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l‘latc. 1 
 
A 
 
 TREATISE 
 
 ON 
 
 MATHEMATICAL INSTRUMENTS, 
 
 INCLUDING 
 
 HOST OF THE INSTRUMENTS EMPLOYED IN DRAWING, 
 
 FOR ASSISTING THE VISION, IN SURVEYING AND LEVELLING, IN PRACTICAL 
 ASTRONOMY, AND FOR MEASURING THE ANGLES OF CRYSTALS : 
 
 IN WHICH 
 
 THEIR CONSTRUCTION, AND THE METHODS OF 
 
 TESTING, ADJUSTING, AND USING THEM, 
 
 ARE CONCISELY EXPLAINED. 
 
 BY J. F. HEATHER, M.A. 
 
 OF THE ROYAL MILITARY ACADEMY, WOOLWICH. 
 
 SECOND EDITION, 
 
 F r 
 
 WITH CORRECTIONS. 
 
 LONDON: 
 
 JOHN WEALE, 59, HIGH HOLBORN. 
 
 1853. 
 
PREFACE. 
 
 An attempt has been made in the following pages to put 
 within the reach of all a short and compendious treatise 
 upon some of the ingenious instruments by which the 
 scientific practitioner is aided in his observations, and in the 
 delineation of the results obtained from them. 
 
 The instruments treated of have been divided into five 
 classes, to each of which a part of the work has been devoted. 
 The first part treats of Mathematical Drawing Instruments; 
 the second, of Optical Instruments; the third, of Surveying 
 Instruments; the fourth, of Astronomical Instruments ; and 
 the fifth, and last, of Goniometrical Instruments, for measur¬ 
 ing the angles of crystals. 
 
 The greater part of the Wood Engravings, and some parts 
 of the Text, of Simms’s Mathematical Drawing Instruments, 
 have been pressed into the service of the present work; and 
 the works of the best writers upon the several parts of the 
 subject have been consulted, and much valuable matter has 
 been extracted from them, particularly from Pearson’s Astro¬ 
 nomy. 
 
 The limits of the bulk and cost of the work have forbidden 
 any extensive excursion into the sciences in which the instru¬ 
 ments are used; but it is hoped that a large mass of informa¬ 
 tion has here been placed in a small compass without sacri¬ 
 ficing perspicuity to undue compression 
 
 R. M. A. 
 
 March, 1849. 
 

 ' i 
 
CONTENTS 
 
 PART I. 
 
 ON MATHEMATICAL DRAWING INSTRUMENTS. 
 
 Page 
 
 Compasses.—Hair Compasses.—Compasses with Movable Points. 
 
 —Bow Compasses ........ 
 
 Wholes and Halves ........ 
 
 Proportional Compasses ....... 
 
 Triangular Compasses ........ 
 
 Drawing Pen. 
 
 Road Pen. 
 
 Pricking Point ......... 
 
 Straight Edge. 
 
 Scales of Equal Parts ........ 
 
 Protracting Scales ........ 
 
 Gunter’s Lines ......... 
 
 The Plain Scale ......... 
 
 Sector .......... 
 
 Marquois’s Scales ........ 
 
 Vernier .......... 
 
 Micrometer ......... 
 
 Beam Compasses ......... 
 
 Plotting Scales ......... 
 
 Pantagraph ......... 
 
 Coggeshall’s Sliding Rule. 
 
 Management of Drawing Paper, &c. ..... 
 
 General Rules in Geometrical Construction .... 
 
 PART II. 
 
 ON OPTICAL INSTRUMENTS. 
 
 Prism .... 
 
 Lenses .... 
 
 Reflectors, Plane 
 
 -Curvilinear 
 
 Microscopes 
 
 Telescopes, Refracting . 
 
 -Reflecting i 
 
 -Adjusted and Tested 
 
 Solar Microscope 
 Camera Obscura . 
 
 --Lucida • . . 
 
 © OO OO CO CO OO •'I M M -T O 
 
 ©H»-VlOO00SClt0WW0uf*'«0(»(»00*taiMM-‘ 
 
CONTENTS. 
 
 vi 
 
 PART III. 
 
 SURVEYING INSTRUMENTS. 
 
 Page 
 
 Land Chain..92 
 
 Off-setting Staff .......... 95 
 
 Spirit Level.96 
 
 Y Level ........... 97 
 
 Levelling Staff .. . 104 
 
 Trough ton’s Level . . . . . . . . .104 
 
 Gravatt’s ........... 105 
 
 Levelling, Remarks on ... 106 
 
 Water Level . . ... .112 
 
 Reflecting Level .. 113 
 
 Prismatic Compass . ... .115 
 
 Box Sextant.117 
 
 Artificial Horizon ......... 121 
 
 Theodolite. 122 
 
 Surveyin'?, Remarks on . ..126 
 
 Circular Protractor . . . . . , . .129 
 
 T Square and Semicircular Pfotractor . . . . . .131 
 
 Plotting Scale of best Form ........ 133 
 
 Computation of the Area of a Plot.135 
 
 Station Pointer. 135 
 
 PART IV. 
 
 ASTRONOMICAL INSTRUMENTS. 
 
 Hadley’s Sextant.137 
 
 Trough ton’s Reflecting Circle . . . . . . .141 
 
 The Dip Sector .. 144 
 
 The Transit Instrument.144 
 
 Method of Observing with the Transit . . . . . .151 
 
 The Altitude and Azimuth Instrument.153 
 
 The Reading Microscope . . . . . . . *. 160 
 
 The Collimator.. • • • .162 
 
 PART V. 
 
 The Common Goniometer 
 Wollaston’s Goniometer 
 
 ON GONIOMETERS. 
 
 164 
 
 165 
 
A TREATISE 
 
 ON 
 
 MATHEMATICAL INSTRUMENTS. 
 
 PART I.—ON MATHEMATICAL DRAWING INSTRUMENTS. 
 
 In this branch of the subject the limits of our little work will 
 not permit us to enter upon all the beautiful contrivances which 
 have been invented for facilitating the operations of the 
 draughtsman; but we shall endeavour to describe the con¬ 
 structions and applications of such as are in most general use, 
 and, as far as our space will allow, to exhibit the principles 
 upon which they are founded, so that the student may readily 
 extend his views, after having completely mastered the matter 
 here presented to him, to the principles of any other instru¬ 
 ments, which may be useful to him in whatever particular 
 professional branch of practical mathematics he may wish to 
 employ himself. With this view we shall describe the instru¬ 
 ments in the ordinary case of drawing instruments, as sold by 
 any mathematical instrument maker; viz., 
 
 Compasses with moveable point, 
 ink point, and pencil point. 
 Hair compasses. 
 
 Bow compasses. 
 
 Drawing pen and pricking 
 point. 
 
 Plain scale. 
 
 Sector. 
 
 And we shall also give some account of the following; 
 
 Whole and halves. 
 Proportional compasses. 
 Triangular compasses. 
 Marquois’s scales. 
 
 Beam compasses. 
 Plotting scales. 
 The pantograph. 
 Sliding Pule. 
 
 viz., 
 
 i 
 
 ON DRAWING COMPASSES. 
 
 This instrument Consists of two legs moveable about a joint, 
 so that the points at the extremities of the legs may be set at 
 any required distance from one another; it is used to transfer 
 and measure distances, and to describe arcs and circles. 
 
 The points of the compasses should be formed of well-tem¬ 
 pered steel, that cannot be easily bent or blunted, the upper 
 part being formed of brass or silver. The joint is framed 
 of two substances; one side being of the same material as 
 
 B 
 
2 
 
 MATHEMATICAL INSTRUMENTS. 
 
 the upper part of the compasses, either brass or silver, and 
 the other of steel. This arrangement diminishes the wear 
 of the parts, and promotes uniformity in their motion. If 
 this uniformity be wanting, it is extremely difficult to set the 
 compasses at any desired distance, for, being opened or closed 
 by the pressure of the finger, if the joint be not good, they will 
 move by fits and starts, and either stop short of, or go beyond 
 the distance required; but, when they move evenly, the pressure 
 may be regulated so as to open the legs to the desired extent, 
 and the joint should be stiff enough to hold them in this posi¬ 
 tion, and not to permit them to deviate from it in consequence 
 of the small amount of pressure which is inseparable from their 
 use. When greater accuracy in the set of the compasses is 
 required than can be effected by the joint alone, we have 
 recourse to the 
 
 Hair Compasses , in which the upper part of one of the steel 
 points is formed into a bent spring, which, being fastened at one 
 extremity to the leg of the 
 
 Fig. 1 
 
 Fig . 
 
 Fig. 3 
 
 compasses almost close 
 up to the joint, is held at 
 the other end by a screw. 
 
 A groove is formed in 
 the shank, which receives 
 the spring when screwed 
 up tight; and, by turning 
 the screw backwards, the 
 steel point may be gradu¬ 
 ally allowed to be pulled 
 backwards by the spring, 
 and may again be gradu¬ 
 ally pulled forwards by 
 the screw being turned 
 forwards. 
 
 Fig. 1 represents 
 these compasses when 
 shut; fig. 2 represents 
 them open, with the 
 screw turned backwards, 
 and the steel point p, in consequence moved backwards by its 
 spring s, from the position represented by the dotted lines, 
 which it would have when screwed tight up. 
 
 Fig. 3 represents a key, of which the two points fit into the 
 two holes seen in the nut, n, of the joint; and by turning this 
 nut the joint is made stiffer or easier at pleasure. 
 
COMPASSES. 
 
 3 
 
 To take a Distance with the Hair Compasses .—Open them 
 as nearly as you can to the required distance, set the fixed leg 
 on the point from which the distance is to be taken, and make 
 the extremity of the other leg coincide accurately with the 
 end of the required distance, by turning the screw. 
 
 COMPASSES WITH MOVEABLE POINTS. 
 
 If an arc or circle is to be described faintly, merely as a 
 guide for the terminating points of other lines, the steel points 
 are generally sufficient for 
 the purpose, and are suscep¬ 
 tible of adjustment with 
 greater accuracy than a pen¬ 
 cil point; but, in order to draw 
 arcs or circles with ink or 
 black lead, compasses w r ith a 
 moveable point are used. In 
 the best description of these 
 compasses the end of the 
 shank is formed into a strong 
 spring, which holds firmly 
 the moveable point, or a pen¬ 
 cil or ink point, as may be 
 required. A lengthening bar 
 may also be attached be¬ 
 tween the shank and the 
 moveable point, so as to 
 strike larger circles, and 
 measure greater distances. 
 
 The moveable point to be at¬ 
 tached to the lengthening 
 bar, as also the pen point and 
 pencil point, are furnished 
 with a joint, that they may be 
 set nearly perpendicular to 
 the paper. 
 
 a, the compasses, with a moveable point at b. 
 
 c and d, the joints to set each point perpendicular to the 
 paper. 
 
 e, the pencil point. 
 
 f, the pen point. (This is represented with a dotting 
 w r heel, the pen point and the dotting point being similar in 
 shape to each other.) 
 
 g, the lengthening bar., 
 
 b 2 
 
4 
 
 MATHEMATICAL INS.TRUMENTS. 
 
 To describe small arcs or 
 circles a small pair of com¬ 
 passes, called boiv compasses, 
 with a permanent ink or pen¬ 
 cil point, are used. They are 
 formed with a round head, 
 which rolls with ease be¬ 
 tween the fingers. The ad¬ 
 joining figures represent two 
 constructions of pen bows, 
 fig. 1 being well adapted to 
 describe arcs of not more 
 than one inch radius, and 
 fig. 2 to describe arcs of 
 small radii with exactness by 
 means of the adjusting screw 
 c. 
 
 For copying and reducing 
 drawings, compasses of a pe¬ 
 culiar construction are used ; 
 the simplest form of which 
 is that called wholes and 
 halves, because the longer 
 legs being twice the length 
 of the shorter, when the former are opened to any given line, 
 the shorter ones will be opened to the half of that line. By 
 their means, then, all the lines of a drawing may be reduced 
 to one-half, or enlarged to double their length. These com¬ 
 passes are also useful for dividing lines by continual bisec¬ 
 tions. 
 
 PROPORTIONAL COMPASSES. 
 
 By means of this ingenious instrument drawings may be 
 reduced or enlarged, so that all the lines of the copy, or the 
 areas or solids represented by its several parts, shall bear 
 any required proportion to the lines, areas, or solids of the 
 original drawing. They will also serve to inscribe regular 
 polygons in circles, and to take the square roots and cube roots 
 of numbers. In the annexed figure the scale of lines is placed 
 on the leg a e, on the left-hand side of the groove, and the 
 scale < f circles, on the same leg, on the right-hand side of the 
 groove. The scales of plans and solids are on the other face 
 of the instrument. 
 
 To set the instrument it must first be accurately closed, so 
 
COMPASSES. 
 
 5 
 
 that the two legs appear but as one; the nut c 
 being then unscrewed, the slider mav be moved, 
 until the line across it coincides with any required 
 division upon any one of the scales. Now tighten 
 the screw, and the compasses are set. 
 
 To reduce or enlarge the Lines of a Drawing .— 
 
 The line across the slider being set to one of the 
 divisions, 2, 3, 4, &c., on the scale of lines, the 
 points a, b will open to double, triple, four times, 
 
 &c., the distances of the points d, e (Euc. vi. 
 prop. 4). If, then, the points a and b be opened 
 to the lengths of the lines upon a drawing, the 
 points d and e will prick off a copy with the lines 
 reduced in the proportions of % to 1, % to 1, £ to 1, 
 
 &c.; but, if the points d and e be opened to the 
 lengths of the lines upon a drawing, the points 
 a and b will prick off a copy with the lines enlarged 
 in the proportions of 2 to 1,3 to 1, 4 to 1, &c. 
 
 To inscribe in a Circle a regular Polygon of 
 any number of Sides from 6 to 20.—The line 
 across the slider being set to any number on the 
 scale of circles, and the points a and b being opened 
 to the length of any radius, the points d and e will 
 prick off a polygon of that number of sides, in 
 the circle described with this radius; thus, if the line across 
 the slider be set to the division marked 12 on the scale of cir¬ 
 cles, and a circle be described with the radius a b, d e will 
 be the chord of a T Vth part of the circumference, and will prick 
 off a regular polygon of 12 sides in it. 
 
 To reduce or enlarge the Area of a Drawing. —The numbers 
 upon the scale of plans are the squares of the ratios of the 
 leugths of the opposite ends of the compasses, when the line 
 across the slider is set to those numbers; and, the distances 
 between the points being in the same ratio as the lengths of 
 the corresponding ends (Euc. vi. prop. 4), the areas of the 
 drawings, and of the several parts of the drawings, pricked off 
 by these points, will have to one another the ratio of 1 to the 
 number upon the scale of plans to which the instrument is 
 set (Euc. vi. props. 19, 20; and xii. prop. 2). Thus, if the 
 line across the slider be set to 4 on the scale of plans, the 
 distance between the points a and b will be twice as great as 
 the distance between d and e ; and, if a and b be opened out 
 to the lengths of the several lines of a drawing, d and e will 
 ;rick off a copy occupying £th the area; if the line across the 
 
6 
 
 MATHEMATICAL INSTRUMENTS. 
 
 slides be set to 5 on the same scale, the distances between 
 
 the points will be in the ratio of 1 to \/5, and the area of the 
 copy pricked off by the points d and e will be -£th of the area 
 of the drawing, of which the lines are taken off by a and b • 
 conversely, if the lines of the drawing be taken off by the 
 points d and e, the points a and b will prick off a copy, of 
 which the area will be 4 times or 5 times as great, according 
 as the line across the slider is set to the division marked 4 or 
 5 on the scale. 
 
 To take the Square Root of a Number .—The line across the 
 slider being set to the number upon the scale of plans, open 
 the points a and b to take the number from any scale of 
 equal parts (see page 9), then the points d and e applied to 
 the same scale of equal parts will take the square root of the 
 number. Thus, to take the square root of 3, set the line 
 across the slider to 3, open out the compasses, till a and b 
 take off 3 from any scale of equal parts, and the points d and e 
 will take off 1.73, which is the square root of 3 from the same 
 scale of equal parts. A mean proportional between two num¬ 
 bers, being the square root of their product, may be found by 
 multiplying the numbers together, and then taking the square 
 root of the product in the manner explained above. 
 
 The numbers of the scale of solids are the cubes of the 
 ratios of the lengths of the opposite ends of the compasses, 
 when the line across the slider is set to those numbers; so 
 that, when this line is set to the division marked 2 upon the 
 scale of solids, the distance between the points a and b will 
 give the side of a solid of double the content of that, of which 
 a like side is given by the distance of the points d and e when 
 the line is set to 3, the respective distances of the points will 
 give the like sides of solids, the contents of which will be in 
 the proportion of 3 to 1; and so on. 
 
 The Cube Root of a given number may be found by setting 
 the line across the slider to the number upon the scale of 
 solids, and, opening the points a, b, to take off the number 
 upon any scale of equal parts, the points d, e, will then take 
 off the required cube root from the same scale. 
 
 THE TRIANGULAR COMPASSES. 
 
 One of the best forms of these instruments is represented 
 in the annexed figure, a b c is a solid tripod, having at the 
 extremity of the three arms three limbs, d, e, and /, moving 
 freely upon centers by which they may be placed in any po- 
 
DRAWING-PEN. 
 
 7 
 
 sition with respect to the tripod and each 
 other. These limbs carry points at right 
 angles to the plane of the instrument, 
 which may be brought to coincide, in the 
 first instance, with any three points on 
 the original, and then transferred to the 
 copy. After this first step two of these 
 points must be set upon two points of 
 the drawing already copied, and the third made to coincide 
 with a new point of the drawing, that is, one not yet copied: 
 then, by placing the two first points oil the corresponding 
 points in the copy, the third point of the compasses will 
 transfer the new point to the copy. 
 
 Another form of triangular compasses is represented in the 
 annexed figure. 
 
 
 THE DRAWING-PEN. 
 
 This instrument is used for drawing straight lines. It 
 
 Fir/. 1. 
 
 a 
 
 Fig. 2. 
 n 
 
 tli 1 j 
 
 Fig. 3. 
 
 consists of two blades with steel 
 points fixed to a handle; and 
 they are so bent, that a suffi¬ 
 cient cavity is left between them 
 for the ink, when the ends of 
 the steel points meet close to¬ 
 gether, or nearly so. The blades 
 are set with the points more or 
 less open by means of a mill¬ 
 headed screw, so as to draw lines 
 of any required fineness or thick¬ 
 ness. One of the blades is framed 
 with a joint, so that by taking 
 out the screw the blades may be 
 completely opened, and the points 
 effectively cleaned after use. The 
 ink is to be put between the 
 blades by a common pen, and in 
 using the pen it should be slightly 
 inclined in the direction of the 
 line to be drown, and care should 
 be taken that both points touch 
 the paper; and the observations 
 
8 
 
 MATHEMATICAL INSTRUMENTS. 
 
 equally apply to the pen points of the compasses before 
 described. The drawing-pen should be kept close to the 
 straight edge (see straight edge), and in the same direction 
 during the whole operation of drawing the line. 
 
 For drawing close parallel lines in mechanical and architec¬ 
 tural drawings, or to represent canals or roads, a double pen 
 (fig. 2) is frequently used, with an adjusting screw to set the 
 pen to any required small distance. This is usually called 
 the road pen. The best pricking point is a fine needle held 
 in a pair of forceps (fig. 3). It is used to mark the intersec¬ 
 tions of lines, or to set off divisions from the plotting scale 
 and protractor (p. 33). This point may also be used to prick 
 through a drawing upon an intended copy, or, the needle being 
 reversed, the eye end forms a good tracing point. 
 
 a straight edge. 
 
 As many instruments are required to have straight edges 
 for the purpose of measuring distances, and of drawing straight 
 lines, it may be considered important to test the accuracy of 
 such edges. This may be done by placing two such edges 
 in contact and sliding them along each other, wdiile held up 
 between the eye and the light: if the edges fit close in some 
 parts, so as to exclude the light, but admit it to pass between 
 them at other parts, the edges are not true: if, however, the 
 edges appear, as far as the test has now proceeded, to be true, 
 still this may arise from a curvature in one edge fitting into an 
 opposite curvature in the other; the final step then is to take a 
 third edge, and try it in the same manner with each of the other 
 tw T o. and if in each case the contact be close throughout the 
 whole extent of the edges, then they are all three good*. 
 
 u To draw a straight line between two points upon a plane, we lay a rule 
 so that the straight edge thereof may just pass by the two points ; then moving 
 a fine-pointed needle, or drawing-pen, along this edge, we draw a line from 
 one point to the other, which, for common purposes, is sufficiently exact; but, 
 where great accuracy is required, it will be found extremely difficult to lay 
 the rule equally with respect to both the points, so as not to be nearer to one 
 point than the other. It is difficult also so to carry the needle, or pen, that 
 it shall neither incline more to one side than the other of the rule; and, 
 thirdly, it is very difficult to find a rule that shall be perfectly straight. 
 
 “ If the two points be very far distant, it is almost impossible to draw the 
 line with accuracy and exactness; a circular line may be described more 
 easily, and more exactly, than a straight or any other line, though even then 
 many difficulties occur, when the circle is required to be of a large radius. 
 
 “ And let no one consider these reflections as the effect of too scrupulous 
 exactness, or as an unnecessary aim at precision ; for, .as the foundation of 
 
 * Euc. bk. i. dof. 10. Peacock’s Algebra, 1st edition art. 532. p. 420. 
 
SCALES OF EQUAL FARTS. 
 
 9 
 
 all oiir knowledge in geography, navigation, and astronomy, is built on obser¬ 
 vations, and all observations are made with instruments, it follows that the 
 truth of the observations, and the accuracy of the deductions therefrom, will 
 principally depend on the exactness with which the instruments are made 
 and divided, and that those sciences will advance in proportion as these are 
 less difficult in their use, and more perfect in the performance of their respec¬ 
 tive operations.” * 
 
 ON SCALES. 
 
 Scales of equal parts are used for measuring straight lines, 
 and laying down distances, each part answering for one foot, 
 one yard, one chain, &c., as may be convenient, and the plan 
 will be larger or smaller as the scale contains a smaller or a 
 greater number of parts in an inch. 
 
 Scales of equal parts may be divided into three kinds; simply- 
 divided scales, diagonal scales, and vernier 
 scales. 
 
 Simply-divided Scales. —Simply-divided 
 scales consist of any extent of equal divisions, 
 which are numbered 1, 2, 3, &c., beginning 
 from the second division on the left hand. 
 
 The first of these primary divisions is sub¬ 
 divided into ten equal parts, and from these 
 last divisions the scale is named. Thus it 
 is called a scale of 30, when 30 of these 
 smdl parts are equal to one inch. If, then, 
 these subdivisions be taken as units, each to 
 represent one mile, for instance, or one chain, 
 or one foot, &c., the primary divisions will be 
 so many tens of miles, or of chains, or of feet, 
 
 &c.; if the subdivisions are taken as tens, 
 the primary divisions will be hundreds ; 
 and, if the primary divisions be units, the 
 subdivisions will be tenths. 
 
 The accompanying drawing represents 
 six of the simply-divided scales, which are 
 generally placed upon the plain scale. To 
 them to feet and inches, the first pri¬ 
 mary division is divided duodeeimally upon 
 an upper line. To lay down 360, or 36, 
 or 3 6, &c., from any one of these scales, 
 extend the compasses from the primary 
 division numbered 3 to the 6th lower sub- 
 
 * Geometrical and Geographical Essays, by the late George Adams, edited 
 by William Jones, F.Am.P.S. 
 
 
 w 
 
 -L 
 
 'j-i 
 
 
 03 
 
 t\3 
 
 03 
 
 VJ 
 
 
 5c 
 
 V) 
 
 Co 
 
 v> 
 
 30 
 
 
 N 
 
 Co 
 
 
 
 
 to 
 
 
 Co 
 
 vO 
 
10 
 
 MATHEMATICAL INSTRUMENTS. 
 
 division, reckoning backwards, or towards the left band. To 
 take off any number of feet and inches, 6 feet 7 inches for 
 instance, extend the compasses from the primary division 
 numbered 6, to the 7th upper subdivision, reckoning back¬ 
 wards, as before. 
 
 Diagonal Scales .—In the simply-divided scales one of the 
 primary divisions is subdivided only into ten equal parts, and 
 the parts of any distance which are less than tenths of a pri¬ 
 mary division cannot be accurately taken off from them; but, 
 by means of a diagonal scale, tbe parts of any distance which 
 are the hundredths of the primary divisions are correctly in¬ 
 dicated, as will easily be understood from its construction, 
 which we proceed to describe. 
 
 Draw eleven parallel equidistant lines; 
 divide the upper of these lines into equal 
 parts of the intended length of the primary 
 divisions; and through each of these di¬ 
 visions draw perpendicular lines, cutting all 
 the eleven parallels, and number these pri¬ 
 mary divisions, 1, 2, 3, &c., beginning from 
 the second. 
 
 Subdivide the first of these primary di¬ 
 visions into ten equal parts, both upon the 
 highest and lowest of the eleven parallel 
 lines, and let these subdivisions be reckoned 
 in the opposite direction to the primary di¬ 
 visions, as in the simply-divided scales. 
 
 Draw the diagonal lines from the tenth 
 subdivision below to the ninth above; from 
 the ninth below to the eighth above; and 
 so on; till we come to a line from the first 
 below to the zero point above. Then, since 
 these diagonal lines are all parallel, and 
 consequently everywhere equidistant, the 
 distance between any two of them in suc¬ 
 cession, measured upon any of the eleven 
 parallel lines which they intersect, is the 
 same as this distance measured upon the 
 highest or lowest of these lines, that 
 is, as one of the subdivisions before 
 mentioned: but the distance be ween the 
 perpendicular, which passes through the 
 zero point, and the diagonal through the 
 same point, being nothing on the highest line, and equal to 
 
SCALES OF EQUAL PARTS. 
 
 11 
 
 one of the subdivisions on the lowest line, is equal (Euc. 
 vi. prop. 4) to one-tenth of a subdivision on the second line, 
 to two-tenths of a subdivision on the third, and so on; so that 
 this, and consequently each of the other diagonal lines, as it 
 reaches each successive parallel, separates further from the 
 perpendicular through the zero point by one-tenth of the ex¬ 
 tent of a subdivision, or one-hundredth of the extent of a pri¬ 
 mary division. Our figure represents the two diagonal scales 
 which are usually placed upon the plane scale of six inches in 
 length. In one, the distances between the primary divisions 
 are each half an inch, and in the other a quarter of an inch. 
 The parallel next to the figures numbering these divisions 
 must be considered the highest or first parallel in each of 
 these scales to accord with the above description. 
 
 The primary divisions being taken for units, to set off the 
 numbers 5-74 by the diagonal scale. Set one foot of the 
 compasses on the point where the fifth parallel cuts the 
 eighth diagonal line, and extend the other foot to the point 
 where the same parallel cuts the sixth vertical line. 
 
 The primary divisions being reckoned as tens, to take off 
 the number 46‘7. Extend the compasses from the point where 
 the eighth parallel cuts the seventh diagonal to the point 
 where it cuts the fifth vertical. 
 
 The primary divisions being hundreds, to take off the 
 number 253. Extend the compasses from the point where 
 the fourth parallel cuts the sixth diagonal to the point where 
 it cuts the third vertical. 
 
 Now, since the first of the parallels, of the diagonals, and 
 of the verticals indicate the zero points for the third, second, 
 and first figures respectively, the second of each of them 
 stands for, and is marked, 1, the third, 2, and so on, and we 
 have the following 
 
 General Buie .—To take off any number to three places of 
 figures upon a diagonal scale. On the parallel indicated by 
 the third figure, measure from the diagonal indicated by the 
 second figure to the vertical indicated by the first. 
 
 Vernier Scales .—The nature of these scales will be under¬ 
 stood from their construction. To construct a vernier scale, 
 which shall enable us to take off a number to three places of 
 figures: divide all the primary divisions into tenths, and 
 number these subdivisions, 1, 2, 3, etc., from the left hand 
 towards the right throughout the whole extent of the scale. 
 
 Take off now, with the compasses, eleven of these subdi¬ 
 visions, set the extent off backwards from the end of the first 
 
12 
 
 MATHEMATICAL INSTRUMENTS. 
 
 primary division, and it will reach beyond the beginning of 
 this division, or zero point, a distance equal to one of the sub¬ 
 divisions. Now divide the extent thus set off into 
 ten equal parts, marking the divisions ou the opposite 
 side of the divided line to the strokes marking the 
 primary divisions and the subdivisions, and number 
 them 1,2, 3, &c., backwards from right to left. Then, 
 since the extent of eleven subdivisions has been di¬ 
 vided into ten equal parts, so that these ten parts 
 exceed by one subdivision the extent of ten subdivi¬ 
 sions, each one of these equal parts, or, as it may be 
 called, one division of the vernier scale, exceeds one 
 of the subdivisions by a tenth part of a subdivision, 
 or a hundredth part of a primary division. In our 
 figure the distances between the primary divisions are 
 each one inch, and. consequently, the distances be¬ 
 tween the subdivisions are each one-tenth of an 
 inch, and the distances between the divisions of the 
 vernier scale each one-tenth and one-hundredth of 
 an inch. 
 
 To take off the number 253 from this scale. In¬ 
 crease the first figure 2 by 1, making it 3; because 
 the vernier scale commences at the end of the first 
 primary division, and the primary divisions are mea¬ 
 sured from this point, and not from the zero point*. 
 
 The first thus increased with the second now represents 
 35 of the subdivisions from the zero point, from 
 which the third figure, 3, must be subtracted, leaving 
 32; since three divisions of the vernier scale will 
 contain three of these subdivisions, together with 
 three-tenths of a subdivision. Place, then, one point 
 of the compasses upon the third division of the vernier 
 scale, and extend the other point to the 32nd sub¬ 
 division, or the second division beyond the 3rd pri¬ 
 mary division, and laying down the distance be¬ 
 tween the points of the compass, it will represent 
 253, or 25’3, or 2 53, according as the primary divi¬ 
 sions are taken as hundreds, tens, or units. 
 
 General Rule .—To take off any number to three 
 places of figures upon this vernier scaleu Increase 
 the first figure by one; subtract the third figure from the 
 second, borrowing one from the first increased figure, if ne- 
 
 * If the vernier scale were placed to the left of the zero point, a distance 
 less than one primary division could not always be found upon the scale. 
 
scales of Equal parts. 
 
 1 . 0 ' • 
 
 O 
 
 cessary, and extend the compasses from the division upon 
 the vernier scale, indicated by the third figure, to the subdivi¬ 
 sion indicated by the number remaining after performing the 
 above subtraction. 
 
 Suppose it were required to take off the number 253 5. By 
 extending the compasses from the third division of the 
 vernier scale to the 32nd subdivision, the number 253 is taken 
 off, as we have seen. To take off, therefore, 253 - 5, the com¬ 
 passes must be extended from one of these points to a short 
 distance beyond the other. Again, by extending the com¬ 
 passes from the 4th division of the vernier scale to the 31st 
 subdivision, the number 254 would be taken off. To take off 
 253-5, then, the compasses must be extended from one of these 
 points to within a short distance of the other; and by setting 
 the compasses so that, when one point of the compasses is set 
 successively on the 3rd and 4th division of the vernier scale, 
 the other point reaches as far beyond the 32nd subdivision as 
 it falls short of the 31st, the number 253'5 is taken off. If 
 the excess in one case be twice as great as the defect in the 
 other, the distance represents the number 253|, or 253-66; 
 and if the excess be half the defect, the distance represents 
 253^, or 253-33. Thus distances may be set off with an. 
 accurately-constructed scale of this kind to within the three- 
 hundredth part of a primary division, unless these divisions 
 be themselves very small. 
 
 We are not aware that a scale of this kind has been put 
 upon the plain scales sold by any of the instrument makers; 
 but, during the time occupied in plotting an extensive survey, 
 the paper which receives the work is affected by the changes 
 which take place in the hygrometrical state of.the air, and the 
 parts laid down from the same scale, at different times, will 
 not exactl}- correspond, unless this scale has been first laid 
 down upon the paper itself, and all the divisions have been 
 taken from the scale so laid down, which is always in the same 
 state of expansion as the plot. For plotting, then, an extensive 
 survey, and accuratel} 1, filling in the minutiae, a diagonal, or 
 vernier scale may advantageously be laid down upon the 
 paper upon which the plot is to be made. A vernier scale is 
 preferable to a diagonal scale, because in the latter it is ex¬ 
 tremely difficult to draw the diagonals with accuracy, and we 
 have no check upon its errors ; while in the former the uniform 
 manner in which the strokes of one scale separate from those 
 of the other is some evidence of the truth of both*. 
 
 * In Mr. Bird’s celebrated scale, by means of which he succeeded in di-. 
 
14 
 
 MATHEMATICAL INSTRUMENTS. 
 
 ON THE PROTRACTING SCALES. 
 
 The nature of these scales will he understood from the 
 following construction (plate 1, fig. 1): 
 
 With centre o, and radius o a, describe the circle abcd; 
 and through the centre o draw the diameters a c, and b d, at 
 right angles to each other, which will divide the circle into 
 four quadrants, a b, b e, c d, and d a. 
 
 Divide the quadrant c d into nine equal parts, each of 
 which will contain ten degrees, and these parts may again be 
 subdivided into degrees, and, if the circle be sufficiently large, 
 into minutes. 
 
 Set one foot of the compasses upon c, and transfer the di¬ 
 visions in the quadrant c d to the right line c d, and we shall 
 have a scale of chords*. 
 
 From the divisions in the quadrant c d, draw right lines 
 parallel to d o, to cut the radius o c, and, numbering the divi¬ 
 sions from o, towards c, we shall have a scale of sines. 
 
 If the same divisions be numbered from c, and continued 
 to a, we shall have a scale of versed sines. 
 
 From the centre o, draw right lines through the divisions 
 of the quadrant c d, to meet the line c t, touching the circle 
 at c, and, numbering from c, towards t, we shall have a scale 
 of tangents. 
 
 Set one foot of the compasses upon the center o, and transfer 
 the divisions in c t into the right line o s, and we shall have 
 a scale of secants. 
 
 Right lines, drawn from a to the several divisions in the 
 quadrant c d, will divide the radius o d into a line of semi¬ 
 tangents, or tangents of half the angles indicated by the 
 numbers; and the scale may be continued by continuing the 
 divisions from the quadrant c d, through the quadrant d a, 
 
 viding, with greatly-improved accuracy, the circles of astronomical instru¬ 
 ments, the inches are divided into tenths, as in the scale described in the 
 text, and 100 of these tenths are divided into 100 parts for the vernier 
 scale. 
 
 * We give the constructions in the text to show the nature of the scales; 
 but in practice a scale of chords is most accurately constructed by values 
 computed from tabulated arithmetical values of sines, which computed 
 values are set off from a scale of equal parts; and the circle is divided most 
 accurately by means of such computed chords. The limits of our work 
 forbid our entering further upon this interesting subject. All the other 
 scales will also be most accurately constructed from computed arithmetical 
 values, taken off by means of the beam compasses hereafter described, and 
 corrected by the aid of a good Bird’s vernier scale. 
 
PROTRACTING SCALES. 
 
 15 
 
 and drawing right lines from a, through these divisions, to 
 meet the radius o d, produced. 7 
 
 Divide the quadrant a d into eight equal parts, subdivide 
 each of these into four equal parts, and, setting one foot of 
 the compasses upon a, transfer these divisions to the right 
 line a d, and we shall have a scale of rhumbs. 
 
 Divide the radius a o into 60 equal parts, and number them 
 from o towards a ; through these divisions draw right lines 
 parallel to the radius o b, to meet the quadrant a b ; and, with 
 one foot of the compasses upon a, transfer these divisions from 
 the quadrant to the right line a b, and we shall have a scale 
 of longitudes. 
 
 Place the chord of GO 0 , or radius*, between the radii o c 
 and o b, meeting them at equal distances from the center; di¬ 
 vide the quadrant c b into six equal parts, for intervals of 
 hours, subdividing each of these parts into 12 for intervals of 
 5 minutes, and further subdividing for single minutes if the 
 circle be large enough; and from the center o draw right 
 lines to the divisions and subdivisions of the quadrant, inter¬ 
 secting the chord or radius placed in the quadrant, and we 
 shall have a scale of hours. 
 
 Prolong the touching line t c to l ; set off the scale of sines 
 from c to l; draw right lines from the center o to the divisions 
 upon c l, and from the intersections of these lines with the 
 quadrant c b draw right lines parallel to the radius o c, to 
 meet the radius o b, and we shall have a scale of latitudes f. 
 
 Corresponding lines of hours and latitudes may also be con¬ 
 structed (as represented in our figure) more simply, and on a 
 scale twice as large as by the preceding method, as follows:— 
 
 With the chord of 45° set olf from b to e, and again from 
 b to f, we obtain a quadrant e f bisected in b ; and, the 
 chord of 60° or radius being set off from a, c, f, and e, this 
 quadrant is divided into six equal parts. From the center o, 
 draw straight lines through these divisions to meet the line 
 touching the circle at b, and we shall have the line of hours. 
 
 From the point d, draw right lines through the divisions 
 upon the line of sines o c, to meet the circumference b c, and 
 
 * Chord of 60° is equal to radius. Euc. book iv. prop. 15, Cor. 
 
 f The line of latitudes is a line of sines, to radius equal the whole length 
 of the line of hours, of the angles, of which the tangents are equal to the 
 sines of the latitudes. The middle of the hour line being numbered for three 
 o’clock, the divisions for the other hours are found by setting off both ways 
 from the middle the tangents of n. 15°, n. being the number of hours from 
 three o’clock, that is. one for two o’clock and four o’clock, two for one o’clock 
 
16 
 
 MATHEMATICAL INSTRUMENTS. 
 
 transferring these divisions from b, as a center to the chord 
 b c, we shall have the corresponding line of latitudes. 
 
 It is not necessary that these scales should all be projected 
 to the same radius; but those which are used together, as the 
 rhumbs and chords, the chords and longitudes, the sines, tan¬ 
 gents, secants, and semitangents, and, lastly, the hours and 
 latitudes, must be so constructed necessarily. In the accom¬ 
 panying diagram (plate 1, fig 2) we have laid down the hours 
 and latitudes to a radius equal to the whole length of the scale, 
 the other lines being laid down to the radius used in the fore¬ 
 going construction. 
 
 The Line of Chords is used to set off an angle, or to measure 
 an angle already laid down. 
 
 1st. To set off an angle , 
 which shall contain d° from 
 the point a, in the straight line 
 a b. Open the compasses to 
 the extent of GO 0 upon the 
 line of chords, which equals the 
 radius to which this line has 
 been laid down (Euc. iv. prop. 
 
 15, Cor.), and, setting one foot upon a, with this extent 
 describe an arc cutting ab in b; then, taking the extent of d c 
 from the same line of chords, set it off from b to c; and, join¬ 
 ing ac, b a c is the angle required. Thus to set oft' an angle 
 of 41°, having described the arc b c, as directed, with one foot 
 of the compasses on b, and the extent of 41° on the line of 
 chords, intersect b c in c, and join a c. 
 
 2nd. To measure the angle contained by the straight lines 
 a b and a c already laid down. Open the compasses to the ex¬ 
 tent of 60° on the line of chords, as before, and with this ra¬ 
 dius describe the arc b c, cutting a b and a c, produced, if 
 necessary, in the points b and c ; then, extending the com¬ 
 passes from b to c, place one point of the compasses on the 
 beginning, or zero point, of the line of chords, and the other 
 point will extend to the number upon this line, indicating the 
 degrees in the angle bac. If, for instance, this point fall on 
 the 41st divisio?), or the first division beyond that marked 40 
 in the figure (plate 1, fig, 2), the angle bac will contain 41°. 
 
 The Line of Rhumbs is a scale of the chords of the angles of 
 deviation from the meridian denoted by the several points 
 and quarter points of the compass, enabling the navigator, 
 without computation, to lay down or measure a ship's course 
 upon a chart. Thus, supposing the ship’s course to be 
 
 \r 
 
PROTRACTING SCALES. 
 
 It 
 
 N.N.E. f E. Through the point a, repre¬ 
 senting the ship’s place upon the chart, 
 draw the meridian a b, and with center a 
 and distance equal to the extent of 60° upon 
 the line of chords describe an arc cutting 
 A b in b ; then on the line of rhumbs take 
 the extent to the third subdivision beyond 
 the division marked 2, because N.N.E. is 
 the second point of the compass from the 
 north, and with one foot of the compasses 
 on b describe an arc intersecting b c in c : 
 join a c. and the angle b a c will represent 
 the ship’s course. On the other hand, if a 
 ship is to be sailed from the point a to a point on the line 
 A c on a chart, draw the meridian a b, describe the arc b c with 
 radius equal to chord of 60°, as before, and the extent from b 
 to c, applied to the line of rhumbs, will give 2 pts. 3 qrs., de¬ 
 noting that the ship must be sailed by the compass N.N.E. f E. 
 
 The Line of Longitudes shows the number of equatorial 
 miles in a degree of longitude on the parallels of latitude in¬ 
 dicated by the degrees on the corresponding points of the 
 line of chords. Example. —A ship in latitude 60° N. sailing 
 E. 79 miles, required the difference of longitude between the 
 beginning and end of her course. Opposite 60 on the line of 
 chords stands 30 on the line of longitudes, which is, therefore, 
 the number of equatorial miles in a degree of longitude at 
 that latitude. Hence, as 30 : 79 :: 60 : 158 miles, the 
 required difference of longitude. 
 
 The Lines of Sines, Secants, Tangents, and Semitangents are 
 principally used for the several projections, or perspective 
 representations, of the circles of the sphere, by means of 
 which maps are constructed. Thus, the meridians and paral¬ 
 lels of latitude being projected, the countries intended to be 
 represented are traced out according to their respective 
 situations and extent, the position of every point being deter¬ 
 mined by the intersection of its given meridian and parallel of 
 latitude. 
 
 The plane upon which the circles are to be delineated is 
 called the primitive, and the circumference of a circle, de¬ 
 scribed with a radius, representing, upon the reduced scale of 
 the drawing, the radius of the sphere, is called the circum¬ 
 ference of the primitive. Lines, drawn from all the points of 
 the circles to the eye, by their intersection with the primitive 
 form the projection. 
 
18 
 
 MATHEMATICAL INSTKUMENTS. 
 
 When the eye is supposed to be infinitely distant, so that 
 the lines of vision are parallel to one another and'perpendi¬ 
 cular to the primitive, the projection is called orthographic. 
 When the primitive is a tangent plane to the sphere, and the 
 eye is supposed to be at the center of the sphere, the projec¬ 
 tion is called gnomonic. When the eye is supposed to be at 
 the surface of the sphere, and the primitive to pass through 
 the center, so as to have the eye in its pole, the projection is 
 called stereographic. 
 
 The projection is further termed equatorial, meridional, or 
 horizontal, according as the primitive coincides with, or is 
 parallel to, the equator, or the meridian or horizon of any 
 place. 
 
 To delineate the Orthographic Projection of the Circles of 
 the Terrestrial Sphere upon the Plane of the Meridian of any 
 place. —With a radius according to 
 the contemplated scale of the pro¬ 
 jection, describe the circle w n e s 
 for the circumference of the pri¬ 
 mitive, and draw the vertical and 
 horizontal diameters n s and w e, 
 which will be the projections of a 
 meridian perpendicular to the pri¬ 
 mitive, and of the equator, respect¬ 
 ively. Take out from the line of 
 sines the sines of the latitudes 
 through which the parallels are to 
 be drawn, and, reducing these sines to the radius of the pri¬ 
 mitive*, setoff these reduced distances both ways from the 
 center upon the line n s ; and also both ways from the center 
 upon the line w e, for the sines of the angles which the 
 meridians, to be drawn at the same intervals as the parallels, 
 make with the meridian n s. Through the divisions thus set 
 off', upon the line n s draw straight lines parallel to w e, and 
 such straight lines will be the projections of the several paral 
 lels of latitude, which are to be numbered 0 to 90, from the 
 equator to either pole for the latitudes. With distances from 
 the center to the divisions set off upon w e as semi-minor axes, 
 and the distance from c to n or s, equal to radius of primitive, 
 
 * If the proportional compasses he set in the proportion of the sine 90° 
 on the line of sines to the radius of the primitive, one pair of points will give, 
 reduced to this radius, the sines taken off by the other pair of points. The 
 manner of taking from the sector a sine to any radius will be hereafter pointed 
 out. 
 
PROJECTION OF THE SPHERE. 
 
 19 
 
 as a common major axis, describe semi-ellipses*, and they will 
 be the projections of the several meridians, which are to be 
 numbered either way from the first meridian for the longitudes. 
 In the figure the primitive coincides with the plane of the 
 meridian of a place in 30° west longitude, or 150° east longi¬ 
 tude, the sum of these two being 180°, as must always be the 
 case. 
 
 To delineate the Gnomonic Projection of the Circles of the 
 Terrestrial Sphere upon a Plane parallel to the Equator. —In 
 this case the meridians 
 will all be projected 
 into straight lines, 
 making the same angles 
 one with another that 
 their originals do on 
 the surface of the 
 sphere; the projection 
 of the pole will be the 
 center of the primitive, 
 and the projections of 
 the parallels of lati¬ 
 tude will be circles de¬ 
 scribed from the pro¬ 
 jection of the pole, as 
 center, with distances 
 equal to the tangents 
 of the respective co¬ 
 latitudes reduced to the radius of the primitive. The parallel 
 of 45° will, therefore, coincide with the circumference of the 
 primitive; the parallels of latitudes greater than 45° will lie 
 within the primitive; and for latitudes less than45° the paral¬ 
 lels will fall without the primitive, the radii of their projec¬ 
 tions increasing as the latitude decreases, until the radius for 
 projecting the equator becomes infinite. Describe, then, a 
 
 * These semi-ellipses may be thus de¬ 
 scribed. From any point p upon the 
 straight edge of a piece of paper set off p c 
 equal to the major axis, and p b equal to 
 the minor axis : then move the paper into 
 various positions, but so that the point c 
 may always be upon the line w* E, and 
 the point p upon the line N s, and the 
 point p will, in every such position, coin¬ 
 cide with a point in the required ellipse. 
 
 S 
 
so 
 
 MATHEMATICAL instruments, 
 
 circle for the primitive; draw straight lines radiating from its 
 center, and equally inclined one to another for the projections 
 of equidistant meridians; and number them 0 to 180 both 
 ways from the first meridian for the longitudes. With the tan¬ 
 gents of the colatitudes, taken at intervals equal to the angle 
 between two successive meridians, and reduced to the radius 
 of the primitive, as distances, describe from the center of the 
 primitive concentric circles ; and number them 90 to 45 from 
 the pole to the primitive for the latitudes, continuing the gra¬ 
 duation beyond for the lower latitudes. 
 
 The gnomonic projection affords a good representation of 
 the polar regions, but all places in latitudes lower than 60° 
 appear greatly distorted. The gnomonic projection enlarges 
 the representations of places at a distance from the center of 
 projection beyond their proportionate true dimensions ; and 
 the orthographic, on the contrary, unduly contracts them; 
 while both are adapted for representing best the countries at 
 only a moderate distance from the center of projection. 
 
 To delineate the Stereographic Projection of the Circles of the 
 Terrestrial Sphere upon the Horizon of anyplace. —With radius 
 determined upon describe a circle for the primitive, and draw 
 its vertical and horizontal diameters, n s and w e, which 
 will be the projections of the meridian of the place and of the 
 prime vertical respectively. From the center c set off upon 
 the radius c s, produced, if necessary, the distance c a, equal 
 to the tangent of the latitude of the place reduced to the 
 radius of the primitive; and with center a and distance a w 
 or a e describe the circle w n e, which will be the projec¬ 
 tion of the meridian at right angles to n s, the meridian of 
 the place ; and, consequently, n will be the projection of the 
 pole. Through a draw the right line a d at right angles to 
 a c, and another line a d making any convenient angle with 
 a b, and, setting off a b equal to the radius of the primi¬ 
 tive, and a d equal to the sine of the colatitude, taken from 
 the line of sines, join b d. Now take from the line of tan¬ 
 gents the angles which the other meridians to be drawn are 
 to make with the meridian w n e, or the complements of 
 the angles which they are to make with n s, and set them 
 off both ways from a upon the line a d ; through each of the 
 divisions l, thus found, draw l o, parallels to b d, and we 
 have at o the centers of the circles for describing the meri¬ 
 dians*. With centers o and distances o n, describe the 
 
 * 
 
 r cot. L .... 
 
 - j where r represents the radius of the pri- 
 
 COS* L 
 
 The distance AO — 
 
PROJECTION OF THE SPHERE. 
 
 21 
 
 meridians, and 
 number them 0 
 to 180, both 
 ways, from the 
 first meridian, 
 for the longi¬ 
 tudes. For a 
 parallel through 
 any given lati¬ 
 tude, take the 
 difference of the 
 complement of 
 the given lati¬ 
 tude and of the 
 colatitude of the 
 place from the 
 line of serni- 
 tan gents, and, 
 having reduced 
 it to the radius 
 of the primitive, 
 set it off at r 
 from c towards 
 n for latitudes 
 greater than the 
 latitude of the 
 place, and from 
 c towards s for 
 latitudes less 
 than the lati¬ 
 tude of the 
 
 place: — again, 
 
 take 'the sum of the complement of the given latitude and 
 of the colatitude of the place from the line of semitangents, 
 and set it off at s from c upon c n produced: then the circle 
 described upon r s* as diameter will be the parallel required. 
 Draw these parallels for intervals of latitude equal to the 
 angles made by two successive meridians, and number them 
 90 to 0 from the pole n for the north latitudes, and again in¬ 
 creasing from 0 on the other side of the equator for the south 
 
 mitive, l the latitude of the place, and L the angle at which the meridian 
 is inclined to the meridian of the place. 
 
 * Diameter of parallel = r tan. ^ (c — 3) -j- r tan. £ (c -J- 3) where e 
 — colatitude of place, and 3 = colatitude of parallel. 
 
MATHEMATICAL INSTRUMENTS. 
 
 as 
 
 latitudes, if the place be in north latitude—or the converse, if 
 the place be in south latitude. 
 
 The practical application of the preceding methods of pro¬ 
 jection is usually confined to the representation of an entire 
 hemisphere, or at least of a considerable portion of a sphere; 
 but for laying down smaller portions of the sphere the method 
 of development may be advantageously adopted. In this me¬ 
 thod the portion of the sphere to be represented is considered 
 as coincident with a portion of a cone, touching the sphere in 
 a circle which is the middle parallel of latitude of the country 
 to be represented, and this portion of the cone when developed 
 forms a portion of a sector of a circle. 
 
 To lay down the meridians and parallels of latitude for this 
 development. 1. Take a straight line, b c a, for the middle 
 meridian of the intended map, and divide it into equal parts, 
 to represent degrees and minutes of latitude according to the 
 scale determined upon for the map. Q. From one of these 
 divisions, a, which is conveniently situated to form the center 
 of the map, set off from a to c the cotangent of the middle 
 latitude, reduced to a radius equal to 57-3 of the divisions pre¬ 
 viously marked off as degrees, or to 3438 of those marked off 
 as minutes. 3. With c as a center and radius c a, describe 
 the arc d a e for the middle parallel of latitude, and divide it 
 into equal parts to represent 
 degrees and minutes of longi¬ 
 tude, the lengths of these 
 parts having, to the lengths 
 of the parts previously set 
 off on the meridian for 
 degrees and minutes of 
 latitude, the ratio cosine 
 of middle latitude : ra¬ 
 dius. 4. With c as center, 
 describe concentric arcs, 
 through the divisions on 
 c e, for the parallels of lati¬ 
 tude; and draw straight lines, 
 radiating from c, through the 
 divisions on d a e for the me¬ 
 ridians. 
 
 In our figure the middle 
 latitude is 55°; a b is equal 
 to the length of 57’3°, or 
 the radius of the sphere; 
 
CONSTRUCTION OF DIALS. 
 
