GIFT OF Mrs. S. S. Montague A TREATISE ON THE PRINCIPAL 1 MATHEMATICAL INSTRUMENTS EM PLOY BD IN SURVEYING, LEVELLING, AND ASTRONOMY, tfe. tp. A TREATISE ON THE PRINCIPAL MATHEMATICAL INSTRUMENTS EMPLOYED IN SURVEYING, LEVELLING, AND ASTRONOMY; EXPLAINING THEIR CONSTRUCTION, ADJUSTMENTS, AND USE : aitb FREDERICK W. SIMMS, F.R.A.S., F.G.S., M.INS.C.E, CIVIL ENGINEER, AUTHOR OF "A TREATISB ON PRACTICAL IUNX ELLIVG," ETC. ETC. EIGHTH EDITION v LONDON : TROUGHTON AND SIMMS, 138, FLEET STREET. M.DCCC.L. entmi at >tatfonere WS. 8. S.- - --7, 7 fe ^ / V PALMER & HOBY, 17, Brownlow Street, Holbofn. 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 requsite information with respect to such instruments can only be obtained by con- sulting various expensive publications, which are not within the 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. ft VI 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only have been described which are applied in modern practice, no reference being made to those which, having been superseded by better ones, may be said to be out of use.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 formulae. 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 preceding part of this treatise. PREFACE 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 formulae have likewise been given in the account of the Portable Transit Instrument, which it is hoped will not be without their use. Vlll 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, P. W. S. Feb. 17th, 1836. CONTENTS, SURVEYING INSTRUMENTS. The Land Chain Page 1 The Surveying Cross and Optical Square 8 The Prismatic Compass ib. The Vernier 5 Tfie 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 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 ib. CONTENTS. ASTRONOMICAL INSTRUMENTS. The Sextant Page 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'8 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 60 The 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 92 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 109 Method of finding Differences of Latitude and Longitude by Trigono- metrical Measurement Ill Method of finding the Longitude by the Eclipses of Jupiter's Satellites . . 112 CONTENTS. XI APPENDIX. On Protracting and Plotting, $c., 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 129 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 different 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'sMeridian 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, contain 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. 2 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 be 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 be 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 to 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. 6 being the hypothenuse, and the horizontal line the base ; or it may be more expeditiously accomplished by our Table IX., which shews 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 B2 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 prism 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 the azimuth of the sun, a dark glass must be interposed ; and the coloured glasses, represented at G, arc intended for that pur- THE VERNIER. 5 pose, the joint upon which they act allowing them to be turned down over the sloping side of the prism-box. At e is shewn 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 be required to measure to hundredths of an inch the parts of a scale which is graduated to tenths, it may be done by means of a scale whose length is nine-tenths of an inch, and divided into ten equal parts ; or by one whose length is eleven- 6 SURVEYING INSTRUMENTS. tenths of an inch, and divided into ten 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 be 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 be 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 be divided into 20' the vernier is divided into 20'' ; if the former be 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 shewing 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. 7 coincide with the second, it will be 60 2O, 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 be 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. Fig. 1. I I I I T Fig. 2. 1-1 Illll 1 1 6 1 1 2 1 I j 6 1 1 I 1 I I I I I I 1111111 Fig. 3. 1 1 1 1 rn t 1 2 1 1 1 1 J r Fig. 4. 1 1 ".I ' ' I ' ' I 420 30 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 rabetted, 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 compliment 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 10 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, like 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. 11 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 be 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 be fixed, the distance, IK, 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 clamp 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 be fixed, the exact length must be laid off, otherwise the point K may be assumed at pleasure on the line so drawn. p c 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 farther 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 PLANE TABLE. 13 the fiducial edge of the index. Then direct in the same manner to B, and draw the line I B ; and so proceed with whatever objects 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 A B C, &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 ABC, &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, Testing 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, , 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 vernier. 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. 15 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 staff head, which is connected by brass j oints 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 16 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 shewn 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, fj which, being raised or lowered, will set the level parallel to the optical axis of the telescope, or line of collimation ; the screw, ff, 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 clipsf 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 adjustment. 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. 17 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 shewn 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 be 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 c 18 SURVEYING INSTRUMENTS. must be repeated carefully, until the adjustment is satisfactory. 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 retire to one end, bring it back one half, by the screw f, 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 return 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 point 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 THE THEODOLITE. 19 after having made the adjustment, or, what is perhaps better, 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 proportionably 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 15), 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 along 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 be 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 circum- THE THEODOLITE. 21 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 clamp 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, moveable 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 flat 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, 1 1, 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, Gr, 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 be 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, 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 = 142 36' 30" B = 37 Mean = 142 36 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 = 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, arid 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 be 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 hah 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 proof of the accuracy of a number of horizontal angles, * 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. 25 if they quite surround the station from whence they are taken, is to add them altogether, and their sums, if correct, will be 360. If they be 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 have 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 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 shews 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 shews 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 foot-screws 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 shewn 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. 26 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, 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, shewn 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 requires repeated trial 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 distinct vision 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 and from them. 28 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 instrument, 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. 