
 2) 152 36 47 
 
 = 76 18 23 
 
 2 _ 
 
 ~ z , = 19 58 13 sine = 9-5334322 
 2 
 
 L X . = 37 47 2 sine = 9-7872371 
 
 A = 9-3206693 
 = 76 18 23 sine = 9*9874766 
 
 18 33 7 sine = 9-5026514 
 
 B = 9-4901280 
 A 9-3206693 
 
 2) 9-8305413 = A B
 
 THE ALTITUDE AND AZIMUTH INSTRUMENT. 
 
 39 26 45-5 ^ star's azimuth, tan. = 9'91 52706 
 9. 
 
 78 53 31-0 = azimuth of star 
 
 4'2 3 4*0 object nearer meridian 
 
 36 50 27'0 = azimuth of object east of south. 
 
 The verification of the meridional position of an instrument by 
 observing the passage of a circumpolar star at both its upper 
 and lower culminations, as well as the method by high and low 
 stars, has been fully explained, when speaking of the transit ; 
 and as the altitude and azimuth circle, when firmly clamped in 
 the plane of the meridian, becomes a complete transit instrument, 
 and may be employed precisely in the same manner and for the 
 same purposes, we refer for this use of it to the account which 
 we have given of that instrument. 
 
 In addition to the method of determining differences of longi- 
 tude by the observed transits of the moon and moon-culminating 
 stars, (page 88,) we subjoin the following as applicable to the 
 use of the instrument which we are now describing. The latitudes 
 and longitudes of a great number of the most conspicuous places 
 in this country, as church steeples, &c., having been determined, 
 and published in the account of the Ordnance Survey, they 
 afford a ready means of finding both the latitude and longitude 
 of places adjacent to them, by means of trigonometrical mea- 
 surement. The process may be understood from the following 
 example. 
 
 Let A represent a place, the longitude and 
 latitude of which are known ; B the station, the If 
 
 situation of which we wish to determine ; C any 
 point to form the triangle ; N S the direction of 
 the meridian. 
 
 First, the angles at the three points must be 
 observed, and one of the sides measured, when 
 the distance A B must be computed by plane 
 trigonometry. Suppose it to be = 6040'6 feet. 
 Then the azimuth of A, from the meridian, or 
 the angle, A B N, must be determined, which 
 may be done by any of the methods we have 
 described ; suppose it = 56 58' 40"; now the 
 line A D perpendicular to the meridian, and 
 B D the difference of latitude of B and A, may be computed 
 from the right-angled triangle A B D, having A B 6040'6 feet, 
 and the angle A B D = 56 58' 40" ; A D comes out = 5064'8 
 feet, and B D == 3292'2 feet. 
 
 With the latitude of A, which suppose = 51 21' 44", enter 
 Table VIII. and take out the length of a second, both of latitude
 
 ASTRONOMICAL INSTRUMENTS. 
 
 and longitude; divide the distances AD andBD by those num- 
 bers, and the quotients will be the difference of longitude and 
 latitude (in arc) required. Thus : 
 
 A D =5064- 8 
 
 T hi 63.31 = 80,44 = 1 20,00 diff. of long, in space. 
 
 = 5 8 ,33 in time. 
 
 B D - 3292*2 
 
 ,p , , "IAO.AO 32,27 difference of latitude. 
 
 M. s. 
 
 Longitude of A = 21 10,40 West. 
 Difference 5,33 East. 
 
 Longitude of B = 21 5,07 West. 
 
 Latitude of A =51 27 44,00 North. 
 Difference 32,27 South. 
 
 Latitude of B =51 27 11,73 North. 
 
 Lastly, we shall give the method of finding the longitude by 
 observations of the eclipses of Jupiter's satellites. 
 
 The Nautical Almanac contains the Greenwich mean time when 
 the phenomena happen, consequently the estimated longitude of 
 the place being applied to the time therein given, will be the time 
 at which an eclipse may be expected to happen at the station 
 of the observer, who, being at his telescope a few minutes before, 
 should steadily watch the spot near the body of the planet where 
 the phenomenon is expected, till he discovers the first glimpse or 
 point of light appear, if it be an emersion from the shadow, or 
 of the final disappearance, if an immersion; noting, by a pre- 
 viously regulated time-keeper, the exact mean time (at his own 
 station) when this happens. The difference between this time 
 and the Greenwich time given in the Nautical Almanac, will be 
 the longitude in time ; east, if the Greenwich time is less than 
 that observed, otherwise west. Before the opposition of the 
 planet to the sun, the eclipses always happen on the west side 
 of the planet, and afterwards on the east. But when using an 
 inverting telescope, the appearance will be reversed. The situa- 
 tion of the satellite with respect to the planet where the eclipse 
 takes place, is given in the Nautical Almanac.
 
 113 
 
 APPENDIX. 
 
 ON PROTRACTING AND PLOTTING, &c., AND THE 
 INSTRUMENTS EMPLOYED. 
 
 IN the execution of extensive surveys upon scientific princi- 
 ples the accurate measurement of angles is of the utmost import- 
 ance, requiring the employment of instruments of a superior 
 construction, as well as considerable care and skill in their 
 management. And one great object of such surveys being the 
 formation of correct maps and charts, it is no less essential, that 
 the angles, when accurately measured, should be accurately laid 
 down. We therefore purpose to describe briefly, in this Ap- 
 pendix, the most approved methods of laying down angles, &c., 
 as supplementary to our account of surveying instruments. 
 
 Extensive surveys are best performed, by extending a series 
 of triangles over the country to be delineated ; and from the 
 length of a side of one triangle measured or otherwise deter- 
 mined, as a base, and the angles found by means of appropriate 
 instruments, the lengths of the various lines forming the sides of 
 the several triangles throughout the series are computed. The 
 accuracy of the distances thus obtained, will depend on the cor- 
 rect measurement of the angles ; and the distance assumed as a 
 base, provided due attention be paid in the first instance to the 
 judicious dispositions of the triangles, which ought to be as 
 nearly equilateral as circumstances will admit. The accurate 
 protracting of the triangles thus determined, is of the next im- 
 portance. They can be more correctly laid down by means of 
 their sides than by their angles ; and one side only, for measures 
 of length, can be taken from a scale and transferred to paper, 
 with more exactness than an angle can be pricked off* from a 
 protractor. But it being in most cases requisite, in plotting a 
 survey, to show the direction of the meridian with regard to the 
 triangulation, it becomes necessary to lay down, from one of the 
 principal stations, the azimuthal angle subtended by some other 
 (remote) station and the meridian : now this angle cannot be laid 
 off from a protractor, even of the most approved construction, 
 so accurately as the plotting of the triangulation may be made 
 from the measured or computed sides of the triangles. To ob-
 
 H4 APPENDIX. 
 
 tain a corresponding degree of exactness, recourse must be had to 
 some other method, and the following is the best that we have 
 seen practised. 
 
 Let A and B represent 
 two stations of a trigono- 
 metrical survey ; and let it 
 be required to lay off the 
 direction of the true meri- 
 dian, N S, with regard to 
 the line A B, the azimuth 
 of which, west of north, be- 
 ing 40 30' 30". Take from 
 an accurately divided dia- 
 gonal scale, exactly five 
 inches as a radius, and 
 from A, as a centre, de- 
 scribe an arc C D ; now the 
 chord of an arc being equal 
 to twice the sine of half 
 that arc, it follows that the 
 chord C D is equal to twice 
 the natural sine of half the 
 angle C A D or B AD, viz., 20 15' 15" ; but the radius of the 
 tables of natural sines being =1 or 10, and taking but the half 
 of 10, or 5 inches for our radius, we must take from the table 
 the natural sine of half the angle BAD, which will, to radius 5, 
 be equal to C D, the chord of the whole angle ; and having taken 
 that distance from the same scale of inches as the radius, place one 
 foot in the point C, and with the other mark the point D on the 
 arc CD, then through D and A draw the line N S, which will 
 represent the meridian. But instead of employing a pair of com- 
 passes and a scale for this purpose, it is better to use a beam- 
 compass, graduated to inches, and having a vernier for minute 
 subdivision, as a measure of length can be taken by its means 
 with greater exactness, than by a pair of compasses from a 
 scale. 
 
 This method of laying off angles may be conveniently em- 
 ployed in dividing a circle to be used as a protractor, when the 
 work is to be laid down to a scale not exceeding six inches to 
 a mile. The protractor may be made either on the same sheet 
 of paper intended to receive the drawing, or on a separate sheet 
 of card-board, when it may be preserved and used on after occa- 
 sions. During the time which must necessarily be occupied in 
 plotting an extensive and minute survey, the paper which re- 
 ceives the work is often sensibly affected by the changes which 
 take place in the hygrometrical state of the air, causing much 
 annoyance to the draftsman, as the parts laid down from the 
 same scale at different times will not exactly correspond. To
 
 APPENDIX. 
 
 remedy in some measure this inconvenience, it has been recom- 
 mended that the apartments appropriated to the purposes of 
 drawing should be constantly kept in as nearly the same tem- 
 perature as possible, and also that the intended scale of the 
 plan should be first accurately laid down upon the paper itself; 
 and from this scale all dimensions for the work should invari- 
 ably be taken, as the scale would always be in the same state 
 of expansion as the plot, though it may no longer retain its 
 original dimensions. The protractor may also be laid down 
 upon the paper ; and when a great many angles are to be 
 plotted, as in a road or town survey, made with a theodolite 
 and chain, especially if done by traversing, or what is fre- 
 quently called surveying by the back angle, this kind of pro- 
 tractor will enable the draftsman to plot the work with great 
 rapidity, and with less chance of error, when the scale is small, 
 than by the method of laying off angles by placing the centre of 
 a metallic protractor at every angular point, and pricking off 
 the angle from its circular edge. The application of the theo- 
 dolite to surveying by a traverse, as well as the method of 
 protraction, we shall endeavour to explain by means of an 
 example. 
 
 .1ST 
 
 A 
 
 Let the above plan represent a survey of roads to be performed 
 with a theodolite and chain. Commencing on a conspicuous 
 spot, , near the place at which two roads meet, the theodolite 
 must be set up and levelled, the upper and lower horizontal 
 plates clamped at zero, and the whole instrument turned^ about 
 until the magnetic needle steadily points to the N S line of the 
 compass-box, and then fixed in that position by tightening the 
 clamp-screw, H, (see page 15.) Now release the upper plate, 
 and direct the telescope to any distant conspicuous object within 
 or near the limits of the survey, such as a pole purposely erected 
 in an accessible situation, that it may be measured to, and the 
 instrument placed upon, the same spot at a subsequent part of 
 the operation, as A and B, and after bisecting it with the cross 
 wires, read both the verniers of the horizontal limb, and enter
 
 H(J APPENDIX. 
 
 the two readings in the field-book ; likewise in the same manner 
 take bearings, or angles, to all such remarkable objects as are 
 likely to be seen from other stations, as the tree situated on a 
 hill ; and lastly, take the angle to your forward station, b, where 
 an assistant must hold a staff for the purpose, on a picket driven 
 into the ground,* in such a situation as will enable you to take 
 the longest possible sight down each of the roads that meet 
 there. In going through the above process, at this and every 
 subsequent station, great caution must be used to prevent the 
 lower horizontal plate from having the least motion after being 
 clamped in its position by the screw H. 
 
 Next measure the distance from a to b, and set up the instru- 
 ment at b, release the clamp-screw H only, not suffering the 
 upper plate to be in the least disturbed from the reading it had 
 when directed at a to the forward station b, with the instrument 
 reading this forward angle ; turn it bodily round, till the tele- 
 scope is directed to the station a (which is now the back station) 
 where an assistant must hold a staff; tighten the clamp-screw H, 
 and by the slow-motion screw, I, (page 15) bisect the staff as near 
 the ground as possible, and having examined the reading, to see 
 that no disturbance has taken place, release the upper plate, and 
 setting it to zero, see if the magnetic needle coincides, as in the 
 first instance, with the N S line of the compass-box ; if it does, 
 all is right ; if not, an error must have been committed in taking 
 the last forward angle, or else the upper plate must have moved 
 from its position before the back station had been bisected : when 
 this is the case it is necessary to return and examine the work at 
 the last station. If this is done every time the instrument is set 
 up, a constant check is kept upon the progress of the work ; and 
 this indeed is the most important use of the compass. Having 
 thus proved the accuracy of the last forward angle, release the 
 upper plate, and measure the angles to the stations m and r, and 
 as before, to whatever objects you may consider will be conspi- 
 cuous from other places ; and lastly, observe the forward angle to 
 the station c, where the theodolite must next be set up, and mea- 
 sure the distance b c. 
 
 At c, and at every succeeding station, a similar operation must 
 be performed, bisecting the back station with the instrument 
 reading the last forward angle ; then take bearings to every con- 
 spicuous object, as the tree on the hill, the station, A, &c., which 
 will fix their relative situations on the plan, and they afterwards 
 serve as fixed points to prove the accuracy of the position of 
 such other stations as may have bearings taken from them to the 
 same object ; for, if the relative situations of such stations are 
 
 * A picket should always be left in the ground at every station, in order to 
 recognise the precise spot, should it afterwards be found necessary to return to it 
 again.
 
 APPENDIX. H7 
 
 not correctly determined, these bearings will not all intersect in 
 the same point on the plan. The last operation at each station 
 is to measure the forward angle. In this manner proceed to the 
 stations d, e, f, g, Bcc., and having arrived at g, measure an angle 
 to the pole, A, as to a forward station, and placing the theodo- 
 lite upon that spot, direct the telescope to g, as a back station, 
 in the usual way ; this done, release the upper plate, and direct 
 the telescope to the first station a, from which A had been ob- 
 served, and if all the intervening angles have been correctly 
 taken, the reading of the two verniers will be precisely the same 
 as when directed to A from the station a : this is called closing 
 the work, and is a test of its accuracy so far as the angles are 
 concerned, independent of the compass-needle. If the relative 
 situation of the conspicuous points, A B, &c., were previously 
 fixed by triangulation, there would be no necessity to have re- 
 course to the magnetic meridian at all, as a line connecting the 
 starting point a with any visible Jlxed object, may be assumed as 
 a working meridian, and if it be thought necessary, the reading 
 of the compass-needle may be noted at a, when such fixed object 
 is bisected, and upon the theodolite being set to the reading of 
 this assumed meridian, at any subsequent station, the compass- 
 needle will also point to the same reading as it did at first, if 
 the work is all correct, and no local attraction influences the 
 compass. 
 
 While the instrument is at A, take angles to all the conspi- 
 cuous objects, particularly to such as you may hereafter be able 
 to close upon, which will (as in the above instance) verify the 
 accuracy of the intervening observations ; having done this, re- 
 turn to g and /, &c. and proceed with the survey in the same 
 manner as before, setting the instrument up at each bend in the 
 road, and taking offsets to the right and left of the station lines ; 
 arriving at i, survey up to, and close upon B ; then return to i, 
 and proceed from station to station till you arrive at m, where, if 
 the whole work is accurate, the forward angle taken to b will be 
 the same as was formerly taken from b to m, which will finish 
 the operation. 
 
 The next step is to lay down the lines and angles thus sur- 
 veyed; and first, the protractor must be constructed. The great 
 difficulty of dividing a circle accurately is well known, but if the 
 arcs are laid off by means of their chords, the division may be 
 performed with sufficient exactness for the purpose in handv 
 The length of the chords should be taken from an accurately 
 divided beam-compass, which, to insure success, should be set 
 with the utmost possible exactness. 
 
 With a radius of five inches describe a circle, and immedi- 
 ately, without altering the compasses, step round the circle> 
 making a fine but distinct mark at each step : this will divide 
 the circle into six parts of 60 each.
 
 APPENDIX. 
 
 Next set the compasses to the natural sine of 15, which to 
 radius jive, will be equal to the chord of 30, and this distance 
 will bisect each 60 and divide the circle into arcs of 30 each. 
 A proof may be obtained of the accuracy of the work as it 
 proceeds, by setting the succeeding chords off each way, from 
 those points which they are intended to bisect ; for if any inac- 
 curacy exists, the bisection will not be perfect, and if the error 
 proves inconsiderable the middle point may be assumed as 
 correct. 
 
 Each sixty degrees may next be trisected, by setting off the 
 natural sine of 10 (equal the chord of 20 to our radius) which 
 will divide the circle to every ten degrees. 
 
 Next the natural sine of 7 30' (equal the chord of 15) 
 stepped from the points already determined, will divide the circle 
 to every fifth degree. 
 
 The natural sine of 3 (equal the chord of 6) being laid off, 
 divides 30 into five parts, and set off from the other divisions, 
 divides the circle to single degrees. 
 
 Fifteen degrees bisected, or the natural sine, 3 45' (equal 
 the chord of 7 SO'} set off from the other divisions, divides the 
 circle into half degrees. 
 
 The natural sine of 3 20' (equal the chord of 6 40') divides 
 20 into three parts, and set off from the rest of the divisions, 
 divides the whole circle to every ten minutes, which is as minute 
 a subdivision as such a circle will possibly admit of; smaller 
 quantities must therefore be estimated by the eye. The divi- 
 sions should be numbered from 0, 10, 20, &c. quite round the 
 circle to 360, the same as the theodolite, which the protractor 
 represents. 
 
 It may be considered troublesome to lay down a protractor of 
 this kind upon every sheet of paper to be plotted on, but having 
 done one, several copies may be obtained from it, by pricking 
 through the divisional points upon paper placed under it for the 
 purpose. Or, if made upon a sheet of card-board, the paper 
 within the graduated circle must be cut out, as the work is plot- 
 ted within the circle forming the protractor. 
 
 Suppose, with a protractor of the latter kind, we proceed to 
 lay down the work of our survey. First draw a line through 
 the assumed starting point, a, across the paper, to represent the 
 magnetic meridian ; or, if the points, A B, &c. have been fixed 
 by previous triangulation, they should be laid down and a line 
 drawn through a, and any one of them (which has been observed 
 from a) may be assumed as a working meridian ; then across the 
 protractor draw a line, through the same divisions that were 
 noted on the theodolite for the reading of the meridian, which 
 in our example was zero, or the divisions marked 180 and 360 
 on the protractor. 
 
 Place the protractor upon the paper, so that the line drawn on
 
 APPENDIX. 
 
 119 
 
 the former shall coincide with the meridian-line drawn upon the 
 latter, and to prevent it shifting, lay weights on its corners. 
 Place the edge of a large parallel ruler on the divisions which 
 were read off for the forward angle to b, and slide the ruler 
 parallel to itself till its edge passes through the station, a, and 
 draw a line from a in the direction, a 6, then with a pair of 
 compasses, and from the scale of the plot, lay off the distance, 
 a b, which will determine the point b. Next place the edge 
 of the ruler on the angles taken at 6, to the stations r and c 
 respectively, and slide it parallel to itself, till its edge passes 
 through b, then draw the lines, b r, and b c, and lay off those 
 distances from the scale of the plot, and the stations r and c will 
 be fixed. Next set the ruler to the forward angle taken at c to 
 the station, d, and move it till its edge passes through c, and 
 draw the line, c d ; lay off the distance, c d, and the station d 
 will be determined. In like manner proceed with the remaining 
 stations of the survey, until you come to the point, m, when, if 
 the lines have been correctly measured and protracted, the for- 
 ward angle will pass through the station, &, and the distance 
 exactly correspond. If the lines have been measured on very 
 uneven ground, each of them must be reduced to the horizontal 
 measure, which may be done at the time of measuring them, by 
 the vertical arc of the theodolite, (see pages 2 and 17.) 
 
 The bearings taken at different stations to various conspicuous 
 objects, are to be laid down as the plotting of the forward angles 
 proceed, for when several bearings have been taken to the same 
 object, the crossing of such lines in the same point is a proof of 
 the relative accuracy of the work, and if these objects have been 
 independently fixed and laid down by triangulation, the bearings 
 will then prove the accuracy of the work with respect to these 
 fixed points. 
 
 We have remarked that the plotting must be performed 
 within the circle forming the protractor, which direction is to be 
 understood as applicable only when the protractor is not on the 
 same paper with the plot, for when it is on the same paper, the 
 angles may be transferred by the parallel ruler to any part of the 
 sheet ; but care must be taken in numbering the divisions of the 
 protractor, so that the working meridian may be in the best direc- 
 tion for getting into the sheet the greatest portion of the survey, 
 if the protractor is on a separate sheet, arid the work has pro- 
 ceeded to its edge, it must be shifted on the paper in the direc- 
 tion of the survey, but must be moved exactly parallel to itself, 
 which may easily be done by drawing more meridian lines 
 parallel to the first meridian, on which to place the protractor, 
 as in the first instance. 
 
 When a survey is to be plotted upon a very large scale, it is 
 necessary, to insure the greatest accuracy in laying down the 
 angles, to protract them by their chords, or by means of a cir-
 
 APPENDIX. 
 
 120 
 
 cular metallic protractor; as the kind of protractor we have just 
 been describing would not answer the purpose, its chief use being, 
 as has been already described, to plot a traverse upon a moder- 
 ately small scale. There are several constructions of the pro- 
 tractor adapted to the purpose now under consideration, but the 
 most approved is represented in the subjoined figure. It consists 
 of an entire circle, A A, connected with its centre by four radial 
 bars, a a, &c. The centre of the metal is removed, and a cir- 
 cular disk of glass fixed in its place, on which are drawn 
 two lines crossing each other at right angles, and dividing the 
 small circle into four quadrants, the intersection of the lines 
 denoting the centre of the protractor. When the instrument is 
 used for laying down an angle, the protractor must be so placed 
 on the paper, that its centre exactly coincides with, or covers the 
 angular point, which may easily be done, as the paper can be 
 seen through the glass centre-piece. 
 
 Round the centre, and concentric with the circle, is fitted a 
 collar, 6, carrying two arms, c c, one of which has a vernier at 
 its extremity, adapted to the divided circle, and the other a 
 milled head, d, which turns a pinion, working in a toothed rack 
 round the exterior circle of the instrument ; sometimes a third 
 arm is applied at right angles to the other two, to which the 
 pinion is attached, and a vernier can then (if required) be applied 
 to each of the other two, and it also prevents the observer dis- 
 turbing that part of the instrument with his hand when moving 
 the pinion. The rack and pinion give motion to the arms, which 
 can be thus turned quite round the circle for setting the vernier 
 to any angle that may be required. Upon a joint near the ex- 
 tremity of the two arms (which form a diameter to the circle) 
 turns a branch, e e, which for packing may be folded over the 
 face of the instrument, but when in use must be placed in the 
 position shown in the figure : these branches carry, near each of 
 their extremities, a fine steel pricker, the two points of which,
 
 APPENDIX. 
 
 and the centre of the protractor, must (for the instrument to be 
 correct) be in the same straight line. The points are prevented 
 from scratching the paper as the arms are moved round, by steel 
 springs, which lift the branches a small quantity, so that, after 
 setting the centre of the protractor over the angular point, and 
 the vernier in its required position, a slight downward pressure 
 must be given to the branches, and each of the points will make 
 a fine puncture in the paper ; a line drawn through one of these 
 punctures and the angular point will be the line required to form 
 the angle. 
 