 23 
 
 A c is equal to the cotangent of 55, or the tangent of 35 
 reduced to this radius; and c, consequently, is the center 
 for describing the parallels, and the radiating point for the 
 meridians. 
 
 In drawing a map of small extent, it is usual to make all 
 the meridians and parallels of latitude straight lines; and to 
 make the extreme parallels, and the meridian passing through 
 the center of the map, proportional to their real magnitude. 
 
 Another and more exact method is to make the meridian 
 passing through the center of the map, and all the parallels 
 of latitude, straight lines, as in the last method. Then all the 
 degrees on each of the parallels are made proportional to their 
 magnitude, and the lines passing through the corresponding 
 points of division on the parallels will represent the meri¬ 
 dians. These will be curved lines, and not straight, as in the 
 last method. This is usually called Flamstead's Projection, 
 as it was first used by that astronomer in constructing his 
 “ Celestial Atlas;” and it is extremely useful in geographical 
 maps for countries lying on both sides of the equator. 
 
 A considerable improvement of this method, for countries 
 of large extent, is to represent all the parallels of latitude by 
 concentric circles, according to the principles of the conical 
 development; and then to lay off the degrees on each parallel, 
 proportional to their magnitude *, and draw lines through 
 the corresponding divisions of these parallels to represent 
 the meridians. This delineation, perhaps, will give the dif¬ 
 ferent parts of a map of some extent in as nearly their due 
 proportions as the 
 nature of the case 
 will admit. 
 
 We will now 
 briefly explain the 
 manner of con¬ 
 structing some of 
 the simplest dials 
 by means of the 
 dialling scales. 
 
 To construct a 
 Horizontal Dial .— 
 
 Draw on your dial 
 plate two parallel 
 
 * That is, the degrees on each parallel must have to a degree of latitude 
 the ratio of radius: cosine of the latitude of the parallel. 
 
Q-t MATHEMATICAL INSTRUMENTS. 
 
 lilies, a b, c il, as a double meridian line, at a distance apart 
 equal to the thickness of the intended style, or gnomon. 
 Intersect them at right angles by another line, ef, called the 
 six o’clock line. From the scale of latitudes take the latitude 
 of tne place with the compasses, and set that extent from c to e 
 and from a to f on the six o'clock line, and then, taking the 
 whole of six. hours between the parts of the compasses from 
 the scale, with this extent set one foot in the point e, and with 
 the other intersect the meridian line c d at d. Do the same 
 from / to b, and draw the right lines e d and / b, which are of 
 the same lengths as the scale of hours. Place one foot of the 
 compasses on the beginning of the scale, and, extending the 
 other to any hour on the scale, lay these extents off from d to 
 e for the afternoon hours, and from b to / for the forenoon. 
 In the same manner the quarters or minutes may be laid down, 
 if required. The edge of a ruler being now placed on the 
 point c, draw the first five afternoon hours from that point 
 through the marks on the line d e, and continue the lines of 
 4 and 5 through the center c to the other side of the dial for 
 the like hours of the morning. Lastly, lay a ruler on the 
 point a, and draw the last five forenoon hours through the 
 marks on the line/ b, continuing the hour lines of 7 and 8 
 through the center a to the other side of the dial, for 
 the evening hours, and figure the hours to the respective 
 lines. 
 
 To make the Gnomon .—From the line of chords, always 
 placed on the same dialling scale, take the extent of 60°, and 
 describe from the center a the arc 
 g n. Then with the extent of the 
 latitude of the place, suppose Lon¬ 
 don, 5i|°, taken from the same line 
 of chords, set one foot in n, and cross 
 the arc with the other at g. From 
 the center at a draw the line a g 
 for the axis of the gnomon a g i, 
 and from g let fall the perpendi¬ 
 cular g i upon the horizontal meri¬ 
 dian line a n, and there will be formed a triangle a g i. A 
 plate or triangular frame similar to this triangle, and of the 
 thickness of the interval of the parallel lines a c and b d , 
 being now made and set upright between them, touching at a 
 and b, its hypothenuse or axis a g will be parallel to the axis 
 of the earth when the dial is fixed truly, and will cast its 
 shadow on the hour of the day. 
 
CONSTRUCTION OF LOGARITHMIC, OR GUNTERS LINES. 25 
 
 To make an erect 
 South Dial. —Take 
 the complement of • 
 the latitude of the 
 place, which for Lon¬ 
 don is 90° less 51^ 
 
 = 38|, from the 
 scale of latitudes, 
 and proceed in all 
 other respects for 
 the hour lines, as 
 above, for the hori¬ 
 zontal dial; only re¬ 
 versing the hours, 
 and limiting them to the 7 ; and 
 for the gnomon making the angle 
 of the style’s height equal to the 
 colatitude 38£. 
 
 To construct an East or West 
 Dial. —Draw the two meridian 
 lines as before, and intersect it at 
 right angles by another line, upon which set off, from the meri¬ 
 dian lines, the tangents of 15°, 30°, 45°, &c., for every 15°, re¬ 
 duced to a radius equal to the intended height of the style. 
 The hour lines are to be drawn through the divisions thus 
 marked, parallel to the meridian lines, and the meridian lines 
 themselves are six o’clock hour lines. The gnomon is a plate 
 in the form of a parallelogram, the breadth of which forms the 
 height of the style or gnomon, and must be equal to the radius 
 to which the tangents have been set off on the dial plate. It 
 is set up between the meridian lines, perpendicular to the dial 
 plate; and the dial is set up, so that the meridian lines, and 
 consequently the edge of the gnomon, may he parallel to the 
 earth’s axis. As the sun onlv shines on the dial during half the 
 day, if the dial fronts the east, it points out the time from sun- 
 rise’to noon, or, if the dial fronts the west, from noon to night. 
 
 GUNTERS LINES. 
 
 These lines are graduated so as to form a scale of the loga¬ 
 rithms of numbers, sines, and tangents; to which are some¬ 
 times added, for the use of the navigator, lines of the loga¬ 
 rithms of the sine rhumbs and tangent rhumbs. They may 
 be constructed as follows :— 
 
 1. To construct the Line of Logarithmic Numbers marked N 
 
 c 
 
26 
 
 MATHEMATICAL INSTRUMENTS. 
 
 — Having fixed upon a convenient length for the entire scale, 
 which must be exactly equal to the length of twenty of the 
 primary divisions of the diagonal or vernier scale, or ol the 
 beam compasses (p. 48), by which it is to be divided, bisect it, 
 and figure it 1 at the commencement on the left hand, 1 again 
 in the middle, and 10 at the end. The half line, then, is taken 
 for unity, or the logarithm of 10, and, consequently, the 
 whole line represents 2, or the logarithm of 100. The 
 lengths corresponding to the three first figures of the loga¬ 
 rithms of 2, 3, &c., up to 9, as found in the common table of 
 logs., may now be taken off from the diagonal scale, or the 
 length corresponding to four or even five figures may be esti¬ 
 mated upon a vernier scale, or upon the beam compasses, if 
 the scale be not less than twenty inches in length. These 
 lengths are to be set off from the 1 at the commencement of 
 the line for the logarithms of 2, 3, &c., to 9, and again from 
 the 1 at the middle of the line for the logarithms of 20, 30, 
 &c., to 90. The divisions thus formed are to be subdivided 
 by setting off, in the same manner, the three, four, or five 
 first figures of the logarithms of IT, T2, T3, &c., to T9; of 
 2T, 2‘2, 23, &c., to 2'9, and so on, each of the primary divi¬ 
 sions being thus subdivided into ten; and these again are to 
 be subdivided each into ten, or five, or two, as the length of 
 the secondary divisions may admit, by setting off the loga¬ 
 rithms of 1 11, 1T2, 1T3, &c.; or of T12, 1T4, &c.; or of 
 1T5, 1-25, &c.; and the scale is completed. 
 
 .'T'p 
 
 s 
 
 o M 
 
 9. To construct the Line of Logarithmic Sines marked S .— 
 The whole length of the scale is taken as the logarithm of 
 the radius, and, since this extent upon the line of numbers 
 represents 2, or the logarithm of 100, it follows that the lines 
 of sines, tangents, &c., are to be scales of the logarithms of 
 the sines, tangents, &c., to radius 100, of which the logarithm 
 is 2: whereas the logarithmic tables of sines, tangents, &c., 
 are set down to a radius, of which the logarithm is 10. By 
 taking 8, then, from each of the tabulated values of the loga¬ 
 rithmic sines, tangents, &c., we should obtain the logarithmic 
 sines, tangents, &c., to radius 100, and the three, four, or five 
 first figures of these reduced values are to be set off, from the 
 left hand towards the right, by one of the scales, or by the 
 
 
 f [5 is [4 rs \c 17 13 \'M\o 2 l o TkT 4 
 
 
 i ' i i i r i i 1 i 1 i 
 
 
 / 12 Itf If 15 \6 1 7 i R \9f\0 2 0 3 
 
 O 4 
 
 <> soe\om 
 
 
 i i i i i i i i i 
 
 
 111 
 
 r 1 la 1 13 1+ is lb \7 \K i 9 
 
 > 1 12 l 
 
 
 3 ,6 
 
 i7i i6- nr) 
 
 :in Miiuiiiiniii il l Liiijiiiliiiiii in NT 
 
 rrn urnirnnrnr 
 
 mi ITi it 
 
 MIL 
 
 1 1 1 XL 
 
CONSTRUCTION OF LOGARITHMIC, OR GUNTERS LINES. 27 
 
 beam compasses, as explained in the construction of the line 
 of numbers; 1st, for every 10 degrees, then for every degree, 
 and then for every half degree, every 10 minutes, and every 
 5 minutes, as far as the length of the several primary divi¬ 
 sions will admit. The line is then numbered 1, 2, 3, &c., at 
 every degree to 10, and afterwards 20, 30, 40, &c., at every 
 ten degrees to 90, which stands at the extreme right, since 
 sine 90° equals radius. 
 
 The tabulated logarithmic sine of 34' 23 7 ', being 8 , 0000669, 
 will coincide, or nearly so, with the zero point upon our scale, 
 and consequently angles smaller than this cannot be taken off 
 from the sines. This remark applies equally to the line of 
 tangents, the tabulated logarithmic tangent of 34' 23 7/ being 
 8 0000886. 
 
 By taking the extents backwards from right to left, and 
 reckoning them as forward distances, the line of sines be¬ 
 comes a line of cosecants *, giving us, in fact, the excesses of 
 the logarithmic cosecants above the logarithmic radius; and, 
 by taking the complements of the required angles, the line 
 of sines becomes a line of cosines when measured forwards 
 from left to right, and a line of secants when measured back¬ 
 wards from right to left. 
 
 3. To construct the Line of Logarithmic Tangents marked T. 
 —8 being taken from each of the tabulated values of the 
 logarithmic tangents up to 45°, the extents corresponding to 
 these values are to be set off upon the scale, and numbered 
 from left to right, in a similar manner to that in which the 
 logarithmic sines were set off and numbered upon the line of 
 logarithmic sines. The logarithmic tangent of 45° extends to 
 the extreme right of the scale, coinciding in extent with the 
 sine of 90, since-tangent 45° equals radius, and the logarithmic 
 tangents of the angles from 45° to 90 are measured back¬ 
 wards from the extreme right to the complement of the angle 
 required, these extents giving us, in fact, the excesses of the 
 logarithmic tangents sought above the logarithmic radius f. 
 
 r l r i 
 
 * Cosecant = -—, and sec. = — 
 
 sin. cos. 
 
 and, therefore, log. cosecant = 2 log. rad. — log. sine; 
 or, log. cosecant — log. rad. = log. rad. — log. sine; 
 and log. secant = 2 log. rad. — log. cos.; 
 or, log. secant — log. rad. = log. rad. — log. cos. 
 r 2 r 2 
 
 f Tan. = —-— = -- f --S 
 
 1 cotan. tan. ol compt. 
 
 log. tan. = 2 log. rad. — log. tan. of compt.; 
 
 or, log. tan. — log. rad. = log. rad. — log. tan. of compt. 
 
28 
 
 MATHEMATICAL INSTRUMENTS. 
 
 When, then, the angle is greater than 45, the distance from 
 radius to the angle, though measured backwards upon the 
 scale, must be reckoned a forward distance, and vice versa. 
 
 The lines of logarithmic sine rhumbs, marked S.R., and 
 tangent rhumbs, marked T.R., are formed in the same way as 
 the lines of logarithmic sines and tangents, but are set off 
 for the angles corresponding to the points and quarter points 
 of the compass, instead of for degrees and minutes. 
 
 We shall now proceed to explain the uses of Gunter’s lines. 
 
 1. The Line of Logarithmic Numbers .—The primary divi¬ 
 sions upon this line, as explained in its construction, repre¬ 
 sent the logarithms of all the integers from 1 to 100, while 
 the extents to the first subdivisions will indicate tenths of an 
 unit from the beginning of the scale to 1 in the middle, and 
 units from 1 in the middle to 10 at the end, where the figures 
 2, 3, &c., stand for 20, 30, &c., as has been explained in the 
 construction. If any of the subdivisions be further subdivided 
 into ten parts, each of these last divisions will indicate hun¬ 
 dredths of an unit from 1 at the beginning to 1 in the middle, 
 and tenths of an unit from 1 in the middle to 10 at the end. 
 Upon pocket sectors (p. 34}, however, upon which Gunter’s 
 lines are now usually placed, affording a greater extent for the 
 purpose than the six-inch plain scale (p. 33), only the part 
 from 1 in the middle to 2 towards the right is a second time 
 divided, and that but into five parts instead of ten, every one 
 of which must be accounted as two-tenths. By this line the 
 multiplication and division of numbers of any denomination 
 either whole or fractional may be readily accomplished, ques¬ 
 tions in proportion solved, and all operations approximatively 
 performed with great rapidity, which can be performed by the 
 common table of logarithms; but the numbers sought must 
 always be supposed to be divided or multiplied by 10 as many 
 times as will reduce them to the numbers, the logarithms of 
 which are actually set off upon the line of numbers, and these 
 tens must be mentally accounted for in the result. 
 
 Multiplication is performed by extending from 1 on the 
 left to the multiplier; and this extent will reach forwards 
 from the multiplicand to the product. Thus, if 125 were 
 given to be multiplied by 250, extend the compasses from 1 
 at the left hand to midway between the second and third sub¬ 
 division, in the first primary division from 1 to 2, for the 
 125. This extent is really the logarithm of T25. Set off 
 this extent towards the right from the fifth subdivision after 
 the primary division marked 2, which is taken to represent 
 
USES OF GUNTERS LINES. 
 
 29 
 
 the log. of 250, but is really the log. of 2-5, and the compasses 
 will reach to a quarter of the next subdivision beyond the first 
 subdivision after the primary division marked 3. The extent 
 to this point is really the logarithm of 3T25; but in this case 
 it represents the number 31250, because two powers of ten have 
 been cast out from both the multiplier and multiplicand, and 
 therefore the product must be multiplied by the product of four 
 tens, or ten thousand; or, in other words, the first figure of the 
 product must be reckoned as so many tens of thousands. 
 
 Division , being the reverse of multiplication, is performed 
 by extending from 1 on the left to the divisor; and this ex¬ 
 tent will reach backwards from the dividend to the quotient. 
 Thus, if 31250 were to be divided by 250, extend the com¬ 
 passes from 1 on the left to 2’5, and this extent will reach 
 backwards from 3T25 to 125. Then, since the divisor con¬ 
 tained 2 powers of ten and the dividend 4, the quotient must 
 contain 2, and therefore the result is 125. 
 
 Proportion being performed by multiplication and division, 
 extend the compasses from the first term to the second, and 
 this extent will reach from the third to the fourth, taking 
 care to measure in the same direction, so that, if the first be 
 greater than the second, the third may be greater than the 
 fourth, and vice versa. Example .—If the diameter of a circle 
 be 7 inches, and the circumference 22, what is the circum¬ 
 ference of another circle, the diameter of which is 10? Ex 
 tend the compasses from 7 to 10, and this extent will reach 
 from 22 to 31 4, or nearly 311 inches, the circumference 
 required. 
 
 The same thing may also be performed by extending from 
 the first term to the third, and this extent will reach from 
 the third term to the fourth (Euc. v. prop. 16). Thus, the 
 extent from 7 to 22 will reach from 10 to 3T4, as before. 
 
 To measure a Superficies , extend from 1 to either the 
 breadth or length, both being reduced to the same denomina¬ 
 tion, and this extent will reach forwards from the length or 
 breadth to the superficial content. Example. —Required the 
 superficial content of a plank 27 feet long by 15 inches broad. 
 Extend from 1 to T25, for 15 inches equals 1 25 feet, and 
 this extent will reach from 27 feet to 3375 feet, the super¬ 
 ficial content required. 
 
 Second Method. —Extend from 12 to the number of inches 
 in the breadth, and this extent will reach in the same direc¬ 
 tion from the number of feet in the length to the number of 
 square feet in the superficial content. Thus the extent for- 
 
MATHEMATICAL INSTRUMENTS 
 
 30 
 
 wards from 12 to 15 will reach forwards from 27 to 33-75, as 
 before; while the extent backwards from 12 to 9 will reach 
 backwards from 27 to 20-25 or 20£, showing the superficial 
 content of a plank 27 feet long by 9 inches broad to be 20-25 
 or 20£ feet. 
 
 To measure a Solid Content. —The breadth, depth, and length 
 being all reduced to the same denomination, extend from 1 to 
 either the breadth or depth, and this extent will reach from 
 the depth or breadth forwards to a fourth number, which will 
 represent the superficial content of the section at the place 
 measured: then, if the breadth and depth be the same 
 throughout the entire length, the extent from l to the super¬ 
 ficial content thus found will reach forwards from the length 
 to the solid content. Example. —What is the solid content of 
 a pillar 1 foot 3 inches square, and 21 feet 9 inches long? 
 The extent from 1 to T25 reaches forward from T25 to T56, 
 the superficial content of a section of the pillar; and the ex¬ 
 tent from 1 to 1-50 reaches from 21 "75 to 34, or more accu¬ 
 rately to 33-93, the solid content in feet*. 
 
 2. The Lines of Logarithmic Sines and Tangents. —These 
 lines are generally used, in connection with the line of num¬ 
 bers, for solving all proportions in which any of the terras are 
 functions of angles, as sines, tangents, &c., and, in fact, all 
 questions in which such quantities appear as factors or divi¬ 
 sors. We will exemplify their use by giving the solution, by 
 their aid, of the several cases of right-angled trigonometry. 
 
 Case If. The hypothenuse and angles being given, to find 
 the perpendicular and base. 
 
 * Our limits forbid us from entering further upon the uses of the line of 
 logarithmic numbers; but the student will, we hope, from what he sees 
 here, be easily enabled to apply it to every case of mensuration, and, in 
 short, to almost every arithmetical operation. Additions and subtractions, 
 however, cannot be performed by it. 
 
 4 These cases are, in fact, the solutions, by the aid of Gunter’s lines, of 
 the following proportions, which will 
 be obvious to the student upon inspec¬ 
 tion of the accompanying figure. 
 
 llad. : sin. A 
 Had. : sin. B 
 Sin. B : rad. 
 Sin. b : sin. a : 
 
 ac : BO 
 
 Had. 
 
 : tan. 
 
 a:: ac : bi 
 
 3 
 
 ab : 
 
 ac : 
 
 : rad. 
 
 : sin. 
 
 B 
 
 Had. 
 
 : sin. 
 
 a : : 
 
 ab : 
 
 BO 
 
 ao : 
 
 bo : 
 
 : rad. 
 
 : tan. 
 
 A 
 
 bo : 
 
 ao : 
 
 : rad. 
 
 : tan. 
 
 B 
 
 Sin. 
 
 a : rad. : : 
 
 bo : 
 
 AB 
 
 Case 1. 
 
 Case 2. 
 
 Case 3. 
 
 Case 4. 
 
USES OF GUNTERS LINES. 31 
 
 Note. —One acute angle of a right-angled 
 triangle being the complement of the other, 
 or the sum of the two acute angles being 
 equal to 90°, when one of the acute angles 
 is given, the other is also given. 
 
 Solution. —Extend the compass 
 from 90°, or radius, on the line of a 
 sines to the number of degrees in 
 either of the acute angles, and that extent -will reach back 
 wards, on the line of numbers, from the hypotlienuse to the 
 side opposite this angle. Example. —Given the hypotlienuse 
 ab = 250, and the angle a = 35° 30'. 
 
 90° O' 
 
 Extend from 90° to 35° 30' on the line of sines, and bac = 35 30 
 this extent will reach from 250 to 145 on the line of abc = 54 30 
 
 numbers . ..*. bo = 145 
 
 Extend from 90° to 54° 30' on the line of sines, and this 
 
 extent will reach from 250 to 203*5 on the line of numbers \ a c = 203*5 
 
 Case 2. The angles, and one. side being given, to find the 
 hypotlienuse, and the other side. 
 
 Solution. —Extend from the angle opposite the given side 
 to 90°, or radius, on the line of sines, and this extent will reach 
 forwards from the given side to the hypotlienuse on the line 
 of numbers. Again, extend from the angle opposite the given 
 side to the angle opposite the required side, and this extent 
 will reach in the same direction on the line of numbers, from 
 the given side to the required side. Or, extend from radius, 
 or 45°, on the line of tangents, to the angle opposite the re¬ 
 quired side, and the extent will reach, in the same direction on 
 the line of numbers, from the given side to the required side; 
 recollecting that, when the angle is greater than 45°, the ex¬ 
 tent is to be taken on the scale backwards from rad. or 45° to the 
 complement of the angle, but is to be reckoned a forward dis¬ 
 tance, the logarithmic tangents of angles greater than 45° ex¬ 
 ceeding the logarithmic tangents of 45°, or radius, by as much as 
 the logarithmic tangents of their complements fall short of it. 
 Example. —Given the angle a = 35° 30' and side ac = 203 5. 
 
 90° O' 
 
 Extend from 54° 30' to 90°, or rad., upon the line of sines, bac = 35 30 
 and this extent will rcacli forwards from 203*5 to 250 ABC = 54 30 
 
 on the line of numbers.*. ab = 250 
 
 Again, extend from 54° 30' backwards to 35° 30', on the 
 line of sines, and this extent will reach backwards from 
 203*5 to 145 on the line of numbers. .*. bc = 145 
 
 Or extend backwards from 45°, rad., to 35° 30' on the line of tangents. 
 
32 
 
 MATHEMATICAL INSTRUMENTS. 
 
 and this extent will reach backwards from 203'5 to 145 on the line of num¬ 
 bers, as before *. 
 
 Case 3. The hypotlienuse and one side being given, to find 
 the angles and the other side. 
 
 Solution. — Extend from the hypotlienuse to the given side 
 on the line of numbers, and this extent will reach from 90 or 
 rad. to the angle opposite the given side upon the line of 
 sines. The other angle is the complement of this. Extend 
 upon the line of sines from the rad. to the angle last found, 
 which is opposite the required side, and this extent will reach 
 from the hypotlienuse to the required side. Example .—Given 
 the hypotlienuse ab = 250, and the side ac=203’5. 
 
 Extend backwards from 250 to 203*5 on the line of num¬ 
 bers, and this extent will reach from 90° to 54° 30' on 90° O' 
 
 the line of sines.\ abc = 54 30 
 
 Extend from 90 to 35° 30' on the line of sines, and this BAC = 35 30 
 extent will reach backwards from 250 to 145 on the 
 line of numbers.\ bc = 145 
 
 Case 4. The two sides being given, to find the angles and 
 the hypotlienuse. 
 
 Solution. —Extend from one side to the other upon the 
 line of numbers, and this extent will reach backwards upon 
 the line of tangents from rad. to the least angle, and to the 
 same point, considered as a forward distance, representing the 
 greatest angle, which is the complement of the least. Again, 
 extend on the line of sines from one of the angles just found 
 to rad., and this extent will reach from the side opposite the 
 angle taken to the hypotlienuse. Example .—Given ac=203*5 
 and bc = 145. 
 
 Extend backwards upon the line of numbers from 203'5 
 to 145, and this extent will reach backwards from 45° 
 to 35° 30' on the line of tangents, which is the angle 90° O' 
 
 opposite the side 145.%.\ bac = 35 30 
 
 If we measure forwards from 145 to 203*5, then from rad. 
 to 35° 30' is to be considered a forward distance, and the 
 angle to be taken as the complement of 35° 30', that 
 is, 54° 30', which is the angle opposite the side 
 
 203 5.% abo = 54 30 
 
 Again, extend from 33° 38' to 90° on the line of sines, 
 and this extent will reach from 145 to 250 upon the 
 line of numbers.*. A B = 250 
 
 * The property that tan. : rad. : : sine : cosine, may be made a test of 
 the accuracy of the scale, since the distance from 45 to any angle upon the 
 line of tangents ought to be the same as the distance from the angle to its 
 complement upon the line of sines. * 
 
THE SECTOR 
 
 33 
 
 THE PLAIN SCALE. 
 
 One of these instruments is represented in the annexed 
 figure, being such a one as is usually supplied with a pocket 
 case of instruments. It is made of ivory, six inches long, and 
 one inch and three quarters broad. On the face of the in¬ 
 strument represented in the engraving, a protractor is formed 
 round three of its edges for readily setting off angles. In 
 
 using this protractor, the fourth edge, which is quite plain, 
 with the exception of a single stroke in the middle, is to be 
 made to coincide with the line from which the angle is to be 
 set off, and the stroke in the middle with the point in this 
 line, at which the angle is to be set off; a mark is then to be 
 made with the pricking point, at the point of the paper which 
 coincides with the stroke on the protractor, marked with the 
 number of degrees in the angle required to be drawn; and, 
 the protractor being now removed, a straight line is to be 
 drawn through the given point in the given line and the 
 point thus pricked off. The instrument has on the same face 
 the two diagonal scales already described (p. 10), and on the 
 opposite face scales of equal parts, and several of the protract¬ 
 ing scales already described (pp. 14-16), according to the pur¬ 
 poses to which the scale is to be applied: thus, for laying 
 down an ordinary survey, we merely require scales of equal 
 parts, and a line of chords, and these consequently are all the 
 lines placed on many of the instruments in the pocket cases; 
 but for projecting maps, lines of sines, tangents and semitan¬ 
 gents are required ; for dialling, the dialling lines ; and for the 
 purposes of the navigator, the lines of rhumbs, and longitudes, 
 the whole of Gunter’s lines already described, and two lines of 
 meridional, and equal parts to be used together in laying down 
 distances, &c., upon Mercator’s charts. The plain scale is 
 sometimes fitted with rollers, as represented in our engraving, 
 making it at the same time a convenient small parallel rule. 
 
 THE SECTOR. 
 
 This valuable instrument may well be called an universal' 
 
 c 3 
 
31 
 
 MATHEMATICAL INSTRUMENTS. 
 
 scale. By its aid all questions in proportion may be solved ; 
 lines may be divided either equally or unequally into any 
 number of parts that may be desired; the angular functions, 
 viz., chords, sines, tangents, &c., may be setoff or measured to 
 any radius whatever; plans and drawings may be reduced or 
 enlarged in any required proportion; and, in short, every ope¬ 
 ration in geometrical drawing may be performed by the aid of 
 this instrument and the compasses only. 
 
 The name sector is derived from the tenth definition of the 
 third Book of Euclid, in which this name is given to the 
 figure contained by two radii of a circle, and the circumference 
 between them. The instrument consists of two equal rulers, 
 called legs, which represent the two radii, moveable about 
 the center of a joint, which center represents the center of the 
 circle. The legs can consequently be opened so as to contain 
 any angle whatever, or completely opened out until their edges 
 come into the same straight line. 
 
 Sectors are made of different sizes, and their length is 
 usually denominated from that of the legs when shut together. 
 Thus, a sector of six inches, such as is supplied in the com¬ 
 mon pocket cases of instruments, forms a rule 
 of twelve inches, when opened; and this cir¬ 
 cumstance is taken advantage of, by filling up 
 the spaces not occupied by the sectoral lines 
 with such lines as it is most important to lay 
 down upon a greater length, than the six-inch 
 plain scale will admit. Among these the most 
 usual are (1) the lines of logarithmic numbers, 
 sines, and tangents already described (pp. 25- 
 28); (2) a scale of 12 inches, in which each 
 inch is divided into ten equal parts; and (3) a 
 foot divided into ten equal primary divisions, 
 each of which is subdivided into ten equal 
 parts, so that the whole is divided into 100 
 equal parts. The last-mentioned is called the 
 decimal scale, and is placed on the edge of the 
 instrument. 
 
 The sectoral lines proceed in pairs from the 
 center, one line of each pair on either leg, and 
 are, upon one face of the instrument, a pair of 
 scales of equal parts, called the line of lines , 
 and marked l ; a pair of lines of chords, 
 marked c; a pair of lines of secants, marked 
 s; a pair of lines of polygons, marked pol. 
 
DESCRIPTION OF THE SECTORAL LINES. 
 
 - 35 
 
 Upon the other face, the sectoral lines are—a pair of lines of 
 sines, marked s; a pair of lines of tangents up to 45°, marked 
 t ; and a second line of tangents to a lesser radius, extending 
 from 45° to 75°. 
 
 Each pair of sectoral lines, except the line of polygons, 
 should be so adjusted as to make equal angles at the center, 
 so that the distances from the center to the corresponding 
 divisions of any pair of lines, and the transverse distance 
 between these divisions, may always form similar triangles. 
 On many instruments, however, the pairs of lines of secants, 
 and of tangents from 45° to 75°, make angles at the center 
 equal to one another, but unequal to the angle made by all 
 the other pairs of lines. 
 
 The solution of questions on the sector is said to be simple , 
 when the work is begun and ended upon the same pair of lines; 
 compound , when the operation is begun upon one pair of lines 
 and finished upon another. 
 
 • In a compound solution the two pairs of lines used must 
 make equal angles at the center, and, consequently, in the 
 exceptional case mentioned above, the lines of secants and of 
 tangents above 45° cannot be used in connection with the 
 other sectoral lines*. 
 
 When a measure is taken on any of the sectoral lines be¬ 
 ginning at the center, it is called a lateral distance; but, 
 when a measure is taken from any point on one line to its 
 corresponding point on the line of the same denomination on 
 the other leg, it is called a transverse or parallel distance. 
 
 The divisions of each sectoral line are 
 contained within three parallel lines, 
 the innermost being the line on which 
 the points of the compasses are to be 
 placed, because this is the only line of 
 the three which goes to the center, and 
 is therefore the sectoral line. 
 
 On the Principle of the Use of the 
 Sectoral Lines. —1st. In the case of a 
 Simple Solution. — Let the lines a b, 
 a c represent a pair of sectoral lines, 
 
 * Since, however, secant : rad. :: rad. : cosine; 
 
 and tangent : rad. :: rad. : cotangent; 
 
 the line of sines may be used with the other sectoral lines in place of the line 
 of secants, and the line of tangents less than 45° in place of the line of tan¬ 
 gents greater than 45°, the complements of the angles being taken upon these 
 lines, in either case, instead of tiie angles themselves. See page 41. 
 
36 ’ 
 
 MATHEMATICAL INSTRUMENTS. 
 
 and b c, d e, any two transverse distances taken on this pair 
 of lines; then, from the construction of the instrument, we 
 have a b equal to a c, and a d equal to a e, so that a b : a c : : 
 a d : a e, and the triangles abc and ade have the angle at 
 a common, and the sides about this common angle propor¬ 
 tional (Euc. vi. prop. 6); they are, therefore, similar and— 
 
 ab : b c : : ad : de. 
 
 In the case of a compound solution, the angles at a are equal, 
 hut not common, and the reasoning is, in all other respects, 
 exactly the same. 
 
 USES OF THE LINE OF LINES. 
 
 To find ' a Fourth Proportional to three given Lines. —Set 
 off from the center a lateral distance equal to the first term, 
 and open the sector till the transverse distance at the division 
 thus found, expressing the first term, is equal to the second 
 term; again, extend to a point whose lateral distance from the 
 center is equal to the third term, and the transverse distance 
 at this point will he the fourth term required. 
 
 If the legs of the sector will not open far enough to make 
 the lateral distance of the second term a transverse distance 
 at the division expressing the first term, take any aliquot part 
 of the second term, which can conveniently be made such 
 transverse distance, and the transverse distance at the third 
 term will be the same aliquot part of the fourth proportional 
 required. 
 
 A third proportional to two given lines is found by taking 
 a third line equal to the second, and finding the fourth pro¬ 
 portional to the three lines. 
 
 Example. —To find a fourth proportional to the numbers 2, 
 5, and 10. Open the sector till the lateral distance of the 
 second term 5 becomes the transverse distance at 2, the first 
 term; then the transverse distance at 10 will extend, as a 
 lateral distance, from the center to 25, the fourth proportional 
 required. 
 
 To bisect a given Straight Line. —Take the extent of the 
 line in the compasses, and open the sector till this extent is a 
 transverse between 10 and 10 on the line of lines: then the 
 transverse distance from 5 to 5, on the same pair of sectoral 
 lines, gives the half of the line, and this extent set off from 
 either end will bisect it. 
 
 To divide a Straight Line into any Number of equal Parts .— 
 1. When the number of parts are a power of 2, the operations 
 
USES OF THE SECTOKAL LINES 
 
 87 
 
 ore best performed by continual bisection. Thus, let it be 
 required to divide the line a b into sixteen equal parts. 
 1. Make a b a transverse distance between 10 and 10 on the 
 
 
 J J. 
 
 10 1L 72 Vi TV 
 
 
 a 
 
 line of lines ; then take off the transverse distance of 5 and 5, 
 and set it off from a or* b to 8, and a b will be divided 
 into two equal parts at the division 8. 2. Make a 8 a 
 
 transverse distance at 10, and then the transverse distance at 
 5, set off from a or* 8 at 4, and from b or* 8 at 12, will 
 divide the line into four equal parts at the divisions 4, 8, and 
 12. 3. Make the extent a 4 a transverse distance at 10, and 
 
 the transverse distance at 5 will again bisect each of the parts 
 last set off, and divide the whole line into eight equal parts 
 at the divisions 2, 4, 6, 8, 10, 12, and 14. Each of these 
 may be again bisected by taking the transverse distance at 2^ 
 or 2 5, that is, at the middle division between the 2 and the 
 3 upon the line of lines, and the line will be divided as 
 required. 
 
 When the divisions become smaller than can be conve¬ 
 niently bisected by the method just explained, the operation 
 may still be continued to any required extent by taking the 
 extent of an odd number of the divisions already obtained as 
 the transverse distance of 10 and 10, and setting off the half 
 of this extent, or the transverse distance at 5, from the several 
 divisions already obtained. Thus, in the preceding example, 
 by making the extent of three of the divisions, or five, or seven, 
 a transverse distance at 10, the transverse distance at 5, set 
 off from the several divisions already obtained, will divide a b 
 into 32 equal parts. 
 
 2. When the number of parts is not a power of 2, the opera¬ 
 tions cannot all be performed by bisections; but still, by a 
 judicious selection of the parts into which the line is first 
 divided, many of the after operations may be performed by 
 bisections. Example .—Let it be required to divide the line 
 
 k i j i -i-1 
 
 a b into seven equal parts. 1. Make the whole extent, a b, a 
 transverse distance between 7 and 7 on the line of lines; then 
 
 * Greater accuracy is obtained by setting off the distance from both 
 ends of the extent to be bisected, and then, in case the two points so found 
 do not accurately coincide, taking the middle point between them, as near 
 as the eye can judge, for the true point of bisection. 
 
38 
 
 MATHEMATICAL INSTRUMENTS.. 
 
 take off the transverse distance of 4 and 4, and set it off from a 
 and b to 4 and 3. 2. Make this extent from a to 4 a trans¬ 
 
 verse distance at 10 ; then the transverse distance at 5 bisects 
 a 4 and 3 b in 2 and 5 ; set off from 3, gives 1, and from 4 
 gives 6; and thus the line a b is divided into seven equal 
 parts as required. 
 
 To open the Sector so that the Line of Lines may answer for 
 any required Scale of equal Parts. —Take one inch in the com¬ 
 passes, and open the sector, till this extent becomes a trans¬ 
 verse distance at the division indicating the number of parts 
 in an inch of the required scale; or, if there be not an inte¬ 
 gral number of parts in one inch, it will be better to take 
 such a number of inches as will contain an integral number 
 of parts, and make the extent of this number of inches, if it 
 be not too great, a transverse distance at the division indi¬ 
 cating the number of parts of the required scale in this extent. 
 
 Example.—To adjust the Sector as a Scale of One Inch to 
 Four Chains. —Make one inch the transverse distance of 4 
 and 4 ; then the transverse distances of the other correspond¬ 
 ing divisions and subdivisions will represent the number of 
 chains and links indicated by these divisions: thus, the trans¬ 
 verse distance from 3 to 3 will represent three chains; the 
 transverse distance at 4 - 7, or the seventh principal subdivi¬ 
 sion after the primary division marked 4, will represent 4 
 chains 70 links, and so on. 
 
 To construct a Scale of Feet and Inches in such a manner 
 that an extent of Three Inches shall represent Twenty Inches .— 
 
 1. Make three inches a transverse distance between 10 and 
 10, and the transverse distance of 8 and 8 will represent 16 
 inches. 2. Set off this extent from a to b, divide it by con- 
 tinual bisection into 16 equal parts, and place permanent 
 strokes to mark the first 12 of these divisions, which will 
 represent inches. 3. Place the figure 1 at the twelfth stroke, 
 and set off again the extent of the whole 12 parts, from 1 to 
 
 2, 2 to 3, &c., to represent the feet. 
 
 As an Example of the Use of the Line of Lines in reducing 
 Lines, let it be required to reduce a drawing in the Proportion 
 of 5 to 8.—Take in the compasses the distance between two 
 points of the drawing, and make it a transverse distance at 
 8 and 8; then the transverse distance of 5 and 5 will be the 
 distance between the two corresponding points of the copy. 
 2. These two points having been laid down, make the dis¬ 
 tance between one of them and a third point a transverse dis¬ 
 tance at 8, and with the transverse distance at 5 describe, from 
 
USES OF THE SECTORAL LINES. 
 
 39 
 
 that point as center, a small arc. 3. Repeat the operation 
 with the other point, and the intersection of the two 
 small arcs will give the required position of the third 
 point in the copy. In the same manner all the other 
 points of the reduced copy may be set off, each one from 
 two points previously laid down. 
 
 LINE OF CHORDS. 
 
 SI 
 
 The double scales of chords upon the sector are more 
 generally useful than the single line of chords described 
 on the plain scale; for, on the sector, the radius with 
 which the arc is to be described may be of any length 
 between the transverse distance of 00 and 00 when the 
 legs are close, and that of the. transverse of 60 and 60 
 when the legs are opened as far as the instrument will 
 admit of: but, with the chords on the plain scale, the 
 arc described must be always of the same radius. 
 
 To protract or lay down a right-lined Angle bac, 
 which shall contain a given number of Degrees , suppose 
 46°.— Case 1 . When the angle contains less than 60°, 
 make the transverse distance of 60 and 60 equal to the 
 length of the radius of the circle, and with that opening 
 describe the arc b c. (Fig. at page 40.) Take the 
 transverse distance of the given degrees 40, and lay 
 this distance on the arc from the point b to c. From 
 the center a of the arc draw two lines a c, a b, each 
 passing through one extremity of the distance b c laid 
 on the arc; and these two lines will contain the angle 
 required. Case 2. When the angle contains more than 
 60°. Suppose, for example, we wish to form an angle 
 containing 148°. Describe the arc b c d, and make 
 the transverse distance of 60 and 60 equal the radius 
 as before. Take the transverse distance of ^ or &c., 
 of the given number of degrees, and lay this distance 
 on the arc twice or thrice, as from b to a, a to b, and 
 from b to d. Draw two lines connecting b to a, and a 
 to d, and they will form the angle required. 
 
 When the required angle contains less than 5°, sup¬ 
 pose 8J, it will be better to proceed thus. With the given 
 radius, and from the center a, describe the arc d g ; and from 
 some point, d, lay off the chord of 60°, which suppose to give 
 the point g, and also from the same point d lay off in the same 
 direction the chord of 56^° (= 60° — 3^ c ), which would give 
 the point e. Then through these two points e and g, draw 
 
40 
 
 MATHEMATICAL INSTRUMENTS. 
 
 lines to the point a, and they 
 will represent the angle of 3^° 
 as required. 
 
 From what has been said 
 about the protracting of an 
 angle to contain a given number 
 of degrees, it will easily be seen 
 how to find the degrees (or 
 measure) of an angle already 
 laid down. 
 
 LINE OF POLYGONS. 
 
 The line of polygons is chiefly useful for the ready division 
 of the circumference of a circle into any number of equal 
 parts from 4 to 13; that is, as a ready means to inscribe 
 regular polygons of any given number of sides, from 4 to 13, 
 within a given circle. To do which, set off the radius of the 
 given circle (which is always equal to the side of an inscribed 
 hexagon) as the transverse distance of 6 and C, upon the line 
 of polygons. Then the transverse distance of 4 and 4 will be 
 the side of a square; the transverse distance between 5 and 
 5, the side of a pentagon; between 7 and 7, the side of a 
 heptagon ; between 8 and 8, the side of an octagon; between 
 9 and 9, the side of a nonagon, dc., all of which is too plain 
 to require an example. 
 
 If it be required to form a polygon, upon a given right 
 line set off the extent of the given line, as a transverse dis¬ 
 tance between the points upon the line of polygons, answering 
 to the number of sides of which the polygon is to consist; as 
 for a pentagon between 5 and 5 ; or for an octagon between 
 8 and 8; then the transverse distance between 6 and 6 will 
 be the radius of a circle whose circumference would be divided 
 by the given line into the number of sides required. 
 
 The line of polygons may likewise be used in describing, 
 upon a given line, an isosceles triangle, whose angles at the 
 base are each double that at the vertex. For, taking the 
 given line between the compasses, open the sector till that 
 extent becomes the transverse distance of 10 and 10, then 
 the transverse distance of 6 and 6 will be the length of each 
 of the two equal sides of the isosceles triangle. 
 
 All regular polygons, whose number of sides wall exactly 
 divide 300 (the number of degrees into which all circles are 
 supposed to be divided) without a remainder, may likewise be 
 set off upon the circumference of a circle by the line of chords. 
 
USES OF THE SECTORAL LINES. 
 
 41 
 
 Thus, take tbe radius of the circle between the compasses, and 
 open the sector till that extent becomes the transverse distance 
 between 60 and 60 upon the line of chords; then, having di¬ 
 vided 360 by the required number of sides, the transverse 
 distance between the numbers of the quotient will be the side 
 of the polygon required. Thus for an octagon, take the dis¬ 
 tance between 45 and 45; and for a polygon of 36 sides take 
 the distance between 10 and 10, &c. 
 
 LINES OF SINES, TANGENTS, AND SECANTS. 
 
 Given the Radius of a Circle (suppose equal to two inches); 
 required the Sine and Tangent of 28° 30' to that Radius .— 
 Open the sector, so that the transverse distance of 90 and 90 
 on the sines, or of 45 and 45 on the tangents, may be equal 
 to the given radius, viz., two inches; then will the transverse 
 distance of 28° 30', taken from the sines, be the length of 
 that sine to the given radius, or, if taken from the tangents, 
 will be the length of that tangent to the given radius. 
 
 But, if the Secant of 28 J 30' is required, make the given 
 radius of two inches a transverse distance of 0 and 0, at the 
 beginning of the line of secants, and then take the transverse 
 distance of the degrees wanted, viz., 28° 30'. 
 
 A Tangent greater than 45° ( suppose 60°) is thus found :— 
 Make the given radius, suppose two inches, a transverse dis¬ 
 tance to 45 and 45, at the beginning of the line of upper tan¬ 
 gents, and then the required degrees (60) may be taken from 
 the scale. 
 
 The tangent, to a given radius, of any number of degrees 
 greater than 45° can also be taken from the line of lower tan¬ 
 gents, if the radius can be made a transverse distance to the 
 complement of those degrees on this line (see note, page 35). 
 
 Example. —To find the tangent of 78° to a radius of two 
 inches. Make two inches a transverse distance at 12 on the 
 lower tangents, then the transverse distance of 45 will be the 
 tangent of 78°. 
 
 In like manner the secant of any number of degrees may 
 be taken from the sines, if the radius of the circle can 
 be made a transverse distance to the complement of those 
 degrees upon this line. Thus making two inches a transverse 
 distance to the sine of 12°, the transverse distance of 90 and 
 90 will be the secant of 78°. 
 
 To find, by means of the lower tangents and sines, the de¬ 
 grees answering to a given line, greater than the radius which 
 expresses the length of a tangent or secant to a given radius. 
 
42 
 
 MATHEMATICAL INSTRUMENTS. 
 
 For a tangent, make the given line a transverse distance at 
 45 on the lower tangents; then take the extent of the given 
 radius, and apply it to the lower tangents; and the comple¬ 
 ment of the degrees at which it becomes a transverse 
 distance will be the number of degrees required. For a 
 secant make the given line a transverse distance at 90 on 
 the sines; then the extent of the radius will be a trans¬ 
 verse distance at the complement of the number of degrees 
 required. 
 
 Given the Length of the Sine, Tangent, or Secant of any 
 Degrees, to find the Length of the Radius to that Sine, Tangent, 
 or Secant. —Make the given length a transverse distance to its 
 given degrees on its respective scale. Then, 
 
 If a sine 'l trine 190 and 90 on the sines ^ will be 
 
 If a tangent under 45" ( ; 45 an( t 45 on tan S en t s ^the ra- 
 
 If a tangent above 45° i f 45 and 45 on the upper tangents ( dius 
 
 If a secant J 1 fo and 0 on the secants J sought. 
 
 To find the Length of a versed Sine, to a given Number of 
 Degrees, and a given Radius. —1. Make the transverse dis¬ 
 tance of 90 and 90 on the sines equal to the given radius. 
 2. Take the transverse distance of the sine of the com¬ 
 plement of the given number of degrees. 3. If the given 
 number of degrees be less than 90, subtract the distance 
 just taken, viz., the sine of the complement, from the radius, 
 and the remainder will be the versed sine: but, if the 
 given number of degrees are more than 90, add the com¬ 
 plement of the sine to the radius, and the sum will be the 
 versed sine. 
 
 To open the legs of a Sector, so that the corresponding 
 double Scales of Lines, Chords , Sines, and Tangents may make 
 each a right Angle.—On the line of lines make the lateral dis¬ 
 tance 10, a transverse distance between 8 on one leg, and 6 
 on the other leg. 
 
 On the line of sines make the lateral distance 90, a trans¬ 
 verse distance from 45 to 45 ; or from 40 to 50; or from 30 
 to 60; or from the sine of any degree to their complement. 
 
 On the line of sines make the lateral distance of 45 a trans¬ 
 verse distance between 30 and 30. 
 
 marquois's scales. (Plate II. Fig. 3.) 
 
 These scales consist of a right-angled triangle, of which 
 the hypothenuse or longest side is three times the length of 
 the shortest, and two rectangular rules. Our figure, which is 
 drawn one-third the actual «izc of the instruments from which 
 
MARQIJOts’s SCALES. 
 
 43 
 
 it is taken, represents the triangle and one of the rules, as 
 being used to draw a series of parallel lines. Either rule is 
 one foot long, and has, parallel to each of its edges, two scales 
 one placed close to the edge and the other immediately within 
 this, the outer being termed the artificial, and the inner, the 
 natural scale. The divisions upon the outer scale are three 
 times the length of those upon the inner scale, so as to bear 
 the same proportion to each other that the longest side of the 
 triangle bears to the shortest. Each inner, or natural scale, 
 is, in fact, a simply divided scale of equal parts (see p. 9), 
 having the primary divisions numbered from the left hand to 
 the right throughout the whole extent of the rule. The first 
 primary division on the left hand is subdivided into ten equal 
 parts, and the number of these subdivisions in an inch is 
 marked underneath the scale, and gives it its name. On 
 one of the pair of Marquois’s scales now before us, we have, 
 on one face, scales of 30 and 60, on the obverse scales of 
 25 and 50, and on the other we have on one face scales of 35 
 and 45, and on the obverse scales of 20 and 40. In the arti¬ 
 ficial scales the zero point is placed in the middle of the edge 
 of the rule, and the primary divisions are numbered both 
 ways from this point to the two ends of the rule, and are, 
 every one, subdivided into ten equal parts, each of which is, 
 consequently, three times the length of a subdivision of the 
 corresponding natural scale. 
 
 The triangle has a short line drawn perpendicular to the 
 hypothenuse near the middle of it, to serve as an index or 
 pointer; and the longest of the other two sides has a sloped 
 edge. 
 
 To draw a IAne parallel to a given Line , at a given Distance 
 from it. —1. Having applied the given distance to the one of 
 the natural scales which is found to measure it most con¬ 
 veniently, place the triangle with its sloped edge coincident 
 with the given line, or rather at such small distance from it, 
 that the pen or pencil passes directly over it when drawn along 
 this edge. 2. Set the rule closely against the hypothenuse, 
 making the zero point of the corresponding artificial scale 
 coincide with the index upon the triangle. 3. Move the tri¬ 
 angle along the rule, to the left or right, according as the re¬ 
 quired line is to be above or below the given line, until the 
 index coincides with the division or subdivision corresponding 
 to the number of divisions or subdivisions of the natural 
 scale, which measures the given distance; and the line drawn 
 
44 
 
 MATHEMATICAL INSTRUMENTS. 
 
 along the sloped edge in its new position will be the line 
 required*. 
 
 Note. —The natural scale may be used advantageously in setting off the 
 distances in a drawing, and the corresponding artificial scale in drawing 
 parallels at required distances. 
 
 To draw a Line 'perpendicular to a given Line from, a given 
 Point in it. —1. Make the shortest side of the triangle coin¬ 
 cide with the given line, and apply the rule closely against 
 the hypothenuse. 2. Slide the triangle along the rule until 
 a line drawn along the sloped edge passes through the given 
 point; and the line so drawn will be the line required. 
 
 The advantages of Marquois’s scales are : 1st, that the sight 
 is greatly assisted by the divisions on the artificial scale 
 being so much larger than those of the natural scale to which 
 the drawing is constructed ; 2nd, that any error in the setting 
 of the index produces an error of but one-third the amount 
 in the drawing. 
 
 If the triangle be accurately constructed, these scales may 
 be^ advantageously used for dividing lines with accuracy and 
 despatch; our figure, as well as the sliding rule (fig. 4j, was 
 drawn b} r the aid of Marquois’s scales alone. 
 
 We proceed to explain a method which we have found to 
 answer well for dividing lines with accuracy and despatch, 
 and which is altogether independent of any error in the con¬ 
 struction of the triangle. 
 
 Let a b represent the d 
 
 line to be divided into any 
 number, n, of equal parts ; 
 select one of the natural 
 scales, of w’hich n divisions 
 or some multiple p n of 
 n divisions, are nearly a 
 
 equal to a b, but less than it; then setting the sloped edge of 
 the triangle perpendicular to a b, so that a line drawn along 
 it passes through a. place the rule closely against the hypo¬ 
 thenuse, making a division of the artificial scale, corresponding 
 
 * If A B c represent the triangle in its new position, and the dotted lines 
 represent its original position, we have, 
 by similar triangles, abc,a'ad, 
 a d : a a' : : b c : b a 
 : : 1 : 3; 
 
 and therefore a p contains as many 
 divisions of the natural as a a' con¬ 
 tains of the artificial scale. 
 
THE VERNIER. 
 
 45 
 
 to the selected natural scale, coincide with the index upon 
 the triangle ; and, moving the triangle along to the rule to 
 the right, until the index coincides with the p w-th division from 
 that to which it was set, draw a line along the sloped edge 
 intersecting a b in c. Upon a b as hypothenuse describe the 
 right-angled triangle abd, having the side ad equal to ac, 
 and placing the sloped edge of the triangle perpendicular to 
 a d, so that a line drawn along it passes through a or d ; slide 
 the triangle along the rule to the right or left, and drawing a 
 line as the index comes into contact with every pth division of 
 the artificial scale, the line will be divided as required. 
 
 the vernier. (Plate II. Figs. 1 and 2.) 
 
