iVEASlIt OF CUIFOnili VERSITI OF CtLIFOmi THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA PRESENTED BY PROF. CHARLES A. KOFOID AND MRS. PRUDENCE W. KOFOID FORNIA VERSITY OF CALIFORNIA /Th ,.••::!""•;-.. LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRAR' { UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LI U iNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA if H'^ UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA The GREi^T THEODOLITE.^ Hamsdcn U.rtti hif ihc hit<: (Ji'N . Roif kc. in the oretit Kinflish Trufmurmehiad Opertitiott.^ GEOMETRICAL AND GRAPHICAL ESSAYS; Containing, A GENERAL DE;sCRIPTION OF THE MATHEMATICAL INSTRUMENTS VSED IN GEOMETRY, CIVIL AND MILITARY SURVEYING, LEVELLING AND PERSPECTIVE; WITH MANY NEW PRACTICAL PROBLEMS. ILLVSTRATED BY THIRTY-FOUR COPPER PLATES. BY THE LATE GEORGE, ADAMS, lATHEMATlCAL INSTRUMENT MAKER TO HIS MAJESTY, &C. THE THIRD EDITION, CORRECTED AND ENLARGED By WILLIAM JONES, F. Am. P. S, LONDON: PRINTED BY \V. GLENDINNING, 25, HATTON-GARDEN } FOR, AND SOLD BY, W. AND S. JONES, OPTICIANS, HOLBOBN. 1803. [Price 14f. In Boards^ TO A3 THE I^.IO^T NOBLE CHARLES, DUKE of RICHMOND, AND LENNOX, Master General of the Ordnance, &c. THESE ESSAYS ARE WITH GREAT RESPECT, JUSTLY INSCRIBED, BY HIS grace's Most obedient, Humble Servant, GEORGE ADAMS. PREFACt: Those who have had much occasion to use the mathematical instruments constructed to faeihtate the arts ot' drawing, surveying, &c. have long com- plained that a treatise was wanting to explain their use, describe their adjustments, and give such an idea of their construction, as might enable them to select those that are best adapted to their re- spective purposes. This complaint has been the more general, as there are few active stations in life whose professors are not often obliged to have recourse to mathema- tical instruments. To the civil, the military, and the naval architect, their use must be familiar ; and they are of equal, ifnotof more importance to the engineer, and the surveyor ; they arc the means by which the abstract parts of the mathematics are ren- dered useful in life, they connect theory with prac- tice, and reduce speculation to use. Monsieur Bmis treatise on the construction of mathematical instruments, which was translated into English by Mr. .S/owr, and published in 1723, is the only regular treatise* we have upon this subject ; the numerous improvements that have been made in instruments since that time, have rendered this work but of little use. It has been my endeavour by the following Essays to do away this complaint ; * I do not speak «f Mr. Robertson's work, as it is confined wholly to the instniments contained in a case of drawing instru- ments. 6 PKEPAGE. and I have spared no pains to render them intelligi- ble, and make them useful. Though the materials, of which they are composed, lie in common, yet it is presumed, that essential improvements will be found in aliHost every part. These Essays begin by defining the necessary terms, and stating a few of those first principles on which the whole of the work is founded : they then proceed to describe the mathematical drawing instruments ; among these, the reader will find an account of an improved pair of triangular com- passes, a small pair of beam compasses with a mi- crometer screw, four new parallel rules, and other articles not hitherto described : these are followed by a large collection of useful geometrical problems ; I flatter myself, that the practitioner will find many that are new, and which are well adapted to lessen labour and promote accuracy. In describino- the manner of dividing large quadrants, I have first given the methods used by instrument makers, previous to the publication of that of Mr. BirJ, subjoining his mode thereto, and endeavouring to render it more plain to the artist by a difFerent arrangement. This IS succeeded by geometrical and mechanical methods of describino- circles of every possible mag- nitude ; for the greater part of which 1 am indebted to Josej^k Pries/Icy, Esq. of l^rodforc), Yorkshire whose merit has been already noticed by an abler pen than mine.* From this, I proceed to give a short view of elliptic and other compasses, and a description of Siiardi's geowelric pe}i, an instrument not known m this country, and whose curious pro- perties will exercise the ingenuity of mechanics and mathematicians. Trigonometry is the next subject ; but as this work was not designed to teach the elements of Miis art, I have contented myself with stating the * I'liestley's Perspective. PREFACE. general principles, and giving the canons for calcu- lation, subjoinino; some useful and curious problems, which! though absolutely necessary in many cases that occur in countv and marine surveymg, have been neglected by every practical wriieron this sub- ject, except Mr.'Machnzie* and B. DowiA Some will' also be found, that arc even unnoticed by the above-mentioned authors. _ Our next article treats of surveying-, and it is presumed the reader will find it a complete, though concise system thereof The several instruments now in use, and the methods of adjusting them, are described in order ; and I think it will appear evi- dent, from a view of those of the best construction, that 'large estates may be surveyed and plotted with o-rcater accuracy than heretofore. ^ The great improvements that have been made within these few years in the art of dividing, have rendered observers more accurate and more atten- tive to the necessary adjustments of their instru- ments which are not now considered as perfect, unless they are so constructed, that the person who uses them can either correct or allow for the errors to which they are liable. Among the improvements which the instruments of science have received from the late Mr. Ramsden, we are to reckon those of the theodolite here described; the surveyor will find also the description of a small quadrant that should be constantlv used with the chain, improve- ments in the circumferentor, plain table, protractor, &c. In treating of surveying, I thought to have met with no difficulty ; having had however no op- portunity of practice myself, I had recourse to books; a multiplicity have been written upon this subject, but they are for the most part imperfect, irregular, and obscure. I have endeavoured (with what suc- cess must be left to the reader's judgment.) to * Treatise onMarilimc Surveying. t Donn's Geometrician, 8 PREFACE. remove their obscurities, to rectify their errors, and supply their deficiencies ; but whatever opinion he may form of my endeavours, I can venture to say, he will be highly gratified with the valuable communi- cations of Mr. Gale^^ and Mr. Milne, here inserted^ and which I think will contribute more to the im- provement of the art of surveying, than any thing it has received since its original invention. The reader will, I hope, excuse me, if I stop a mo- ment to give him some account of Mr. Gales im- provements; they consist, first, in a new method of plotting, which is performed by scales of equal parts, without a protractor, from the northings and south- ings, castings and westings, taken out of the table which forms the appendix to this work ;'i~ this method is much more accurate than that in com- mon use, because any small inaccuracy that might happen in laying down one line is naturally cor- rected in the next ; whereas, in the common me- thod of plotting by scale and protractor, any in- accuracy in a former line is naturally communicated to all the succeeding lines. The next improvement consists in a new method of determining the area, with superior accuracy, from the northings, south- ings, eastings, and westings, without any regard to the plot or draught, by an easy computation. As to the measuring a straight line with exactness is one of the greatest difficulties in surveying, I was much surprised to find many land surveyors using only a chain ; a mode in which errors are mul- tiplied without a possibility of their being discovered, or corrected. I must not forget to mention here, that I have inserted in this part Mr. Brealis, me- thod of surveying and planning by the plain table, the bearings being taken and protracted at the same * A gentleman well known for his ingenious publication on finance. f Tlie table is printed separate, that it may be purchased, or not as the surveyor sees convenient. PRRPACE. Q instant in the field upon one sheet of paper ; thus avoiding the trouble and inconvenience of sliifting the ])apcr : this is followed by a small sketch of ma- ritime surveying ; the use ot the pentographcr, or pentagraph : the art ot levelling, and a few astrono- mical problems, with the manner of using //^^//^-s quadrant and sextant ; even here some suggestions will be tbnnd that are new and useful. 