 Any inaccuracy in placing the centre of the protractor over the 
 angular point may easily be discovered, for, if incorrectly done, 
 a straight line drawn through the two punctures in the paper 
 will not pass through the angular point, which it will do, if all 
 be correct. 
 
 The face of the glass centre-piece on which the lines are 
 drawn is placed as nearly even with the under surface of the 
 instrument as possible, that no parallax may be occasioned by a 
 space between the lines and the surface of the paper. 
 
 By help of the vernier the protractor is graduated to single 
 minutes, which, taking into consideration the numerous sources 
 of inaccuracy in this kind of proceeding, is the smallest angular 
 quantity that we can pretend to lay down with certainty. Greater 
 accuracy may perhaps be obtained by the help of a table of 
 natural sines and a well-graduated beam-compass, as explained 
 at page 114. 
 
 For plotting offsets, measured to the right and left of the 
 station lines, ivory scales with fiducial edges are usually em- 
 ployed. The figure in the following page represents an ingeni- 
 ous contrivance for an offset scale, extensively employed on the 
 Ordnance Survey of Ireland. 
 
 The graduated scale, A A, is perforated nearly its whole length 
 by a dove-tail shaped groove, for the reception of a sliding piece 
 which is fastened to the cross-scale, B B, by the screw, C. It 
 will readily be understood from an inspection of the figure, that 
 the cross scale, B, slides along the scale, A, the whole length of 
 the groove, and at right angles to it. The graduations on both 
 the scales represent either feet or links, &c., or whatever length 
 may have been assumed as the unit in the operation of measuring. 
 The mode of its application is simply this : place the scale, A A, 
 on the paper, parallel to the line on which the offsets are to be 
 plotted, and at such a distance, that the zero division on the cross 
 scale, B, (which is placed about its middle) may coincide with it 
 as the scale slides along, and also that the zero of the scale, A, 
 may be exactly opposite that end of the line at which the measure- 
 ment commenced; then in sliding the scale, B, from the begin- 
 ning of the line, stop it at every divisional line on A, correspond- 
 ing to the distance on the station line, at which an offset was taken, 
 and lay off the exact length of the offset from the edge of the scale,
 
 APPENDIX. 
 
 122 
 
 B, either to the right or left of the station line, to which it will be at 
 right angles as taken in the field ; the instrument thus gives both 
 dimensions^ the same time. It is perhaps needless to add, that 
 the extremities of the offsets being connected, will represent 
 the curved line, &c., to which they were measured ; weights may 
 be placed at the two ends of the scale, A A, to keep it steadily 
 in its position. In our figure, the instrument is represented as 
 in the act of plotting offsets upon a station line. 
 
 Sbabion Line 
 
 Station 
 
 It very frequently happens that a surveyor requires copies to 
 be made of his plans, and these occasionally on an enlarged or 
 diminished scale. There are various methods of accomplishing 
 this purpose, some of which we shall here enumerate. 
 
 When a copy is to be made of the same size as the original, 
 it is a common practice to lay the plan upon the sheet of paper 
 intended for the copy, and press them close together by means 
 of weights ; then with a fine needle prick through all the corners 
 and leading points on the plan, making corresponding punctures 
 in the paper beneath, which may then be connected by lines to 
 complete the copy. But when the lines on the original are very 
 crooked, this method cannot be successfully applied without the 
 aid of a pair of compasses and tracing paper ; when, having 
 pricked off the principal points, the remainder may be found by 
 the compasses, and the curved lines transferred by drawing them 
 on tracing paper, the back of which being rubbed over with 
 powdered black lead, and placed in its correct relative situation 
 on the copy, a blunt point* may be drawn along the lines which 
 will leave corresponding lines on the copy beneath. 
 
 * The point of a porcupine's quill, or the edge of the eye-end of a fine needle, 
 make good tracing instruments.
 
 APPENDIX. 
 
 Tracing paper is sometimes thus used for making a copy of 
 the whole plan, but as this process occupies so much time, it is 
 frequently applied in the following manner : A sheet of tracing, 
 or hank-post paper, having one side covered with powdered black- 
 lead, is laid between the original and the copy, the former being 
 uppermost; a tracing point is then carefully passed over all the 
 lines on the plan with a slight pressure, depending upon the 
 thickness of the paper, the sheet beneath will receive correspond- 
 ing marks forming an exact copy, which may be inked in at 
 leisure. 
 
 Another method is by means of a large piece of plate glass, called 
 a copying glass, upon which the plan is placed with a fair sheet of 
 paper uppermost, the glass being then fixed in such a position 
 as to have a strong light fall upon it behind, the whole plan be- 
 comes visible through the sheet of paper, upon which a fair copy 
 can be made without the danger of soiling or injuring the origi- 
 nal by pricking through, &c. 
 
 When a plan is to be copied upon a reduced or enlarged scale, 
 other means must be resorted to, which are also applicable to 
 copying upon the same scale. One of them is by the use of pro- 
 portional and triangular compasses, which it is only necessary to 
 mention ; another is by dividing the surface of the original into a 
 great number of small squares, and drawing a similar number 
 upon the copy, which must be formed larger or smaller than those 
 on the original, in the exact proportion of the required difference 
 of the scales ; the squares in the latter may then be filled up 
 with the same detail of the plan as is contained in the corre- 
 sponding squares on the former. When, from the great value of 
 the original, it becomes improper to draw lines upon or otherwise 
 deface it, recourse has been had to a frame of wood or metal, 
 having fine threads stretched across it each way, forming a series 
 of squares : this being laid upon the plan, will, if accurately 
 done, answer the same purpose. 
 
 The last method we shall speak of is by means of a well-known 
 instrument called a pentagraph. 
 
 The subjoined engraving represents the pentagraph, which con- 
 sists of four flat rulers, made either of wood or brass ; the two 
 outside ones are generally from 15 to 24 inches long, and the 
 others about half that length ; the longer ones, A B, and A C, 
 are united together at A, by a pivot about which they turn, and 
 the two smaller rulers are similarly attached to each other at D, 
 and to the longer rulers at E and F. A sliding box is placed on 
 each of the arms, A B and E D, which may be fixed by a clamp 
 screw at any part of the ruler ; these slides carry a tube to con- 
 tain either a blunt tracing point, a pencil, or pen, or the fulcrum 
 G, which is a heavy weight of lead, having a point on the under 
 side to pierce the drawing board and remain immovable in its 
 proper position, it being the centre upon which the whole instru-
 
 APPENDIX. 
 
 ment turns. Several ivory castors support the surface of the 
 machine parallel to the paper, as well as facilitate its motions. 
 
 The arms, ED, and EB, are graduated and marked with the 
 ratios, ^, , &c. so that when a copy of a plan is required to be 
 made in any of these proportions, it is only requisite to fix, at 
 the required ratio, the slides carrying the fulcrum, Gr, and the 
 tube at D, with a pencil or pen, and the instrument will be 
 ready for operation. Thus, suppose it were required to make a 
 copy of a plan exactly one half the size of the original, our 
 engraving represents the pentagraph so employed ; the slide 
 carrying the pencil at D, and that working on the fulcrum, G, 
 are each fixed by their respective clamp-screws at the divisions 
 marked %, the original plan is placed under the tracing point, C, 
 and exactly parallel to it is placed a sheet of paper under the 
 pencil, D ; the pentagraph being first spread out so as to give 
 room for the tracing point to be passed over every line on the 
 plan, whilst the pencil at D is making corresponding marks on 
 the copy, which it is evident will be equal to one half the size 
 of the original. A fine string is attached to the pencil-holder, 
 and passed round by E A, &c. to the tracing point, the pulling 
 at which raises the pencil a small quantity above the paper, to 
 prevent false or improper marks upon the copy. It should also 
 be remarked, that the cup represented on the top of the pencil- 
 holder is intended to receive a weight, to keep the pencil down 
 upon the paper, or when a stronger mark is required. 
 
 When the instrument is set for work, the tracing point, the 
 pencil, and the fulcrum, must in all cases be in a straight line, 
 which may be proved by stretching a fine string over them..
 
 APPENDIX. 
 
 When it is required to make an enlarged copy of a plan, the 
 setting of the instrument is precisely the same as above stated, 
 only the tracing point and pencil must change places, the original 
 being placed under D, and the copy under C. But when a copy 
 is to be made of the same size as the original, the fulcrum must 
 be placed in the middle at D, and the pencil at B, under which 
 will be the copy, whilst the original must be placed under the 
 tracing point at C. 
 
 When a survey is to be made for the purposes of a line of 
 railway or turnpike-road, it is necessary to delineate not only the 
 fields through which it is contemplated the line would pass, but 
 also one or more fields on each side, to the extent of full one 
 hundred yards, for the purpose of admitting hereafter, if neces- 
 sary, an alteration 10 that extent at any point on the line. The 
 instrument usually employed on such surveys is, the prismatic 
 compass, described at page 3 (or else a circumferentor,) together 
 with a land-chain. 
 
 To execute a survey of this kind, supposing the line to have 
 been previously chosen, the surveyor must set up his compass at 
 one extremity of the work, and take the bearing of some distant 
 object situated in the direction of the intended line of railway or 
 road; having done which, and entered it in his field-book, he 
 must commence chaining in that direction, taking offsets to the 
 fences of the fields, and every remarkable object within a short 
 distance to the right and left of his line ; he must also note the 
 point at which his chain crosses the various fences, and at the 
 same time and place set up his compass, to observe the bearing 
 of such fences, or, in other words, the angle their direction makes 
 with the meridian : this angle is at once given by the compass, 
 and furnishes data for laying down their position with regard 
 to the main line which crosses them, but does not determine 
 their respective lengths ; the surveyor must therefore measure 
 along the side of each fence, both to the right and left of the 
 point at which he crosses it, till he comes to their extremity, or 
 the points where such fence meets the other, or side fences of the 
 field ; these now become known or fixed points, from whence 
 the bearing of every fence which diverges from this may be 
 taken, giving the means of laying down their several directions 
 on the general plan. 
 
 If the sui'veyor should require to represent the boundaries of 
 the fields which are still more remote from his main line, he must 
 similarly measure the lengths and curves of the fences he has pre- 
 viously taken the bearings of ; and then again the bearings, &c. 
 of others, till he possesses sufficient data for his purpose ; but he 
 will occasionally find it more convenient to measure secondary or 
 side lines branching from the main line, which, by crossing a
 
 APPENDIX. 
 
 number of fences, give so many fixed points to take bearings 
 from, as frequently to reduce his labour materially, both in the 
 field, and afterwards in plotting the work. 
 
 Having proceeded onward with the measurement of his first 
 main line, as far as may be convenient for his purpose, and also 
 completed the measurements branching therefrom, the surveyor 
 must again set up his compass at the point where he wishes to 
 change the direction of his course, or commence a second main 
 line, when having taken the bearing of some natural conspicuous 
 object in the required direction, he must proceed to measure 
 such second line, and all its subsidiary dimensions, in the same 
 manner as before, completing as much as possible all the minor 
 measurements depending on each main line before he commences 
 a new one. 
 
 Such is the general method of procedure ; but as every thing 
 depends upon the experience and tact of the surveyor, it is im- 
 possible to give more than a general description: particular 
 rules for surveying are useless, as new cases, and sometimes 
 difficult ones, are hourly occurring, which the experience of the 
 surveyor alone will enable him to overcome, and suggest at the 
 time a method which no book, in all probability, could inform 
 him of. 
 
 The protraction of a railway survey is the most easily per- 
 formed by having a protractor laid down upon the plot itself, 
 from which the angles can be transferred by a parallel ruler to 
 any part of the work as described at page 118 ; but instead of 
 constructing a protractor in the manner there directed, it may 
 be done by laying on the paper a metallic circular protractor, 
 placing the zero divisions, (180 and 360) in the exact position 
 it may be necessary to have the meridian represented, then prick 
 off, and mark all the degrees, and if sufficiently large, the half 
 degrees also ; thus a protractor may be drawn on the paper ready 
 for use in a very few minutes ; the plotting is then performed in 
 a manner so precisely similar to that described for the traverse of 
 a road survey, at page 118, that to enter upon the subject here 
 would be merely a recapitulation of what is there stated, to 
 which we accordingly refer. 
 
 " Maritime surveying is of a mixed kind : it not only deter- 
 mines the positions of the remarkable headlands and other con- 
 spicuous objects that present themselves along the vicinity of a 
 coast, but likewise ascertains the situations of the various inlets, 
 rocks, shallows, and soundings, which occur in approaching the 
 shore. To survey a new or inaccessible coast, two boats are 
 moored at a suitable interval, which is carefully measured or 
 otherwise determined ; and from each boat, the bearings of all 
 the prominent points of land are taken by means of an azimuth
 
 APPENDIX. 
 
 compass, or the angles subtended by these points and the other 
 boat are measured by a sextant. Having now on paper drawn 
 the base to any scale, straight lines radiating from each end at 
 the observed angles, will, by their intersections, give the positions 
 of the several points from which the coast may be sketched. But 
 a chart is more accurately constructed, by combining a survey 
 made on land, with observations taken on the water. A smooth 
 level piece of ground is chosen, on which a base of considerable 
 length is measured, and station staves are affixed at its extremi- 
 ties. If no such place can be found, the mutual distance and 
 position of two points conveniently situated for planting the 
 staves, though divided by a broken surface, are determined from 
 one or more triangles, connected with a shorter and temporary 
 base measured near the beach. A boat then explores the offing, 
 and at every rock, shallow, or remarkable sounding, the bearings 
 of the station staves are noticed. These observations furnish so 
 many triangles, from which the situation of the several points 
 are easily ascertained. When a correct map of the coast can be 
 procured, (or previously constructed) the labour of executing a 
 maritime survey is materially shortened. From each important 
 point on the water, the bearings of two known objects on the 
 land are taken, or the intermediate angles subtended by three 
 such objects are observed." The situation of the observer at the 
 time such angles are taken, may then be laid down by means of 
 an instrument called a station-pointer, which is represented in 
 the annexed figure, and which we shall now describe.
 
 APPENDIX. 
 
 This instrument consists of three rulers, A B C, (fig. 1,) con- 
 nected together by a common centre upon which they turn, and 
 can be opened to form two angles of any inclination. The ruler 
 B is connected with the circular arc, b, the ruler, C, with the arc, 
 c, and the middle ruler, A, with the two verniers, a a, adapted to 
 the two arcs. The middle ruler is double, and has a fine wire or 
 thread stretched along its opening ; the other rulers have like- 
 wise a fine wire stretched from end to end, and so adjusted by 
 the little projecting pieces which carry them, that all the three 
 wires tend to the centre of the instrument, where they would 
 meet if produced. The graduated circular arcs, b and c, are for 
 setting the rulers, or rather the fine wires, at whatever angles 
 they may be required to form at the centre of the instrument. 
 Through the centre is an opening sufficiently large to admit a 
 steel pricker, (fig. 2) to be gently pressed into the paper, when 
 the instrument is adjusted in its position : the puncture thus 
 made will represent the station required. 
 
 That the application of the instrument may the more readily 
 be understood, we have represented it in the act of being used. 
 Suppose the points marked DEF to be three conspicuous ob- 
 jects on the coast, whose relative situations are known and laid 
 down upon the map ; and that, on exploring the offing in a boat, 
 a remarkable sounding occurred, which it was necessary should 
 be marked in the chart, the situation of the boat, at the time the 
 sounding was taken, with regard to the shore, must therefore be 
 determined : with a sextant measure the angle, subtended by 
 the objects F and E, likewise the angle subtended by D and E, 
 which suppose to be 35 10', and 20 50', then to lay down on 
 the chart the position of the boat, open the rulers of the station- 
 pointer, and by the circular arcs, set them to the observed angles ; 
 lay the instrument on the paper, and move it till the three wires 
 pass through the three fixed objects ; the centre of the instru- 
 ment will then occupy the relative situation of the boat, and by 
 the steel pricker, the place may be marked on the paper. When 
 several soundings have been taken, and angles observed at the 
 time to any three fixed objects, the station-pointer affords great 
 facility in laying them down : thus the position of shoals, sunken 
 rocks, &c., may be correctly determined.* 
 
 In the absence of the station-pointer, a substitute may be ob- 
 tained, by drawing on a piece of tracing-paper three lines form- 
 ing the observed angles, and moving them about till they pass 
 through the three fixed objects, and the angular point of these 
 angles will then occupy the position of the boat. A very good 
 station-pointer may be made by graduating an arc of a circle on 
 a piece of plate glass, one side of which must be ground, to re- 
 
 * It will readily be perceived, that the station-pointer may be successfully 
 employed in land surveys of considerable extent.
 
 APPENDIX. 129 
 
 ceive the lines forming the observed angles, and it may be ap- 
 plied to the paper as above described, the centre of the gradu- 
 ated arc showing the situation of the boat on the chart. 
 
 The position of the boat may also be determined geometri- 
 cally, as follows ; (but this would be too tedious a process where 
 a great number of stations are to be determined.) Let ABC 
 be three fixed objects on shore, and from the boat at D, suppose 
 the angles C D B and B D A were found = 40 and 60. Sub- 
 tract double the angle C D B from 1 80, and take half the re- 
 mainder = 50, and lay off this angle from C and B, the two 
 lines will meet in E, which will be the centre of a circle passing 
 through B, C, and the place of the boat, which will be somewhere 
 
 on this circle. To find the ~.., 
 
 exact point, take double the 
 
 angle, B D A, from 180, and 
 
 lay off half the remainder = 
 
 ,30 from B and A; these 
 
 lines will meet in a point, F, 
 
 which will also be the centre 
 
 of a circle passing through 
 
 A, B, and the place of the 
 
 boat; consequently, where 
 
 these two circles intersect 
 
 each other, viz. at D, will be '**... 
 
 the situation of the boat on 
 
 the plan, with regard to the shore, as required. 
 
 In conclusion, it may be useful to add a few remarks on the 
 scales used in plotting the work of a survey. 
 
 One chain to an inch (80 inches to 1 mile) is perhaps the 
 largest scale used in plans of land and road surveys, and is 
 adopted only when great clearness is required, and when the 
 work is of limited extent. It is a very useful scale for plans 
 of building or pleasure grounds. 
 
 Two chains to an inch (40 inches to 1 mile) is a very clear 
 scale for land surveys, the extent of which is not very great. It 
 may likewise be used with advantage for gardens and building 
 grounds. 
 
 Three chains to an inch (26- inches to 1 mile) has hitherto 
 not been in very general use, but has lately been adopted by 
 the Tithe Commissioners for the scale of their plans, consider- 
 ing it as " the smallest scale that can with safety be used, in 
 all cases for plans from which the contents are to be com- 
 puted." 
 
 Four chains to an inch (20 inches to 1 mile) is a scale fre- 
 quently employed in plotting surveys of estates, and is very 
 convenient for either enlargement or reduction. 
 
 K
 
 130 APPENDIX. 
 
 Smaller scales are usually employed in extensive operations : 
 six inches to 1 mile is a large scale for the survey of a county, 
 and is the one employed in drawing the plans of the Ordnance 
 Survey of Ireland. The English survey is published on a scale 
 of one inch to a mile. 
 
 The plans and sections for projected railways, &c. deposited 
 with the Houses of Parliament, to obtain the sanction of the 
 Legislature, are required to be drawn on scales, not less than 
 4 inches to the mile for the plan, and one hundred feet to the 
 inch for the section. 
 
 Tyler & Heed, Printers, 5, Bolt-court, Fleet-street.
 
 TABLE I. 
 
 To reduce the Apparent to the True Level. 
 Argument = the Distance in Feet. 
 
 Dist. 
 in 
 Feel. 
 
 Correct" in 
 Decimals 
 of a Foot. 
 
 Dist. 
 in 
 Feet. 
 
 Correct" iu 
 Decimils 
 of a Foot. 
 
 Dist. 
 in 
 Feet. 
 
 Correct" in 
 Decimals 
 of a Foot. 
 
 Dist. 
 
 in 
 Feet. 
 
 Correct" in 
 Decimals 
 of a Foot. 
 
 Dist. 
 iu 
 Feet. 
 
 Correct" in 
 Decimals 
 of a Foot. 
 