 The property of this ingenious little subsidiary instrument 
 will be readily comprehended from what has been already said 
 of the construction and use of a vernier scale (p. 11). It is 
 so constructed as to slide evenly along the graduated limb of 
 an instrument, and enables us to measure distances, or read 
 off observations, with remarkable nicety. In the vernier scale 
 before described, the divisions on the lower or subsidiary scale 
 'vvere longer than those on the upper or primary scale; but 
 in the vernier now to be described the divisions are usually 
 shorter than those upon the limb to which it is attached, the 
 length of the graduated scale of the vernier being exactly 
 equal to the length of a certain number (n — 1) of the divisions 
 upon the limb, and the number (n) of divisions upon the 
 vernier being one more than the number upon the same 
 length of the limb. 
 
 Let, then, l represent the length of a division upon the 
 limb, and v, the length of a division upon the vernier: 
 
 so that (n — 1) l = n v; 
 
 and therefore l — v = l 
 
 n n 
 
 or the defect of a division upon the vernier from a division 
 upon the limb is equal to the ?ith part of a division upon the 
 limb, n being the number of divisions upon the vernier*. 
 
 * If n divisions of the vernier were equal to (n -f 1) divisions of the limb, 
 
 or (n + 1) L = n v, 
 
 , i j n-\-\ 1 
 
 then would v — l =- l — L = — l ; 
 
 n n 
 
 or the excess of a division upon the vernier above a division upon the limb 
 would be equal to the nth part of a division upon the limb. With this 
 arrangement, however, we should nave the inconvenience of reading the 
 vernier backwards. 
 
46 
 
 MATHEMATICAL INSTRUMENTS. 
 
 In fig. 1, six divisions of the vernier are equal to five divi¬ 
 sions of the limb, and, consequently, the above defect, or 
 l — v, is equal to a sixth part of a division upon the limb, or 
 to 20', since a division of the limb is equal to 2°. 
 
 In fig. 2, ten divisions of the vernier are equal to nine divi 
 sions of the limb, and, consequently, l — v is equal to a tenth 
 part of a division upon the limb, or to the hundredth part of 
 an inch, a division of the limb being equal to the tenth part 
 of an inch. 
 
 In reading off we must first look to the lozenge, as point¬ 
 ing out the exact place upon the limb at which the required 
 measurement is indicated. If, then, the stroke upon the 
 vernier at the lozenge exactly coincides with a stroke upon 
 the limb, the reading at this stroke gives the measurement 
 required; but, if the stroke at the lozenge be a distance 
 beyond a stroke upon the limb, then will this distance be 
 equal to once, or twice, or thrice, &c., the difference of a 
 division upon the limb and upon the vernier, according as the 
 stroke at the end of the first, or second, or third, &c., divi¬ 
 sion upon the vernier coincides with a stroke upon the limb. 
 
 In fig. 1 the stroke upon the vernier at the lozenge falls 
 beyond the stroke indicating 22° upon the limb, and the 
 stroke at the end of the second division upon the vernier 
 coincides with a stroke upon the limb; the reading therefore 
 is 22° 40'. 
 
 In fig. 2, the stroke upon the vernier at the lozenge falls 
 beyond the stroke indicating one inch and three-tenths 
 upon the limb, and the stroke at the end of the sixth division 
 upon the vernier coincides with a stroke upon the limb: the 
 reading, therefore, is T36 inches, or one inch three tenths 
 and six hundredths. 
 
 The limbs of the best sextants are now divided at every 10 
 minutes, and 59 of these parts are made equal to 60 divisions 
 of their verniers. In this case 
 
 so that these instruments can be read off by the aid of their 
 verniers to an accuracy of 10 seconds. The verniers occupy 
 on the limbs spaces equal to 9° 50'*. 
 
 * That is, according to the graduation of the instrument; but, as the 
 angles observed by a sextant are double the angles moved over by the in¬ 
 dex, the limb of the instrument is graduated, as though it were double th® 
 size; so that the verniers really occupy an arc of 4° 55' only. 
 
MICROMETER. 
 
 47 
 
 The limbs of small theodolites are generally divided at 
 every 30 minutes, and 29 of these parts are made equal to 30 
 divisions of their verniers, which therefore enables us to read 
 
 oil to an accuracy ot —, or 1. 
 
 In the mountain barometer the scale being divided into 
 f f^ths of an inch, 9 of these parts are made equal to 10 
 divisions of the vernier, which therefore enables us to read 
 off to an accuracy of ^f^ths of an inch. 
 
 In the above explanations we have only considered the 
 case of an exact coincidence between some one of the strokes 
 upon the vernier and a stroke upon the limb. Suppose now 
 that in fig. 1 the stroke at the end of the second division, in¬ 
 stead of coinciding with a stroke upon the limb, fell a little 
 beyond it, while the stroke at the end of the third division 
 fell a little short of a stroke upon the limb ; then the measure¬ 
 ment indicated would be something between 22° 40' and 23°, 
 which the observer, should there be no other mechanism at¬ 
 tached to the vernier, must estimate by guess, according to 
 the best of his judgment. By the aid, however, of a piece of 
 mechanism, which is called a micrometer, and which we pro¬ 
 ceed to describe, the excess of the angle indicated above 
 22° 40' might be exactly computed. 
 
 The instrument having been nearly set by the hand alone, 
 the vernier is fixed in this position by turning a screw, called 
 the clamping screw, which is shown on the top of the vernier 
 in fig. 2, but is not seen in fig. 1, being at the back of the 
 instrument. The instrument is then set more accurately by 
 the screw at the side of the vernier, shown in both figures, 
 which gives a slow motion to the vernier plate, and to the 
 limb or index bar attached to the vernier. This screw is 
 called a tangent or slow motion screw, and the micrometer 
 consists of a graduated cylindrical 
 head, bb, attached to this screw, and 
 an index, i, attached to the vernier. 
 
 Suppose, now, the tangent screw to 
 be of that fineness that, whilst it is 
 turned once round, by means of the milled head h, so that 
 the graduated head bb makes one complete revolution, the 
 vernier is advanced on the limb of the instrument, a distance 
 
/ 
 
 48 MATHEMATICAL INSTRUMENTS. 
 
 equal to the difference of a division of the limb, and of the 
 vernier: then, in fig. 1, one revolution of the screw advances 
 the vernier a distance equal to 20'; and, if the cylindrical 
 head bb be divided into 60 equal parts, a revolution of the 
 screw through one of these parts would advance the vernier a 
 distance equal to 20". 
 
 .Suppose, then, that in the illustration above given the screw 
 has to be turned back, so that 14 of these graduations pass 
 the index i, in order to bring the stroke at the end of the 
 second division upon the vernier into coincidence with a 
 stroke upon the limb ; then the corresponding space moved 
 through by the vernier would be equal to 20" x 14, or 4' 40", 
 and the reading of the instrument would be 22° 44' 40". 
 
 Similarly, by means of a micrometer divided into ten equal 
 parts, a distance to the thousandth part of an inch may be 
 read off by the vernier in fig. 2. If the micrometer in this 
 case were divided into one hundred equal parts, a distance 
 might be read off to the ten-thousandth part of an inch; or 
 the same effect may be produced by dividing the micrometer 
 into ten equal parts, and making the screw of such fineness 
 that ten complete revolutions move the vernier through a 
 distance equal to the difference of a division of the limb and 
 of the vernier, or the hundredth part of an inch. 
 
 THE BEAM COMPASSES. 
 
 sists of a beam, a a, of any length required, generally made 
 of well-seasoned mahogany. Upon its face is inlaid through¬ 
 out its whole length a slip of holly, or boxwood, a a, upon 
 which are engraved the divisions or scale, either feet' and 
 decimals, or inches and decimals, or whatever particular scale 
 may be required. Those made for the use of the persons en¬ 
 gaged on the Ordnance survey of Ireland were divided to a 
 scale of chains, 80 of which occupied a length equal to six 
 inches, which, therefore, represented one mile, six inches to 
 
THE BEAM COMPASSES. 
 
 49 
 
 the mile being the scale to which that important survey is 
 plotted *. Two brass boxes, b and c, are adapted to the 
 beam ; of which the latter may be moved, by sliding, to any 
 part of its length, and fixed in position by tightening the 
 clamp screw e. Connected with the brass boxes are the two 
 points of the instrument, g and h, which may be made to have 
 any extent of opening by sliding the box c along the beam, 
 the other box, b, being firmly fixed at one extremity. The 
 object to be attained, in the use of this instrument, is the 
 nice adjustment of the points g, h. to any definite distance 
 apart. This is accomplished by two vernierf or reading 
 plates, b, c, each fixed at the side of an opening in the brass 
 boxes to which they are attached, and affording the means of 
 minutely subdividing the principal divisions, a a, on the 
 beam, which appear through those openings, d is a clamp 
 screw for a similar purpose to the screw e, namely, to fix the 
 box b, and prevent motion in the point it carries after ad¬ 
 justment to position, f is a slow motion screw, by which the 
 point g may be moved any very minute quantity for perfecting 
 the setting of the instrument, after it has been otherwise set 
 as nearly as possible by the hand alone. 
 
 The method of setting the instrument for use may be un¬ 
 derstood from the above description of its parts, and also by 
 the following explanation of the method of examining and 
 correcting the adjustment of the vernier, b, which, like all 
 other mechanical adjustments, will occasionally get deranged. 
 This verification must be performed by means of a detached 
 scale. Thus, suppose, for example, that our beam compass is 
 divided to feet, inches, and tenths, and subdivided by the 
 vernier to hundredths, &c. First set the zero division of the 
 vernier to the zero of the principal divisions on the beam, by 
 means of the slow motion screw f. This must be done very 
 nicely. Then slide the box c, with its point g, till the zero 
 on the vernier c exactly coincides with any principal division 
 on the beam, as twelve inches, or six inches, &c. To enable 
 us to do this with extreme accuracy some superior kinds of 
 beam compasses have the box c also furnished with a tangent 
 or slow motion screw, by which the setting of the points or 
 divisions may be performed with the utmost precision. Lastly, 
 apply the points to a similar detached scale, and, if the 
 adjustment be perfect, the interval of the points gh will 
 
 * The survey of the metropolis is plotted to a scale of 60 inches to the 
 mile. 
 
 + For a description of the vernier, see preceding article. 
 
 D 
 
50 
 
 MATHEMATICAL INSTRUMENTS. 
 
 measure on it the distance to which they were set on the 
 beam. If they do not, by ever so small a quantity, the ad¬ 
 justment should be corrected by turning the screw f till the 
 points do exactly measure that quantity on the detached scale ; 
 then, by loosening the little screws which hold the vernier b 
 in its place, the position of the vernier may be gradually 
 changed, till its zero coincides with the zero on the beam ; 
 and, then tightening the screws again, the adjustment will be 
 complete. 
 
 PLOTTING SCALES. 
 
 Plotting scales, also called feather-edged scales, are straight 
 rulers, usually about ten or twelve inches long. Each ruler 
 has scales of equal parts, decimally divided, placed upon its 
 edges, which are made sloping, so that the extremities of the 
 strokes marking the divisions lie close to the paper. The 
 primary divisions represent chains, and the subdivisions, con¬ 
 sequently, ten links each, as there are 100 links on the sur¬ 
 veying chain. Plotting scales may be procured in sets, each 
 with a different number of chains to the inch. 
 
 The advantages of this arrangement are, that the distances 
 required can be transferred with great expedition from the 
 scale to the paper by the aid of the pricking-point alone, and 
 the marks denoting the divisions are in no danger of be¬ 
 coming defaced, as upon the plain scale, by the frequent 
 application of the compasses. 
 
 One of the best plotting scales consists of two feather-edged 
 rulers, one sliding along the other in a dovetailed groove, so 
 that the two are always at right angles to each other. We 
 shall describe this instrument more particularly when we 
 come to speak of plotting, after describing the instruments 
 used in surveying. 
 
 THE PANTAGRAPH. 
 
 The pantagraph consists of four rulers, ab, ac, df, and 
 ef, made of stout brass. The two longer rulers, ab and ac, 
 are connected together by, and have a motion round a center 
 at a. The two shorter rulers are connected in like mariner 
 with each other at f, and with the longer rulers at d and e, 
 and, being equal in length to the portions ad and ae of the 
 longer rulers, form with them an accurate parallelogram, 
 a d f e, in every position of the instrument. Several ivory 
 castors support the machine parallel to the paper, and allow 
 it to move freely over it in all directions The arms, ab and 
 
PANTAGRAPH. 
 
 51 
 
 D f, are graduated and marked J, &c., and have each a 
 sliding index, which can be fixed at any of the divisions by a 
 milled-headed clamping screw, seen in the engraving. The 
 sliding indices have each of them a tube, adapted either to 
 slide on a pin rising from a heavy circular weight called the 
 fulcrum, or to receive a sliding holder with a pencil or pen, 
 or a blunt tracing point, as may be required. 
 
 When the instrument is correctly set, the tracing point, 
 pencil, and fulcrum will be in one straight line, as shown by 
 the dotted line in the figure. The motions of the tracing 
 point and pencil are then each compounded of two circular 
 motions, one about the fulcrum, and the other about the joints 
 at the ends of the rulers upon which they are respectively 
 placed. The radii of these motions form sides about equal 
 angles of two similar triangles, of which the straight line b c, 
 passing through the tracing point, pencil, and fulcrum, forms 
 the third sides. The distances passed over by the tracing 
 point and pencil, in consequence of either of these motions, 
 have then the same ratio, and, therefore, the distances passed 
 over in consequence of the combination of the two motions 
 have also the same ratio which is that indicated by the set¬ 
 ting of the instrument. 
 
 Our engraving represents the pantagraph in the act of 
 reducing a plan to a scale of half the original. For this pur¬ 
 pose the sliding indices are first clamped at the divisions 
 upon the arms marked \; the tracing point is then fixed in a 
 
 d 2 
 
52 
 
 MATHEMATICAL INSTRUMENTS. 
 
 socket at c, over the original drawing; the pencil is next 
 placed in the tube of the sliding index upon the ruler d f, 
 over the paper to receive the copy; and the fulcrum is fixed 
 to that at b, upon the ruler a b. The machine being now 
 ready for use, if the tracing point at c be passed delicately 
 and steadily over every line of the plan, a true copy, but of 
 one-half the scale of the original, will be marked by the pencil 
 on the paper beneath it. The fine thread represented as 
 passing from the pencil quite round the instrument to the 
 tracing point at c, enables the draughtsman at the tracing 
 point to raise the pencil from the paper, whilst he passes the 
 tracer from one part of the original to another, and thus to 
 prevent false lines from being made on the copy. The pencil 
 holder is surmounted by a cup, into which sand or shot may 
 he put, to press the pencil more heavily on the paper, when 
 found necessary. 
 
 If the object were to enlarge the drawing to double its 
 scale, then the tracer must be placed upon the arm d f, and 
 the pencil at c; and, if a copy were required of the same scale 
 as the original, then, the sliding indices still remaining at the 
 same divisions upon df and ab, the fulcrum must take the 
 middle station, and the pencil and tracing point those on the 
 exterior arms, ab and ac, of the instrument. 
 
 The successful use of the pantagraph in copying very 
 minute and complicated drawings can only be attained by 
 perseverance and experience, and we therefore proceed to 
 mention some of the other means employed for the attain¬ 
 ment of the same object. In fact, while the pantagraph affords 
 the most rapid means of reducing a drawing, we cannot re¬ 
 commend its use for enlarging a copy, or even for copying 
 upon the same scale. 
 
 To produce a Copy of the same Size as the Original. First 
 Method .—Lay the original drawing upon the sheet of paper 
 intended for the copy, and fix them together by means of 
 weights or drawing pins*. 2. With a fine needlef prick 
 through all the angles and principal points, making corre¬ 
 sponding punctures in the paper beneath. 3. Draw upon 
 the copy such of the lines on the original as are straight, or 
 nearly so, by joining the points thus marked upon the paper. 
 
 * The drawing pin consists of a brass head, 
 with a steel point at right angles to its plane. A 
 represents it as seen edgewise, and B as seen from 
 above. 
 
 + See pricking point, page 8. 
 
METHODS OF COPYING AND REDUCING DRAWINGS. 
 
 53 
 
 4 . Set off such other points upon the copy, by means of the 
 compasses, as may be desirable, and draw the curved lines 
 upon tracing paper placed over the drawing. 5. Fill in the 
 lines indicated by the points set off by the compasses, and 
 transfer the curved lines from the tracing paper to the copy, 
 by rubbing the back of the tracing paper with powdered black 
 lead, placing it in its correct situation upon the copy, and 
 passing a blunt tracing point* over the lines drawn upon it. 
 
 Second Method .—A sheet of tracing paper having the under 
 side rubbed over with powdered black lead may be placed 
 upon the paper intended for the copy. The original being 
 then placed over this, the tracing point may be carefully and 
 steadily passed over all the lines of the drawing with a pres¬ 
 sure proportioned to the thickness of the paper; and the 
 paper beneath will receive corresponding marks, forming an 
 exact copy, which is afterwards to be inked in. 
 
 Third Method. —The drawing is placed upon a large sheet 
 of plate glass called a copying glass, and the paper to receive 
 the copy placed over the drawing. The glass is then fixed in 
 such a position as to have a strong light fall upon it from be¬ 
 hind, and shine through it. By this means the original 
 drawing becomes visible through the paper placed over it, 
 and a copy can be made with precision and ease, without any 
 risk of soiling or injuring the original. 
 
 To copy with nicety upon a reduced or enlarged Scale .— 
 For this purpose we may have recourse to the method of 
 squares, by which, with the aid of the proportional compasses, 
 the most minute detail may be copied with great accuracy. 
 This may, perhaps, be best shown by an example. 
 
 Let figure 1 in the following engraving represent a plan of an 
 estate, which it is required to copy upon a reduced scale of one- 
 half. The copy will therefore be half the length and half the 
 breadth, and, consequently, will occupy but one-fourth of the 
 space of the original. Our subject is a map of an estate, but 
 the process would be precisely the same if it were an archi¬ 
 tectural, mechanical, or any other drawing. 
 
 1. Draw the lines fi and fg at right angles to each other. 
 2. From the point f towards i and g, set off any number of 
 equal parts, as f a, a b, b c, &c., on the line fi, and f i, i k, 
 k l , &c., on the line fg. 3. From the points on the line f t 
 draw lines parallel to the other line fg, as a a, b b. c c, &c., 
 
 * The eye end of the pricking needle, or the fine point of a porcupine a 
 quill, may be used for this purpose. 
 
51 
 
 MATHEMATICAL INSTRUMENTS. 
 
 Fig. 1. Fig. 2. 
 
 F a b c d e / g h i f i 
 
 and from the points on f g draw lines parallel to f i, as i i, 
 k k, 11 , &c., which being sufficiently extended towards g and i, 
 the whole of the original drawing will be covered with a reti¬ 
 cule of small but equally-sized squares. 
 
 This done, draw upon the paper intended for the copy a 
 similar set of squares, but having each side only one-half the 
 length of the former, as is represented in figure 2. It will 
 now be evident that, if the lines of the plan a b, b c, c d, 
 &c., figure 1, be drawn in the corresponding squares of figure 
 2, a correct copy of the original will be produced, and of half 
 the original scale. Commencing then at a, observe where, in 
 the original, the angle a falls, which is towards the bottom of 
 the square marked on the top d e. In the corresponding 
 square, therefore, of the copy, and in the same proportion to¬ 
 wards the left-hand side of it, which should be determined by 
 the use of the proportional compasses, described at page 4, 
 place the same point in the copy. From thence, finding by 
 the proportional compasses the point on the bottom line of that 
 square, where the curved line a f crosses, which is about two- 
 fifths from the left-hand corner towards the right, cross it simi¬ 
 larly in the copy. Again, as it crosses the right-hand bottom 
 corner of the second square below d e, describe it so in the 
 copy; and by means of the proportional compasses find the 
 points where it crosses the lines// and g g, above the line 11, 
 by taking the distances of such crossings from the nearest cor¬ 
 ner of a square in the original, between the large points of the 
 proportional compasses, and with the small points at their op¬ 
 posite end, setting off the required crossing on the correspond 
 ing lines on the copy. Lastly, determine the place of the 
 

 
 
 ' • 1 . • '" ■ 
 
a[P ' _ Q\Zj)\ _£_!T__ o\i 
 
 London. John, Wt'ale.1848 
 
THE SLIDING EULE 
 
 55 
 
 point b, in the third square below g h on the top line; and a 
 line drawn from a in the copy, through these several points 
 to b, will be a correct reduced copy of the original line. Pro¬ 
 ceed in like manner with every other line on the plan, and 
 its various details, and you will have the plot or drawing, laid 
 down to a small scale, yet bearing all the proportions in itself 
 exactly as the original. 
 
 It may appear almost superfluous to remark, that the process 
 of enlarging drawings by meaus of squares is a similar opera¬ 
 tion to the above, except that the points are to be determined 
 in the smaller squares of the original, and transferred to the 
 larger squares of the copy. The process of enlarging, under 
 any circumstances, does not, however, admit of the same ac¬ 
 curacy as that of reducing. 
 
 Having now completed the description of those instruments, 
 applicable to the purposes of geometrical drawing, to the con¬ 
 sideration of which we propose for the present to limit our¬ 
 selves, in accordance with the plan of our little work, we now 
 propose to add thereto a description of CoggeshalVs Sliding Rale, 
 and then to conclude this part of our subject with some practi¬ 
 cal hints*, which we think may prove acceptable to the com¬ 
 mencing student. 
 
 COGGESHALLS SLIDING RULE.— (Plate II. Fig. 4.) 
 
 Coggeshall’s, or the Carpenters Sliding Rule, is the instru¬ 
 ment most commonly used for taking the dimensions and 
 finding the contents of timber. It consists of a rule one foot 
 long, having on its face a groove throughout its entire length, in 
 which a second rule of the same length slides smoothly. On 
 the face of the rule are four logarithmic lines marked at one 
 end a, b, c, and d. The three lines a, b, c, are called double 
 lines, because the figures from 1 to 10 are contained twice in 
 the length of the rule, and are, in fact, repetitions of the loga¬ 
 rithmic line of numbers already described (p. 25). The fourth 
 line, d, is a single line numbered from 4 to 40, and is called 
 the Girt Line, because the girt dimensions are estimated upon 
 it in computing the contents of trees and timber. The lengths 
 upon this line denote the logarithms of the squares of the 
 numbers, from 4 to 40, placed against the several divisions; 
 and enable us, as will be seen, to obtain approximately the 
 contents of a solid by a single operation. 
 
 * Extracted from a treatise on drawing instruments, by F. W. Simms, 
 Civil Engineer and Surveyor. 
 
56 
 
 MATHEMATICAL INSTRUMENTS. 
 
 The line c is used with the girt line d, and the two lines a and 
 b, enable us to perform more readily all such operations as have 
 been already described as being performed by the logarithmic 
 line of numbers with the aid of the compasses, the second line 
 B, upon the slider, supplying the place of the compasses. 
 
 On the girt line is a mark at the point 18*79, lettered g 
 ( gallons ), which is the imperial gauge point*, enabling us to 
 compute contents in imperial gallons. 
 
 The back of the rule has a decimal scale of one foot divided 
 into one hundred equal parts, by which dimensions are taken 
 in decimals of afoot; and also a scale of inches, numbered 
 from 1 to 12, which scale is continued on the slider and num¬ 
 bered from 12 to 24, so that, when the slider is pulled out, a 
 two feet rule is formed, divided into inches.—The vacant 
 spaces on the rule are filled up with various other scales and 
 tables, which may be selected to suit the purposes of the 
 various purchasers. 
 
 The method of notation on the rule, and the manner of es¬ 
 timating any number upon it, are the same as have already 
 been fully explained, when treating of the line of logarithmic 
 numbers (p. 28). 
 
 Problem 1. To multiply two Numbers together. —Set 1 on b 
 to the multiplier on a, and against the multiplicand on b will 
 be found the required product on a. Example. —To multiply 
 33 by 23. Set 1 on B to 2*3 on a, and against 3*3 on b will 
 be found 7*59 on a, and 759 is therefore the product re¬ 
 quired f. 
 
 Problem 2. To divide one Number by another. — Set 1 on b 
 to the divisor on a, and against the dividend on a will be found 
 the required quotient on b. Example. —To divide 759 by 23. 
 Set 1 on b to 2*3 on a, and against 75*9 on a will be found 
 33 on b, which is the quotient required. 
 
 Problem 3. To find a Fourth Eroportional to three given 
 Numbers. —Set the first term on b to the second term on a, 
 and against the third term on b will be found the required 
 fourth term on a. Or, against the first term on a, set the se¬ 
 cond term on b, and against the third term on a will be found 
 the required fourth term on b. Example. — To find a fourth 
 
 * 18*79 is the diameter of a cylindrical vessel to contain one gallon 
 for each inch of depth. The gauge point for the old wine gallon was at 
 17*15, lettered \V. Gr., and for tlie old ale gallon at 18*95, lettered A. G. 
 These marks are consequently found upon rules constructed prior to 
 January, 1826. 
 
 f The tens must be supplied mentally, as explained at page 28. 
 
USES OF THE SLIDING RULE. 
 
 57 
 
 proportional to the three numbers 3^. 11, and 14. Set 3|, 
 or 3 5, on b to 11 on a, and against 14 on b will be found 
 on a, 4.4, the fourth proportional required. 
 
 Problem 4. To find a Third Proportional to two given 
 Numbers. — This is the same problem as the preceding, re¬ 
 peating the second number for the third term of the proportion. 
 Example. —To find a third proportional to the two numbers 
 3£ and 11. This is to find a fourth proportional to the three 
 numbers 3^, 11, and 11. Set therefore 3^, or 3'5, on b to 11 
 on a, and against 11 on b will be found on a 34'6, the third 
 proportional required. 
 
 Problem 5. To square a given Number. First Method, bg 
 means of the Lines a and b. —Set 1 on b to the given number 
 on a, and against the given number on b will be found its 
 square upon a. Example. —Required the square of 23. Set 
 1 on b to 23 on a, and against 23 on b will be found its square 
 529 on a. Second Method, by means of the Lines c and D. — If 
 the number to be squared lie between 1 and 4, or 10 and 40, 
 or 100 and 400, &c., so that its first significant digit is less 
 than 4 ; set the 1 on c to 10 on d. and against the digits on 
 d, expressing the given number, will be found on e the digits 
 expressing the required square. Then, the square of 1 being 
 1, of 10, 100, of 100, 10,000, &c., and of T being -01, of *01 
 being ‘0001, &c., the digits upon c must be estimated at the 
 actual values represented by them as numbered upon the 
 scale, viz., 1, 2, &c., to 16, or at 100 times their values, from 
 100 up to 1600, or at 10,000 times their values from 10,000 up 
 to 160.000, &c., or, again, at the T ^ 0 th part of these values 
 from ’01 up to T6, or at -y^^th part of these values from 
 •0001, up to '0016, &c., according as the highest denomination 
 in the number to be squared is units, or tens, or hundreds, &c., 
 or, again, tenths, hundredths, &c. Example. —Required the 
 square of 23. The 10 on d being set against the l on c, against 
 23 on d will be found 5'29 on c, and, the highest denomina¬ 
 tions in 23 being tens, the square required is 529. Also the 
 squares of 2'3, 230, 2300, '23, *023, would be 5 29, 52,900, 
 5,290,000, -0529, '000529, respectively, the highest denomina¬ 
 tions in the proposed numbers being respectively units, hun¬ 
 dreds, thousands, tenths, and hundredths. Third Method, by 
 means of the Lines c and d. —If the number to be squared lie 
 between 4 and 10, or 40 and 100, or 400 and 1000, &c., so that 
 its first significant digit is not less than 4 ; set the 100 on c 
 against the 10 on d, and against the digits on d, expressing 
 
 D 3 
 
58 
 
 MATHEMATICAL INSTRUMENTS. 
 
 the given number, will be found on c the digits expressing 
 the required square. Example. —Required the square of 15. 
 The 100 on c being set against the 10 on d, against 5 - l on d 
 will be found 26 on c, and, the highest denomination in 51 
 being tens, the square required is 2600 *. 
 
 Problem 6. To extract the Square Root of a given Number. 
 —This problem being the converse of the preceding, set the 
 rule in the same manner, with the 1 on c against the 10 
 on d, if the given square be between 1 and 16, or 100 and 
 1600, or 10,000 and 160,000, &c., or again between '01 and 
 T6, or -0001 and '0016, &c., and with the 100 on c against the 
 10 on d, if the given square be between 16 and 100, or 1600 and 
 10,000 &c., or again between T6 and l,or -0016 and *01, &c. ; 
 and then against the given number on c will stand its square 
 root on d. Example 1.—Required the square root of 529. 
 The given number being between 100 and 1600, set the 1 on 
 c against the 10 on d, and against 5*29 on c will be found 
 23 on d, the square root required. Example 2.—Required the 
 square root 2601. The given number being between 1600 
 and 10,000, set the 100 on c against the 10 on d, and 
 against 26 on c will be found 5T on d, and 51 is therefore 
 the root sought. 
 
 Problem 7. To find a mean Proportional between two given 
 Numbers.—Set one of the numbers upon c to the same number 
 on d, and against the other number on c will be found upon 
 d the mean proportional required f. Example. —Required a 
 mean proportional between 4 and 49. Set 4 on c to 4 on d, 
 and against 49 on c will be found on d 14, the mean propor¬ 
 tional required. 
 
 If one number exceed the other so much that they cannot 
 both be taken off from the line c, the T ^ 0 th part of the larger 
 may be taken, and the mean proportional then found, multi¬ 
 plied by 10, will give the mean proportional required. Also 
 if the second number on c be situated beyond the scale d, the 
 1 i^th part of such second number may be substituted for it, 
 and the result multiplied by 10; or 100 times such number may 
 
 * The accurate square is 2601, hut the fourth figure cannot be esti¬ 
 mated upon a foot rule, and the third figure only approximately. The 
 solution, in fact, may be considered as obtained to within the 200th part of 
 the whole, but, if greater accuracy is required, arithmetical methods must 
 be resorted to. 
 
 f If a b, then a : b : : a* : *-; and, therefore, 
 
 log. b — log. a = log. x 2 — log. a' 2 ; whence the rule given in the text. 
 
C/SES OF THE SLIDING RULE. 
 
 59 
 
 be taken, and the result divided by 10; or, again, such second 
 number may be multiplied or divided by 4, 9, or any square 
 number, and the result divided or multiplied by 2, 3, or the 
 square root of this number; or, again, the numbers may be both 
 multiplied and divided by any the same number, and the 
 result divided or multiplied also by the same number, and, in 
 each case, the required mean proportional will be correctly de¬ 
 termined. 
 
 Problem 8. To find the Area of a Board or Plank. First 
 Method .—Set 12 on b to the mean breadth in inches on a, 
 and against the length in feet on b will be found upon a the 
 required area in feet and decimals of a foot. If the plank 
 taper regularly, the mean breadth is half the sum of the ex¬ 
 treme breadths; but, if the plank be irregular, several 
 breadths should be measured at equal distances from each 
 other, and their sum divided by their number may be taken 
 as the mean breadth. In the latter case, however, greater 
 accuracy would be obtained by finding separately the areas of 
 portions of the plank, and adding them together for the whole 
 area, or by the following. Second Method .—Take the measure 
 in inches of several breadths at equal distances from each 
 other, and add together half the two extreme breadths, and 
 the sum of all the intermediate breadths. Set 12 on b to the 
 sum thus found upon a, and against the distance in feet, at 
 which the breadths have been measured, upon b will be found 
 upon a the required area in feet and decimals of a foot. Ex¬ 
 ample 1.—A board, 15 feet long, being 14 inches broad at one 
 end, and 8 inches broad at the other, required its area. The 
 mean breadth is 11 inches, half the sum of 8 and 14. Set, 
 then, 12 on b against 11 on a, and against 15 on b will be 
 found upon a 13-75 or 13f feet, the area required. Ex¬ 
 ample 2.—An irregular board, 18 feet long, being 7 inches 
 broad at one extremity, 11 inches broad at the other, and the 
 intermediate breadths at each 3 feet of the length being 13 
 inches, 25 inches, 23 inches, 32 inches, and 22 inches, re¬ 
 quired its area. By the first method, the sum of the seven 
 breadths divided by 7, gives 19 inches for the mean breadth ; 
 and, setting 12 on b against 19 on a, against 18 on b will be 
 found upon a 28*5 or 28| feet, the area required. By the 
 second method, half of the two extreme breadths added to 
 the intermediate breadths, gives the sum, 123 inches ; and 
 setting 12 on b against 123 upon a, against 3 on b will be 
 found upon a 30f, the area required, a more accurate result 
 than the preceding. 
 
60 
 
 MATHEMATICAL INSTRUMENTS. 
 
 Problem 9. To find the solid Content of squared or four- 
 sided Timber, of the same Size throughout its entire Length. 
 First Method. —Multiply the breadth by the thickness, and 
 their product again by the length (Problem 1), and the result 
 will be the content required. Second Method. —Set the length 
 on c against 1*2 on d, and against the quarter girt, measured 
 in inches, on D, will be found the approximated content on o 
 in cubic feet: or set the length on c against 10 on d, and 
 against the quarter girt, measured in tenths of feet, on d will 
 be found the approximate content on c. The approximate 
 content thus found is greater than the true content, and the 
 correction to be subtracted to leave the true content is given 
 in the following Table :— 
 
 TABLE I. 
 
 Fraction of 
 breadth equal 
 to excess of 
 breadth over 
 thickness. 
 
 Excess in 
 inches for each 
 12 inches of 
 breadth. 
 
 Fractional por¬ 
 tion of approxi¬ 
 mate content to 
 be subtracted. 
 
 Percentage 
 mate content 
 traded. 
 
 of 
 
 to 
 
 approx i- 
 be sub- 
 
 k bre 
 
 6 
 
 o app. 
 
 cont. 
 
 
 
 11 
 
 per cent. 
 
 b 
 
 77 
 
 4 
 
 l 
 
 25 
 
 77 
 
 
 
 4 
 
 
 77 
 
 k 
 
 77 
 
 3 
 
 TO 
 
 77 
 
 
 
 2 
 
 
 77 
 
 1 
 
 -r 
 
 O 
 
 77 
 
 9 * 
 
 L TO 
 
 ST 
 
 77 
 
 n 
 
 or 
 
 1*23 
 
 
 77 
 
 1 
 
 5 
 
 77 
 
 2 
 
 1 
 
 T3T 
 
 77 
 
 5 
 
 6 
 
 or 
 
 •83 
 
 
 77 
 
 ! 
 
 77 
 
 i 9 
 
 T50 
 
 77 
 
 1? 
 
 or 
 
 •59 
 
 
 77 
 
 1 
 
 8 
 
 77 
 
 i b 
 
 1 
 
 2*25 
 
 77 
 
 4 
 
 0 
 
 or 
 
 •44 
 
 
 77 
 
 b 
 
 77 
 
 i b 
 
 2S0 
 
 77 
 
 
 or 
 
 •35 
 
 
 77 
 
 i 
 
 77 
 
 1^5 
 
 sin 
 
 77 
 
 2 
 
 7 
 
 or 
 
 •28 
 
 
 77 
 
 TT 
 
 7 } 
 
 It't 
 
 i 
 
 Tit 
 
 77 
 
 5 
 
 22 
 
 or 
 
 •23 
 
 
 77 
 
 1 
 
 13 
 
 77 
 
 1 
 
 520 
 
 77 
 
 lt> 
 
 S3 
 
 or 
 
 •19 
 
 
 77 
 
 The fractional portion of the approximate contents in column 
 3 may be found by dividing the approximate contents by the 
 denomination of the fractions. (Problem 2.) 
 
 If the excess of the breadth over the thickness be compared 
 with the quarter girt, the correction has to the approximate 
 content the duplicate ratio of half the excess to the quarter 
 girt, as shown in the following Table :— 
 
USES / OF THE SLIDING RULE. 
 
 61 
 
 TABLE II. 
 
 Fraction of 
 quarter girt 
 
 Half the ex¬ 
 cess of 
 
 
 Correction to be subtracted. 
 
 
 equal to 
 
 breadth over 
 
 
 
 
 
 
 half 
 
 the ex- 
 
 thickness 
 
 
 
 
 
 
 CPSS of 
 
 for each 12 
 
 
 
 
 
 
 breadth over 
 thickness. 
 
 inches of 
 quarter girt. 
 
 Fractional portion of 
 
 Percentage of approxi- 
 
 approximate content. 
 
 mate content. 
 
 k 
 
 qr. girt. 
 
 6 inches. 
 
 X 
 
 4 
 
 approx, cont. 
 
 
 25 per cent. 
 
 
 
 5 
 
 yy 
 
 b nearly or „ 
 
 171 or 
 
 17-36 
 
 yy 
 
 4 
 
 »» 
 
 4 
 
 yy 
 
 4 
 
 yy 
 
 
 11 
 
 yy 
 
 4 
 
 >9 
 
 3 
 
 yy 
 
 A 
 
 yy 
 
 61 or 
 
 6-25 
 
 yy 
 
 4 
 
 }> 
 
 
 yy 
 
 A 
 
 yy 
 
 
 4 
 
 yy 
 
 4 
 
 )9 
 
 2 
 
 yy 
 
 A 
 
 yy 
 
 % or 
 
 2-78 
 
 yy 
 
 4 
 
 
 If 
 
 yy 
 
 A 
 
 yy 
 
 
 2-04 
 
 yy 
 
 8 
 
 )> 
 
 n 
 
 yy 
 
 A 
 
 yy 
 
 1? or 
 
 1-56 
 
 
 5 
 
 1) 
 
 n 
 
 yy 
 
 BT 
 
 yy 
 
 11 or 
 
 1-23 
 
 yy 
 
 A 
 
 
 Rg 
 
 yy 
 
 Ti>5 
 
 yy 
 
 
 1 
 
 yy 
 
 A 
 
 yy 
 
 1A 
 
 yy 
 
 At 
 
 yy 
 
 * or 
 
 •83 
 
 >y 
 
 A 
 
 yy 
 
 l 
 
 yy 
 
 1 
 
 T4T 
 
 
 A °r 
 
 •69 
 
 yy 
 
 A 
 
 yy 
 
 
 yy 
 
 Tib 
 
 yy 
 
 i, or 
 
 •6 
 
 yy 
 
 A 
 
 >> 
 
 i 
 
 yy 
 
 JL-, 
 
 Tg3 
 
 yy 
 
 k °r 
 
 •51 
 
 yy 
 
 i 
 
 To 
 
 yy 
 
 T(i 
 
 yy 
 
 l 
 
 622 
 
 yy 
 
 s or 
 
 •44 
 
 yy 
 
 The correction may also be found as follows: —Set the 
 length upon c against 12 upon d, and against half the excess 
 of the breadth over the thickness upon d will be found upon 
 c the required correction in cubic feet. 
 
 As the error of the result obtained with the rule may 
 amount to the -g^th part of the whole, the correction given 
 above may always be neglected, whenever the excess of the 
 breadth over the thickness does not exceed the £th part of 
 the breadth, or 1| inch for each 12 inches of breadth, and 
 the result may be depended upon to as great an accuracy as 
 can be obtained by the rule. When, however, the excess is 
 more than two inches for each 12 inches of breadth, either 
 the correction should be applied or the first method be used. 
 
 Example l.—Required the content of a piece of timber 10 
 inches broad, 8 inches thick, and 18 feet long. 
 
 10 8 80 
 
 Since — x — = ■—-> set 80 on b against 144 on a, and 
 12 12 144 
 
 against 18 on a will be found 10 on b, and the content re¬ 
 quired is 10 cubic feet. Example 2.—Required the content 
 of a piece of timber 15 inches broad, 10 inches thick, and 24 
 
62 
 
 MATHEMATICAL INSTRUMENTS. 
 
 feet long. Set 24 on c against 12 on d, and against 12 5, or 
 12|, the quarter girt on d, will be found on c 26 04, the ap¬ 
 proximate content. The excess of 15 over 10 being |rd of 
 15, Table I. shows the required correction to -J^th of 26 04. 
 Set then 25 on b against 26 04 on a, and against 1 on b will 
 be found T04 on a, which subtracted from 26*04 cubic feet 
 leaves 25 cubic feet, the true content. 
 
 Problem 10. To find the Content of a Piece of Square 
 Timber , which tapers from end to end. —Set the length in feet 
 upon c against 12 upon d, and against half the sum in inches 
 of the quarter girts at the two ends upon d will be found a 
 content in cubic feet upon c. Again, set one-third of the 
 length in feet upon c against 12 upon d, and against half the 
 difference, in inches, of the quarter girts at the two ends upon 
 d will be found a second content in cubic feet upon c. Add 
 together the two contents thus found for the content required. 
 If the breadth exceed the thickness considerably, the same 
 part of the result must be subtracted, as in Problem 9. 
 
 Example. —The quarter girts at the ends of a piece of tim¬ 
 ber 21 feet long, being 22 inches and 10 inches respectively, 
 and the breadth not much exceeding the thickness, required 
 the content. Set 21 upon c against 12 on d, and against 16 
 upon d will be found 37^ or 37'3 upon c. Again, set 7 upon 
 o against 12 upon d, and against 6 upon d will be If or 175 
 upon c. The sum of 37^ cubic feet and If cubic foot is then 
 SOAj or 39T cubic feet, the whole content required. 
 
 Problem 11. To find the Content of a Pound Piece of 
 Timber of the same Size throughout its entire Length. — Set the 
 length in feet upon c against 10*63 * upon d, and against the 
 quarter girt in inches upon d will be found the content upon 
 e. Example. —Required the content of a round piece of tim¬ 
 ber 32 feet long, the quarter girt being 11 inches. Set 32 
 upon c against 10*63 upon d, and against 11 upon d will be 
 found upon c 34*25 or 32^, the content required. 
 
 Problem 12. To find the Content of a Bound Piece of 
 Timber , which tapers from end to end. —Set the length in feet 
 upon c against 10*63 upon d, and against half the sum in 
 inches of the quarter girts at the two ends upon d will be 
 found a content in cubic feet upon c. Again, set one-third of 
 the length in feet upon c against 10*63 upon d, and against 
 half the difference in inches of the quarter girts at the two 
 
 * A mark is placed upon the rule at this point, 10*63 being the quarter 
 girt in inches of the circle, whose area is a square foot. 
 
USES OF THE SLIDING RULE. 
 
 63 
 
 ends upon d will be found a second content in cubic feet upon 
 
 c. Add together the two contents thus found for the content 
 required. 
 
 Note .—In buying rough or unsquared timber, an allowance of about 
 fjth should be made for the bark. A further allowance should also be 
 made for the loss in squaring down the tree to make useful shaped timber. 
 The whole amount of timber to be taken off to make a square piece from a 
 round piece of timber will be 36 per cent., or more than a third of the whole. 
 The timber so taken off must not, however, be considered completely value¬ 
 less. If the length upon c be set against 12 upon n, instead of upon 10"63 
 in the two preceding problems, this will be equivalent to an allowance of 
 about 21 2 per cent., which may be considered a just allowance. 
 
 Example 1.—A piece of round tapering timber measures 23 
 feet in length, the quarter girt at the larger end is 23^ inches, 
 and at the smaller end the quarter girt is 15| inches. Re¬ 
 quired the true content. Set 23 upon c against 10.63 upon 
 
 d, and against 19| or 19*5 upon d will be found 77'5 upon c. 
 Again, set 7| or 7-66 upon c against 10 63 upon d, and against 
 8 upon d will be found 4-3 upon c. Then the sum of 77*5 
 cubic feet and 4-3 cubic feet is 81'8 cubic feet, the content 
 required. Example 2.—Required the content of a piece of 
 unsquared timber of the same dimensions as in the preceding 
 example, making allowance of 21^ per cent, for loss in squaring 
 down into a useful shape. Set 23 upon "c against 12 upon d, 
 and against 19^, or 19-5, upon d will be found 60-75 upon c. 
 Again, set 7| or 7-66 upon c against 12 upon d, and against 
 8 upon d will be found 34 upon c. Then the sum of 60-75 
 cubic feet and 3‘4 cubic feet is 64-15 cubic feet, the content 
 required. 
 
 Problem 13. To find the Content of a Cylindrical Vessel 
 in Gallons. — Set the length of the cylinder ip inches upon c 
 against the gauge mark at 18-79, marked g, upon d, and 
 against the diameter of the cylinder in inches upon d will be 
 found the required content in gallons upon c. If the number 
 of inches in the diameter lie beyond c, or if this number be 
 greater than 40, so as not to be contained upon d, the -J^th 
 part, or any part that may be convenient, of the number of 
 inches in the diameter may be taken, and the result thus ob¬ 
 tained, multiplied by 100, or the square of the divisor made 
 use of, will give the content required. Example. —A circular 
 vat 5 feet in diameter being filled to the depth of four feet, 
 required the quantity of liquor in it. Set 48 upon c against 
 the gauge mark at 18-79 upon d, and against 6, the -j-^tli 
 part of the diameter in inches, upon d will be found upon 
 c 4 - 9; and consequently 4-9 x 100 or 490 gallons is the 
 quantity of liquor in the vat. 
 
64 
 
 MATHEMATICAL INSTRUMENTS 
 
 PRACTICAL HINTS, ETC. * 
 
 ON THE MANAGEMENT OF DRAWING PAPER. 
 
 The first thing to be done, preparatory to the commencement 
 of a drawing, is to stretch the paper evenly upon the smooth 
 and flat surface of a drawing board. The edges of the paper 
 should first be cut straight, and, as nearly as possible, at right 
 angles with each other; also the sheet should be so much 
 larger than the intended drawing and its margin, as to ad¬ 
 mit of being afterwards cut from the board, leaving the 
 border by which it is attached thereto by glue or paste, as we 
 shall next explain. 
 
 The paper must first be thoroughly and equally damped 
 with a sponge aud clean water, on the opposite side from that 
 on which the drawing is to be made. When the paper ab¬ 
 sorbs the water, which may be seen by the wetted side be¬ 
 coming dim, as its surface is viewed slantwise against the 
 light, it is to be laid on the drawing board with the wetted 
 side downwards, and placed so that its edges may be nearly 
 parallel with those of the board; otherwise, in using a T 
 square, an inconvenience may be experienced. This done, lay 
 a straight flat ruler on the paper, with its edge parallel to, and 
 about half an inch from, one of its edges. The ruler must 
 now be held firm, while the said projecting half inch of paper 
 is turned up along its edge; then, a piece of solid glue (com¬ 
 mon glue will answer the purpose), having its edge partially 
 dissolved by holding it in boiling water for a few seconds, 
 must be passed once or twice along the turned edge of the 
 paper, after which, this glued border must be again laid flat 
 by sliding the rule over it, and, the ruler being pressed down 
 upon it, the edge of the paper will adhere to the board. If 
 sufficient glue has been applied, the ruler may be removed 
 directly, and the edge finally rubbed down by an ivory book- 
 knife, or any clean polished substance at hand, which will then 
 firmly cement the paper to the board. Another but adjoining 
 edge of the paper must, next, be acted upon in like manner, 
 and then the remaining edges in succession; we say the ad¬ 
 joining edges, because we have occasionally observed that, 
 when the opposite and parallel edges have been laid down 
 first, without continuing the process progressively round the 
 board, a greater degree of care is required to prevent undula¬ 
 tions in the paper as it dries. 
 
 * Extracted from a Treatise on Drawing Instruments by F. W. Simms, 
 Civil Engineer and Surveyor. 
 
DRAWING PAPER. 
 
 65 
 
 Sometimes strong paste is used instead of glue ; but, as this 
 takes a longer time to set, it is usual to wet the paper also on 
 the upper surface to within an inch of the paste mark, care 
 being taken not to rub or injure the surface in the process. 
 The wetting of the paper in either case is for the purpose 
 of expanding it; and the edges being fixed to the board 
 in its enlarged state, act as stretchers upon the paper, while 
 it contracts in drying, which it should be allowed to do gradu¬ 
 ally. All creases or undulations by this means disappear 
 from the surface, and it forms a smooth plane to receive the 
 drawing. 
 
 TABLE OF DIMENSIONS OF DRAWING PAPER. 
 
 Demy . . . 
 
 20 in. by 151 in. 
 
 Columbier . . 
 
 35 
 
 in. by 23 \ in. 
 
 Medium. . . 
 
 2-f » 
 
 171 „ 
 
 Atlas .... 
 
 34 
 
 » 26 ,, 
 
 Royal . . . 
 
 24 „ 
 
 191 „ 
 
 Double Elephant 
 
 40 
 
 „ 27 „ 
 
 Super Royal . 
 
 271 „ 
 
 191 „ 
 
 Antiquarian 
 
 53 
 
 )> 31 „ 
 
 Imperial . . 
 
 Elephant . . 
 
 30 „ 
 
 28 „ 
 
 22 „ 
 23 „ 
 
 Emperor . . 
 
 68 
 
 „ 48 „ 
 
 MOUNTING PAPER AND DRAWINGS, VARNISHING, ETC. 
 
 In mounting paper upon canvas, the latter should be well 
 stretched upon a smooth flat, surface, being damped for that 
 purpose, and its edges glued down as was recommended in 
 stretching drawing paper. Then with a brush spread strong 
 paste upon the canvas, beating it in till the grain of the can 
 vas be all filled up; for this, when dry, will prevent the can¬ 
 vas from shrinking when subsequently removed; and, having 
 cut the edges of the paper straight, paste one side of every 
 sheet, and lay them upon the canvas, sheet by sheet, overlap¬ 
 ping each other a small quantity. If the drawing paper is 
 strong, it is best to let every sheet lie five or six minutes after 
 the paste is put on it; for, as the paste soaks in, the paper 
 will stretch, and may be better spread smooth upon the can¬ 
 vas ; whereas, if it be laid on before the paste has moistened 
 the paper, it will stretch afterwards and rise in blisters when 
 laid upon the canvas. The paper should not be cut off from 
 its extended position till thoroughly dry; and the drying 
 should not be hastened, but gradually take place in a dry 
 room, if time permit; if not, the paper may be exposed to 
 the sun, unless in the winter season, when the help of a fire 
 is necessary, care being had that it is not placed too near a 
 scorching heat. 
 
 In joining two sheets of paper together by overlapping, it 
 
60 
 
 MATHEMATICAL INSTRUMENTS. 
 
 is necessary, in order to make a neat joint, to feather edge 
 each sheet; this is done by carefully cutting with a knife half 
 way through the paper near the edges, and on the sides, which 
 are to overlap each other; then strip off a feather-edged slip 
 from each, which, being done dexterously, the edges will form 
 a very neat and efficient joint when put together. 
 
 The following method of mounting and varnishing drawings 
 or prints was communicated some years ago by Mr. Peacock, 
 an artist of Dublin. Stretch a piece of linen on a frame, to 
 which give a coat of isinglass or common size. Paste the 
 back of the drawing, leave it to soak, and then lay it on the 
 linen. When dry, give it at least four coats of well-maae 
 isinglass size, allowing it to dry between each coat. Take 
 Canada balsam diluted with the best oil of turpentine, and 
 with a clean brush give it a full flowing coat 
 
 GENERAL RULES APPLICABLE IN ALL GEOMETRICAL 
 
 CONSTRUCTIONS. 
 
 In selecting black-lead pencils for use, it may be remarked 
 that they ought not to be very soft, nor so hard that their 
 traces cannot be easily erased by the India rubber. Great 
 care should be taken, in the pencilling, that an accurate out¬ 
 line be drawn; the pencil marks should be distinct yet not 
 heavy, and the use of the rubber should be avoided as much 
 as possible, for its frequent application ruffles the surface of 
 the paper, and will destroy the good effect of shading or 
 colouring, if any is afterwards to be applied. 
 