1 have now to name another gentleman, who has contributed to render this work more perfect than it would otherwise have been, and it is with plea- sure I return my best thaiiks to Mr. Landman, Professor of Fortification and Artillery to the Royal Academy at Woolwich, tor his communications, more particularly for the papers from which the course of practical geometry on the ground was extracted. If the professors of useful sciences would thus liberally cooperate f;r their advance- ment, the progress thereof would be rapid and ex- tensive. Ibis course will be found useful not only to the military ofHcer, but would make a useful and entertaining part of every gentleman's educa- tion. I found it necessary to abridge the papers Mr. Landman lent me, and leave out the calcula- tions, as the work had already swelled to a larger size than was originally intended^ though printed on a page unusually fall. The work linishes with a small tract on perspec- tive, and a description of two instruments designed to promote and facilitate the practice of that useful art. It is hoped, that the publication of these will prevent the public from being imposed upon bv men, who, under the pretence of secrecy, enhance the value of their contrivances. I knew an in- stance where -iOl. was paid for an instrument infe- rior to the most ordinary of the kind that are sold in the shops. Some pains have been taken, and no small expence incurred, to offer something to the public superior in construction, and easier in use. LIST OF AUTHORS, CONSULTED FOR THIS WORK. Blon Construction, &:c. of Mathematical Instruments, London, 1/23 Break System of Land Surveying, . . . . < . . . London, 1778 Bonnycastle. . Introduction to Mensuration, London, 1787 Cunu Treatise on the Sector, London, 1729 Clavius Astrolabium Trlbus Libris Explica- tum, Moguntiae, 161 1 Cagnoli Traite de Trigonometrie, Paris, 1786 De la Grive . . Manuel de Trigonometrie, Paris, 1754 Donn Geometrician London, 177^ Davidet Introduction a la Geometric, Paris, 178O Dalrj^mple . . . Essay on Nautical Surveying, London, I786 Eckhardt .... Description d'un Graphometre,. ... A la Haye, 1778 Gardner Practical Surv^ing, London, 1737 Gibson Treatise of Surveying, Dublin, 1763 Hutton Treatise on Mensuration, London, 1803 Hume Art of Surveying, London, 1763 Hammond . . Practical Surveyor, London, 1765 Le Febvre . . Oeuvres Comptettes, Maastricht, 1778 Love Art of Surveying, London, 1786 Mackenzie . . Treatise of Maritime Surveying, London, 1774 Mandey .... Marrow of Measuring, London, I717 Nicholson. . . Navigator's Assistant, London, 1784 Noble Essay on Practical Surveying, London, 177-* Payne Elements of Geometry, London, 1 ']Q'] Trigonometry, London, 1772- Picard Traite du Nivelment, Paris, 1784 Robertson. . . Treatise of Mathematical Instruments, London, 177^ Spiedell .... Geometrical Extraction, London, 1(557 Talbot Complete Art of J^and-Measuring London, 1784 J-Iutton's Mathematical and Philosophical Dictionary, 4to. 1/95* WORKS B)- the hU- Author of these Essays, now puhlisbcd and sold hy W. and S. Jones, Holbom, London. I. AN ESSAY ON ELECTRICITY, explaining clearly and hilly the Principles of that useful Science, describing the various In- .^truments that have been contrived either to illustrate the Theory, or render the Practice of it entertaining. The difterent Modes in which the Electrical Fluid may be applied to the human Frame for jMedlcal Purposes, are distinctly and clearly pointed out, and the necessary Apparatus explained. To which is added, A Letter to the Author, from Mr. John Birch, Surgeon, on the subject of MEDICAL ELECTRICITY. Fifth Edition, 8vo. with Correc- tions and Improvements^ by W. Jones, F. Am. P. S. Price 8s. illustrated with six Plates. H. AN ESSAY ON VISION, briefly explaining the Fabric of the Eye, and the nature of vision > intended for the Service of those whose Eyes are weak and impaired, enabling them to form an ac- curate Idea of the State of their Sight, the Means of preserving it, totrether with proper Rules for ascertaining when Spectacles are ne- cessary, and how to choose them without injuring the Sight. III. ASTRONOMICAL AND GEOGRAPHlCAi. ESSAYS, containing, I . A full and comprehensive View, on a new Plan, of the general Principles of Astronomy, with a large Account of the Discoveries of Mr. Herschel. 2. The Use of the Celestial and Terrestrial Globes, exemplified in a greater Variety of Problems than are to be found in any other Work ; they are arranged under dis- tinct Heads, and interspersed with much curious but relative Infor- mation, 3. The Description and Use of small Orreries or Plane- taria, &c. 4. An Intrcduction to Practical Astronomy, by a Set of easy and entertaining Problems. Fourth Edition, 8vo. corrected by the same Editor, Price 10s. 6d. In Koards, illustrated by Sixteen Plates. IV. AN INTRODUCTION TO PRACTICAL ASTRO- NOMY, or the Cse of the Quadrant and Equatorial, being extracted from the preceding Work. Sewed, with two Plates, 2s. (id. V. AN APPENDIX to the GEOxMETRICAL AND GRA- PHICAL ESSAYS, containing the following Table by Mr. John Galc,vlz. a Table of the Northings, Southings, Eastings, and West- G. ADAMS S WORKS. ings, to every Degree and fifteenth Minute of the Quadrant, Radius from 1 to 100, with all the intermediate Numbers, computed to three places of Decimals. VI. ESSAYS ON THE MICROSCOPE, containing a parti- cular Description of the most improved Microscopes 3 a general History of Insects, their Transformations, peculiar Habits, and Eco- nomy j an Account of the various Species and singular Properties of the Hydrae and Vorticellae ; a Description of 3/9 Animalcula ; a View of the Organization of Timber, and the Configurations of Salts when under the Microscope, &:c. &"c. Second Edition, with considerable Corrections, Augmentations, and Improvements, and occasional Notes j together with Instructions for Procuring and Collecting Insects, and a new copious List of the most curious and interesting microscopic Objects 3 by Frederick Kanmacher, F. L. S. In one large Volume 4to, and illustrated by thirty-three folio Plates and Frontispiece. Price ll, 12s. in Boards. — — ••♦©(CC>)d©»«— — In the Press y AND SPEEDILY WILL BE PUBLISHED, By the same EDITOR, LECTURES IN NATURAL AND EXPERIMENTAL PHILOSOPHY, In Five Volumes, 8vo. The Third Edition, with many Plates, considerable Alterations and Improvements 3 containing more com- plete Explanations of the Instruments, Machines, &c. and the De- scriptions of many others not inserted in the first Edition. In this Third Edition the Article of CHEMISTRY will under- go a complete Revision, and contain the most important and recent Improvements in that Science, A TABLE ov CONTENTS. Page Necesfary Definitions and First Principles 1 Of Mathematical Drawing Instrumenls. 10 Of Drawing Compasses 14 Of Parallel Rules 20 Of the Protractor 28 Of the Plain Scale 30 Of the Sector 39 Select Greometrical Problems 51 Of the Division of Straight Lines 5J Of Proportional Lines 64 Of the Transformation and Reduction of Figures 77 Curious Problems on the Division of Lines and Circles. ... 89 Mr. Bird's Method of Surveying IO9 Methods of describing Arcs of Circles of large Magnitude . ISQ To describe an Ellipse, &c 144 Suardl's Geometric Pen 143 Of the division of Land 151 Of Plain Trigonometry l6j Curious Trigonometrical Problems 1G5 Of Surveying 183 Of the Instruments used in Surveying 1C)0 < )f the Chain ibid. Of King's Surveying Quadrant 193 Of the Perambulator 195 Of the Surveying Cross 19S Of the Optical Square 200 Of the Circumferentor 202 Of the Plain Table 212 'Of Improved Theodolites 217 Tbe complete Theodolite 224 Riimfdtn's great Theodolite 229 TABLE OF CONTENTS, Page Of Hadley's Quadrant and Sextant 237 Of the Artificial Horizon 266 To Survey with the Chain 267 by the Plain Table 269 by the Common Circumferentor 287 Of Mr. Gale's improved Method of Surveying 2Q0 Mr, Milne's Method of Surveying with the best Theodolite 2p8 Of Plotting 305 Mr. Gale on Plotting 311 Mr. Milne on Plotting 318 Of determining the Area of Land 323 Of Maritime Surveying 3.12 To transfer one Plan from another 354 Description of the Pentagraph 355 Of Levelling and the best Spirit Level 358 Astronomical Problems 379 Military Geometry 414 Essays on Perspective * 438 Of Instruments for Drawing in Perspective 458 ADDENDA, BY THE EDITOR. Emerson's Method of Surveying a large Estate 461 Rodham's new Method of Surveying and keeping a Field Book 467 ICeith's improved Parallel Scale 47O Gunner's Callipers • - . . 475 Quadrant , . . . 484 . Perpendiculars ibid Shot Gauges ibid. Description of the Carpenter's Sliding Rule 485 Of Timber Measure 489 British and Foreign Tables, Measures, &c 408 Of French Measures 500 Of the New French Measures 501 Various Standard National Measures 512 Useful Graphical Tables, &;c 513 ^ List of the Principal Instruments and their Prices, as made and sold byW. and S. Jones, Holborn, London. GEOMETRICAL AND GRAPHICAL ESSAYS. NECESSARY DEFINITIONS, and FIRST PRINCIPLES. vJEo:\iETRY originally signified, according to the etymology of tlie name, the art of measuring the earth ; but is now the science that treats of, and considers the properties of magnitude in ge- neral. In other words, extension and figure are the ohjeets of geometry. It is a science, in which Jiuman reason has the most ample field, and can go deeper, and with more certainty, than in any other. It is divided into two parts, theoretical and ])ractical. Theoretical geometry considers and treats of first j)rinciples abstractedly. Practical geometry applies tliesc considerations to the purposes of life. By practical geometry many operations are performed of the utmost importance to society and the arts. " The effects thereof are extended through the principal operations of human skill : it conducts the B 2 NECESSARY DEFINITION, soldier in the field, the seaman on the ocean : it gives strength to the fortress, and elegance to the palace." The invention of geometry has been, by all the most eminent writers on the sciencCj attributed to the Egyptians ; and, that to the frequent inunda- tions