29 Y level and the theodolite). On the top of the telescope, and partly imbedded within its tube, is the spirit-level, c d, 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, a b, 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 shewn 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 hundredths or two-hundredths of an inch ; and it is so fixed, that the divided edge intersects the line of collimatiou, 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 shewn hereafter. It is also very useful in roughly estimating equal distances from the instrument in any direction. Thus, if a man in attendance hold 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 move 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 shew objects in- verted, and as such a telescope requires fewer glasses than one which shews 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, e, f, which attach the telescope to the horizontal bar, and the other half by the parallel plate-screws : the capstan-screws, e, /, 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 shews 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 give 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 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 be correct ; but if it be incorrect, the direction of the line of sight will be either above or below c d, as is shewn by the dotted lines. If above it, the difference will be greater than two feet, and the TROUGHTON'S IMPROVED LEVEL. 31 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 may be 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 TROUGH-TON'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 adjustment, 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 collimation 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 lengh 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. MIL GRAVATTS LEVEL. The following 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 shewn in the cut,) secures to the slide and the eye-piece a steady and parallel motion when adjusting GRAVATTTS LEVEL. 33 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 staff, and adjusted for distinct vision, the two levels at once shew 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, but in all cases the requisite marks or divisions are made on the bubble-tube. 34 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 shewn 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 nse, 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 0, 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 , and also the staff 35 B, placed on the stake b; call the two readings, A' and B': then, although the instrument be out of adjustment, yet the points read 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 t then ought C" C' (A" A') be equal 2 [B"-B'-(A"-A')]j 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 T^yth of a foot." EXAMPLE. The instrument being placed half-way between two stakes, a and b (at one chain from each,) the staff on a or A' read 6*53, and staff on b or B' read 3'34 ; placing the instrument half-way between 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 stake 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 , but the closer the better,) the staff on a or A" read 4*01, on b or B", 1-03 and on c or C ', 3'07. Now, had the instrument been D2 36 LEVELLING INSTRUMENTS. in complete adjustment (under which term curvature and refrac- tion are included), when the reading on staff a was 4-01, the readings on b and c should have been respectively 0'82 aud 2-12. The instrument therefore points upwards, the error at b being 0*21, 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 0*21 = 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, 1'84. Now 0-82-075=0-07, and 2'12-l'84=0-28. And as 4x0-07=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 shewn 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- 1 LEVELLING STAVES. 37 ment be great, requires to be corrected for the curvature of the earth. The method of computing this correction will be pre- sently shewn. 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 shewn 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. THE NEW LEVELLING-STAVES. Several years ago, WILLIAM GBAVATT, 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 most practical purposes, thus securing greater certainty and 38 LEVELLING INSTRUMENTS. expedition in the work : for it not ^infrequently 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, a the distance B C or B D ; also F E is the excess of height a F ; G H, that at G, &c. ON LEVELLING. 39 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, 2 A C + C D : BD: : B D : 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 A C may be taken for 2 A C -f- C D in this proportion without sensible error. The propor- tion then will be 2 A C : B D : : B D : CD; B D 2 B C 2 whence D C is = 2 A C or % ^Q 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 T> r\% TJ excess 9 An is Tgjg f a m il e ^ which is 8,004 inches for the 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 shews 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 shew 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, 11 '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 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 shews 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 urifrequently 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 shews 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 Station. Back Station. Forward Station. Reduced Levels. Bearing, Distance. Staff. Staff. Distance. Bearing. 1 30020 140 Ft. 2-15 Ft. 14-97 358 o 120-10 100-00 14-97 85-03 2-15 2 300-40 89 0-50 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 0-90 215 120-0 29-28 3-78 2550 10-50 7 300-30 221 8-84 128 119-40 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 t 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, 7072, 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 shews 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 shewn 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. 43 known hereafter. Their use is chiefly for future reference, in the event of its being necessary either to check the levels by repetition, to 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 shewn 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. No. of Back Station. Back Station. Dist. of Instru- ment from Correct for Curvat. Height of Instru- Forward Station. Dist. of Instru- ment from Correct for Cur vat. Remarks. Station. Station. Ft.InJOec. 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. Ft. In. Sum of forward stations, corrected for curva- 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 = 18957 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 that 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*8195430, when the sum, rejecting 10 from the index, will be the log. of tire difference of level, in feet. ON LEVELLING. 45 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 theo- dolite 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 46 LEVELLING INSTRUMENTS. the height 38 feet, 760 divided by 38 gives 20, shewing the rate ofin clination 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, shews 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 oflice ; 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 barometer, to indicate its temperature ; while another, detached LEVELLING WITH THE MOUNTAIN BAROMETER. 47 from the instrument, is employed at the same time, to shew 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 ' 'te , . ; . V. ;: , ; 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 noarly as well as if simultaneous observations were made by a barometer at each station. 49 ASTRONOMICAL INSTRUMENTS. THE SEXTANT. IT was our intention, before describing the 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 $ s an index, A G, (to which is attacked a mirror at A,) movable about A as a cen- \ i 1 tre, and denoting the angle it has moved through on the arc, B C; also let the ?.?. -\- 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, B A G (or D A F) = the incident an- gle, S A I, and the reflected angle, b D E = the incident a D A = D A E = DEA, because a b is parallel to A C. Now H A I = D F A = (F A E + F E A,) and D A E being equal to D E A, it follows that H A I = (D A E + F A E.) From H A I and (DAE + F A E ) take the equal angles, S A I and D A F, 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 G AC. 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 the 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. 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 shewing the objects erect, and the other inverting them ; the last has a greater magnifying power, shewing 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 the 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 shewn at O, is fixed at the back of the instrument. The hole in the middle 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. E 2 52 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 division 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 require 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 be 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. 53 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 observation. 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 ; .' , . off the arc jj^ . Difference . , . ...,, , Index error , ? ," > . 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 semidiameter, 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 from 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 remain 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. A s 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 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 be 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. Obs. alt. of the sun's lower limb Index error \" V = 61 i 13 M 5 i 7 I/ Apparent altitude f\i 19 4- : . . ~~61 12 46,47 Semidiameter .