 The following seven useful rules are taken from Mr. 
 Thomas Bradley’s valuable work on Practical Geometry:— 
 
 “1. Arcs of circles, or right lines by which an important point is to be 
 found, should never intersect each other very obliquely, or at an angle of 
 less than 15 or 20 degrees; and, if this cannot be avoided, some other pro¬ 
 ceeding should be had recourse to, to define the point more precisely. 
 
 “2. When one arc of a circle is described, and a point in it is to be deter- 
 j mined by the intersection of another arc, this latter need not be drawn at 
 j all, but only the point marked off on the first, as it is always desirable to 
 avoid the drawing of unnecessary lines. The same observation applies to a 
 point to be determined on one straight line by the intersection of another. 
 
 "3. Whenever the compasses can be used in any part of a construction, or 
 to construct the whole problem, they are to be preferred to the rule, unless 
 the process is much more circuitous, or unless the first rule (above) forbids. 
 
 “4. A right line should never be obtained by the prolongation of a very 
 short one, unless some point in that prolongation is first found by some other 
 means, especially in any essential part of a problem. 
 
 “5. The larger the scale on which any problem, or any part of one, is con¬ 
 structed, the less liable is the result to error; hence all angles should be set 
 off on the largest circles which circumstances will admit of being desetibed. 
 
THE PRISM. 
 
 67 
 
 and the largest radius should be taken to describe the arcs by which a point 
 is to be found through which a right line is to be drawn; and the greater 
 attention is to be paid to this rule, in proportion as that step of the problem 
 under consideration is conducive to the correctness of the final result. 
 
 4< 6. All lines, perpendicular or parallel to another, should be drawn long 
 enough at once, to obviate the necessity of producing them. 
 
 “ 7. "Whenever a line is required to be drawn to a point, in order to insure 
 the coincidence of them, it is better to commence the line from the point; 
 and if the line is to pass through two points, before drawing it the pencil 
 should be moved along the rule, so as to ascertain whether the line will, 
 when drawn, pass through them both. Thus, if several radii to a circle were 
 required to pass through any number of points respectively, the lines should 
 be begun from the center of the circle; any error being more obvious when 
 several lines meet in a point. 
 
 PART II.—ON OPTICAL INSTRUMENTS. 
 
 Under this head our principal object will be to consider the 
 construction, and principles of action, of such instruments as 
 are indispensable to assist the vision in making observations 
 upon distant objects, whether upon terrestrial objects for the 
 purposes of the surveyor, or upon celestial objects for the pur¬ 
 poses of astronomy and navigation. We propose, however, to 
 add a few words upon such other optical instruments, as by 
 their utility, or by the frequency with which they are brought 
 before us, appear to demand our attention. 
 
 We shall thus be led, in the first place, to review 7 briefly 
 the properties of prisms, lenses, and plane and curvilinear 
 reflectors, and shall then proceed to give descriptions of the 
 following instruments, viz., 
 
 Microscopes. "1 Such as are adapted to surveying and astronomical 
 
 Telescopes. J instruments, rather fully. 
 
 The Camera Lucida. 1 , r , . n 
 The Camera Obscura. j Ver y br,tfl y- 
 
 THE PRISM. 
 
 A collection of straight lines, either conical or cylindrical, 
 representing rays of light, is called a pencil of light, and the 
 axis of the cylinder or cone is called the axis of the pencil. 
 
 The term medium is used in optics to signify any trans¬ 
 parent substance, that is, any substance into which a portion 
 of the light falling upon it can pass. 
 
 The term prism in optics is used to signify a portion of 
 any medium bounded by plane surfaces which are inclined to 
 one another. The bounding surfaces are called the faces of 
 
68 
 
 MATHEMATICAL INSTRUMENTS. 
 
 the prism ; the line in which the faces intersect is called tho 
 edge of the prism; and the angle at which the faces are in¬ 
 clined is called the refracting angle. 
 
 The prism is to be placed so that the axis of the pencil, by 
 which an object is seen through it, be in a plane perpendicular 
 to the edge of the prism; and the axis of the pencil during 
 aud after its passage through the prism still remains in this 
 plane. 
 
 One effect of a prism of denser material than the surround¬ 
 ing medium is to bend every ray of light passing through it, 
 and, consequently, the whole pencil, further from the edge of 
 the prism. 
 
 Another effect of such a prism is to decompose each single 
 ray of white light into several rays of different colours, which 
 rays are bent at different angles, so as to form a lengthened 
 image of different colours, of the point from which the ray 
 proceeds. This image is called the spectrum, aud these 
 colours the colours of the spectrum. 
 
 When, then, any object is viewed through a prism, the two 
 following effects 
 are produced, 
 lstly. The ap¬ 
 parent position of 
 the ob ect is 
 
 o 
 
 changed, so that, 
 if the prism be 
 held with its edge 
 downwards, as in the accompanying figure, the object appears 
 lower than it really is, while, if the prism were held with its 
 edge upwards, the object would appear in a position higher 
 than its actual position. 2ndly. The boundaries of the object 
 are indistinctly defined, and fringed with colours. 
 
 Our figure represents the section of the prism made by the 
 plane of incidence, that is, by the plane which is perpendicu¬ 
 lar to the edge of the prism, and contains the incident ray of 
 light P Q, forming the axis of the pencil under consideration, 
 which proceeds from one point of an object p. aq and aii 
 are sections of the faces of the prism ; a is a point in its 
 edge; and the angle qar is its refracting angle. Now the 
 ray of light, pq, proceeding from the object at p through the 
 medium of the atmosphere, is bent, upon entering the denser 
 medium of the prism, from the direction q t into the direction 
 q r, nearer to lqk, the perpendicular, at the point of inci¬ 
 dence q, to the face a q of the prism ; and, upon emerging 
 
THE PRISM. 
 
 69 
 
 from the prism into the rarer medium of the atmosphere, is 
 again bent from the direction q r into the direction rs, further 
 from m r n, the perpendicular to the face a r at the point of 
 emergence r. The eye, being placed at s, sees the point p, 
 therefore, by means of a pencil of light of which s r is the 
 axis, and p consequently appears at p' on the prolongation of 
 the line sr. A similar effect being produced upon every 
 other point, the entire object is apparently depressed, as re¬ 
 presented in the figure. 
 
 The angle tes, or pep', between ers, the direction of 
 emergence, and pet, the direction of incidence, is called the 
 angle of deviation *. 
 
 The consideration of the properties of the prism is of great 
 importance, as exhibiting in the simplest manner the princi¬ 
 ples of the refraction and dispersion of light. The prism is 
 also used in optical instruments, to change the direction of 
 the pencils of light by which an object is observed, in order to 
 make the apparent place of this object, as viewed through the 
 prism, coincide with the actual place of other objects seen 
 directly, as in the prismatic compass f, or for the mere pur¬ 
 pose of convenient observation, as in the Newtonian tele¬ 
 scope f. 
 
 * The amount of refraction when a ray of light passes from one medium 
 into another varies with the angle of incidence, so that the sine of the angle 
 of refraction bears a constant ratio to the sine of the angle of incidence. 
 This ratio varies for each different medium, and is called the refracting 
 power of the medium. The deviation of a ray in passing through a prism 
 varies also with the angle of incidence, and has a minimum value when 
 the angles of incidence and emergence are equal: and the refracting 
 power can be determined by finding practically this minimum deviation, 
 as follows :—Place the prism with its edge downwards, so as to receive a 
 small beam of solar light, admitted into a dark room through a hole in a 
 shutter, and let the beam of light, after refraction, be received upon a 
 screen behind the prism. The prism must then be turned round an axis 
 parallel to its edge, so as to vary the angle of incidence, and, consequently, 
 the position of the bright spot upon the screen; and. in one particular 
 position, we shall find the bright spot to remain stationary for an instant, 
 though the motion of the prism is continued. The deviation will then be 
 a minimum, and will be equal to the sum of the sun’s altitude and the in¬ 
 clination of the emergent beam to the horizon. Let s represent this 
 minimum deviation, and A the refracting angle of the prism, and let the 
 
 . A -(- S 
 
 refracting power be represented by p : then p 
 
 + These instruments will be described hereafter. 
 
70 
 
 MATHEMATICAL INSTRUMENTS. 
 
 LENSES. 
 
 A portion of any medium bounded by two spherical sur¬ 
 faces having a common axis, or by a spherical surface and a 
 plane one, is called a lens. 
 
 The effects produced by lenses upon pencils of light de¬ 
 pend both upon the form of the lens itself, and upon the 
 direction in which the pencil is proceeding with respect to 
 the lens. Lenses consequently receive distinguishing names, 
 to mark either different forms, or different positions with 
 respect to the light falling upon them. These distinguishing 
 names are the following, and the forms and positions of the 
 corresponding lenses are represented in the accompanying 
 diagram, the light being considered to be proceeding from 
 left to right. 
 
 1. Convex meniscus. 3. 
 
 2. Concave meniscus. 4. 
 
 7. Plano-concave. 
 
 8. Concavo-plane. 
 
 Double convex. 5. Plano-convex. 
 
 Double concave. 6. Convexo-plane. 
 
 9. Concavo-convex. 
 
 10. Convexo-concave. 
 
 The rays forming any pencil of light must either be diver¬ 
 gent, parallel, or convergent; and when a pencil of light 
 passes through an essentially convex lens, that is, one which 
 is thicker in the middle than at the edges, as 1, 2, 3, 5, 6, 
 the rays are made more convergent, so that a pencil of con¬ 
 verging rays becomes still more convergent, a pencil of paral¬ 
 lel rays becomes convergent, and a pencil of diverging rays 
 becomes either less divergent, parallel, or convergent: but 
 when a pencil of light passes through an essentially concave 
 lens, that is, one which is thinner in the middle than at the 
 -edges, as 4, 7, 8, 9, 10, the rays are made more divergent, so 
 that a pencil of converging rays becomes either less conver¬ 
 gent, parallel, or divergent, a pencil of parallel rays becomes 
 divergent, and a pencil of diverging rays becomes still more 
 divergent. 
 
 The sensation of vision is produced by pencils of rays pro¬ 
 ceeding from every point of the visible object, and entering 
 
EFFECTS PRODUCED BY LENSES. 
 
 71 
 
 the pupil of the eye; and in order to produce distinct vision 
 the rays of each such pencil must either be parallel or slightly 
 divergent. Thus the sun, moon, and planets are seen dis¬ 
 tinctly, although so distant, by parallel rays ; and the least 
 distance from the eye at which an object can be seen distinctly 
 varies in different persons, according to the power of the 
 natural lenses which, in fact, form the eye. 1. When any 
 object is brought nearer to the eye than this without the inter¬ 
 vention of a lens, the vision becomes confused. If, however, 
 the rays of light proceeding from the object were, by the 
 interposition of a convex lens, rendered less divergent or 
 parallel, the vision would be again distinct. 2. It is further 
 necessary for distinct vision that the intensity of the light be 
 not less than a certain intensity, as may easily be exemplified 
 by gradually closing the shutters of a room, and thus dimi¬ 
 nishing the intensity of the light proceeding from the objects 
 in the room, when they will grow more and more indistinct. 
 3. Lastly, it is also necessary, for distinct vision of any object, 
 that the axes of the pencils proceeding from its extreme parts 
 enter the eye under an angle not less than a certain angle, so 
 that, however strong a light be thrown upon an object, if this 
 object be very minute, or removed to a distance very great 
 with respect to its magnitude, it will not be seen by the 
 naked eye. If, however, by the assistance of a lens, or 
 combination of lenses, a sufficient number of rays to produce 
 the required intensity of light can be collected from each 
 point of an object, and passed through the pupil of the eye, 
 and if at the same time the axes of the extreme pencils are 
 bent by this lens, or combination of lenses, so as to enter the 
 eye under a sufficiently large angle, while the rays of each 
 pencil are made parallel, or but slightly divergent, then vision 
 will ensue, no matter how minute, how distant, or liow T im¬ 
 perfectly illuminated the object may be. 
 
 When a pencil of rays proceeding from a point of an object 
 passes through a lens, the rays which pass through at different 
 distances from its center will diverge from or converge to dif¬ 
 ferent points, so that the whole pencil will not any longer 
 diverge from or converge to a single point; and from this 
 cause the image of one point will overlap the image of an¬ 
 other, and an indistinctness of the object will arise. This 
 source of indistinctness is called the aberration. A combina¬ 
 tion of lenses may, however, be formed, so that the aberration 
 of one shall be corrected by the aberration of the others. 
 Such a combination is said to be aplanatic. 
 
72 
 
 MATHEMATICAL INSTRUMENTS. 
 
 Since rays of light proceed in every direction from tne 
 points of visible objects, the pencils of light intercepted by a 
 lens for the first time are all centrical, that is, their axes all 
 pass through the center of the lens; but the pencils, upon 
 emerging from this first lens, are already determinate both in 
 extent and direction, and consequently will fall, some of them 
 at least, eccentrically upon a second lens, that is, their axes 
 will meet this lens at different distances from its center. The 
 axes of the most eccentrical pencils will then, after emer¬ 
 gence, cross the axis of vision nearer to the lens than will 
 those of the more nearly central pencils; and thus, while the 
 center of the object is seen distinctly, the parts at a distance 
 from the center will be distorted, or vice versa. This source 
 of error is called the spherical confusion. The spherical con¬ 
 fusion is diminished by dividing the desired deviations, or 
 bendings of the axes, between two or more lenses; and it is 
 found by opticians to be a good rule to divide the deviations 
 equally amongst the lenses employed. 
 
 The most important source of indistinctness, however, is the 
 dispersion of each ray into rays of different colours refracted 
 at different angles (p. 68) which is called the chromatic dis¬ 
 persion. The effect of this dispersion upon a centrical pencil 
 is partly analogous to the spherical aberration, causing the 
 images of neighbouring points to be of finite extent and over¬ 
 lap one another; and it, moreover, fringes the image with 
 colour. An eccentrical pencil is separated by this dispersion 
 into pencils of rays of different colours, the axes of which are 
 bent at different angles; and the imperfection arising from this 
 cause is far greater than that from both the spherical aberra¬ 
 tion and spherical confusion. 
 
 Before stating the maimer in which the imperfections 
 arising from the chromatic dispersion are remedied, it will be 
 expedient to explain what is understood by the focal length of a 
 lens. Now a pencil of parallel rays, after passing through a 
 lens, becomes either convergent or divergent, as the lens is 
 convex or concave, and the distance from the point to which 
 the pencil converges, or from which it diverges, to the surface 
 of the lens, is called the focal length of the lens. 
 
 To findl practically the Focal Length of a Convex Lens .— 
 Place a lighted candle at one extremity of a scale of inches 
 and parts, with which the lens has been connected in such a 
 manner as to slide along, and always have its axis parallel to 
 the scale. A flat piece of card is also to be made to slide 
 along, so as to be always in a line with the light and the lens, 
 
FOCAL LENGTH OF LENSES. 
 
 73 
 
 the lens being between the light and the card. The lens and 
 card are then to be moved along, backwards and forwards, till 
 the least distance between the card and light is discovered at 
 which a clear image of the light is formed upon the card : and 
 this distance is four times the focal length. 
 
 The imperfection arising from the chromatic dispersion is 
 remedied, for the centrical pencil, by making a compound lens 
 of two or more lenses of different substances, as flint glass and 
 crown glass, which are fitted close together, and are of such focal 
 lengths that the chromatic dispersion of one is counteracted 
 by the chromatic dispersion of the other. The effect of the 
 chromatic dispersion upon an eccentrical pencil is remedied 
 by setting two or more lenses at proper distances depending 
 upon their focal lengths. Such a combination of lenses is 
 called an achromatic eye-piece. 
 
 When an object is placed at a distance from a convex lens 
 greater than its focal length, the divergent pencils of rays, 
 proceeding from every point of the object, become, after pass¬ 
 ing through the lens, convergent, and at a certain distance 
 from the lens *, having converged nearly to points again, form 
 there an inverted image of the object. The essential differ¬ 
 ence between any point in this image, and the corresponding 
 point in the object itself, is, that the latter emits light in all 
 directions, while the light from the former is limited to the 
 pencil which has been transmitted through the lens, and is 
 consequently determinate both in magnitude and direction. 
 If, however, a screen be placed at the required * distance from 
 the lens, a picture of the object in an inverted position will 
 be formed upon this screen, and from each point of this 
 picture light will be emitted in all directions in the same 
 manner as from the points of the object itself. The single 
 pencil of light from any point of the object, transmitted 
 through the lens, supplies, however, in this case, the light for 
 all the pencils emitted from the corresponding point of the 
 image; and a very strong light must therefore be thrown 
 upon the object to give a moderate brightness to the picture ; 
 more especially if the picture be of larger dimensions than the 
 
 * If u be the distance from the lens at which the object is placed,/the 
 focal length of the lens, then v, the distance from the lens at which the 
 
 111 
 
 image is formed, is determined from the equation — =-. The linear 
 
 v f u 
 
 magnitude of the image is to that of the object as v to u. 
 
 £ 
 
74 
 
 MATHEMATICAL INSTRUMENTS. 
 
 object. A portion of the light is also absorbed by the lens 
 itself. 
 
 REFLECTORS. 
 
 When a ray of light is reflected at a plane surface, the re¬ 
 flection takes place in a plane perpendicular to the reflecting 
 surface, and the incident and reflected rays make equal angles 
 -with this surface. Thus, if qa represent 
 a ray of light incident upon a piane re¬ 
 flector at the point a, and the plane of 
 the paper represent the plane which con¬ 
 tains q a, and is perpendicular to the re¬ 
 flecting surface, intersecting it in the line r a r', then making 
 the angle q'ar' in the plane qar' equal to the angle qar, aq' 
 will represent the course of the reflected ray. 
 
 The effect of a plane reflector upon the pencils of light which 
 fall upon it is to change the direction of all the rays forming 
 each pencil without altering the angles at which the several 
 rays of the pencil are inclined to one an¬ 
 other, so that the divergency or con- 
 vergency of the pencils remains the same 
 after reflection as before, and the objects 
 from which they proceed appear to be at the 
 same distances behind the mirror as they 
 really are in front of it. Thus, a pencil of light diverging 
 from a point of an object at p, after reflection at the point r 
 of a plane mirror, appears to proceed from the point p' on the 
 line pmp', perpendicular to the mirror, at the distance mp' 
 behind the mirror, equal to the distance m p. The point p', 
 from which, after reflection, the pencil appears to have di¬ 
 verged, is called a virtual focus; and the apparent image of the 
 object behind the mirror is called a virtual image. 
 
 The uses of a single plane reflector in mathematical instru¬ 
 ments are nearly the same as the uses of a prism: viz., either 
 to alter the apparent position of an object, so as to make its 
 visual image coincide with the real image of some other ob¬ 
 ject, as in the prismatic compass (described hereafter), or 
 merely to change the direction of the pencils for the purposes 
 of more convenient observation, as in the Newtonian telescope, 
 (see page 84), the diagonal eye-piece (see page 83). &c. 
 
 When a ray of light , proceeding in a plane at right angles 
 to each of two plane mirrors , which are inclined to each other 
 at any angle whatever, is successively reflected at the plane sur - 
 
REFLECTORS. 
 
 75 
 
 faces of each of the mirrors , the total deviation of the ray is 
 double the angle of inclination of the 
 mirrors . For let i i and h h represent 
 sections of the two mirrors made by 
 the plane of incidence at right angles 
 to each of them, and let si represent 
 the course of the incident ray: then the 
 ray s i is reflected at i into the direction 
 ih, making with i i the angle hi a, 
 equal to the angle si i, and is again 
 reflected at h into the direction h e, 
 making, with h li, the angle eh a equal 
 to the angle ih h. Now the angle ah v, 
 being equal to the exterior angle i h h, 
 is equal to the two angles hia and 
 h a i; and because the vertical angles 
 a v h and i v e are equal, and that the three angles of every 
 triangle are equal to two right angles, therefore the two angles 
 vie and s eh are, together, equal to the two angles ahv and 
 ha i, and therefore to the angle hia and twice the angle 
 ha i (since ahv has been proved equal to hia and hai); 
 but vie, being equal to the vertical angle si i, is equal to the 
 angle hia: therefore, taking away these equals, the remainder, 
 the angle seh, is equal to the remainder, twice the angle 
 hai. Q.E.D. 
 
 This property of two plane reflectors enables us by their 
 aid to measure the angle subtended at the eye by any two ob¬ 
 jects whatever, and is the foundation of the construction of 
 Hadley’s Quadrant, and the improvements upon it: viz., Had¬ 
 ley’s Sextant, and Troughton’s Reflecting Circle, hereafter to 
 be described. 
 
 Note .—Plane reflectors are usually made of glass silvered at the back; 
 and, as reflection takes place at each surface of the glass, there is formed a 
 secondary image, which must not be mistaken for the primary and distinct 
 image intended to be observed. 
 
 ON CURVILINEAR REFLECTORS. 
 
 Spherical reflectors , or specula, as they are called, produce 
 upon pencils of rays results precisely similar, with one ex¬ 
 ception, to those produced by lenses. Thus, a concave reflector 
 makes the rays of the pencils incident upon it more con¬ 
 vergent, and corresponds in its uses with a convex lens; while 
 a convex reflector makes the rays of the incident pencils more 
 divergent after reflection, and corresponds in its uses with a 
 
 e 2 
 
76 
 
 ft 
 
 MATHEMATICAL INSTRUMENTS. 
 
 concave lens. The exception to the similarity of the results 
 produced by lenses and reflectors is, that with the latter there 
 is no chromatic dispersion, and the only sources of error are 
 the aberration and spherical confusion, which are common 
 to both spherical reflectors and lenses. For astronomical 
 observations, however, in which case the rays incident upon 
 the object-speculum are parallel, these sources of error are 
 removed by making this speculum of a parabolic form, and 
 another speculum, if it be used, of the form of the vertex of a 
 prolate spheroid. There is great difficulty in procuring flint 
 glass in pieces of large size without flaws, and we are con¬ 
 sequently limited as to the size of the lenses of good quality 
 that can be formed with such glass; and, without its use, we 
 have not hitherto been able to form available achromatic ob¬ 
 ject-glasses. Recourse is, therefore, had to parabolic or 
 spherical specula in the formation of telescopes of large power 
 for examining the heavens*. These specula are formed of 
 metal, and the chief objection to them is the impossibility of 
 producing an accurate surface. Even supposing its general 
 form to be correct, there are always minute inequalities arising 
 from the nature of the substance, which cause a waste or dis¬ 
 persion of light. Great pains are, consequently, taken in 
 their construction to obtain the form and surface of the best 
 possible quality f. 
 
 MICROSCOPES. 
 
 The microscope is an instrument for magnifying minute, 
 but accessible objects. A convex lens is a microscope, but 
 the imperfections of such an instrument have been already 
 explained (p. 71), and the greater the power of the lens the 
 
 * Sir William Herschel’s largest telescope was 40 feet long, and the 
 mirror 4 feet wide. Lord Rosse’s largest telescope is 56 feet long, and the 
 mirror 6 feet wide. 
 
 t The following description of the methods employed in forming and 
 polishing parabolic reflectors is extracted in an abridged form from an 
 account of Skerryvore Lighthouse, by Alan Stevenson, LL.B. F.R.S.E. 
 M.I.C.E., Engineer of the Northern Light Board. 
 
 “ The reflector plate is formed of virgin silver and the purest copper, 
 from the ingot, in the proportion of 6 oz. silver to 16 oz. of copper. The 
 two metals are formed into pieces of the form of rectangular pnrallelopipeds 
 about 3 inches in length, and the same in breadth, and are then tied together 
 with wire, placed in the furnace, and united with a flux of burnt borax and 
 nitre, mixed to the consistence of cream. The metal thus united is re¬ 
 peatedly passed through the rolling mill, and annealed in the furnace after 
 each time of passing through, until it comes out a plate 28 inches square. 
 It is then cut into a circular disc ready for hammering; and great care must 
 
 
MICROSCOPES. 
 
 77 
 
 greater will De theso imperfections. For small magnifying 
 powers, then, convex lenses may be used, as they are lor spec- 
 
 be taken to keep the metal clean during the processes of hammering and 
 polishing now to be performed. 
 
 “The hammering is F{ L F{% F i g . & 
 
 commenced by placing 
 the plate with the cop¬ 
 per side upon a block 
 slightly concave, and 
 beating it on the inner 
 or silver side with a box¬ 
 wood mallet, rounded 
 at each end, c and d 
 (fig. 1). The beating 
 is commenced on the edge and continued round and round till the center is 
 gradually reached. After the disc has been raised sufficiently by this means, 
 it is taken to a machine called the horse, and beaten with a wooden mallet 
 upon the copper side, its concave face being turned about upon a bright steel 
 head a (fig. 2), until it has nearly reached the proper height for the reflector, 
 which is ascertained by a mould m (fig. 3). 
 
 “ After each course of raising with the wooden mallet the reflector is an¬ 
 nealed, as follows : first damped with clean water, and dusted over with 
 powder, composed of one pint of powdered charcoal to one ounce of saltpetre; 
 then put on a clear charcoal fire, till the powder Hies off and shows when it 
 is duly heated. It is next plunged into a pickle, composed of one quart of 
 vitriol in five or six gallons of water; and, lastly, washed with clean water 
 and scoured with Calais sand. 
 
 “ The next step is to put the reflector into an iron stool, and, having drilled 
 a small hole in its vertex, to describe a circle from this point with beam com¬ 
 passes, and cut the paraboloid to the proposed size. 
 
 “ The reflector is now hard-hammend with a planishing hammer, or 
 planished, as it is called, on the bright steel head a; and then smoothed 
 with a lighter hammer muffled with parchment. Then comes *the finishing, 
 called also, filling up to the mould, which is thus performed. It is con¬ 
 stantly tried with the mould m, and those portions which do not meet it are 
 marked with fine slate pencil, and then gone over with the muffled hammer, 
 till every point touches the mould. Great care must be taken in this pro¬ 
 cess that no part of the surface be raised above the gauge, or the reflector 
 would have to be re-formed with the wooden mallet, and the whole process 
 repeated. The reflector is then tried with a lamp brought to its focus, and, 
 if the whole surface is luminous, it is fit for polishing; but, if not, it must 
 be again tested by the mould, and carefully filled up with the muffled ham¬ 
 mer, till the result of the lamp test is perfectly satisfactory. 
 
 ** The edge of the reflector is next turned over to stiffen it, and the bizzle 
 w (fig. 1), and back belt g (fig. 2), having been soldered on, the final 
 process of polishing may be proceeded with. This process is commenced 
 by scouring, first with a piece of pure charcoal of hard wood, and next with 
 a mixture of Florence oil and finely-washed rotten stone, applied by means 
 of a large ball of superfine cloth. The reflector is then cleansed with a 
 fine flannel dipped in Florence oil, and afterwards dusted over with powder 
 of well-washed whiting, and wiped out with a soft cotton cloth. Lastly, 
 it is carefully rubbed by the naked hand with finely-washed rouge and 
 
78 
 
 MATHEMATICAL INSTRUMENTS. 
 
 tacles; but for obtaining good images with high magnifying 
 powers a combination of lenses must be used. 
 
 Small glass spheres are used as microscopes of high powers; 
 but a thin lens composed of any more highly refractive sub¬ 
 stance is preferable; because, the focal length of the sphere 
 measured from its center being but three semi radii, the dis¬ 
 tance of the object from the surface is only one semi-radius, 
 which prevents its being used in the examination of delicate 
 objects. The refracting sphere is much improved as a micro¬ 
 scope by cutting a groove round it in a diametrical plane, 
 and filling it up with some black opaque substance. By 
 this contrivance the aperture is diminished, without con¬ 
 tracting the field of view, and all the pencils are necessarily 
 centrical. 
 
 Microscopes have been made of diamond and sapphire, and 
 the aberration is much less than with glass. Dr. Brewster 
 
 clean water, and wiped with a smooth chamois skin. In all the polishing 
 and cleansing processes some skill is required in the manipulation, as the 
 hand must be moved in successive circles parallel to the lips of the reflector, 
 and having their centers on the axis of the generating curve.” 
 
 Fig. 1. Fig. 2. 
 
 The speculum of Lord Rosse’s great telescope is composed of 1264 parts 
 of copper and 589 of tin, fused together and cast in a mould, the bottom 
 of which is formed of hoop iron bound closely together with the edges up¬ 
 permost. By this means the heat is conducted away through the bottom 
 so as to cool the metal towards the top, while the interstices between the 
 hoops, though small enough to prevent the metal from running out, are suf¬ 
 ficiently open to allow the air to escape. After casting, the speculum is 
 annealed in a brick oven, which is heated almost to a red heat, and shut up 
 with the speculum in it, and allowed to cool gradually. The speculum is 
 then placed with its face upwards upon a turning apparatus, and the grind¬ 
 ing and polishing performed entirely by the aid of mechanical contrivances, 
 so that the proper parabolic form is accurately given to it. To test the work, 
 the dial-plate of a watch is placed upon the top of a mast at 90 feet distance 
 from the speculum, and the image of this dial-plate formed by the speculum, 
 being viewed through an eye-glass properly placed the distinctness cf this 
 image denotes the accuracy of the speculum. 
 
MICROSCOPES 
 
 T9 
 
 employed, as a microscope, a drop of Canada balsam or tur¬ 
 pentine varnish upon a thin plate of glass, of which the 
 surfaces were exactly parallel. This is a very ready way of 
 forming a plano-convex lens, and if kept free from dust will 
 last some time. 
 
 The compound or achromatic microscope consists of four 
 lenses and a diaphragm, placed in the following order : the 
 object-lens ; the diaphragm ; the amplifying lens, so called be¬ 
 cause it amplifies or enlarges the field of view; the field-lens; 
 and the eye-lens. The relations between the focal lengths 
 and intervals of the lenses, and the distance of the diaphragm 
 from the object-lens, are determined, so that the combination 
 may be achromatic, aplanatic, and free from spherical con¬ 
 fusion. The field-lens and eye-lens form what is called the 
 eye-piece; and the object-lens and amplifying lens form, or 
 tend to form, an enlarged image of the object, in the focus 
 of the eye-piece, which image is viewed through the eye¬ 
 piece. When the focus of the eye-piece is beyond the field- 
 lens, so that the image is formed between the amplifying lens 
 and the field-lens, the eye-piece is called a positive eye-piece; 
 but when the focus of the eye-piece is between the two lenses 
 of which it is composed, in which case its effect corresponds 
 with that of a concave lens, it is called a negative eye-piece . 
 With a negative eye piece the pencils proceeding from the 
 amplifying lens are intercepted by the field-lens before form¬ 
 ing an image, and the image is formed between the field-lens 
 and the eye-lens, in the focus of the latter. 
 
 The best microscopes are constructed with compound object- 
 lenses, which are both achromatic and aplanatic; and by this 
 means the aperture, and consequently the quantity of light, is 
 much increased. Good compound lenses possessing the re¬ 
 quired properties have been formed of a concave lens of flint 
 glass, placed between two convex lenses, one of crown glass, 
 and the other of Dutch plate. 
 
 The magnifying power of any refracting microscope or tele¬ 
 scope may be practically found, by pointing the object-end of 
 the instruments towards the light, and receiving the image of 
 the object-glass formed by the other lenses upon a screen 
 placed at the eye-end of the instrument, and at a proper dis¬ 
 tance from it, which may be determined by trial. Then the 
 ratio of the diameter of the object-glass, or of the diaphragm, 
 in the case of the compound microscope, to the diameter of its 
 image upon the screen, gives the magnifying power of the 
 telescope or microscope. In all microscopes it is necessary 
 
80 
 
 MATHEMATICAL INSTRUMENTS. 
 
 to illuminate the object strongly, in consequence both of the 
 diffusion of the small portion of light, received from the 
 object, over the magnified image, and of the absorption of the 
 light by the several lenses. 
 
 The Reflecting Microscope .—In this instrument a concave 
 speculum of short focal length is substituted for the object- 
 lens. The object is placed on one side of the axis of the 
 instrument, so that its perpendicular distance from the axis, 
 together with the distance from the speculum of the poin 
 where this perpendicular meets the axis, may be a littl 
 greater than the focal length of the speculum. A small plan 
 reflector is placed upon the axis of the instrument at th 
 point where the perpendicular from the object meets it. Thi 
 reflector is set at an angle of 45° to the axis, and having it 
 plane perpendicular to the plane through the object and tin 
 axis. The object being strongly illuminated, the pencils o 
 rays proceeding from it, after reflection at the plane reflectoi 
 and concave speculum, tend to form a magnified image, bu; 
 are intercepted by the field-glass of the negative achromatic 
 eye-piece, called the Huyghenian eye-piece (p. 82); and the 
 image formed after the transmission of the rays through the 
 field-glass is viewed through the eye-glass. 
 
 In the examination of small objects with a high power, it is 
 necessary that the microscope should be perfectly free from 
 all tremor, the slightest motion being so magnified as to pre¬ 
 vent a good view from being obtained. Regard must be had, 
 therefore, to solidity and accuracy in the fitting of all the 
 joints and screws: in the choice of an instrument, and for a 
 first-rate instrument, recourse should be had only to a maker 
 of well-known talent, as many so-called opticians are mere 
 sellers of articles of the qualities of which they are totally 
 ignorant. The adjustment of the eye-piece should be ob¬ 
 tained through the medium of a clamp and slow motion screw 
 of the best kind *, in which the screw acts upon a spiral 
 spring, and by means of which the adjustment for a good focus 
 may be obtained with the greatest possible accuracy, and 
 without the slightest tremor. If the workmanship and fittings 
 of the instrument appear to be satisfactory, a few test objects 
 should be examined with it, to try the quality of the combina¬ 
 tion of lenses. Two of the best test objects are the Podura 
 plombea, or Skiptail, a small wingless insect, the size of a 
 flea, found in damp cellars, and the Navicula Sigma, a small 
 
 * The best clamp, referred to in the text, is Dollond’s, or a modification of 
 Ibllond’s clamp. < •* 
 
TELESCOPES. 
 
 81 
 
 shell found in fresh water pools. The surface of the scales of 
 the Podura plombea should appear covered with a great 
 number of delicate marks, like notes of admiration. The 
 Navicula Sigma should appear completely chequered with a 
 number of longitudinal and transverse lines. Should the 
 instrument show these test objects well, it may at once be 
 deemed a good one. 
 
 TELESCOPES. 
 
 The Refracting Telescope consists of a convex object-glass, 
 which forms an image of a distant object, and an eye-piece of 
 one or more lenses, which per¬ 
 forms the office of a microscope /fff 
 
 for viewing this image. The 
 most simple form of the tele 
 scope is that called the astronomical telescope , and consists of 
 two convex lenses, the object-glass o, of as great focal length, 
 and, consequently, low magnifying power, as the size of the 
 telescope will permit, and the eye-glass e, of small focal length, 
 and, consequently, high magnifying power. When arranged 
 for distinct vision of a distant object, the distance between 
 the two lenses is equal to the sum of their focal lengths: an 
 inverted image, i, of the object is, consequently, formed in 
 the common focus of the two lenses, and the pencils pro¬ 
 ceeding from the image consist, after refraction at the eye¬ 
 glass, of parallel rays, which are the most favourable for 
 distinct vision. 
 
 The magnifying power of this instrument is represented by 
 the ratio of the focal length of the object-glass to that of the 
 eye-glass, and may therefore be increased either by increasing 
 the focal length of the object-glass, or by diminishing that of 
 the eye-glass. The latter means, however, cannot be resorted 
 to without increasing both the chromatic dispersion and the 
 spherical aberration. Hence, before the means were dis¬ 
 covered of forming achromatic and aplanatic object-glasses, 
 the only unobjectionable way of increasing the power of the 
 telescope was by increasing the focal length of the object-glass, 
 and astronomers used to attach the object-glass to the end of 
 a long pole. This contrivance was called an aerial telescope. 
 Huyghens used one of 123 feet in length, and Cassini one of 
 150 feet. 
 
 That the field of view should be as bright as possible, the 
 image of the object-glass formed by the eye-glass at the place 
 of the eye should not be larger than the pupil of the eye; and 
 
 e 3 
 
•V 
 
 82 MATHEMATICAL INSTRUMENTS. 
 
 the brightness will then vary directly as the square of the dia¬ 
 meter of the object-glass, and inversely as the square of the 
 magnifying power. The brightness is also diminished by 
 passing through the refracting media; and hence it is always 
 an object to employ as few lenses as possible, consistently with 
 the attainment of the other requisites of a good telescope. 
 
 Refracting telescopes for astronomical observations are now 
 constructed with achromatic object-glasses, and eye-pieces of 
 two lenses, called celestial eye-pieces, which are of one or the 
 other of the two following constructions : 
 
 1. The Huyghenian Eye-piece consists of two convexo-plane 
 lenses, with their plane sides, consequently, turned towards 
 the eye, their focal lengths and the interval between them 
 being as 3, 1, and 2. The lens of greatest focal length, /, is 
 
 next the object-glass, and is called the field-lens, because it 
 enlarges the field of view. When the telescope is arranged 
 for distinct vision of a distant object, the field-lens is placed 
 between the object-glass and its focus, at a distance from the 
 latter equal to half its own focal length. The pencils of rays 
 from the object-glass, tending to form an image at a distance 
 from the field-lens equal to three-fourths of the interval be¬ 
 tween the two lenses of the eye-piece, are intercepted by the 
 field-lens and brought sooner to a focus so as to form the 
 image i, half way between the two len?°-s, and consequently 
 in the focus of the eye-lens e. In this eye-piece the refrac¬ 
 tions of the axes of the pencils are equally divided between 
 the two lenses, by which the spherical confusion is much 
 diminished; the forms of the lenses are also such as to dimi¬ 
 nish the spherical aberration, and the relation between the 
 focal lengths of the lenses and the interval between them is 
 such as to satisfy the conditions of achromatism. This eye¬ 
 piece, called a negative eye-piece (p. 79), is always to be pre¬ 
 ferred, when we are only seeking to obtain the best defined 
 and most distinct view of an object, and is the best eye-piece 
 for all reflecting telescopes; but when it is necessary to place 
 cross-wires or spider-lines at the place of the image in the 
 field of view, for the purpose of accurately measuring the 
 position of an object, at the time of observation, or to apply an 
 apparatus, called a micrometer, for measuring the dimensions 
 
REFLECTING TELESCOPES. 
 
 83 
 
 of an image, the Huyghenian eye-piece can no longer be em¬ 
 ployed. 
 
 We have then recourse to Ramsdens Eye-piece , called a 
 Positive Eye-piece (p. 79). This consists of two lenses of equal 
 
 focal lengths, one piano-convex, and the other convexo-plane, 
 so that the convex sides are turned towards one another, the 
 interval between them being equal to two-thirds of the focal 
 length of either. When the telescope is arranged for distinct 
 vision of a distant object, the field-lens /, is placed at a dis¬ 
 tance from the object-glass o, greater than the focal length of 
 this glass by one-fourth of its own focal length. The focus of 
 the object-glass is then also the focus of the entire eye-piece, 
 and the rays proceeding from the image at i, emerge from the 
 eye-lens e, parallel, or in the condition best adapted for dis¬ 
 tinct vision. This eye-piece is not achromatic, but the spherical 
 aberration is less with it than with the Huyghenian eye-piece. 
 Whether the eye-piece be positive or negative, a diaphragm is 
 placed at the place of the image so as to intercept ail the ex¬ 
 traneous light. 
 
 With the eye-pieces of which we have been speaking, the 
 object appears inverted, which is no inconvenience when this 
 object is one of the heavenly bodies. These eye-pieces are 
 consequently called celestial eye-pieces. For the convenient 
 observation of stars near the zenith, a plane reflector or prism 
 is placed in the eye-piece, by which the directions of the 
 pencils are turned, so that the axis of the eye-lens is at right 
 angles to the axis of the instrument. Such an eye-piece is 
 called a diagonal eye-jiiece. 
 
 When terrestrial objects are to be viewed, it is generally 
 necessary that they should appear erect, for which purpose 
 the inverted image formed by the object-glass must be again 
 inverted by the eye-piece. The terrestrial, or erect eye-piece, 
 used for this purpose, is coincident with the compound micro¬ 
 scope already described (p. 79), consisting of an object-lens, a 
 diaphragm, amplifying lens, field-lens, and eye-lens, the two 
 latter forming either a negative or positive eye-piece. In con¬ 
 sequence of the loss of light consequent upon this construc¬ 
 tion, portable telescopes with celestial eye-pieces are used by 
 navigators for descrying objects at night, and these telescopes 
 are, consequently, called night-glasses. 
 
84 
 
 MATHEMATICAL INSTRUMENTS. 
 
 By substituting for the convex eye-lens of the astronomical 
 telescope a concave eye-lens of the same focal length, a simple 
 telescope is formed with only two lenses, which shows objects 
 erect. This is called the Galilean 
 telescope, and is the construction 
 used for opera glasses. When ~AL 
 arranged for distinct vision of a 
 distant object, the object-glass and eye-lens are separated 
 by a distance equal to their focal lengths. The pencils 
 of light proceeding from the object, after refraction at the 
 object-glass o, tend to form an image of the object in the 
 common focus of the two lenses; but, being intercepted by 
 the concave eye-lens e, their rays are rendered parallel, and, 
 consequently, adapted to produce distinct vision to an eye 
 placed behind this lens. 
 
 The magnifying power, as in the astronomical telescope, is 
 represented by the ratio of the focal length of the object-glass 
 to that of the eye-lens. 
 
 Reflecting Telescopes .—Since the discovery of the methods 
 of forming achromatic and aplanatic object-glasses, the magni¬ 
 tude and available magnifying powers of refracting telescopes 
 are theoretically unlimited ; but the difficulty of procuring 
 flint glass of even texture and free from flaws, in pieces of 
 any considerable magnitude, has hitherto practically placed a 
 limit upon the magnitude and available power of refracting 
 telescopes. By the substitution, however, of reflectors, which 
 are always achromatic, for the object-glasses, telescopes of 
 colossal magnitude have been most successfully constructed. 
 Of reflecting telescopes there are four kinds—the Newtonian, 
 the Gregorian, the Cassegrainian, and the Herschelian. 
 
 The Newtonian tele¬ 
 scope consists of a con¬ 
 cave object-speculum, s, 
 a plane reflector m, 
 making an angle of 45° 
 with the axis of the 
 telescope, placed between the object-speculum and its focus, 
 and an eye-piece. The pencils of light proceeding from a 
 distant object tend to form an image after reflection at the 
 object-speculum, />ut are bent by the plane reflector, so that 
 the image is formed at i, on the axis of the eye-piece, and in 
 the focus of the eye-lens. 
 
 The Gregorian telescope consists of a concave object-spe¬ 
 culum, s, a small concave speculum, r, whose focal length is 
 
REFLECTING TELESCOPES. 
 
 85 
 
 short compared 
 with that of the 
 object - specu¬ 
 lum, and an__ 
 
 eye-piece. The 
 
 small speculum is placed so that its focus is near the focus 
 of the object-speculum, but a little further from this speculum. 
 The pencils of light proceeding from a distant object, after 
 inflection at the object-speculum, form an inverted image, h, 
 of the object at the focus of this speculum, and after reflection 
 again at the small speculum form a second image, i, inverted 
 with respect to the former, and, consequently, erect with 
 respect to the object. 
 
 This telescope has, for terrestrial purposes, the advantage 
 over the Newtonian telescope, of showing objects erect, but 
 yields to it both in the brightness and perfection of the image, 
 because the second mirror increases the spherical aberration 
 produced by the first, and it is extremely difficult to give the 
 mirrors the proper curvature to remedy this evil. 
 
 The Cassegrainian telescope consists of two specula and an 
 eye-piece, like the Gregorian, but the second speculum is con¬ 
 vex instead of concave, and is placed between the object- 
 speculum and its principal focus, at a distance from this focus 
 somewhat less than its own focal length. The pencils of light 
 proceeding from a distant object, after reflection at the object- 
 speculum, tend to form an inverted image of the object, but 
 are intercepted, before doing so, by the convex speculum, and 
 made to form the image still inverted, in the focus of the eye- 
 lens. Objects, therefore, are still inverted; but the spherical 
 aberration of the convex speculum being opposite to that of 
 the concave object-speculum, the whole spherical aberration is 
 diminished. This telescope is also shorter than the Gregorian. 
 It is, however, inferior to the Newtonian telescope for celestial 
 observations, and not well adapted for terrestrial purposes on 
 account of the inversion of the ol ject. 
 
 When light is reflected at a mirror or speculum, there will 
 always be a waste and dispersion ; and in consequence of the 
 two reflections, and also of the light intercepted by the plane 
 mirror, or second speculum, the loss of light in all the re¬ 
 flecting telescopes hitherto described is considerable. Sir W. 
 Herschel, by a very simple contrivance, obtained what is called 
 the front view ; but this construction is only applicable to in 
 struments of very large dimensions. In the Herschelian tele¬ 
 scope the axis of the object-speculum, s, is slightly inclined to 
 
86 
 
 MATHEMATICAL INSTRUMENTS. 
 
 the axis of the tube, and the image i , being thus thrown to 
 one side of the tube, is there viewed by the eye-piece. 
 
 We shall now proceed to explain the best methods of 
 adjusting and testing telescopes, as given by Pearson in his 
 valuable work on Practical Astronomy. 
 
 Methods of Adjusting and Testing Refracting Telescopes. —Let us suppose 
 that we have a refracting telescope of 3j feet focal length, and inches 
 aperture. Then, to test the object-glass, lay the tube of the telescope in a 
 horizontal position upon some fixed support about the height of the eye, and 
 place a printed card vertically, but for a celestial eye-piece in an inverted 
 position, against some wall or pillar at thirty or forty yards' distance, so as 
 to be exposed to a clear sky; then, when the telescope is directed to this 
 object, and adjusted by the sliding tube for distinct vision, the letters on the 
 card should appear clearly and sharply defined, without any colouration or 
 mistiness ; and, if very small points appear well defined, the object-glass may 
 be deemed a pretty good one for viewing terrestrial objects. If the glass 
 be intended for astronomical observations, fix at the same distance a black 
 board, or one-half of a sheet of black paper, and a circular disc of whitepaper, 
 about a quarter of an inch or less in diameter, upon the center of the black 
 ground; then having directed the telescope to this object, and adjusted for 
 distinct vision, mark with a black-lead pencil the slicing eye-tube, at the 
 end of the main tube, so that this position can always be known ; and if this 
 sliding tube be gradually drawn out, or pushed in, while the eye beholds the 
 disc, it will gradually enlarge and lose its colour, till its edges cease to be well 
 defined. Now, if the enlarged misty circle is observed to be concentric with 
 the disc itself, the object-glass is properly centered, as it has reference to the 
 tube; but, if the misty circle goes to one side of the disc, the cell of the 
 object-glass is not at right angles to the tube, and must have its screws re¬ 
 moved, and its holes elongated by a rat-tailed file, small enough to enter the 
 holes. When this has been done, replace the cell, and examine the disc a 
 second time, and a siight stroke on the edge of the cell by a wooden millet 
 will show, by the alteration made in the position of the misty portion of the 
 disc, how the adjustment is to be effected, which is known to be right when 
 a motion in the sliding tube will make the disc enlarge in a circle concentric 
 with the disc itself. When, then, the disc will enlarge so as to make a ring 
 of diluted white light round its circumference, as the sliding tube holding the 
 eye-piece is pushed in, or drawn out, the cell may be finally fixed by the 
 screws passing through its elongated holes. When the object-glass is thus 
 adjusted, we can proceed to ascertain whether the curves of the respective 
 lenses composing the object-glass are well formed and suitable for each other. 
 
REFRACTING TELESCOPES. 
 
 87 
 
 If a small motion of the sliding tube of about one-tenth of an inch from the 
 point of distinct vision, in a Sj-feet telescope, will dilute the light of the disc 
 and render the appearance confused, the figure of the object-glass is good; 
 particularly if the same effect will take place at equal distances from the 
 point of good vision, when the tube is alternately drawn out and pushed in. 
 Such an object-glass is said to be aplanatic. A telescope that will admit of 
 much motion in the sliding tube without affecting sensibly the distinctness 
 of vision will not define an object well at any point of adjustment, and must 
 be considered as having an imperfect object-glass in which the spherical aber¬ 
 ration is not duly corrected. The achromatism of the object-glass is to be 
 judged of by the absence of colouration round the enlarged disc. When an 
 object-glass is free from imperfection both in respect of its aplanatism and 
 achromatism, it may be considered a good glass for all terrestrial purposes. 
 
 How far an object-glass is good for astronomical observations can only be 
 determined by actual observation of a heavenly body. When a good telescope 
 is directed to the Moon, or to Jupiter, the achromatism may be judged of by 
 alternately pushing in, and drawing out, the eye-piece, from the place of dis¬ 
 tinct vision; in the former case a ring of purple will be formed round the 
 edge; and, in the latter, a ring of light green, which is the central colour of 
 the prismatic spectrum; for these appearances show that the extreme colours, 
 red and violet, are corrected. Again, if one part of a lens employed have a 
 different refractive power from another part of it, that is, if the glass, parti¬ 
 cularly flint glass, be more dense in one part than another, a star of the first, 
 or even of the second magnitude will point out the natural defect by the 
 exhibition of an irradiation, or what opticians call a wing at one side, which 
 no peifection of figure or adjustment will banish; and, the greater the 
 aperture, the more liable is the evil to happen. 
 
 Another method of determining both the figure and quality of the object- 
 glass is by first covering its center by a circular piece of paper, as much as 
 one-half of its diameter, and adjusting it for distinct vision of a given object, 
 which may be the disc above mentioned, when the central rays are inter¬ 
 cepted, and then trying if the focal length remains unaltered, when the paper 
 is taken away, and an aperture of the same size applied, so that the extreme 
 rays may in their turn be cut off. If the vision remains equally distinct in 
 both cases, without any new adjustment for focal distance, the figure is good, 
 and the spherical aberration cured; and it may be seen, by viewing a star of 
 the first magnitude successively in both cases, whether the irradiation is pro¬ 
 duced more by the extreme, or by the central parts of the glass; or, in case 
 one-half of the glass be faulty and the other good, a semicircular aperture, by 
 being turned gradually round in trial, will detect what semicircle contains 
 the defective portion of the glass; and, if such portion should be covered, the 
 only inconvenience that would ensue would be the loss of so much light as is 
 thus excluded. 
 
 The smaller a large star appears in any telescope, the better is the figure 
 of the object-glass ; but, if the image of the star be free from wings, the size 
 of its disc is not an objection in practical observations, as it may be bisected 
 by the small line by which the measure is to be taken. When, however, an 
 object glass produces radiations in a large star, it is unfit for the nicer pur¬ 
 poses of astronomy. In testing a telescope, if a glass globe be placed at 40 
 yards distance when the sun is shining, the speck of light reflected from this 
 globe forms a good substitute for a large star, as an object to be viewed. 
 
 Whenever an object-glass is under examination, it will be proper to have 
 the object examined by it in the center of the field of view; and, when an 
 object-glass is tested for astronomical purposes by the methods described 
 
88 
 
 MATHEMATICAL INSTRUMENTS. 
 
 above, it is necessary to employ a good negative eye-piece, which generally 
 gives a better field of view than the positive. 
 
 If any fringes of red or yellow are observed on the edges of a white disc 
 placed on a black ground, when the telescope is adjusted for distinct vision, 
 and the disc carried too near the edges of the field, this species of colouration 
 indicates that the eye-piece is not sufficiently free from spherical aberrations ; 
 and, if the curves of the lenses are suitable for each other, the cure is effected 
 by an alteration in the distance between them, which mus be finally adjusted 
 by trial with a good object-glass. 
 