of the river Nile upon the country, we owe the rise of this sublime branch of human knowledge; the land-marks and boundaries being in this way destroyed, the previous knowledge of the figure and dimensions was the only method of ascertaining individual property again. But, surely, it is not necessary to gratify learned curiosity, by such ac- counts as these ; for geometry is an art that must have grown with man ; it is in a great measure, na- tural to the human mind ; we were born spectators of the universe, which is the kingdom of geometry, and are continually obliged to judge of heights, measure distances, ascertain the figure, and estimate the bulk of bodies. The first definition in geometry, is a foint, which is considered by geometricians, as that which has no j)arts or magnitude. A line is length without breadth. A straight, line is that which lies evenly between its extreme points or ends. A superficies is that which has only length and breadth. A plane angle is an opening, or corner, made by two straight lines meeting one another. When a straight line A ^,fig. l^ plate 4, standing upon another C D, makes angles ABC, A B D, on each side equal to one another ; each of these angles is called a right angle-, and the line AB is said to be perpendicular to the line CD. It is usual to express an angle by three letters, that placed at the angular point being always in the mid- dle ; as B is the angle of A B C. AND I-nisT PRINCIPLES. 3 An ohluse angle is that which is greater than a right angle. An acitlc angle is that whicli is less than a right angle. A line A B, fg. 2, flute 4, eiitling another line CD in E, will make the opposite angles equal, namely, the angle AE C equal to BED, and AE D equal to B EC. A line AB, Jig. 3, plate 4, standing any-way upon another CD, makes two angles CB A, ABD, which, taken together, are equal to two right angles. A plane triangle is a figure bounded by three right lines. An equilateral triangle is that which has three equal sides. An isosceles triangle is that which has only two equal sides. A scalene trian(rle is that which has all its sides unequal. A right-angled triangle is that which lias one right angle. In a right-angled triangle, the side opposite to the right ang;e is called the hypothenuse. An ohliqne-avgled triangle is that which has no right angle. In the same triangle, opposite to the greater side is the greater angle; and op[)Obite to the greater angle is the greater side. If any side of a plane triangle be produced, the B2 NECESSARY DEFINITIONS, outward angle will be equal to both the inward re- mote angles. The three angles of any plane triangle taken toge- ther, are equal to two right angles. Parallel lines are those which have no inclina- tion towards each other, or which are every where equidistant. All plane figures, bounded by four right lines, are called quadrarigles, or quadrilaterals. A square is a quadrangle, whose sides are all equal, and its angles all right angles. A rliomlms is a quadrangle, whose sides are all equal, but its angles not right angles. K parallelogram is a quadrangle, whose [" opposite sides are parallel I A rectangle is a parallelogram, whose angles are all right angles. A rhomboid is a parallelogram, whose /'' 7 angles are not right angles. / / All other four-sided figures besides these, are called trapeziums. A right line joining any two opposite angles of a four-sided figure, is called the diagonal. All plane ligures contained under more than four sides, are called polygons. Polygons having five sides, arc called pentagons ; those having six sides, hexagons; with seven sides, heptagons-, and so on. A regular polygon is that whose angles and sides are all equal. AND FFRST PRINCIPLES. 5 The base of any figure is that side on which it is supposed to stand, and the altitude is the perpendicular falling thereon from the opposite angle. Parallelograms upon the same base, and between the same parallels, are equal. Parallelograms having the same base, and equal altitudes, are equal. Parallelograms having equal bases, and equal alti- tudes, are equal. If a triangle and parallelogram have equal bases and equal altitudes, the triangle is half the parallelo- gram. A circle is a plane figure, bounded by a curve line called the clrcumj'erence, every part whereof is equally distant from a point within the same figure, called the centre. Any part of the circumference of a circle is called an arch. Any right line drawn from the centre to the cir- cumference of a circle, is called a radius. All the radii of the same circle are equal. The circumference of every circle is supposed to be divided into 36o equal parts, called degrees; each degree into 6o equal parts, called vi'imites^ &c. A ^zWr^/// of a circle will therefore contain 90 de- grees, being a fourth part of 360. Equal angles at the centres of all circles, will in- tercept equal numbers of degrees, minutes, &c. in their circumferences. The measure of every plane angle is an arch of a circle, whose centre is the angular point, and is said to be of so many degrees, minutes, &c. as are con- tained in its measuring arch. All right angles, therefore, are of r)0 degrees, or contain 90 degrees, because their measure is a quadrant. The three angles of every plane triangle taken to- B3 NECESSARY DEFINITIONS, gether contain ] 80 degrees, being equal to two right angles. In a right-angled plane triangle, the sum of its two acute angles is QO degrees. The complement of an arch, or of an angle, is its dif- ference from a quadrant or a right angle. The supplement of an arch, or of an angle, is its difference from a semicircle, or two right angles. The magnitudes of arches and angles are deter- mined by certain straight lines, appertaining to a circle called chords, sines, tangents, &c. The ^>^07y/ of an arch is a straight line, joining its extreme points. A diameter is a chord passing through the centre. A seg77ie?it is any part of a circle bounded by an arch and its chord. A sector is any part of a circle bounded by an arch, and two radii drawn to its extremities. The sine of an arch is a line drawn from either end of it, perpendicular to a diameter meeting the other end. The versed sine of an arch is that part of the dia- meter intercepted between the sine and the end of the said arch. The tangent of an arch is a line proceeding from either end. perpendicular to the radius joining it; its length is limited by a line drawn from the centre, through the other end. The secayit of an arch is the line proceeding from the centre, and limiting the tangent of the same arch. The co-sine and co-tangent^ &c, of any arch is the sine and tangent, &c. of its complement. Thus in fg.^^, plate \,YO is the chord of the arch FVO, and FR is the sine of the arches FV, FAD; RV, RD arc the veri;cd sines of the arches FV,FAD. AND FIRST PRINCIPLES. 7 VT is the tangent of the arch FV, and its supple- ment. CTis the secant of the arch FV. A I is the co-tangcnt, and CI the co-secant of the arch FV. The chord of 6o°, the sine of 90°, the versed sine of 90^, the tangent of 45, and the secant of 0.0, are all equal to the radius. It is obvious, that in making use of these lines, we must always use the same radius, odicrwise there would be no settled proportions between them. Whosoever considers the whole extent and depth of geometry, will find that the main design of all its speculations is mensuration. To this the Elements of Euclid are almost entirely devoted, and this has been the end of the most laboured geometrical disquisitions, of either the ancients or moderns. Now the whole mensuration of figures may be re- duced to the measure of triangles, which are always the half of a rectangle of the same base and alti- tude, and, conscquentlv, their area is obtained by taking the half of the product of the base multiplied by the altitude. By dividing a polygon into triangles, and taking the value of these, that of the polygon is ob- tained ; by considering the circle as a polygon, with an infinite number of sides, we obtain the measure thereof to a sufficient degree of accu- racy. The theory of triangles is, as it were, the hinge upon which all geometrical knowledge turns. All triangles are more or less similar, according as their angles are nearer to, or more remote from, equality. The similitude is perfect, when all the angles of the one are equal respectively to those of the other; the sides are then also proportional. B4 8 NECESSARY DEFINITIONS. The angles and the sides determine the relative and absolute size, not only of triangles, but of all things. Strictly speaking, angles only determine the rela- tive size; equiangular triangles may be of very une- qual magnitudes^ yet perfectly similar. But, when they are also equilateral, the one having its sides equal to the homologous sides of the other, they are not only similar and equiangled, but are equal in every respect. The angles, therefore, determine the relative spe- cies of the triangle, the sideSj its absolute size, and, consequently, that of every other figure, as all are re- solvable into triangles. Yet the essence of a triangle seems to consist inuch more in the angles than the sides ; for the an- gles are the true, precise, and determined boundaries thereof: their equation is always fixed and limited to two right angles. The sides have no fixed equation, but may be extended from the infinitely little, to the infinitely great, without the triangle changing its nature and kind. It