-e.a*.^. uirf.1 + 15 46,91 Parallax ' ;i>>O I cT r-*G r- CO CO <* r-i 80 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 be 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 to, or subtracting from, as the case may be, 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 interval 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 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. co-sine 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 From the first wire to the third = 36,647 second = 18,305 fourth = 18,309 fifth = 36,606 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. 81 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 be 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 shewn by the clock will be the right ascension of the star as given in the Almanac. The difference therefore between the time shewn 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) when it is less. , Thus, on March 18th, 1834, (page 79,) H. 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 shewn, 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 sun's 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 on each day, the mean of the whole will pro- G 82 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 have 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, shewing 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) for the interval j 5^,78 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. 8. Acceleration for 5 hours 49,148 22 minutes .... 3,604 9 seconds .... 0,025 65 hundredths . . . 0,003 For the whole interval = 52,780 * 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. 83 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. Mean time H. . . . 5 M. 91 s. 16,87 Table IV . . . 4- 52,78 Sidereal interval . . . 5 9,9, 9,65 Sidereal time at mean noon 23 49 15 64 29 4 25,29 - 24 Sidereal time of the star's passage, or | ~ 4 25 9Q its right ascension J " 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 shew 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 shewn 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 84 . 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 of 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, ir 15 c = the error of collimation, + if the deviation be 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 wa*nt 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, shewn by the air-bubble receding from the centre towards that pivot which is the highest : but, as it is necessary, when correcting for the adjustment, 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. 85 Readings of the Scale. East End. West End. tt n 109,0 .... 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 J difference = 0,23 = the amount of deviation in arc, shewing 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. (n X) 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. X = 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 X) co-sec. IT 96 ASTRONOMICAL INSTRUMENTS. in which a = the factor for azimuthal deviation, + when the instrument deviates to the eastward of the south meridian. TT = the polar distance of the star. X = 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. 8. Observed time of transit . . . = 14 35 4, 86 Error of collimation . . 12" or = + 0, 80 Inclination of the axis . . . = 1, 75 Deviation of instrument in azimuth . = 4,737 The errors are in units, each of which =15" Polar distance . . . = 62 12 Co-latitude . . . . = 38 27 The Correction for the Collimation. Deviation = + 0,80 log. . . . . = + 9-90309 Polar dist. 62 12' co-secant . = -I- 0-05326 Correction = + 8 ,904 . . log. = + 9'95635 The Correction for the Level. Deviation = -1,75 log = - 0-24304 Polar dist. 62 12' co-secant . . . = + 0-05326 Polar dist. minus co-lat. 23 45' co-s. . . = + 9-96157 Correction =-! 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 8 ,157 . log. = - 0-33379 Now apply the sum of these corrections to the observed time of the star's transit, and the actual time of transit will be obtained THE PORTABLE TRANSIT-INSTRUMENT. 87 as correctly as if the instrument had been in a state of perfect adjustment when the observation was made. H. M. S. Observed time of transit = 14 35 4,860 Correction for the collimation . . = + 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 interval of her transit over two different meridians her right ascension varies, and the difference between the two compared differences exhibits the amount of this variation, which, added to the difference of meridians, shews 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 ascension, 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 Eoyal 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. 88 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 be 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 shew the signs with which these corrections are to be applied. Sign of First Correction. Limb Moon. Obsd. -, T , r preceding iW M ? ns star IE semldlam - i following iW star 3 E i ^ Cor- rect. + + Moon. Limb Cor- Obsd. rect. _ decreasing star 8 | E + Sign of Second Correction. Moon's declination increasing Limb Cor- Moon. Obsd. rect. preceding! W + star JE - following ) W star JE + Limb Cor- Moon. Obsd. rect. Moon's f Feeding W ~ 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; or, THE PORTABLE TRANSIT-INSTRUMENT. 89 if more than one star have 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 360th 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 F., 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 f parts ' opposite the corresponding tenths. Thus, for l m 42 8 ,57, the log. for l m 42 8 ,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 at Greenwich. Rate of Clock. Clock Transit observed at Cambridge. Rate of Clock. 96 Aquarii .... n Piscium .... j) 's 1st Limb . . s Piscium .... nCoeti , H. M. 8. 23 10 58,18 23 39 35,32 23 47 29,86 23 57 1,18 21 45 08 s. 0,68 H. M. S. 23 10 14,40 23 38 51,72 23 46 45,52 23 56 17,53 21 1 32 s. 2,56 First find the mean intervals between the passage of the stars and the moon at both places, thus : Greenwich Intervals. M. 36 6 9 34 36 7 9 34 31,68 54,54 31,32 15,22 Cambridge Intervals. 36 31,12 7 53,80 9 32,01 34 15,80 Intervals corrected for Rate. 31,70 36 31,18 54,54 7 53,81 31,32 9 32,03 15,23 34 15,85 90 ASTRONOMICAL INSTRUMENTS. Mean On December 8th, 1834, the moon passed the meridian of Greenwich at 6 h 40 m , the decimation being then about 7, and it would be about one-thousandth of a degree different at Cam- bridge ; and the 1000th 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. 8. Hourly change of j) 'a R.A. from 5 hours to 6 . 1 50,19 6 7 . 1 50,03 7 8 . 1 49,87 Hence, at 6 h 40 m the hourly rate of change would be about l m 50 3 ,08. 1* 50,08 Table V 1-502334 ,65 log =9-812913 Longitude of Cambridge in time 21 a ,3 1-315247 The longitude of the Cambridge Observatory has been deter- mined by Professor AIRY to be23",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 observation 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 y Draconis, 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 more 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 <-- 1 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 frpm the Nautical Almanac) . X = the co-latitude of the place, then tan. X = tan. -n 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 IJZ ASTRONOMICAL INSTRUMENTS. on various nights, the telescope should be reversed. With these precautions, and a level of the best kind, the latitude may be obtained within a second or two, if the place of the star be sufficiently 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 preceding diagram) the angle, S P Z ; which time, subtracted from that of the meridian passage, will give the time of the star being on the prime vertical, or in the required position for making the first observation. Formula, co-s. S P Z (or A") = tan. P.Z, co-t. 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 j 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. 93 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 centered 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 shewn at I). This motion should, however, be omitted in instruments destined for exact work, as the bringing the zero into the meridian is not requisite 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 shew the de- gree and nearest five minutes to be read off on the azimuth circle j 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 being 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 bearing by two friction rollers, on which it rests; one of which is shewn 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 shewn at L. A similar contrivance for the azimuth circle is re- presented at M. The reading microscopes for the vertical circle, 94 ASTRONOMICAL INSTRUMENTS. are carried by two arms, bent upwards near their extremities, and attached towards the top of one of the pillars. The projecting arms are shewn 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 moveable 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 : and 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 horizont jl 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 shews when the vertical axis is set perpendicular to the horizon. A scale, usually shewing 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 have 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 moveable, up or down, by means of the screw, /, for the purpose of adjusting the bubble. A striding-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 shewn by the divisions on the scale attached to the level. Having done this, turn the instrument half round in azimuth, and if the axis be truly perpendicular the bubble will again settle in the middle of the tube, but if not, the amount of de- viation will shew 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 have 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." 