 Methods of Adjusting and Testing Reflecting Telescopes .—To adjust 
 the specula of a Cassegrainian or Gregorian instrument procure a Ramsden's 
 eye-piece, which will render an object visible in the compound focus of the 
 two lenses of which it is composed ; then hold this eye-piece in front of the 
 Iduyghenian eye-piece of the telescope, and, by varying the distance, find the 
 position in which the image of the large speculum is seen, well defined through 
 both eye-pieces, and, if the image of the small speculum is seen precisely on 
 the center of the large one, the metals may be considered as rightly placed ; 
 but, if not, the proper screws must be used in succession, till the required 
 position is determined. When the face of the large metal stands at right 
 angles to the length of the tube, the adjustment may generally be finished 
 without disturbing it; and, when the bed that receives it has once been pro¬ 
 perly finished, it will be advisable not to alter it, unless some accident should 
 render such alteration indispensable. 
 
 To try whether the figures of the metals are adapted for each other.—Let 
 the instrument be directed to some luminous point, as a white disc on a black 
 ground, or, what is better, to a star : then having adjusted for distinct vision, 
 firstly observe if the disc or star is well defined, and free from irradiations; 
 secondly, carrying the small speculum short distances beyond, and short of, 
 the place for distinct vision, examine if the disc or star enlarges alike in 
 similar changes of position : if the result be satisfactory, the metals may be 
 considered as well placed, and well adapted for each other. 
 
 To try whether the large speculum partake of the parabolic form, let the 
 aperture be partially covered, first at the central part, and then round the 
 circumference by tin, pasteboard, or stiff paper; and if on trial the same 
 adjustment for distinct vision be good in both these cases, and also when the 
 speculum is all exposed, the figure may be considered good. If these effects 
 be not produced, the instrument will be incompetent to perform several of the 
 nicer observations in astronomy. When a mistiness appears in the field, it 
 is a proof that the aberrations are not corrected, and that the figure of at 
 least one of the specula is not perfect. 
 
 If a telescope is not good with its full aperture, its effect may be greatly 
 improved, by putting a cover on the mouth, with a circular aperture, of 
 about one-half the diameter that the tube has, in such a way that the dimi¬ 
 nished aperture may fall entirely at one side of the opening of the tube. 
 
 THE SOLAR MICROSCOPE. 
 
 In this instrument the object itself is not viewed through 
 a combination of lenses, as in the microscopes already de¬ 
 scribed (pp. 76-81), but a magnified image of the object is 
 formed by a combination of lenses, and received upon a screen. 
 The term solar is applied to the instrument, because the light 
 of the sun, concentrated by a lens, is made use of to illuminate 
 
CAMERA OBSCUIiA. 
 
 89 
 
 the object to be observed, and the construction is in all other 
 respects identical with the common magic lantern, and the 
 oxy-hydrogen microscope. In the case of the microscope, how¬ 
 ever, whether illuminated by the sun or the brilliant oxy- 
 hydrogen light, great regard must be had to the forms of the 
 lenses and the perfection of the setting; while a comparatively 
 very rough instrument forms a very amusing toy as a magic 
 lantern, exhibiting grotesque figures and scenes, which are 
 painted in transparent colours upon glass slides. 
 
 The arrangement of the apparatus will be understood from 
 the annexed diagram; r is a reflector for turning the sun's 
 
 rays in a direction parallel to the axis of the instrument: c 
 is the lens for concentrating these rays upon the object placed 
 at o, a little further from the first lens, p, of the magnifier, 
 than the focal length of this magnifier, which is one-fourth the 
 focal length of p ; then we have p and in, the two lenses form¬ 
 ing the magnifier, which are of equal focal length, and sepa¬ 
 rated by an interval equal to two-thirds of the common focal 
 length, as in Ramsden’s positive eye-piece: lastly comes the 
 diaphragm, d , placed at a distance from in, the second lens 
 of the magnifier, equal to the focal length of this magnifier, 
 which is one-fourth the focal length of in or p. 
 
 The best forms of the two lenses are, for the first, a plano¬ 
 convex, and, for the second, a convex meniscus, the radii of 
 whose surfaces are as 1 to 15; and the advantage aimed at 
 in this construction is to render the image flat, and conse¬ 
 quently capable of coinciding with the plane screen upon which 
 it is to be received. A similar purpose is the object of the 
 construction of Ramsden’s eye-piece, viz., to obtain, as it is 
 there called, a flat field. 
 
 The object being placed a little further from p than the 
 focal length of the magnifier, the pencils of rays from each 
 point of the object, after passing through the two lenses, be¬ 
 come slightly convergent, and, at a distance from the diaphragm 
 depending upon the distance of the object from the lens p, the 
 magnified image is formed inverted with respect to the object. 
 
 THE CAMERA OBSCURA. 
 
 This instrument consists of a plane reflector, upon which 
 
90 
 
 MATHEMATICAL INSTRUMENTS. 
 
 pencils of light from the various points of a landscape are re¬ 
 ceived and reflected, so as to pass first through a diaphragm, 
 and then through a plano-convex lens, after which the rays of 
 the pencils become convergent, and form an image upon a 
 screen in a darkened chamber placed to receive it. The dia¬ 
 phragm and lens are placed in a tube, which is passed through 
 a hole in the chamber just large enough to receive it, so that 
 no extraneous light mav be admitted. The distance of the 
 lens from the diaphragm is determined upon the condition 
 that the image shall be distinct. The form of the screen also, 
 that the image may be distinct, is a paraboloid of revolution, 
 or figure formed by the revolution round its axis of a para¬ 
 bola, whose radius of curvature at the vertex is /*/,/* being the 
 refracting power of the medium of which the lens is formed, 
 and / the focal length of the lens. A curved surface of this 
 form is, therefore, made of plaster of Paris, , and placed at a 
 distance from the lens rather greater than the focal length, 
 the exact distance depending upon the nearness or remote¬ 
 ness of the landscape to be depicted, and being easily found 
 by trial. If the camera be set up in the neighbourhood of a 
 well-frequented thoroughfare, we have then an agreeable suc¬ 
 cession of distinct and vividly-coloured pictures, differing from 
 finely-executed paintings only by exhibiting the actual motious 
 of the objects viewed, men walking, horses trotting, soldiers 
 marching, banners streaming, and foliage shaking in the 
 breeze. 
 
 THE CAMERA LUCIDA. 
 
 This ingenious instrument, the invention of Dr. Wollaston, 
 consists of a quadrilateral prism, of which abcd represents a 
 section made by a plane at right angles 
 to each of its edges, mounted upon an 
 axle parallel to its edges. This axle 
 is attached to the end of a rod sliding 
 in a tube, which has at the other end 
 a clamp for fixing it to the edge of a 
 table, so that the distance of the prism 
 from the table can be shortened or 
 lengthened at pleasure, a b is equal 
 to b c, and a d to d c, and the angles of the prism are a right 
 angle at b, an angle of 135° at d, and angles each 67° 3u' 
 at a and c. Over the face b a, and projecting beyond a, is a 
 plate of metal having in it a narrow longitudinal aperturo, 
 which is just bisected by the edge a of the prism. 
 
SU UVEY1NG INSTRUMENTS. 
 
 91 
 
 The axis q r, of a small pencil of light from an object q, 
 directly in front of the face b c, passes straight through this 
 face, and falls upon the face d c, making with it an angle of 
 22° 30'. It is ihere reflected * into the direction r s, and falling 
 upon the face d a, at the same angle, is again reflected into 
 the direction st, perpendicular to the face ab, and conse¬ 
 quently passes straight through this face without refraction. 
 Looking down through the aperture in the metal plate, an image 
 of the object q is seen at p, at a distance from a b equal to 
 the distance of the object itself from b c; and if a b be placed, 
 by means of the sliding rod before mentioned, at a distance 
 from the table equal to the distance of the object from the 
 prism, and a sheet of paper be laid upon the table at p, the 
 apparent place of the object, as seen through the prism, 
 will coincide with the actual place of the paper, seen through 
 the projecting part of the aperture, and an accurate drawing 
 of the object may be traced upon the paper. If the object q 
 be distant, its image may be brought nearer, and thus made 
 to coincide with the place of the paper, by placing a concave 
 lens before the face b c of the prism. 
 
 PART III.—SURVEYING INSTRUMENTS. 
 
 Surveying instruments may be divided into three classes: 
 1. Instruments for measuring distances. 2. Instruments for 
 measuring angles. 3. Instruments for laying down the survey 
 upon paper, or, as it is called, plotting the survey. 
 
 Under the first of these classes we propose to describe— 
 
 1. The chain. 
 
 2. The spirit level and levelling staves. 
 
 * "When a ray of light passes from a denser into a rarer medium it is re¬ 
 fracted farther from the perpendicular to the refracting surface, so that, if 
 <p he the angle which the ray in the denser medium makes with the perpen¬ 
 dicular to this surface, and <p' the angle wiiich the ray in the rarer medium, 
 after refraction, makes with the same perpendicular, ft sin. <p — sin. <p r , the 
 refracting power tt being greater than unity. If, then, the angle <p be in¬ 
 creased, the angle <p' is also increased, and becomes a right angle, when <p 
 
 becomes equal to the angle whose sine is equal . The ray then is re- 
 
 fracted directly along the surface, and neither emerges, nor is reflected ; but, 
 if <p be still farther increased, the ray of light is reflected back into the 
 denser medium, according to the ordinary law of ri flection. With ordinary 
 crown glass, for which ft = ij, this takes place when <p exceeds 41° 49', or 
 the ray makes with the surface an angle less than 4S° 11'. 
 
92 
 
 MATHEMATICAL INSTRUMENTS. 
 
 Under the second we shall include— 
 
 1. The prismatic compass. 3. The optical square. 
 
 2. The box sextant. 4. The theodolite. 
 
 And under the third, in addition to the instruments already described in 
 Part I. of this Work, we shall say something of— 
 
 1. The large circular protractor. 
 
 2. The T square and semicircular protractor. 
 
 3. The best form of plotting scale. 
 
 4. The station pointer. 
 
 THE LAND CHAIN. 
 
 Gunter’s chain is the instrument used almost universally 
 for measuring the distances required in a survey. For exten¬ 
 sive and important surveys, however, such as those carried on 
 under the Board of Ordnance, a base of about 5 or 6 miles 
 in length is first measured by some more accurate instru¬ 
 ment, and all the principal lines, and the distances of the 
 extreme points, are calculated from triangles connecting them 
 with this base. An instrument, which has been known to 
 answer well for this purpose is a steel chain 100 feet long, 
 constructed by Ramsden, jointed like a watch chain. This 
 chain is always stretched to the same tension, supported on 
 troughs laid horizontally, and allowances are made for changes 
 in its length made by temperature, at the rate of -0075 of an 
 inch for each degree of heat from 62° of Fahrenheit. 
 
 To return, however, to Gunter's chain;—it is 66 feet, or four 
 poles in length, and is divided into 100 links, which are joined 
 together by rings. The length of each link, together with the 
 
 . . , , 66 x 12 
 
 rings connecting it with the next, is consequently - ^ —• 
 
 inches, or 7 - 92 inches. To every tenth link are attached 
 pieces of brass of different shapes for more readily counting 
 the links in distances less than a chain. 
 
 The following tables exhibit the number of chains and links 
 in the different units of lineal measure, and the number of 
 square chains and links in the different units of square mea¬ 
 sure, made use of in this country :— 
 
 A TABLE OF LINEAR MEASURES. 
 
 Links. 
 
 25 
 
 Feet. 
 
 16* 
 
 Y ards. 
 
 5i 
 
 Poles. 
 
 1 
 
 
 100 
 
 66 
 
 22 
 
 4 
 
 Chains. 
 
 1 
 
 1,000 
 
 660 
 
 220 
 
 40 
 
 10 
 
 8,000 
 
 5,280 
 
 1,760 
 
 320 
 
 80 
 
 Furlongs. 
 
 1 
 
 8 
 
 Mile 
 
 1 
 
USE OF THE LAND CHAIN 
 
 93 
 
 A TABLE OF SQUARE MEASURES. 
 
 Sq. Links. 
 
 Sq. Feet. 
 
 Sq. Yards. 
 
 Sq. Pole, 
 or Perch. 
 
 
 
 
 625 
 
 272f 
 
 301 
 
 1 
 
 Sq. Ch. 
 
 
 
 10,000 
 
 4,356 
 
 484 
 
 16 
 
 1 
 
 Roods. 
 
 
 25,000 
 
 10,890 
 
 1,210 
 
 40 
 
 2] 
 
 1 
 
 Acres. 
 
 100,000 
 
 43,560 
 
 4,840 
 
 160 
 
 10 
 
 4 
 
 1 
 
 64,000,000 
 
 2,787,400 
 
 3,097,600 
 
 102,400 
 
 6,400 
 
 2,560 
 
 640 
 
 Sq.Mile. 
 
 1 
 
 As, then, an acre contains 100,000 square links, if the con¬ 
 tent of a survey, cast' up in square links, be divided by 100,000, 
 the quotient gives at once the content in acres, and decimals 
 of an acre. But the division by 100,000 is performed by 
 merely pointing off the five last figures towards the right 
 hand for the decimals of an acre, and the remaining figures 
 towards the left hand are the acres in the content required. 
 
 The decimals thus pointed off being then multiplied by 4, 
 and the five last figures pointed off as before, the 1175 
 remaining figures are the roods; and the five deci- H75 
 mals cut off from this product, multiplied by 40, 
 give the poles, or perches, and decimals of a pole, 
 the same number, 5, of digits being again pointed 
 off, including the zero, which arises from the mul¬ 
 tiplication by 40. Thus, if the side of a square 
 field measured 11 chains, 75 links, or 1175 links, 
 the area of the field would contain 1175 x 1175, or 
 1,380,6*25 square links, which is equivalent to 
 
 13-80625 acres. Then *80625 acres is equivalent- 
 
 to-80625 x 4, or 3-22500 roods; and, again, *22500 9-00000 
 
 roods is equivalent to *22500 x 40, or 9-00000 poles. The 
 field consequently would measure 13 acres, 3 roods, 9 poles. 
 
 Ten arrows must be provided with the chain, about 12 
 inches long, pointed at one end, so as to be easily pressed 
 into the ground, and turned at the other end, so as to form a 
 ring to serve for a handle. 
 
 In using the chain marks are first to be set up at the ex¬ 
 tremities of the line to be measured. Two persons are then 
 required to perform the measurement. The chain leader starts 
 with the ten arrows in his left hand, and one end of the chain 
 in his right, while the follower remains at the starting point, 
 and, looking at the mark, or staff, at the other extremity of 
 the line to be measured, directs the leader to extend the 
 
 5875 
 
 8225 
 
 12925 
 
 13-80625 
 
 4 
 
 3*22500 
 
 40 
 
04 
 
 MATHEMATICAL INSTRUMENTS. 
 
 chain in the direction of this mark. The leader then puts 
 down one of his arrows, and proceeds a second chain’s length 
 towards the end of the line, while the follower comes up to 
 the arrow first put down. A second arrow being now put 
 down by the leader, the first is taken up by the follower, and 
 the same operation is repeated till the leader has expended 
 all his arrows. Ten chains, or 1000 links, have now been 
 measured, and, this measurement having been noted in the 
 field book, the follower returns the ten arrows to the leader, 
 and the same operations are repeated. When the leader 
 arrives at the end of the line, the number of arrows in the fol¬ 
 lower’s hand shows the number of chains measured since the 
 last exchange of arrows, noted in the field book, and the 
 number of links extending from the last arrow to the mark, or 
 staff, at the end of the line being also added, gives the 
 entire measurement of the line. Thus, if the arrows 9000 
 have been exchanged 9 times, and if the follower have 
 
 4 arrows, and from the arrow last laid down to the end _ 
 
 of the line measure 63 links, the whole measurement 9463 
 will be 9463 links. 
 
 To assist in preserving the line straight, as well as to serve 
 for a check upon the number of chains measured, it is a good 
 method to set up a staff at each ten chains, when the arrow's 
 are exchanged. 
 
 In using the chain care must be taken to stretch it always 
 with the same tension, and, as it will give when much used, it 
 must occasionally be examined, and shortened if necessary. 
 
 When the ground over which the measurement is taken 
 rises or falls, or both alternately, the horizontal distances are 
 what we require for plotting the survey, and not the actual 
 distances measured along the line of the ground. 
 
 For many ordinary purposes the horizontal measurement 
 may be obtained by holding one end of the chain up, so as to 
 keep it, as nearly as can be judged, horizontal, the arrow being 
 placed vertically under the end so held up; but, when a more 
 accurate survey is required, the distances must be measured 
 along the line of ground, and, the angles of elevation and de- 
 pressibn of the several inclined parts of the line being after¬ 
 wards taken with the theodolite, or the vertical risings and 
 fallings being taken by the process of levelling with the spirit 
 level and staves, the correct horizontal distances must thence 
 be computed. The following table shows the number of links 
 to be subtracted from every chain, or 100 links, for the angles 
 there set down, being in fact the versed sines of those angles 
 
USE OF THE LAND CHAIN. 
 
 95 
 
 to a radius .of 100. The correction for each 100 links, for any 
 angle whatever, may at once be taken from a table of natural 
 versed sines, by considering the first two figures as integers. 
 The correction may also be taken from a table of natural co¬ 
 sines, by subtracting each of the first four figures from 9, and 
 reckoning the first two figures as integers, and the last two as 
 decimals: thus, to find the correction for an inclination of 8° 19', 
 take the first four figures of the cosine of 8° 19', which will 
 be 9894, and, subtracting each of these four figures from 9, 
 we obtain 0105: then, considering the first two figures of this 
 result as integers, and the last two as decimals, we have T05 
 for the correction due to the inclination 8° 19' for every 100 
 links. If the last figure in the correction thus found be in¬ 
 creased by 1, whenever the fifth figure of the cosine is less than 
 5, the result will be more accurate. 
 
 Table showing the Reduction in Links and Decimals of a Link upon 100 
 Links for every half Degree of Inclination from 3° to 20° 30'. 
 
 Angle. 
 
 Reduction. 
 
 Angle. 
 
 Reduction. 
 
 Angle. 
 
 Reduction. 
 
 3° O' 
 
 0-15 
 
 9° O' 
 
 1-23 
 
 15° 0' 
 
 3-41 
 
 30 
 
 0 19 
 
 30 
 
 1-37 
 
 30 
 
 3-64 
 
 4 0 
 
 0-24 
 
 10 0 
 
 1-53 
 
 16 0 
 
 3-87 
 
 30 
 
 031 
 
 30 
 
 1-67 
 
 30 
 
 442 
 
 5 0 
 
 0-38 
 
 11 0 
 
 1-84 
 
 17 0 
 
 4-37 
 
 30 
 
 0-46 
 
 30 
 
 2-01 
 
 30 
 
 4-63 
 
 6 0 
 
 0.55 
 
 12 0 
 
 2-19 
 
 18 0 
 
 4-89 
 
 30 
 
 0-64 
 
 30 
 
 2-37 
 
 30 
 
 5*17 
 
 r 0 
 
 0-75 
 
 13 0 
 
 2-56 
 
 19 0 
 
 5-45 
 
 30 
 
 0.86 
 
 30 
 
 2-76 
 
 30 
 
 5-74 
 
 8 0 
 
 0-97 
 
 14 0 
 
 2-97 
 
 20 0 
 
 6-03 
 
 30 
 
 1-10 
 
 30 
 
 319 
 
 30 
 
 633 
 
 The advantage of Gunter’s chain is its adaptation to the 
 superficial measure of land in acres, &c.; but, when a survey 
 is to be made for the purpose of linear measurements only, or 
 when it may be more convenient to compute the area in 
 square feet, a chain 100 feet long, divided into links of a foot 
 long, is to be preferred. Such a chain is best adapted to 
 military surveying. 
 
 Offsets, perpendicular to the main line, to hedges and re¬ 
 markable objects on either side of it, are measured from the 
 chain as it lies stretched upon the ground, by means of an 
 offsetting staff. This staff should be 10 links in length, and 
 
MATHEMATICAL INSTRUMENTS. 
 
 % 
 
 divided into links. With Gunter’s chain the staff, then, will 
 be 6 - 6 feet, or 6 feet 7'2 inches long, while with the 100 feet 
 chain it will be 10 feet in length. 
 
 THE SPIRIT LEVEL. 
 
 Certain parts of the capital instruments used in surveying, 
 and in astronomical observations, require to be adjusted in truly 
 horizontal positions; and, to arrive at this adjustment, one or 
 more subsidiary instruments, called spirit levels, are attached 
 to such principal instruments. The spirit level attached to a 
 good telescope, furnished with a compass, and such means of 
 correct adjustment as we shall presently describe, becomes 
 also itself a capital instrument, being used in that depart¬ 
 ment of surveying, termed levelling, which consists in 
 measuring the vertical distances between various stations. 
 
 The spirit level consists of a glass tube, differing from the 
 cylindrical form by having its diameter largest in the middle, 
 and decreasing slightly and with great regularity from the 
 middle to the ends. The tube is nearly but not quite filled 
 with spirits of wine, thus leaving in it a bubble of air, b b, which 
 rises to the highest part of the tube, so as to have its two ends 
 equally distant from the middle, when the instrument is in 
 adjustment, as represented in the annexed figure. 
 
 The tube is generally fitted into another tube of 
 metal, and attached to a frame terminating in 
 angular bearings, by which the level can either be suspended 
 from, or else be stood upon, cylindrical pivots. When, how¬ 
 ever, the level forms a permanent part of any instrument, 
 the manner of attaching it is modified to suit the particular 
 form of the instrument to which it is attached. A small and 
 accurately-divided scale is attached to the best instruments, 
 or otherwise a scale is scratched upon the glass tube itself, as 
 represented in the figure given above. 
 
 The annexed figure 
 is a representation of 
 such a level as is used 
 for levelling the axis of 
 the best astronomical 
 instruments. It is 
 provided with a fixed 
 scale, seen in the figure, and is suspended by means of ac¬ 
 curately constructed angular bearings. 
 
 The following criteria of a good level are extracted from 
 
97 
 
 THE Y LEVEL. 
 
 Dr. Pearson’s valuable work on Practical Astronomy, before 
 referred to. 
 
 " Firstly, the bubble must be long enough, compared with the whole tube, 
 to admit of quick displacement, and yet not too long to admit of its proper 
 elongation by low temperature. 
 
 “ Secondly, the curve must be such, that the sensibility and uniform run 
 of the bubble will indicate quantities sufficiently minute, while those quan¬ 
 tities correspond exactly to the changes of inclination, as read on the graduated 
 limb of the instrument of which it forms a part. 
 
 “ Thirdly, the bubble must keep its station when the angles are moved a 
 little round the pivots of suspension. 
 
 ** Fourthly, the opposite ends of the bubble must vary alike in all changes 
 of temperature, or, in other words, the ends of the bubble must elongate or 
 contract alike in opposite directions, so that the middle point may always be 
 stationary. 
 
 “ Fifthly, the angles of the metallic end-pieces must be so nicely adjusted 
 that reversion on horizontal pivots that are equal will not alter the place of 
 the bubble. 
 
 “ Sixthly, the distance between the two zeros of a fixed scale, when such 
 a graduated scale is used, should be equal to the length of the bubble at the 
 temperature of 60° of Fahrenheit’s scale, and should be marked at equal dis¬ 
 tances from the visible ends of the glass tube. Then, as the bubble 
 lengthens by cold, or shortens by heat, its extreme ends may always be 
 referred to these fixed marks, 0 0, on the scale, and will fall either with¬ 
 in, upon, or beyond them, according to the existing temperature. The num¬ 
 ber of subdivisions of the scale that each end of the bubble is standing at, 
 counted from the fixed zero marks, at the instant of finishing an observation, 
 must always be noted, that an allowance may be made for the value of the de¬ 
 viation in seconds, or as the case may require. 
 
 “ Seventhly, when the two ends of the bubble are not alike affected by a 
 change of temperature, the scale should be detached, and adjustable to the 
 new zero points, by an inversion of the level. 
 
 "* Eighthly, when the scale has only one zero at its center, which is a mode 
 of dividing the least liable to misapprehension, the positions must be reversed 
 at each observation, and both ends of the bubble read in each position ; for 
 in this case, if any change has taken place in the true position of this zero, 
 the resulting error will merge in the reduction of the observation. This mode 
 of graduating is generally practised on the continent.” 
 
 We proceed now to the description of the roost accurate in¬ 
 struments for measuring the differences of level, or vertical dis¬ 
 tances, between different stations. 
 
 Of spirit levels for this purpose there are now three ip use, 
 namely, the Y level, Troughton’s improved level, and Gravatt’s 
 level. 
 
 THE Y LEVEL. 
 
 The following figure represents this instrument, a is an 
 achromatic telescope, resting upon two supporters, which in 
 shape resemble the letter Y, and are consequently called the 
 Ys. The lower ends of these supporters are let perpendicu- 
 
 F 
 
98 
 
 MATHEMATICAL INSTRUMENTS. 
 
 larly into a strong bar, which carries a compass box, c. This 
 compass box is convenient for taking bearings, and has a con¬ 
 trivance for throwing the needle off its center, when not in 
 use. One of the Y supporters is fitted into a socket, and can 
 be raise or lowered by the screw b. 
 
 Beneath the compass box, which is generally in one piece 
 with the bar, is a conical axis passing through the upper of 
 two parallel plates, and terminating in a ball supported in a 
 socket. Immediately above this upper parallel plate is a 
 collar, which can be made to embrace the conical axis tightly 
 by turning the clamping screw e, and a slow horizontal mo¬ 
 tion may then be given to the instrument by means of the 
 tangent screw d. The two parallel plates are connected to¬ 
 gether by the ball and socket already mentioned, and are set 
 firm by four milled-headed screws, which turn in sockets fixed 
 to the lower plate, while their heads press against the under 
 side of the upper plate, and thus serve the purpose of setting' 
 the instrument up truly level. 
 
 Beneath the lower parallel plate is a female screw, adapted 
 to the staff-head, which is connected by brass joints with three 
 mahogany legs, so constructed as, when shut together, to form 
 one round staff, a very convenient form for portability, and, 
 w ? hen opened out, to make a very firm stand, be the ground 
 ever so uneven. 
 
 The spirit level 11 is fixed to the telescope by a joint at 
 one end, and a capstan-headed screw at the other, to raise or 
 depress it for adjustment. 
 
 In looking through a telescope a considerable field of view 
 is embraced; but the measurements indicated by any instru- 
 
ADJUSTMENT OF COLLIGATION. 
 
 90 
 
 m'ent, of which the telescope may form a part, will only have 
 reference to one particular point in this field of view, which 
 particular point is considered as the center of this field of 
 view. We must therefore place some fixed point in the field 
 of view, and in the focus of the eye-piece, and the point to 
 which the measurement will have reference will be that point 
 of the object viewed, which appears to be coincident with this 
 fixed point, or which, as the technical phrase is, is bisected 
 by the fixed point. 
 
 The intersection of two fixed lines will furnish us with such 
 a fixed point, and consequently two lines of spider’s thread 
 are fixed at right angles to each other in the focus of the 
 eye-piece. They are attached by a little gum to a brass ring 
 of smaller dimensions than the tube of the telescope, and which 
 is fixed to the tube by four small 
 screws, a, b, c, d . If the screw d 
 be eased, while at the same time c 
 is tightened, the ring will be 
 moved to the right; but, if c be 
 eased and d tightened, the ring 
 will be moved to the left; and in 
 a like manner it may be moved up 
 or down by means of the screws 
 a and b. 
 
 When the instrument is in adjustment, the axis of the tube 
 of the telescope is set truly horizontal by means of the level 
 beneath it, and the line of observation ought consequently to 
 be parallel to this axis. Let A represent the proper position 
 
 of the intersection of the cross _ 
 
 wires, and o a, the direction jg l— Q 
 
 of the axis of a pencil of light A \ 1 
 
 passing through the object- 
 
 glass and coming to its focus at a. Then, the axis of the 
 tube of the telescope being set truly horizontal, the line a o 
 is also truly horizontal, and every point bisected by the inter¬ 
 section of the cross wires will be situated on the prolongation 
 of the horizontal line a o. 
 
 Suppose now the position of the diaphragm carrying the cross 
 wires to have become deranged, so that the point of inter¬ 
 section is moved to b, then every point bisected by the inter¬ 
 section of the cross wires will be on the prolongation of the 
 line b o, and will consequently be below the true level point 
 on the line a o. 
 
 ‘ Let now the telescope be turned half round' in the Ys, and 
 
 f 2 
 
 
100 
 
 MATHEMATICAL INSTRUMENTS. 
 
 let the annexed figure represent it in its new position ; then, 
 
 in this new position of the 
 
 telescope, the prolongation 
 
 of the line b o will rise above & 
 
 the prolongation of the level 
 
 line ao, and, at the same distance from the telescope, the 
 point now bisected by the intersection of the cross wires w r ill be 
 as much above the true level point on the line a o as the point 
 before bisected by them was below it. The true level point is 
 therefore midway between the two points observed in the two 
 positions of the telescope, and the diaphragm carrying the 
 cross wires is to be moved by means of the screws a, b, c, d, 
 till their point of intersection coincides with that true 
 level point. The telescope is then to be again turned round 
 upon the Ys, and, if the same point be still bisected by the 
 intersection of the cross wires, they are in their proper posi¬ 
 tion ; but, if not, the same method of adjustment must be re¬ 
 peated till the same point is bisected by the intersection of the 
 cross wires in every position of the telescope. 
 
 This error of derangement has a technical denomination. 
 The line o a, or o b, from o to the point of intersection of the 
 cross wires, is called the line of collimation, and the error 
 arising from their derangement, which we have shown the 
 method of detecting and correcting, is called the error of 
 collimation. 
 
 When the image of the object viewed, formed by the object- 
 glass, either falls short of, or beyond the place of the cross 
 wires, the error arising from this cause is called parallax. 
 The existence of parallax is determined by moving the eye 
 about when looking through the telescope, observing whether 
 the cross wires change their position, and are flittering and 
 undefined. 
 
 To correct this error, first adjust the eye-piece, by means of 
 the moveable eye-piece tube, till you can perceive the cross 
 wire clearly defined, and sharply marked against any white 
 object. 
 
 Then by moving the milled-headed screw a, at the side of the 
 telescope, the internal tube a is thrust outwards or drawn in¬ 
 wards, until you obtain the properfocus, according to the distance 
 of the object, and you are enabled at once to see clearly the 
 object, and the intersection of the wires, clearly and sharply 
 defined, before it. The existence of parallax is very incon¬ 
 venient, and, where disregarded, has frequently been productive 
 of serious error. It will not always be found sufficient to set 
 
ADJUSTMENTS OF BUBBLE-TUBE, AND OF AXIS. 101 
 
 the eye-glass first, and the object-glass afterwards. The set¬ 
 ting of the object-glass, by introducing more distant rays of 
 light, will affect the focus of the eye-glass, and produce parallax 
 or indistinctness of the wires, when there was none before; 
 the eye-piece must, in this case, be adjusted again. 
 
 Generally, when once set for the day, there is no occasion 
 for altering the eye-glass, but the object-glass will of course 
 have to be altered at every change of distance of the object. 
 
 In adjusting the instrument, the parallax should be first 
 corrected, and then the error of collimation. The line of colli- 
 mation being thus brought to coincide with the axis of the tube of 
 the telescope, twofurt.heradjustments are necessary : the first to 
 adjust the bubble-tube, so that it may truly indicate when the 
 axis of the telescope is horizontal; and the second to set the 
 axis of the telescope perpendicular to the vertical axis round 
 which the instrument turns. 
 
 To adjust the Bubble-Tube. —Move the telescope till it lies 
 in the direction of two of the parallel plate screws, and by 
 giving motion to these screws bring the air bubble to the 
 center of its run. Now reverse the telescope carefully in the 
 Ys, that is, turn it end for end; and, should the bubble not 
 settle at the same point of the tube as before, it shows that 
 the bubble-tube is out of adjustment, and requires correcting. 
 The end to which the bubble retires must then be noticed, 
 and the bubble made to return one-half the distance by turn¬ 
 ing the parallel plate screws, and the other half by turning 
 the capstan-headed screw at the end of the bubble-tube. The 
 telescope must now again be reversed, and the operation be 
 repeated, until the bubble settles at the same point of the 
 tube, in the center of its run, in both positions of the instru¬ 
 ment. The adjustment is then perfect, and the clips which 
 serve to confine the telescope in the Ys should be made fast. 
 
 Lastly, to set the Axis of the Telescope perpendicular to the 
 Vertical Axis round which the Instrument turns. — Place the 
 telescope over two of the parallel plate screw's, and move them, 
 unscrewing one while screwing up the other, until the bubble 
 of the level settles in the center of its run; then turn the 
 instrument half round upon the vertical axis, so that the con¬ 
 trary ends of the telescope may be over the same two screws, 
 and, if the bubble does not again settle at the same point as 
 before, half the error must be corrected by turning the screw' 
 B, and the other half by turning 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, 
 
102 
 
 MATHEMATICAL INSTRUMENTS. 
 
 and repeat the process to bring these two screws also into 
 adjustment; and when, after a few trials, the bubble main¬ 
 tains exactly the same position in the center of its run, while 
 the telescope is turned all round upon the axis, this axis will 
 be truly vertical, and the axis of the telescope, being hori¬ 
 zontal by reason of the previous adjustment of the bubble- 
 tube, will be perpendicular to that vertical axis, and remain 
 truly horizontal, while the telescope is turned completely round 
 upon the staves. The adjustment is therefore perfect. 
 
 The object of the above adjustments is to make the line of 
 collimation move round in a horizontal plane, when the in¬ 
 strument is turned round its vertical axis, and the methods 
 above explained suppose that the telescope itself is constructed 
 with the utmost perfection, so that the axis of the tube carry¬ 
 ing the object-glass is always in the same straight line with 
 the axis of the main tube, which carries the diaphragm with the 
 cross wires. If this perfection in th^ construction of the in¬ 
 strument does not exist, the line of collimation will vary, as 
 the tube carrying the object-glass is thrust out, and drawn in, 
 to adjust the focus for objects of different distances. What is 
 really required, then, is that the cross wires be so adjusted 
 that the line of collimation may be in the same straight line 
 with the line in which the center of the object-glass is moved, 
 and that the bubble of the level be at the center of its run, when 
 this line of collimation is directed to view objects, at the same 
 level, or at the same distance from the center of the earth. 
 
 We are indebted to Mr. Gravatt, of whose level we shall 
 hereafter speak, for a method of collimating, which satisfies 
 the above requirements, and removes any error arising from 
 imperfection in the slide of the telescope, while at the same 
 time the line of collimation is set with the end at the object- 
 glass, slightly depressed, instead of exactly horizontal, so as 
 to remove, or nearly so, the errors arising from the curvature 
 of the earth, and the horizontal refraction. 
 
 To examine and correct the Collimation by Mr. Gravatt's Method .—“On 
 a tolerably level piece of ground drive in three stakes at intervals of about 
 four or five chains, calling the first stake a, the second b, and the third c. 
 
 “ Place the instrument half way between the stakes a and b, and read the 
 staff A, placed on the stake a, and also the staff b, placed on the stake b ; call 
 the two readings, a' and b'; then, although the instrument be out of adjustment *, 
 
 * The axis of the instrument is to be set vertical by means of the parallel 
 plate screws, by placing the telescope over each pair alternately, and moving 
 them, until the air bubble remains in the same position, when the instru¬ 
 ment is turned half round upon its axis. - 
 
gravatt’s adjustment. 
 
 100 
 
 .yetThe points read off will be equidistant from the earth's .center, and con¬ 
 sequently 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 read¬ 
 ing on c, we get three points, say a', b', and c', equidistant from the earth's 
 center, or in the same true level. 
 
 “ Place the instrument at any short distance, say half a chain beyond it, 
 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', o '. Now, if the stake b 
 be half way between a and c * , then ought o"—o'— (a" — a') to be equal to 
 2 [b" — b'—(a" — a')] ; but if not, alter the screws which adjust the diaphragm, 
 and consequently the horizontal spider line, or wire, until such be the case; 
 and then the instrument will be adjusted for collimation. 
 
 “ To adjust the spirit bubble without removing the instrument, read tlje 
 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 center 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 -j-J^th of a foot.” 
 
 Before making observations with this instrument, the ad¬ 
 justments should be carefully examined and rectified, after 
 which the screw b should never be touched; but at each station 
 the parallel plate screws alone should be used for setting the 
 axis round which the instrument turns truly vertical, when, 
 in consequence of the adjustments previously made, the line 
 ofcollimation will be truly level. For this purpose the tele¬ 
 scope must be placed over each pair of the parallel plate 
 screws alternately, and they must be moved till the air 
 bubble settles in the middle of the level, and the operation 
 being repeated till the telescope can be turned quite round 
 upon the staff-head, without any change taking place in the 
 position of the bubble, the instrument will be ready to read 
 off the graduations upon the levelling staves, which we pro¬ 
 ceed to describe. 
 
 * Whatever be the distances between the stakes a, l, and c, the follow¬ 
 ing proportions ought to hold, viz.: — 
 
 The distance from a : b : the distance a to c :: b" — b' — (a"—a') : c" — o' 
 
 + If this adjustment be made by the screw b, instead of the parallel plate 
 screws, the line of collimation will be brought into its proper position with 
 respect to the vertical axis. 
 
 i 
 
104 
 
 MATHEMATICAL INSTRUMENTS. 
 
 The best constructed levelling 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. 
 
 In all work where great accuracy is required, the Y level* 
 above described, is preferable to either of the others; but both 
 Troughton’s level and Gravatt’s level are calculated, by their 
 lightness, and by their being less liable to derangement when 
 once properly adjusted, to get rapidly over the ground 
 
 troughton’s level. 
 
 In this level the telescope, T, rests close down upon the 
 horizontal bar, b b, the spirit level, 11, is permanently fixed to 
 the top of the telescope, and does not, therefore, admit of ad¬ 
 justment, and the compass box, c, is supported over the level 
 by four small pillars attached to the horizontal bar. This 
 construction makes the instrument very firm and compact. 
 The staves, staff-head, and parallel plates by which the in¬ 
 strument is supported, and the vertical axis upon which it 
 turns, are of exactly the same construction as has been 
 already described as used for supporting the Y level. 
 
 The diaphragm is furnished with three threads, two of 
 them vertical, between which the levelling staff may be seen, 
 and the third, horizontal, gives the reading of the staff by its 
 coincidence with one of the graduations marked upon it. Some¬ 
 times a pearl micrometer-scale is fixed on the diaphragm, in¬ 
 stead of the wires. The central division on the scale, then, 
 indicates the collimating point, and by its coincidence with a 
 division of the levelling staff gives the required reading of 
 
 * This staff was first introduced into use by William Gravatt, Ssq. 
 
 
GRAY ATT S 1.EVEL. 
 
 105 
 
 this staff; and the scale serves the purpose of measuring dis¬ 
 tances approximately, and of determining stations nearly equi¬ 
 distant from the instrument, since at such equal distances the 
 staff will subtend the same number of divisions upon the mi¬ 
 crometer-scale. 
 
 In selecting a level of Troughton’s construction, and also 
 in testing and adjusting the collimation subsequentl}% Mr. 
 Gravatt’s method, already described, is the best to be used; 
 and, when the line of collimation is thus brought into adjust¬ 
 ment, if the bubble be far from the center of its run, the fault 
 can only be remedied by the maker; but, if the bubble settle 
 very nearly in the center of its run, the instrument may be 
 deemed a good one, and, the divisions on the glass tube which 
 coincide with the ends of the bubble being noted, the instru¬ 
 ment must be set up for use with the bubble in this position. 
 
 The line of collimation is set perpendicular to the vertical 
 axis, in the same manner as in the Y level, by means of the 
 capstan screws, b b, the bubble being made to maintain the 
 requisite position, as above determined, while the instrument 
 is turned completely round on its axis. 
 
 MR. GRAVATTS LEVEL. 
 
 This instrument is furnished with an object-glass of large 
 aperture and short focal length ; and, sufficient light being 
 
 thus obtained to admit of a higher magnifying power in the 
 eye-piece, the advantages of a much larger instrument are 
 obtained, without the inconvenience of its length. The dia¬ 
 phragm is carried by the internal tube a a , which is nearly 
 equal in length to the external tube. The external tube t t is 
 sprung at its aperture, and gives a steady and even motion to 
 the internal tube a a, wdiich is thrust out, and drawn in, to 
 adjust the focus for objects at different distances by means of 
 the milled-headed screw a. The spirit level is placed above 
 the telescope, and attached to it by capstan-headed screws, 
 
 f 3 
 
106 
 
 MATHEMATICAL INSTRU ME NTS. 
 
 one at either end, by means of which the bubble can he 
 brought to the center of its run, as in the case of the Y level, 
 when the line of collimation is brought to the proper level 
 by Mr. Gravatt’s method of adjustment, already explained. 
 
 The telescope is attached to a horizontal bar, in a similar 
 manner to Troughton’s level, but room is just left between 
 the telescope and the bar for the compass-box. 
 
 A cross level, k, is placed upon the telescope at right angles 
 to the principal level 11 , by which we are enabled to set the 
 instrument up at once with the axis nearly vertical. A mirror, 
 m, mounted upon a hinge-joint, is placed at the end of the level 
 11 , so that the observer, while reading the staff, can at the 
 same time see that the instrument retains its proper position 
 — a precaution by no means unnecessary in windy weather, or 
 on bad springy ground. 
 
 The telescope is attached to the horizontal bar by capstan- 
 headed screws, bb, as in Troughton’s level, by which the lino 
 of collimation is set perpendicular to the vertical axis; and 
 the instrument is set up upon parallel plates, as before de¬ 
 scribed, for the Y level. 
 
 The operation of determining the difference of level be¬ 
 tween two stations by observations made at a single station is 
 called simple levelling, and is performed as follows:— 
 
 Let a and h be two points whose difference of level is re¬ 
 quired. Plant the instrument at d, and adjust it to a hori¬ 
 zontal position. Read off a c, the height of a staff held at a, 
 then turn the telescope round and read off e h, the height of 
 a similar staff, or of the same staff held at h. Then a c is the 
 height of k, the axis of the telescope above the point a, and 
 e h is the height of k above another point, h ; and it is 
 clear that eh — ac = gh, the difference of level between a, 
 and h. 
 
 In this operation the station a, at which the staff is first read 
 off, is called the back station, and the station h is called the 
 fore station; and, if the reading of the staff at the back station 
 be greater than that at the fore station, the difference of level 
 is called a rise; but, if the reading at the back station be less 
 
LEVELLING. 
 
 lor; 
 
 than (hat at the fore, as in the example just given, the differ¬ 
 ence of level is called a fall. > 
 
 -When, from the nature of the ground, or the great distance 
 between the two points, they cannot both be observed from a 
 single spot, a series of simple levels must be taken, the fore 
 station at each operation being made the back station at 
 the next operation; and from the combination of all the re¬ 
 sults thus obtained the required difference of level is obtained. 
 In these operations care must be taken, in going over soft 
 ground, lest the staff at the fore station, when turned round 
 to be read as the staff at the back station in the next opera¬ 
 tion, should sink further into ground; and, to prevent this, 
 the foot of the staff must be placed upon a flat, hard substance, 
 as a piece of slate or tile. There is a simple instrument called 
 a tripod, sold for this purpose by the instrument makers, being 
 simply a plate of iron with a small rounded projection in the 
 center, two small spikes at the side to fix it in its place, and 
 a short chain to lift it by, when the staff-holder wishes to re¬ 
 move from his place. 
 
 In determining by this method the difference of level be¬ 
 tween two distant points, it is immaterial by what route we 
 proceed from one to another, so that such spots may be selected 
 for the intermediate stations as are most convenient for the 
 purpose. The bearings of the stations from the instrument 
 are also matter of indifference; but, the more nearly the in¬ 
 strument is equidistant from the two stations observed at each 
 operation, the more correct will be the result obtained, the 
 errors in the back readings compensating, for the most part, 
 the errors in the fore readings, whether the errors arise from 
 refraction* and curvaturef, or from the imperfect adjustment 
 of the instrument. 
 
 If, then, the object be only to obtain the difference of level 
 of two points, we have only to record in two separate columns 
 the readings of the staff at the back stations and fore stations 
 respectively, and the difference of the sums of these readings 
 
 * The error of refraction is that arising from the bending of the rays of 
 light during their passage through the atmosphere, and makes all objects 
 appear higher than they really are. « 
 
 t The object of levelling is to determine points upon a spherical surface 
 or equally distant from the earth’s center, or to determine the differences of 
 the distances of a series of points from the earth’s centre. The line of sight, 
 or prolongation of the line of collimation, however, is a tangent to the spheri¬ 
 cal surface, and therefore the points observed upon this line are really above 
 the level of the point of observation. The correction for curvature is there¬ 
 fore additive, while that for refraction is subtractive. 
 
108 
 
 MATHEMATICAL INSTRUMENTS. 
 
 will be the difference of level required.—Thus, if the differ¬ 
 ence of level between two points a and b, be required, and if 
 the readings at a and b, and three intermediate stations © 1, 
 © 2, © 3, be recorded as follows, viz.:— 
 
 Back © 
 
 Feet. 
 
 Beading of staff at A . . 3 65 
 ,, © 1 . . 2 05 
 
 „ © 2 . . 3-89 
 
 „ 0 3.. 5*28 
 
 14-87 
 
 Then 22-18 feet is the fall from 
 
 Fore 0 
 
 Feet. 
 
 5 80 Reading of staff at 0 1 
 8-50 „ © 2 
 
 8-40 „ © 3 
 
 14-35 „ b 
 
 37-05 
 
 14-87 
 
 22-18 
 
 to b, or A is 22-18 feet above b. 
 
 When, however, it is required not only to find the difference 
 of level between two distant points, but to make such obser¬ 
 vations as shall enable us to draw a section exhibiting the un¬ 
 dulations of the ground along some specified route from the 
 one point to the other, then the stations must be so chosen that 
 one of them shall be at the commencement of each change in 
 the inclination of the ground; the distances between the sta¬ 
 tions must also be carefully measured; and it is further advis¬ 
 able to note the distances and bearings of the stations from 
 the instrument, which it will be more convenient now to place 
 on a point in the specified route between the stations. 
 
 In drawing the section, it is the horizontal distances be¬ 
 tween the several stations that must be laid down. For 
 short distances, or over very irregular ground, such horizontal 
 measurements may be obtained by bidding an assistant hold 
 one end of a measuring tape close to the ground at the 
 highest end of the distance, and holding the other end above 
 the ground, stretching the tape in a horizontal line, a stone 
 let fall from this end then marking upon the ground the point 
 to which the measurement reaches. But, when the ground 
 rises and falls in long regular slopes, the measurements should 
 be taken along the slopes, and then be reduced to horizontal 
 distances by calculation. If the rise or fall is but slight, this 
 reduction r» ay be altogether disregarded, the difference between 
 the horizontal and hypothenusal measurements not exceeding 
 the limits of error in the measure itself. 
 
 Care should be taken to record all the observations in a 
 clear and intelligible form, and for this purpose a field book 
 may be prepared of the following form •— 
 
FIELD BOOK 
 
 100 
 
 
 Distance 
 
 to 
 
 stations. 
 
 Bearings. 
 
 Staff 
 
 readings. 
 
 Heights 
 
 above datum. 
 
 Reduction. 
 
 1 
 
 Reduced horizontal i 
 
 distances. 1 
 
 1 
 
 REMARKS. 
 
 1 
 
 | From start- 
 | ing point. 
 
 From in¬ 
 struments. 
 
 X 
 
 o 
 
 a 
 
 Q 
 
 6 
 
 o 
 
 B ck 0 
 
 feet. 
 
 feet. 
 
 210 
 
 300-00 
 
 100-00 
 
 3-65 
 
 
 feet. 
 
 100-00 
 
 
 
 Back © 300 feet 
 from hedge, wind-i 
 mill bearing 125»: 
 from instrument, 
 church-spirebear¬ 
 ing 223-5°. 
 
 Fore 0 
 
 46’» 
 
 250 
 
 12010 
 
 
 5 80 
 
 3*65 
 
 103-65 
 
 5-80 
 
 9785 
 
 
 
 Back © 
 
 460 
 
 180 
 
 300-00 
 
 2-05 
 
 
 97-85 
 
 
 
 Road to lime kilns. 
 
 Fore 0 
 
 320 
 
 780 
 
 140 
 
 120-00 
 
 
 8-50 
 
 2-05 
 
 90- 90 
 8-50 
 
 91- 40 
 
 
 
 Back © 
 
 780 
 
 180 
 
 209*75 
 
 3-89 
 
 
 91-40 
 
 
 
 
 Fore 0 
 
 380 
 
 1160 
 
 200 
 
 119-40 
 
 
 8-40 
 
 3-89 
 
 95-29 
 
 8-40 
 
 86-89 
 
 
 
 Back 0 
 
 1160 
 
 180 
 
 300-25 
 
 528 
 
 
 8689 
 
 
 
 
 Fore © 
 
 360 
 
 1520 
 
 180 
 
 120-00 
 
 
 14-35 
 
 5-28 
 
 92-17 
 
 14-35 
 
 77-82 
 
 
 
 Back © 
 
 1320 
 
 300 
 
 300-00 
 
 12-25 
 
 
 77-82 
 
 
 
 1 
 
 1 
 
 Bottom of canal] 
 distant 150 feet. 
 
 Fore ® 
 
 
 
 
 
 15-78 
 
 12-25 
 
 90-07 
 
 1578 
 
 74-29 
 
 ! 
 
 
 
 Back © 
 
 
 
 
 15-78 
 
 
 74-29 
 
 
 
 
 Fore © 
 
 580 
 
 2100 
 
 280 
 
 120"00 
 
 
 9*21 
 
 1578 
 
 90-07 
 
 9-21 
 
 80-86 
 
 
 
 Back © 
 
 2100 1 205 
 
 300-15 
 
 11-05 
 
 
 80*86 
 
 
 
 
 Fore © 
 
 400 
 
 2500 
 
 195 
 
 120-no 
 
 
 M2 
 
 11-05 
 
 91-91 
 
 1-12 
 
 90-79 
 
 
 
 
 
 2500 
 
 ! 
 
 i 
 
 153 95' 
 63-161 
 
 63*16 
 
 
 
 
 
 90-791 
 
 In the first column are entered the distances between the 
 several stations, which, being successively added to the pre¬ 
 ceding total, give the total distances of each station from the 
 starting point: in the next column are entered the distances 
 of the stations from the instrument; and in the third are 
 
no 
 
 MATHEMATICAL INSTRUMENTS. 
 
 entered the hearings of the stations from the instrument. In 
 the fourth and fifth columns are entered the readings, of the 
 staves ; and in the sixth column the heights above datum of 
 the several stations are computed by adding the back reading 
 to the height last found, and subtracting the fore reading from 
 the sum. The seventh and eighth columns are added for per¬ 
 forming the reduction of the measured distances to horizontal 
 distances, when the slope is sufficient to render this reduction 
 necessary. In carrying forward the distances to the next 
 page of the book, the total reduced horizontal distance should 
 be carried to the top of the first and second columns instead 
 of the total measured distance along the slope; but such sub¬ 
 stitutions should not be made at any other part of the page, 
 as it would interfere with the proof of the distances by adding 
 up the second column, which ought to produce the last dis¬ 
 tance entered in the first. The levels are proved by subtract¬ 
 ing the sum of the numbers in the sixth column from the 
 sum of the numbers in the fifth, when the remainder should 
 be the height above datum of the last station recorded at the 
 bottom of the page. 
 
 To facilitate the reduction of the measured distances to the 
 corresponding horizontal distances, the following table show¬ 
 ing the reduction upon each 100 feet for each foot difference 
 of level should be inserted in the field book:— 
 
 Difference of 
 Level for 100 
 feet distance. 
 
 Reduction 
 upon 100 feet 
 of distance. 
 
 Difference of 
 Level for 100 
 feet distance. 
 
 Reduction 
 upon 100 feet 
 of distance. 
 
 Difference of 
 Level for 100 
 feet distance. 
 
 Reduction 
 upon 100 feet 
 of distance. 
 