is in the theory of isopcrimctrical figures* that we feci how efiicacious angles arc, and how ineffica- cious lines, to determine not only the kind, but the size of the triangles, and all kinds of figures. For, the lines still subsisting the same, we see bow a square decreases, in proportion as it is changed into a more oblique rhomboid : and thus acquires more acute angles. The same observation holds good in all kinds of figures, whether plane or solid. Of all isoperimetrical figures, the plane triangle and solid triangle, or pyramid, are the least capa^• * Isoperimetrical figures arc such as have equal circumferences. AND FIRST PRINCIPLES. Q cious ; and, amongst these, those have the least ca- pacity, whose angles are most acute. But curved s^urfaces, and curved bodies, and, among curves, the circle and sphere, arc those whose capacity are the largest, being formed, if we may so speak, of the most obtuse angles. The theory of geometry may, therefore, be re- duced to the doctrine of angles, for it treats only of the boundaries of things, and by angles the ulti- mate bounds of all things are formed. It is the angles which give them their figure. Angles are measured by the circle ; to these we may add parallels, which, according to the significa- tion of the term, are the source of all geometrical similitude and comparison. The taking and measuring of angles is the chief operation in practical geometry, and of great use and extent in surveying, navigation, geography, astrono- my, &c. and the instruments generally used for this purpose, are quadrants, sextants, theodolites, circum- ferentors, &c. as described in the following pages. It is necessary for the learner first to be acquainted with the names and uses of the drawing instruments ; which are as follow. 10 DRAWING INSTRUMENTS. OF MATHEMATICAL DRAWING INSTRUMENTS. Common Names of the principal Instruments, as represented in Flutes \, 2, and 3. Vlate ^,fg' A, is a pair of proportional compasses, without an adjusting screw. B, a pair of best drawing compasses : b, the plain point with a joint; c, the ink point ; d, the dotting point ; e, the pencil or crayon point ; PQ, additional pieces fitting into the place of the moveable point b, and to which the other parts are fitted. F, a pair of bow compasses for ink ; G, a ditto for a pencil ; H, a pair of ditto with a plain point for stepping minute divisions ; h, a screw to one of the legs thereof, which acts like the spring leg of the hair compasses. L, the hair compasses ; n, the screw that acts upon the spring leg. I K, the drawing pen ; I, the upper part ; k, the protracting pin thereof; K, the lower, or pen part. N, a pair of triangular compasses. V, a pair of very portable compasses which contains the ink and pencil points within its two legs, at a. O, the feeder and tracing point. K, a pair of bisecting compasseSj called wholes and halves. S, a small protracting pin. T, a knife, screw-driver, and key, in one piece. PJale 2, fig. A, the common parallel rule. B, the double barred ditto. C, the improved double barred parallel rule. D, the cross barred parallel rule. Of these rules, that figured at C is the most perfect. DRAWING INSTRUMENTS. 11 E, Eckhardts, or the rolling parallel rule. F G H, the rectangular parallel rule. I K L, the protracting parallel rule. M M O, riaywood's parallel rule. Plate 3, fi^. 1 , the German parallel rule. Fig. 2, a semicircular protractor ; fg. 3, a rectan- gular ditto, Ffg. 4 and 5, the two faces of a sector. Fig. 6, Jackson's parallel rule. Ftg. 7 and 8, two views of a pair of proportionable compasses, with an adjusting screw. Fig. g, a pair of sectoral compasses. In this in- strument are combined the sector, beam elliptical, and calliper compasses : Jig. Q, a, the square for ellif)- ses ; /' c, the points to work therein ; d e, the calli- per points. Fig. JO, a pair of beam compasses. Fig. 11, Sisson's protracting scale. Fig. 12, improved triangular compasses. Fig. 13, a pair of small compasses with a beam and micrometer head adjusting same. The strictness of geometrical demonstration ad- mits of no other instruments, than a rule and a pair of compasses. But, in proportion as the practice of geometry was extended to the different arts, either connected with, or dependent upon it, new instru- ments became necessary, some to answer peculiar purposes, some to facilitate operation, and others to promote accuracy. It is the business of this work to describe these instruments, and explain their various uses. In per- forming this task, a difficulty arose relative to the arrangement of the subject, whether each instrument, with its application, should be described separately, or whether the description should be introduced under those problems, for whose performance they were peculiarly designed. After some consideration, I determined to adopt neither rigidly, but to use 12 DKAWING IXSTRUMENTS. either the one or the other, as they appeared to an- swer best the purposes of science. As ahnost every artist, whose operations are con- nected with mathematical designing, furnishes him- self with a case of drawing instruments suited to his peculiar purposes, they are fitted up in various modes, some containing more, others, fewer instruments. The smallest collection put into a case, consists of a plane scale, a pair of compasses with a moveable leg, and two spare points,which maybe applied occasion- ally to the compasses ; one of these points is to hold ink ; the other, a porte crayon, for holding a piece of black-lead pencil. What is called a full pocket case, contains the fol- lowing instruments. A pair of large compasses with a moveable point, an ink point, a pencil point, and one for dotting ; either of these points may be inserted in the compas- ses, instead of the moveable leg. A pair of plain compasses somewhat smaller than those with the moveable leg. A drawing pen with a protracting pin in the upper part. A pair of bow compasses. A gector. A plain scale. A protractor. A parallel rule. A pencil. The plain scale, the protractor, and parallel rule, are sometimes so constructed, as to form but one instrument ; but it is a construction not to be re- commended, as it injures the plain scale, and lessens the accuracy of the protractor. In a case with the best instruments, the protractor and plain scale are always combined. The instruments in most general use are those of six inches; instruments are seldom made longer, but often smaller. Those of six inches DRAWING INSTRUMENTS. 13 arc, however, to be preferred, in genera], before any other size ; they will effect all that can be performed with the shorter ones, while, at the same time, they are better a(laj)ted to large work. Large collections arc called, }?iuguzlne cases of iii- sfna/iefi/s ; these generally contain. A pair of six inch compasses with a moveable leg, an ink point, a dottinn; point, the crayon point, so contrived as to hold a whole pencil, two additional pieces to lengthen occasionally one leg of the com- passes, and thereby enable them to measure greater extents, and describe circles of a larger radius. A pair of hair compasses. A pair of bow compasses. A pair of triangular compasses. A sector. A parallel rule. A protractor. A pair of proportional compasses, either with or without an adjusting screw. A pair of wholes and halves. Two drawing pens, and a pointrll. A pair of small hair compasses, with a head simi- lar to those of the bow compasses. A knife, a file, key, and screw-driver for the com- passes, in one piece. A small set of fine water colours. To these some of the following instruments arc often added. A pair of beam compasses. A pair of gunners callipers. A pair of elliptical compasses. A pair of spiral ditto. A pair of perspective compasses. A pair of compasses with a micrometer screw. A rule for drawing lines, tending to a centre at a great distance. A protractor and parallel rule, such as is repre- sented at Ji^. I K L, plate 2. 14 DRAWING COMPASSES. One or more of the parallel rules represented, flate 2. A pantographer, or Pentagraph. A pair of sectoral compasses, forming, at the same time, a pair of beam and calliper compasses. OF DRAWING COMPASSES. Compasses are made either of silver or brass, but with steel points. The joints should always be framed of different substances ; thus, one side, or part, should be of silver or brass, and the other of steel. The difference in the texture and pores of the two metals causes the parts to adhere less together, diminishes the wear, and promotes uniformity in their motion. The truth of the work is ascertained by the smoothness and equality of the motion at the joint, for all shake and irregularity is a certain sign of im- perfection. The points should be of steel, so tem- pered, as neither to be easily bent or blunted ; not too fine and tapering, and yet meeting closely when the compasses are shut. As an instrument of art, compasses are so well known, that it would be superfluous to enumerate their various uses ; suffice it then to say, that they are used to transfer small distances, measure given spaces, and describe arches and circles. If the arch or circle is to be described obscurely, the steel points are best adapted to the purpose ; if it is to be in ink or black lead, either the drawing pen, or crayon points are to be used. To use a 'pair of compasses. Place the thumb and middle finger of the right hand in the opposite hol- lows in the shanks of the compasses, then press the compasses, and the legs will open a little way ; this being done, push the innermost leg with the third finger, elevating, at the same time, the furthermost, with the nail of the middle finger, till the compas- ses are sufficiently opened to receive the middle and DRAWING COMPASSES. 