96 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 be 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 microscopes into a diameter, and finally level again. The adjustment for the line 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 Z3nith 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 be 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 the maker, and cannot be adjusted by the observer. THE ALTITUDE AND AZIMUTH INSTRUMENT. 97 justed by their proper screws, so as to shew 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 effect 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 H 98 ASTRONOMICAL INSTRUMENTS. 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 y 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, b, working in the box, c c'. Fig. 2 shews the field of view, with the magnified divisions on the instrument, 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 shewn in the figure. On the left hand of the diaphragm appears the comb or scale of minutes, each of the teeth representing one minute. Moveable with the wires along the comb is a small index or pointer, e y which in the figure is represented at zero, the centre of the scale, as is shewn 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 shewn 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 shewn 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 be also well defined, and 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 have 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' 10" 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 versd. 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 geodetical 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 2 100 ASTRONOMICAL INSTRUMENTS. ing the same angle upon different parts of the arc ; thus, Af te r 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 before 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 versd 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 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 422, of the star Arcturus, as observed at the Dublin Observatory. Face of Inst. Observed Alt. Hour Angle in Time, Versed Sine. East of meridian -I E. E. o t 'I 56 40 5,2 42 -22,9 m. 8. 10 35 3 6 35 3 1067 0413 West of meridians W. W. 45 10,0 43 23,1 2 47 7 7 48 7 0074 0580 Means 564245,3 533,5 Reduction.. + 1 52,4 / 56 44 37,7 Declination !! 20 7 37,8 Mer. Z. D.= 33 16 Refraction , , . . Meridian Alt. . 56 43 59,9 log. cos. cos. Constant log. ' " it Reduction.. 152,4=112,4 = 2-7271 = 9-7756 = 9-9727 = 0-2608 = 9-3144 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 102 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 to 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 refractions. The latitude thus obtained does not depend on a previous 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. WOLLASTON, 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 towards 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 shewn 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 j ust 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. 103 upper limb arrives at the fourth horizontal one : setting that time clown 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 have 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 at 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 have 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 shew 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:* T 1440 tan. T =B _ 2 correction = + A . 8 . tan. L + B . 5 . tan. D in which L = the latitude of the place (minus when south). 8 the double daily variation in the srn'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 Kule. 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. BAILY'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. fji, in the page relating to true noon. THE ALTITUDE AND AZIMUTH INSTRUMENT. 105 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, B, 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 SUIT'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, fyc., p. 227.) On July 25, 1823, in N. Lat. 54 20' at 8 h 59 m 4 s A.M., and at 3 h O m 40 3 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 6h im 3 g8^ wmcn converted into arc = 90 24', and by the Nau- tical Almanac the declination of the sun at noon on that day was + 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 T 2 = 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 6 = - 1529" log. = - 3*1844 log. - 3*1844 L = + 54 20' tan. + 0*1441 D = 19 48' tan. -f 9*5563 + 12 8 ,57 = + 1*0992 . . - 2 8 ,29 = - 0*3594 correction = + 12 8 ,57 - 2 8 ,29 = + 10 8 ,28 106 ASTRONOMICAL INSTRUMENTS. This value being added to the mean of the times of the observed altitudes, or J (20 h 59 m 4" + 27 h O m 40") = 23 h 59 m 52 s , will give O h O m 2 8 ,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. This 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. /rp/ rm Correction= J (D' D) sec. Lat. co-sec.-^ a 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. 107 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 = 5 6", 77, 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. = 51 28' 39" . . . .sec. = 0-2056388 co-sec.= 0-2155529 Correction = 374",31 . . . log. = 2'5732353 = 6' 14",31 ' '' Reading of the middle point . . . . = 81 23 15 Correction - 6 14,31 Reading of the instrument when set ( QI 1 7 n o to the true meridian . . . . i Another, and an easy method of finding a meridian line, where dependence can be placed upon the time shewn 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 be 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 riot be visible through telescopes of small size. To make a single observation available for the purpose, the azi- 108 ASTRONOMICAL INSTRUMENTS. miith (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 be 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 enable us to correct its position ; thus, Suppose the polar-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 be perfectly level, will also appear bisected ; if it do 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. 109 carefully elevate the telescope to the star itself, which will also be bisected if the estimation of half the amount of deviation have 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. Tang. \ azimuth = \ 8 s sin. <,; _ _ -v 2 & sin. 2 A, s . s sin. -. sin.- - TT In which - = half the sum of the polar distance, the co-latitude and the zenith distance, TT and A = the polar distance and co- latitude, Z = the zenith distance of the object. 110 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. O I II IT (from the N.A.) = 57 45 16 X = 38 31 21 z . = 56 20 10 2) 152 36 47 -TT-= 76 IS 23 19 58 13 sine = 9-5334322 37 47 2 sine = A = 9-7872371 9-3206693 76 18 23 sine = 9-9874766 18 33 7 sine = B = A = 9-5026514 9-4901280 9-3206693 2) 9-8305413 = A - B THE ALTITUDE AND AZIMUTH INSTRUMENT. Ill o / n 39 26 45-5 4 star's azimuth, tan. = 9*9152706 2 78 53 31-0 = azimuth of star 42 3 4-0 = object near 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 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 27' 44", enter Table VIII. and take out the length of a second, both of latitude 112 ASTRONOMICAL INSTRUMENTS. and longitude ; divide the distances A D and B D by those num- bers, and the quotients will be the difference of longitude and latitude (in arc) required. Thus : A D = 5064- 8 Table = 63-31 = 80,44 = 1 20,00 = diff. of long, in space. = 5 8 ,33intime. B B = 3292*2 Table 102'02 == ^ffl 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 . v . 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 be 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 principles 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 shew 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- 114 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 A D, 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 loot in the point C, and with the other mark the point D on the arc C D, 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. 115 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. Let the above plan represent a survey of roads to be performed with a theodolite and chain. Commencing on a conspicuous spot, a, 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 i2 116 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 do, 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 be 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 measure 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 be * A picket should always be left in the ground at every station, in order to recognize the precise spot, should it afterwards be found necessary to return to it again. APPENDIX. 