 4 
 
 0-08 
 
 13 
 
 0-85 
 
 22 
 
 2-45 
 
 5 
 
 0-13 
 
 14 
 
 0-98 
 
 23 
 
 2 68 
 
 6 
 
 0-18 
 
 15 
 
 1-13 
 
 24 
 
 2-92 
 
 7 
 
 0 25 
 
 16 
 
 1*29 
 
 25 
 
 3T8 
 
 8 
 
 0-32 
 
 17 
 
 1-46 
 
 26 
 
 344 
 
 9 
 
 0-41 
 
 18 
 
 T63 
 
 27 
 
 3*71 
 
 10 
 
 0-50 
 
 19 
 
 1-82 
 
 28 
 
 4*00 
 
 11 
 
 0-61 
 
 20 
 
 202 
 
 29 
 
 4-30 
 
 12 
 
 0-72 
 
 21 
 
 2 23 
 
 30 
 
 4-61 
 
 When it is required to plot the section on a large scale,' 
 and to show every undulation of the surface, it is not neces¬ 
 sary to remove and re-set the instrument to obtain the height, 
 above datum of every point necessary to be known for this 
 purpose; but, besides reading the staff at the back and fore 
 station, it may be read off from the same place of the instru¬ 
 ment, at as many intermediate points as may be deemed 
 
BENCH MARKS. 
 
 m 
 
 desirable; and these readings, being entered both as back and 
 fore readings, will produce the same effect as back and fore 
 readings of the same points obtained in different positions of 
 the instrument. The distances from the instrument of these 
 points should be omitted from the second column; but, the 
 distances between them being entered successively in the first 
 column, their respective distances from the instrument may at 
 any time be determined, if required. The height of the in¬ 
 strument itself may be entered in this way as an intermediate 
 sight; and, as the same height that is added as a back reading 
 is subtracted again as a fore reading, any error in this reading 
 will not at all affect the levels afterwards taken, and, provided 
 it be not greater than the limit within which distances can be 
 laid down and estimated upon the plot, is of no moment. 
 Now, in taking the section of a line of any considerable extent, 
 the scale is seldom sufficiently large to admit of less than six 
 inches being laid down or estimated upon the plot, and conse¬ 
 quently an error of two or three inches in the intermediate 
 sights would be immaterial. When observations are made out 
 of the line, to be levelled, in order, for instance, to obtain the 
 height of this line above neighbouring rivers, canals, roads, 
 &c., the readings are to be entered in the same manner as 
 for other intermediate sights; and, the column of bearing and" 
 distance being left blank, no mistake can be made in drawing 
 the section. The bearing and distance of such points, if 
 desirable to be noted, must be entered in the space left for 
 remarks. 
 
 For the purpose of reference on any future occasion, in 
 order either to check the accuracy of the levels already ob¬ 
 tained, or for the convenience of commencing a new series 
 in some other direction, marks should be left upon some con¬ 
 venient fixed points upon which the staff has been held, and 
 the reading noted with the greatest possible care. These bench 
 marks, as they are called, should ordinarily be left at about 
 every half-mile of distance, and may be either on or off the 
 line. In the latter case the readings are to be recorded in 
 the manner already explained for points out of the line. The 
 hooks and tops of gates, copings, sills, or steps of doors, &.c ., 
 are commonly used for bench marks, and the mark must be 
 made exactly on the point upon which the staff has been held. 
 A stout stake may be driven into the ground for a bench mark, 
 and is by many persons preferred to any other. 
 
 When a section of considerable length is to be plotted, the 
 horizontal distances cannot be laid down on as large a scale 
 
m 
 
 MATHEMATICAL INSTRUMENTS. 
 
 as is necessary for the vertical heights above datum, in order 
 that the section may be of any practical use, without making 
 the plot of most unwieldy dimensions. It is therefore usual 
 to make the vertical scale much larger than the horizontal 
 one: thus 4 inches to a mile for the horizontal distances, 
 with one inch to 100 feet for the vertical distances, is a usual 
 combination. In the accompanying figure we have drawn the 
 portion of a section from the portion of the field book at page 
 109, making use of a scale of 1 inch to 800 feet for the hori¬ 
 zontal distances, and of a scale of 1 inch to 200 feet for the 
 vertical distances 
 
 A g is ruled for the datum line; on it are set off from a, 
 the horizontal distances at the points b, c, d, e, f, g, according 
 to the horizontal scale of 1 inch for each 800 feet, and through 
 the points a, b, c, d, e, f, and g, are drawn lines a a, b b, &c., 
 perpendicular to ag; on these lines are set off the vertical 
 distances to the points a , b, c, &c., according to the vertical 
 scale of 1 inch for each 200 feet; and the line a g, passing 
 through all the points a, b, c , &c., will represent the required 
 section. A line is drawn between the stations e, f, at the 
 proper distance from the datum line to represent the level of 
 the canal; and proceeding in this manner, and making any 
 remarks that may seem desirable, opposite the corresponding 
 points of the section, the work will be completed. 
 
 Having now explained the construction and use of the most 
 accurate instruments for tracing the level of any portion of 
 country, we proceed to notice the water level, a very simple in¬ 
 strument, adapted to give a rapid delineation of any portion of 
 country, an object frequently of greater importance than ac¬ 
 curacy. It can be made by any workman, will cost but a few 
 shillings, and requires no adjustment when using it. 
 
 “ a b is a hollow tube of brass, about half an inch in diameter, 
 
WATER LEVEL 
 
 113 
 
 and about 3 feet long; c and d are short pieces of brass tube 
 of larger diameter, into which the long tube is soldered, and 
 are for the purpose of receiving the two small bottles, e and/, 
 the ends of which, after the bottoms have been cut off, by 
 tying a piece of string round them when heated, are fixed in 
 their positions by putty or white lead; the projecting short 
 
 axis, g, w'orks (in the instrument from which the sketch was 
 taken) in a hollow brass cylinder, h, which forms the top of a 
 stand used for observing with a repeating circle; but it may 
 be made in a variety of ways, so as to revolve on any light 
 portable stand. The tube, when required for use, is filled with 
 water (coloured with lake or indigo), till it nearly reaches to 
 the necks of the bottles, which are then corked for the con¬ 
 venience of carriage. On setting the stand tolerably level by 
 the eye, these corks are both withdrawn, which must be done 
 carefully, and when the tube is nearly level, or the w r ater 
 will be ejected with violence; and the surface of the water in 
 the bottles, being necessarily on the same level, gives a hori 
 zontal line in whatever direction the tube is turned, by which 
 the vane of a levelling staff is adjusted.” 
 
 The instrument, however, with which observations upon 
 the level of a country may be most 
 expeditiously made, and generally with 
 greater correctness than with the water 
 level, is the reflecting level. This in¬ 
 strument consists merely of a piece of 
 common looking-glass, l l, one inch 
 square, set in a frame fixed against a 
 plate of metal weighing about a pound, 
 and suspended from a ring, r, by a 
 twisted wire, w, so that it may swing 
 freely, but not turn round on its axis 
 of suspension. A fine silk thread, tt, 
 is stretched across the center of the 
 mirror, and a small opening, o, at one 
 side of the mirror. 
 
MATHEMATICAL INSTRUMENTS. 
 
 3T4 
 
 The instrument is adjusted as follows It is suspended ip 
 a frame, constructed to hold it, and bring it soon to rest, at 
 about 50 yards in front of a wall. The observer looks into 
 the mirror, and brings his eye into such a position that its 
 image is bisected by the silk thread, t t; and the point upon 
 the wall, seen through the opening, o, which coincides with 
 the silk thread, is marked upon the wall. The mirror‘is then 
 turned round, and the point is also marked upon the wall, the 
 reflection of which in the mirror now coincides with the silk 
 thread, when this thread again bisects the image of the ob¬ 
 server’s eye as before. 
 
 Lastly, the middle point, between the two thus found, is 
 marked upon the wall; and by turning a screw, s, the 
 center of gravity of the instrument is altered, till the mirror 
 hangs so as to bring the reflection of this last mark upon the 
 thread, when the observer’s eye is bisected by it. The - in¬ 
 strument will now be in perfect adjustment, and, when the 
 image of the eye is brought upon the thread, all points bi¬ 
 sected by the thread, whether seen by reflection, or directly 
 through the opening, o, will be on the same level with the 
 eye of the observer. The observations may be made either 
 by holding the instrument at arm’s length, or by suspending 
 it from the branch of a tree, or from any post or rail of a con¬ 
 venient height. Greater accuracy is obtained by suspending 
 it by means of a frame fitting on a three-legged stand, such 
 as already described as used for supporting the more accurate 
 instruments; but it must not be forgotten that this instru¬ 
 ment is not to be at all compared with them for minute accu¬ 
 racy; but that its advantages are the great rapidity with which 
 it can be used, whether in a very confined space, or in an open 
 country. 
 
 INSTRUMENTS FOR MEASURING ANGLES. 
 
 In every map and plan the distances and angles laid down 
 are not the actual distances and angles between the points of 
 which the relative positions are intended to be represented, 
 but they are the distances and angles between the projections* 
 of those points upon the same horizontal plane, and are culled 
 the horizontal angles and distances between the points. Now, 
 if our surveying instruments were-constructed to measure the 
 actual angles subtended by different objects, the process of 
 calculating all the horizontal angles from these observed 
 angles "would be very laborious; but, by having such instru- 
 
 * The projection of a point upon a horizontal plane is the point in 
 which a vertical line through that point meets the horizontal plane. . 
 
THE PRISMATIC COMPASS. 
 
 115 
 
 ments as will at once determine bj observation the horizontal 
 angles, we are saved a vast amount of labour, and also from 
 any errors which might otherwise creep into the calculations. 
 
 THE PRISMATIC COMPASS. 
 
 With this instrument horizontal angles can be observed 
 with great rapidity, and, when used with a tripod stand, with 
 a considerable degree of accuracy. It is, consequently, a very 
 valuable instrument to the military surveyor, who can make 
 his observations with it, while holding it in his hand, with all 
 the accuracy necessary for a military sketch. It is also a 
 useful instrument for filling in the detail of an extensive sur¬ 
 vey*, after the principal points have been laid down by means 
 of observations made with the theodolite, hereafter to be de¬ 
 scribed, and for any purpose, in short, in which the portability 
 of the instrument and rapidity of execution are of more im¬ 
 portance than extreme accuracy. 
 
 c is a compass 
 card divided usually 
 to every 20', or 
 third part of a 
 degree, and having 
 attached to its under 
 side a magnetic 
 needle, which turns 
 upon an agate 
 center, o, fixed in 
 the box b ; n is a 
 spring, which, being 
 touched by the 
 finger, acts upon 
 the card, and checks 
 its vibrations, so as to bring it sooner to rest, when making an 
 observation, s is the sight-vane, having a fine thread stretched 
 along its opening, by which the point to be observed with the 
 instrument is to be bisected. The sight-vane is mounted upon 
 a hinge-joint, so that it can be turned down flat in the box 
 when not in use. p is the prism attached to a plate sliding in 
 a socket, and thus admitting of being raised or lowered at 
 pleasure, and also supplied with a hinge-joint, so that it can 
 be turned down into the box when not in use. In the plate 
 to which the prism is attached, and which projects beyond the 
 
 * The prismatic compass was used for this purpose by the gentlemen en¬ 
 gaged in making the ordnance surveys. 
 
Ii6 
 
 MATHEMATICAL INSTRUMENTS 
 
 prism, is a narrow slit, forming the sight through which the 
 vision is directed when making an observation. On looking 
 through this slit, and raising or lowering the prism in its 
 socket, distinct vision of the divisions on the compass card 
 immediately under the sight-vane is soon obtained, and these 
 divisions, seen through the prism, all appear, as each is suc¬ 
 cessively brought into coincidence with the thread of the sight- 
 vane by turning the instrument round, as continuations of the 
 thread, which is seen directly through the part of the slit that 
 projects beyond the prism. 
 
 The method of using the instrument is as follows:—The 
 sight-vane s, and the prism p, being turned up upon their 
 hinge-joints as represented in our figure, hold the instrument as 
 nearly in a horizontal position as you can judge, or, if it be used 
 with a tripod stand, set it as nearly as you can in a horizontal 
 position by moving the legs of the stand, so that the card may 
 play freely. Raise the prism in its socket till the divisions 
 upon the card are seen distinctly through the prism, and, turn¬ 
 ing the instrument round, until the object to be observed is 
 seen through the portion of the slit projecting beyond the 
 prism in exact coincidence with the thread of the sight-vane, 
 tiring the card to rest by touching the spring n ; and then the 
 reading at the division upon the card, which appears in coin¬ 
 cidence with the prolongation of the thread, gives the mag¬ 
 netic azimuth of the object observed, or the angle which a 
 straight line, drawn from the eye to the object, makes with 
 the magnetic meridian*. The magnetic azimuth of a second 
 object being obtained in the same manner, the difference be¬ 
 tween these two aeimuths is the angle subtended by the ob¬ 
 jects at the place of the eye, and, which is an important point, is 
 independent of any error in the azimuths, arising from the slit 
 in the prism not being diametrically opposite to the thread of 
 the sight-vane. 
 
 For the purpose of taking the bearings of objects much 
 
 * The magnetic meridian now makes an angle of 24° with the true 
 meridian, at London, the north point of the compass being 24° west 
 of the true north point. This angle is called the variation of the com¬ 
 pass, and is different at different places, and also at the same place at dif¬ 
 ferent times. Since this variation will affect equally, or nearly so, all 
 azimuths observed within a limited extent and during a limited time, the 
 angles subtended by any two of the objects observed, being the difference 
 of their azimuths, will not be affected by the variation, and hence the map, 
 or plan, may be constructed with all the objects in their proper relative posi¬ 
 tions : but the true meridian must be laid down, if required, by observations 
 made for the purpose. 
 
THE BOX SEXTANT. 
 
 117 
 
 above or below the level of the observer, a mirror, n, is sup¬ 
 plied with the instrument, which slides on and off the sight- 
 vane s, with sufficient friction to remain at any part of the 
 vane that may be desired. It can be put on with its face 
 either upwards or downwards, so as to reflect the images of 
 objects considerably either above or below the horizontal plane 
 to the eye of the observer; and, if the instrument be used for 
 obtaining the magnetic azimuth of the sun, it must be sup¬ 
 plied with dark glasses, d, to be interposed between the sun s 
 image and the eye. 
 
 There is a stop in the side of the box, not shown in our 
 figure, by touching which a little lever is raised and the card 
 thrown off its center; as it always should be when not in use, 
 or the constant playing of the needle would wear the fine 
 agate point upon which it is balanced, and the sensibility of 
 the instrument would be thereby impaired. The sight-vane 
 and prism being turned down, a cover fits on to the box, which 
 is about three inches in diameter, and one inch deep; and 
 the whole, being packed in a leather case, may be carried in 
 the pocket without inconvenience *. 
 
 THE BOX SEXTANT. 
 
 ♦ 
 
 This instrument, which is equally portable with the prismatic 
 compass, forming, when shut up, a box of about three inches in 
 diameter, and an inch and a half deep, will measure the actual 
 angle between any two objects to a single minute. It requires 
 no support but the hand, is easily adjusted, and, when once 
 adjusted, but seldom requires re-adjusting. 
 
 When the sex¬ 
 tant is to be used, 
 the lid, e, of the 
 box is taken off 
 and screwed on 
 to the bottom, 
 where it makes 
 a convenient han¬ 
 dle for holding 
 the instrument. 
 
 The telescope, t, 
 being then drawn 
 
 * For much valuable information respecting the use of the prismatic com¬ 
 pass, especially in military surveying and sketching, we can refer our 
 readers to a Treatise on Military Surveying, &c., by Lieutenant-Colonel 
 Basil Jackson, in which the subject is handled with great ability. 
 
TI3 
 
 MATHEMATICAL INSTRUMENTS. 
 
 out, the instrument appears as represented in our figure. 
 a is an index arm, having at its extremity a vernier, 
 of which 30 divisions coincide with 29 of the divisions 
 upon the graduated limb, 11 ; and the divided spaces upon 
 the limb denoting each 30 minutes, or half a degree, the 
 angles observed are read off by means of the vernier to 
 a single minute. The index is moved by turning the milled 
 head, b, which acts upon a rack and pinion within the box. 
 To the index arm is attached a mirror, called the index 
 glass, which moves with the index arm, and is firmly fixed 
 upon it by the maker, so as to have its plane accurately 
 perpendicular to the plane in which the motion of the index 
 arm takes place, and which is called the plane of the in¬ 
 strument. This plane is evidently the same as the plane 
 of the face of the instrument, or of the graduated limb, L l. 
 In the line of sight of the telescope is placed a second glass, 
 called the horizon glass, having only half its surface silvered, 
 and which must be so adjusted that its plane may be perpen¬ 
 dicular to the plane of the instrument, and parallel to the 
 plane of the index glass when the index is at zero. The in¬ 
 strument is provided with two dark glasses, which can be 
 raised or lowered by means of the little levers seen at d, so 
 as to be interposed, when necessary, between the mirrors and 
 any object too bright to be otherwise conveniently observed, as 
 the sun. The eye-end of the telescope is also furnished with 
 a dark glass, to be used when necessary. 
 
 The principle upon which the sextant is constructed has 
 been proved at page 75 ; viz. that the total deviation of a ray of 
 light, after reflection successively at the index glass and hori¬ 
 zon glass, is double the inclination of the two glasses. Now 
 the limb, 11, being divided into spaces, each of 15' extent, and 
 these spaces being figured as 30' each, the reading of the 
 limb gives double the angle moved over by the index arm 
 from the position in which the reading is zero, or double the 
 angle of inclination of the two mirrors, if these mirrors be 
 parallel when the reading is zero. If, then, the instrument 
 be in perfect adjustment, and any object be viewed by it after 
 reflection at both the mirrors, the reading of the instrument 
 gives the total deviation of the rays of light, by which the 
 vision is produced, or the angle between the bearing of the 
 object from the center of the index mirror, and the bearing of 
 the reflected image from the place of the eye, that is, between 
 lines drawn respectively from the object to the center of the 
 index glass, and from the reflected image in the horizon glass 
 
THE BOX SEXTANT. 
 
 II£T 
 
 to the eye. This angle is very nearly equal to the angle 
 subtended by the object and its image at the place of the eye, 
 differing from it only by the small angle subtended at the 
 object by the place of the eye and the center of the index glass. 
 This small angle is called the parallax of the instrument, and 
 is scarcely perceptible at the distance of a quarter of a mile, 
 \yhile for distances greater than that it is so small that it 
 may be considered to vanish. It also varies with the amount 
 of deviation, and vanishes altogether whenever the center of 
 the index glass is in a direct line between the object and 
 the eye*. 
 
 To see if the instrument be in perfect adjustment, place 
 the dark glass before the eye-end of the telescope, and looking at 
 the sun, and moving the index backwards and forwards a little 
 distance on either side of zero, the sun’s reflected image will 
 he seen to pass over the disc as seen directly through the 
 horizon glass, and if in its passage the reflected image com¬ 
 pletely covers the direct image, so that but one perfect orb is 
 seen, the horizon glass is perpendicular to the plane of the 
 instrument; but, if not, the screw at a must be turned by the 
 key, k , till such is the case. The key, /c, fits the square heads 
 of both the screws seen at a and b, and fits into a spare part 
 of the face of the instrument, so as to be at hand when wanted. 
 This adjustment being perfected, bring the reflected image of 
 the sun’s lower limb in exact contact with the direct image of 
 his upper limb, and note the reading of the vernier; then 
 move the index back beyond the zero division of the limb, till 
 the reflected image of the sun’s upper limb is in exact contact 
 with the direct image of his lower limb, and, if the zero of 
 the vernier be now exactly as far behind the zero of the limb 
 as it was at the former reading in front of it, so that the read¬ 
 ing now on the part of the limb called the arc of excess, be¬ 
 hind its zero divisionf, be the same as the former reading, 
 
 * We have seen a method given for what is called correcting the parallax, 
 when an observation is made at a short distance, by finding the deviation 
 at this distance, when the angle between the object and its image is equal to 
 fcero; this deviation being given by the reading of the instrument, when 
 the reflected image of the object observed exactly coincides with the object 
 itself, seen through the unsilvered part of the horizon glass. This deviation, 
 however, is not the parallax, even for a small angle between the object and 
 its image, and, if the angle be not very small, the error introduced by the 
 method will be greater than the parallax itself. 
 
 1* In reading an angle upon the arc of excess, the division to read on the 
 .limb is that next in front of the zero of the vernier, or between the zero of 
 the vernier and the zero of the limb, and the divisions of the vernier itself 
 are to be read from the end division, marked 30, and not, as usually, from 
 
120 
 
 MATHEMATICAL INSTRUMENTS. 
 
 the instrument is in perfect adjustment; but, if not, half the 
 difference of the two readings is the amount of the error, and 
 is called the index error, being a constant error, for all angles 
 observed by the instrument, of excess, if the first reading be 
 the greatest, and of defect, if the second reading on the arc of 
 excess be the greatest. 
 
 In the former case, then, the true angle will be found by 
 subtracting the index error from, and in the latter by adding 
 it to, the reading of the instrument at every observation. 
 
 This method of correcting for the index error is to be used 
 with the larger instruments, hereafter to be described under 
 the head of Astronomical Instruments; but in the box sex¬ 
 tant this error should be removed by applying the key, k, to 
 the screw at b, and turning it gently till both readings are 
 alike, each being made equal to half the sum of the two read¬ 
 ings first obtained. When this adjustment is perfected, if 
 the zeros of the vernier and limb are made exactly to coin¬ 
 cide, the reflected and direct image of the sun will exactly 
 coincide, so as to form but one perfect orb, and the reflected 
 and direct image of any line, sufficiently distant not to be 
 affected by parallax, as the distant horizon, or the top or 
 end of a wall more than half a mile off, will coincide so as to 
 form one unbroken line. 
 
 To obtain the angle subtended by two objects situated 
 nearly or quite in the same vertical plane, hold the instru¬ 
 ment in the right hand, and bring down the reflected image 
 of the upper object by turning the milled head b, till it ex¬ 
 actly coincides with the direct image of the lower object, and 
 the reading of the instrument will give the angle between the 
 two objects. 
 
 To obtain the angle subtended by two objects nearly in the 
 same horizontal plane, hold the sextant in the left hand, and 
 bring the reflected image of the right-hand object into coin¬ 
 cidence with the direct image of the left-hand object. 
 
 It will be seldom that the surveyor need pay any attention 
 to the small error arising from parallax; but, should great 
 accuracy be desirable, and one of the objects be distant while 
 the other is near, the parallax will be eliminated by observ¬ 
 ing the distant object by reflection, and the near one by 
 
 the zero division : thus, if the zero division of the vernier were a little 
 further from the zero division of the limb, then the first division on the 
 arc of excess; and if the twenty-seventh division on the vernier, or the third 
 from the end division, marked 30, coincided with a division upon the limb, 
 then th$ reading would be S3 7 . 
 
THE BOX SEXTANT. 
 
 1*21 
 
 direct vision, holding the instrument for this purpose with its 
 face downwards if the distant object be on the left hand. If 
 both objects be near, the reflected image of a distant object, in 
 a direct line with one of the objects, must be brought into coin¬ 
 cidence with the direct image of the other object, and the 
 parallax will thus be eliminated. 
 
 For the purposes of surveying, the horizontal angles between 
 different objects are required, and the reduction of these angles 
 from the actual oblique angles subtended by the objects, 
 would be a troublesome and laborious process. If the angle 
 subtended by two objects be large, and one be not much 
 higher than the other, the actual angle observed will be, how¬ 
 ever, a sufficient approximation to the horizontal angle re¬ 
 quired ; and, if the angle between the two objects be small, 
 the horizontal angle will be obtained with sufficient accuracy 
 by taking the difference of the angles observed between each of 
 the objects, and a third object at a considerable angular dis¬ 
 tance from them. With a little practice the eye will be able 
 to select an object in the same direction as one of the objects, 
 and nearly on a level with the other object, and the angle be¬ 
 tween this object and the object selected will be the horizontal 
 angle required. 
 
 At sea the altitude of an object may be determined by ob¬ 
 serving the angle subtended by it and the verge of the horizon ; 
 but upon land a contrivance, called an artificial horizon, be¬ 
 comes necessary for correctly determining altitudes. The best 
 kind of artificial horizon 
 consists of an oblong 
 trough, t t , filled with 
 mercury, and protected 
 from the wind by a roof, 
 rr, havingin either slope 
 a plate of glass with its 
 two surfaces ground into 
 perfectly parallel planes. 
 
 The angle s e s' between 
 the object and its re¬ 
 flected image seen in the 
 mercury is double the 
 angle of elevation s e h, 
 and, the angle s e 3' 
 being observed, its half 
 will consequently be the 
 angle of elevation re- 
 
 G 
 
MATHEMATICAL INSTRUMENTS 
 
 123 
 
 quired. If the angle of elevation be greater than 60°, the 
 angle s e s' will be greater than 120°, and cannot be observed 
 with the sextant we have been describing. 
 
 The pocket sextant is a most convenient instrument for 
 laying off offsets or perpendicular distances from a station 
 line; for by setting the index at 90°, and walking along the 
 station line, looking through the horizon glass directly at the 
 further station staff, or any other remarkable object upon the 
 station line, any object off the station line will be seen by re¬ 
 flection when the observer arrives at the point where the per¬ 
 pendicular from this object upon the station line falls, and 
 the distance from this point to the object being measured, is 
 its perpendicular distance from the station line. 
 
 For the mere purpose of measuring offsets an instrument 
 called an optical square is now very generally employed, which 
 consists of the two glasses of the sextant fixed permanently 
 at an angle of 45°, so that any two objects seen in it, the one 
 by direct vision, and the other by reflection, subtend at the 
 place of the observer an angle of 90°. 
 
 THE THEODOLITE. 
 
 The theodolite is the most important instrument used by- 
 surveyors, and measures at the same time both the horizontal 
 angles subtended by each two of the points observed with it, 
 and the angles of elevation of these points from the point of 
 observation. 
 
 This instrument may be considered as consisting of three 
 parts; the parallel plates with adjusting screws fitting on to 
 the staff head, of exactly the same construction as already 
 described for supporting the Y, and other, levels ; the horizon¬ 
 tal limb, for measuring the horizontal angles ; and the ver¬ 
 tical limb, for measuring the vertical angles, or angles of 
 elevation. 
 
 The horizontal limb is composed of two circular plates, l 
 and v, which fit accurately one upon the other. The lower 
 plate projects beyond the other, and its projecting edge is 
 sipped off, or chamfered, as it is called, and graduated at 
 every half degree. The upper plate is called the vernier 
 plate, and has portions of its edge chamfered off, so as to 
 form with the chamfered edge of the lower plate continued 
 portions of the same conical surface. These chamfered por¬ 
 tions of the upper plate are graduated to form the verniers, 
 by which the limb is subdivided to single minutes. The five- 
 inch theodolite represented in our figure has two such ver- 
 
THE THEODOLITE.- 
 
 123 
 
 plate, and terminating in a ball fitting in a socket upon the 
 lower parallel plate, exactly as the vertical axis of the Y level 
 already described. This axis is, however, hollowed to receive 
 a similar conical axis ground accurately to fit it, so that the 
 axes of the two cones may be exactly coincident, or parallel*. 
 
 * Upon this depends, in a great measure, the perfection of the instru¬ 
 ment, as far as the horizontal measurements are concerned; and, whe we 
 
MATHEMATICAL INSTRUMENTS. 
 
 124 
 
 To the internal axis the upper, or vernier, plate of the hori¬ 
 zontal limb is attached, and thus, while the whole limb can 
 be moved through any horizontal angle desired, the upper 
 plate only can also be moved through any desired angle, when 
 the lower plate is fixed by means of the clamping screw, c, 
 which tightens the collar, d. t is a slow-motion screw, which 
 moves the whole limb through a small space, to adjust it more 
 perfectly, after tightening the collar, d, by the clamping screw, 
 c. There is also a clamping screw, c , for fixing the upper, 
 
 or vernier, plate to the lower plate, and a tangent screw, t , 
 
 for giving the vernier plate a slow motion upon the lower 
 
 plate, when so clamped. Two spirit levels, b, b, are placed 
 
 upon the horizontal limb, at right angles to each other, and a 
 compass, o, is also placed upon it in the center, between the 
 supports, f, f, for the vertical limb. 
 
 The vertical limb, n n, is divided upon one side at every 
 80 minutes, each way from 0° to 90°, and subdivided by the 
 vernier, which is fixed to the compass box, to single minutes. 
 Upon the other side are marked the number of links to be 
 deducted from each chain, for various angles of inclination, 
 in order to reduce the distances, as measured along ground 
 rising or falling at these angles, to the corresponding hori¬ 
 zontal distances. The axis, a, of this limb must rest, in a 
 position truly parallel to the horizontal limb, upon the sup¬ 
 ports, ff, so as to be horizontal when the horizontal limb is 
 set truly level, and the plane of the limb, n n, should be 
 accurately perpendicular to its axis. To the top of the ver¬ 
 tical limb, n n, is attached a bar which carries two Ys for 
 supporting the telescope, which is of the same construction 
 as that before described for the Y spirit level, and under¬ 
 neath the telescope is a spirit level, s s, attached to it at one 
 end by a joint, and at the other end by a capstan-headed 
 screw as in the Y level. The horizontal axis, a, can be fixed 
 by a clamping screw, c, and the vertical limb can then be 
 moved through a small space by a slow-motion screw, i. 
 
 Before commencing observations with this instrument, the 
 following adjustments must be attended to:— 
 
 1. Adjustments of the telescope: viz., 
 
 the adjustment for parallax. 
 
 •-for collimation. 
 
 2. Adjustment of the horizontal limb : viz., 
 
 to set the levels on the horizontal limb to indicate the verticality of 
 the azimuthal axis. 
 
 describe presently the adjustments of the instrument, we shall explain the 
 method of detecting an inaccuracy in the grinding of the axes. 
 
THE THEODOLITE. 
 
 125 
 
 3. Adjustment of the vertical limb : viz., 
 
 to set the level beneath the telescope to indicate the horizontality of 
 the line of collimation. 
 
 1. Parallax and Collimation .—These adjustments have 
 already been described (p. 100) under the head of the Y level. 
 
 2. Adjustment of the Horizontal Limb .—Set the instrument 
 up as accurately as you can by the eye, by moving the legs of 
 the stand. Tighten the collar, d, by the clamping screw, c, 
 and, unclamping the vernier plate, turn it round till the tele¬ 
 scope is over two of the parallel plate-screws. Bring the 
 bubble, b , of the level, s s, beneath the telescope to the center 
 of its run by turning the tangent screw, i. Turn the vernier 
 plate half round, bringing the telescope again over the same 
 pair of the parallel plate screws; and, if the bubble of the 
 level be not still in the center of its run, bring it back to the 
 center, half way by turning the parallel plate screws over 
 which it is placed, and half way by turning the tangent screw, 
 i. Repeat this operation till the bubble remains accurately 
 in the center of its run in both positions of the telescope; 
 and, then turning the vernier plate round till the telescope is 
 over the other pair of parallel plate screws, bring the bubble 
 again to the center of its run by turning these screws. The 
 bubble will now retain its position while the vernier plate is 
 turned completely round, showing that the internal azimuthal 
 axis about which it turns is truly vertical. The bubbles of 
 the levels on the vernier plate, being now, therefore, brought 
 to the centers of their tubes, will be adjusted to show the 
 vertically of the internal azimuthal axis. Now, having 
 clamped the vernier plate, loosen the collar, d, by turning 
 back the screw, c, and move the whole instrument slowly 
 round upon the external azimuthal axis, and if the bubble of 
 the level s s, beneath the telescope, maintains its position 
 during a complete revolution, the external azimuthal axis is 
 truly parallel with the internal, and both are vertical at the 
 same time ; but, if the bubble does not maintain its position, it 
 shows that the two parts of the axis have been inaccurately 
 ground, and the fault can only be remedied by the instrument- 
 maker. 
 
 3. Adjustment of the Vertical Limb .—The bubble of the 
 level, s s, being in the center of its run, reverse the telescope 
 end for end in the Ys, and, if the bubble does not remain in 
 the same position, correct for one-half the error by the capstan¬ 
 headed adjusting screw at one end of the level, and for the 
 other half by the vertical tangent screw, i. Repeat the opera- 
 
126 
 
 MATHEMATICAL INSTRUMENTS. 
 
 lion till the result is perfectly satisfactory. Next turn the 
 telescope round a little both to the right and to the left, and, 
 if the bubble does not still remain in the center of its run, 
 the level, s s, must be adjusted laterally by means of the 
 screw at its other end. This adjustment will probably dis- 
 turb the first, and the whole operation must then be carefully 
 repeated. By means of the small screw fastening the vernier 
 of the vertical limb to the vernier plate over the compass 
 box, the zero of this vernier may now be set to the zero of 
 the limb, and the vertical limb will be in perfect adjustment. 
 
 With an increase in the size of the theodolite a second 
 telescope is placed beneath the horizontal limb, which serves 
 to detect any accidental derangement of the instrument dur¬ 
 ing an observation, by noting whether it is directed to the 
 same point of a distant object at the end of the observation 
 to which it has been set at the commencement of the observa¬ 
 tion. Also the vertical limb, in the larger theodolites, admits 
 of an adjustment to make it move accurately in a vertical 
 plane, when the horizontal limb has been first set in perfect 
 adjustment. This adjustment is important, and should be 
 examined with great care; and in the small theodolites, when 
 the vertical limb is permanently fixed to the horizontal limb 
 by the maker, an instrument which will not bear the test of 
 the examination which we proceed to describe must be con¬ 
 demned, till set in better adjustment by the maker. The 
 azimuthal axis having been set truly vertical, direct the tele¬ 
 scope to some well-defined angle of a building, and making 
 the intersection of the wires exactly coincide with this angle 
 near the ground, elevate the telescope by giving motion to 
 the vertical limb, and, if the adjustment be perfect, the inter¬ 
 section of the cross wares will move accurately along the angle 
 of the building, still continuing in coincidence with it. A 
 still more perfect test will be to make the intersection of the 
 cross wires coincide with the reflected image of a star in an 
 artificial horizon, and elevating the telescope, if the adjust¬ 
 ment be perfect, the direct image of the 6tar itself will again 
 be bisected by the cross wires. 
 
 In the conduct of an extensive survey, the two principal 
 desiderata are accuracy and despatch, neither of which should 
 be unduly sacrificed to the other. To obtain both these ends, 
 the principal points of the survey should be determined by a 
 system of triangles proceeding from an accurately-measured 
 base of considerable length. The angles of these triangles 
 should be observed with a large and perfect theodolite con- 
 
THE THEODOLITE. 
 
 127 
 
 structed for the purpose, or with an altitude and azimuth in¬ 
 strument ; and numerous corrections should be applied for 
 the spherical form of the earth, the refraction of the atmo¬ 
 sphere, the errors due to the imperfect graduation of the in¬ 
 strument, &c. 
 
 The boundaries of the entire country to be surveyed being 
 thus determined with the greatest possible accuracy, and a 
 series of stations laid down throughout, the spaces included 
 between these stations may be subdivided into spaces of 
 smaller extent, the boundaries of which may be surveyed with 
 considerable despatch by means of the chain, and a portable 
 theodolite, such as we have been describing above, and lastly, 
 the details of the country within these spaces may be sketched 
 with still greater rapidity by means of the prismatic compass. 
 
 The boundaries of the spaces to be surveyed by the chain 
 and small theodolite should not exceed three or four miles in 
 extent, and the following is the manner of proceeding. 
 
 Let a , b, c, d, e,f, represent the boundary to be surveyed, 
 and let a, b, and c, be three stations which have been accu¬ 
 rately laid down by the previous triangulation, of which both 
 b and c can be seen from a, and a can be seen from c. First 
 measure with the chain the lengths of the several lines a b, 
 b c, c d, &c., taking offsets to all remarkable points on either 
 side of these lines in the usual manner, and driving pickets at 
 a, b, c, &c. Measure also the distance from a to a, and from 
 d to c. These measurements having been made, set up the 
 theodolite at a, level it, and clamp the vernier plate to the 
 low'er plate of the horizontal limb at zero, or so that the read¬ 
 ings of the two verniers may be 360° and 180 respectively, this 
 adjustment being perfected by the slow r -motion screw, t. Next 
 move the whole instrument round upon the azimuthal axis, 
 till the object b is accurately bisected by the cross wires, 
 clamp it firmly in this position by the screw c, tightening the 
 collar d, and enter in the field book the reading of the com¬ 
 pass. Now release the vernier plate, and turning it round, 
 bisect the object c, by the cross wires, and enter the readings 
 of both verniers in the field book. Observing, in like manner, 
 the bearings of any other remarkable objects, and, entering 
 the readings in the same way, direct the telescope lastly to a, 
 at which station an assistant must be placed, with a staff held 
 upon the picket there driven into the ground, and, entering the 
 reading of the vernier as before, clamp the vernier plate care¬ 
 fully, and remove the instrument to a. Level the instrument 
 at a, unclamp the collar d, and, turning round the whole in- 
 
123 
 
 MATHEMATICAL INSTRUMENTS. 
 
 strument upon the azimuthal axis, direct the telescope to the 
 last station a, tighten the collar d, and perfect the adjustment, 
 if necessary, by the slow-motion screw t. Now release the 
 vernier plate, and, bringing it back to zero, if the reading of 
 the compass be the same as the reading previously entered in 
 the field book, we assume our work, as far as it has gone, to 
 be correct; but, if not, we must go back to a, and go over the 
 work again*. Next release the vernier plate, and enter the 
 readings, when the telescope is directed to the several remark¬ 
 able points visible from a , and lastly direct the telescope to 
 the next forward 
 station b, as be¬ 
 fore. In the 
 same manner 
 proceed from b 
 to c, c to d, and 
 d to c; and, hav¬ 
 ing directed the 
 telescope at c to 
 the last back sta¬ 
 tion d, and re¬ 
 leased the ver¬ 
 nier plate, direct 
 the telescope to 
 a; and, if all the 
 angles have been 
 correctly mea¬ 
 sured up to this d 
 time, the reading of the vernier w r ill now be the same as 
 when the telescope was directed to c from the point a. 
 If then we have not been able to make all the compass 
 readings agree at the previous stations, after going twice over 
 the work at such stations w T e may now consider that our work 
 was correct, and that the error in the compass reading arose 
 from some local attraction, or extraordinary variation of the 
 needle. This verification of the work at c is called closing the 
 work. We now come back again to d, and proceed from d to e , 
 and so on, as before, till we come to some other station, which 
 has been observed either from a or c, and which we again close 
 upon, and at last arriving at /, if the readings agree within a 
 minute or two with the readings for /, previously observed at 
 
 * If the same result is again arrived at, we may presume that the com¬ 
 pass is acted on by some local attraction, and proceed with the work; and 
 the accuracy of this presumption will be further tested as we go on. 
 
THE CIRCULAR PROTRACTOR. 1*20 
 
 a y the whole work may be considered to have been performed 
 with a sufficient degree of accuracy; but, if the error amount 
 to more than a minute or two, we must proceed back again 
 from / to 0 , and so on till we find out the station at which the 
 error has occurred. If the ground along any of the lines a b , 
 b c, &c., rise or fall, suppose, for instance, along b c, then we 
 must direct the telescope from b, so as to make the cross wires 
 bisect upon the staff, held upon the picket at c , a point at the 
 same distance from the ground as the center of the telescope, 
 and then upon one side of the vertical limb is pointed out the 
 number of links to be deducted for each chain from the mea¬ 
 sured distance b c, to reduce it to the required horizontal dis¬ 
 tance. This reduction is then to be entered in the field book*. 
 
 INSTRUMENTS FOR PLOTTING THE SURVEY. 
 
 In plotting the survey, as in taking it, due regard must be 
 had to both accuracy and despatch, and we should aim to lay 
 down the various points observed with an accuracy propor¬ 
 tionate to the accuracy of the survey itself. To this end the 
 principal points should be laid down by setting off with the 
 beam compasses the computed sides of the triangles, the 
 angles of which have been accurately observed with the large 
 theodolite; and the direction of the meridian is to be laid 
 down from an observation of the angle which it makes, with a 
 side of one of these triangles, by means of the computed 
 chordsf, which chord is also to be set off with the beam com¬ 
 passes. 
 
 THE CIRCULAR PROTRACTOR. 
 
 The principal points having thus been laid down, the bound¬ 
 aries observed by the small theodolite may be put in by first 
 laying down upon the paper a large circular protractor. This 
 protractor may be pricked off by means of the circular metallic 
 protractor represented in the accompanying figure, and the 
 lines can then be transferred to any part of the paper by 
 means of a large ruler and triangle, or by any parallel ruler. 
 The circular protractor is a complete circle, a a, connected 
 with its center by four radii, a a a a. The center is left open, 
 and surrounded by a concentric ring, or collar, b, which carries 
 
 * The method of surveying with the chain and theodolite, explained 
 above, is called surveying by a traverse. 
 
 + If a table of chords be not at hand, take out the sine of half the angle 
 from a table of natural sines, and, reckoning the first figure as integral, this 
 will be the chord of the whole angle to radius 5, or, reckoning the first two 
 figures integral, it will be the chord to radius 50. 
 
 G 3 
 
130 
 
 MATHEMATICAL INSTRUMENTS. 
 
 two radial bars, cc. To the extremity of one bar is a pinion, 
 d, working in a toothed rack quite round the outer circumfer¬ 
 ence of the protractor. To the opposite extremity of the 
 other bar, c, is fixed a vernier, which subdivides the primary 
 divisions on the protractor to single minutes, and by estima¬ 
 tion to 30 seconds. This vernier, as may readily be under¬ 
 stood from the engraving, is carried round the protractor by 
 turning the pinion d. Upon each radial bar, c c, is placed a 
 branch e e , carying at its extremity a fine steel pricker, whose 
 point is kept above the surface of the paper by a spring placed 
 
 ture in the paper. The branches e e are attached to the bars 
 c c , with a joint which admits of their being folded backwards 
 over the instrument when not in use, and for packing in its 
 case. The center of the instrument is represented by the in¬ 
 tersection of two lines drawn at right angles to each other on 
 a piece of plate glass, which enables the person using it to 
 place it so that the center, or intersection of the cross lines, 
 may coincide with any given point on the plan. If the instru¬ 
 ment is in correct order, a line connecting the fine pricking 
 points with each other would pass through the center of the in¬ 
 strument, as denoted by the before-mentioned intersection of 
 the cross-lines upon the glass, which, it may be observed, are 
 drawn so nearly level with the under surface of the instrument 
 as to do away with any serious amount of parallax, when set¬ 
 ting the instrument over a point from which any angular lines 
 are intended to be drawn. In using this instrument the ver 
 
THE T SQUARE AND PROTRACTOR. 
 
 131 
 
 nier should first be set to zero (or the division marked 360) 
 on the divided limb, and then placed on the paper, so that the 
 two fine steel points may be on the given line (from whence 
 other and angular lines are to be drawn), and the center of the 
 instrument coincide with the given angular point on such line. 
 This done, press the protractor gently down, which will fix it 
 in position by means of very fine points on the under side. 
 It is now ready to lay off the given angle, or any number of 
 angles that may be required, which is done by turning the 
 pinion d till the opposite vernier reads the required angle. 
 Then press downwards the branches e e, which will cause the 
 points to make punctures in the paper at opposite sides of the 
 circle; which being afterwards connected, the line will pass 
 through the given angular point, if the instrument was first 
 correctly set. In this manner, at one setting of the instru¬ 
 ment, a great number of angles, or a complete circular protrac¬ 
 tor, may be laid off from the same point. 
 
 THE T SQUARE AND SEMICIRCULAR PROTRACTOR. 
 
 We t cannot speak too highly of a method by which a tra¬ 
 verse can be most expeditiously as well as accurately plotted, 
 
 by means of the t square and semicircular protractor, the 
 manner of using which is thus described by Mr. Howlett, 
 chief draughtsman, Royal Ordnance Office, in vol. i. of Papers 
 
132 
 
 Mathematical instruments. 
 
 on Subjects connected with the Duties of the Royal En 
 gineers:—- 
 
 “ As, when away from home, it seldom happens that the surveyor can 
 obtain a good drawing board, or even a table with a good straight edge, I 
 fix a flat ruler, A, to the table bbb, by means of a pair of clamps, o D, and 
 against this ruler I work the pattern square E, one side of which has the 
 stock flush with the blade; or, if a straight-edged board be at hand, then the 
 square may be turned over, and used against that edge instead of the ruler A. 
 Here, then, is the most perfect kind of parallel ruler that art can produce, 
 capable of carrying the protractor over the whole of a sheet of plotting paper 
 of any size, and may be used upon a table of any form. It is convenient to 
 suppose the north on the left hand, and the upper edge of the blade to repre¬ 
 sent the meridian of the station. 
 
 “ This protractor is held in the hand while the vernier is set, which is an 
 immense comfort to the sight; and it will be seen that, as both sides of the 
 arm are parallel with the zero and center, the angle may be drawn on the 
 paper against either side, as the light or other circumstances may render 
 desirable.” 
 
 From this description and a mere glance at the engraving, it 
 is clear that angles taken with the theodolite can be transferred 
 to the plot as accurately as the protractor can be set, namely, 
 to a single minute, and that, too, in a rapid and pleasant 
 manner. 
 
 Another most admirable and expeditious method of plotting, 
 especially useful when it is a principal point to obtain the 
 area of an estate or parish, &c., is to procure or form a table 
 of northings, southings, eastings, and westings *, for all 
 angles made with a meridian line, and for all distances 
 from 1 to 100. These distances may be either links, feet, 
 chains, or estimated in any denomination whatever, and 
 the corresponding northings, southings, eastings, and west¬ 
 ings will be in the same denomination. This table is in fact 
 nothing more than the products of the sines and cosines 
 of the angles, made with the meridian line, multiplied 
 by the several distances, and the following is the method 
 of using it. Take out from this table the northings, south¬ 
 ings, eastings, and westings made on each of the lines of the 
 survey, the line from which the angles have been measured 
 being for this purpose assumed as the meridian, no matter 
 in what direction it may lie, and place them in a table, which 
 we may call a traverse table, in four separate columns, being 
 the third, fourth, fifth, and sixth columns of the tablet, headed 
 
 * This table is the same as the table given in books on navigation, and 
 then called a table of latitude and departure. 
 
 f The first and second columns of the traverse table contain the courses 
 and distances. 
 
THE T SQUARE AND PROTRACTOR. 
 
 133 
 
 N., S., E., and W. respectively. Add up these several 
 columns, and, if the work is so far correct, the sum of the 
 northings will equal the sum of the southings, and the sum 
 of the eastings will equal the sum of the westings. Then 
 in two additional columns enter the whole quantities of north¬ 
 ing and easting, made at the termination of each of the seve¬ 
 ral bounding lines of the survey; which quantities will be 
 determined by putting zero for the greatest southing, and 
 adding or subtracting the northing or southing made 011 each 
 particular line to or from the whole quantity of northing made 
 at the beginning of this line, or at the termination of the 
 preceding line ; and again, by putting zero for the greatest 
 
 westing, and adding 
 
 or subtracting the 
 
 easting 
 
 or westing 
 
 
 
 e- e 
 
 
 made on each line to or from the whole quantity of easting 
 at the beginning of the line. 
 
 This preparatory table having been formed, the plot may be 
 laid down with great ease and accuracy by means of a plotting 
 scale, formed of two ivory graduated rules, one of which, n n, 
 represents the assumed meridian along which the northings 
 
 are to be measured, 
 and the other, e e, re¬ 
 presents the east and 
 west line, and serves 
 to set off the east¬ 
 ings. The rule, n n, 
 is perforated through¬ 
 out nearly its whole 
 length with a dove¬ 
 tail groove, receiving 
 an accurately fitted 
 sliding piece, to which 
 the rule, ee, is fixed 
 
 wnmTmnp^mjwmmT- 
 
 n 
 
 ~ | " .. .. ■ 1 ■ ■ i 
 
 by the screw s , so as to slide along, and always have its edges 
 at right angles to the edges of the rule nn. The rule, nn, is 
 to be placed on the paper with its zero division opposite that 
 point of the line assumed as a meridian, at which the plotting 
 is to be commenced, and with its edges parallel to this line, 
 and at such distance from it, that the zero division on the rule, 
 e e , may be upon the assumed meridian. The rule, n n, is then 
 to be fixed by placing weights upon its extremities, or by clamps. 
 The scale, e e, being now slid along till either of its edges coin¬ 
 cides with the divisions upon the scale, n n , answering to the 
 whole quantities of northing at the termination of each line of 
 the survey, the divisions upon the scale, e e, answering to the 
 
134 
 
 MATHEMATICAL INSTRUMENTS. 
 
 
 i 
 
 CO O O'! 
 
 OO iO 
 
 co r- 
 
 o io 
 
 O 
 
 
 
 (N N ® 
 
 co oo 
 
 O rH 
 
 CM O 
 
 CM 
 
 
 
 03 -H > V 
 
 CM 03 
 
 CO £- 
 
 o o 
 
 O 
 
 rH 
 
 •*-> 3 
 
 1- GO O 
 
 co co 
 
 (N r-H 
 
 ->H CO 
 
 
 rH 
 
 OO 00 03 
 
 CO l'- 
 
 03 03 
 
 CO 03 
 
 0° 
 
 
 3-C 
 
 O 
 
 c« 
 
 6 M O) 
 
 CM CO 
 
 ^ i) 
 
 CO 
 
 ib o 
 lO i £3 
 
 rH t- 
 rH CM 
 rH 
 
 do 
 
 o 
 
 13 
 
 North Pro¬ 
 ducts. 
 
 acres. 
 
 6-04996 
 
 •94512 
 
 •83505 
 
 •90972 
 
 lO 
 
 g ^ 
 
 co : : 
 
 i> • • 
 
 GO l l 
 
 12 
 
 West Pro¬ 
 ducts. 
 
 acres. 
 
 5- 23770 
 
 6- 66832 
 
 •84091 
 
 12-74693 
 
 Acres . 
 
 Hoods . 
 
 11 
 
 East Pro¬ 
 ducts. 
 
 acres. 
 
 19-14980 
 
 41-78504 
 
 6-32730 
 
 1-40196 
 
 6866410 
 
 12-74693 
 
 .ZUI6-SS 
 
 10 
 
 Sums ot 
 Total 
 East¬ 
 ings. 
 
 3 NOO r-1 O T)C ICJ r)( 
 
 .£ 03 rH CO lO --H OO OO 
 
 A CO rH CM — < t- CM O 
 
 O CM CO CO rH 
 
 9 
 
 Sums of 
 Total 
 North¬ 
 ings. 
 
 A O CM O O 0O CO 03 
 
 ■A (N W CO rH CO 03 CO 
 
 A CO lO 03 CM U3 CM CO 
 
 C3 CO CO CM rH 
 
 8 
 
 Total 
 
 East¬ 
 
 ings. 
 
 •-^COIOCOOSIOOhH 
 .5 CO rH CO 03 lO CO O 00 
 
 .cOOiO'O^^OO 
 
 CJ rH rH r-H 
 
 7 
 
 Total 
 
 North¬ 
 
 ings. 
 
 A CO rH CO CM 00 O CO CO 
 £t-^c0^©001> 
 
 3 CO CM CM CO lO O M CO 
 
 O rH CM CM rH rH 
 
 t 
 
 00 ^ 
 
 <o gj> 
 
 A t- -h r- 
 
 .5 co t- oo 
 
 '“■* • • • • • 
 
 • • • • • 
 
 • • • • • 
 
 A (MH(N 
 
 U rH 
 
 16 98 
 
 5 
 
 East¬ 
 
 ings. 
 
 A 03 03 H rH 
 
 .3 CM CM CO CO 
 
 • • • • 
 
 • • • • 
 
 ,c * * * O 
 
 o 
 
 16 96 
 
 4 
 
 South¬ 
 
 ings. 
 