15 third finger ; the}' mny then be extended at pleasure, by pushing the furlhcrmost leg outwards witli the middle, or pressing it inwards with the fore finger. In deseribing circles, or arches, set one foot of the compasses on* the centre, and then roll the head of the compasses between the middle and fore finger, the other point pressing at the same time upon the paper. They should be held as upright as possible, and care should be taken not to press forcibly upon them, but rather to let them act by their own weight; the legs should never be so far extended, as to form an obtuse angle with the paper or plane, on which, they are used. The ink and crayon points have a joint just under that part which fits into the compasses, by this they may be always so placed as to be set nearly perpen- dicular to the paper ; the end of the shank of the best compasses is framed so as to form a strong spring, to bind firmly the moveable points, and pre- vent them from shaking. This is found to be a more effectual method than that by a screw. Fig. B, plate 1, represents a pair of the best com- passes, with the plain point, c, the ink, d, the dotting, e, the crayon point. In small cases, the crayon and ink points arc joined by a middle piece, with a &;ocket at each end to receive the points, which, by this means, only oc- cupy one place in the case. Two additional pieces, 7?^. P. Qi, pJate 1, are often applied to these compasses ; these, by lengthening the leg A, enable them to strike larger circles, or measure greater extents, than they would otherwise perform, and that without the inconveniences attend- ing longer compasses. When compasses arc fur- nished with this additional piece, the moveable leg has a joint, as at /', that it may be placed perpendicu- lar to the paper. Of the hair eonifdsses., fig. L, pla/e 1. They are so named, on account of a contrivance iu the shank l6 DRAWING COMPASSES. to set them with greater accuracy than can be effec- ted by the motion of the joint alone. One of the steel points is fastened near the top of the compasses^ and may be moved very gradually by turning the screw «, either backwards or forwards. To 2ise these compasses. 1. Place the leg, to which the screw is annexed, outermost ; i2. Set the fixed leg on that point, from whence the extent is to be taken ; 3. Open the compasses as nearly as possible to the required distance, and then make the points accurately coincide therewith by turning the screw. Of the bow compasses, Jig. F, plate 1 . These are a small pair, usually with a point for ink ; they are used to describe small arches or circles, which they do much more conveniently than large compasses, not only on account of their size, but also from the shape of the head, which rolls with great ease between the fingers. It is, for this reason, customary to put into magazine cases of instruments, a stuaU pah- of hair compasses. Jig . Yi, plate 1, with a head similar to the bows ; these are principally used for repeating divi- sions of a small but equal extent, a practice that has acquired the name of stepping. Of the drawing pen and protracting pin. Jig. I K, plate 1. The pen part of this instrument is used to draw strait lines ; it consists of two blades with steel points fixed to a handle, the blades are so bent, that the ends of the steel -points meet, and yet leave a suf- ficient cavity for the ink ; the blades may be opened more or less by a screw, and, being properly set, will draw a line of any assigned thickness. One of the blades is framed with a joint, that the points may be separated, and thus cleaned more conveniently ; a small quantity only of ink should be put at one time into the drawing pen, and this should be placed in the cavity, between the blades, by a common pen, or the feeder ; the drawing pen acts better, if the feeder, or pen, by which the ink is inserted, be made to pass through the blades. To use the drawing pen, first DRAWING COMPASSES. ]/ feed it with ink, then rco^ulatc it to the thickness of the rcijiiirccl line by the screw. In drawing lines, in- cline the pen a small degree, taking care, however, tliat the edges of both the blades touch the paper, keeping the pen close to the rule and in the same tlirection during the whole operation : the blades i^hould always be wiped very clean, before the pen is put away. These directions are equally apj^licable to the ink. point of the compasses, only observing, that when an arch or circle is to be described, of more than an inch radius, the point should be so bent, that the blades of the pen may be nearly perpendicular to the paper, and both of them touch it at the same time. The protracting p'm k, is onlv a short piece of steel wire, with a very tine point, fixed at one end of the upper part of the handle of the drawing pen. It is used to mark the intersection of lines, or to set off divisions from the plotting scale, and protractor. The feeder^ jig. O, plate 1 , is a thin flat piece of metal ; it sometimes forms one end of a cap to fit on a pencil, or it is framed at the top of the tracing point, as in the figure. It serves to place the ink be- tween the blades of the drawing pens, or to pass be- tween them when the ink does not How freely. The I racing pointy or point rel, is a pointed piece of steel fitted to a brass handle; it is used to draw obscure liiic^, or to trace the out-lines of a drawing or print, when an exact copy is required, an article that will be fully explained in the course of this work; it forms the bottom part of the feeder O. Of inangular compasses. A pair of these are re- prtuiited at 7?^. N, plate 1. They consist of a pair of compasses, to whose head a joint and socket is fitted for the reception of a third leg, which may be moved in almost every direction. Thc^e compasses, tiiough exceedingly useful, are but little known ; they arc very serviceable in copy- ing all kinds of drawings, as from two fixed points C 18 DRAWING COMPASSES. they will always ascertain the exact position of a third point. Fig. 12, plate 3, represents another kind, which has some advantages over the preceding. 1. 'Ihat there are many situations so oblique, that the third point cannot be ascertained by the former, though it may by these. 2. It extends much further than the other, in proportion to its size. 3. The points are in all positions perpendicular to the paper. Of wholes and halves, fg. R, flate 1. A name given to these compasses, because that when the lon- ger legs are opened to any given line, the shorter ones will be opened to the half of that line ; being al- ways a bisection. Fig. V, represents a new pair of very curious and portable compasses, which may be considered as a case of instruments in itself. The ink and pencil points slide into the legs by spring sockets at a ; the ink, or pencil point, is readily placed, by only sliding either out of the socket, reversing it, and sliding in the plain point in its stead. Proportio7ial compasses. These compasses are of two kinds, one plain, represented j^^. A, flate 1 ; the other with an adjusting screw, of which there are two views, one edgeways, fig. 8, plate 3, the other in the front, fg. 7j plate 3 : the principle on which they both act is exactly the same ; those with an adjusting- screw are more easily set to any given division or line, and are also more firmly fixed, when adjusted. There is a groove in each shank of these compasses, and the centre is moveable, being constructed to slide with regularity in these grooves, and when pro- perly placed, is fixed by a nut and screw ; on one side of these grooves are placed two scales, one for lines, the other for circles. By the scale of lines, a right line may be divided into any number of equal parts expressed on the scale. By the sciile for circles^j a regular polygon may be inscribed in a circle, pro- vided the sides do not exceed the numbers on the DUAWrXG COMPASSES. \Q 5oalc. Sometimes arc aildccl a scale for superficies and a scale for solids. To liivuie a given line into a proposed mmiher ( 1 1 ) fif cqiiiil purls. I . Shut the compasses. 2. Unscrew the milled nut, and move the sHder until the line across it coincide with the \ lih divison on the scale. .1. Tighten the screw, that the slider may be immove- able. 4. Open the compasses, so that the longer points may take in exactly the given line, and the shorter will give you T-',th of that line. To inscribe in a circle a regular polygon of \2 sides. 1. Shut the compasses. 2. Unscrew the milled nut, and set the division on the slider to coincide with the 1 2th division on the scale of circles. 