117 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 y g, &c., 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 , 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 fixed object may be assumed as a working meridian j 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 be 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 be 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 j and, first, the protractor must be constructed. The great difficulty of dividing a circle accurately is well known, but if the arcs be laid off by means of their chords, the division may be performed with sufficient exactness for the purpose in hand. 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. 118 APPENDIX. Next set the compasses to the natural sine of 15, which, to radius five, 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 oft* each way, from those points which they are intended to bisect ; for if any inac- curacy exist, the bisection will not be perfect, and if the error prove 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 7 (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 of 3 45' (equal the chord of 7 30') 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 #, 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 rnler 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 b ; then with a pair of compasses, and from the scale of the plot, lay off the distance a b y which will determine the point b. Next, place the edge of the ruler on the angles taken at b 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 take 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 b } 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 proceeds; 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 be on a separate sheet, and 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- 120 APPENDIX. 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 #, &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, b, 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 shewn in the figure: these branches carry, near each of their extremities, a fine steel pricker, the two points of which, APPENDIX. 121 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, 122 APPENDIX. and lay off the exact length of the offset from the edge of the scale, 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 at 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. B Station Lime Station 35 so A | 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. APPENDJX. 123 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 bank-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 becomes visible through the sheet of paper, upon which a fair copy can be made, without the danger of soiling or injuring the original 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 immoveable in its proper position, it being the centre upon which the whole instru- 124 APPENDIX. ment turns. Several ivory castors support the surface of the machine parallel to the paper, as well as facilitate its motions. The arms, E D, and E B, 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, G, 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. 125 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 to 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 surveyor should require to represent the boundaries of the fields which are still more remote fro m 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 126 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. 127 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 Eoint on the water the bearings of two known objects on the md 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. 128 APPENDIX. This instrument consists of three rulers, ABC, (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 y 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 D E F 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 GX 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 shewing 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 andB D A were found = 40 and 60. Sup- tract double the angle C D B from 180, 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 C 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 one 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. 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 railway s, &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. ._- TABLE I. To reduce the Apparent to the True Level. Argument = the Distance in Feet. Dist. in Feet. Correct "in Decimals of a Foot. Dist. in Feet. Correct" in Decimals of a Foot. Dist. in Feet. Correct" in Decimals of a Foot. Dist. in Feet. Correct" in Decimals of a Foot. Dist. in Feet. Correct" in Decimal* of a Foot. 20 40 60 80 O'OOOOl 00004 00009 00015 1020 40 60 80 -02489 02588 02688 02791 2020 40 60 80 '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 '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 0-00096 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 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 -04689 04824 04961 05100 05241 2400 20 40 60 80 0-13781 14012 14244 14480 14715 3400 20 40 60 80 -27658 27985 28313 28643 28975 4400 20 40 60 80 -46320 46742 47166 47591 48020 500 20 40 60 1 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 a '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 3600 20 40 60 80 '31008 31353 31700 32050 32401 4600 20 40 60 80 -50627 51067 51511 51957 52404 '52852 53302 53755 54211 54667 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 80 800 20 40 60 80 900 20 40 60 80 1000 0-01531 01609 01688 01769 01853 0-01938 02025 02114 02205 02298 -02392 1800 20 40 60 80 1900 20 40 60 80 2000 0-07752 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 0-55124 55586 56048 56512 56978 -57447 57917 i 58388 58860 59337 -59814 -08637 08820 09005 09191 09380 -09570 2900 20 40 60 80 3000 -20121 20400 20681 20962 21247 -21532 3900 20 40 60 80 4000 -36390 36766 37142 37520 37899 '38281 4900 20 40 60 80 5000 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 XXXVIIf. Thermometers in open Air. Thermometers to the Barometer. Latitude of the Place. S A S A S A D B L C 40 [ 41 42 1 43 t 44 4 76891 76940 76989 77039 77089 o 84 85 86 87 88 4 -79019 79066 79113 79160 79207 128 129 130 131 132 4-81048 81093 81138 81183 81228 o 1 2 3 4 o -ooooo 00004 00009 00013 00017 3 6 9 12 0-00117 00116 00114 00111 00107 45 I 46 ! 47 i 48 1 49 4-77138 77187 77236 77286 77335 89 90 91 92 93 4 -79254 79301 79348 79395 79442 133 134 135 136 137 4-81272 81317 81362 81407 81451 5 6 7 8 9 -00022 00026 00030 00035 00039 15 18 21 24 27 0-00101 00095 00087 00078 00069 50 1 51 52 53 i 54 4 -77383 77433 77482 77531 77579 94 95 96 97 98 99" 100 101 102 103 4 -79488 79535 79582 79629 79675 138 139 140 141 142 4-81496 81541 81585 81630 81675 10 11 12 13 14 -00043 00048 00052 00056 00061 30 33 36 39 42 -00059 00048 00036 00024 00012 , 55 [ 56 ' 57 58 59 4 -77628 77677 77726 77774 77823 4. 79722 79768 79814 79860 79907 143 144 145 146 147 4-81719 81763 81807 81851 81896 15 16 17 18 19 '00065 00069 00074 00078 00083 45 48 51 54 57 -00000 9-99988 99976 99964 99952 60 61 1 62 63 1 64 4-77871 77920 77968 78017 78065 104 105 106 107 108 4 -79953 79999 80045 80091 80137 148 149 150 151 152 4-81940 81983 82027 82072 82116 20 21 22 23 24 -00087 00091 00096 00100 00104 60 63 66 69 72 9-99941 99931 99922 99913 99905 65 66 67 68 69 4-78113 78161 78209 78257 78305 109 110 111 112 113 4-80183 80229 80275 80321 80367 153 154 155 156 157 4-82160 82204 82248 82291 82335 25 26 27 28 29 0-00109 00113 00117 00122 00126 75 78 81 84 87 9 -99899 99893 99889 99886 99884 70 71 72 73 74 4 -78353 78401 78449 78497 78544 ^78592 78640 78688 78735 78783 114 115 116 117 118 4 -80412 80458 80504 80549 80595 158 159 160 161 162 4 -82379 82422 82466 82510 82553 30 31 00130 0-00134 90 9 -99883 f the sum of the detached S = < thermometers at the I two stations, fthe difference of the at- D = < tached thermometers (_ at the two stations. L = the latitude. f height of the barometer ^ ~ \ at the upper station. , f height of the barometer = (^ at the lower station. 75 76 77 78 79 119 120 121 122 123 4-80641 80687 80732 80777 80823 163 164 165 166 167 4 -82597 82640 82683 82726 82770 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 je^K""" Make R = log. |8' (B + log. 0) when upper thermometer reads lowest, or R = log. )8' + 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. h m a m m m 8 8 t , s 1 9, 830 1 0,164 21 3, 440 41 6,717 1 0,003 21 0,057 41 0,112 2 19,659 2 0,328 22 3,604 42 6,881 2 0,005 22 0,060 42 0,115 3 29,489 3 0,491 23 3,768 43 7,044 3 0,008 23 0,063 43 0,118 4 39,318 4 0,655 24 3,932 44 7,208 4 0,011 24 0,066 44 0,120 5 49,148 5 0,819 25 4,096 45 7,372 5 0,014 25 0,068 45 0,123 6 58,977 6 0,983 26 4,259 46 7, 536 6 0,016 26 0,071 4(i 0,126 7 1 8, 807 7 1,147 27 4,423 47 7,700 7 0,019 27 0,074 47 0,128 8 1 18,636 8 1,311 28 4,587 48 7,864 8 0,022 28 0, 076 48 0,131 9 1 28,466 9 1,474 29 4,751 49 8,027 9 0,025 29 0,079 4!) 0,134 10 1 38,296 10 1,638 30 4,915 50 8,191 10 0,027 30 0,082 50 0, 137 11 1 48,125 11 1,802 31 5,079 51 8,355 11 0,030 31 0,085 51 0,140 12 1 57,955 12 1,966 32 5,242 52 8,519 12 0,033 32 0,087 52 0,142 13 2 7, 784 13 2,130 33 5,406 53 8,683 13 0, 036 33 0,090 53 0,145 14 2 17,614 14 2,294 34 5,570 54 8,847 14 0,038 34 0,093 54 0,148 15 2 27,443 15 2,457 35 5,734 55 9,010 15 0,041 35 0,096 :>:> 0,150 16 2 37, 273 16 2,621 36 5,898 56 9,174 16 0,044 36 0,098 56 0,153 17 2 47, 103 17 2,785 37 6,062 57 9,338 17 0,047 37 0,101 57 0,156 18 2 56, 932 18 2,949 38 6,225 58 9,502 18 0,049 38 0,104 58 0,159 19 3 6, 762 19 3,113 39 6,389 59 9,666 19 0,052 39 0,106 59 0,161 20 3 16,591 20 3,277 40 6,553 60 9,830 20 0,055 40 0,109 (JO 0,164 21 3 26,421 22 Q Qfi Or.