 3 CO rH OO 
 
 •B -h 1— CO 
 
 ^ • • • • • 
 
 • • • • • 
 
 A ’ ' ’ CO O lO 
 
 u rH 
 
 22 88 
 
 3 
 
 North¬ 
 
 ings. 
 
 3 CO rH CO CO 
 
 .= (O^ 03 OO 
 
 • • • • 
 
 • • • • 
 
 • • • • 
 
 ,C 00 O (M GO 
 
 O rH 
 
 22 88 
 
 2 
 
 Dist¬ 
 
 ances. 
 
 3 O CO CO CO O rH 03 CO 
 
 ^OrllMOO CO 
 
 ^OOOiOHCOilO 
 
 CJ rH rH rH rH 
 
 05 
 
 i 
 
 >- 
 
 3 
 
 O 
 
 u 
 
 p4 p4 m ^ ^ ^ p4 
 
 OOO 
 CM rH O 
 
 O 
 
 rH t— h^< 
 CO CO I—I 
 
 WON® 
 
 WrlrtN 
 
 CM rH 
 rH CO rH 
 
 £i' ^ cd ad ai ^ 
 
 Perches . 37-60800 
 
THE STATION POINTER, 
 
 135 
 
 whole quantities of easting, will give the terminations of these 
 lines, which, being pricked off, have only afterwards to he 
 joined, and the plot will be completed. 
 
 To compute the Area of the Plot .—Rule six additional columns. 
 In the first of these, or ninth column of the traverse table, 
 set the sums of the total northings, and in the tenth, the sums 
 of the total eastings at the beginning and end of each line in 
 the survey, which sums will be found by adding together each 
 pair of succeeding numbers in the two preceding columns. 
 In the eleventh column set the products of the eastings made 
 on the respective lines of the survey, found in the fifth column, 
 multiplied by the corresponding sums of the total northings 
 in the ninth column ; and in the twelfth column set the pro¬ 
 ducts of the "westings found in the sixth column, and the cor¬ 
 responding sums of the total northings in the ninth column. 
 Sum up the eleventh and twelfth columns, and the difference of 
 the totals thus found will be twice the area of the plot. Again 
 in the thirteenth and fourteenth columns set the products of 
 the northings and southings in the third and fourth columns, 
 multiplied by the corresponding sums of the total eastings in 
 the tenth column, and the difference of the sums of the thir¬ 
 teenth and fourteenth columns will again be twice the area of 
 the plot, and, if agreeing nearly with the double area before 
 found, shows the calculations to have been correctly performed. 
 (We give an example in the opposite page.) 
 
 The near agreement of the sums of the third and fourth, and 
 of the fifth and sixth columns is a test of the accuracy of the 
 survey ; in columns 7 and 8 we have the distances to be set off 
 by the plotting scale; in column 9 we have the multipliers by 
 which the east and west products in columns 11 and 12 are 
 found; and in column 10 we have the multipliers for finding 
 ■the north and south products in columns 13 and 14. The 
 difference of the sums of the eleventh and twelfth columns 
 gives double the area, the difference of the sums of the 
 thirteenth and fourteenth gives again double the area, and, 
 taking the mean of these results, by adding them together 
 an’d dividing by 4, we obtain the area most probably to within 
 a quarter of a perch, since the two double areas differ by less 
 than a perch. 
 
 THE STATION FOINTER. 
 
 When the principal features of a country have been laid 
 down by the methods already pointed out, the details may be 
 
186 
 
 MATHEMATICAL INSTRUMENTS. 
 
 put in with great rapidity by means of the instrument which 
 we are now about to describe. 
 
 a a, b b, c c, are three arms moveable about a common 
 center, o, and carrying three fine wires stretched quite tight, 
 the prolongation of which would pass exactly through the 
 center, o. The arms, bb, and c c, carry each a graduated 
 arc, b b, and c c ; and the arm a a carries two verniers, a, a', 
 adapted to the graduated arcs, b 6, and c c , respectively 
 
 The angles l o m, and lon, subtended, at one of the 
 points which we wish to put in, by l, m, and N, three of the 
 principal points of the survey already laid down, having been 
 observed by means of the prismatic compass, or pocket sex¬ 
 tant, the arms of the station pointer are opened out till the 
 verniers point out these angles upon the graduated limbs, b b, 
 and c c, respectively. The station pointer being then placed 
 upon the paper, and moved about till the fine wires pass ex¬ 
 actly over the stations l, m, n, as marked already upon the 
 plot, the center will be exactly over the point to be filled 
 in, and its place is to be marked by passing a pricking point 
 through a small opening, which is made at the center to 
 serve this purpose. 
 
 PART IV.—ON ASTRONOMICAL INSTRUMENTS. 
 
 The space which we have occupied with the former parts 
 of this work will compel us to be more brief than we had con¬ 
 templated in the description of the Astronomical Instruments. 
 We shall, therefore, confine our attention more particularly to 
 Hadley’s Sextant; 
 
THE SEXTANT. 
 
 13 / 
 
 Troughton’s Reflecting Circle; 
 
 The Transit Instrument; 
 
 The Collimator; 
 
 Altitude and Azimuth Instruments; 
 with some brief remarks upon other instruments, used for 
 astronomical purposes. 
 
 hadley’s sextant. 
 
 This instrument differs from the pocket sextant, already de¬ 
 scribed, in its appearance, from the absence of the box in which 
 the pocket sextant is fixed, in its size, varying usually from 4 
 inches to 6 inches radius, and in its requiring and admitting 
 of more perfect and minute adjustment. 
 
 ll is the graduated limb of the instrument, graduated 
 from 0° to 140° at every 10' or 20', according to the size of 
 the instrument, and subdivided by the vernier, v, to 10" or 
 20", thus enabling 
 us to read off an¬ 
 gles by estimation, 
 to 5". The limb 
 is also graduated 
 through a small 
 space, called the 
 arc of excess, on 
 the other side of 
 the zero point. 
 t is the tangent 
 screw for giving 
 a slow motion to 
 the index bar, 
 after it has been 
 clamped by a 
 screw at the back 
 of the instrument, 
 not shown in the 
 figure. m is a 
 microscope, attached to the index bar by an arm moveable 
 round the center, n, so as to command a view of the vernier 
 throughout its entire length. 
 
 i is the index glass, or first reflector, attached to and moving 
 with the index bar ; and h is the horizon glass, having its lower 
 half silvered to form the second reflector, and its upper half 
 transparent. Four dark glasses are placed at c, any one or 
 more of which can be turned down between the index glass 
 
1 JIB MATHEMATICAL INSTRUMENTS. 
 
 and horizon glass to moderate the intensity of the light from 
 any very bright object viewed by reflection ; and at g are three 
 dark glasses, any one or more of which can be turned up to 
 moderate the intensity of the light from any bright object, 
 viewed directly through the transparent part of the horizon 
 glass, d is a ring for carrying the telescope, attached to a stem 
 s, called the up and down piece, which can be raised or 
 lowered by means of a milled-headed screw. The use of this 
 up and down piece is to raise or lower the telescope, till the 
 objects seen directly and by reflection appear of the same 
 brightness, k is the handle by which the instrument is held. 
 
 In selecting an instrument care must be taken that all the 
 joints of the frame are close, without the least opening or 
 looseness, and that all the screws act well, and remain steady, 
 while the instrument is shaken by being carried from place 
 to place. All the divisions on the limb and vernier, when 
 viewed through the microscope, must appear exceedingly fine 
 and distinct, and the inlaid plates, upon which the divisions 
 are marked, must be perfectly level with the surface of the 
 instrument. The index, or zero, of the vernier, should also 
 be brought into exact coincidence successively with each 
 division of the limb, till the last division upon the vernier 
 reaches the last division upon the limb; and, if the last 
 division of the vernier do not in each case also exactly coin¬ 
 cide with a division upon the limb, the instrument is badly 
 graduated, and must be rejected. All the glass used in the 
 instrument should be of the best quality, and the glasses of the 
 reflectors should each have their faces ground and polished 
 perfectly parallel to each other, to avoid refraction. Look, 
 therefore, into each reflector, separately, in a very oblique 
 direction, and observe the image of some distant object; and 
 if the image appears clear and distinct in every part of the 
 reflector, the glass is of good quality; but if the image ap¬ 
 pears notched, or drawn with small lines, the glass is veiny, 
 
 I and must be rejected. Again, if the image appears singly, 
 and well defined about the edges, the two surfaces of the glass 
 are truly parallel; but if the edge of the image appears misty, 
 or separated like two images, the two surfaces are inclined to 
 one another. The examination will be more perfect, if the 
 image be examined with a small telescope. 
 
 A plain tube and two telescopes, one showing objects in¬ 
 verted, and the other erect, are usually supplied with the 
 sextant. The manner of testing the telescopes has already 
 been explained in the part of the work devoted to optical in- 
 
THE SEXTANT. 
 
 130 
 
 Btruments. A dark glass is also supplied to fit on to the eye- 
 end of the telescope, and a key for turning the adjusting 
 screws. 
 
 To examine the Error arising from the Imperfection of the 
 Dark Glasses. —Fit the dark glass to the eye-end of the tele¬ 
 scope, and, all the shades being removed, bring the reflected 
 image of the sun into contact with his image seen directly 
 through the unsilvered part of the horizon glass. Then re¬ 
 move the dark glass from the eye-end of the telescope, and, 
 setting up first each shade separately, and then their various 
 combinations, if the two images do not in any case remain 
 in contact, the angle through which the index must be moved 
 to restore the contact, is the error of the dark glass, or com¬ 
 bination of dark glasses, used in the observation, and which 
 error should be recorded for each glass and each combination 
 of the glasses. 
 
 The adjustments of the instrument consist in setting the 
 horizon glass perpendicular to the plane of the instrument, 
 and in setting the line of collimation of the telescope parallel 
 to the plane of the instrument. 
 
 To adjust the Horizon Glass. —While looking steadily at 
 any convenient object, sweep the index slowly along the limb, 
 and, if the reflected image do not pass exactly over the direct 
 image, but one projects laterally beyond the other, then the 
 reflectors are not both perpendicular to the face of the limb. 
 Now the index glass is fixed in its place by the maker, and 
 generally remains perpendicular to the plane of the instru¬ 
 ment, and, if it be correctly so, the horizon glass is adjusted by- 
 turning a small screw at the bottom of the frame in which it is 
 set, till the reflected image passes exactly over the direct image. 
 
 To examine if the Index Glass be perpendicular to the Plane 
 of the Instrument. —Bring the vernier to indicate about 45^, 
 and look obliquely into this mirror, so as to view the sharp 
 edge of the limb of the instrument by direct vision to the right 
 hand, and by reflection to the left. If, then, the edge and its 
 image appear as one continued arc of a circle, the index 
 glass is correctly perpendicular to the plane of the instrument: 
 but if the arc appears broken, the instrument must be sent to 
 the maker to have the index glass adjusted. 
 
 To adjust the Line of Collimation. — 1 . Fix the telescope in 
 its place and turn the eye-tube round, that the wires in the 
 focus of the eye-glass may be parallel to the plane of the in¬ 
 strument. 2. Move the index till two objects, as the sun and 
 moon, or the moon and a star, more than 90° distant from 
 each other, are brought into contact at the wire of the dia- 
 
140 
 
 MATHEMATICAL INSTRUMENTS. 
 
 phragm, which is nearest the plane of the instrument. 3. Now 
 fix the index, and altering slightly the position of the instru¬ 
 ment, bring the objects to appear on the other wire; and, if 
 the contact still remain perfect, the line of collimation is in 
 correct adjustment. If, however, the two objects appear to 
 separate at the wire that is further from the plane of the in¬ 
 strument, the object-end of the telescope inclines towards the 
 plane of the instrument; but, if they overlap, then the object- 
 end of the telescope declines from the plane of the instru¬ 
 ment. In either case the correct adjustment is to be obtained 
 by means of the two screws, which fasten to the up and down 
 piece the collar holding the telescope, tightening one screw 
 and turning back the other, till, after a few trials, the contact 
 remains perfect at both wires. 
 
 The instrument having been found by the preceding me¬ 
 thods to be in perfect adjustment, set the index to zero, and 
 if the direct and reflected images of any object do not perfectly 
 coincide, the arc, through which the index has to be moved to 
 bring them into perfect coincidence, constitutes what is called 
 the index error, which must be applied to all observed angles 
 as a constant correction. 
 
 To determine the Index Error .—The most approved method 
 is to measure the sun’s diameter, both on the arc of the instru¬ 
 ment, properly so called, to the left of the zero of the limb, and 
 on the arc of excess to the right of the zero of the limb. For 
 this purpose, firstly, clamp the index at about 30' to the left 
 of zero, and, looking at the sun, bring the reflected image of 
 his upper limb into contact with the direct image of his lower 
 limb, by turning the tangent screw, and set down the minutes 
 and seconds denoted by the vernier; secondly, clamp the 
 index at about 30' to the right of zero, on the arc of excess, 
 and, looking at the sun, bring the reflected image of his lower 
 limb into contact with the direct image of his upper limb, by 
 turning the tangent screw, and set down the minutes and 
 seconds denoted by the vernier underneath the reading before 
 set down. Then half the sum of these two readings will be 
 the correct diameter of the sun, and half their difference will 
 be the index error. When the reading on the arc of excess 
 is the greater of the two, the index error, thus found, must be 
 added to all the readings of the instrument; and when the 
 reading on the arc of excess is the less, the index error must 
 be subtracted in all cases. To obtain the index error with 
 the greatest accuracy, it is best to repeat the above operation 
 several times, obtaining several readings on the arc of the 
 instrument, and the same number on the arc of excess; and 
 
 
 l 
 
 
THE REFLECTING CIRCLE. 
 
 141 
 
 the difference of the sums of the readings in the two cases, 
 divided by the whole number of readings, will be the index 
 error; while the sum of all the readings, divided by their 
 number, will be the sun’s diameter. 
 
 EXAMPLE. 
 
 Readings on the Arc of Readings on the Arc of 
 Instrument. Excess. 
 
 35 
 
 
 29 
 
 25 
 
 35 
 
 5 
 
 29 
 
 35 
 
 35 
 
 10 
 
 29 
 
 20 
 
 105 
 
 15 
 
 88 
 
 20 
 
 88 
 
 20 
 
 105 
 
 15 
 
 No. of readings 6) 16 55 Difference. 6) 193 35 Sum. 
 
 2 49 Index error. 32 15'8 Sun’s diameter. 
 
 The readings on the arc of excess being less than those on 
 the arc of the instrument, the index error, 2' 49", is to be sub¬ 
 tracted from all the readings of the instrument. 
 
 Note. —In taking off the readings on the arc of excess, the vernier must 
 be read backwards; that is, the division read off on the limb, being the divi¬ 
 sion next to the left of the zero of the vernier, the divisions of the vernier to 
 be added must be reckoned from the other end of the vernier to the division 
 coinciding with a division upon the limb; or the reading of the vernier for¬ 
 wards, according to the usual method, may be subtracted from 10', the limb 
 being divided to 10’, and the remainder added to the reading of the division 
 upon the limb next to the left of the zero of the vernier, as before. 
 
 The manner of observing with the sextant has been already 
 explained, when treating of the pocket sextant (p. 120). 
 
 troughton’s reflecting circle. 
 
 In this instrument, which is the same in principle as the sex¬ 
 tant, the limb is a complete circle, l l l. It has three verniers, 
 v v v, one of which is furnished with the clamp and tangent 
 screw, s s, for regulating the contacts; and the verniers are 
 read by a magnifier, m, which may be applied successively to 
 all the verniers. In the middle of the frame, and attached to 
 it by a broad base or flanch, is a hollow center, upwards of two 
 inches long, in which an axis revolves. The triple vernier bar, 
 1 11 , is attached at one end of the axis, and the index glass at 
 the other, so that both turn together, but on opposite sides of 
 the instrument. A secondary frame, b b, carries the telescope, 
 t, the horizon glass, and the dark glasses, h h are two 
 handles, one of them bent, and passing round to the center of the 
 instrument on the other side; and there is a third handle, 
 
142 
 
 MATHEMATICAL INSTRUMENTS. 
 
 which can be screwed on perpendicular to the plane of the in¬ 
 strument, either into the handle at c, or upon the other side 
 of the instrument, at its center. The adjustments and man¬ 
 ner of observing with the instrument are explained by the 
 inventor, Mr. Troughton, as follows :— 
 
 Directions for observing with Troughton's Reflecting Circle.—“ Prepare 
 the instrument for observation by screwing the telescope into its place, adjust¬ 
 ing the drawer to focus, and the wires parallel to the plane, exactly as you 
 do with a sextant: also set the index forwards to the rough distance of the 
 sun and moon, or moon and star; and, holding the circle by the short handle, 
 direct the telescope to the fainter object, and make the contact in the usual 
 way. Now read off the degree, minute, and second, by that branch of the 
 index to which the tangent screw is attached; also, the minute and second 
 shown byt he other two branches; these give the distance taken on three 
 different sextants; but as yet it is only to be considered as half an observa¬ 
 tion : what remains to be done, is to complete the whole circle, by measuring 
 that angle on the other three sextants. Therefore set the index backwards 
 nearly to the same distance, and reverse the plane of the instrument, by hold¬ 
 ing it by the opposite handle, and make the contact as above, and read off as 
 before what is shown on the three several branches of the index. The mean 
 of all six is the true apparent distance, corresponding-to the mean of the two 
 times at which the observations were made. 
 
 “ When the objects are seen very distinctly, so that no doubt whatever re¬ 
 mains about the contact in both sights being perfect, the above may safely be 
 relied on as a complete set;, but if, from the haziness of the air, too much 
 motion, or any other causes, the observations have been rendered doubtful, 
 it will be advisable to make more : and if, at such times, so many readings 
 should be deemed troublesome, six observations, and six readings, may be 
 conducted in the manner following :—Take three successive sights forwards, 
 exactly as is done with a sextant; only take care to read them off on different 
 
THE REFLECTING CIRCLE. 
 
 143 
 
 branches of the index. Also make three observations backwards, using the 
 same caution ; a mean of these will be the distance required. When the num¬ 
 ber of sights taken forwards and backwards are unequal, a mean between the 
 means of those taken backwards and those taken forwards, will be the true 
 angle. 
 
 “ It need hardly be mentioned, that the shades, or dark glasses, apply like 
 those of a sextant, for making the object nearly of the same brightness ; but 
 it must be insisted on, that the telescope should, on every occasion, be raised 
 or lowered, by its proper screw, for making them perfectly so.” 
 
 The foregoing instructions for taking distances apply 
 equally for taking altitudes by the sea or artificial horizon, 
 they being no more than distances taken in a vertical plane. 
 Meridian altitudes cannot, however, be taken both backwards 
 and forwards the same day, because there is not time; all, 
 therefore, that can be done is, to observe the altitude one way, 
 and use the index error; but, even here, you have a mean of 
 that altitude, and this error taken on three different sextants. 
 Both at sea and land, where the observer is stationary, the 
 meridian altitude should be observed forwards one day, and 
 backwards the next, and so on alternately from day to day; 
 the mean of latitudes, deduced severally from such observa¬ 
 tions, will be the true latitude; but in these there should be 
 no application of index error, for that being constant, the re¬ 
 sult would in some measure be vitiated thereby. 
 
 “ When both the reflected and direct images require to be darkened, as is 
 the case when the sun’s diameter is measured, and when his altitude is taken 
 with an artificial horizon, the attached dark glasses ought not to be used: in¬ 
 stead of them, those which apply to the eye-end of the telescope will answer 
 much better; the former having their errors magnified by the power of the 
 telescope, will, in proportion to this power, and those errors, be less distinct 
 than the latter. 
 
 “ In taking distances, when the position does not vary from the vertical 
 above thirty or forty degrees, the handles which are attached to the circle are 
 generally most conveniently used; but in those which incline more to the 
 horizontal, that handle which screws into a cock on one side, and into the 
 crooked handle on the other, will be found more applicable. 
 
 “ When the crooked handle happens to be in the way of reading one of the 
 branches of the index, it must be removed, for the time, by taking out the 
 finger screw, which fastens it to the body of the circle. 
 
 “ If it should happen that two of the readings agree with each other very 
 well, and the third differs from them, the discordant one must not on any ac¬ 
 count be omitted, but a fair mean must always be taken. 
 
 " It should be stated, that when the angle is about thirty degrees, neither 
 the distance of the sun and moon, nor an altitude of the sun, with the sea 
 horizon, can be taken backwards; because the dark glasses at that angle 
 prevent the reflected rays of light from falling on the index glass; whence it 
 becomes necessary, when the angle to be taken is quite unknown, to observe 
 forwards first, where the whole range is without interruption ; whereas in 
 that backwards you will lose sight of the reflected image about that angle. 
 But in suoh distances, where the sun is out of the question, and when his 
 
144 
 
 MATHEMATICAL INSTRUMENTS. 
 
 altitude is taken with an artificial horizon (the shade being applied to the 
 end of the telescope), that angle may be measured nearly as well as any 
 other; for the rays incident on the index glass will pass through the trans¬ 
 parent half of the horizon glass without much diminution of their brightness. 
 
 “ The advantages of this instrument, when compared with the sextant, 
 are chiefly these : the observations for finding the index error are rendered 
 useless, all knowledge of that being put out of the question, by observing 
 both forwards and backwards. By the same means the errors of the dark 
 glasses are also corrected; for if they increase the angle one way, they must 
 diminish it the other way by the same quantity. This also perfectly corrects 
 the errors of the horizon glass, and those of the index g'ass very nearly. But 
 what is of still more consequence, the error of the center is perfectly corrected 
 by reading the three branches of the index; while this property, combined 
 with that of observing both ways, probably reduces the errors of dividing to 
 one-sixth part of their simple value. Moreover, angles may be measured as 
 far as one hundred and fifty degrees, consequently the sun’s double altitude 
 may be observed when his distance from the zenith is not less than fifteen 
 degrees ; at which altitude the head of the observer begins to intercept the 
 rays of light incident on the artificial horizon; and, of course, if a greater 
 angle could be measured, it would be of no use in this respect. 
 
 “ This instrument, in common with the sextant, requires three adjustments: 
 first, the index glass perpendicular to the plane of the circle. This being 
 done by the maker, and not liable to alter, has no direct means applied to 
 the purpose; it is known to be right when, by looking into the index glass, 
 you see that part of the limb which is next you reflected in contact with the 
 opposite side of the limb as one continued arc of a circle : on the contrary, 
 when the arc appears broken where the reflected and direct parts of the limb 
 meet, it is a proof that it wants to be rectified. The second is, to make the 
 horizontal glass perpendicular. This is performed by a capstan screw, at the 
 lower end of the frame of that glass; and is known to be right when by a sweep 
 of the index the reflected image of any object will pass exactly over, or cover 
 the image of that object seen directly. The third adjustment is for making 
 the line of collimation parallel to the plane of the circle. This is performed by 
 two small screws, which also fasten the collar into which the telescope screws 
 to the upright stem on which it is mounted ; this is known to be right when 
 the sun and moon, having a distance of one hundred and thirty degrees, or 
 more, their limbs are brought in contact, just at the outside of that wire which 
 is next to the circle, and then examining if it be just the same at the outside 
 of the other wire : its being so is the proof of adjustment.” 
 
 Another instrument of Troughtons construction upon the 
 principle of the sextant is the dip sector, for measuring the 
 dip of the horizon. Any person who is thoroughly acquainted 
 with the sextant will find no difficulty in using it, after a few 
 words of explanation from the maker. 
 
 THE TRANSIT INSTRUMENT. 
 
 The reflecting instruments, which we have just described, 
 from their portability and the promptitude and facility with 
 which they may be used in all situations, and upon all occa¬ 
 sions, a-re most useful instruments to the surveyor. The 
 sextant or reflecting circle, with an artificial horixon, and a 
 
T1IE PORTABLE TRANSIT. 
 
 145 
 
 good chronometer, forms, in fact, a complete observatory, with 
 which the latitudes and longitudes of places may be deter¬ 
 mined to a great degree of accuracy; while to the navigator 
 a reflecting instrument is indispensable; all other instruments 
 requiring to be supported upon a stand perfectly at rest*, 
 while the sextant and similar instruments are held in the 
 hand, and perform their duty well on the deck of a rolling 
 ship. In permanent observations, however, the capital an¬ 
 gular instruments are placed permanently in the plane of the 
 meridian, and the measurements sought for by their aid are 
 the exact times at which the observed objects pass the meri¬ 
 dian, and their angular altitudes or zenith distances, when 
 upon the meridian. The instrument with which the first of 
 these measurements are obtained is called a transit instru¬ 
 ment, transit telescope , or merely a transit. Transits of port¬ 
 able dimensions, besides their use in small or temporary ob¬ 
 servatories, are also found serviceable to the surveyor, for 
 determining, with the greatest possible accuracy, the true 
 north point, and thence setting out a line in any required 
 direction; and to the scientific traveller, for determining the 
 longitude of any place from astronomical observations, and for 
 adjusting his time-keepers with greater accuracy than can be 
 obtained by his sextant or reflecting circle. The annexed 
 figure represents a portable transit. 
 
 t t is a telescope formed of two parts, connected by a 
 spherical center-piece, into which are fitted the larger ends 
 of two cones, the common axis of which is placed at right 
 angles to the axis of the telescope, to serve as the horizontal 
 axis of the instrument. The two small ends of these cones 
 are ground into two perfectly equal cylinders, called pivots. 
 The pivots rest upon angular hearings or Ys. The Ys are sup¬ 
 ported upon the standards e and w, of which e may be called 
 the eastern, and w the western standard ; and one of the Ys is 
 fixed in a horizontal groove, on the western standard^ so that, 
 by means of the screw s, one end of the axis may be pushed a 
 little fonvards or backwards, and a small motion in azimuth 
 be thus communicated to the telescope f. The standards, e 
 
 * In observatories the instruments are supported by stone walls, or pil¬ 
 lars, which pass below the floors, without touching them, or any port of the 
 building, and are consequently independent of any tremor, communicated to 
 the floor or walls of the buildings. It was considered that the passage of a 
 railway through Greenwich Park would impair the observations at the lloyal 
 Observatory, by communicating a tremor to the ground. 
 
 + The large transits in permanent observations have their Ys placed in 
 two dove-tailed grooves, one horizontal, and the other vertical. By means 
 
 H 
 
146 
 
 MATHEMATICAL INSTRUMENTS. 
 
 and w, are fixed by screws upon a brass circle, o o, and steadied 
 by oblique braces, b b, which spring from the cross-piece, c. 
 
 On one end of the axis is fixed, so as to revolve with the 
 axis, a vertical circle, v v; and a double index bar, furnished 
 
 niers, n ?i, adapted to the vertical circle, and showing the 
 angle of elevation of the telescope. The index-bar is fixed in 
 its position by the clamping screw, c, and can be fixed upon 
 
 of the latter one end of the axis may be raised or depressed; but in 
 the portable transit the same object is attained by turning one of the foot 
 screws upon which the entire instrument rests. 
 
ADJUSTMENTS OF THE TRANSIT 
 
 147 
 
 either the eastern or western standard, at pleasure, while the 
 telescope, with its attached circle, can also be lifted out of, and 
 have its position reversed in, the Ys. The pivot, which does 
 not carry the vertical circle, is pierced, and allows the light 
 from a lamp to fall upon a plane speculum, fixed, in the 
 spherical center piece, on the axis of the telescope, and in¬ 
 clined to this axis at an angle of 45°. The light is thus 
 thrown directly down the telescope, and illuminates the wires 
 of the diaphragm, placed in the principal focus of the tele¬ 
 scope. Of these wires, one is horizontal; and a vertical wire, 
 intersecting it in the center of the field of view, gives, by its 
 intersection with it, the collimating point. There are, then, 
 other vertical wires arranged in pairs equidistant from the 
 central vertical wires, so that we have either three, or five, or 
 seven vertical wires, the most common number being five. 
 The lamp has a contrivance for regulating the quantity of 
 light thrown into the telescope, by turning a screw, so that 
 the light from a small “star may not be overpowered by the 
 superior light of the lamp. 
 
 The requisites of a good instrument, are—lstly, that the 
 telescope be of the best quality, which is to be tested by the 
 methods already given (pp. 86-88); 2ndly, that the feet 
 screws act well and remain steady; 3rdly, that all the 
 screws, by which the instrument is put together, are turned 
 home, and remain so, after the instrument has been shaken 
 by carriage; 4thly, that the length of the axis be just suffi¬ 
 cient to reach from one Y to the other, without either friction 
 or liberty; 5thly, that the lamp be held so as not to require 
 adjustment for position ; 6thly, that the screws of adjustment 
 of the diaphragm, and Ys, be competent to give security of po¬ 
 sition to the parts adjusted by them; 7thly, that the metal¬ 
 lic parts be free from flaws in casting, and that the pivots be 
 formed of hard bell metal and incapable of rusting. 
 
 The principal adjustments of the transit are three ':— 
 
 1st. To make the axis on which the telescope moves horizontal. 
 
 2nd. To make the line of collimation move in a great vertical circle, by 
 setting it perpendicular to the horizontal axis. 
 
 3rd. To make it move in that vertical circle, which is the meridian. 
 
 To make the Axis Horizontal .—Apply to the pivots the 
 large level, ll, which is supplied with the instrument for this 
 purpose, and is either constructed to stand upon the pivots, 
 in which case it is called a striding level, or of the form 
 shown at page 96, in which case it is suspended from the 
 pivots, and is called a hanging level. Bring the air bubble 
 r n 2 
 
MATHEMATICAL INSTRUMENTS 
 
 U8 
 
 to the center of its run, by turning the foot screw,/. Turn 
 the level end for end, and, if the air bubble retains its posi¬ 
 tion, the axis is horizontal, but, if not, it must be brought back 
 half by the foot screw,/, and the other half by turning the 
 small screw at one end of the level. Repeat the operation 
 till the bubble retains the same position in both positions of 
 the level, and the axis will be horizontal. 
 
 To adjust the Line of Collimation in Azimuth .—Direct the 
 telescope to some distant, small, and well-defined object, and 
 bisect it by one extremity of the middle vertical wire, giving 
 the telescope the azimuthal motion necessary for this purpose 
 by turning the screw s. By elevating or depressing the tele¬ 
 scope, examine whether the object is bisected by every part 
 of the middle vertical wire; and if not, loosen the screws 
 which hold the eye-end of the telescope in its place, and turn 
 the end round verv carefully till the error is moved. Lift 
 
 r r 
 
 the transit off the Ys, and reverse it, so that the end of the 
 axis, which was upon the eastern Y, may now be upon the 
 western, and vice versa; and, if the object is still bisected by 
 the central vertical wire, the collimation in azimuth is per¬ 
 fect ; but, if not, move the center of the cross wires half 
 way towards the object by turning the small screws which 
 hold the diaphragm, and, if this half distance has been cor¬ 
 rectly estimated, the adjustment will be accomplished. Again, 
 bisect the object by the center of the cross wires by turning the 
 azimuthal screw s, and repeat the operation, till the object is 
 bisected by the center of the cross wires in both positions 
 of the instrument, and the adjustment will be known to be 
 perfect *. 
 
 To adjust the Transit to the Meridian .—The line of collima¬ 
 tion by reason of the previous adjustment describes a vertical 
 circle, and, therefore, bisects the zenith, which is one point in 
 the meridian. If, then, we can make it also bisect another poiut 
 in the meridian, it will move entirely in the meridian. Com¬ 
 pute from the tables in the Nautical Almanack, the time of 
 Polaris coming to the meridian, and at the computed time 
 bisect the star by the middle vertical wire, and the transit 
 will be very nearly adjusted to the meridian. 
 
 To make the great vertical circle described by the line of 
 collimation more nearly coincident with the meridian, let the 
 intervals between the successive passages of Pularis across 
 
 * The horizontal motion given to the Y, by the azimuthal screw s, forms, 
 evidently, no p.irt of the adjustment for collimation, but only enables us to 
 examine if the adjustment has been made with sufficient exactness. 
 
ADJUSTMENTS OF THE TRANSIT. 
 
 149 
 
 the meridian be observed, as indicated by the instrument. 
 Then, if the interval between the inferior and superior pas¬ 
 sage be equal to the interval between the superior and infe¬ 
 rior, the adjustment to the meridian is perfect; but if the 
 interval between the inferior and superior passage be less than 
 the interval between the superior and inferior, the circle 
 described by the line of collimation deviates to the eastward of 
 the true meridian, from the zenith to the north point of the 
 horizon, and to the westward, from the zenith to the south 
 point of the horizon; while if the interval between the infe¬ 
 rior and superior passage be the greater, the deviation is in 
 the contrary directions. 
 
 Let £ be the observed difference of the intervals from twelve 
 hours, or half the difference between the two intervals in 
 seconds, t the polar distance of the star Polaris, and l the 
 latitude of the place, then, z representing the deviation from 
 the meridian in time, the value of z will be given by the 
 logarithmic formula, 
 
 log. z = log. i 0 g. sec . l log. tan. r — 20. 
 
 A 
 
 EXAMPLE. 
 
 Place of observation, Cambridge, latitude 52° 12' 38 r/ . 
 
 Polar distance of Polaris, 1° 39' 25 '05. 
 
 Difference of intervals from 12 hours 7'" 22* = 442*. 
 
 $ 
 
 - = 221.log. = 2-3443923 
 
 l = 52° 12' 36".log. sec. = 10 2127030 
 
 *-= 1° 39' 25 '‘05...log. tan. = 84513064 
 
 z = 10*495.log. = 21 0084017 
 
 To determine the value of a revolution of the azimuthal screw, 
 s, the time* of passage of an equatorial star across the middle 
 vertical wire must be noted one day; and then, turning the 
 screw, s, once round, the time of passage* must be noted 
 again ; and the difference of these times will be the value in 
 time of a revolution of the screw. Suppose the difference 
 thus observed to amount to two seconds, then the value of 
 one complete revolution of the screw, s, is two seconds, and 
 the value of the motion of the adjusting screw being thus ob¬ 
 tained. must be reduced to the horizon, by increasing it in 
 the ratio of cosine of latitude to radius, and may then be ap¬ 
 plied to correct the error of deviation as found above. 
 
 * The time here spoken of, and throughout the description of this instru¬ 
 ment, unless otherwise expressly stated, is sidereal, and not mean time. 
 
150 
 
 MATHEMATICAL INSTRUMENTS. 
 
 A second method, founded on the same principles as the 
 preceding, consists in observing the pole star, and another 
 star, which crosses the meridian near the zenith of the place 
 of observation. The time of passage of such a star, Capella, 
 for instance, when near its superior transit, across the middle 
 wire of the telescope, will differ but very little from the time 
 I of passing the true meridian, if the deviation of the instru¬ 
 ment from the meridian be but small. Assume the two 
 times to agree exactly, and the difference between the times 
 of superior transit of Capella and Polaris will be the differ¬ 
 ence of the observed right ascensions of these two stars. 
 From this difference subtract the difference of the computed, 
 or catalogued, right ascensions of the two stars, and call the 
 result d ; and the deviation will be given by the formula, 
 log. z = log. d -f- log. sin. 7T -f log. sec. (l + t) ; 
 
 7r being the polar distance of Polaris, and l the latitude of 
 the place of observation. From Capella not having been 
 exactly on the meridian, when on the middle vertical wire, 
 the value of d, as above obtained, is only an approximation 
 to the error of the observed right ascension of Polaris, and 
 the deviation computed from it will be only approximately 
 correct; but, by repeating the operation, the adjustment may 
 be completely perfected. 
 
 d is actually the value of the sum of the errors of the ob¬ 
 served right ascensions of Capella and Polaris, and hence 
 the value of z will be correctly given, by so considering it, 
 instead of supposing as above, that this error for Capella is 
 zero. The true deviation then is given by the formula, 
 
 log. z = log. d -f- log. sin. or -j- log. sin. or' -)- log. cosec. (or' — or) -f- log. sec. L ; 
 
 7 / being the polar distance of Capella. 
 
 Using this last formula, the method may be applied to 
 Polaris, and any star distant from the pole, or to any two stars 
 differing from each other not less than 40° in declination. If, 
 however, the transit of one star is observed above, and of the 
 other, below the pole, the formula will be 
 
 log. z = log. D -f log. sin. !T -f- log. sin. <r -f log. cosec. («r' + «•) + log. sec. L. 
 
 Considerable advantage may be obtained by selecting two 
 stars that differ but little in right ascension, as there is then 
 the less probability of error from a change in the rate of the 
 clock, or in the position of the instrument, on which account 
 such methods are to be preferred in temporary observatories, 
 where the stability of the instrument is not to be depended 
 upon for any length of time. 
 
USE ‘OF THE TRANSIT. 
 
 151 
 
 . In all the preceding formulae, the deviation from the me¬ 
 ridian is given in time; but, to convert it into angular measure, 
 if desirable, we have only to multiply by 15, and the seconds 
 of time will be converted into seconds of a degree. 
 
 When the instrument is by any of the methods explained 
 above brought into the meridian, a distant mark may be set 
 up in the plane of the meridian, by which the adjustment to 
 the meridian may afterwards be tested. 
 
 METHOD OF OBSERVING WITH THE TRANSIT. 
 
 The adjustments having been completed, in making obser¬ 
 vations with the instrument, the instant of a star’s passing 
 the middle vertical wire will be the time of the stars transit; 
 but the time of the star’s passing all the five wires must be 
 noted, and the mean of the times, taken as the time of transit, 
 will be a more accurate result than the time observed at the 
 middle wire only. 
 
 When the sun is the object observed, the time of the center 
 of his disc passing the middle wire is the time of transit; but, 
 as it would be impossible to estimate the center with accu¬ 
 racy, the time of both his limbs coming into contact with each 
 wire in succession is to be noted, and a mean of all these times 
 will be the time of transit required. This mean may be con¬ 
 veniently taken, by writing the observed times of contact of the 
 first and second limbs underneath each other in the reverse 
 order, when the sums of each pair will be nearly equal *. 
 
 EXAMPLE. 
 
 1826, Sept. 23 
 
 20-4 
 
 38*7 
 
 h 
 
 11 
 
 m 
 
 58 
 
 57-0 
 
 S 
 
 15*5 
 
 33-7 
 
 Q 1 Limb. 
 
 
 42-3 
 
 24-0 
 
 12 
 
 1 
 
 5-7 
 
 47-2 
 
 28-7 
 
 0 2 Limb. 
 
 
 2-7 
 
 2-7 
 
 24 
 
 0 
 
 2-7 
 
 2-7 
 
 2-4 
 
 The sum = 13*2 
 
 The time of either limb passing the center wire is recorded 
 in full, but for the other wires, the seconds only are recorded, 
 as the sums of the several pairs only differ by decimals of a 
 second. Half the sum of the times at the middle gives, then, 
 the correct time of transit as far as the seconds, and the deci¬ 
 mals are found by removing the decimal point one place to 
 the left in the sum 13*2, which is equivalent to dividing by 
 10. Then the time of transit, or mean of observations in the 
 above example, is J.2 h 0 m l s *32. This example is taken from 
 observations made with a large transit; and, if with a smaller 
 
 * This is Dr. Pearson’s method. . 
 
152 
 
 MATHEMATICAL INSTRUMENTS. 
 
 instrument the sums of the several pairs of observations 
 should differ by more than a second, it will be necessary to 
 take the sums of both figures of the seconds, and the division 
 by 10, performed as above, will give the last figure of the se¬ 
 conds, as well as the decimals. 
 
 Jn taking transits of the moon the luminous edge alone can 
 be observed, from which the time of transit of the center must 
 be deduced by the aid of Lunar tables. 
 
 In observing the larger planets, one limb may be observed 
 at the first, third, and fifth wires, and the other at the second 
 and fourth, and the mean of these observations will give the 
 transit of the planet's center. 
 
 It will sometimes happen that from the state of weather, or 
 from some other cause, a heavenly body may not have been 
 observed at all the wires; but, if the declination of the body 
 be known, au observation at any one of the wires may be re¬ 
 duced to the central wire, so as to give the time of transit, as 
 deduced from this observation. If an observation be ob¬ 
 tained at more than one wire, the mean of the times of pass¬ 
 ing the center, as deduced from each wire observed, is to be 
 taken as the time of transit. The reduction to the center 
 wire is given by the formula, 
 
 r = v cosec. it, 
 
 or log. r = log. v -flog, cosec. it; 
 
 in which R represents the reduction, it the polar distance of 
 the body observed, and v the equatorial interval from the 
 wire, at which the observation has been made, to the central 
 wire. The equatorial intervals for each side wire must, there¬ 
 fore, be carefully observed, and tabulated for the purpose of 
 this reduction. The formula r=v cosec. it is only an approxi¬ 
 mate value of the reduction, and with large instruments capable 
 of giving results within 0"-05, a further correction is necessary 
 for bodies within 10° of the pole. The whole reduction in 
 this case is given by the formula, 
 
 b' = — s in. 1 (cosec. it sin. 15 v). 
 
 11 ) 
 
 The time of any star’s passage from one of the side wires to 
 the center wire being observed, the equatorial interval from 
 that wire to the center is obtained by multiplying the ob¬ 
 served interval bv the sine of the star’s polar distance; and 
 the equatorial intervals being deduced in this manner from a 
 great many stars, the mean of the results may be considered 
 as very correct values of the equatorial intervals required 
 
THE ALTITUDE AND AZIMUTH INSTRUMENT. 
 
 153 
 
 No star very near the pole should, however, be taken for this 
 purpose. 
 
 USE OF THE PORTABLE TRANSIT. 
 
 The large transits in permanent observatories are used to 
 obtain, with the greatest possible accuracy, the right ascen¬ 
 sions of the heavenly bodies, from which, and the meridian 
 altitudes observed by a mural circle, an instrument consisting 
 of a telescope attached to a large circle, and placed in the 
 plane of the meridian, nearly all the data necessary for every 
 astronomical computation are obtained. For such purposes 
 the small portable transit is not adapted; but it is competent 
 to determine the time to an accuracy of half a second, to 
 determine the longitude by observations of the moon and 
 moon culminating stars, and to determine the latitude by- 
 placing it at right angles to the meridian, or in the plane of 
 the prime vertical*. 
 
 The transit of the sun’s center gives the apparent noon at 
 the place of observation, and the mean time at apparent noon is 
 found by subtracting or adding the equation of time, as found 
 in the Nautical Almanack, to *24 hoursf. The difference 
 between the mean time, thus found, and the time of the sun’s 
 transit, as shown by a clock or chronometer, is the error of 
 the clock or chronometer for mean time at the place of obser¬ 
 vation. 
 
 The time shown by a sidereal clock when any heavenly 
 body crosses the meridian should coincide with the right 
 ascension of that body, as given in the Nautical Almanack. 
 The difference between the time shown by the sidereal clock, 
 at the transit, and the right ascension of the body, taken 
 from the almanack, will, therefore, be the error of the clock, 
 -{-, or too fast, when the clock time is greater than the right 
 ascension, —, or too slow, when it is less. 
 
 THE PORTABLE ALTITUDE AND AZIMUTH INSTRUMENT. 
 
 The bending of an unbraced telescope renders it unfit for 
 the determination of altitudes; but by placing the telescope 
 
 * The prime vertical is the great circle which passes through the zenith 
 and the east and west points of the horizon. 
 
 + The astronomical day commences at noon, and conta.ns 24 hours, the 
 hours after midnight being called 13, 14, &c., and the day ends at the next 
 noon. The equation of time is given in the Nautical Almanack for ap¬ 
 parent noon at the meridian of Greenwich, and the correction to give the 
 equation of time at any other meridian will be found by multiplying the 
 difference for one hour, as given in the almanack, by the longitude of the 
 place, estimated in time. 
 
 H 3 
 
154 
 
 MATHEMATICAL INSTRUMENTS. 
 
 between two circles braced together, an instrument maybe 
 formed capable pf observing both the meridian altitudes and 
 times of transit of the heavenly bodies. The increased weight 
 of the instrument, however, must now be prevented from pro¬ 
 ducing flexure in the horizontal axis, and this has been very 
 ingeniously accomplished by Troughton. By mounting the 
 transit and altitude instrument, as Troughton’s transit-circle 
 may be called, upon a horizontal plate or circle having an azi 
 muthal motion round a vertical axis, an instrument is formed 
 by which observations may be made either in or out of the 
 meridian. When constructed of a portable size, the altitude 
 and azimuth instrument may also be used in important sur¬ 
 veying operations; for, in fact, it may be considered as a 
 rather large theodolite of superior construction. 
 
 The altitude and 
 azimuth instrument 
 may be considered as 
 consisting of three 
 parts: 1, the tripod 
 carrying the vertical 
 axis about which the 
 instrument turns; 2, 
 the horizontal revolv¬ 
 ing plate carrying the 
 vertical pillars, with 
 their appendages; and 
 3, the vertical circles 
 with the telescope. 
 
 The tripod, a a, is 
 supported by three 
 footscrews, by which 
 the vertical axis is 
 brought into adjust¬ 
 ment, and carries the 
 lower horizontal plate, 
 which is graduated to 
 show the azimuths or 
 horizontal angles. The 
 vertical axis is a solid 
 metallic cone rising 
 from the center of the 
 tripod to a height about 
 equal to the radius of 
 the horizontal circle. 
 
THE ALTITUDE AND AZIMUTH INSTRUMENT. 
 
 153 
 
 The upper horizontal plate, or horizontal revolving plate, 
 vv, carries an index, to point out the graduation, upon the 
 lower horizontal plate, or azimuth circle, which denotes nearly 
 the angle to be read off. The graduations upon the azimuth 
 circle, as well as upon the vertical circle, are subdivided by 
 reading microscopes, the construction and adjustments of 
 which we shall presently explain. The reading microscopes 
 of the azimuth circle are attached to the revolving plate, v v, 
 which also carries two upright pillars. From the center 
 of the upper horizontal plate, vv, rises a hollow brass cone 
 which just fits over, and moves smoothly upon the solid 
 metallic vertical axis rising from the tripod stand. A hori¬ 
 zontal brace connects the two upright pillars with one an¬ 
 other and with the top of the hollow brass cone, and keeps 
 the pillars firm and parallel to one another. On the top of 
 each pillar a gibbet piece is fixed, projecting beyond the 
 pillars, and upon the extreme ends of these pieces are carried 
 the Ys for supporting the pivots of the horizontal, or transit 
 axis. The Ys are each capable of being raised or lowered by 
 turning a milled-headed screw. The top of one of the pillars 
 carries a cross-piece for supporting the two reading micro¬ 
 scopes of the vertical circle; and to this cross-piece is attached 
 the level, ll, by which the adjustment of the vertical axis is 
 denoted. 
 
 The third portion of the instrument consists of the vertical 
 circle and its telescope. This circle consists of two limbs 
 firmly braced together, and preventing any tendency to flexure 
 in the tube of the telescope, by affording it support at the op¬ 
 posite ends of a diameter. One of the limbs only is gradu¬ 
 ated, and the graduated side is called the face of the instru¬ 
 ment, and the clamp and tangent screw, for giving a slow 
 motion to the vertical circle, act upon the ungraduated limb, 
 and are fixed to the vertical pillar on the side of that limb. 
 The horizontal axis which supports the telescope and vertical 
 circle is constructed exactly as the axis of the transit instru¬ 
 ment already described; but, as it might press too heavily on 
 the Ys from the increased load of the vertical circle, a spiral 
 spring, fixed in the body of each pillar, presses up a friction 
 roller against the conical axis with a force which is nearly a 
 counterpoise to its weight. The adjustment of the horizontal 
 axis is denoted by a striding level, as in the portable transit 
 already described. 
 
156 
 
 MATHEMATICAL INSTRUMENTS 
 
 ADJUSTMENTS. 
 
 Adjustments of the Vertical Axis. —Turn the instrument 
 round till the level, ll, is over two of the foot-screws, and 
 adjust the level, so that its bubble may retain the same 
 position, when the instrument is turned half round, so that 
 the level is again over the same foot-screws, but in the re¬ 
 verse position. The error at each trial is corrected, as nearly 
 as can be judged, half by the foot-screws, and half by the 
 adjusting screw of the level itself. 
 
 Next turn the instrument round 90° in azimuth, so that 
 the level, l l, may be at right angles to its former positions, 
 and bring the bubble to the same position as before, by turn¬ 
 ing the third foot-screw. Repeat the whole operation till the 
 result is satisfactory. 
 
 Adjustment of the Horizontal Axis. —This adjustment is 
 performed in the same manner, as already described for the 
 transit instrument (p. 147), with the single exception that one 
 end of the axis is to be raised or lowered, if necessary, by the 
 screw acting upon its Y, and not by moving a foot-screw, 
 which would derange the previous adjustment. 
 
 Adjustment of the Circle to its Reading Microscopes. —This 
 is performed by raising or lowering both the Ys equally, so as 
 not to derange the previous adjustment, till the microscopes 
 are directed to opposite points in its horizontal diameter. 
 
 Adjustment of Col Umation in Azimuth. —Instead of taking 
 the axis out of its bearings and turning it end for end, the 
 whole instrument is turned round in azimuth; but in all 
 other respects the method of performing this adjustment 
 is the same as that already described for the transit instru¬ 
 ment (p. 148). 
 
 Adjustment of Collimatiofl in Altitude. —Point the tele¬ 
 scope to a very distant object, or star, and, bisecting it by the 
 cross wires, read off the angle upon the vertical circle denoted 
 by the reading microscopes. Turn the instrument half round 
 in azimuth, and, again bisecting the same object by the cross 
 wires, read off the angle. One of these readings will be 
 an altitude, and the other a zenith distance *, and their sum, 
 therefore, when there is no error of collimation in altitude, 
 will be 90°. If the sum is not 90°, half its difference from 90° 
 
 * Both the horizontal and vertical circles are usually divided alike into 
 four quadrants, and each quadrant graduated from 0' to 90°, proceeding in 
 the same direction all round the circles. 
 
THE ALTITUDE AND AZIMUTH INSTRUMENT. 157 
 
 will be the error of collimation in altitude, and this error 
 being added to, or subtracted from, the observed angles, ac¬ 
 cording as the sum of the readings is less or greater than DO 0 , 
 will give the true zeniih distance and altitude. The error of 
 collimation in altitude may then be corrected by adjusting 
 the microscopes to read the true zenith distance and altitude, 
 thus found, while the object is bisected by the cross wires of 
 the telescope. The error of collimation of this and other 
 astronomical instruments may also be found, or corrected, by 
 the collimator. 
 
 Use of the Altitude and Azimuth Instrument .—In using the 
 altitude and azimuth instrument, for astronomical purposes, 
 double observations should always be made, with the face first 
 to the east, and then to the west, or vice versa , or several ob¬ 
 servations may be made with the face to the east, and as many 
 with the face to the west, and the mean of the results, reduced 
 to the meridian, taken as the true results. The place for a 
 meridian mark may be determined by the methods already ex¬ 
 plained when describing the transit instrument, or by observing 
 the readings of the azimuthal circle, or noting the times, when 
 any celestial object has equal altitudes. Since the diaphragm 
 of the telescope is furnished not only with the central hori¬ 
 zontal wire, but with other horizontal wires at equal distances 
 above and below it, so that there may be altogether either three, 
 or five, or seven horizontal wires, the azimuths and times mav 
 be observed, when the object observed is bisected by each of 
 these wires. If a fixed star be the object observed, the mean 
 of the times will give the time of the star’s passing the 
 meridian, and the mean of the azimuths will give the reading 
 of the azimuth circle when the star was on the meridian, or the 
 correction to be applied to the readings of the azimuth circle 
 to give the true azimuths. If the sun be the body observed, 
 a correction is necessary on account of the change of his de¬ 
 clination, during the intervals between the observations. 
 
 The correction for the time, as deduced from a pair of equal 
 altitudes of the sun, is given by the formula, 
 
 t 
 
 Correction = _L v ?-(tan. D X cos. tan. l.) 
 
 720 x sin. 15°. t_ * 
 
 2 
 
 in which $ represents the variation in the sun’s declination from the noon of 
 the day preceding the observations to the noon of the day suc¬ 
 ceeding ; 
 
 t represents the interval between the observations expressed in 
 hours and decimals of an hour; 
 
158 
 
 MATHEMATICAL INSTRUMENTS. 
 