 20 
 40 
 60 
 80 
 
 O'OOOOl 
 00004 
 00009 
 00015 
 
 1020 
 40 
 60 
 80 
 
 '02489 
 02588 
 02688 
 02791 
 
 2020 
 40 
 60 
 80 
 
 0-09762 
 09957 
 10153 
 10351 
 
 3020 
 40 
 60 
 80 
 
 0-21821 
 22111 
 22403 
 22697 
 
 4020 
 40 
 60 
 80 
 
 -38665 
 39050 
 39439 
 39828 
 
 100 
 20 
 40 
 60 
 80 
 
 0-00024 
 00034 
 00047 
 00061 
 00077 
 
 1100 
 20 
 40 
 60 
 80 
 
 -02895 
 03001 
 03109 
 03219 
 03331 
 
 2100 
 20 
 40 
 60 
 80 
 
 0-10551 
 10753 
 10956 
 11162 
 11370 
 
 3100 
 20 
 40 
 60 
 80 
 
 -22993 
 23290 
 23590 
 23892 
 24195 
 
 4100 
 20 
 40 
 60 
 80 
 
 0-40218 
 40613 
 41008 
 41404 
 41805 
 
 200 
 20 
 40 
 60 
 80 
 
 -00096 
 00116 
 00138 
 00162 
 00187 
 
 1200 
 20 
 40 
 60 
 80 
 
 '03445 
 03561 
 03679 
 03798 
 03920 
 
 2200 
 20 
 40 
 60 
 80 
 
 0-11580 
 11792 
 12005 
 12220 
 12437 
 
 3200 
 20 
 40 
 60 
 80 
 
 '24500 
 24807 
 25117 
 25427 
 25740 
 
 4200 
 20 
 40 
 60 
 80 
 
 42205 
 42607 
 43014 
 43420 
 43827 
 
 300 
 20 
 40 
 60 
 80 
 
 0-00215 
 00245 
 00276 
 00310 
 00345 
 
 1300 
 20 
 40 
 60 
 80 
 
 -04043 
 04169 
 04296 
 04425 
 04556 
 
 -04689 
 04824 
 04961 
 05100 
 05241 
 
 2300 
 20 
 40 
 60 
 80 
 
 0-12657 
 12878 
 13101 
 13326 
 13553 
 
 3300 
 20 
 40 
 60 
 80 
 
 '26055 
 26372 
 26691 
 27011 
 27334 
 
 4300 
 20 
 40 
 60 
 
 80 
 
 -44239 
 44650 
 45066 
 45483 
 45899 
 
 400 
 20 
 40 
 60 
 80 
 
 '00383 
 -00422 
 00462 
 00506 
 00551 
 
 1400 
 20 
 40 
 60 
 80 
 
 2400 
 20 
 40 
 60 
 80 
 
 0-13781 
 14012 
 14244 
 14480 
 14715 
 
 3400 
 20 
 40 
 60 
 80 
 
 0-27658 
 27985 
 28313 
 28643 
 28975 
 
 4400 
 20 
 
 40 
 60 
 80 
 
 -46320 
 46742 
 47166 
 47591 
 48020 
 
 500 
 20 
 40 
 60 
 
 80 
 
 '00598 
 00647 
 00697 
 00750 
 00805 
 
 1500 
 20 
 40 
 60 
 80 
 
 '05383 
 05528 
 05674 
 05822 
 05972 
 
 2500 
 20 
 40 
 60 
 80 
 
 0-14954 
 15194 
 15436 
 15680 
 15926 
 
 3500 
 20 
 40 
 60 
 80 
 
 0-29309 
 29644 
 29982 
 30323 
 30664 
 
 4500 
 20 
 40 
 60 
 80 
 
 -48449 
 48881 
 49316 
 49749 
 50189 
 
 600 
 20 
 40 
 60 
 80 
 
 0-00861 
 00920 
 00980 
 01042 
 01106 
 
 1600 
 20 
 40 
 60 
 80 
 
 0-06125 
 06279 
 06435 
 06593 
 06753 
 
 2600 
 20 
 40 
 60 
 80 
 
 0-16173 
 16423 
 
 16675 
 16929 
 17184 
 
 ,5600 
 20 
 40 
 60 
 80 
 
 '31008 
 31353 
 31700 
 32050 
 32401 
 
 4600 
 20 
 40 
 60 
 80 
 
 -50627 
 51067 
 51511 
 51957 
 52404 
 
 700 
 20 
 40 
 60 
 80 
 
 0-01172 
 01240 
 01310 
 01382 
 01456 
 
 1700 
 20 
 40 
 60 
 
 80 
 
 0-06914 
 07078 
 07244 
 07411 
 07581 
 
 2700 
 20 
 40 
 60 
 80 
 
 0-17441 
 17701 
 17962 
 18225 
 18490 
 
 3700 
 20 
 40 
 60 
 80. 
 
 -32754 
 33110 
 33466 
 33825 
 34186 
 
 4700 
 20 
 40 
 60 
 80 
 
 '52852 
 53302 
 53755 
 54211 
 54667 
 
 800 
 20 
 40 
 60 
 80 
 
 0-01531 
 01609 
 01688 
 01769 
 01853 
 
 1800 
 20 
 40 
 60 
 
 80 
 
 -07752 
 07925 
 08100 
 08277 
 08456 
 
 2800 
 20 
 40 
 60 
 80 
 
 0-18758 
 19026 
 19298 
 19571 
 19844 
 
 3800 
 20 
 40 
 60 
 80 
 
 '34548 
 34913 
 35280 
 35650 
 36018 
 
 4800 
 20 
 40 
 60 
 80 
 
 -55124 
 55586 
 56048 
 56512 
 56978 
 
 900 
 20 
 40 
 60 
 80 
 1000 
 
 0-01938 
 02025 
 02114 
 02205 
 02298 
 '02392 
 
 1900 
 20 
 40 
 60 
 80 
 2000 
 
 -08637 
 08820 
 09005 
 01)191 
 09380 
 -09570 
 
 2900 
 20 
 40 
 60 
 80 
 3000 
 
 0-20121 
 20400 
 20681 
 20962 
 21247 
 -21533 
 
 3900 
 20 
 40 
 60 
 80 
 1000 
 
 -36390 
 36766 
 37142 
 
 37520 
 37899 
 -38281 
 
 4900 
 20 
 40 
 60 
 80 
 5000 
 
 57447 
 57917 : 
 58388 
 58860 
 59337 
 -59814 
 
 The correction to be subtracted from the apparent (or observed) to obtain 
 the true level.
 
 TABLE II. 
 
 For determining Altitudes with the Barometer. 
 
 Computed by Mr. BAILY'S Formula XXXVIII. 
 
 
 
 
 Thermometers in open Air. 
 
 to the Barometer 
 
 of the Place. 
 
 S 
 
 A 
 
 S 
 
 A 
 
 S 
 
 A 
 
 D 
 
 B 
 
 L 
 
 C 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 
 o 
 
 
 o 
 
 
 o 
 
 
 o 
 
 
 40 
 
 4-76891 
 
 84 
 
 4-79019 
 
 128 
 
 4-81048 
 
 
 
 o -ooooo 
 
 
 
 0-00117 
 
 41 
 
 76940 
 
 85 
 
 79066 
 
 129 
 
 81093 
 
 1 
 
 00004 
 
 3 
 
 001 Hi 
 
 42 
 
 7K989 
 
 86 
 
 79113 
 
 130 
 
 81138 
 
 2 
 
 00009 
 
 6 
 
 00114 
 
 43 
 
 77039 
 
 87 
 
 79160 
 
 131 
 
 81183 
 
 3 
 
 00013 
 
 9 
 
 00111 
 
 44 
 
 77089 
 
 88 
 
 79207 
 
 132 
 
 81228 
 
 4 
 
 00017 
 
 12 
 
 00107 
 
 45 
 
 4-77138 
 
 89 
 
 4 -79254 
 
 133 
 
 4-81272 
 
 5 
 
 '00022 
 
 15 
 
 0-00101 
 
 46 
 
 77187 
 
 90 
 
 79301 
 
 134 
 
 81317 
 
 6 
 
 00026 
 
 18 
 
 00095 
 
 47 
 
 77236 
 
 91 
 
 79348 
 
 135 
 
 81362 
 
 7 
 
 00030 
 
 21 
 
 00087 
 
 48 
 
 77286 
 
 92 
 
 79395 
 
 136 
 
 81407 
 
 8 
 
 00035 
 
 24 
 
 00078 
 
 49 
 
 77335 
 
 93 
 
 79442 
 
 137 
 
 81451 
 
 9 
 
 00039 
 
 27 
 
 00069 
 
 50 
 
 4 -77383 
 
 94 
 
 4 -79488 
 
 j 138 
 
 4-81496 
 
 10 
 
 '00043 
 
 31) 
 
 0-00059 
 
 51 
 
 77433 
 
 95 
 
 79535 
 
 139 
 
 81541 
 
 11 
 
 00048 
 
 33 
 
 00048 
 
 52 
 
 77482 
 
 96 
 
 79582 
 
 140 
 
 '81585 
 
 12 
 
 00052 
 
 36 
 
 00036 
 
 53 
 
 77531 
 
 97 
 
 79629 
 
 141 
 
 81630 
 
 13 
 
 00056 
 
 39 
 
 00024 
 
 54 
 
 77579 
 
 98 
 
 79675 
 
 142 
 
 81675 
 
 14 
 
 00061 
 
 42 
 
 00012 
 
 55 
 
 4 -77628 
 
 99 
 
 4.79722 
 
 143 
 
 4-81719 
 
 15 
 
 '00065 
 
 45 
 
 -00000 
 
 56 
 
 77677 
 
 100 
 
 79768 
 
 144 
 
 81763 
 
 16 
 
 00069 
 
 48 
 
 9 -99988 
 
 57 
 
 77726 
 
 101 
 
 79814 
 
 145 
 
 81807 
 
 17 
 
 00074 
 
 51 
 
 99976 
 
 58 
 
 77774 
 
 102 
 
 79860 
 
 146 
 
 81851 
 
 18 
 
 00078 
 
 54 
 
 99964 
 
 59 
 
 77823 
 
 103 
 
 79907 
 
 147 
 
 81896 
 
 19 
 
 00083 
 
 57 
 
 99952 
 
 60 
 
 4-77871 
 
 104 
 
 4 -79953 
 
 148 
 
 4-81940 
 
 20 
 
 '00087 
 
 60 
 
 9-99941 
 
 61 
 
 77920 
 
 105 
 
 79999 
 
 149 
 
 81983 
 
 21 
 
 00091 
 
 63 
 
 99931 
 
 62 
 
 77968 
 
 106 
 
 80045 
 
 150 
 
 82027 
 
 22 
 
 00096 
 
 66 
 
 99922 
 
 63 
 
 78017 
 
 107 
 
 80091 
 
 151 
 
 82072 
 
 23 
 
 00100 
 
 69 
 
 99913 
 
 ' 64 
 
 78065 
 
 108 
 
 80137 
 
 152 
 
 82116 
 
 24 
 
 00104 
 
 72 
 
 99905 
 
 65 4 78113 
 
 109 
 
 4-80183 
 
 153 
 
 4-82160 
 
 25 
 
 0-00109 
 
 75 
 
 9 '99899 
 
 66 
 
 78161 
 
 110 
 
 80229 
 
 154 
 
 82204 
 
 26 
 
 00113 
 
 78 
 
 99893 
 
 67 
 
 78209 
 
 111 
 
 80275 
 
 155 
 
 82248 
 
 27 
 
 00117 
 
 81 
 
 99889 
 
 68 
 
 78257 
 
 112 
 
 80321 
 
 156 
 
 82291 
 
 28 
 
 00122 
 
 84 
 
 99886 
 
 69 
 
 78305 
 
 113 
 
 80367 
 
 157 
 
 82335 
 
 29 
 
 00126 
 
 87 
 
 99884 
 
 70 
 
 4 -78353 
 
 114 
 
 4-80412 
 
 158 
 
 4 -82379 
 
 30 
 
 00130 
 
 90 
 
 9 -99883 
 
 71 
 
 78401 
 
 115 
 
 80458 
 
 159 
 
 82422 ! 
 
 31 
 
 0-00134 
 
 
 
 72 
 
 78449 
 
 116 
 
 80504 
 
 160 
 
 82466 
 
 
 
 
 
 73 
 
 78497 
 
 117 
 
 80549 
 
 161 
 
 82510 
 
 
 74 
 
 78544 
 
 118 
 
 80595 
 
 162 
 
 82553 
 
 f the sum of the detached 
 S = < thermometers at the 
 (^ two stations. 
 
 75 
 76 
 
 4 -78592 
 78640 
 
 119 
 120 
 
 4-80641 
 80687 
 
 163 
 164 
 
 4 -82597 
 82640 
 
 77 
 78 
 79 
 
 78688 
 78735 
 
 78783 
 
 121 
 122 
 123 
 
 80732 
 80777 
 80823 
 
 165 
 166 
 167 
 
 82683 
 82726 
 82770 
 
 {the difference of the at- 
 tached thermometers 
 at the two stations. 
 
 80 
 81 
 82 
 83 
 
 84 
 
 4 -78830 
 78878 
 78925 
 78972 
 4-79019 
 
 124 
 125 
 126 
 127 
 128 
 
 4 -80869 
 80914 
 80958 
 81003 
 4 -81048 
 
 168 
 169 
 170 
 171 
 172 
 
 4-82813 ! 
 82857 
 82900 
 82943 
 4 -82986 
 
 L = the latitude. 
 ( height of the barometer 
 = | at the upper station. 
 , ( height of the barometer 
 = \ at the lower station. 
 
 Make R = log. /3' (B + log. 0) when upper thermometer reads lowest, 
 
 or R _ ] O g. j3' + B log. B when upper thermometer reads highest. 
 
 Then the log. diff. of altitude in English feet = A + C + log. of R.
 
 TABLE III. 
 
 For converting Intervals of Sidereal into corresponding Intervals of Mean 
 Solar Time. 
 
 Hours. 
 
 Minutes. 
 
 Seconds. 
 
 i 
 0,112 
 0,115 
 0,118 
 0, 120 
 0, 123 
 0,126 
 0,128 
 0,131 
 0,134 
 0,137 
 
 h 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 
 9, 830 
 19,659 
 29,489' 
 39,318] 
 49,148 
 58,977 
 1 8, 807 
 1 18,636 
 1 28,466 
 1 38,296 
 
 1 
 2 
 3 
 
 t 
 
 6 
 
 7 
 8 
 9 
 10 
 
 0,164 
 0, 328 
 0,491 
 0, 655 
 0,819 
 0, 983 
 1,147 
 1,311 
 1,474 
 1,638 
 
 21 
 22 
 23 
 24 
 
 25 
 26 
 27 
 28 
 29 
 30 
 
 3, 440 
 3,604 
 3, 768 
 3,932 
 4,096 
 4,259 
 4,423 
 4,587 
 4, 751 
 4,915 
 
 41 
 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 50 
 
 6,717 
 6,881 
 7,044 
 7,208 
 7,372 
 7,536 
 7,700 
 7,864 
 8,027 
 8,191 
 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 
 s 
 
 0,003 
 0, 005 
 0, 008 
 0,011 
 0,014 
 0,016 
 0,01!) 
 0,022 
 0, 025 
 0,027 
 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 30 
 
 0,057 
 0,060 
 0, 063 
 0,066 
 0,068 
 0,071 
 0,074 
 0,076 
 0,079 
 0,082 
 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 4!) 
 50 
 
 11 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 
 1 48,125 
 1 57,955 
 2 7, 784 
 2 17,614 
 2 27,443 
 2 37,273 
 2 47, 103 
 2 56,932 
 3 6, 762 
 3 16,591 
 
 11 
 12 
 13 
 14 
 15 
 l(i 
 17 
 18 
 19 
 20 
 
 1,802 
 1,966 
 2,130 
 2,294 
 2, 457 
 2,621 
 2,785 
 2, 949 
 3,113 
 3,277 
 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 
 5,079 
 5,242 
 5,406 
 5,570 
 5, 734 
 5,898 
 6, 062 
 6,225 
 6,389 
 6, 553 
 
 51 
 52 
 53 
 54 
 55 
 56 
 57 
 58 
 59 
 60 
 
 8,355 
 8,519 
 8, 683 
 8,847 
 9,010 
 9,174 
 9,338 
 9, 502 
 9,666 
 9,830 
 
 11 
 12 
 13 
 14 
 
 15 
 16 
 17 
 18 
 19 
 20 
 
 0,030 
 0,033 
 0,036 
 0, 038 
 0,041 
 0,044 
 0,047 
 0,049 
 0,052 
 0, 055 
 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 
 0,085 
 0,087 
 0,090 
 0,093 
 0,096 
 0,098 
 0,101 
 0, 104 
 0,106 
 0,109 
 
 51 
 
 52 
 53 
 54 
 55 
 56 
 57 
 58 
 59 
 60 
 
 0,140 
 0, 142 
 0, 145 
 0,148 
 0,150 
 0,153 
 0, 156 
 0, 159 
 0,161 
 0,164 
 
 21 
 22 
 23 
 24 
 
 3 26,421 
 3 36,250 
 3 46,080 
 3 55,909 
 
 The quantities taken from this Table must be subtracted from 
 a sidereal interval, to obtain the corresponding interval in mean 
 solar time. 
 
 TABLE IV. 
 
 For converting Intervals of Mean Solar into corresponding Intervals of 
 Sidereal Time. 
 
 Hours. 
 
 Minutes. 
 
 Seconds. 
 
 h 
 
 m 
 
 m 
 
 . 
 
 m 
 
 8 
 
 m 
 
 . 
 
 
 
 
 
 , 
 
 
 
 , 
 
 . 
 
 1 
 
 9, 856 
 
 1 
 
 0,164 
 
 21 
 
 3,450 
 
 41 
 
 6, 735 
 
 1 
 
 0,003 
 
 21 
 
 0,057 
 
 41 
 
 0,112 
 
 2 
 
 19,713 
 
 2 
 
 0,329 
 
 22 
 
 3,614 
 
 42 
 
 6,900 
 
 2 
 
 0,005 
 
 22 
 
 0,060 
 
 42 
 
 0,115 
 
 3 
 
 29,569 
 
 3 
 
 0, 493 
 
 23 
 
 3,778 
 
 43 
 
 7,064 
 
 3 
 
 0,008 
 
 23 
 
 0, 063 
 
 43 
 
 0,118 
 
 4 
 
 39, 426 
 
 4 
 
 0,657 
 
 24 
 
 3,943 
 
 44 
 
 7,228 
 
 4 
 
 0,011 
 
 24 
 
 0,066 
 
 44 
 
 0,120 
 
 5 
 
 49,282 
 
 5 
 
 0,821 
 
 25 
 
 4,107 
 
 45 
 
 7,392 
 
 5 
 
 0,014 
 
 25 
 
 0,068 
 
 45 
 
 0,123 
 
 6 
 
 59, 139 
 
 6 
 
 0,986 
 
 26 
 
 4,271 
 
 46 
 
 7, 557 
 
 6 
 
 0,016 
 
 26 
 
 0,071 
 
 46 
 
 0, 126 
 
 7 
 
 1 8, 995 
 
 7 
 
 1, 150 
 
 27 
 
 4,436 
 
 47 
 
 7,721 
 
 7 
 
 0,019 
 
 27 
 
 0,074 
 
 47 
 
 0,128 
 
 H 
 
 1 18,852 
 
 8 
 
 1,314 
 
 28 
 
 4,600 
 
 48 
 
 7, 885 
 
 8 
 
 0, 022 
 
 28 
 
 0,076 
 
 48 
 
 0, 131 
 
 9 
 
 1 28,708 
 
 9 
 
 1,478 
 
 29 
 
 4,764 
 
 49 
 
 8,050 
 
 9 
 
 0, 025 
 
 29 
 
 0, 079 
 
 49 
 
 0,134 
 
 10 
 
 1 38,565 
 
 10 
 
 1,643 
 
 30 
 
 4,928 
 
 50 
 
 8,214 
 
 10 
 
 0,027 
 
 30 
 
 0,082 
 
 50 
 
 0, 137 
 
 11 
 
 1 48,421 
 
 11 
 
 1,807 
 
 31 
 
 5, 092 
 
 51 
 
 8,378 
 
 11 
 
 0, 030 
 
 31 
 
 0, 085 
 
 51 
 
 0,140 
 
 12 
 
 1 58,278 
 
 12 
 
 1,971 
 
 32 
 
 5,257 
 
 52 
 
 8,542 
 
 12 
 
 0, 033 
 
 32 
 
 0,087 
 
 52 
 
 0,142 
 
 13 
 
 2 8, 134 
 
 13 
 
 2,136 
 
 33 
 
 5,421 
 
 53 
 
 8,707 
 
 13 
 
 0,036 
 
 33 
 
 0,090 
 
 53 
 
 0, 145 
 
 14 
 
 2 17,991 
 
 14 
 
 2,300 
 
 34 
 
 5,585 
 
 54 
 
 8,871 
 
 14 
 
 0,038 
 
 34 
 
 0, 093 
 
 54 
 
 0,148 
 
 15 
 
 2 27,847 
 
 15 
 
 2,464 
 
 35 
 
 5,750 
 
 55 
 
 9,035 
 
 15 
 
 0,041 
 
 35 
 
 0,096 
 
 55 
 
 0,150 
 
 16 
 
 2 37,704 
 
 16 
 
 2,628 
 
 36 
 
 5,914 
 
 56 
 
 9,199 
 
 16 
 
 0,044 
 
 36 
 
 0,098 
 
 56 
 
 0, 153 
 
 17 
 
 2 47,560 
 
 17 
 
 2,793 
 
 37 
 
 6,078 
 
 57 
 
 9, 364 
 
 17 
 
 0,047 
 
 37 
 
 0,101 
 
 57 
 
 0, 156 
 
 18 
 
 2 57,416 
 
 18 
 
 2,957 
 
 38 
 
 6,242 
 
 58 
 
 9,528 
 
 18 
 
 0,049 
 
 38 
 
 0,104 
 
 58 
 
 0,159 
 
 19 
 
 3 7, 273 
 
 19 
 
 3,121 
 
 39 
 
 6,407 
 
 59 
 
 9,692 
 
 19 
 
 0, 052 
 
 39 
 
 0,106 
 
 59 
 
 0,161 
 
 20 
 
 3 17,129 
 
 20 
 
 3,285 
 
 40 
 
 6,571 
 
 60 
 
 9,856 
 
 20 
 
 0,055 
 
 40 
 
 0,109 
 
 60 
 
 0,164 
 
 21 
 
 3 26,986 
 
 
 
 
 
 
 
 
 
 
 
 
 
 22 
 
 3 36,842 
 
 
 23 
 
 3 46,699 
 
 The quantities taken from this Table must be added to a mean 
 
 24 
 
 3 56,555 
 
 interval, to obtain the corresponding interval in sidereal time.
 
 TABLE V. 
 
 
 Logarithms to compute the Longitude from the Difference between the Intervals 
 
 of the Transit of the Moon's bright Limb and a Star. 
 
 
 1 
 
 
 
 1 
 
 
 
 1 
 
 
 
 2 
 
 
 
 Min. 
 
 Log. 
 
 'arts 
 
 Min. 
 
 Log. 
 
 Paris 
 
 Min. 
 
 Log. 
 
 Parts 
 
 Min. 
 
 Log. 
 