(\ 23 24 o ou, ou 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 oiuereai J.1U1C. Hours. Minutes. Seconds. h m t m t in i m B i t a s 8 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 ,150 27 4,436 47 7,721 7 0,019 27 0,074 47 0,128 8 1 18,852 8 ,314 28 4,600 48 7,885 8 0,022 28 0,076 48 0,131 9 1 28,708 9 ,478 29 4,764 49 8,050 9 0,025 29 0,079 49 0,134 10 1 38,565 10 ,643 30 4,928 50 8,214 10 0,027 30 0,082 50 0,137 11 1 48,421 11 ,807 31 5,092 51 8,378 11 0,030 31 0,085 51 0,140 12 1 58,278 12 ,971 32 5,257 52 8,542 12 0,033 32 0,087 f,2 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 (it) 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 Sl,ar. 1 1 1 2 Mia. Log. 'arts Min, Log. ?arls Min. Log. Parts Min. Log. Parts 42-0 1 '536442 8-0 510875 4-0 i -486648 o-o 1 -463627 1 1 -536004 43 1 510461 41 1 1 -486255 39 1 1-463253 37 2 1 '535566 87 2 -510047 82 2 1 -485862 78 2 1 -462879 75 3 I -535130 130 3 509633 123 3 1 '485469 118 3 1 '462505 112 4 1 -534693 174 4 509220 165 4 1 -485077 157 4 1 -462132 149 5 1 -534256 217 5 508808 206 5 1 -484685 196 5 1-461759 187 6 1 -533821 261 6 508395 247 6 1 -484294 235 6 1 -461386 224 7 1 -533385 304 7 507983 288 7 1 -483903 274 7 1 -461014 261 ' '8 i -532950 348 8 507571 330 8 1 '483512 313 8 1 -460642 298 9 1 -532515 391 9 507161 371 9 1 '483121 353 9 1 -460270 336 !43'0 1 -532081 9-0 506749 J5-0 1 -482731 1-0 1 -459898 i ! 1 -531647 43 j 506337 41 1 1-482341 39 1 1 -459527 37 2 1-531214 86 2 505928 82 2 1 -481952 78 2 1 -459156 74 3 1 -530781 130 3 505518 123 3 1 -481562 117 3 1 -458785 111 4 1 -530348 173 4 505108 164 4 1-481173 155 4 1-458414 148 5 1 -529916 216 5 504699 205 "5 1 -480785 194 5 1 '458044 185 6 1 -529484 259 6 504290 245 6 1 '480397 233 6 1 '457674 222 7 1 -529053 302 7 1 '503881 286 T / 1 '480009 272 7 1 -457305 259 8 1 -528622 346 8 I -503473 327 8 1 '479621 311 8 1 -456936 296 9 1-528191 389 9 1 -503065 368 9 1 '479234 350 9 1 -456567 333 44-0 1 '527761 o-o I -502658 D6-0 1 '478847 2'0 1-456198 1 1 -527331 43 1 1 -502251 41 1 1 '478460 39 1 1 -455830 37 2 1 -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 5 1 -525616 214 5 1 -500626 203 5 1 -476917 193 5 1 -454360 184 6 1-525188 257 6 1 '500221 243 6 1 -476532 231 6 1 -453993 220 "7 1 -524761 300 7 1 '499816 284 7 1 -476147 270 7 1 '453627 257 8 1 -524334 342 8 1-499411 324 8 1.475763 308 8 1 '453261 294 9 i -523907 385 9 1 -499007 365 9 1.475379 347 9 1 -452895 330 45'0 1 -523481 ol -0 1 -498603 57-0 1 -474995 3'0 1 -452529 1 1 -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 153 411 '451069 146 5 1 -521355 212 5 1 -496589 201 1 -473080 191 5 1 '450705 182 6 i -520932 254 6 I '496188 241 1 -472699 229 6 1 -450341 218 '7 1 -520508 297 *7 1 -495786 281 *7 1 -472317 267 711 '449977 255 8 1 -520085 339 8 I -495385 322 8 471936 306 8 1 '449614 291 9 1 -519662 382 t 1 -494984 362 t 471555 344 91 -449251 328 46-0 1 -519240 52-0 1 -494584 58-0 471174 4-01 -448888 1 1 -518820 42 1 1 -494184 40 1 470794 38 1 ! 1 -448525 36 2 1 -518400 84 r 1 -493784 80 .c 1 -470414 76 21-448163 72 3 1 -517980 126 J 1 -493385 120 j 470034 114 3 1 -447801 108 4 1 -517560 168 "4 1 -492986 159 '4 469655 152 4 1 -447439 144 5 1 -517140 210 .c t. 1 -492587 199 5 469276 190 5 1 -447078 181 6 1-516719 252 6 1 -492189 239 6 468897 227 61 -446717 217 7 1 -516299 294 *r 1-491791 279 7 1 '468518 265 71-446356 253 8 1-515879 336 8 1 -491393 319 8 1 -468140 303 81 -445995 289 9 1 -515459 378 < 1 -490996 359 9 1 -467762 341 9 1 -445635 325 47-0 1-515039 53-0 490599 59-0 467385 5 -Oil -445275 1 1 -514621 42 490202 39 '1 467008 38 111-444915 36 2 1 -514203 83 *2 48980f 79 2 466631 75 21 -444556 71 3 1 -513786 125 jj 489410 119 3 466255 113 Si 1-444 196 107 4 1 -513369 167 ^ 489014 158 4 465878 150 4< 1-443837 143 5 1 -512952 208 t 1 -488619 198 5 -465502 188 5 1 '443478 179 fi 1-512536 250 *i 1 -488224 237 6 -465127 226 6 1 -443120 215 7 1-512120 292 *7 1 -487830 277 7|1 -464751 63 7 1 -442762 251 8 1-511705 333 1 -487436 316 8 1 -464376 301 8 1 -442404 286 s 1-511290 375 '! 1 -487042 356 9.1 '464001 338 9 1 -442046 322 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. J arts Min. Log. Parts Min. Log. Parts Min. Log. Parts 6-0 1-441689 2-0 420737 8-0 400682 4-0 1 -381448 1 1 -441332 36 1 420396 34 1 400355 33 1 1 -381134 31 2 1 -440975 71 2 420055 68 2 400028 65 2 I -380820 63 3 1 '440619 107 3 419715 102 3 399701 98 3 1 -380506 94 4 1 -440263 142 4 419373 136 4 399375 130 4 1 -380193 125 5 1 -439907 178 5 419033 170 5 399049 163 5 1 -379880 156 6 1 -439551 214 6 418693 204 6 398723 196 6 1 -379567 188 7 1 -439196 249 7 1 '418354 238 7 398397 228 7 1 -379254 219 8 1 -438840 285 8 1 -418014 272 8 398071 261 8 1 '378941 250 9 1 -438486 320 9 1 '417674 306 9 397746 293 9 1 -378629 282 7-0 1-438131 3-0 1 -417335 9-0 397421 !VO 1-378317 1 -437777 35 1 1 -416996 34 1 397096 32 1 1 -378005 31 2 1 -437423 71 2 1 -416657 67 2 396772 65 2 1 -377693 62 3 1 -437069 106 3 1 -416319 101 3 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 6 1 -436009 212 6 1 -415306 202 -6 1 -395476 194 6 1 -376449 186 7 1 -435650 247 "7 1 -414969 236 7 1-395152 226 7 1 '376138 218 8 I -435304 282 '8 1 -414631 276 8 1 -394829 258 8 1 -375827 249 'i! 1 -434952 318 9 I -414294 303 9 1 -394506 291 9 1 -375517 280 8-0 1 '434600 14-0 1 -413957 20-0 1 -394183 26-0 1 -375207 1 1 -434248 35 1 1 -413621 33 1 1 '393860 32 1 1 -374897 31 *r; 1 '433897 70 2 1-413285 67 2 1 -393538 64 2 1/374587 62 V 1 -433546 105 3 1 -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 *3 1 -373659 154 i '6 1 -432494 210 6 1-411942 201 6 1 -392251 193 6 1 -373351 185 t ; 1 -432144 245 "7 1 '411607 234 *7 1 -391930 225 *7 1 -373042 216 8 1 -431795 280 8 1-411273 268 8 1 -391608 257 8 1 -372733 246 t 1 -431445 315 (j 1 -410938 301 1 -391288 289 c 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 r 1 -430398 70 '2 1 -409936 67 .r 1 -390327 64 f 1 '371501 61 .^ 1 -430050 104 ^ 1 -409602 100 1 -390007 96 \ I -371194 92 "^ 1 -429701 139 '4 1 -409269 133 '4 1 '389687 128 4 1 -370887 122 "5 1 -429354 174 5 1 -408935 166 c O 1 '389367 159 | . 1 -370579 153 6 1 -429006 209 6 1 -408602 200 6 1 -389048 191 6 1 -370273 184 .7 1 -428659 244 "7 1 -408270 233 ", 1 -388729 223 "7 1 -369966 214 8 1 -428312 278 8 1 -407937 266 8 1 -388410 255 8 1 -369659 245 i 1 -427965 313 ( 1 -407605 300 ( 1 -388091 287 .( 1 -369353 275 lO'O 1 -427618 16-0 1 -40727; 22-0 1 -387773 28-0 1 -369047 1 -427272 35 40694 33 1 1 -387454 32 ] 1 -368741 30 .< 1 -426925 69 2 406610 66 J 1 -387137 63 J 1 -368435 61 j 1 -426580 103 < 406278 99 j 1 -386819 95 *; 1 -368130 91 n 1 -426234 138 t 405947 132 "4 1 '386501 127 */ 1 -367825 122 .5 1 -425888 172 i 405617 165 '5 1 '386183 158 '5 1 -367520 152 ( 1 -425543 207 ( 405286 198 '6 1 -385867 190 ( 1 -367215 183 *7 1 -425198 242 '7 404956 231 "7 1 '385550 222 J 1 -366910 213 "8 1 -424854 276 8 404626 264 8 1 -385233 254 1 -366605 244 ( 1 -424509 310 9 40429f 297 "9 1 -384916 285 9 1 -366301 274 ll'O 1 -42416o 17-0 403966 23-0 1 -384600 29-0 1 -365997 1 -42382 34 40363 33 j 1 -384284 31 ] 1 -365693 30 1 -423477 69 < 1 -40330 66 j 1 -383968 63 2 1 -365389 61 1 1 -423134 103 1 -402978 98 j 1 -383652 94 3 1 -365086 91 / 1 -42279 137 -i 1 -402650 131 'i 1 -383337 126 4 1 -364782 121 1 1 -422448 171 i 1 -40232 164 5 1 -38302 157 5|1 -364479 151 { 1-422105 206 *6 1 -40199. 197 I 1 -382706 189 6 1-364176 182 *7 1-421763 240 "7 1 -40166j 230 "7 1 -38239 220 7 1 -363873 212 '8 1-42142 274 '8 1 -40133 262 c 1 -382077 252 e 1 -363571 242 < 1 '421079 309 *! 1-401009 295 9 1 -381762 283 s 1 -363268 273 TABLE V. Logarithm s 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 Log. Mm. Log. Paits Min. Log. Parts Miu. Log. Parts Min. Paits 1 s 30-0 1 -362966 34-0 1 '351035 38 S -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 326912 112 5 1 -361458 150 5 1 '349564 147 5 1 -337961 143 5 326632 140 6 1 -361157 181 6 1 -349271 176 6 1 -337675 172 6 326352 168 7 1 -360856 211 7 1 -348977 206 7 1 -3373881 201 7 326073 196 8 1 -360556 241 8 1 -348684 235 8 1 -337102J230 8 325793 224 ; 9 1 -360255 271 9 1 -348391 265 9 1-336816 258 9 325514 252 31'0 1 -359955 35-0 1 '348098 39-0 336530 43-0 325235 1 1 '359655 30 j 1 -347805 29 1 336244 28 1 324956 28 2 1 '359355 60 2 1 "347513 58 2 335959 57 2 324677 56 3 1 -359055 90 3 1 -347220 88 3 335674 85 3 1- 324399 83 4 1 '358756 120 4 1 -346928 117 4 335389 114 4 1- 324120 111 I '5 1 -358457 149 5 1 '346636 146 5 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 1% 8 1 -357560 239 8 1 -345761 234 8 1 -334250 228 8 1- 323008 222 9 I '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 j 1 -356665 30 1 1 '344888 29 1 1 -333398 28 *1 1- 322176 28 2 1 -356367 59 2 1 '344598 58 2 1-333114 57 -2 1- 321898 55 3 1 -356069 89 3 1 '344307 87 3 1 -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 261 911-331132 255 9 1 -319963 249 33 1 -353990 37-0 1 '342279 41-0 1 -330850 45'0 319687 1 353694 29 1 1 '341990 29 1 1 '330568 28 1 319411 27 2 353397 59 2 1 -341701 58 2 330285 56 2 319136 55 3 353102 88 3 1 -341412 86 3 330003 84 3 318860 82 4 352806 118 4 1-341124 115 4 329722 112 4 318585 110 5 352510 147 5 1 -340835 144 5 329440 140 5 318310 137 6 1 -352215 177 6 1 -340547 173 6 329158 169 6 318035 165 7 1 '351920 206 7 1 -340259 202 7 328877 197 ~ 1 -317760 192 8 1 -351624 