3. Fasten the milled nut. 4. Open the compasses, so that the lon- ger legs may take the radius, and the distance be- tween the shorter legs will be the side of the required polygon. To use the frGporlional cojv passes ivith an adjusting scre-iv. The application being exactly the same as tkc simple one, we have nothing more to describe than the use and advantage of the adjusting screw. ]. Shut the legs close, slacken the screws of the nuts j^and /; move the nuts and slider k to the division wanted, as near as can be readily done by the hand, and screw fast the nut /: then, by turning the ad- juster /;, the mark on the slider k may be brought exactly to the division : screw fast the nut g. 2. Open the compasses ; gently lift the end e of the screw of the luit/out of the hole in the bottom of the nut g ; move the beam round its pillar a, and slip the point e into the hole in the pin ;/, whicli is fixed to the under leg; slacken the screw of the nut/; take the given line between the longer points of the com- passes, and screw fast the nut/: then may the shorter points of the compasses be used, without any danger of the legs changing their position ; this being one of the inconveniences that attended the proportional compasses, before this ingenious contrivance. C^ 2 20 PARALLEL RULES. Fi^. 10, pidte 3, represents a pair of l^ea^n compasses; they are used for taking off and transferring divisions from a diagonal or nonius scale, describing large arches, and bisecting lines or arches. It is the in- strument upon which Mr. Bird principally depended, in dividing those instruments, whose accuracy has so much contributed to the progress of astronomy. These compasses consist of a long beam made of brass or wood, furnished with two brass boxes, the one fixed at the end, the other sliding along the beam, to any part of which it may be firmly fixed by the screw P. An adjusting screw and micrometer are adapted to the box A at the end of the beam; by these, the point connected therewith may be moved with ex- treme regularity and exactness, even less than the thousandth part of an inch. Fig. \3, pidle 3, is a small pair of beam co??/passes, with a micrometer and adjusting screw, for accu- rately ascertaining and laying down small distances. Fig. 11, pla/e 3, represents a scale of equal par^s, constructed by Mr. Sisson; that figured here con- tains two scales, one of three chains, the other of four chains in an inch, being those most frequently used ; each of these is divided into 10 links, which are again subdivided by a nonius into single links ; the in- dex carries the protracting pin for setting off the lengths of the several station lines on the plan. By means of an instrument of this kind, the length of a station line may be laid down on paper with as much exactness as it can be measured on land. OF PARALLEL RULES. Parallel lines occur so continually in every specie? of mathematical drawing, that it is no wonder so many instruments have been contrived to delineate them with more expedition than could be effected by the general geometrical methods : of the various con- PARALLEL JRULES. 21 trivanccs for this purpose, the following are those most approved. 1, The common par aU el rule. Jig. A, plate 1. This consists of two straight rules, which are connected together, and always maintained in a parallel position by the two equal and parallel bars, which move very freely on the centre, or rivets, by which they are fas- tened to the straight rules. 2. The douhlc parallel rule, fg. B, pJate 2. This instrument is constructed exactly upon the same principles as the foregoing, but with this advantage, that in using it, the moveable rule may always be so placed, that its ends may be exactly over, or even with, the ends of the fixed rule, whereas in the for- mer kind, they are always shifting away from the ends of the fixed rule. This instrument consists of two equal flat rules, and a middle piece ; they arc connected together by four brass bars, the ends of two bars are rivettcd on the middle line of one of the straisfht rules ; the ends of the other two bars are rivetted on the middle line of the other straight rule ; the other ends of the brass bars are taken two and two, and rivetted on the mid- dle piece, as is evident from the figure ; it would be needless to observe, that the brass bars move freely on their rivets, as so many centres. 3. Of the improved double parallel ru'e, fg. C, plate 2. The motions of this rule are more regular than those of the preceding one, but with somewhat more friction ; its construction is evident from the figure ; it was contrived by the ingenious mechanic, Mr. Haywood. 4. The cross barred parallel rule, fg. D, plate 2. In this, two straight rules are joined by two brass bars, which cross each other, and turn on their inter- section as on a centre ; one end of each bar moves on a centre, the other slides in a groove, as the rules recede from each other. C 3' 12 PARALLEL EULES. As the four parallel rules above described, are all used in the same way, one problem will serve for them all ; ex. gr. a right line being green to draw a line farallel thereto by either of the foregoing instrii- ments : Set the edge of the uppermost rule to the given line ; press the edge of the lower rule tight to the paper with one hand, and, with the other, move the upper rule, till its edge coincides with the given point ; and a line drawn along the Q(}igQ, through the point, is the line required. 5. Of the rolling 'parallel rule. Ibis instrument was contrived by Mr. Eckhardt, and the simplicity of the construction does credit to the inventor; it must, however, be owned, that it requires some prac- tice and attention to use it with success. Fig. E, flate 2, represents this rule ; it is a rec- tangular parallelogram of black ebony, with slips of ivory laid on the edges of the rule, and divided into inches and tenths. The rule is supported by two small wheels, which are connected together by a long axis, the wheels being exactly of the same size, and their rolling surfaces being parallel to the axis ; when they are rolled backwards or forwards, the axis and rule will move in a direction parallel to themselves. The wheels are somewhat indented, to prevent their sliding on the paper ; small ivory c)iinders are some- times affixed to the rollers, as in this figure ; they are called rolling scales. The circumferences of* these are so adjusted, that they indicate, with exactness, the parts of an inch moved through by the rule. In rolling these rules, one hand only must be used, and the fingers should be placed nearly in the mid- dle of the rule, that one end may not have a tendency 10 move faster than the other. The wheels only should touch the paper when the rule is moving, and the surface of the paper should be smooth and tlat. In using the rule with the rolling scales^ to draw a PARALLEL RULES, 23 line parallel to a given line at any determined (WsiSLUCQ, adjust the edge of the rule to the given line, and pressing the edge down, raise the wheels a little from the paper, and you may turn the cylinders round, to bring the first division to the index : then, if you move the rule towards you, look at the ivory cylinder on the left hand, and the numbers will shew in tenths of an inch, how much the rule moves. If you move the rule from you, then it will be shewn by the num- bers on the right hand cylinder. To raise a perpendicidar from a given point on a give7i line. Adjust the edge of the rule to the line, placing any one of the divisions on the edge of the rulfe to the given point ; then roll the rule to any distance, and make a dot or point on the paper, at the same di- vision on the edge of the rule; through this point draw the perpendicular. To let fall a per pendicidar from any given point to a given line. Adjust the rule to the given line, and roll it to the given point ; then, observing what division, or point, on the edge of the rule the given point comes to, roll the rule back again to the given line, and the division, or point, on the edge of the rule will shew the point on the given line, to which the per- pendicular is to be drawn. By this method of drawing perpendiculars, squares and parallelograms may be easily drawn of any di- mensions. To divide any given line into a numher of equal parts. Draw a right line from either of the extreme points of the given line, making any angle with it. By means of the rolling scales, divide that line into as many inches, or parts of an inch, as will equal the number of parts into which the given line is to be divided. Join the last point of division, and the extreme point of the given line : to that line draw ()arallel lines through the other points of division, and they will divide the given line into equal parts. d. Of the square parallel rule. The evident ad- C A 24 PARALLEL RULES, vantages of the T square and drawing board over every other kind of parallel rule, gave rise to a variety of contrivances to be used, when a drawing board was not at hand, or could not, on account of the size of the paper, be conveniently used ; among these are, 1. The square parallel rule. 1. The parallel rule and protractor, both of which I contrived some years since, as substitutes to the T square. The square- parallel rule, besides its use as a parallel rule, is pe- culiarly applicable to the mode of plotting recom- mended by Mr. Gale. Its use, as a rule for drawing parallel lines at given distances from each other, for raising perpendiculars, forming squares, rectangles, &c. is evident from a view of the figure alone ; so that what has been already said of other rules, will be sufficient to explain how this may be used. It is also evident, that it will plot with as much accuracy as the beam,j?