 Parts 
 
 42-0 
 
 1 -536442 
 
 
 
 48-0 
 
 1 -510875 
 
 
 
 a4 -0 
 
 1 -486648 
 
 
 
 o-o 
 
 1 -463627 
 
 
 
 1 
 
 1 -536004 
 
 43 
 
 1 
 
 1 -510461 
 
 41 
 
 1 
 
 1 -486255 
 
 39 
 
 1 
 
 1 -463253 
 
 37 
 
 2 
 
 1 '535566 
 
 87 
 
 2 
 
 1 -510047 
 
 82 
 
 2 
 
 1 -485862 78 
 
 2 1 -462879 
 
 75 
 
 3 
 
 I -535130 
 
 130 
 
 3 
 
 1 -509633 
 
 123 
 
 3 
 
 1-485469 118 
 
 3 
 
 1 -462505 
 
 112 
 
 4 
 
 1 '534693 
 
 174 
 
 4 
 
 1 -509220 
 
 165 
 
 4 
 
 1-485077 157 
 
 4 
 
 1 '462132 
 
 149 
 
 5 
 
 1 -534256 
 
 217 
 
 "5 
 
 1 -508808 
 
 206 
 
 5 
 
 1 -484685 196 
 
 "5 
 
 1 -461759 
 
 187 
 
 6 
 
 1 -533821 
 
 261 
 
 6 
 
 1 -508395 
 
 247 
 
 6 
 
 1-484294 235 
 
 6 
 
 1 -461386 
 
 224 
 
 7 
 
 1 -533385 
 
 504 
 
 7 
 
 1 -507983 
 
 288 
 
 "7 
 
 1 "483903 274 
 
 7 
 
 1 -461014 
 
 261 
 
 8 
 
 1 -532950 
 
 348 
 
 8 
 
 1-507571 
 
 330 
 
 8 
 
 1 -483512 313 
 
 8 
 
 1 -460642 
 
 298 
 
 9 
 
 1 -532515 
 
 391 
 
 9 
 
 1 -507161 371 
 
 9 
 
 1-483121 353 
 
 9 
 
 1 -460270 
 
 336 
 
 43-0 
 
 1-532081 
 
 
 
 49-0 
 
 1 -506749 
 
 
 
 >5-0 
 
 1-482731 
 
 
 
 1 -0 
 
 1 -459898 
 
 
 
 1 
 
 1 -531647 
 
 43 
 
 1 
 
 1 -506337 
 
 41 
 
 1 
 
 1-482341 
 
 39 
 
 1 
 
 1 -459527 
 
 37 
 
 2 
 
 1-531214 
 
 86 
 
 2 
 
 I -505928 
 
 82 
 
 2 
 
 1 -481952 
 
 78 
 
 2 1 -459156 
 
 74 
 
 3 
 
 1 -530781 
 
 130 
 
 3 
 
 1 -505518 
 
 123 
 
 3 
 
 I -481562 117 
 
 3 
 
 1 -458785 
 
 111 
 
 4 
 
 I -530318 
 
 173 
 
 4 
 
 1 -505108 
 
 164 
 
 4 
 
 1-481173 
 
 155 
 
 4 
 
 1 -458414 
 
 148 
 
 "5 
 
 1-529916 
 
 216 
 
 5 
 
 1 -504699 
 
 205 
 
 "5 
 
 1 -480785 
 
 194 
 
 5 
 
 1 -458044 185 
 
 -6 
 
 1 '529484 
 
 259 
 
 6 
 
 1 -504290 
 
 245 
 
 6 
 
 1 -480397 
 
 233 
 
 6 
 
 1 -457674 
 
 222 
 
 7 
 
 1 -529053 
 
 302 
 
 "7 
 
 I -503881 286 
 
 y 
 
 I '480009 
 
 272 
 
 7 
 
 1 -457305 
 
 259 
 
 8 
 
 I 528622 
 
 346 
 
 8 
 
 1 -503473 
 
 327 
 
 8 
 
 1 -479621 
 
 311 
 
 8 
 
 1 -456936 
 
 296 
 
 D 
 
 1-528191 
 
 389 
 
 9 
 
 1 -503065 
 
 368 
 
 9 
 
 1 -479234 
 
 350 
 
 9 
 
 1 '456567 
 
 333 
 
 44-0 
 
 1 '527761 
 
 
 
 ;>o -o 
 
 I -502658 
 
 
 
 ">6 () 
 
 I -478347 
 
 
 
 2-0 
 
 1 -456198 
 
 
 
 1 
 
 1 -527331 
 
 43 
 
 1 
 
 1 -502251 
 
 41 
 
 1 
 
 1 -478460 
 
 39 
 
 1 
 
 1 -455830 
 
 37 
 
 2 
 
 I -526902 
 
 86 
 
 2 
 
 1 -501844 
 
 81 
 
 "2 
 
 1 -478074 
 
 77 
 
 2 
 
 1 -455462 
 
 73 
 
 3 
 
 1 -526473 
 
 128 
 
 3 
 
 1 -501438 
 
 122 
 
 3 
 
 1 -477688 
 
 116 
 
 3 
 
 1 '455094 
 
 110 
 
 4 
 
 1 -526044 
 
 171 
 
 4 
 
 1 -501032 
 
 162 
 
 4 
 
 1 -477302 
 
 154 
 
 4 
 
 1 '454727 
 
 147 
 
 ,") 
 
 1-525616 
 
 214 
 
 -5 
 
 1 -500626 
 
 203 
 
 "5 
 
 1-476917 
 
 193 
 
 "5 
 
 1 '454360 
 
 184 
 
 6 
 
 1 -525188 
 
 257 
 
 6 
 
 1 -500221 
 
 243 
 
 6 
 
 I -476532 
 
 231 
 
 (i 
 
 1 '453993 
 
 220 
 
 "7 
 
 1 -524761 
 
 300 
 
 7 
 
 1 -499816 
 
 284 
 
 " 
 
 1 -476147 
 
 270 
 
 '7 
 
 1 '453627 
 
 257 
 
 8 
 
 1 -524334 
 
 342 
 
 8 
 
 1-499411 
 
 324 
 
 8 
 
 1.475763 
 
 308 
 
 8 
 
 1 -453261 
 
 294 
 
 9 
 
 1 -523907 
 
 385 
 
 9 
 
 1 '499007 
 
 365 
 
 J) 
 
 1.475379 
 
 347 
 
 9 
 
 1 -452895 
 
 330 
 
 45-0 
 
 1 -523481 
 
 
 
 ol -0 
 
 1 -498603 
 
 
 
 57-0 
 
 1 -474995 
 
 
 
 3-0 
 
 1 -452529 
 
 
 
 1 
 
 I -523055 
 
 42 
 
 1 
 
 1 '498200 
 
 40 
 
 1 
 
 1-474612 
 
 38 
 
 1 
 
 1 452164 
 
 36 
 
 2 
 
 1 -522630 
 
 85 
 
 "2 
 
 1 -497797 
 
 80 
 
 2 
 
 1 -474228 
 
 76 
 
 2 
 
 1 -451799 
 
 73 
 
 3 
 
 1 '522205 
 
 127 
 
 3 
 
 1 '497393 
 
 121 
 
 3 
 
 1 -473845 
 
 115 
 
 3 
 
 1 -451434 
 
 109 
 
 4 
 
 1 -521780 
 
 170 
 
 4 
 
 1 -496991 
 
 161 
 
 4 
 
 1 -473462 
 
 153 
 
 4 
 
 1 -451069 
 
 146 
 
 5 
 
 1 '521355 
 
 212 
 
 "5 
 
 1 '496589 
 
 201 
 
 "5 
 
 1 -473080 
 
 191 
 
 "5 
 
 1 -450705 
 
 IS- 1 
 
 6 
 
 1 -520932 
 
 254 
 
 6 
 
 I -496188 
 
 241 
 
 6 
 
 1 -472699 
 
 229 
 
 6 
 
 1 -450341 
 
 218 
 
 '7 
 
 1 -520508 
 
 297 
 
 7 
 
 1 -495786 
 
 281 
 
 7 
 
 1 -472317 
 
 267 
 
 7 
 
 1 "449977 
 
 255 
 
 8.1 -520085 
 
 339 
 
 8 
 
 I -495385 
 
 322 
 
 8 
 
 1 -471936 
 
 306 
 
 8 
 
 1 "449614 
 
 291 
 
 9 T519662 
 
 382 
 
 9 
 
 1 -494984 
 
 362 
 
 9 
 
 1 -471555 
 
 344 
 
 9 
 
 1 -449251 
 
 328 
 
 46-0 1-519240 
 
 
 
 52-0 
 
 1 -494584 
 
 
 
 58-0 1-471174 
 
 
 
 4-0 
 
 1 -448888 
 
 
 
 1 1 '518820 
 
 42 
 
 1 
 
 1-494184 
 
 40 
 
 1)1 -470794 
 
 38 
 
 1 
 
 1 -448525 
 
 36 
 
 2 1-518400 
 
 84 
 
 5 
 
 1 -493784 
 
 80 
 
 2|1 -470414 
 
 76 
 
 2 
 
 1-448163 
 
 72 
 
 3 1 -517980 
 
 126 
 
 .J 
 
 1 -493385 
 
 120 
 
 3 I -470034 
 
 114 
 
 3 
 
 447801 
 
 108 
 
 41 -517560 
 
 168 
 
 '4 
 
 1 -492986 
 
 159 
 
 4 1-469655 
 
 152 
 
 4 
 
 447439 
 
 144 
 
 5 1 -517140 
 
 210 
 
 5 
 
 1 -492587 
 
 199 
 
 5 1 -469276 190 
 
 5 
 
 447078 
 
 181 
 
 6 
 
 1-516719 
 
 252 
 
 6 
 
 1 -492189 
 
 239 
 
 6 1 -468897 
 
 227 
 
 6 
 
 446717 
 
 217 
 
 7 
 
 1 -516299 
 
 294 
 
 "7 
 
 1-491791 
 
 279 
 
 7 1 -468518 265 
 
 "7 
 
 446356 
 
 253 
 
 8 
 
 1 -515879 
 
 336 
 
 8 
 
 1 -491393 
 
 319 
 
 81 -468140 
 
 303 
 
 8 
 
 445995 
 
 289 
 
 9 1 -51545f 
 
 378 
 
 c 
 
 1 -490996 
 
 359 
 
 9 1 -467762 
 
 341 
 
 9 
 
 445635 
 
 325 
 
 47-0 
 
 1 -515039 
 
 
 
 53-0 
 
 1 -490599 
 
 
 
 59-0 1-467385 
 
 
 
 5-0 
 
 1 -445275 
 
 
 
 1 
 
 1 -514621 
 
 42 
 
 1 
 
 1 -490202 
 
 39 
 
 1 1-467008 
 
 38 
 
 1 
 
 1-444915 
 
 36 
 
 2 
 
 1-514203 
 
 83 
 
 "2 
 
 1 -489806 
 
 79 
 
 2 1 -466631 
 
 75 
 
 2 1 -444556 
 
 71 
 
 3 
 
 1 -51378f 
 
 125 
 
 i 
 
 1-489410 
 
 119 
 
 3 1 -466255 
 
 113 
 
 3 
 
 1-444196 
 
 107 i 
 
 4 
 
 1 -513369 
 
 167 
 
 'f. 
 
 1-489014 
 
 158 
 
 4 1 -465878 150 
 
 4 
 
 1 -443837 
 
 143 
 
 5 
 
 1 -512952 
 
 208 
 
 5 
 
 I -4886H 
 
 198 
 
 5 1 -465502 188 
 
 5 
 
 1 -443478 
 
 179 
 
 fi 
 
 1-5 1253ft 
 
 250 
 
 6 
 
 1 -488224 
 
 237 
 
 6 1 -465127 226 
 
 6 1 -443120 
 
 215 
 
 "7 
 
 1 -512120 
 
 292 
 
 j 
 
 1 -487830 
 
 277 
 
 7 1 -464751 263 
 
 7 1 -442762 
 
 251 
 
 8 
 
 1-51170. 
 
 333 
 
 8 1 -487436 
 
 316 
 
 8 1 -464376 301 
 
 8 1 -442404 
 
 286 
 
 ! 
 
 1 -511291 
 
 375 
 
 9 1 -487042 
 
 356 
 
 9 1 -464001 338 
 
 9 
 
 1 -442046 
 
 322 
 
 
 
 i 
 
 
 
 

 
 TABLE V. 
 
 Logarithms to compute the Longitude from the Difference between the Intervals 
 of the Transit of the Moon's bright Limb and a Star. 
 
 2 
 
 
 
 2 
 
 
 
 2 
 
 
 
 2 
 
 
 
 Min 
 
 Log. 
 
 Paris 
 
 Min. 
 
 Log. 
 
 Parts 
 
 Min. 
 
 Log. 
 
 Paris 
 
 Min. 
 
 Log. 
 
 Parts 
 
 s 
 6-0 
 
 1 -441689 
 
 
 
 S 
 
 12-0 
 
 1 -420737 
 
 
 
 18-0 
 
 1 -400682 
 
 
 
 24-0 
 
 1 -381448 
 
 
 
 1 
 
 1 -441332 
 
 36 
 
 1 
 
 1 -420396 
 
 34 
 
 1 
 
 1 -400355 
 
 33 
 
 1 
 
 1 "381134 
 
 31 
 
 2 
 
 1 -440975 
 
 71 
 
 2 
 
 1 -420055 
 
 68 
 
 2 
 
 1 -400028 
 
 65 
 
 * A 
 
 1 -380820 
 
 63 
 
 3 
 
 1 -440619 
 
 107 
 
 3 
 
 1 -419715 
 
 102 
 
 3 
 
 1 -399701 
 
 98 
 
 3 
 
 1 -380506 
 
 94 
 
 4 
 
 1 -440263 
 
 142 
 
 4 
 
 1 -419373 
 
 136 
 
 4 
 
 1 -399375 
 
 130 
 
 4 
 
 1 -380193 
 
 125 
 
 5 
 
 1 -439907 
 
 178 
 
 "5 
 
 1 -419033 
 
 170 
 
 5 
 
 1 '399049 
 
 163 
 
 "5 
 
 1 -379880 
 
 156 
 
 6 
 
 1-439551 
 
 214 
 
 6 
 
 I -418693 
 
 204 
 
 6 
 
 1 -398723 
 
 196 
 
 6 
 
 1 "379567 
 
 188 
 
 7 
 
 1 -439196 
 
 249 
 
 "7 
 
 1 -418354 
 
 238 
 
 7 
 
 1 -398397 
 
 228 
 
 7 
 
 1 -379254 
 
 219 
 
 8 
 
 1 -438840 
 
 285 
 
 8 
 
 1-418014 
 
 272 
 
 8 
 
 I -398071 
 
 261 
 
 8 
 
 1-378941 
 
 250 
 
 9 
 
 1 -438486 
 
 320 
 
 9 
 
 1 '417674 
 
 306 
 
 9 
 
 1 -397746 
 
 293 
 
 9 
 
 1 -378629 
 
 282 
 
 7-0 
 
 1-438131 
 
 
 
 13 -0 
 
 1-417335 
 
 
 
 19-0 
 
 1 -397421 
 
 
 
 25-0 
 
 1 -378317 
 
 
 
 1 
 
 1 '437777 
 
 35 
 
 1 
 
 1 -416996 
 
 34 
 
 1 
 
 1 -397096 
 
 32 
 
 -1 
 
 1 -378005 
 
 31 
 
 2 
 
 1 -437423 
 
 71 
 
 2 
 
 1 -416657 
 
 67 
 
 2 
 
 1 -396772 
 
 65 
 
 2 
 
 I -377693 
 
 62 
 
 3 
 
 1 '437069 
 
 106 
 
 3 
 
 I -416319 
 
 101 
 
 3 
 
 1 -396447 
 
 97 
 
 3 
 
 1 '377382 
 
 93 
 
 4 
 
 1 -436715 
 
 141 
 
 4 
 
 1 -415981 
 
 135 
 
 4 
 
 1 -396123 
 
 129 
 
 4 
 
 1 -377071 
 
 124 
 
 5 
 
 1 -436362 
 
 177 
 
 5 
 
 1 -415643 
 
 168 
 
 5 
 
 1 -395799 
 
 161 
 
 5 
 
 1 -376759 
 
 155 
 
 (i 
 
 1 -436009 
 
 212 
 
 6 
 
 1-415306 
 
 202 
 
 6 
 
 1 -395476 
 
 194 
 
 6 
 
 1 -376449 
 
 186 
 
 7 
 
 1 '435650 
 
 247 
 
 7 
 
 1-414969 
 
 236 
 
 "7 
 
 T 395 152 
 
 226 
 
 7 
 
 1 -376138 
 
 218 
 
 8 
 
 I -435304 
 
 282 
 
 8 
 
 1-414631 
 
 276 
 
 8 
 
 1 '394829 
 
 258 
 
 8 
 
 1 -375827 
 
 249 
 
 9 
 
 1 -434952 
 
 318 
 
 9 
 
 1 '414294 
 
 303 
 
 9 
 
 1 -394506 
 
 291 
 
 9 
 
 1-375517 
 
 280 
 
 8-0 
 
 1 '434600 
 
 
 
 14-0 
 
 1 -413957 
 
 
 
 20 
 
 1 -394183 
 
 
 
 26-0 
 
 1 -375207 
 
 
 
 1 
 
 1-434248 
 
 35 
 
 1 
 
 1 -413621 
 
 33 
 
 1 
 
 1 -393860 
 
 32 
 
 1 
 
 1 -374897 
 
 31 
 
 2 
 
 1 -433897 
 
 70 
 
 2 
 
 1-413285 
 
 67 
 
 2 
 
 1 -393538 
 
 64 
 
 2 
 
 1 '374587 
 
 62 
 
 3 
 
 1 -433546 
 
 105 
 
 3 
 
 I -412949 
 
 100 
 
 3 
 
 1 -393216 
 
 96 
 
 3 
 
 1 -374278 
 
 92 
 
 4 
 
 1 -433195 
 
 140 
 
 4 
 
 1 -412613 
 
 134 
 
 4 
 
 1 -392894 
 
 128 
 
 4 
 
 1 -373969 
 
 123 
 
 5 
 
 1 -432845 
 
 175 
 
 "5 
 
 1 -412278 
 
 167 
 
 '5 
 
 1 -392572 
 
 160 
 
 5 
 
 1 -373659 
 
 154 
 
 I -6 
 
 1 -432494 
 
 210 
 
 6 
 
 1 -411942 
 
 201 
 
 6 
 
 1-392251 
 
 193 
 
 6 
 
 1 -373351 
 
 185 
 
 7 
 
 I -432144 
 
 245 
 
 "7 
 
 1 -411607 
 
 234 
 
 7 
 
 1-391930 
 
 225 
 
 '7 
 
 1 -373042 
 
 216 
 
 8 
 
 1 -43179:; 
 
 280 
 
 8 
 
 1-411273 
 
 268 
 
 8 
 
 1-391608 
 
 257 
 
 8 
 
 1 -372733 
 
 246 
 
 9 
 
 1 -431445 
 
 315 
 
 9 
 
 1 -410938 
 
 301 
 
 9 
 
 1 -391288 
 
 289 
 
 9 
 
 1 -372425 
 
 277 
 
 9-0 
 
 1 -431096 
 
 
 
 15-0 
 
 1 -410604 
 
 
 
 21-0 
 
 1 -390967 
 
 
 
 27-0 
 
 1-372117 
 
 
 
 1 
 
 1 -430747 
 
 35 
 
 1 
 
 1-410270 
 
 33 
 
 1 
 
 1 -390647 
 
 32 
 
 1 
 
 1 -371809 
 
 31 
 
 2 
 
 1 -430398 
 
 70 
 
 2 
 
 1 -409936 
 
 67 
 
 2 
 
 1 -390327 
 
 64 
 
 2 
 
 1-371501 
 
 61 
 
 3 
 
 1 -430050 
 
 104 
 
 3 
 
 1 -409602 
 
 100 
 
 3 
 
 1 '390007 
 
 96 
 
 3 
 
 1 -371194 
 
 92 
 
 4 
 
 1-429701 
 
 139 
 
 4 
 
 I -409269 
 
 133 
 
 4 
 
 1 -389687 
 
 128 
 
 4 
 
 1 -370887 
 
 122 
 
 5 
 
 1 -429354 
 
 174 
 
 5 
 
 1 -408935 
 
 166 
 
 5 
 
 1 -389367 
 
 159 
 
 5 
 
 1 -370579 
 
 153 
 
 6 
 
 1 -429006 
 
 209 
 
 6 
 
 1 -408602 
 
 200 
 
 6 
 
 1 -389048 
 
 191 
 
 6 
 
 1 -370273 
 
 184 
 
 7 
 
 I -428659 
 
 244 
 
 -7 
 
 1 -408270 
 
 233 
 
 7 
 
 1 -388729 
 
 223 
 
 7 
 
 1 -369966 
 
 214 
 
 8 
 
 1 -428312 
 
 278 
 
 8 
 
 1 -407937 
 
 266 
 
 8 
 
 1 -388410 
 
 255 
 
 8 
 
 1 -369659 
 
 245 
 
 9 
 
 1 -427965 
 
 313 
 
 9 
 
 1 -407605 
 
 300 
 
 9 
 
 1 -388091 
 
 287 
 
 9 
 
 1 '369353 
 
 275 
 
 10-0 
 
 1 -427618 
 
 
 
 16-0 
 
 1 '407273 
 
 
 
 22-0 
 
 1 -387773 
 
 
 
 28-0 
 
 1 -369047 
 
 
 
 1 
 
 1 -427272 
 
 35 
 
 1 
 
 1 -406941 
 
 33 
 
 1 
 
 1 -387454 
 
 32 
 
 1 
 
 1 -368741 
 
 30 
 
 2 
 
 1 -426925 
 
 69 
 
 2 
 
 1 -406610 
 
 66 
 
 2 
 
 1 -387137 
 
 63 
 
 2 
 
 1 -368435 
 
 61 
 
 3 
 
 1 -426580 
 
 103 
 
 3 
 
 1 -406278 
 
 99 
 
 3 
 
 1 -386819 
 
 95 
 
 3 
 
 1-368130 
 
 91 
 
 4 
 
 1 -426234 
 
 138 
 
 4 
 
 1 -405947 
 
 132 
 
 4 
 
 I -386501 
 
 127 
 
 4 
 
 1 -367825 
 
 122 
 
 5 
 
 1 -425888 
 
 172 
 
 "5 
 
 1 -405617 
 
 165 
 
 5 
 
 1 -386183 
 
 158 
 
 5 
 
 1 -367520 
 
 152 
 
 6 
 
 1 -425543 
 
 207 
 
 6 
 
 1 -405286 
 
 198 
 
 6 
 
 1 -385867 
 
 190 
 
 6 
 
 1 -367215 
 
 183 
 
 7 
 
 1 -425198 
 
 242 
 
 7 
 
 1 -404956 
 
 231 
 
 7 
 
 1 '385550 
 
 222 
 
 7 
 
 1 -366910 
 
 213 
 
 8 
 
 1 -424854 
 
 276 
 
 8 
 
 1 -404626 
 
 264 
 
 8 
 
 1 -385233 
 
 254 
 
 8 
 
 1 -366605 
 
 244 
 
 9 
 
 1 -424509 
 
 310 
 
 9 
 
 1 -404296 
 
 297 
 
 9 
 
 1 -384916 
 
 285 
 
 9 
 
 1-366301 
 
 274 
 
 11-0 
 
 1-424165 
 
 
 
 17-0 
 
 1 -403966 
 
 
 
 23-0 
 
 1 -384600 
 
 
 
 29-0 
 
 1 -365997 
 
 
 
 1 
 
 1 '423821 
 
 34 
 
 1 
 
 1 -403636 
 
 33 
 
 1 
 
 1 -384284 
 
 31 
 
 1 
 
 1 -365693 
 
 30 
 
 2 
 
 1 -423477 
 
 69 
 
 2 
 
 1 -403307 
 
 66 
 
 2 
 
 1 -383968 
 
 63 
 
 2 
 
 1 -365389 
 
 61 
 
 3 
 
 1 -423134 
 
 103 
 
 3 
 
 1 -402978 
 
 98 
 
 3 
 
 1 -383652 
 
 94 
 
 3 
 
 1 -365086 
 
 91 
 
 4 
 
 1 -422791 
 
 137 
 
 4 
 
 1 -402650 
 
 131 
 
 4 
 
 1 -383337 
 
 126 
 
 4 
 
 1 -364782 
 
 121 
 
 5 
 
 1 -422448 
 
 171 
 
 '5 
 
 1 -402321 
 
 164 
 
 "5 
 
 1 -383021 
 
 157 
 
 5 
 
 1 -364479 
 
 151 
 
 6 
 
 1 -422105 
 
 206 
 
 6 
 
 1 -401993 
 
 197 
 
 6 
 
 1 -382706 
 
 189 
 
 6 
 
 1-364176 
 
 182 
 
 7 
 
 1-421763 
 
 240 
 
 7 
 
 1 -401665 
 
 230 
 
 "7 
 
 1 -382391 
 
 220 
 
 7 
 
 1 -363873 
 
 212 
 
 8 
 
 1 -421421 
 
 274 
 
 8 
 
 1 -401337 
 
 262 
 
 8 
 
 1^382077 
 
 252 
 
 8 
 
 1 -363571 
 
 242 
 
 9 
 
 1 -421079 
 
 309 
 
 9 
 
 1 -401009 
 
 295 
 
 9 
 
 1 -381762 
 
 283 
 
 9 
 
 1 -363268 
 
 273
 
 
 
 
 TABLE V. 
 