236 8 1 '339671 230 8 1 -328596 225 8 1 '317486 220 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 Semidiameter on the Time of its passing the Meridian. Change of Moon's Declination. }) 's Semidiam 8 16 22 28 1 8 07 8 07 8 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 62 9 62 63 65 68 70 10 '69 70 72 75 78 p" TABLE VII. Reduction to the Meridian. Argument = the Hour Angle from the Meridian, J ID l m 2 ra 3 m 4'" 5 ,n flm 7 m gm ym 10 m ll m 12 m 13 14m y 38 86 152 238 343 466 609 771 952 1152 1370 1608 1865 1 10 39 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 1 352 478 622 785 963 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 253J 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 14 47 99 171 261 370 498 645 811 997 1201 1424 1667 1928 15 15 48 100 172 262 372 500 648 814 1000 1205 1428 1671 1932 16 15 49 102 173 264 374 503 650 817 1003 1208 1432 1675 1937 17 16 50 103 175 266 376 505 653 820 1006 1212 1436 1679 1941 18 16 50 104 176 267 378 507 656 823 1010 1215 1440 1683 1946 19 17 51 105 177 269 380 510 658 826 1013 1219 1444 1688 1950 20 T7 52 106 179 271 382 512 661 829 1016 1222 1448 1692 1955 21 17 53 107 180 272 384 514 664 832 1020 1226 1451 1696 1960 22 18 53 108 182 274 386 517 666 835 1023 1230 1455 1700 1964 23 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 23 62 120 197 293 408 543 696 868 1059 1270 1499 1747 2014 34 23 63 121 198 295 410 545 699 871 1062 1273 1503 1751 2019 35 24 64 122 200 297 412 547 701 874 1066 1277 1507 1756 2024 36 24 64 123 201 299 415 550 704 877 1069 1281 1511 1760 2028 37 25 65 124 203 300 417 552 707 880 1073 1284 1515 1764 2033 38 25 66 126 204 302 419 555 709 883 1076 1288 1519 1769 2038 39 26 67 127 206 304 421 557 712 886 1079 1292 1523 1773 2042 40 26 68 128 207 306 423 559 715 889 1083 1295 1527 1777 2047 41 27 68 129 209 307 425 562 718 892 1086 1299 1531 1782 2052 42 28 69 130 210 309 427 564 720 896 1090 1303 1535 1786 2056 43 28 70 131 212 311 429 567 723 899 1093 1307 1539 1790 2061 44 29 71 133 213 313 432 569 72e 902 1096 1310 1543 1795 2066 45 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 i 30 74 136 218 318 438 577 734 911 1107 1321 1555 1808 2080 48 ( 31 75 137 219 320 440 579 737 914 1110 1325 1559 1812 2084 49 ( 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 1336 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 35 81 146 230 333 455 597 757 936 1134 1352 1588 1843 2117 56 3 36 82 147 232 335 458 599 760 939 1138 1355 1592 1847 2122 57 8 36 83 14S 233 337 460 602 763 942 1141 1359 1596 1852 2127 58 ; 37 84 150 235 339 462 604 765 945 1145 1363 1600 185( 2132 59 i 37 85 151 236 341 464 607 768 949 1148 1367 1604 1861 2136 60 ] 38 86 152 238 343 46f 609 771 952 1152 1370 1608 1865 2141 TABLE XI. Correction of Moon's Meridional Passage. The application of this and the following Table is explained at page 80. Long, in Time. h m o. o 20 40 1. 20 40 2. 20 40 Argument Daily Change of Mer. Passage. Lung, in Arc. o 5 10 15 20 25 30 35 40 40'" in 1 1 2 2 3 3 4 4 42" m 1 1 2 2 3 3 4 4 44m m 1 1 2 2 3 46"' in 1 2 2 3 48 m 111 1 1 2 3 3 50 m 1 1 2 3 3 52 m in 1 1 2 3 3 54 m m 1 1 2 3 4 56 m 1 1 2 3 4 58"' m 1 2 2 3 4 5 5 6 60"' m 2 2 3 4 5 6 6 62" m 1 2 2 3 4 5 6 7 64m m 1 2 3 3 4 5 6 7 66* in 1 2 3 4 4 5 6 7 4 5 4 5 5 5 5 5 5 6 5 6 5 6 3. 20 40 4. 20 40 5. 20 40 5 5 6 6 7 7 8 9 9 5 6 6 7 7 8 9 9 10 5 6 7 7 8 8 9 9 10 6 6 7 7 8 9 9 10 11 6 6 7 8 8 9 10 10 11 6 7 7 8 9 9 10 11 11 6 7 8 8 9 10 10 11 12 7 7 8 9 9 10 11 12 12 7 7 8 9 10 10 11 12 13 7 8 9 9 10 11 12 12 13 7 8 9 10 10 11 12 13 14 7 8 9 10 11 12 12 13 14 8 9 9 10 11 12 13 14 14 8 9 10 11 11 12 13 14 15 45 50 55 60 65 70 75 80 85 6. 20 40 7. 20 40 8. 20 40 10 10 11 11 12 12 13 13 J4 10 11 11 12 12 13 14 14 15 11 11 12 12 13 14 14 15 15 11 12 12 13 14 14 15 15 16 12 12 13 14 14 15 15 16 17 12 13 13 14 15 15 16 17 17 13 13 14 15 15 16 17 17 18 13 14 14 15 16 17 17 18 19 13 14 15 16 16 17 18 19 19 14 15 15 16 17 18 19 19 20 14 15 16 17 18 18 19 20 21 15 16 17 17 18 19 20 21 21 15 16 17 18 19 20 20 21 22 16 17 17 18 19 20 21 22 23 90 95 100 105 110 115 120 125 130 9. 20 40 10. 20 40 11. 20 40 12. 14 15 15 16 16 17 17 18 18 19 15 16 16 17 18 18 19 19 20 20 16 17 17 18 18 19 20 20 21 21 17 17 18 19 19 20 20 21 22 22 17 18 19 19 20 21 21 22 23 23 18 19 19 20 21 21 22 23 23 24 19 20 20 21 22 22 23 24 24 25 20 20 21 22 22 23 24 25 25 26 20 21 22 22 23 24 25 25 26 27 21 22 22 23 24 25 26 26 27 28 22 22 23 24 25 26 26 27 28 29 22 23 24 25 26 26 27 28 29 30 23 24 25 26 26 27 28 29 33 31 24 25 25 26 27 28 29 30 31 32 135 140 145 150 155 160 165 170 175 180 TABLE XII. Effect of a Change of 1 in Declination on the Moon's Semidiameter (as given in the Naut. Aim.) | Dec. Corr. Dec. Corr. Dec. Corr. Dec. Corr. < ^ 2 3 4 5 6 7 s 000 019 036 057 076 095 114 134 o 8 9 10 11 12 13 14 S 134 154 173 194 214 235 256 278 14 15 16 17 18 19 20 21 278 300 323 346 368 394 419 445 o 21 22 23 24 25 26 27 28 S 445 472 499 527 557 587 619 652 CATALOGUE INSTRUMENTS, MADE BY TROUGH TON AND SIMMS, OPTICIANS. MATHEMATICAL INSTRUMENT MAKERS TO THE asoarU of rirname, 138, FLEET STREET, LONDON. TROUGHTON & SIMMS beg to caution those who may have occasion to write from abroad, tftat no reliance can be placed on the genuineness of the Instruments they obtain, unless the application be made DIRECT, or through the most respectable channels. A CATALOGUE, esc. SPECTACLES AND OPERA GLASSES. s. d. Gold Spectacles, Single Joint from 3/. 85. to 5 5 Ditto, Double Joint from 41. 4s. to 6 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 . . . 1 1 Tortoiseshell Spectacles, Single Joint . . 7 6 Ditto, Double Joint . 12 6 Ditto, Hand Frame ... from 7s. 6d. to 16 Fine Blue Steel Spectacles, Single Joint . 10 6 Ditto, Double Joint .... . 12 6 Eye Glasses, Gold Frame from 11. Is. to 2 2 Ditto, Silver Frame .... from 6s. to 8 Ditto, Shell Frame from 4s. 6d. to 6 (If with Brazilian Pebbles, & ?. per pair extra.) Spectacle Cases .... each 1 Opera Glasses, not Achromatic, in Plated Mountings, from 65. to ... 1 11 6 Ditto, Achromatic, with Ivory Bodies and Gilt Mount- ings . from 2/. 2s. to 5 5 TELESCOPES. One-foot Achromatic (Camp) Telescope, having one Drawer 1 5 Ditto ditto, Portable, with Drawers . . 1 9 Ditto ditto, in Electrum . 2 2 Ditto ditto, larger aperture in Brass . 1 15 Eighteen-inch ditto .... . 2 12 6 Two-feet ditto, Reconnoitering . 3 13 6 4 TROUGHTON AND SIMMs's CATALOGUE. s. d. Two-feet Achromatic, Reconnoitring, in Electrum . 550 Ditto ditto, in Electrum, with Compass, Agate Cap, Needle, &c. . . ' ^ . ' . . 660 Ditto ditto, ditto, with four Drawers in Brass ;. . 3180 Thirty-inch ditto ditto . , v 550 Three-feet ditto ditto . . . 660 Military Case and Sling * . from 105. 6d. to 12 6 Portable Brass Stand for any of the above Telescopes, from 2 1*. to v ( , ; , v . . . 2 12 6 Walking Stick Telescope, Portable, with Compass . 3136 Ditto, without Compass '';./ . > . .1 '."." 330 Ditto, in one length, with Compass . . . . 2126 Ditto, without ditto . "V . ^ -,^ Vv : >-.;,... 220 Dumpy Navy ^ "".' . . . . -'5* '"-/ 440 Ditto, with Pancratic Eye-piece . . . *.;'' ^ ' 4 14 6 Two-feet Navy Telescope ; - . ;;-,-' 212 6 Ditto, ditto, Brass Body, covered with Leather t* . ^ *' 2 15 Ditto, ditto, with Spray Shade . . <>, ." . . ', 330 Three-feet ditto ' > ;, . ". '' " '."> '".'/ '*.../ 550 Ditto, ditto, with Spray Shade '; .. , s , . ^ :> . 5 15 6 Four feet ditto, with Two Powers, in Case . v .1212 Day or Night Telescope (Deck Glass) . ;.;U- V 440 Ditto, ditto, with Spray Shade . . . -. v ' ,i 4 14 6 Night Glass . . '.;. .- V*aiir 1 5'M^>- 330 Ditto, large size v **$ .'., . ; ., . '; .' .; . . ,\-^\\- 440 Ordnance Signal Station Telescope . . ~. . '>. 6 16 6 Thirty-inch Achromatic Telescope, two-and~a-quarter- inch Object Glass, mounted on Pillar- and -claw Stand, with aTerrestrial and an Astronomical Eye- piece, in a Mahogany Case . .. * . 10 10 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 Mahogany 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 TROUGHTON AND SIMMS's CATALOGUE. 5 s. d. Forty-five-inch Achromatic Telescope, three-and-three- quarter-inch Ohject 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 Completely mounted Equatorial, with Clock Movement, Micrometers, &c., 5 feet focus, three and three- quarter-inch Object Glass j 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 carrying Telescopes from three- and- a-half to seven feet . . . 12 12 MICROSCOPES, &c. Solar Microscopes . . . from 6/. 16*. 6rf. to 21 Botanic ditto, small size 15 Ditto, ditto 150 Compound ditto . . . . 2 12 6 Ditto ditto 2 15 Ditto ditto 3100 Ditto ditto 4100 Ditto ditto 5 10 Ditto, ditto, larger, from 10Z. 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 I/. 10s. to 5 Common Azimuth Compass . . . . 4 14 6 Azimuth (Prismatic) Compass, large size, best construc- tion, with Tripod Stand complete . . 16160 Ditto ditto, smaller size 10 10 Pocket Compasses in Wood, Brass, Metal, Gilt, Silver, and Gold .... from 3s. 6d. to 5 5 Ebony Quadrant with Tangent Screw . . . 3 13 6 6 TROUQHTON AND SIMMS's CATALOGUE. S. d. Ebony Quadrant, with Tangent Screw and Telescope 4 14 6 Ditto ditto, best . . . . . 5 15 6 Ebony Sextant, with Telescopes . ( V;> ^ . , 880 Ditto ditto, with Brass Arch, &c. . >?v>- 10 10 Optical Square ,r^t;;>^<;, I 1 Box Sextant, plain ". ^ . . . . U$&3 . 3 13 6 Ditto, with Telescope . -' ^*?- . . . w&ai 4 14 6 Box Sextant, Ordnance Pattern .... 550 Ditto, with Supplementary Arc . . . . . 