^. \\, plate 3. Fig. Y G H, plate 2, represents this instrument ; the two ends F G are lower than the rest of the rule, tjbat weights may be laid on them to steady the rule, M'iien both hands are wanted. The two rules are fastened together by the brass ends, the frame is made to slide regularly between the two rules, carrying at the same time, the rule H in a position at right angles to the edges of the rule E G. There are slits in the frame a, b, with marks to coincide with the respective scales on the rules, while the frame is moving up c)i- clown : c is a point for pricking off with certainty di- visions from the scales. The limb, or rule H, is made to take off, that other rules with different scales of equal parts may be applied ; when taken off, this jnstrumenl has this further advantage;, that if the dis- tances, to which the parallels are to be drawn, exceed the limits of the rules, -the square part, when taken off, may be used with any straight rule, by applying the perpendicular part against it. 7. Fig. I K L, plate '1, represents the /)/-o//v7r//V/§f parallel rule i the uses of this in drawing parallel lines PARALLEL RULES. 25 in diftcrent directions, arc so evident from an inspec- tion of the lijjure, as to render a particular description unnecessary. It answers all the purposes of the T scjuarcand bevil, and is peculiarly useful to surveyors for plotting and protracting, which will be seen, when we come to treat of those branches. M N O is a parallel rule upon the same principle as the former: it was also contrived by Mr. Haywood. It is made either of wood edged with ivory, or of brass, and any scales of equal parts placed on it to the con- venience of the purchaser. Each of the rules M N turns upon a centre ; its use as a parallel rule is evi- dent from the figure, but it would require more pages than can be spared to describe all the uses of which it is capable ; it forms the best kind of callipers, or guage ; serves for laying down divisions and angles with peculiar accuracy ; answers as a square, or bevil ; indeed, scarce any artist can use it, without reaping considerable advantage from it, and finding uses peculiar to his own line of business. 8. Of the German parallel rule, fig. 1, flate 3. This was, probably, one of the first instruments in- vented to facilitate the drawing of parallel lines. It has, however, only been introduced within these few years among the English artists ; and, as this intro- duction probably came from some German work, it has thence acquired its name. It consists of a square and a straight rule, the edge of the square is moved in use by one hand against the rule, which is kept steady by the other, the c<\^q. having been previously set to the given line : its use and construction is obvious from fig. 1, plate 3. Simple as it is in its principle, it has undergone some variations, two of which I shall mention ; the one by Mr. Jackson, of Totten- ham ; the other by Mr. Marquois, of London. ;]//-. Jacho?is^fig. C), pla/e 3, consists of two equal triangular pieces of brass, ivory, or wood, A B C, D E F, right angled at B and E ; the edges A B and A C are divided into any convenient number of ^6 . PARALLEL RULES. equal parts, the divisions in each equal ; B C and D E are divided into the same number of equal parts as A C, one side of D E F may be divided as a pro- tractor. To draw a line G H, Jig. V, plate 2, parallel to a riven line, through a pnint P, or at a given distance. Place the edge D E upon the given line I K, and let the instrument form a rectangle, then slide the up- per piece till it come to the given point or distance, keeping the other steady with the left hand, and draw the line GH. By moving the piece equal distances by the scale on B C, any number of equidistant pa- rallels may be obtained. If the distance, fg. X, plate 2, between the giv-en and required lines be considerable, place A B upon I K, and E F against A C, then slide the pieces al- ternately till D E comes to the required point ; in this manner it is easy also to construct any square or rectangle, &c. From any giv 672 point or angle V,Jig. S, plate 2, to let fall or raise a perpendicular on a given litie. Place either of the edges A C upon G P, and slide A B upon it, till it comes to the point P, and draw P PI., To divide a line into any proposed number f.f equal parts, Jig. T, plate 2. Find the proposed number in the scale B C, and let it terminate at G, then place the rules in a rectangular form, and move the whole about the point G, till the side D E touches H ; now move D one, two, or three divisions, according to the number and size of the required divisions, down BC, and make a dot at I, where D E cuts the line for the first part ; then move one or more divisions as before, make a second dot, and thus proceed till the whole be completed. Of Murquois''s parallel scales. These consist of a right angled triangle, whose hypothenuse, or longest side, is three times the length of the shortest, and two rectangular scales. It is from this relative length of the hypothenuse that these scales derive their pc- PARALLEL RULES. 27 culiar advantages, and it is this alone that renders them different from the common German parallel rule : for this we are much indebted to Mr. Mar- quois. What has been already said of the German rule, applying equally to those of Mr. Marquois's, I shall proceed to explain their chief and peculiar excellence. On each edge of the rectangular rule are placed two scales, one close to the edge, the other within this. The outer scale, Mr. Marquois terms the artificial scale, the inner one, the natural scale : the divisions on the outer arc always three limes longer than those on the inner scale, as, to derive any advantage from this invention, they must always bear the same pro- portion to each other, that the shortest side of the right angled triangle does to the longest. The tri- angle has a line near the middle of it, to serve as an index, or pointer ; when in use, this point should be placed so as to coincide with the O division of the scales ; the numbers on the scales are reckoned both ways from this division ; consequently, by confining the rule, and sliding the triangle either way, parallel lines may be drawn on either side of a given line, at any distance pointed out by the index on the triangle. The advantages of this contrivance are : 1. That the sight is greatly assisted, as the divisions on the outer scale arc so much larger than those of the inner one, and yet answer the same purpose, for the edge of the right angled triangle only moves through one third of rhe space passed over by the index. 2. That it promotes accuracy, for all error in the 5 ferpendicidar to that Vine. Apply the protractor to the line A B^ so that the centre may coincide with the point A, and the divi- sion marked gO may be cut by the line A B, then a line D A drawn against the diameter of the protrac- tor will be perpendicular to A B. Further uses of this instrument will occur in most parts of this work, particularly its use in constructing polygons, which will be found under their respective heads. Indeed, the general use being explained, the particular application must be left to the practitioner, or this work would be unnecessarily swelled by a te- dious detail and continual repetitions. OF THE PLAIN SCALE. The divisions laid down on the plain scale are of two kinds, the one having more immediate relation to the circle and its properties, the other being mere- ly concerned with dividing straight lines. It has been already observed, that though arches of a circle are the most natural measures of an angle, yet in many cases right lines were substituted, as being more convenient ; for the comparison of one right line with another, is more natural and easy, than the comparison of a right line with a curve ; hence it is usual to measure the quantities of angles not by the arch itself, which is described on the angular point, but by certain lines described about that arch. See definitions. The lines laid down on the plain scales for the measuring of angles, or the protracting scales, are, 1. A line of chords marked cho. 2. A line of sines marked sin. o( tancrents marked tan. o( serjiitantan^ gents marked st. and of secants marked sec this last is often upon the same line as the sines, because its gradations do not begin till the sines end. THE PLAIN SCALE. 31 There arc two other scales, namely, the rhiimhs, marked ru. and long^ marked lon". Scales of lati- tude and hours are sometimes put upon the plain scale ; but, as diallini^ is now but seldom studied, they are only made to order. The divisions used for measuring straight lines are called iCiitt's of equal parts, and are of various lengths for the convenience of delineating any figure of a large or smaller size, according to the fancy or pur- poses of the draughts-man. They are, indeed, no- thing more than a measure in miniature for laying down upon paper, &c. any known measure, as chains, yards, feet, &c. each part on the scale answering to one foot, one yard, &c. and the plan will be larger or smaller, as the scale contains a smaller or a greater number of parts in an inch. Hence a variety of scales is useful to lay down lines of any required length, and of a convenient proportion with respect to the size of the drawing. If none of the scales happen to suit the purpose, recourse should be had to the line of lines on the sector; for, by the differ- ent openings of that instrument, a line of any length may be divided into as many equal parts as any per- son chooses. Scales of equal parts are divided into two kinds, the one simple, the other diagonally divided. Six of the simj)Iy divided scales are generally placed one above another upon the same rule ; they are divided into as many equal parts as the length of the rule will admit of ; the numbers placed on the right hand, shew how many parts in an inch each scale is divided into. The upper scale is sometin-es shortened for the sake of introducing another, called the line of chords. The first of the larger, or primary divisions, on every scale is subdivided into 10 equal ])arts, which small parts are those which give a name to the scale; thus it is called a scale of 20, when 20 of these di- 32 THE PLAIN SCALE, visions are equal to one inch. If, therefore, these lesser divisions be taken as units, and each repre- sents one league, one mile, one chain, or one yard, &c. then will the larger divisions be so many tens ; but if the subdivisions are supposed to be tens, the larger divisions will be hundreds. To illustrate this, suppose it were required to set off from either of the scales of equal parts 44, 36, or 360 parts, either miles or leagues. Set one foot of your compasses on 3, among the larger or primary divisions, and open the other point till it falls on the 6th subdivision, reckoning backwards or towards the left hand. Then will this extent represent 44? 