 
 
 Logarithms to compute the Longitude from the Difference between the Intervals 
 of the Transit of the Moon's bright Limb and a Star. 
 
 2 
 
 Log. 
 
 
 2 
 
 
 
 
 2 
 
 
 
 
 2 
 
 Log. 
 
 
 Min. 
 
 Paits 
 
 Min. 
 
 Log. 
 
 I' arts 
 
 Win. 
 
 Log. 
 
 Parts 
 
 Min. 
 
 i'ait; 
 
 30-0 
 
 1 -362966 
 
 
 
 34-0 
 
 1 -351035 
 
 
 
 38-0 
 
 1 -339396 
 
 
 
 42-0 
 
 1 -328034 
 
 
 
 "1 
 
 1 -362664 
 
 30 
 
 1 
 
 1 -350740 
 
 29 
 
 1 
 
 I -339109 
 
 29 
 
 1 
 
 1 '327753 
 
 28 
 
 2 
 
 1 -362362 
 
 60 
 
 2 
 
 1 -350446 
 
 59 
 
 2 
 
 1 -338821 
 
 57 
 
 2 
 
 1 -327473 
 
 56 
 
 3 
 
 1 -362061 
 
 90 
 
 3 
 
 1 -350152 
 
 88 
 
 3 
 
 1 -338534 
 
 86 
 
 3 
 
 1 -327192 
 
 84 
 
 4 
 
 1 -361759 
 
 120 
 
 4 
 
 1 '349858 
 
 118 
 
 4 
 
 1 -338248 115 
 
 4 
 
 1 -326912 
 
 112 
 
 5 
 
 1 -361458 
 
 150 
 
 5 
 
 1 -349564 
 
 147 
 
 5 
 
 1 -337961; 143 
 
 5 
 
 1 -326632 
 
 140 
 
 6 
 
 1-361157 
 
 181 
 
 6 
 
 1 -349271 
 
 176 
 
 6 
 
 1 -337675 172 
 
 6 
 
 I -326352 
 
 168 
 
 7 
 8 
 
 1 -360856 
 1 -360556 
 
 211 
 241 
 
 7 
 8 
 
 1 -348977 
 1 -348684 
 
 206 
 235 
 
 7 
 8 
 
 1 -337388 201 
 1 -337102 230 
 
 7 
 8 
 
 1 -326073 
 1 -325793 
 
 196 | 
 
 224 
 
 9 
 
 1 -360255 
 
 271 
 
 9 
 
 1 -348391 
 
 265 
 
 9 
 
 1 -336816 258 
 
 9 
 
 1 -325514 
 
 252 
 
 31 -0 1 -359955 
 
 
 
 35-0 
 
 1 "348098 
 
 
 
 39-0 
 
 1 -336530 
 
 43-0 
 
 1 -325235 
 
 
 
 1 
 
 1 -359655 
 
 30 
 
 1 
 
 1 -347805 
 
 29 
 
 1 
 
 1 -336244 28 
 
 1 
 
 1 -324956 
 
 28 
 
 2 1 -359355 
 
 60 
 
 2 
 
 1 -347513 
 
 58 
 
 2 
 
 1 -335959 57 
 
 2 
 
 1 -324677 
 
 56 
 
 3 1 -359055 
 
 90 
 
 3 
 
 1 -347220 
 
 88 
 
 3 
 
 1 -335674 85 
 
 3 
 
 1 -324399 
 
 83 
 
 4 
 
 1 -358756 
 
 120 
 
 4 
 
 1 -346928 
 
 117 
 
 4 
 
 1-335389 114 
 
 4 
 
 1-324120 
 
 111 
 
 i '5 
 
 1 -358457 
 
 149 
 
 5 
 
 1 -346636 
 
 146 
 
 5 
 
 1 -335104 
 
 142 
 
 5 
 
 1 -323842 
 
 139 
 
 6 
 
 1 358157 
 
 179 
 
 6 
 
 1 '346344 
 
 175 
 
 6 
 
 1 -334819 171 
 
 6 
 
 1 -323564 
 
 167 
 
 7 
 
 1 -357859 
 
 209 
 
 '7 
 
 1 -346053 
 
 204 
 
 '7 
 
 1 -334534; 199 
 
 7 
 
 1 -323286 
 
 195 
 
 8 1 357560 
 
 239 
 
 8 
 
 1 -345761 
 
 234 
 
 8 
 
 1 -334250 228 
 
 8 
 
 1 -323008 
 
 222 
 
 9 
 
 1 -357261 
 
 269 
 
 9 
 
 1 '345470 
 
 263 
 
 9 
 
 1 -333966 256 
 
 9 
 
 1 -322730 
 
 250 
 
 32-0 
 
 1 -356963 
 
 
 
 36-0 
 
 1 -345179 
 
 
 
 40-0 
 
 1 -333682 
 
 
 
 44'0 
 
 1 -322453 
 
 
 
 1 
 
 1 -356665 
 
 30 
 
 1 1 -344888 
 
 29 
 
 1 
 
 1 -333398 
 
 28 
 
 1 
 
 1-322176 
 
 28 
 
 2 
 
 1 -356367 
 
 59 
 
 2 1 -344598 
 
 58 
 
 21-333114 
 
 57 
 
 2 
 
 1 -321898 
 
 55 
 
 3 
 
 1 -356069 
 
 89 
 
 3 1 -344307 
 
 87 
 
 3il -332831 
 
 85 
 
 3 
 
 1 -321621 
 
 83 
 
 4 
 
 1 -355772 
 
 119 
 
 4 1-344017 
 
 116 
 
 4 1 -332547 
 
 113 
 
 4 
 
 1 -321345 
 
 111 
 
 5 
 
 1 -355474 
 
 148 
 
 5 1 -343727 
 
 145 
 
 5 1 -332264 
 
 141 
 
 '5 
 
 1 -321068 
 
 138 
 
 6 
 
 1-355177 
 
 178 
 
 6 1 -343437 
 
 174 
 
 6 1 -331981 
 
 170 
 
 6 
 
 1 -320791 
 
 166 
 
 7 
 
 1 -354880 
 
 208 
 
 7 1 -343147 
 
 203 
 
 7 1 -331698 
 
 198 
 
 7 
 
 1-320515 
 
 194 
 
 8 
 
 1 -354583 
 
 238 
 
 8 1 -342858 
 
 232 
 
 81-331415 
 
 226 
 
 8 1 -320239 
 
 222 
 
 9 
 
 1 -354286 
 
 267 
 
 9 1 -342568 
 
 261 
 
 91-331132 
 
 255 
 
 9 1-319963 
 
 249 
 
 33 
 
 1 -353990 
 
 
 
 37-0 
 
 1 -342279 
 
 
 
 41-01 -330850 
 
 
 
 45-0 1-319687 
 
 
 
 1 
 
 1 -353694 
 
 29 
 
 1 
 
 1-341990 
 
 29 
 
 1 1 -330568 
 
 28 
 
 1 1-319411 
 
 27 
 
 2 1 '353397 
 
 59 
 
 2:1-341701 
 
 58 
 
 2| 1-330285 
 
 56 
 
 21 -319136 
 
 55 
 
 3 
 
 1 -353102 
 
 88 
 
 3 1 -341412 
 
 86 
 
 31-330003 
 
 84 
 
 3 1 -318860 
 
 82 
 
 4 1 -352806 
 
 118 
 
 4 1-341124 
 
 115 
 
 4 1 -329722 
 
 112 
 
 4 1 -318585 
 
 110 
 
 5 
 
 1 -352510 
 
 147 
 
 *5 
 
 1 -340835 
 
 144 
 
 5 1 -329440 
 
 140 
 
 5 1 -318310 
 
 137 
 
 6 
 
 1 -352215 
 
 177 
 
 6 
 
 1 -340547 
 
 173 
 
 61-329158 
 
 169 
 
 6 
 
 1 -318035 
 
 165 
 
 7 
 
 1 -351920 
 
 206 
 
 7 
 
 1 -340259 
 
 202 
 
 71-328877 
 
 197 
 
 '7 
 
 1-317760 
 
 192 
 
 8 1 -351624 
 
 236 
 
 8 1 -339671 
 
 230 
 
 81-328596 
 
 225 
 
 8 
 
 1-317486 
 
 220 
 
 1 -9 
 
 1 -351330 
 
 265 
 
 9 1 -339683 
 
 259 
 
 9 1 -328315 
 
 253 
 
 9 
 
 1-317211 
 
 247 
 
 
 
 
 TABLE VI. 
 
 
 
 Effect of a Change in the Moon's Seir idiameter on the Time of its passing 
 
 
 
 
 the Meridian. 
 
 
 
 
 
 Change of 
 ^\ >_ 
 
 Moon's Declination. 
 
 
 
 
 J) 8 
 Sctnidiam 
 
 
 
 8 
 
 16 
 
 22 
 
 28 
 
 
 
 
 i' 
 
 07 
 
 07 
 
 07 
 
 07 
 
 07 
 
 
 
 
 2 
 
 14 
 
 14 
 
 14 
 
 15 
 
 15 
 
 
 
 
 3 
 
 21 
 
 21 
 
 22 
 
 23 
 
 23 
 
 
 
 
 4 
 
 28 
 
 28 
 
 29 
 
 30 
 
 31 
 
 
 
 
 5 
 
 34 
 
 35 
 
 36 
 
 38 
 
 39 
 
 
 
 
 6 
 
 41 
 
 42 
 
 43 
 
 45 
 
 47 
 
 
 
 
 7 
 
 -48 
 
 49 
 
 50 
 
 53 
 
 '55 
 
 
 
 
 8 
 
 55 
 
 56 
 
 58 
 
 60 
 
 fi2 
 
 
 
 
 9 
 
 62 
 
 63 
 
 65 
 
 68 
 
 70 
 
 
 
 
 10 
 
 69 
 
 70 
 
 72 
 
 75 
 
 78 

 
 TABLE VII. 
 
 Reduction to the Meridian. 
 
 Argument = the Hour Angle from the Meridian. 
 
 
 
 On, 
 
 I" 1 
 
 2 ... 
 
 .") 
 
 4m 
 
 5 ln 6 nl 
 
 7,,, 
 
 gm 
 
 ym 
 
 10 m 
 
 11 12'" 
 
 13" 
 
 14 
 
 
 
 
 
 y 
 
 38 
 
 86 
 
 152 
 
 238 343 
 
 466 
 
 609 
 
 771 
 
 952 
 
 1152 
 
 1370 
 
 1608 
 
 1865 
 
 1 
 
 
 
 10 
 
 3!, 
 
 87 
 
 153 
 
 240 345 
 
 469 
 
 612 
 
 774 
 
 955 
 
 1155 
 
 1374 
 
 1612 
 
 1870 
 
 2 
 
 
 
 10 
 
 39 
 
 88 
 
 155 
 
 241 346 
 
 471 
 
 614 
 
 777 
 
 958 
 
 1159 
 
 1378 
 
 1617 
 
 1874 
 
 3 
 
 
 
 10 
 
 40 
 
 89 
 
 156 
 
 243' 348 
 
 473 
 
 617 
 
 780 
 
 961 
 
 1162 
 
 1382 
 
 1621 
 
 1879 
 
 4 
 
 
 
 11 
 
 41 
 
 90 
 
 157 
 
 244 350 
 
 475 
 
 619 
 
 782 
 
 964 
 
 1166 
 
 1386 
 
 1625 
 
 1883 
 
 5 
 
 
 
 11 
 
 41 
 
 91 
 
 159 
 
 246 352 
 
 478 
 
 622 
 
 785 
 
 968 
 
 1169 
 
 1390 
 
 1629 
 
 1887 
 
 6 
 
 
 
 12 
 
 42 
 
 92 
 
 160 
 
 248 354 
 
 480 
 
 624 
 
 788 
 
 971 
 
 1173 
 
 1393 
 
 1633 
 
 1892 
 
 7 
 
 
 
 12 
 
 43 
 
 93 
 
 161 
 
 249 356 
 
 482 
 
 627 
 
 791 
 
 974 
 
 1176 
 
 1397 
 
 1637 
 
 1896 
 
 8 
 
 
 
 12 
 
 43 
 
 93 
 
 163 
 
 251 358 
 
 484 
 
 630 
 
 794 
 
 977 
 
 1180 
 
 1401 
 
 1641 
 
 1901 
 
 9 
 
 
 
 13 
 
 44 
 
 94 
 
 164 
 
 253 360 
 
 487 
 
 632 
 
 797 
 
 981 
 
 1183 
 
 1405 
 
 1646 
 
 1905 
 
 10 
 
 
 
 13 
 
 45 
 
 95 
 
 165 
 
 254 362 
 
 489 
 
 635 
 
 800 
 
 984 
 
 1187 
 
 1409 
 
 1650 
 
 1910 
 
 11 
 
 
 
 13 
 
 45 
 
 96 
 
 167 
 
 256 364 
 
 491 
 
 637 
 
 803 
 
 987 
 
 1190 
 
 1413 
 
 1654 
 
 1914 
 
 12 
 
 
 
 14 
 
 46 
 
 97 
 
 168 
 
 257 
 
 366 
 
 493 
 
 640 
 
 806 
 
 990 
 
 1194 
 
 1416 
 
 1658 
 
 1919 
 
 13 
 
 
 
 14 
 
 47 
 
 98 
 
 169 
 
 259 
 
 368 
 
 496 
 
 643 
 
 809 
 
 993 
 
 1197 
 
 1420 
 
 1662 
 
 1923 
 
 14 
 
 1 
 
 14 
 
 47 
 
 99 
 
 171 
 
 261 
 
 370 
 
 498 
 
 645 
 
 811 
 
 997 
 
 1201 
 
 1424 
 
 1667 
 
 1928 
 
 15 
 
 1 
 
 IS 
 
 48 
 
 100 
 
 172 
 
 262 
 
 372 
 
 500 
 
 648 
 
 814 
 
 1000 
 
 1205 
 
 1428 
 
 1671 
 
 1932 
 
 16 
 
 1 
 
 15 
 
 49 
 
 102 
 
 173 
 
 264 
 
 374 
 
 503 
 
 650 
 
 817 
 
 1003 
 
 1208 
 
 1432 
 
 1675 
 
 1937 
 
 17 
 
 1 
 
 Hi 
 
 50 
 
 103 
 
 175 
 
 266 
 
 376 
 
 505 
 
 653 
 
 820 
 
 1006 
 
 1212 
 
 1436 
 
 1679 
 
 1941 
 
 18 
 
 1 
 
 16 
 
 50 
 
 104 
 
 176 
 
 267 
 
 378 
 
 507 
 
 656 
 
 823 
 
 1010 
 
 1215 
 
 1440 
 
 1683 
 
 1946 
 
 19 
 
 1 
 
 17 
 
 51 
 
 105 
 
 177 
 
 269 
 
 380 
 
 510 
 
 658 
 
 826 
 
 1013 
 
 1219 
 
 1444 
 
 1688 
 
 1950 
 
 20 
 
 1 
 
 17 
 
 52 
 
 106 
 
 179 
 
 271 
 
 382 
 
 512 
 
 661 
 
 829 
 
 1016 
 
 1222 
 
 1448 
 
 1692 
 
 1955 
 
 21 
 
 1 
 
 17 
 
 53 
 
 107 
 
 180 
 
 272 
 
 384 
 
 514 
 
 664 
 
 832 
 
 1020 
 
 1226 
 
 1451 
 
 1696 
 
 1960 
 
 22 
 
 1 
 
 18 
 
 53 
 
 108 
 
 182 
 
 274 
 
 386 
 
 517 
 
 666 
 
 835 
 
 1023 
 
 1230 
 
 1455 
 
 1700 
 
 1964 
 
 23 
 
 1 
 
 18 
 
 54 
 
 109 
 
 183 
 
 276 
 
 388 
 
 519 
 
 669 
 
 838 
 
 1026 
 
 1233 
 
 1459 
 
 1705 
 
 1968 
 
 24 
 
 2 
 
 19 
 
 55 
 
 110 
 
 184 
 
 278 
 
 390 
 
 521 
 
 672 
 
 841 
 
 1029 
 
 1237 
 
 1463 
 
 1709 
 
 1973 
 
 25 
 
 2 
 
 19 
 
 56 
 
 111 
 
 185 
 
 279 
 
 392 
 
 524 
 
 674 
 
 844 
 
 1033 
 
 1241 
 
 1467 
 
 1713 
 
 1978 
 
 26 
 
 2 
 
 20 
 
 56 
 
 112 
 
 187 
 
 281 
 
 394 
 
 526 
 
 677 
 
 847 
 
 1036 
 
 1244 
 
 1471 
 
 1717 
 
 1982 
 
 27 
 
 2 
 
 20 
 
 57 
 
 113 
 
 188 
 
 283 
 
 396 
 
 528 
 
 680 
 
 850 
 
 1039 
 
 1248 
 
 1475 
 
 1722 
 
 1987 
 
 28 
 
 2 
 
 20 
 
 58 
 
 114 
 
 190 
 
 284 
 
 398 
 
 531 
 
 682 
 
 853 
 
 1043 
 
 1251 
 
 1479 
 
 1726 
 
 1992 
 
 29 
 
 2 
 
 21 
 
 59 
 
 116 
 
 191 
 
 286 
 
 400 
 
 533 
 
 685 
 
 856 
 
 1046 
 
 1255 
 
 1483 
 
 1730 
 
 1997 
 
 30 
 
 3 
 
 21 
 
 59 
 
 117 
 
 193 
 
 288 
 
 402 
 
 535 
 
 688 
 
 859 
 
 1049 
 
 1259 
 
 1487 
 
 1734 
 
 2001 
 
 31 
 
 3 
 
 22 
 
 60 
 
 118 
 
 194 
 
 289 
 
 404 
 
 538 
 
 690 
 
 862 
 
 1053 
 
 1262 
 
 1491 
 
 1739 
 
 2005 
 
 32 
 
 3 
 
 22 
 
 61 
 
 119 
 
 196 
 
 291 
 
 406 
 
 540 
 
 693 
 
 865 
 
 1056 
 
 1266 
 
 1495 
 
 1743 
 
 2010 
 
 33 
 
 3 
 
 23 
 
 62 
 
 120 
 
 197 
 
 293 
 
 408 
 
 543 
 
 696 
 
 868 
 
 1059 
 
 1270 
 
 1499 
 
 1747 
 
 2014 
 
 34 
 
 3 
 
 23 
 
 63 
 
 121 
 
 198 
 
 295 
 
 410 
 
 545 
 
 699 
 
 871 
 
 1062 
 
 1273 
 
 1503 
 
 1751 
 
 2019 
 
 35 
 
 3 
 
 24 
 
 64 
 
 122 
 
 200 
 
 297 
 
 412 
 
 547 
 
 701 
 
 874 
 
 1066 
 
 1277 
 
 1507 
 
 1756 
 
 2024 
 
 36 
 
 3 
 
 24 
 
 64 
 
 123 
 
 201 
 
 299 
 
 415 
 
 550 
 
 704 
 
 877 
 
 1069 
 
 1281 
 
 1511 
 
 1760 
 
 2028 
 
 37 
 
 4 
 
 25 
 
 65 
 
 124 
 
 203 
 
 300 
 
 417 
 
 552 
 
 707 
 
 880 
 
 1073 
 
 1284 
 
 1515 
 
 1764 
 
 2033 
 
 38 
 
 4 
 
 25 
 
 66 
 
 126 
 
 204 
 
 302 
 
 419 
 
 555 
 
 709 
 
 883 
 
 1076 
 
 1288 
 
 1519 
 
 1769 
 
 2038 
 
 39 
 
 4 
 
 26 
 
 67 
 
 127 
 
 206 
 
 304 
 
 421 
 
 557 
 
 712 
 
 886 
 
 1079 
 
 1292 
 
 1523 
 
 1773 
 
 2042 
 
 40 
 
 4 
 
 26 
 
 68 
 
 128 
 
 207 
 
 306 
 
 423 
 
 559 
 
 715 
 
 889 
 
 1083 
 
 129, r ) 
 
 1527 
 
 1777 
 
 2047 
 
 41 
 
 4 
 
 27 
 
 68 
 
 129 
 
 209 
 
 307 
 
 425 
 
 562 
 
 718 
 
 892 
 
 1086 
 
 1299 
 
 1531 
 
 1782 
 
 2052 
 
 42 
 
 5 
 
 28 
 
 69 
 
 130 
 
 210 
 
 309 
 
 427 
 
 564 
 
 720 
 
 896 
 
 1090 
 
 1303 
 
 1535 
 
 1786 
 
 2056 
 
 43 
 
 5 
 
 28 
 
 70 
 
 131 
 
 212 
 
 311 
 
 429 
 
 567 
 
 723 
 
 899 
 
 1093 
 
 1307 
 
 1539 
 
 1790 
 
 2061 
 
 44 
 
 5 
 
 29 
 
 71 
 
 133 
 
 213 
 
 313 
 
 432 
 
 569 
 
 726 
 
 902 
 
 1096 
 
 1310 
 
 1543 
 
 1795 
 
 2066 
 
 45 
 
 5 
 
 29 
 
 72 
 
 134 
 
 215 
 
 315 
 
 434 
 
 572 
 
 729 
 
 905 
 
 1100 
 
 1314 
 
 1547 
 
 1799 
 
 2070 
 
 46 
 
 6 
 
 30 
 
 73 
 
 135 
 
 216 
 
 316 
 
 436 
 
 574 
 
 732 
 
 908 
 
 1103 
 
 1318 
 
 1551 
 
 1804 
 
 2075 
 
 47 
 
 6 
 
 30 
 
 74 
 
 136 
 
 218 
 
 318 
 
 438 
 
 577 
 
 734 
 
 911 
 
 1107 
 
 1321 
 
 1555 
 
 1808 
 
 2080 
 
 48 
 
 6 
 
 31 
 
 75 
 
 137 
 
 21!) 
 