 D represents the sun’s declination at noon on the day on which the 
 observations are made; 
 
 L represents the latitude of the place. 
 
 5 is to be reckoned positive when the sun’s declination is increasing, and 
 negative when it is decreasing. 
 
 The correction for azimuth is given by the formula, 
 
 15 
 
 Correction = \ (d' — d) sec. lat. cosec. — (t' — t). 
 
 in which D represents the change of the sun’s declination, 1 between the 
 and t' — t represents the interval in time, J observations. 
 
 When the sun is advancing towards the North Pole, this 
 correction will carry the middle point towards the west of the 
 approximate south point; but when he is approaching the 
 South Pole, it will carry the same point towards the east, and 
 must be applied accordingly. 
 
 The altitude and azimuth instrument being adapted to ob¬ 
 serve the heavenly bodies in any part of the visible expanse 
 of the heavens, its powers may be applied at any time to de¬ 
 termine the data from which the time, the latitude of the 
 place of observation, or the declination of the body observed, 
 may be at once determined. We subjoin some of the formulae, 
 adapted to logarithmic computation, connecting the parts of 
 what may be called the astronomical triangle, of which the 
 angular points are, the pole, p, the zenith, z, and the appa¬ 
 rent place of the body observed, s. 
 
 Let p z, the colatitude of the place, be represented by A. 
 
 p s, the polar distance of the body observed. tt. 
 
 zs, the zenith distance of the body observed .z. 
 
 zps, the hour angle from the meridian . h. 
 
 p z s, the azimuthal angle . a. 
 
 z 
 
 Then we have the following formulae for determining the 
 time, the latitude, and the declination of the body observed. 
 
THE ALTITUDE AND AZIMUTH INSTRUMENT. 
 
 159 
 
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ICO 
 
 M ATHEMATICAL INSTRUMENTS. 
 
 THE READING MICROSCOPE. 
 
 The first of the annexed figures represents a longitudinal 
 section of this instrument, and the second represents the 
 
 field of view, showing the magnified divisions of the limb of 
 the instrument to which the microscope is applied, and the 
 diaphragm, d d, of the microscope, with 
 its comb, c c, and cross wires, w w 
 The diaphragm is contained in the box, 
 
 11, and consists of two parts moving 
 one over the other, the comb, c c, 
 which is moved by the screw, i, at the 
 bottom of the box. for the purpose of 
 adjustment, and the cross wires, tv w, 
 and index, i, which are moved over the 
 comb and the magnified image of the 
 limb, by turning the milled head, h. The micrometer head, 
 m, is attached by friction to the screw turned by the milled 
 head, so that, by holding fast the milled head, the micrometer 
 head can be turned round for adjustment. 
 
 e is the eye-piece, which slides with friction into the cell, c, 
 so as to produce distinct vision of the spider s lines of the micro¬ 
 meter. The object-glass, o, is held by a conical piece, d d, which 
 screws further into or out of the body of the instrument, so 
 as to produce distinct vision of the divided limb to be read by 
 the microscope, and, when adjusted, is held firmly in its place 
 by the nut, b b. The microscope screws into a collar, so as to 
 be capable of adjustment with respect to its distance from the 
 divided limb, and, when so adjusted, is held firmly in its place 
 by the nuts, n n , n! n. 
 
THE HEADING MICROSCOrE. 
 
 161 
 
 Adjustments of the Heading Microscope .—Screw the object- 
 glass home. Insert the body of the microscope into the collar 
 destined to receive it, and screw home the nuts, nn and n' n'. 
 Make the diaphragm and spiders lines visible distinctly, by 
 putting the eye-piece, e , the proper depth into the cell, c. 
 Then make the graduated limb also distinctly visible without 
 parallax by turning the nuts, nn, and n'n', unscrewing one 
 and screwing up the other till the desired object is attained. 
 
 Now bring the point of intersection of the spider’s lines 
 upon a stroke of the limb, and turn the micrometer head, m, 
 to zero ; then, turning the screw through five revolutions, if 
 the point of intersection of the spider’s lines has not moved 
 over the whole of one of the divided spaces on the limb, the 
 object lens must be screwed up to diminish the power by turn¬ 
 ing the cone, dd; and if it has moved over more than one of 
 the divided spaces, it must be unscrewed to increase the 
 power, and then altering the position of the microscope, by 
 turning the nuts, nn and n'n' } till distinct vision of the limb 
 is again obtaiued, the measure of the space, moved over by 
 five revolutions of the screw, must be repeated, as before. 
 When, after repeated trials, the result is satisfactory, the 
 three nuts, n n, n'n\ and hh , must be screwed tight home, to 
 render the adjustment permanent. 
 
 When the microscope has been thus adjusted for distance, 
 the zero of the division on the limb must be brought to the 
 point of intersection of the spider’s lines, and the divided 
 head, m, turned, till its zero is pointed to by its index, and 
 then, if the zero on the comb, cc, be not covered exactly by 
 the index, i, the comb must be moved by turning the screw, i , 
 which enters the bottom of the micrometer box, till its zero 
 is covered by the index pin. The adjustment of the reading 
 microscope will now be perfect; and the graduated limb to be 
 read by it, being divided at every five minutes, the degree 
 and nearest five minutes of an observed angle will be shown 
 by the pointer or index to this graduated limb; while the 
 number of complete revolutions, and the parts of a revolution, 
 of the screw, in the order of the numbers upon the micro¬ 
 meter head, m, required to bring the point of intersection of 
 the spider's lines upon a division of the graduated limb, will 
 be the number of minutes ar.d seconds, respectively, to be 
 added to the degrees and minutes shown by the index of the 
 circle. The complete revolutions, or minutes, to be added, 
 are shown by the number of teeth the index, i, has passed 
 over from zero, and the parts of a revolution, or seconds and 
 
16-2 
 
 mathematical instruments. 
 
 tenths to be added, are pointed out upon the micrometer 
 head m, by its index. 
 
 THE COLLIMATOR. 
 
 b b, is a rectangular mahogany box partly filled with mer¬ 
 cury. f f, is a float of cast iron partly immersed in the mer¬ 
 cury. b b, are two iron-bearing pieces, screwed to the bottom 
 of the box by short iron screws ; and each of these pieces has 
 two vertical plates turned up, the inner one of which has a 
 longitudinal slit in it, into which slits iron pivots, screwed 
 into the sides of the float, are admitted. The use of these 
 parts is to keep the sides of the float parallel to the sides of 
 the box, and at an inch, or more, from contact with any part of 
 the box, that the mercury may assume a flat surface, h and 
 k are two holding pieces of metal cast along with the float, and 
 are perforated, to receive each a socket. The socket at h re¬ 
 ceives an achromatic object-glass, and is adjustable by a screw 
 for its focal distance, and the socket at k holds two cross 
 wires; while another socket, let into the end of the box at l, 
 carries a lens forming an 
 eye-piece; so that the col¬ 
 limator is in fact an astro¬ 
 nomical telescope with a 
 system of cross wires in 
 the common focus of the 
 object-glass and eye-lens. 
 
 The inclination, as com¬ 
 pared with the surface of the fluid, of the optical axis of this 
 telescope, or of the line joining the center of the object-glass 
 and the intersection of the cross wires, can be modified by the 
 addition of perforated pieces of iron, held steady by the 
 vertical pin, p, and by their weight depressing the end of the 
 float. The mercury must be as pure as can be obtained, and 
 particles of dust must be constantly excluded by a lid that 
 covers over the top of the box. At m is a circular hole, closed 
 when the instrument is not in use, through which the tele¬ 
 scope, of which the error of collimation is sought, is to be 
 directed; and a lamp is placed behind the eye-lens at l, to 
 illuminate the cross wires. 
 
 Use of the Collimator with an Altitude and Azimuth Instru¬ 
 ment .—Place the collimator in the plane of the meridian on 
 the south side of the observatory, and direct it so that the 
 cross wires of the telescope of the altitude and azimuth instru- 
 
THE COLLIMATOR. 
 
 163 
 
 ment may be seen through it, in the center of the field of 
 view; then also will tfie cross wires of the collimator be seen 
 through the telescope in the center of its field of view. Read 
 off the altitude'of the cross wires of the collimator, and then, 
 turning the instrument half round in azimuth, observe again 
 the cross wires of the collimator, and read off the angle upon 
 the vertical limb, which will now be a zenith distance. The 
 difference between the sum of these readings and 90°, is the 
 correction which is to be applied to the altitudes and zenith 
 distances observed with the instrument. 
 
 Example .—The sun’s meridian altitude had been observed 
 on the 20th December, 1826, and the following determination 
 of the error was made immediately after the observations 
 were finished; viz :— 
 
 Before reversion the apparent altitude of the cross 
 wires was ........ 
 
 After reversion the apparent zenith distance of the cross 
 wires was * . 
 
 Sum. 
 
 Defect from 90° 
 
 Correction of errors of collimation, &c. . 
 
 Altitude of cross wires corrected = -j + 1 23 *66 } “ 
 
 Zenith distance . . . = { 58 23* *66 } = 89 58 34,00 
 
 0° 
 
 1' 
 
 T 
 
 •33 
 
 89 
 
 58 
 
 10 
 
 •33 
 
 89 
 
 59 
 
 12 
 
 ■66 
 
 
 
 47 
 
 •33 
 
 
 
 23 
 
 •66 
 
 * 
 
 1 
 
 26 
 
 •00 
 
 89 
 
 58 
 
 34 
 
 •00 
 
 90 
 
 0 
 
 0 
 
 •00 
 
 The collimator may also be used for a meridian mark with 
 the transit instrument. When used with a circle for measur¬ 
 ing altitudes and zenith distances, which has no motion in 
 azimuth, the collimator must be moved from the north to the 
 south side of the observatory, and the mean of the observa¬ 
 tions in each of these two positions will give the correction 
 for the errors of collimation, &c., as above. 
 
 PART V.—ON THE GONIOMETER. 
 
 The last instrument to which we shall call attention in this 
 little work, is Wollaston’s Goniometer, used for measuring 
 the angles of crystals. The following lucid description of the 
 construction and method of using this instrument is extracted 
 from the able article on Crystallography in the “ Encyclo¬ 
 pedia Metropolitan. ” 
 
164 
 
 MATHEMATICAL INSTRUMENTS. 
 
 Fig. 1. 
 d 
 
 1 
 
 6 
 
 The instruments used for measuring the angles at which the planes of 
 crystals incline to each other, are called Goniometers. 
 
 The mutual inclination of any two planes, as of 
 a and b, fig. 1, is indicated by the angle formed by 
 two lines, ed, ef, drawn upon them from any point, 
 t, on the edge at which they meet, and perpen¬ 
 dicular to that edge. 
 
 “Now it is known that if two right lines, as 
 gf, d h, fig. 2, cross each other at any point, e, 
 the opposite angles, def, geh, are equal. If, 
 therefore, the lines, gf, d h, are supposed to be 
 very thin and narrow plates, and to be attached 
 together by a pin at e, serving as an axis to permit 
 the point, /, to be brought nearer either to d, or to 
 h, and that the edges, e d, ef, of those plates, are 
 applied to the planes of the crystal, fig. 1, so as to 
 rest upon the lines, ed, ef, it is obvious that the 
 angle, geh,oi the moveable plates would be exactly 
 equal to the angle, d ef, of the crystal. 
 
 lig. 2. 
 
 d — 
 
 “ The common goniometer is a small instrument for measuring this angle, 
 geh, of the moveable plates. It consists of a semicircle, fig. 3, divided into 
 360 equal parts, or half degrees, and a pair of moveable arms, d h, gf, fig. 4, 
 the semicircle having a pin at i, which fits into a hole in the moveable arms 
 at e. 
 
 Fig. 3. Fig. 4. 
 
 “The method of using this instrument is to apply the edges, d e, ef, of 
 the moveable arms to the two adjacent planes of any crystal, so that they 
 shall actually touch or rest upon those planes in directions perpendicular to 
 their edge. The arm, d h, is then to be laid on the plate, m n, of the semi¬ 
 circle, fig 3, the hole at e being suffered to drop on the pin at i, and the 
 edg > nearest to h of the arm will then indicate on the semicircle, as in fig. 6, 
 the number of degrees which the measured angle contains. 
 
 “ When this instrument is applied to 
 the planes of a crystal, the points, d and 
 f, fig. 4, should be previously brought 
 sufficiently near together for the edees, 
 d e, ef, to form a more acute angle than 
 that about to be measured. The edges 
 being then gently pressed upon the 
 cr\ stal, the points, d and f. will be 
 gradually separat' d, until the edges co¬ 
 incide so accurately with the places 
 that no light can be perceived between 
 them. 
 
 Fig. 5. 
 
THE GONIOMETER. 
 
 1C5 
 
 “ The common goniometer is. however, incapable of affording very precise 
 results, owing to the occasional imperfection of the planes of crystals, their 
 frequent minuteness, and the difficulty of applying the instrument with the 
 requisite degree of precision. 
 
 “ The more perfect instrument, and one of the highest value to crystallo¬ 
 graphy, is the reflecting goniometer, invented by Dr. Wollaston, which will 
 give the inclination of planes whose area is less than an inch, to 
 
 less than a minute of a degree. 
 
 “ This instrument has been less resorted to than might, from its import¬ 
 ance to the science, have been expected, owing, perhaps, to an opinion of its 
 use being attended with some difficulty. But the observance of simple rules 
 will render its application easy. 
 
 “ The principle of the instrument may be thus explained :— 
 
 “ Let a b, fig. 6, represent a 
 crystal, of which one plane only is 
 visible in the figure, attached to a 
 circle, graduated on its edge, and 
 moveable on its axis at o ; and let 
 a and b mark the position of the 
 two planes whose mutual inclina¬ 
 tion is required. 
 
 “And let the lines, oe,og , 
 represent imaginary lines, resting 
 on those planes in directions per¬ 
 pendicular to their common edge, 
 and the dots at i and h, some per¬ 
 manent marks in a line with the 
 center, o. 
 
 “ Let the circle be in such a position that the line, oe, would pass through 
 the dot at h, if extended in that direction, as in fig. 6. 
 
 “ If the circle now be turned round with its attached crystal, as in fig. 7, 
 until the imaginary line, og, is brought into the position of the line, o e, in 
 fig. 6, the number 120 will stand opposite the dot at i. This is the number 
 of degrees at which -the planes a 
 and b incline to each other. For 
 if the line o g be extended in the 
 direction o i, as in fig. 7, it is ob¬ 
 vious that the lines, oe, oi, which 
 are perpendicular to the common 
 edge of the planes, a and b, would 
 intercept exactly 120° of the circle. 
 
 “ Hence an instrument construct¬ 
 ed upon the principle of these dia¬ 
 grams is capable of giving with ac¬ 
 curacy the mutual inclination of any 
 two planes which reflect objects 
 with sufficient distinctness, if the 
 means can be found for placing 
 them successively in the relative 
 figures. 
 
 “ This purpose is effected by causing an object, as the line at m. fig. 8, to 
 be reflected successively from the two planes, a and b, at the same angle. It 
 is well known that the images of objects are reflected from hright planes at 
 the same angle as that at which their rays fall on those planes; and that 
 
 Fig. 7. 
 
 positions shown in the two preceding 
 
 Fig. 6. 
 
166 
 
 MATHEMATICAL INSTRUMENTS. 
 
 Fig. 8, 
 
 when the image of an object 
 reflected from a horizontal 
 plane is observed, it appears 
 60 much below the reflecting 
 surface as the object itself is 
 above. 
 
 “ If, therefore, the planes a 
 and b, fig. 8, are successively- 
 brought into such positions as 
 will cause the reflection of the 
 line at nt, from each plane, to 
 appear to coincide with an¬ 
 other line at n, both planes 
 will be successively placed in 
 the relative positions of the 
 corresponding planes in figs. 
 
 6 and 7. To bring the planes of any crystal successively into these relative 
 positions, the following directions will be found useful. 
 
 “ The instrument, as shown in the sketch, 
 fig. 9, should be first placed on a pyramidal 
 stand, and the stand on a small steady table, 
 about six or ten or twelve feet from a flat window. 
 
 The graduated circular plate should stand perpen¬ 
 dicularly from the window, the pin x being hori¬ 
 zontal, not in the direction of the axis, as it is 
 usually figured, but with the slit end nearest to 
 the eye. 
 
 “ Place the crystal which is to be measured on 
 the table, resting on one of the two planes whose 
 inclination is required, and with the edge at which 
 those planes meet, nearest and parallel to the 
 window. 
 
 “ Attach a portion of wax, about the size of 
 d, to one side of a small brass plate, e, fig. 10 ; 
 lay the plate on the table with the edge, f, 
 parallel to the window, the side to which the 
 wax is attached being uppermost, and press 
 the end of the wax against the crystal until it 
 adheres ; then lift the plate with its attached 
 
 crystal, and place it in the slit of the pin, x , with that side uppermost which 
 rested on the table. 
 
 “ Bring the eye now so near the crystal, as, without perceiving the crystal 
 itself, to permit the images of objects reflected from its planes to be distinctly 
 observed, and raise or lower that end of the pin, x, which has the small 
 circular plate on it, until one of the horizontal upper bars of the window is 
 seen reflected from the upper or first plane of the crystal, corresponding with 
 the plane a, fig. 6, and until the image of the bar appears to touch some line 
 below the window, as the edge of the skirting-board where it joins the floor. 
 
 “ Turn the pin, x, on its own axis also, if necessary, until the reflected 
 image of the bar of the window coincides accurately with the observed line 
 below the window. 
 
 “ Turn now the small circular handle, a, on its axis, until the same bar 
 of the window appears reflected from the second plane of the crystal corre¬ 
 sponding with plane b , figs. 6 and 7, and until it appears to touch the line 
 
 Fig. 10. 
 
 
 
THE GONIOMETER. 
 
 167 
 
 below ; and having, in adjusting first plane, turned the pin, x, on its axis, 
 to bring the reflected image of the bar of the window to coincide accurately 
 with the line below, now move the Lower end of the pin laterally, either 
 towards or from the instrument, in order to make the image of the same bar, 
 reflected from the second plane, coincide with the same line below. 
 
 “ Having ascertained by repeatedly looking at, and adjusting both planes, 
 that the image of the horizontal- bar, reflected successively from each plane, 
 coincides with the observed lower line, the crystal may be considered ready 
 for measurement. 
 
 “Let the 180° on the graduated circle be now brought opposite the 0 of 
 the vernier at c, by turning the handle, b ; and while the circle is retained 
 accurately in this position, bring the reflected image of the bar from the first 
 plane, to coincide with the line below, by turning the small circular handle, a. 
 Now turn the graduated circle, by means of the handle, l, until the image of 
 the bar, reflected from the second plane, is also observed to coincide with the 
 same line below. In this state of the instrument the vernier at c will indi¬ 
 cate the degrees and minutes at which the two planes are inclined to each 
 other. 
 
 “ The accuracy of the measurements taken with this instrument will 
 depend upon the precision with which the image of the bar, reflected suc¬ 
 cessively from both planes, is made to appear to coincide with the same line 
 below; and also upon the 0, or the 180 J , on the graduated circle, being made 
 to stand precisely even with the lower line of the vernier, when the first 
 plane of the crystal is adjusted for measurement. A wire being placed hori¬ 
 zontally between two upper bars of the window, and a black line of the same 
 thickness being drawn parallel to it below the window, will contribute to 
 the exactness of the measurement, by being used instead of the bar of the 
 window and any other line. 
 
 “ Persons beginning to use this instrument are recommended to apply it 
 first to the measurement of fragments at least as large as that represented 
 in fig. 10, and of some substance whose planes are bright. Crystals of car¬ 
 bonate of lime will supply good fragments for this purpose, if they are merely 
 broken by a slight blow of a small hammer. . 
 
 “ For accurate measurement, however, the fragments ought not, when the 
 planes are bright, to exceed the size of that shown in fig. 9, and they ought 
 to be so placed on the instrument, that a line passing through its axis should 
 also pass through the center of the small minute fragment which is to be 
 measured. This position on the instrument ought also to be attended to 
 when the fragments of crystal are large. In which case the common edge of 
 the two planes, whose inclination is required, should be brought very nearly 
 / to coincide with the axis of the goniometer; and it is frequently useful to 
 blacken the whole of the planes to be measured, except a narrow stripe on 
 each close to the edge over which the measurement is to be taken.” 
 
INDEX 
 
 Aberration and aplnnatism of lenses 
 explained, 71. 
 
 Achromatic eye-piece, 73. 
 
 Aerial telescope, 81. 
 
 Altitude and azimuth instrument, 
 description of, 153; adjustments of, 
 156 ; use of, 157. 
 
 Angles, methods of setting off, 16, 39; 
 measuring, 16. 
 
 Area, of a drawing, reduced or en¬ 
 larged, 52; of a board or plank 
 found, 59 ; of a survey computed, 
 135. 
 
 Artificial horizon, 121. 
 
 Astronomical triangle, 158; table of 
 formulae connecting the parts of, 
 159. 
 
 Bench marks, 111. 
 
 Bisection of a straight line, 36. 
 
 Bow compasses, 4. 
 
 Box sextant, described, 117; its pa¬ 
 rallax explained, 119; and elimi¬ 
 nated, 121. 
 
 Camera obscura, 89; lucida, 90. 
 
 Cassini, his aerial telescope, 81. 
 
 Cassegrainian telescope, 84. 
 
 Chains, Gunter’s and land, 92. 
 
 Chords, line of, constructed, 14; uses 
 of, 16. 
 
 Chromatic dispersion of light ex¬ 
 plained, 72. 
 
 Clamping screw, 47. 
 
 Collimation, line of, 100. 
 
 Collimator, description of, 162; use 
 of, 162. 
 
 Compasses, 1; beam, 48; bow, 4; 
 hair, 2; with mo\able points, 3; 
 proportional, 4 ; trangular, 6. 
 
 Compass, prismatic, 115. j* 
 
 Content of a drawing reduced or en¬ 
 larged, 5 ; solid, measured, 30. 
 
 Copying drawings, 52. 
 
 Course, method of setting off a, 16. 
 
 Crystals measured, 165. 
 
 Cube root, extraction of, 6. 
 
 Cylindrical vessel, its content in gal¬ 
 lons measured, 63. 
 
 Development of a portion of a sphere, 
 
 22 . 
 
 Diagonal scale, 10. 
 
 Dial, horizontal, 23; erect south, 25; 
 east and west, 25. 
 
 Dip sector. 144. 
 
 Division, arithmetical, 29, 56; of a 
 line into equal parts, 36. 
 
 Drawing, a, its linear dimensions re¬ 
 duced or enlarged, 5, 38, 51, 53; 
 its area reduced or enlarged, 5; its 
 solid content reduced or enlarged, 6. 
 
 Drawing-paper, management of, 64 ; 
 dimensions of, 65. 
 
 Drawing pen, 7. 
 
 Edge, straight, 8. 
 
 Equal parts, scales of, 9. 
 
 Eye-piece, celestial negative, 82; 
 positive, 83; terrestrial, or erect, 
 83. 
 
 Flam stead’s projection, 23. 
 
 Geometrical construction, general 
 rules, 66. 
 
 Gnomon, 24. 
 
 Gnomonic projection, 19. 
 
 Goniometer, common, 164; Wollas¬ 
 ton’s, 165. 
 
 Gravatt, his adjustment # of the level, 
 102; his level, 105. 
 
INDEX. 
 
 169 
 
 Gregorian telescope, 84. 
 
 Gunter’s chain, 92; Gunter’s lines, 
 their construction, 25; their use, 
 28. 
 
 Hadley’s sextant, described, 137; its 
 adjustments, 139; its index error 
 determined, 140. 
 
 Herschel, his telescope, 84. 
 
 Horizon, artificial, 121. 
 
 Huyghens, his aerial telescope, 81; 
 his eye-piece, 82. 
 
 Land chain, 92. 
 
 Lenses, various forms of, and effects 
 produced by them, 70 ; their focal 
 lengths defined, 72 ; focal length 
 of a convex lens practically deter¬ 
 mined, 72 ; inuiges formed by len- 
 
 . ses, 73. 
 
 Level, spirit, 96; the T level, 97; 
 its adjustments, 101; Troughtnn’s 
 level, 104; Gravatt’s level, 105; 
 water level, 112; reflecting level, 
 113. 
 
 Levelling staff, 104. 
 
 Levelling, remarks on, 106; for sec¬ 
 tions, 108; field-book, 109. 
 
 Light, pencils of, defined, 67; chro¬ 
 matic dispersion of, 72. 
 
 Linear measures, table of, 92. 
 
 Marquois’s scales, 42. 
 
 Mean proportional found, 58. 
 
 Micrometer, 47. 
 
 Microscopes, refracting, described, 76; 
 their magnifying powers deter¬ 
 mined, 79; reflecting, described, 
 80; the reading microscope de¬ 
 scribed, 160; its adjustments, 161; 
 the solar microscope, 88. 
 
 Mirrors, glass, secondary images 
 formed in them, 75. 
 
 Mounting, varnishing, &c., paper and 
 drawings, 65. 
 
 Multiplication, 28, 56. 
 
 Newtonian telescope, 84. 
 
 Numbers, line of, 25. 
 
 Offsetting staff, 95. 
 
 Opera-glass, or Galilean telescope, 84. 
 
 Optical square, 122. 
 
 Orthographic projection, 18. 
 
 Pantagraph, 50. 
 
 Parallax, of a telescope, 100; of a 
 sextant, 119. 
 
 Parallel lines drawn, 43. 
 
 Pen. drawing, 7; road, 8. 
 
 Perpendicular lines drawn, 44. 
 
 Plain scale, 33. 
 
 Plotting scales, 50, 133; sections, 
 111; a survey, 129. 
 
 Polygon, inscribed in a circle, 5, 40; 
 formed upon a given line, 40. 
 
 Pricking point, 8. 
 
 Prism, defined, 67; its effects upon 
 light explained, 68 ; its uses in 
 optical instruments, 69. 
 
 Prismatic compass, 115. 
 
 Proportion, 29, 36, 56. 
 
 Protracting scales, 14. 
 
 Protractors, upon plain scales, 33; 
 circular, 129; semicircular, 131. 
 
 Radius found, from the length of the 
 sine, tangent, or secant, of an arc, 
 42. 
 
 Ramsden, his eye-piece, 83. 
 
 Reflectors, plane, their effects and 
 uses in optical instruments, 74; 
 curvilinear reflectors, 75; methods 
 of forming and policing parabolic 
 reflectors, 76, foot note. 
 
 Reflecting circle, described, 141; 
 directions for observing with it, 142. 
 
 Rhumbs, line of, 16,17; sine rhumbs, 
 28 ; tangent rhumbs, 28. 
 
 Scales, of equal parts simply divided, 
 9; diagonal, 10; vernier, 11; 
 formed by sector, 38; of chords, 
 sines, versed sines, tangents, se¬ 
 cants, and semitangents, 14; of 
 rhumbs, chords, and longitudes, 
 15; of hour-lines and latitudes, 
 15; logarithmic, see Gunter's lines; 
 Marquois’s, 42. 
 
 Screw, clamping, 47; tangent, 47. 
 
 Secant, to any desired radius, found, 
 41. 
 
 Sector, 33. 
 
 Sextant, box, 117; Hadley’s, 137. 
 
 Sines, scales of, 14; to any desired 
 radius, found, 41; logarithmic, see 
 
170 
 
 INDEX. 
 
 Gunter’s lines, 25; sine rhumbs, 
 28. 
 
 Sliding rule, 55. 
 
 Solid contents found, 30 ; of squared 
 timber, 60 ; of tapering timber, 59. 
 
 Sphere, its orthographic projection, 
 18 ; gnomonic projection, 19 ; ste¬ 
 reographic projection, 20; develop¬ 
 ment of a portion, 22; Flamstead's 
 projection, 23. 
 
 Spherical confusion of lenses ex¬ 
 plained, 72. 
 
 Spirit level, 96. 
 
 Square, of a number, found, 57; 
 root, 6; measures, table of, 93. 
 
 Station pointer, 135. 
 
 Superficies, or area, measured, 29. 
 
 Surveying, remarks upon, 126. 
 
 T square, 131. 
 
 Tables, giving the correction to be 
 subtracted from the approximate 
 content of timber, as determined 
 by the sliding rule, 61; of dimen¬ 
 sions of drawing paper, 65; of 
 linear measure, 92 ; of square 
 measures, 93; showing the reduc¬ 
 tion in links and decimals of a link 
 upon 100 links-for every half 
 degree of inclination from 3° to 
 20° 30', 95 ; showing the reduction 
 in feet and decimals of a foot upon 
 100 feet for each foot difference of 
 level, 110; of formula for deter¬ 
 mining the time, the latitude of 
 place of observation, and the decli¬ 
 nation of a heavenly body, from 
 observations made with the altitude 
 and azimuth instrument, 159. 
 
 Tangents, scale of, 14; to any de¬ 
 sired radius, found, 42; logarithmic, 
 27 ; tangent rhumbs, 28; tangent 
 screw, 47. 
 
 Telescopes, refracting, astronomical, 
 81; with achromatic object glasses, 
 82 ; Galilean, or opera glasses, 84; 
 refracting, adjusted, and tested, 
 86 ; reflecting, why required, 84 ; 
 Newtonian, 84 ; Gregorian, 84 ; 
 Cassegrainian, 85; Herschelian, 
 85; reflecting, adjusted, and tested, 
 88 . 
 
 Theodolite, described, 122; adjust¬ 
 ments of horizontal limb, 125; 
 adjustments of vertical limb, 125; 
 surveying with, 127. 
 
 Transit instrument, described, 144; 
 its adjustments, 147; method of 
 making and recording observations, 
 151; use of the portable transit, 
 153. 
 
 Triangle, isosceles, having each of 
 the angles at the base double of 
 the third angle, constructed, 40; 
 right-angled, solved, 31. 
 
 Troughton, his level, 104; his reflect¬ 
 ing circle, 141. 
 
 Vernier, described, 45 ; scale, 11. 
 
 Versed sines, scales of, 14. 
 
 Vision by means of lenses, 70. 
 
 Wholes and halves, 4. 
 
 Y level, 97. 
 
LIST 
 
 OF 
 
 MATHEMATICAL, OPTICAL, 
 
 AND 
 
 PHILOSOPHICAL INSTRUMENTS 
 
 iHanufarturelr tg 
 
 WILLIAM ELLIOTT AND SONS, 
 
 56; STRAND; LONDON. 
 
 
 
 
 
 
 
 
 £ 
 
 s. 
 
 d. 
 
 Four-inch plain Theodolite 
 
 • 
 
 • 
 
 • 
 
 
 • 
 
 • 
 
 13 
 
 13 
 
 0 
 
 Four-inch best Theodolite, divided 
 
 on 
 
 Silver 
 
 • 
 
 
 
 • 
 
 15 
 
 15 
 
 0 
 
 Ditto ditto ditto, with Tangent Screw Adjustments . 
 
 
 • 
 
 20 
 
 0 
 
 0 
 
 Five-inch Theodolite 
 
 
 
 
 
 
 
 16 
 
 16 
 
 0 
 
 Ditto, divided on Silver, with Tangent Screw 
 
 Adjustments 
 
 • 
 
 24 
 
 3 
 
 0 
 
 Six-inch Theodolite 
 
 
 
 
 
 
 
 18 
 
 18 
 
 0 
 
 Ditto, divided on Silver to 20 Seconds, with Tangent Screw 
 
 
 
 
 Adjustments 
 
 
 
 
 
 
 
 29 
 
 8 
 
 0 
 
 Ditto ditto, with 2 Telescopes . 
 
 • 
 
 • 
 
 • 
 
 
 
 • 
 
 33 
 
 0 
 
 0 
 
 Seven-inch best Theodolite 
 
 • 
 
 • 
 
 • 
 
 
 
 • 
 
 34 
 
 13 
 
 0 
 
 Ditto ditto, with 2 Telescopes . 
 
 • 
 
 • 
 
 • 
 
 
 
 • 
 
 44 
 
 2 
 
 0 
 
 Five-inch best Transit Theodolite 
 
 • 
 
 • 
 
 • 
 
 
 
 • 
 
 29 
 
 8 
 
 0 
 
 Six-inch ditto ditto. 
 
 
 
 
 
 
 
 34 
 
 13 
 
 0 
 
 Seven-inch Sitto ditto . . 
 
 • 
 
 • 
 
 • 
 
 
 
 • 
 
 40 
 
 19 
 
 0 
 
 Everest Theodolite, 4-inch 
 
 • 
 
 • 
 
 • 
 
 
 
 • 
 
 21 
 
 0 
 
 0 
 
 Ditto ditto, 5-inch . 
 
 • 
 
 • 
 
 • 
 
 
 
 • 
 
 25 
 
 4 
 
 0 
 
 Small Level, with Stand, for drainage 
 
 • 
 
 • 
 
 
 
 • 
 
 6 
 
 6 
 
 0 
 
 Ten-inch Y Level . 
 
 
 
 
 
 
 
 9 
 
 9 
 
 0 
 
 Fourteen-inch Y Level with Compass 
 
 • 
 
 • 
 
 
 
 • 
 
 14 
 
 14 
 
 0 
 
 Elliott's improved Dumpy Level 
 
 • 
 
 • 
 
 • 
 
 
 
 • 
 
 14 
 
 14 
 
 0 
 
 Ditto ditto, with Compass . 
 
 • 
 
 • 
 
 • 
 
 
 
 • 
 
 16 
 
 16 
 
 0 
 
 Ditto ditto, 14-inch 
 
 
 
 
 
 
 
 15 
 
 15 
 
 0 
 
 Ditto ditto ditto, with Compass 
 
 • 
 
 • 
 
 • 
 
 . 
 
 
 • 
 
 17 17 
 
 0 
 
 Ditto, ditto, extra large . 
 
 • 
 
 • 
 
 • 
 
 
 
 • 
 
 21 
 
 0 
 
 0 
 
 Circumferenters . . , 
 
 • 
 
 • 
 
 from 51. 
 
 15s. 6<A to 
 
 15 
 
 15 
 
 0 
 
 Prismatic Compass . 
 
 • 
 
 • 
 
 • 
 
 
 3/. 
 
 3s. to 
 
 3 
 
 13 
 
 6 
 
 Pocket Sextant 
 
 • 
 
 • 
 
 • 
 
 
 • 
 
 • 
 
 4 
 
 4 
 
 0 
 
 Ditto, with Telescope 
 
 • 
 
 • 
 
 • 
 
 
 • 
 
 • 
 
 4 
 
 14 
 
 6 
 
 Ditto, ditto, and Supplementary Arc 
 
 • 
 
 • 
 
 
 • 
 
 • 
 
 5 
 
 15 
 
 6 
 
 Optical Square 
 
 
 
 
 
 
 
 1 
 
 1 
 
 0 
 
 Pentagraphs .... 
 
 • 
 
 • 
 
 • 
 
 51. 
 
 15s. CcZ. to 
 
 10 
 
 10 
 
 0 
 
 Eidograph .... 
 
 
 
 
 
 
 
 10 
 
 10 
 
 0 
 
 Transit Instruments 
 
 • 
 
 • 
 
 • 
 
 16/. 16s. to 
 
 73 10 
 
 0 
 
o 
 
 
 
 - 
 
 
 £ 
 
 s. 
 
 d. 
 
 Cross Staff ..... 
 
 • 
 
 
 10s. 6(7. to 
 
 0 
 
 16 
 
 0 
 
 Ditto, with Compass and Screw Staff 
 
 • 
 
 • • 
 
 21. 2s. to 
 
 2 12 
 
 6 
 
 Miners’ Compass .... 
 
 • 
 
 
 57. 5s. to 
 
 18 
 
 18 
 
 0 
 
 Circular Protractors, with Verniers . 
 
 • 
 
 . 3/. 
 
 13s. 6 cl. to 
 
 7 
 
 17 
 
 6 
 
 Metal Circular and Semicircular Protractors 
 
 
 10s. 6(7. to 
 
 5 
 
 5 
 
 0 
 
 Level Staff, 14-feet, portable . 
 
 • 
 
 . from 
 
 21. 15s. to 
 
 3 
 
 0 
 
 0 
 
 Ditto, 17-feet ..... 
 
 • 
 
 
 • • 
 
 4 
 
 14 
 
 6 
 
 Gravatt’s Level Staff 
 
 • 
 
 
 37. 3s. to 
 
 4 
 
 4 
 
 0 
 
 Papers for Level Staves . 
 
 • 
 
 
 per foot 
 
 0 
 
 0 
 
 2 
 
 Ditto, in sets, for 14-feet Portable Staff 
 
 • 
 
 
 * • 
 
 0 
 
 3 
 
 0 
 
 Chain, 66-feet .... 
 
 • 
 
 
 10s. to 
 
 0 
 
 18 
 
 0 
 
 Standard Chains .... 
 
 • 
 
 
 3/. 3s. to 
 
 5 
 
 , 5 
 
 0 
 
 Arrows, set . 
 
 • 
 
 
 • • 
 
 0 
 
 2 
 
 0 
 
 Tapes, 33-feet to 100-feet 
 
 • 
 
 
 6s. to 
 
 1 
 
 1 
 
 0 
 
 Pocket Tapes. 
 
 • 
 
 
 3s. to 
 
 0 
 
 10 
 
 6 
 
 Four-inch Metal Sextant, divided on Silver 
 
 
 • • 
 
 9 
 
 9 
 
 0 
 
 Five-inch ditto ditto 
 
 
 
 • • 
 
 12 
 
 12 
 
 0 
 
 Six-inch ditto ditto 
 
 
 
 • • 
 
 13 
 
 13 
 
 0 
 
 Seven-inch ditto ditto 
 
 
 
 • • 
 
 15 
 
 15 
 
 0 
 
 Eight-inch ditto ditto 
 
 
 
 • • 
 
 16 
 
 16 
 
 0 
 
 Ditto ditto ditto, with double frame . 
 
 
 
 • • 
 
 17 
 
 17 
 
 0 
 
 Quadrant ..... 
 
 
 . 27. 
 
 12s. 6c7. to 
 
 6 
 
 6 
 
 0 
 
 Artificial Horizon .... 
 
 
 . 21. 
 
 12s. 6c7. to 
 
 4 
 
 14 
 
 6 
 
 1 i 
 
 - 
 
 Brass. 
 
 German Sib 
 
 ’er. 
 
 
 
 £ 
 
 s. 
 
 d. 
 
 £ 
 
 s. 
 
 d. 
 
 Proportional Callipers, 12 inches . 
 
 • • 
 
 2 
 
 0 
 
 0 
 
 2 
 
 12 
 
 6 
 
 Ditto ditto, 9 inches .... 
 
 • • 
 
 1 
 
 14 
 
 0 
 
 2 
 
 2 
 
 0 
 
 Proportional Compass, 12 inches . 
 
 • • 
 
 3 
 
 0 
 
 0 
 
 3 
 
 13 
 
 6 
 
 Ditto ditto, 9 inches .... 
 
 • • 
 
 2 
 
 5 
 
 0 
 
 3 
 
 3 
 
 0 
 
 Ditto ditto, 6 inches .... 
 
 • • 
 
 1 
 
 10 
 
 0 
 
 2 
 
 0 
 
 0 
 
 Ditto ditto, ditto, with Adjustment 
 
 • « 
 
 2 
 
 0 
 
 0 
 
 * 2 
 
 10 
 
 0 
 
 Whole and half ditto .... 
 
 • • 
 
 0 
 
 18 
 
 0 
 
 1 
 
 4 
 
 0 
 
 Triangle ditto. 
 
 • • 
 
 0 
 
 18 
 
 0 
 
 1 
 
 4 
 
 0 
 
 Tube Compass. 
 
 • • 
 
 1 
 
 15 
 
 0 
 
 2 
 
 5 
 
 0 
 
 Ditto, Needle Points .... 
 
 • • 
 
 1 
 
 18 
 
 0 
 
 2 
 
 8 
 
 0 
 
 Tube Beam Compass .... 
 
 • • 
 
 2 
 
 0 
 
 0 
 
 2 
 
 12 
 
 6 
 
 Pocket Turn-in Compass 
 
 • V 
 
 1 
 
 6 
 
 0 
 
 1 
 
 16 
 
 0 
 
 Ditto ditto, with handles 
 
 • • 
 
 1 
 
 10 
 
 0 
 
 2 
 
 2 
 
 0 
 
 Ditto ditto, ditto, and Bars . 
 
 • • 
 
 2 
 
 2 
 
 0 
 
 2 
 
 12 
 
 6 
 
 Pillar Compass, with divided Sheath 
 
 • • 
 
 1 
 
 11 
 
 6 
 
 2 
 
 2 
 
 0 
 
 Napier Compass ..... 
 
 • • 
 
 
 
 
 2 
 
 2 
 
 0 
 
 Pocket Dividers, with Sheath 
 
 • • 
 
 0 
 
 8 
 
 0 
 
 0 
 
 10 
 
 0 
 
 Best double-jointed Compass 
 
 • « 
 
 0 
 
 15 
 
 0 
 
 1 
 
 1 
 
 0 
 
 Ink Pencil, and Wheel Legs for ditto . 
 
 . each 
 
 0 
 
 5 
 
 6 
 
 0 
 
 7 
 
 6 
 
 Lengthening Bar .... 
 
 • • 
 
 0 
 
 5 
 
 0 
 
 0 
 
 7 
 
 0 
 
 Flair Divider . 
 
 • • 
 
 0 
 
 10 
 
 0 
 
 0 
 
 13 
 
 0 
 
 Plain Divider ..... 
 
 • • 
 
 0 
 
 6 
 
 0 
 
 0 
 
 8 
 
 0 
 
 Tate’s Bow ..... 
 
 • • 
 
 1 
 
 4 
 
 0 
 
 1 
 
 11 
 
 6 
 
 Double-jointed Bowpen and Bowpencil . 
 
 . each 
 
 0 
 
 10 
 
 6 
 
 0 
 
 13 
 
 0 
 
 Ditto ditto, Needle-pointed , » , 
 
 . each 
 
 0 
 
 12 
 
 0 
 
 0 
 
 15 
 
 0 
 
2 
 
 
 
 
 
 
 
 Brass. German Silver. 
 
 
 
 
 
 
 £ 
 
 S. 
 
 d. 
 
 £ 
 
 s. 
 
 d. 
 
 Hair Bow Divider 
 
 
 
 • 
 
 « 
 
 0 
 
 9 
 
 0 
 
 0 
 
 12 
 
 0 
 
 Large Needle Bows 
 
 
 
 
 each 
 
 0 
 
 15 
 
 0 
 
 1 
 
 1 
 
 0 
 
 Spring Bowpen and Pencil . 
 
 
 
 
 each 
 
 0 
 
 8 
 
 6 
 
 0 
 
 8 
 
 6 
 
 Spring Divider . 
 
 
 
 
 • 
 
 0 
 
 7 
 
 6 
 
 0 
 
 7 
 
 6 
 
 Ivory Handle Drawing Pen . 
 
 
 
 
 • 
 
 0 
 
 5 
 
 0 
 
 0 
 
 5 
 
 0 
 
 Ditto, ditto, jointed 
 
 
 
 
 • 
 
 0 
 
 5 
 
 6 
 
 0 
 
 7 
 
 0 
 
 Ebony ditto 
 
 
 
 
 • 
 
 0 
 
 3 
 
 6 
 
 0 
 
 3 
 
 6 
 
 Double Pen 
 
 
 
 
 • 
 
 0 
 
 10 
 
 6 
 
 0 
 
 10 
 
 6 
 
 Six Wheels and Box . 
 
 
 
 
 • 
 
 0 
 
 6 
 
 6 
 
 0 
 
 6 
 
 6 
 
 Six Pens and one Handle 
 
 
 
 
 • 
 
 1 
 
 6 
 
 0 
 
 1 
 
 6 
 
 0 
 
 Tracer .... 
 
 
 
 
 • 
 
 0 
 
 2 
 
 6 
 
 0 
 
 2 
 
 6 
 
 Crow Quill Holder 
 
 
 
 
 • 
 
 0 
 
 3 
 
 0 
 
 0 
 
 3 
 
 0 
 
 Needle Holder . 
 
 
 
 
 • 
 
 0 
 
 2 
 
 6 
 
 0 
 
 3 
 
 0 
 
 Knife .... 
 
 
 
 
 • 
 
 0 
 
 1 
 
 6 
 
 0 
 
 1 
 
 6 
 
 Sector jointed Compass, 4^, 5 
 
 or 6 
 
 in. . 
 
 
 • 
 
 0 
 
 10 
 
 0 
 
 0 
 
 15 
 
 0 
 
 Divider 
 
 
 
 
 • 
 
 0 
 
 6 
 
 0 
 
 0 
 
 8 
 
 0 
 
 ink, Pencil, and Wheel Legs 
 
 • 
 
 
 
 each 
 
 0 
 
 5 
 
 6 
 
 0 
 
 7 
 
 6 
 
 Bar. 
 
 • 
 
 
 
 • 
 
 0 
 
 5 
 
 0 
 
 0 
 
 7 
 
 0 
 
 Bowpen and Bowpencil 
 
 • 
 
 
 
 • 
 
 0 
 
 12 
 
 0 
 
 0 
 
 16 
 
 0 
 
 Marlow Compass 
 
 • 
 
 • 
 
 • 
 
 • 
 
 0 
 
 9 
 
 0 
 
 0 
 
 13 
 
 0 
 
 Divider .... 
 
 • 
 
 • 
 
 • 
 
 • 
 
 0 
 
 4 
 
 6 
 
 0 
 
 6 
 
 6 
 
 Ink and Pencil Legs . 
 
 • 
 
 • 
 
 • 
 
 • 
 
 0 
 
 9 
 
 0 
 
 0 
 
 13 
 
 0 
 
 Bar. 
 
 • 
 
 • 
 
 • 
 
 • 
 
 0 
 
 4 
 
 6 
 
 0 
 
 6 
 
 6 
 
 College Compass . 
 
 • 
 
 • 
 
 • 
 
 • 
 
 0 
 
 7 
 
 0 
 
 
 
 
 Ink and Pencil Legs . 
 
 • 
 
 • 
 
 • 
 
 • 
 
 0 
 
 7 
 
 0 
 
 
 
 
 Divider .... 
 
 • 
 
 • 
 
 
 • 
 
 0 
 
 3 
 
 6 
 
 
 
 
 Ink and Pencil Bows . 
 
 • 
 
 • 
 
 • 
 
 • 
 
 0 
 
 7 
 
 0 
 
 
 
 
 Steel Joint Compass, with Pen and Pencil Legs 
 
 0 
 
 7 
 
 6 
 
 
 
 
 Divider .... 
 
 • 
 
 
 • 
 
 • 
 
 0 
 
 2 
 
 6 
 
 
 
 
 Bowpen and Pencil 
 
 • 
 
 • 
 
 • 
 
 
 0 
 
 6 
 
 0 
 
 
 
 
 Drawing Pen 
 
 • 
 
 • 
 
 • 
 
 • 
 
 0 
 
 2 
 
 0 
 
 
 ■ 
 
 
 Case of Instruments, as supplied to the Royal Military Academy 2 12 6 
 
 Ditto, as supplied to the College of Civil Engineers at Putney . 3 10 0 
 
 Ditto, as supplied to King’s College, London . . . . 2 15 0 
 
 Ditto, as supplied to the College, Cheltenham . . . .330 
 
 Ditto, as supplied to the Cadets at Addiseombe, 3/. 3s., 
 
 31. 13s. 6d., and. . . . . . . . .440 
 
 Magazine Cases of Instruments made of Brass, German Silver, 
 
 or Silver, from 10/. 10s. to . . . . . . . 50 0 0 
 
 Sector .... 
 Marquois Scales, Boxwood 
 Ditto, Ivory . 
 
 Ditto, German Silver 
 
 6 s. to 1 1 0 
 
 . 0 12 0 
 . 2 12 6 
 . 3 13 6 
 
4 
 
 Ivory Plotting Scales, 12 inches 
 Ditto ditto, 6 inches 
 Ditto Offset Scales, 2 inches . 
 
 Ditto Architect’s Scales, 12 inches 
 DLto ditto, 6 inches 
 Ditto Protractor, 12 inches 
 Ditto ditto, 6 inches 
 Ditto Parallel Rule, Brass Bars 
 Ditto ditto, German Silver Bars 
 Ditto Pocket Rules . ..... 4s. 
 
 Slide Rules .... 
 
 Routledges’ Slide Rule, with Book 
 Hawthorn’s ditto, with Book . 
 
 Box Plotting Scales, 12 inches 
 Ditto Offsets .... 
 
 Ditto Architect’s Scale 
 Ditto Pocket Rules 
 Ebony Parallel Rule, id. per inch. 
 
 Rolling Parallel Rules, Is. per inch. 
 
 Ditto ditto, with Ivory Edges, Is. 8 d. per inch. 
 
 Rolling Protractor, Ivory 
 
 Metal Standard Scale, stamped by Tithe Commissioners 
 English and Foreign Standard Measures made to Order. 
 Steel Straight Edges, 6s. per foot. 
 
 Lance wood T. Squares, 3s. per foot. 
 
 Hbonv T. Squares, 4s. per foot 
 Computing Scales . 
 
 Oentrolinead . 
 
 Sets of Radii Curves 
 Trammels 
 Beam Compasses 
 
 6d. to 
 
 £ s. d. 
 0 10 0 
 0 5 0 
 0 2 6 
 0 13 0 
 0 7 0 
 1 10 0 
 0 6 0 
 0 4 6 
 0 6 0 
 1 11 6 
 0 5 
 0 9 
 0 12 0 
 0 4 0 
 
 0 1 
 0 6 
 
 2s. 
 
 0 
 
 0 
 
 6 
 
 0 
 
 6d. to 0 14 0 
 
 0 18 0 
 110 
 
 0 16 0 
 
 Twelve-inch Achromatic Telescope . 
 
 Eighteen-inch ditto. 
 
 Twenty-four-inch ditto .... 
 
 Thirty-six-inch ditto ..... 
 
 Twelve-inch Portable Achromatic Telescope 
 Eighteen-inch ditto ditto.... 
 
 Twenty-four-inch ditto ditto 
 Military Telescope, in Leather Case and Strap 
 Naval Telescopes ..... 
 
 Thirty-inch Astronomical Telescope, on Stand 
 Forty-five inch ditto ditto ditto, 2| aperture 
 Ditto ditto ditto, 3| aperture . 
 
 Telescopes, Opera Glasses, Spectacles, Barometers, and all descriptions of 
 Optical and Philosophical Instruments. 
 
 Elliott's Drawing Pencils, and small Pencils to fit Instruments. Tachet’s 
 Patent Drawing Boards, Colours, Drawing Paper, &c., &c. 
 
 . 11. 8s. to 
 
 1 
 
 11 
 
 6 
 
 . 11. Is. to 
 
 5 
 
 5 
 
 0 
 
 21. 12s. 6d. to 
 
 5 
 
 5 
 
 0 
 
 If. to 
 
 5 
 
 5 
 
 0 
 
 . 15s. to 
 
 1 
 
 11 
 
 6 
 
 . If. Is. to 
 
 2 
 
 2 
 
 0 
 
 . 21. 2s. to 
 
 3 
 
 3 
 
 0 
 
 . it. 4s. to 
 
 5 
 
 5 
 
 0 
 
 If. 11s. 6d. to 
 
 2 
 
 2 
 
 0 
 
 . 21. 2s. to 
 
 2 
 
 12 
 
 6 
 
 21. 12s. 6d. to 
 
 3 
 
 3 
 
 0 
 
 • • • 
 
 3 
 
 10 
 
 0 
 
 from If. 4s. to 
 
 6 
 
 6 
 
 0 
 
 • • • 
 
 12 
 
 12 
 
 0 
 
 • • • 
 
 25 
 
 4 
 
 0 
 
 • • • 
 
 42 
 
 0 
 
 0 
 
'