5 15 6 Ditto, with ditto, and Levels .... >uH- 660 Leather Case and Strap for Box Sextant . '>..! ?,l i 9 Metal Sextant, 4-inch Radius,divided on Silver to 20 seconds 10 10 Ditto, 5-inch ditto, to 20 seconds . r yyr.i,v. 14 14 Ditto,[7-inch ditto, to 10 seconds -...< ,.: ,. . 16 16 Ditto, 8-inch ditto, with Double Frames, divided on Silver to 10 seconds . . . ,> r . , .18 18 Ditto ditto, divided on Platina 1 " .' , 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 . . - :,. ' .'' .. ' V 23 2 Six-inch Borda's Repeating Circle by Reflexion .- 21 Eight-inch ditto . . ... . . : . '- 1 ..23 2 Ten-inch ditto ..... ..'-... .- -'-. , -'...;- 25 Brass Counterpoise Stand for Circle or Sextant, in Mahogany Box . . . . . .5156 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, Ordnance 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 5. d. LEVELS, THEODOLITES, &c. Four-inch Pocket Level 096 Six-inch ditto . . . . . . . . 12 6 Eight- inch ditto 15 6 Ten-inch ditto .110 Twelve-inch ditto 1 11 6 Block Level . 110 Level with Sights and Socket, in Box . . 1 11 6 Portable Levelling Instrument, with Telescope, . 880 Ditto, with Compass, &c. . ... . .990 Fourteen-inch improved Level, with Round Legs . 1111 Ditto, with Tripod Stand 12 12 Twenty-inch ditto, with Round Legs . . . 13130 Ditto ditto, with Tripod Stand 14 14 Y Levels, with nine-inch Telescope . . . 10 10 Ditto, with twenty-inch ditto 16160 Ditto, with ditto and Compass . . . . 17170 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 1717 Ditto, large size, complete . . . . . 22 Standard Levelling Instrument 42 Plane Table, with Sights and Round Legs . . 6 16 6 Ditto, with Telescope, &c. . . . . . . 12 12 Cylindrical Cross Staff . . ... . . 16 Ditto, with Compass and Legs . . . 2 1 6 Circumferenter, in Mahogany, without Legs . . 2126 Ditto, ditto, larger size . . . . . .440 Ditto, in Brass, with Round Legs from 41 14s. 6d. to 6 6 Ditto, ditto, with Ball and Socket Joint, and Round Leg 616 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, Ac., and Legs, complete 16 16 Prismatic Compass, plain . . . . . .330 8 TROUGHTON S. d. Prismatic Compass, with Azimuth Glasses . . 3 13 6 Ditto, three-and-a-half inch, plain *, . . . 3 13 6 Ditto, ditto, with Azimuth Glasses . ... -v^ 4 4 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 . - :/ 1414 Four-inch Cradle Theodolite, divided on Silver v :; v 1616 Four-inch best Theodolite (Captain Dawson's) . : .. 21 00 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 . ;v t !> . 31 10 Six-inch ditto, with one Telescope, divided to 20 seconds, complete v- .v ,/> ^^-^ ^ - ? '- 31 10 Six-inch ditto, with two Telescopes . ,W^ . '.40 Six-inch ditto, with Transit Axis and Vertical Circle . 36 15 Six- inch ditto (Captain J. T. Boileau's construction,) with Axis, Level, &c. . .:.*. -,, c;y.a^-. ^ >* r '42 Seven-inch ditto, with one Telescope . . i 35 14 Seven-inch ditto, with two Telescopes . k ' -<'w .*, -V ' 45 Eight-inch ditto, Azimuth and Altitude, with Axis, Level, &c. . . $M&8 . . h^ : ^- 5210 Twelve-inch ditto, for Horizontal Angles only ., ', 42 Four-inch ditto (Colonel Everest's construction) >T v v 22 Five-inch ditto, ditto .*.-.; . f v^ N U 26 5 Seven-inch ditto, ditto . ... ,,...- . . 36 15 Five-and-a-half- inch Kater's Circle, with Stand, complete 35 Small Eater's Circle, with Stand . . . . 16 Level Collimator . . . from 107. 10s. to 15 15 (Larger Theodolites, #c., made to Order.) STATION POINTERS. PROTRACTORS, PENTAGRAPHS, &c. Twelve-inch Station Pointer . . . . .6166 Eighteen-inch ditto 7176 Two-feet ditto 990 Thirty-inch ditto 12 12 Three-feet ditto 18180 Wollaston's Goniometer 313 6 TROUGHTON AND SIMMS's CATALOGUE. 9 a. d. Eight-inch best Brass Circular Protractor, with Clamp and Tangent- screw and folding Arms 770 Ditto, ditto, divided upon 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 3 13 6 Fifteen-inch plain Circular Protractor . . 350 Eight-inch ditto .... from 17. 5s. to 1 11 6 1 1 Semicircular plain Protractors . . . from 1 6s. to 220 Ivory Protractors .... from 6s. to 15 Ditto, upon Parallel Rollers . . . from IBs. to 1 5 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 6 Plain Perambulators (wood) 990 12 12 Best ditto, with Metallic Wheel .... 16 16 Common twelve-feet Levelling-Staff .... 1 11 6 Best ditto 1 15 Troughton's Improved Portable ditto, with Level . 2 12 6 Sopwith's ditto, for Reading without an Assistant 2 12 6 Ditto, stronger, with painted divisions .... 330 Gravatt's Levelling Staff 440 Tape Measure, 25 feet, links . 070 Ditto, ditto, decimals ...... 080 Ditto, 33 feet, links .... 080 Ditto, ditto, decimals . . . . . . 090 Ditto, 50 feet, links 10 Ditto, ditto, decimals ...... 12 Ditto, 66 feet, links 12 Ditto, ditto, decimals ...... 14 Ditto, 100 feet, links ...... 16 Ditto, ditto, decimals ...... 18 10 TROUGHTON AND SIMMS J S CATALOGUE. s. d. Land Chains, 50 feet, and Arrows . . . .0136 Ditto, 100 feet, and ditto . . from II. 3s. 6d. to 150 Ditto, 66 feet, 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 feet . . . . ,. . 8/. 8s. and 9 19 6 (Stronger Chains, #c., made to Order). Set of Marquois Scales, in Box . . . . 12 6 Ditto, in Ivory . '. . =....' . ; . . . 2 5 Ditto, in Brass . . . '- . > . . . 2126 Ditto, in Electrum . . . - r . . .440 Twelve-inch Ivory Plotting Scales --. . from 11 s. to 110 Twelve-inch Boxwood ditto . . from 4s. to 070 Ivory Offsett and Pocket Scales . from 2s. 6d. to 6 6 Gunter's Scale, Brass, 2 feet . . ... 2 2 Ditto Boxwood from 5s. to 9 Ivory Folding Rules .... from 1 2s. to 018 Boxwood ditto . . . . from 6s. 6d. to 15 Gunner's Rules . . . . . from 3s. to 010 6 Camera Lucida . . . from I/. 11s. 6d. to 2 12 6 Stand for ditto from II. Is. to 111 6 Drawing Instruments, in Skin Cases, (Sappers and Miners) 14 Ditto, ditto, East India Company's Pattern . : .150 Ditto, ditto, Woolwich Pattern . . . . 1150 Ditto, ditto, School Pattern 220 Ditto, ditto, Ordnance Pattern .... 330 Ditto, in Mahogany Cases, Addiscombe Pattern . .330 Ditto, ditto, Admiralty Pattern . . . . 3136 Ditto, ditto, Sector-jointed Instruments, Parallel Rulers, Sector and Protractor 3 13 6 Ditto, ditto, with Sector double-jointed Dividers . 440 Ditto, ditto, with proportional Compasses . . . 5 15 6 Ditto, ditto, with Spring Bows . 770 TROUGHTON AND SIMMS's CATALOGUE. 11 *. d. Drawing Instruments, in Mahogany Cases, with Road and Wheel Pens, Needle-holder and small Dividers, &c 990 Ditto, ditto, in Electrum, packed in Rosewood and Ma- hogany 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 51. 5s. to 10 10 Plain Ebony Parallel Rulers, 12 inches . . .046 Ditto, ditto, 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 . . . . 1116 Ditto, ditto, ditto, 3 feet 220 Rolling Ebony Parallel Rulers, with Brass Edges, 12 inches 1 Ditto, ditto, ditto, 15 inches 150 Ditto, ditto, ditto, 18 inches 1 11 6 Ditto, ditto, ditto, 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 . 6 16 6 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). 14 TROUGHTON AND SIMMS J S CATALOGUE. 5 d. 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 . . '/' ,'" 1 ' , . 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 7 17 6 (Larger Machines made to Order). A Universal Discharger and Press . . . .1160 Jointed Discharger, with Glass Handles . . . 0126 Ditto, plain from 4s. 6d. to 86 Exhausted Flask for shewing the Aurora Borealis . 086 Electrical Batteries of combined Jars from 2/. 12s. 6d. to 10 10 Cuthbertson's Improved Electrometer, with grain Weight 2 12 6 Bennet's Gold Leaf Electrometer . . . 18 Cavallo's Bottle Electrometer, for Atmospherical purposes .... from 12s. to 4 14 6 Quadrant Electrometer, with divided Arch . . 096 Kinnersley's Electrometer . . . . . .110 Coulomb's Electrometer 1 16 Pith-Ball ditto 0160 Luminous Conductors .... from 12s. ta 1 A Thunder-house, for shewing the use of Conductors . 080 15 s. d. A Thunder-house, with Drawer . . . . 096 A Powder-house for ditto .0160 An Obelisk or Pyramid for ditto . . . . 10 6 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 . . . .1160 Ditto, with a Dome 280 Luminous Names or Words . from 10s. 6d. to 111 6 A Set of 3 Plain Bells 0106 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 18s. to 180 An Electrical Cannon 0180 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 . . . . . . 1 16 Atwood's Machine for Demonstrating the Law of Accel- eration 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 A Syren 220 A Small Still with Worm Tub and Lamp . . 2 18 Set of Lactometers in Box . . . . . 0180 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 818 6 Hydrostatic Paradox . . . . . . 770 Model of the Centrifugal Pump . from 4/. 10s. to 770 Sectional Model of a Steam Engine . . . 1800 (Models of Steam Engines, Machinery, &c., made to Order.) Bar Magnets for correcting the derangement of the Compass in Iron Vessels, 2 feet each, 21. 2s.; 14 inches each, 17. Is. ; 8 inches . . . 10 6 Magnetometers, Collimating and Referring Telescopes for use in Magnetic and other Observatories 16 TROUGH-TON AND SIMMS's CATALOGUE. 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 Illustrations. By FREDERICK W. SIMMS, F. R. A. S., F. G. S., M. INS. C.E. 1 1 PALMER & HOBV, Printers, 17, Brownlow Street, Holborn. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 5O CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. SEP 5 1933 SEP 6 1'J33 APR 8 Oct'S. 8 Oct'5b * RKC'O U0 SEP 24 1958 LIBRARY USE MAY 5 86 LD 21-50m-l,'3S D I II H7/76 56 / 850 UNIVERSITY OF CALIFORNIA UBRARY