 320 
 
 440 
 
 579 
 
 737 
 
 914 
 
 1110 
 
 1325 
 
 1559 
 
 1812 
 
 2084 
 
 49 
 
 C 
 
 31 
 
 75 
 
 139 
 
 221 
 
 322 
 
 442 
 
 582 
 
 740 
 
 917 
 
 1114 
 
 1329 
 
 1563 
 
 1817 
 
 2089 
 
 50 
 
 7 
 
 32 
 
 76 
 
 140 
 
 222 
 
 324 
 
 444 
 
 584 
 
 743 
 
 920 
 
 1117 
 
 1333 
 
 1567 
 
 1821 
 
 2094 
 
 51 
 
 7 
 
 33 
 
 77 
 
 141 
 
 225 
 
 326 
 
 447 
 
 587 
 
 745 
 
 923 
 
 1120 
 
 133fi 
 
 1571 
 
 1825 
 
 2099 
 
 52 
 
 7 
 
 33 
 
 78 
 
 142 
 
 226 
 
 328 
 
 449 
 
 589 
 
 748 
 
 927 
 
 1124 
 
 1340 
 
 1575 
 
 1830 
 
 2103 
 
 53 
 
 7 
 
 34 
 
 79 
 
 144 
 
 227 
 
 329 
 
 451 
 
 592 
 
 751 
 
 930 
 
 1127 
 
 1344 
 
 1580 
 
 1834 
 
 2108 
 
 54 
 
 8 
 
 34 
 
 80 
 
 145 
 
 229 
 
 331 
 
 453 
 
 594 
 
 754 
 
 933 
 
 1131 
 
 1348 
 
 1584 
 
 1839 
 
 2113 
 
 55 
 
 8 
 
 35 
 
 81 
 
 146 
 
 230 
 
 333 
 
 455 
 
 597 
 
 757 
 
 936 
 
 1134 
 
 1352 
 
 1^88 
 
 1843 
 
 2117 
 
 56 
 
 8 
 
 36 
 
 82 
 
 147 
 
 232 
 
 335 
 
 458 
 
 599 
 
 760 
 
 939 
 
 1138 
 
 1355 
 
 1592 
 
 1847 
 
 2122 
 
 57 
 
 8 
 
 34 
 
 83 
 
 149 
 
 233 
 
 337 
 
 460 
 
 602 
 
 763 
 
 942 
 
 1141 
 
 1359 
 
 1596 
 
 1852 
 
 2127 
 
 58 
 
 ( 
 
 37 
 
 84 
 
 150 
 
 235 
 
 339 
 
 462 
 
 604 
 
 765 
 
 945 
 
 1145 
 
 1363 
 
 1600 
 
 1856 
 
 2132 
 
 59 
 
 i) 
 
 37 
 
 85 
 
 151 
 
 236 
 
 341 
 
 464 
 
 607 
 
 768 
 
 949 
 
 1148 
 
 1367 
 
 1604 
 
 1861 
 
 2136 
 
 60 
 
 i 
 
 38 
 
 86 
 
 152 
 
 238 
 
 343 
 
 466 
 
 609 
 
 771 
 
 952 
 
 1152 
 
 1370 
 
 1608 
 
 1865 
 
 2141
 
 TABLE VII. 
 
 Reduction to the Meridian. 
 
 Argument = the Hour Angle from tbe Meridian. 
 
 
 
 15'" 
 
 16" 
 
 17"' 
 
 18-" 
 
 IS" 
 
 20" 
 
 21- 
 
 22 m 
 
 23 m 
 
 24 m 
 
 
 
 2141 
 
 2436 
 
 2750 
 
 3083 
 
 3434 
 
 3805 
 
 4195 
 
 4604 
 
 5031 
 
 5478 
 
 1 
 
 2146 
 
 2441 
 
 2755 
 
 3088 
 
 3441 
 
 3812 
 
 4202 
 
 4611 
 
 5039 
 
 5486 
 
 2 
 
 2151 
 
 2446 
 
 2761 
 
 3093 
 
 3447 
 
 3818 
 
 4208 
 
 4618 
 
 5046 
 
 5494 
 
 3 
 
 2155 
 
 2451 
 
 2766 
 
 3099 
 
 3453 
 
 3824 
 
 4215 
 
 4625 
 
 5053 
 
 5501 
 
 4 
 
 2160 
 
 2456 
 
 2771 
 
 3106 
 
 3459 
 
 3831 
 
 4222 
 
 4632 
 
 5061 
 
 5509 
 
 5 
 
 2165 
 
 2461 
 
 2777 
 
 3111 
 
 3465 
 
 3837 
 
 4228 
 
 4639 
 
 5068 
 
 5516 
 
 6 
 
 2170 
 
 2466 
 
 2782 
 
 3117 
 
 3471 
 
 3S43 
 
 4235 
 
 4646 
 
 5075 
 
 5524 
 
 7 
 
 2175 
 
 2472 
 
 2788 
 
 3123 
 
 3477 
 
 3850 
 
 4242 
 
 4653 
 
 5083 
 
 5531 
 
 8 
 
 2179 
 
 2477 
 
 2793 
 
 3128 
 
 3483 
 
 3856 
 
 4248 
 
 4660 
 
 5090 
 
 5539 
 
 9 
 
 2184 
 
 2482 
 
 2799 
 
 3134 
 
 3489 
 
 3863 
 
 4255 
 
 4667 
 
 5097 
 
 5547 
 
 10 
 
 2189 
 
 2487 
 
 2804 
 
 3140 
 
 3495 
 
 3869 
 
 4262 
 
 4674 
 
 5105 
 
 5554 
 
 11 
 
 2194 
 
 2492 
 
 2809 
 
 3146 
 
 3501 
 
 3875 
 
 4269 
 
 4681 
 
 5112 
 
 5562 
 
 12 
 
 2198 
 
 2497 
 
 2815 
 
 3151 
 
 3507 
 
 3882 
 
 4275 
 
 4688 
 
 5119 
 
 5570 
 
 13 
 
 2203 
 
 2502 
 
 2820 
 
 3157 
 
 3513 
 
 3888 
 
 4282 
 
 4695 
 
 5127 
 
 5577 
 
 14 
 
 2208 
 
 2507 
 
 2826 
 
 3163 
 
 3519 
 
 3895 
 
 4289 
 
 4702 
 
 5134 
 
 5585 
 
 15 
 
 2213 
 
 2513 
 
 2831 
 
 3169 
 
 3525 
 
 3901 
 
 4295 
 
 4709 
 
 5141 
 
 5593 
 
 16 
 
 2218 
 
 2518 
 
 2837 
 
 3175 
 
 3531 
 
 3907 
 
 4302 
 
 4716 
 
 5149 
 
 5600 
 
 17 
 
 2223 
 
 2f>23 
 
 2842 
 
 3180 
 
 3538 
 
 3914 
 
 4309 
 
 4723 
 
 5156 
 
 5608 
 
 18 
 
 '2-227 
 
 2528 2848 
 
 3186 
 
 3544 
 
 3920 
 
 4316 
 
 4730 
 
 5163 
 
 5616 
 
 19 
 
 2232 
 
 2533 2853 
 
 3192 
 
 3550 
 
 3927 
 
 4322 
 
 4737 
 
 5171 
 
 5623 
 
 20 
 
 2237 
 
 2538 2859 
 
 3198 
 
 3556 
 
 3933 
 
 4329 
 
 4744 
 
 5178 
 
 5631 
 
 21 
 
 2242 
 
 2544 
 
 2864 
 
 3204 
 
 3562 
 
 3940 
 
 4336 
 
 4751 
 
 5186 
 
 5639 
 
 22 
 
 2247 
 
 2549 
 
 2870 
 
 3209 
 
 3568 
 
 3946 
 
 4343 
 
 4758 
 
 5193 
 
 5647 
 
 23 
 
 2252 
 
 2554 
 
 2875 
 
 3215 
 
 3574 
 
 3952 
 
 4350 
 
 4766 
 
 5200 
 
 5654 
 
 24 
 
 2257 
 
 2559 
 
 2881 
 
 3221 
 
 3581 
 
 3959 
 
 4356 
 
 4773 
 
 5208 
 
 5662 
 
 25 
 
 2262 
 
 2564 
 
 2886 
 
 3227 
 
 3587 
 
 3965 
 
 4363 
 
 4780 
 
 5215 
 
 5670 
 
 26 
 
 2266 
 
 2570 
 
 28L.2 
 
 3233 
 
 3593 
 
 3972 
 
 4370 
 
 4787 
 
 5223 
 
 5678 
 
 27 
 
 2272 
 
 2575 
 
 2897 
 
 3239 
 
 3599 
 
 3978 
 
 4377 
 
 4794 
 
 5230 
 
 5685 
 
 28 
 
 2276 
 
 2580 
 
 2903 
 
 3244 
 
 3605 
 
 3l> 85 
 
 4383 
 
 4801 
 
 5237 
 
 5693 
 
 2!) 
 
 2281 
 
 25.85 
 
 2908 
 
 3250 
 
 3611 
 
 3991 
 
 4390 
 
 4808 
 
 5245 
 
 5701 
 
 30 
 
 2286 
 
 2590 
 
 2914 
 
 3256 
 
 3618 
 
 3998 
 
 4397 
 
 4815 
 
 5252 
 
 5709 
 
 31 
 
 2291 
 
 2596 
 
 2919 
 
 3262 
 
 3624 
 
 4004 
 
 4404 
 
 4822 
 
 5260 
 
 5716 
 
 32 
 
 2296 
 
 2601 
 
 2925 
 
 3268 
 
 3630 
 
 4011 
 
 4411 
 
 4829 
 
 5267 
 
 5724 
 
 33 
 
 2301 
 
 2606 
 
 2930 
 
 3274 
 
 3636 
 
 4017 
 
 4418 
 
 4837 
 
 5275 
 
 5732 
 
 34 
 
 2306 
 
 2611 
 
 2936 
 
 3280 
 
 3642 
 
 4024 
 
 4424 
 
 4844 
 
 5282 
 
 5740 
 
 35 
 
 2311 
 
 2617 
 
 2942 
 
 3286 
 
 3648 
 
 4030 
 
 4431 
 
 4851 
 
 5290 
 
 5747 
 
 36 
 
 2316 
 
 2622 
 
 2947 
 
 3291 
 
 3655 
 
 4037 
 
 4438 
 
 4858 
 
 5297 
 
 5755 
 
 37 
 
 2321 
 
 2627 
 
 2952 
 
 3297 
 
 3661 
 
 4043 
 
 4445 
 
 4865 
 
 5305 
 
 5763 
 
 38 
 
 2326 
 
 2632 
 
 2958 
 
 3303 
 
 3667 
 
 4050 
 
 4452 
 
 4872 
 
 5312 
 
 5771 
 
 39 
 
 2331 
 
 2638 
 
 2964 
 
 3309 
 
 3673 
 
 4056 
 
 4459 
 
 4880 
 
 5320 
 
 5779 
 
 40 
 
 2335 
 
 2643 
 
 2970 
 
 3315 
 
 3680 
 
 4063 
 
 4465 
 
 4887 
 
 5327 
 
 5786 
 
 41 
 
 2540 
 
 2648 
 
 2975 
 
 3321 
 
 3686 
 
 4070 
 
 4472 
 
 4894 
 
 5335 
 
 5794 
 
 42 
 
 2345 
 
 2654 
 
 2981 
 
 3327 
 
 3692 
 
 4076 
 
 4479 
 
 4901 
 
 5342 
 
 5802 
 
 43 
 
 2350 
 
 2659 
 
 2986 
 
 3333 
 
 3698 
 
 4083 
 
 4486 
 
 4908 
 
 5350 
 
 5810 
 
 44 
 
 2355 
 
 2664 
 
 2992 
 
 3339 
 
 3705 
 
 4089 
 
 44S3 
 
 4916 
 
 5357 
 
 5818 
 
 45 
 
 2360 
 
 2669 
 
 2998 
 
 3345 
 
 3711 
 
 4096 
 
 4500 
 
 4923 
 
 5365 
 
 5826 
 
 46 
 
 2365 
 
 2675 
 
 3003 
 
 3351 
 
 3717 
 
 4102 
 
 4507 
 
 4930 
 
 5372 
 
 5833 
 
 47 
 
 2370 
 
 2680 
 
 3009 
 
 3357 
 
 3723 
 
 4109 
 
 .4514 
 
 4937 
 
 5380 
 
 5841 
 
 48 
 
 2375 
 
 2685 
 
 3015 
 
 3363 
 
 3730 
 
 4116 
 
 4520 
 
 4944 
 
 5387 
 
 5849 
 
 49 
 
 2380 
 
 2690 
 
 3020 
 
 3369 
 
 3736 
 
 4122 
 
 4527 
 
 4952 
 
 5395 
 
 5857 
 
 50 
 
 2385 
 
 261,6 
 
 3026 
 
 3375 
 
 3742 
 
 4129 
 
 4534 
 
 4959 
 
 5402 
 
 5865 
 
 51 
 
 2390 
 
 2701 
 
 3031 
 
 3380 
 
 3749 
 
 4135 
 
 4541 
 
 4966 
 
 5410 
 
 5873 
 
 52 
 
 2395 
 
 2707 
 
 303' 
 
 3386 
 
 3755 
 
 4142 
 
 4548 
 
 4973 
 
 5417 
 
 5881 
 
 53 
 
 2401 
 
 2712 
 
 3043 
 
 3392 
 
 3761 
 
 4149 
 
 4555 
 
 4981 
 
 5425 
 
 5888 
 
 54 
 
 2406 
 
 2718 
 
 3049 
 
 3398 
 
 3767 
 
 4155 
 
 4562 
 
 4988 
 
 5433 
 
 5896 
 
 55 
 
 2411 
 
 2723 
 
 3054 
 
 3404 
 
 3774 
 
 4162 
 
 4569 
 
 4995 
 
 5440 
 
 5904 
 
 56 
 
 2416 
 
 2728 
 
 3060 
 
 3410 
 
 3780 
 
 4168 
 
 4576 
 
 5002 
 
 5448 
 
 5912 
 
 57 
 
 2421 
 
 2734 
 
 3066 
 
 3416 
 
 3786 
 
 4175 
 
 4583 
 
 5010 
 
 5455 
 
 5920 
 
 58 
 
 2426 
 
 2739 
 
 3071 
 
 3422 
 
 3793 
 
 4182 
 
 4590 
 
 5017 
 
 5463 
 
 5928 
 
 59 
 
 2431 
 
 2744 
 
 3077 
 
 3428 
 
 3799 
 
 4188 
 
 4597 
 
 5024 
 
 5470 
 
 5936 
 
 60 
 
 2436 
 
 2750 
 
 3083 
 
 3434 
 
 3805 
 
 4195 
 
 4604 
 
 5031 
 
 5478 
 
 5944
 
 TABLE VIII. 
 
 TABLE IX. 
 
 Showing the Length of a Second of 
 
 Reduction in Links and Decimals upon 
 
 Longitude and Latitude in English 
 
 each Chain's Length, for the fol- 
 
 Feet, on the Earth's Surface. 
 
 lowing Angles of Elevation and 
 
 
 Depression. 
 
 Compression = ^Q- 
 
 
 Computed by Mr. BAILY'S Formula XLIII 
 
 Angle. 
 
 Re- 
 duct" 
 
 Angle. 
 
 Re- 
 
 dnct 1 
 
 Angle. 
 
 Re- 
 duct 
 
 
 3. 
 
 0.14 
 
 9. ( 
 
 1 1.24 
 
 15. 
 
 3.40 
 
 
 
 
 
 
 
 T !* 
 
 Second 
 
 Second 
 
 
 Second 
 
 Secom 
 
 
 
 3( 
 
 1.38 
 
 :30 
 
 3.- 64 
 
 Ljat. 
 
 of Long, 
 
 of Lat. 
 
 Lat. 
 
 of Long. 
 
 of Lat. 
 
 4. 
 
 0.25 
 
 10. C 
 
 1.52 
 
 16 6 . 
 
 3.88 
 
 o 
 
 Ft 
 
 Ft. 
 
 o 
 
 Ft 
 
 Ft. 
 
 
 
 3C 
 
 1.68 
 
 30 
 
 4.12 
 
 
 
 101.42 
 
 101.42 
 
 35 
 
 83.17 
 
 101.75 
 
 
 
 
 
 
 
 1 
 
 101.40 
 
 
 36 
 
 82.15 
 
 
 5. 
 
 0.38 
 
 11. 
 
 1.84 
 
 17. 
 
 4.37 
 
 2 
 
 101.36 
 
 
 37 
 
 81.10 
 
 
 
 
 3C 
 
 2.01 
 
 30 
 
 4.63 
 
 3 
 
 101.28 
 
 
 38 
 
 80.02 
 
 
 
 
 
 
 
 
 4 
 
 101.17 
 
 
 39 
 
 78.92 
 
 
 6. 
 
 0.55 
 
 12. 
 
 2.19 
 
 18. 
 
 4.90 
 
 5 
 
 101.03 
 
 101.43 
 
 40 
 
 77.80 
 
 101.84 
 
 30 
 
 0.65 
 
 3( 
 
 2.37 
 
 30 
 
 5. 17 
 
 6 
 
 7 
 
 100. 87 
 100.67 
 
 
 41 
 42 
 
 76.65 
 75.48 
 
 
 7. 
 
 0.75 
 
 13. 
 
 2.56 
 
 19. 
 
 5.44 
 
 8 
 
 100.44 
 
 
 43 
 
 74.29 
 
 
 30 
 
 0.86 
 
 30 
 
 2.77 
 
 30 
 
 5.74 
 
 9 
 
 100.18 
 
 
 44 
 
 73. 07 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8. 
 
 n oa 
 
 14. 
 
 2. 97 
 
 20. 
 
 6. 03 
 
 10 
 
 99.89 
 
 101.45 
 
 45 
 
 71.83 
 
 101.93 
 
 30 l!lO 
 
 30 
 
 3.18 
 
 30 
 
 6.33 
 
 11 
 
 99.57 
 
 
 46 
 
 70.57 
 
 
 
 
 
 
 
 12 
 
 99.22 
 
 
 47 
 
 69.29 
 
 
 
 13 
 
 98.84 
 
 
 48 
 
 67.99 
 
 
 The reduction for one chain (from 
 
 14 
 
 98.43 
 
 
 49 
 
 66. 66 
 
 
 the above Table) multiplied by the 
 
 15 
 16 
 17 
 18 
 
 97.99 
 97.52 
 97.02 
 96.49 
 
 101.49 
 
 50 
 51 
 52 
 53 
 
 65.32 
 63.95 
 62.57 
 61.17 
 
 102. 02 
 
 number of chains, will give the quan- 
 tity to be subtracted from the mea- 
 sured length of an inclination, to re- 
 duce it to horizontal measure. 
 
 19 
 
 95.44 
 
 
 54 
 
 59.75 
 
 
 
 20 
 
 95.36 
 
 101.54 
 
 55 
 
 58.30 
 
 102. 11 
 
 
 21 
 
 94.74 
 
 
 56 
 
 56.84 
 
 
 
 22 
 
 94.09 
 
 
 57 
 
 55.37 
 
 
 
 23 
 
 93.41 
 
 
 58 
 
 53.87 
 
 
 TABLE X. 
 
 24 
 
 92.70 
 
 
 59 
 
 52.36 
 
 
 
 25 
 
 91.97 
 
 101.60 
 
 60 
 
 50.84 
 
 102.19 
 
 Shewing the Rate of Inclination of 
 
 26 
 
 91.21 
 
 
 61 
 
 49.30 
 
 
 Inclined Planes, for the following 
 
 27 
 
 90.43 
 
 
 62 
 
 47.74 
 
 
 Angles of Elevation. 
 
 28 
 
 89. 62 
 
 
 63 
 
 46.17 
 
 
 
 29 
 
 88.77 
 
 
 64 
 
 44.58 
 
 
 
 . 
 
 
 Ono in 
 
 
 
 30 
 
 87. 90 
 
 101.67 
 
 65 
 
 42.98 
 
 102. 26 
 
 ingle. 
 
 One ID 
 
 nge. 
 
 lie ii 
 
 Angle. 
 
 DUG in 
 
 31 
 
 87.01 
 
 
 66 
 
 41.37 
 
 
 
 
 
 
 
 
 32 
 
 86.09 
 
 
 67 
 
 39.74 
 
 
 015' 
 
 228 
 
 3 30' 
 
 17 
 
 70' 
 
 8 
 
 33 
 
 85. 14 
 
 
 68 
 
 38.10 
 
 
 0.30 
 
 114 
 
 3.45 
 
 16 
 
 7.30 
 
 n 
 
 34 
 
 84.17 
 
 
 69 
 
 36.45 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0.45 
 
 76 
 
 4. 
 
 15 
 
 8. 
 
 7 
 
 
 1. 
 
 56 
 
 4.15 
 
 14 
 
 9. 
 
 6$ 
 
 " If the equatorial diameter of the 
 
 1.15 
 
 46 
 
 4.30 
 
 13 
 
 10. 
 
 6 
 
 earth be assumed equal to 7924 miles, a 
 
 1.30 
 
 38 
 
 4.45 
 
 12 
 
 11. 
 
 5* 
 
 degree of longitude at the equator will be 
 
 1.45 
 
 32 
 
 5. 
 
 11$ 
 
 12. 
 
 5J 
 
 equal to 69. 15 miles = 305110 feet; and 
 consequently one second in time at the 
 
 2. 
 
 28 
 
 5.15 
 
 11 
 
 13. 
 
 5 
 
 equator will be equal to 1521.3 feet." 
 
 2.15 
 
 26 
 
 5.30 
 
 10$ 
 
 14. 
 
 4$ 
 
 
 2.30 
 
 23 
 
 5.45 
 
 10 
 
 15. 
 
 4 
 
 
 2.45 
 
 21 
 
 6. 
 
 9$ 
 
 16. 
 
 33 
 
 
 3. 
 
 19 
 
 6.30 
 
 9 
 
 17. 
 
 3 
 
 
 3.15 
 
 
 6.45 
 
 8* 
 
 18. 
 
 3*
 
 TABLE XL 
 
 Correction of Moon's Meridional Passage. 
 
 The application of this and the following Table is explained at page 80. 
 
 Long. 
 la 
 
 Argument Daily Change of Mer. Passage. 
 
 Long. 
 in 
 
 Time. 
 
 40 m 
 
 42 m 
 
 44m 
 
 46m 
 
 48"' 
 
 50- 
 
 52" 
 
 54"> 
 
 56" 
 
 58"' 
 
 60> 
 
 62 
 
 64> 
 
 66 
 
 Arc. 
 
 h m 
 
 in 
 
 m 
 
 m 
 
 m 
 
 m 
 
 m 
 
 in 
 
 m 
 
 m 
 
 m 
 
 m 
 
 m 
 
 m 
 
 m 
 
 o 
 
 0. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 20 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 5 
 
 40 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 10 
 
 1. 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 3 
 
 3 
 
 15 
 
 20 
 
 2 
 
 2 
 
 2 
 
 2 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 4 
 
 20 
 
 40 
 
 2(1 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 25 
 
 OA 
 
 V 
 
 20 
 
 4 
 
 4 
 
 4 
 
 4 
 
 5 
 
 5 
 
 5 
 
 5 
 
 5 
 
 5 
 
 6 
 
 6 
 
 6 
 
 6 
 
 OU 
 
 35 
 
 40 
 
 4 
 
 4 
 
 5 
 
 5 
 
 5 
 
 5 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 7 
 
 7 
 
 7 
 
 40 
 
 3. 
 
 5 
 
 5 
 
 5 
 
 6 
 
 6 
 
 6 
 
 6 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 8 
 
 8 
 
 45 
 
 20 
 
 5 
 
 6 
 
 6 
 
 6 
 
 6 
 
 7 
 
 7 
 
 7 
 
 7 
 
 8 
 
 8 
 
 8 
 
 9 
 
 9 
 
 50 
 
 40 
 
 6 
 
 6 
 
 7 
 
 7 
 
 7 
 
 7 
 
 8 
 
 8 
 
 8 
 
 9 
 
 9 
 
 9 
 
 9 
 
 10 
 
 55 
 
 4. 
 
 6 
 
 7 
 
 7 
 
 7 
 
 8 
 
 8 
 
 8 
 
 9 
 
 9 
 
 9 
 
 10 
 
 10 
 
 10 
 
 11 
 
 60 
 
 20 
 
 7 
 
 7 
 
 8 
 
 8 
 
 8 
 
 9 
 
 9 
 
 9 
 
 10 
 
 10 
 
 10 
 
 11 
 
 11 
 
 11 
 
 65 
 
 40 
 
 7 
 
 8 
 
 8 
 
 9 
 
 9 
 
 9 
 
 10 
 
 10 
 
 10 
 
 11 
 
 11 
 
 12 
 
 12 
 
 12 
 
 70 
 
 5. 
 
 8 
 
 9 
 
 9 
 
 9 
 
 10 
 
 10 
 
 10 
 
 11 
 
 11 
 
 12 
 
 12 
 
 12 
 
 13 
 
 13 
 
 75 
 
 20 
 
 9 
 
 9 
 
 9 
 
 10 
 
 10 
 
 11 
 
 11 
 
 12 
 
 12 
 
 12 
 
 13 
 
 13 
 
 14 
 
 14 
 
 80 
 
 40 
 
 9 
 
 10 
 
 10 
 
 11 
 
 11 
 
 11 
 
 12 
 
 12 
 
 13 
 
 13 
 
 14 
 
 14 
 
 14 
 
 15 
 
 85 
 
 6. 
 
 10 
 
 10 
 
 11 
 
 11 
 
 12 
 
 12 
 
 13 
 
 13 
 
 13 
 
 14 
 
 14 
 
 15 
 
 15 
 
 16 
 
 90 
 
 20 
 
 10 
 
 11 
 
 11 
 
 12 
 
 12 
 
 13 
 
 13 
 
 14 
 
 14 
 
 15 
 
 15 
 
 16 
 
 16 
 
 17 
 
 95 
 
 40 
 
 11 
 
 11 
 
 12 
 
 12 
 
 13 
 
 13 
 
 14 
 
 14 
 
 15 
 
 15 
 
 16 
 
 17 
 
 17 
 
 17 
 
 100 
 
 7. 
 
 11 
 
 12 
 
 12 
 
 13 
 
 14 
 
 14 
 
 15 
 
 15 
 
 16 
 
 16 
 
 17 
 
 17 
 
 18 
 
 18 
 
 105 
 
 20 
 
 12 
 
 12 
 
 13 
 
 14 
 
 14 
 
 15 
 
 15 
 
 16 
 
 16 
 
 17 
 
 18 
 
 18 
 
 19 
 
 19 
 
 110 
 
 40 
 
 12 
 
 13 
 
 14 
 
 14 
 
 15 
 
 15 
 
 16 
 
 17 
 
 17 
 
 18 
 
 18 
 
 19 
 
 20 
 
 20 
 
 115 
 
 8. 
 
 13 
 
 14 
 
 14 
 
 15 
 
 15 
 
 16 
 
 17 
 
 17 
 
 18 
 
 19 
 
 19 
 
 20 
 
 20 
 
 21 
 
 120 
 
 20 
 
 13 
 
 14 
 
 15 
 
 15 
 
 16 
 
 17 
 
 17 
 
 18 
 
 19 
 
 19 
 
 20 
 
 21 
 
 21 
 
 22 
 
 125 
 
 40 
 
 )4 
 
 15 
 
 15 
 
 16 
 
 17 
 
 17 
 
 18 
 
 19 
 
 19 
 
 20 
 
 21 
 
 21 
 
 22 
 
 23 
 
 130 
 
 y. o 
 
 14 
 
 15 
 
 16 
 
 17 
 
 17 
 
 18 
 
 19 
 
 20 
 
 20 
 
 21 
 
 22 
 
 22 
 
 23 
 
 24 
 
 135 
 
 20 
 
 15 
 
 16 
 
 17 
 
 17 
 
 18 
 
 19 
 
 20 
 
 20 
 
 21 
 
 22 
 
 22 
 
 23 
 
 24 
 
 25 
 
 140 
 
 40 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 19 
 
 20 
 
 21 
 
 22 
 
 22 
 
 23 
 
 24 
 
 25 
 
 25 
 
 145 
 
 10. 
 
 16 
 
 17 
 
 18 
 
 19 
 
 19 
 
 20 
 
 21 
 
 22 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 26 
 
 150 
 
 20 
 
 16 
 
 18 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 26 
 
 27 
 
 155 
 
 40 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 26 
 
 27 
 
 28 
 
 160 
 
 11. 
 
 17 
 
 19 
 
 20 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 26 
 
 27 
 
 28 
 
 29 
 
 165 
 
 20 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 170 
 
 40 
 
 18 
 
 20 
 
 21 
 
 22 
 
 23 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 175 
 
 12. 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 180 
 
 TABLE XII. 
 
 9 
 
 Effect of a Change of 1 in Declination on the Moon's 
 
 Semidiameter (as given in the Naut. Aim.) 
 
 
 Dec. Corr. 
 
 Dec. 
 
 Con. 
 
 Dec. 
 
 Corr. 
 
 Dec. 
 
 Corr. 
 
 
 o 
 
 
 
 8 
 
 o 
 
 S 
 
 o 
 
 S 
 
 
 o -ooo 
 
 7 
 
 134 
 
 14 
 
 278 
 
 21 
 
 445 
 
 
 1 -019 
 
 8 
 
 154 
 
 15 
 
 300 
 
 22 
 
 472 
 
 
 2 -038 
 
 9 
 
 173 
 
 16 
 
 323 
 
 23 
 
 499 
 
 
 3 '057 
 
 10 
 
 194 
 
 17 
 
 346 
 
 24 
 
 527 
 
 
 4 '076 
 
 11 
 
 214 
 
 18 
 
 368 
 
 25 
 
 557 
 
 
 5 -095 
 
 12 
 
 235 
 
 19 
 
 394 
 
 26 
 
 587 
 
 
 6 '114 
 
 13 
 
 256 
 
 20 
 
 419 
 
 27 
 
 619 
 
 
 7 '134 
 
 14 
 
 278 
 
 21 
 
 445 
 
 28 
 
 652
 
 CATALOGUE 
 
 INSTRUMENTS, 
 
 TROUGHTON AND SIMMS, 
 
 OPTICIANS, 
 
 MATHEMATICAL INSTRUMENT MAKERS 
 
 S?onout:a&fe 33oartf of (Zfkfcnance, 
 
 138, FLEET STREET, 
 LONDON.
 
 A CATALOGUE, 
 
 SPECTACLES AND OPERA GLASSES. 
 
 & s. <L 
 
 Gold Spectacles, Single Joint .... from 3/. 3s. to 5 5 
 
 Ditto, Double Joint from 41. 4s. to G 6 
 
 Ditto, Hand Frames ..... from 41. 4s. to 6 6 
 
 Silver Spectacles, Single Joint .... from 10s. to 12 
 
 Ditto, Double Joint . . . . . from 13s. to 18 
 
 Ditto, Hand Frame 110 
 
 Tortoiseshell Spectacles, Single Joint . . . . .076 
 
 Ditto, Double Joint . 0126 
 
 Ditto, Hand Frame from 7s. 6 d. to 16 
 
 Fine Blue Steel Spectacles, Single Joint . . , . 10 6 
 
 Ditto, Double Joint 0126 
 
 Eye Glasses, Gold Frame .... from 11. Is. to 2 2 
 
 Ditto, Silver Frame from 6s. to 8 
 
 Ditto, Shell Frame from 4*. 6d. to 6 
 
 (If with Brazilian Pebbles, 8s. per pair extra.) 
 
 Spectacle Cases ....... each 010 
 
 Opera Glasses, not Achromatic, in Plated Mountings, from 6s. to 111 6 
 Ditto, Achromatic, with Ivory Bodies and Gilt Mount- 
 ings from 21. 2s. to 5 5 
 
 TELESCOPES. 
 
 One-foot Achromatic (Camp) Telescope, having one Drawer .150 
 Ditto ditto, Portable, with Drawers . . . . . 190 
 
 Ditto ditto, in Electrum 220 
 
 Ditto ditto, larger aperture in Brass . . . . . 1 15 
 
 Eighteen-inch ditto 2 12 6 
 
 Two-feet ditto, Reconnoitering . . . . . . 3 13 6 
 
 Ditto ditto, in Electrum . . . .. . . .550 
 
 Ditto ditto, in Electrum, with Compass, Agate Cap, Needle, &c. 660 
 Ditto ditto, ditto, with four Drawers in Brass .. . . 3 18 
 
 Thirty-inch ditto ditto 550 
 
 Three-feet ditto ditto . 660
 
 4 TROUGHTON AND SIMMS'S CATALOGUE. 
 
 s. (L 
 
 Military Case and Sling . . . from 10s. Cxi. to 012 6 
 
 Portable Brass Stand for any of the above Telescopes, from 21s. to 212 G 
 Walking-Stick Telescope, Portable, with Compass . . 3 13 6 
 
 Ditto, without Compass . . . . . 330 
 
 Ditto, in one length, with Compass . . . .2126 
 
 Ditto, without ditto . . . . . 220 
 
 Dumpy Navy . . . . . .440 
 
 Ditto, with Pancratic eye-piece . . . . 4 14 6 
 
 Two-feet Navy Telescope . . . 2 12 6 
 
 Ditto ditto, Brass Body, covered with Leather . . 2 15 
 
 Ditto ditto, with Spray Shade . . .330 
 
 Three-feet ditto ... 550 
 
 Ditto ditto, with Spray Shade . 5 15 6 
 
 Four-feet ditto, with Two Powers, in Case 1212 
 
 Day or Night Telescope (Deck Glass) . . 440 
 
 Ditto ditto, with Spray Shade . 4 14 6 
 
 Night Glass . . . . . . .330 
 
 Ditto, large size . . . . . . 440 
 
 Ordnance Signal Station Telescope . . . .6166 
 
 Thirty-inch Achromatic Telescope, two-and-a-quartcr-inch 
 Object Glass, mounted on Pillar-and-claw Stand, with a 
 Terrestrial and an Astronomical Eye-piece, in a Maho- 
 gany Case . . . . . .10100 
 
 Ditto, with Vertical Rack Motion . . . 12 12 
 
 Forty-five-inch Achromatic Telescope, two-and-three-quarter- 
 inch Object Glass, on Brass Pillar-and-claw Stand, with 
 a Terrestrial and an Astronomical Eye-piece, in a Maho- 
 gany Case . . . . ... 23 2 
 
 Ditto with Vertical Rack Motion, Finder, and extra Eye- 
 pieces . . . . . . 26 5 
 
 Ditto, with Horizontal Rack, and Steadying Rods, complete 31 10 
 Ditto, three-and-a-quarter-inch Object Glass, with Rack-work 
 Motions, Finder, one Terrestrial and three Astronomical 
 Eye-pieces, in Mahogany Case . . . 42 
 
 Ditto, three-and-three-quarter-inch Object Glass, mounted as 
 
 above . . . . . . 68 5 
 
 Equatorial Stand, instead of Pillar-and-claw to the above 
 Telescopes, constructed to any given Latitude, 30 Guineas 
 extra. 
 Universal Equatorial, with 30-inch Telescope . 80
 
 TllOUGHTON AND SIMMS'S CATALOGUE. 5 
 
 .8. d. 
 
 Completely mounted, Equatorial, with Clock Movement, 
 Micrometers, &c., 5 feet focus, three and three- 
 quarter-inch Object Glass .... 200 
 Ditto ditto, 4-inch Object Glass . . . 230 
 Ditto ditto, 8 feet focus, five-and-a-half-inch ditto . . 400 
 Ditto, 10 feet ditto, 6-inch ditto . . . 600 
 Varley's Stand, Mahogany, with Brass Fittings, capable of car- 
 rying Telescopes from three-and-a-half to seven feet 12 12 
 
 MICROSCOPES, &c. 
 
 Solar Microscopes . . from 61. IGs. 6d. to 21 
 
 Botanic ditto, small size . . . . . 15 
 
 Ditto, ditto . . . . . . 150 
 
 Compound ditto . . . . . . 2126 
 
 Ditto ditto . . . . . . 2 15 
 
 Ditto ditto . . . . ..3100 
 
 Ditto ditto . . . . . . 4 10 
 
 Ditto ditto . . . . . . .5100 
 
 Ditto ditto, larger, from 10/. 10s. upwards. 
 Cloth Microscopes, Diagonal Print Machines, Black Mirrors, 
 Claude Glasses, Magic Lanterns, &c. 
 
 COMPASSES, SEXTANTS, QUADRANTS, &c. 
 
 Binnacle Compasses . . . from 12s. to 018 
 
 Brass Hanging Compasses for Cabins . from II. 10s. to 500 
 
 Common Azimuth Compass . . . . 4 14 6 
 
 Azimuth (Prismatic) Compass, large size, best construction, 
 
 with Tripod Stand complete . . . 16 16 
 
 Ditto ditto, smaller size . . . . .10100 
 Pocket Compasses in Wood, Brass, Metal Gilt, Silver and 
 
 Gold .... from 3s. Gd. to 550 
 
 Ebony Quadrant with Tangent Screw . . . 3 13 6 
 
 Ditto, with ditto and Telescope . . . . 4146 
 
 Ditto ditto, best . . . . . 5 15 6 
 
 Ebony Sextant, with Telescopes . 880 
 
 Ditto, ditto, with Brass Arch, &c. . . 10 10 
 
 Optical Square . . . . . .110 
 
 Box Sextant plain . . . . 3 13 6 
 
 Ditto, with Telescope . 4 14 6
 
 p TROUGHTON AND SIMMS'S CATALOGUE. 
 
 *. d. 
 
 Box Sextant, Ordnance Pattern . . . 550 
 
 Ditto, with Supplementary Arc . . 5 15 6 
 
 Ditto, with ditto, and Levels . . 660 
 
 Leather Case and Strap for Box Sextant . . 090 
 
 Metal Sextant, 4-inch Radius, divided on Silver to 20 seconds 10 10 
 
 Ditto, 5-inch ditto, to 20 seconds . . . . 13 13 
 
 Ditto, 6-inch ditto, to 20 seconds . . . 14 14 
 
 Ditto, 7-inch ditto, to 10 seconds . . . 16 16 
 Ditto, 8-inch ditto, with Double Frames, divided on Silver to 
 
 10 seconds . . . . . 18180 
 
 Ditto ditto, divided on Platina . . . 21 
 
 Ditto ditto, divided on Gold . . . . 23 2 
 
 Dip Sector, as described in ' Treatise on Instruments,' by 
 
 F. W. Simms . 12 12 
 
 Troughton's Reflecting Circle . . . 23 2 
 
 Six-inch Borda's Repeating Circle by Reflexion . 21 
 
 Eight-inch ditto . . 23 2 
 
 Ten-inch ditto ... 25 
 Brass Counterpoise Stand for Circle or Sextant, in Mahogany 
 
 Box . . . . . . 5 15 6 
 
 Glass Plane Artificial Horizons, two-and-a-quarter-inch diameter 1 11 6 
 Ditto, two-and-a-half-inch diameter . . . 220 
 Ditto, three-inch diameter . . 330 
 Ditto, three-and-three-quarter inch . 440 
 Ditto ditto, Ordnance Pattern . . 440 
 Best Mercurial Horizon, with Iron Bottle and Trough, Ord- 
 nance Pattern .... 550 
 
 Ditto ditto, smaller size . 4146 
 
 Ditto ditto, with Brass Folding-roof . 550 
 
 Ditto ditto, with Ebony ditto . . . 4 14 6 
 
 Ditto ditto, with Mahogany ditto . 440 
 
 Marine Horizon (Captain Becher's) . 550 
 
 LEVELS, THEODOLITES, &c. 
 
 Four-inch Pocket Level . . . . "096 
 
 Six-inch ditto . . . . !'> 6 
 
 Eight-inch ditto ... 15 6 
 
 Ten-inch ditto ... 110 
 
 Twelve-Inch ditto . 1116 
 
 Block Level ... 110
 
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 s. d. 
 
 Level with Sights and Socket, in Box . . . 1116 
 
 Portable Levelling Instrument, with Telescope . 880 
 
 Ditto, with Compass, &c. . . . . .990 
 
 Fourteen-inch improved Level, with Round Legs . 11110 
 
 Ditto, with Tripod Stand . . . . 12 12 
 
 Twenty-inch ditto, with Round Legs . . . 13 13 
 
 Ditto ditto, with Tripod Stand . . . 14 14 
 
 Y Levels, with nine-inch Telescope . . . 10 10 
 
 Ditto, with twenty-inch ditto . . . . 16 16 
 
 Ditto, with ditto and Compass . . . . 17 17 
 
 Gravatt's Dumpy Level, without Legs or Compass . 12 12 
 
 Ditto, with Silver Ring Compass and Round Legs . 15 15 
 
 Ditto, with Tripod Stand and Silver Ring Compass . . 16 16 
 
 Ditto, fourteen-inch, with Round Legs and Card Compass 15 15 
 
 Ditto, ditto, with Silver Ring Compass and Tripod Stand . 17 17 
 
 Ditto, large size, complete . . , . 22 
 
 Standard Levelling Instrument . . . 42 
 
 Plane Table, with Sights and Round Legs . . 6166 
 
 Ditto, with Telescope, &c. . . . . . 12 12 
 
 Cylindrical Cross Staff . . . . . 16 
 
 Ditto, with Compass and Leg . , . . 2 12 6 
 
 Circumferenter, in Mahogany, without Legs . . 2 12 6 
 
 Ditto, ditto, larger size . . . . .440 
 
 Ditto, in Brass, with Round Legs from 41. 14s. 6d. to 660 
 
 Ditto, ditto, with Ball and Socket Joint, and Round Legs . 6 16 6 
 Ditto, with Rack Motion, divided to 3 minutes, Ball and 
 
 Socket Joint and Round Legs . . . 10 10 
 
 Ditto, ditto, with Levels and Levelling Plates . . 12 12 
 Best Brass Miners' Compass, with divided Cover, Ball and 
 
 Socket Joint and Legs . . . . 7176 
 Ditto, with Vertical Arc, Telescopic and Plain Sights, Levels 
 
 in Compass, Rack Motion, &c., and Legs complete . 16 16 
 
 Prismatic Compass, plain . . . . 330 
 
 Ditto, with Azimuth Glasses . . . . 3 13 6 
 
 Ditto, three-and-a-half inch, plain . . . 3 13 6 
 
 Ditto, ditto, with Azimuth Glasses . . . 440 
 
 Prismatic Compass, three-and-a-half inch, with Silver Ring 5 5 
 
 Stand for Prismatic Compass, with Ball and Socket Joint 111 6 
 
 Common Theodolite, with Telescope . . 14 14 
 
 Four-inch Cradle Theodolite, divided on Silver . . 16160
 
 g TROUGHTON AND SIMMS'S CATALOGUE. 
 
 8. d. 
 
 Four-inch best Theodolite (Captain Dawson's) . 21~ 
 
 Five-inch Cradle ditto . . . . . 21 
 Five-inch ditto- (best construction,) divided on Silver, with 
 
 Tangent-screw Motions . . . 25 4 
 
 Five-inch ditto ditto, with two Telescopes . . . 31 10 
 Six-inch ditto, with one Telescope, divided to 20 seconds, 
 
 complete ...... 
 
 Six-inch ditto, with two Telescopes 
 
 Six-inch ditto, with Transit Axis and Vertical Circle 
 
 Six-inch ditto (Captain J. T. Botleau's construction,) with 
 
 Axis, Level, &c. . . , . 42 
 
 SBven-inch ditto, with one Telescope . ... 35 14 
 
 Seven-inch ditto, with two Telescopes . . . 45 
 
 Eight-inch ditto, Azimuth and Altitude, with Axis, Level, &c. 5-2 10 
 
 Twelve-inch ditto, for Horizontal Angles only . , 42 
 
 Four-inch ditto (Col. Everest's construction) . . 22- O 
 
 Five-inch ditto, ditto . . . , . 26 5 
 
 Seven-inch ditto, ditto . . . . . 36 15 
 
 Five-and-a-half-inch Kater's Circle, with Stand, complete 35 
 
 Small Kater's Circle, with Stand . . 16 
 
 Level Collimator . . . from 101. 10*. to 15 15 
 (Larger Theodolites, fyc., made to Order.) 
 
 STATION POINTERS, PROTRACTORS, PENTAGRAPHS, ETC. 
 
 Twelve-inch Station Pointer . * . . 6 16 
 
 Eighteen-inch ditto , * 717 
 
 Two^feet ditto . .99 
 
 Thirty-inch ditto . , . . . 12 12 
 
 Three-feet ditto . . . 18 18 
 
 Wollaston's Goniometer . . . . 3 13 
 
 Eight-inch best Brass Circular Protractor, with Clamp and 
 
 Tangent Screw and folding Arms . . 770 
 
 Ditto, ditto, divided vipon Silver . . .880 
 
 Six-inch ditto, with Rack and Pinion . , . 4 14 6 
 
 Ditto, ditto, divided upon Silver . . . . 5 15 6 
 
 Six-inch Semicircular Protractor, with Vernier and Arm 330 
 
 Eight-inch ditto, ditto , . . ' . . 3 13 6 
 
 Fifteen-inch plain Circular Protractor < . . 350 
 
 Eight-inch ditto . . . from I/. 5*. to 1 11 6 
 
 Six-inch ditto 1 1
 
 TROUGHTON AND SIMMS'S CATALOGUE. 9 
 
 s. d. 
 
 Semicircular plain Protractors , . from 1 6s. to 220 
 
 Ivory Protractors .... from 6s. to 15 
 
 Ditto, upon Parallel Rollers . . from 18s. to 150 
 
 Eighteen-inch best Brass Pentagraph . . .550 
 
 Two-feet ditto . . . . . . 660 
 
 Two-and-a-half-feet ditto . . . . .770 
 
 Three-feet ditto . . . . . . 880 
 
 Three-and-a-half-feet ditto . . . . .990 
 
 Trochiameter, for counting the Revolutions of a Carriage-wheel 250 
 
 Leather Case with Strap for ditto . . . 10 
 
 Plain Perambulators (Wood) . . . . 990 
 
 Ditto, Brass-mounted . . . . 12 12 
 
 Best ditto, with Metallic Wheel . . . . 16 16 
 
 Common twelve-feet Levelling-StafF . . .1116 
 
 Best ditto . . . . . . 1 15 
 
 Trough ton's Improved Portable ditto, with Level . . 2 12 6 
 
 Sopwith's ditto, for Reading without an Assistant . 2 12 
 
 Ditto, stronger, with painted divisions . . .330 
 
 Gravatt's Levelling Staff . . . t 440 
 
 Tape Measure, 25 feet, links . . . .070 
 
 Ditto, ditto, decimals . . . . . 080 
 
 Ditto, 33 feet, links < . . . . .080 
 
 Ditto, ditto, decimals . . . . . 090 
 
 Ditto, 50 feet, links . . . . . .0100 
 
 Ditto, ditto, decimals , . . . . 12 
 
 Ditto, 66 feet, links . . . < . 12 
 
 Ditto, ditto, decimals . . . , , 14 
 
 Ditto, 100 feet, links . . . . 16 
 
 Ditto, ditto, decimals . . . . . 18 
 
 Land Chains, 50 feet, and Arrows . . . 13 6 
 
 Ditto, 100 feet, and ditto . from 11. 3s. Gd. to 150 
 Ditto, 66 feut, with two Round Rings between each link and 
 
 Arrows . . . . . . 15 6 
 
 Ditto, ditto, with three Round Rings, &c. . . .0176 
 
 Ditto, ditto, with two Oval Rings, &c. . . 0180 
 
 Ditto, ditto, with three Oval Rings, &c. . . .110 
 
 Standard Chain, 50 feet . . . 41. 4s. and 550 
 
 Ditto, 66 feet . . 51. 5s. and 6 16 6 
 
 Ditto, 100 fcvt . . ... 81. 8s. and 9 19 6 
 
 (Stronger Chains, #c., made to Order.)
 
 JO TROUGHTON AND SIMMS'S CATALOGUE. 
 
 S. d. 
 
 Set of Marquois Scales, in Box . . . 12 6 
 
 Ditto, in Ivory . . . . . . 250 
 
 Ditto, in Brass . . . . . 2 12 6 
 
 Ditto, in Electrum . . . . . 440 
 
 Twelve-inch Ivory Plotting Scales . . from 11*. to 1 1 
 
 Twelve-inch Boxwood ditto . . from 4s. to 070 
 
 Ivory OfFsett and Pocket Scales . from 2s. Gd. to 6 G 
 Gunter's Scale, Brass, 2 feet . . . .220 
 
 Ditto, Boxwood . . . from 5s. to 090 
 
 Ivory folding Rules .... from 12s. to 18 
 
 Boxwood ditto .... from 6s. 6d. to 15 
 
 Gunner's Rules . ... from 3s. to 10 6 
 
 Camera Lucida . . . from I/. 11s. 6d. to 2 12 6 
 
 Stand for ditto from II. Is. to 1 11 6 
 
 Drawing Instruments, in Skin Cases, (Sappers and Miners) 14 
 Ditto, ditto, East India Company's Pattern . . .150 
 
 Ditto, ditto, Woolwich pattern . . . . 1 15 
 
 Ditto, ditto, School Pattern . . . . .220 
 
 Ditto, ditto, Ordnance Pattern . . . . 330 
 
 Ditto, in Mahogany Cases, Addiscombe Pattern . .330 
 
 Ditto, ditto, Admiralty Pattern . . . . 3 13 6 
 
 Ditto, ditto, Sector-jointed Instruments, Parallel Rulers, 
 
 Sector and Protractor . . . 3 13 G 
 
 Ditto, ditto, with Sector double-jointed Dividers . 440 
 
 Ditto, ditto, with proportional Compasses . . 5 15 G 
 
 Ditto, ditto, with Spring Bows . . . . 770 
 
 Ditto, ditto, with Road and Wheel Pens, Needle-holder and 
 
 small Dividers, &c. . . . . .990 
 
 Drawing Instruments in Electrum, packed in Rosewood and 
 
 Mahogany Cases, of the best description, from 51. 5s. to 13 13 
 Ditto, ditto, large Magazine Cases . . 26 5 
 
 Proportional Compasses . . . . .1116 
 
 Ditto, ditto, with Adjusting Screw . . 220 
 
 Plain Beam Compasses . . . . 1 15 
 
 Ditto, with Pen and Pencil Points . .220 
 
 Plain Beam Compasses, Ordnance Pattern . . 2 12 6 
 
 Beam Compasses with Double Adjustments and Divided 
 
 Beam .... from 41. 4s. to 660 
 
 Ditto ditto, Tubular Beam . . from 5/ 5s. to 10 10 
 
 Plain Ebony Parallel Rulers, 12 inches . 046
 
 TROUGHTON AND SIMMSS CATALOGUE. JJ 
 
 *. d. 
 
 Plain Ebony Parallel Rulers, 15 inches . . . 070 
 
 Ditto ditto, 18 inches . . . . . 090 
 
 Ditto ditto, 2 feet . . . . . 12 6 
 
 Ditto, ditto, with Brass Edges, 18 inches . . 14 
 
 Ditto, ditto, ditto, 2 feet ... .110 
 
 Ditto, ditto, ditto, 2 feet 6 inches . . . 1 11 6 
 
 Ditto ditto, ditto, 3 feet . . . . .220 
 
 Rolling Ebony Parallel Rulers, with Brass Edges, 12 inches 100 
 
 Ditto, ditto, ditto, 15 inches . . . .150 
 
 Ditto, ditto, ditto, 18 inches . . . . 1110 
 
 Rolling Ebony Parallel Rulers, with Plain Edges, per inch . 010 
 
 Ditto, ditto, ditto, with Ivory Edges divided . . 016 
 
 HORIZONTAL DIALS MADE TO ANY LATITUDE. 
 Six-inch to 5 minutes . . . .110 
 
 Nine-inch to ditto . . . . . 250 
 
 Twelve-inch to 2 minutes, and Equation Table . . 6166 
 
 Twelve-inch, with Turned Edge, divided to two minutes, and 
 
 Equation Table . . . . .770 
 
 Fifteen-inch, divided to 1 minute, without Turned Edge 717 6 
 
 Eighteen-inch, divided to 1 minute, 32 Points Lettered, 
 
 Equation Table, &c., &c. . . . 18180 
 
 (Larger to Order. ) 
 
 UNIVERSAL JOINT DIALS. 
 
 Two-and-a-half-inch, in Case complete . . . 220 
 
 Three-and-a-hulf-inch ditto . . . . 2 12 6 
 
 Four-and-a-half-inch ditto, with Levels in Compass . 4 14 6 
 
 TRANSITS AND CIRCLES. 
 
 Twenty-inch Transit Instrument, with Iron Stand 21 
 
 Ditto, ditto, with graduated Scale to Level, &c. . . 23 2 
 
 Two-feet ditto, with Portable Brass Stand . . 26 5 
 
 Two-and-a-half-feet ditto, with Iron Stand . . 42 
 
 Ditto, ditto, Improved . 47 5 
 
 Three-and-a-half ditto, constructed for fixing upon Stone 
 
 Piers, complete . . . . 84 
 
 Variation Transit, best construction . . . . 63 
 
 Dipping Needle, ditto . . . . . 30 
 
 Annular Micrometer, with Eye-piece . 150
 
 TROUGHTON AND SIMMS'S CATALOGUE. 
 
 
 
 
 s. 
 
 d. 
 
 Parallel Wire Micrometer, with Eye-piece, 81. 8s., 121. 12s., and 
 
 15 
 
 15 
 
 
 
 Twelve-inch improved Altitude and Azimuth Instrument 
 
 
 
 
 divided on Silver, the Azimuth Circle Reading by 
 
 
 
 
 Verniers, and the Altitude by Micrometers 
 
 105 
 
 
 
 
 
 Fifteen-inch ditto, both Circles with Reading Micrometers 
 
 130 
 
 
 
 
 
 Ditto, the Altitude Circle 18, and the Azimuth 15 inches with 
 
 
 
 
 Micrometers ..... 
 
 150 
 
 
 
 
 
 Ditto, both Circles 1 8 inches .... 
 
 210 
 
 
 
 
 
 Twelve-inch Repeating Circle (Borda's) 
 
 84 
 
 
 
 
 
 Eigh teen-inch ditto (Borda's) .... 
 
 105 
 
 
 
 
 
 (Larger Transits and Circles to Order.) 
 
 
 
 
 Tropical Tempest Sympiesometer .... 
 
 5 
 
 5 
 
 
 
 Sympiesometer ...... 
 
 4 
 
 4 
 
 
 
 Marine Barometers. . . . from 41. 4s. to 
 
 7 
 
 17 
 
 6 
 
 Chamber ditto .... from 31. 3s. to 
 
 6 
 
 C 
 
 
 
 Best ditto ditto, with Float Gauge .... 
 
 8 
 
 8 
 
 
 
 Standard Siphon Barometer .... 
 
 16 
 
 16 
 
 
 
 Wheel Barometer . . . . from 41. 4s. to 
 
 5 
 
 5 
 
 
 
 Mountain Barometer (Englefield's construction) 
 
 4 
 
 14 
 
 6 
 
 Ditto, ditto, with Long Scale .... 
 
 5 
 
 
 
 
 
 Leather Case for ditto ..... 
 
 1 
 
 1 
 
 
 
 Mountain Barometer (Gay Lussac's) 
 
 7 
 
 17 
 
 (i 
 
 Ditto, ditto, with Wire Stand, Ordnance Pattern 
 
 7 
 
 17 
 
 G 
 
 Ditto, ditto, Troughton's best construction 
 
 12 
 
 12 
 
 
 
 Wollaston's Thermometer, with Apparatus for .Boiling Water, &c. 
 
 4 
 
 4 
 
 
 
 Thermometers, various . . . from 5s. to 
 
 1 
 
 15 
 
 
 
 Standard Thermometers . . 21. 12s. 6d. and 
 
 3 
 
 3 
 
 
 
 Six's Self-Registering Thermometers . from 11. 10s. to 
 
 2 
 
 2 
 
 
 
 Horizontal ditto (Maximum and Minimum) from 15s. to 
 
 1 
 
 11 
 
 G 
 
 Day or Night ditto, singly . . from 7*. Gd. to 
 
 
 
 13 
 
 
 
 Hygrometer, Pocket, Brass ..... 
 
 
 
 13 
 
 
 
 Ditto, ditto, Gilt . . 
 
 
 
 18 
 
 
 
 Ditto (Mason's) Wet and Dry Bulb .... 
 
 
 
 18 
 
 
 
 Ditto, ditto, in Case, for Travellers 
 
 1 
 
 5 
 
 
 
 Ditto, Daniell's ...... 
 
 2 
 
 12 
 
 6 
 
 Ditto, ditto, large Size . 
 
 4 
 
 4 
 
 
 
 Rain- Gauge, Common ..... 
 
 1 
 
 15 
 
 
 
 Ditto, Best 
 
 4 
 
 14 
 
 6 
 
 Whewell's Anemometer 
 
 12 
 
 12 
 

 
 TROUGHTON AND SIMMS'S CATALOGUE. J3 
 
 *. d. 
 
 Minimum Thennometer and Parabolic Reflector . 2 10 
 
 Photometer (Leslie's) .... 330 
 
 Geothermometer . ..220 
 
 Professor Leslie's Machine for making Ice . . 78 
 
 A ditto for ditto, with one Plate . . . . 48 
 
 A large Air-Puinp, on a Stand, with Barometer Gauge 22 
 
 A large Table ditto, with Siphon Gauge . . 15 15 
 
 A Middle Size ditto, with ditto ... 990 
 
 A Small ditto, with ditto . . . . 6 16 
 
 A Single Barrel ditto . from II. lls. 6d. to 440 
 
 APPARATUS. 
 
 Guinea and Featfier Experiment, Receiver included . . 2 15 
 
 A set of Windmills . . . from II. 15s. to 2 12 6 
 
 Apparatus for Freezing Water . . . . 140 
 A Bell for proving that without Air there is no sound, 
 
 from 10*. 6d. to . . . . 1 11 6 
 
 Brass Hemispheres, to demonstrate external Pressure, from 20s. to 1 18 
 
 Model of a Water Pump . . . . 1116 
 Double Transferrer ' . . . . .330 
 
 Single Transferrer, with Fountain Pipe . . . 150 
 Glass Vessel for Fountain in Vacuo . . . .070 
 
 Six Breaking Squares, Cage and Cap . . . 0180 
 
 Apparatus for striking Steel and Flint in Vacuo . . 0180 
 Copper Bottle, Beam, and Stand, for weighing Air, and 
 
 other Experiments . . . . 330 
 
 Model of Forcing Pumps for a constant stream, with glass barrels 330 
 Gun Lock Experiment . . . . .110 
 
 Bacchus ditto . . . . . . 1 14 6 
 
 Small-sized Japanned Copper Fountain, with Syringe and 5 Jets 700 
 
 Torricellian Experiments . . . . . 10 6 
 
 A twelve-inch Electrical Plate Machine, packed with Medical 
 
 Apparatus . . . . 7 10 
 
 A fifteen-inch ditto, packed . . . . . 10 10 
 
 An eighteen-inch ditto, packed . . . 12 12 
 
 A two-feet ditto, packed . . . . . 18 18 
 
 A Cylinder Machine, 16 by 10, packed . . 12 12 
 
 A ditto ditto, 14 by 8, packed . . 10 10 
 
 A ditto ditto, 12 by 7, packed . . . 7176 
 (Larger Maehines made to Order.)
 
 14, TROUGHTON AND SIMMS'S CATALOGUE- 
 
 APPARATUS. 
 
 L s. J. 
 
 A Universal Discharger and Press . . . .1100 
 
 Jointed Discharger, with Glass Handles . . 0126 
 
 Ditto, plain .... from 4*. 6J. to 086 
 
 Exhausted Flask for showing the Aurora Borealis . 086 
 
 Efectrical Batteries of combined Jars . from 21. 12s. 6d. to 10 10 
 Cuthbertson's Improved Electrometer, with Grain Weight 2 12 G 
 Bennet's Gold Leaf Electrometer . . . 18 
 
 Cavallo's Bottle Electrometer, for Atmospherical purposes, 
 
 from 12*. to . . . . 4 14 C 
 
 Quadrant Electrometer, with divided Arch . . 090 
 
 Kinnersley's Electrometer . . . . .110 
 
 Coulomb's Electrometer . . . . 1 16 
 
 Pith-Ball ditto . . . . . 16 
 
 Luminous Conductors . . . from 12s. to 100 
 
 A Thunder-house, for showing the use of Conductors . 080 
 
 Ditto ditto, with Draw . ..096 
 
 A Powder-house for ditto . . . . 10 
 
 An Obelisk or Pyramid for ditto . . . 0106 
 
 A Magic Picture for giving Shocks . from 7s. 6d. to 016 6 
 
 Spiral Tubes to illuminate by the Spark . from 8s. to 0106 
 
 A Set of 5 Spiral Tubes on a Stand . . . 1 16 
 
 Ditto, with a Dome . . . . .-280 
 
 Luminous Names or Words . from 10s. Qd. to 1116 
 
 A Set of 3 Plain Bells . . . . . 10 6 
 
 A Set of 8 Bells, containing the Gamut . . 1 14 
 
 Diamond or Spotted Jars . . from 8s. to 016 
 
 A Double Jar for explaining the Franklinian Theory, from 1 8s. to 1 8 
 An Electrical Cannon . . . . 18 
 
 A Brass Electrical Pistol . . . . 096 
 
 Copper Plates and Stand for Dancing Images . . 10 6 
 
 A Small Head with Hair . 080 
 
 An Artificial Spider . . . . .016 
 
 Sportsman and Birds . . . . . 1160 
 
 Atwood's Machine for Demonstrating the Law of Acceler- 
 ation in Falling Bodies . . from 20/. to 30 
 Working Model of Locomotive Engine, 4 wheels . 25 
 Ditto, 6 wheels . . . . 40 
 Model of Bramah's Hydrostatic Press . . . 15 15
 
 TROUGHTON AND SIMMS'S CATALOGUE, J5 
 
 s. d. 
 A Syren . . . . . .220 
 
 A Small Still with Worm, Tub and Lamp . 2 18 
 
 Set of Lactometers in Box . . . . 18 
 
 Glass Model of a Diving Bell . from 21. 12s. 6d. to 770 
 
 Whirling Table complete . . . . . 30 
 
 Hydrostatic Balance . from 21. 2s., 41. 14s. 6d. to 8186 
 
 Hydrostatic Paradox . . . . . 770 
 
 Model of the Centrifugal Pump . . from 4/. 10s. to 770 
 
 Sectional Model of a Steam Engine . . . 18 
 
 (Models of Steam Engines, Machinery, Sec., made to Order.) 
 
 Bar Magnets for correcting the Derangement of the Compass 
 in Iron Vessels, 2 feet each, 11. Is. ; 14 inches each, 
 II. Is.; 8 inches . . . . 10 
 
 Magnetometers, Collimating and Referring Telescopes, for use 
 in Magnetic and other Observatories. 
 
 BOOKS. 
 
 s. d. 
 Dr. Pearson's work on Practical Astronomy, with 31- Copper 
 
 Plates, in 2 vols. 4to. . . . .770 
 
 A Treatise on the Principal Mathematical Instruments 
 employed in Surveying, Levelling, and Astronomy ; 
 explaining their Construction, Adjustments, and Use. 
 With an Appendix and Tables. By FREDERICK W. 
 SIMMS, F.R.A.S., F.G.S., M. INS. C.E. . . 060 
 
 Practical Tunnelling ; explaining the setting out, the con- 
 struction, and cost of such works ; exemplified by the 
 particulars of Blechingley and Saltwood Tunnels. 
 With 12 Copper Plates and 45 Wood-cut Illustra- 
 tions. By FREDERICK W. SIMMS, F.R.A.S., F.G.S., 
 M. INS. C.E. 1 1 
 
 TROUGHTON AND SIMMS beg to caution those who may have occa- 
 sion to write from abroad, that no reliance can be placed on the genuine- 
 ness of the Instruments they obtain, unless the application be made direct, 
 or through the most respectable channels. 
 
 Tyler & Reed, Printers, Bolt court, London.
 
 .*f CALIFORNIA LIBRARY 
 Los Angeles 
 
 University of California 
 
 SOUTHERN REGIONAL LIBRARY FACILITY 
 
 405 Hilgard Avenue, Los Angeles, CA 90024-1388 
 
 Return this material to the library 
 
 from which it was borrowed.
 
 A 000 959 537 2 
 
 TA. 
 562 
 S59t 
 1844
 
 
 Unive: 
 
 Soi 
 
 Li