36, or 360 miles or leagues, &c. and bear the same proportion in the plan as the line measured does ta the thing represented. To adapt these scales to feet and inches, the first primary division is often duodecimally divided by an upper line ; therefore, to lay down any number of feet and inches, as for instance, eight feet eight inches, extend the compasses from eight of the larger to eight of the upper small ones, and that distance laid down on the plan will represent eight feet eight inches. Of the scale of equal 'parts diagonally divided. The use of this scale is the same as those already described. But by it a plane may be more accurate- ly divided than by the former ; for any one of the larger divisions may by this be subdivided into 100 equal parts; and, therefore, if the scale contains 10 of the larger divisions, any number under 1000 may- be laid down with accuracy. The diagonal scale is seldom placed on the same side of the rule with the other plotting scales. The first division of the diagonal scale, if it be a foot long, is generally an inch divided into 100 equal parts, and at the opposite end there is usually half an inch divided into 100 equal parts. If the scale be six THE PLAIN SCALE. 33 inches long, one end has commonly half an Inch, the other a quarter of an inch subdivided into 100 equal parts. The nature of this scale will be better under- stood by considering its construction. For this ])urpo?e : First. Draw eleven parallel lines at equal dis- tances ; divide the upper of these lines into such a number of equal parts, as the scale to be expressed is intended »o contain ; from each of these divi- sions draw perpendicular lines through the eleven pa- rallels. Secondly. Subdivide the first of these divisions into ten equal parts, both in the upper and lower lines, Thirdly. Subdivide again each of these subdivi- sions, by drawing diagonal lines from the 10th below to the <:)th above ; from the Sth below to the 7th above ; and so on, till from the first below to the above: by these lines each of the small divisions is divided into ten parts, and, consequently, the whole first space into 100 equal parts ; for, as each of the subdivisions is one-tenth part of the whole first space or division, so each parallel above it is one-tenth of such subdivision, and, consequently, one-hundreth part of the whole first space ; and if there be ten of the larger divisions, one-thousandth part of the whole space. If, therefore, the larger divisions be accounted as ;inits, the first subdivisions will be tenth parts of an init, and the second, marked by the diagonal upon the parallels, hundrcth parts of the unit. But, if we suppose the larger divisions to be tens, the first sub- divisions will be units, and the second tenths. I{ the larger are hundreds, then will the first be tens, and the second units. The iiuiiibers, therefore, 57(3, 57,(5, 5^76, are all expressible by the same extent of the compasses : thus, setting one foot in the number five of the D 34 THE PLAIN SCALE. larger divisions, extend the other along the sixth pa-* railel to the seventh diagonal. For, if the five larger divisions be taken for 500, seven of the first subdivisions will be 70, which upon the sixth pa- rallel, taking in six of the second subdivisions for units, makes the whole number 576. Or, if the five larger divisions be taken for five tens, or 50, seven of the first subdivisions will be seven units, and the six second subdivisions upon the sixth pa- rallel, will be six tenths or an unit. Lastly, if the five larger divisions be only esteemed as five units, then will the seven first subdivisions be seven tenths, and the six second subdivisions be the six hundredth parts of an unit. Of the use of the scales of equal parts. Though what 1 have already said on this head may be deemed sufScient, I shall not scruple to introduce a few more examples, in order to render the young prac- titioner more perfect in the management of an in- strument, that will be continually occurring to him in practical geometry. He will have already observed, that by scales of equal [)arts lines may be laid down, or geometrical figures constructed, whose right- lined sides shall be in the same proportion as any given numbers. Example ]. To tahe off the numher A.,"]^ from a diagonal scale. Set one foot of the compasses on the point where the fourth vertical line cuts the seventh horizontal line, and extend the other foot to the point where the ninth diagonal cuts the seventh horizontal line. Example 1. To lake off the number "J Q, A. Ob- serve the points where the sixth horizontal cuts the seventh vertical and fourth diagonal line, the extent between these points will represent the num- ber 76,4. In the first example each primary division is taken for one, in the second it is taken for ten. Example 3. To lay ddiz'u a line of 7,85 chains by Tlin PLAIN SCALE. 35 the tl'iiigoiul Si ill e. Set one point of your compasses where the eight j)rirallel, counting upward^;, cuts the seventh vertical line ; and extend the other point to the intersection of the same eighth parallel with the fifth diagonal. Set oiF the extent 7,85 thus found on the line. Kxiimplf A. To rneiisure hy the diagonal scale a line that is already laid do-wn. Take the extent of the line in your coinpas^e^, place one foot on the first vertical line that will bring the other foot among the diagonals ; move both feet upwards till one of them fall into the point where the diagonal from the nearest tenth cuts the same parallel as is cut by the other on the vertical line ; then one foot shews the chain, and the other the hundredth parts or odd links. Tlius, if one foot is on the eighth diagonal of the fourth parallel, while the other is on the same parallel, but coincides with the twelfth vertical, we have 12 chains, 48 links, or 12,48 chains. Example 5. Three adjacent farts of any right- lined triangle being given, to form the flan thereof. Thus J suppose the base of a triangle, fg. 15, plate 3, 40 chains, the angle A JB C equal 36 deg. and angle BAG equal 4 1 dcg. Draw the line A B, and from any of the scales of equal parts take ofF 40, and set it on the same line Irom A to B for the base of the triangle ; at the points A, B, make the angle ABC eqtial to 3G degrees, and BAG to 41, and the triangle will be formed ; then take in your compasses the length of the side A C, and apply it to the same scale, and you will find its length to be 24 chains ; do the same by the side BC. and you will find it measure 27 chains, and the protractor will shew that the angle A G B contains 103 degrees. Example 6. Given the base A B, fg. l6, plate 3, of a Iriafigle 317 yards, ihe angle GAB 44"" 30', arul the side A G, 208 yards, to constitute the said triangle, and fnd the lenalh of the other sides. D2 39 THE PROTRACTING SCALES. Draw the line A B at pleasure, then take 327 parts from the scale, and lay it from A to B ; set the centre of the protractor to the point A, lay off 44° 30', and by that mark draw A C ; then take with the compasses from the same scale 208, and lay it from A to C, and join C B^ and the parts of the triangle in the plan will bear the same proportion to each other as the real parts in the field do. OF THE REMAINING LINES ON THE PLAIN SCALE. OF THE PROTRACTIXG SCALES. 1 . Of the line of chords. This line is used to set off an angle from a given point in any right line, or to measure the quantity of an angle already laid down. Thus to draw a line A B, fig. 14, flate 3, that shall make with the line A B an angle, con- taining a given number of degrees^ suppose 40 de- grees. Open your compasses to the extent of 6o degrees upon the line of chords, (which is always equal to the radius of the circle of projection,) and setting one foot in the angular point, with that extent de- scribe an arch ; then taking the extent of 40 de- grees from the said chord line, set it off from the given line on the arch described ; a right line drawn from the given point, through the point marked upon the arch, will form the required angle. The degrees contained in an angle already laid down, are found nearly in the same manner ; for instance, to measure the angle CAB. From the centre describe an arch with the chord of 6o de- THE PROTRACTIVG SCALES. 37 »rees, and the distance C B, mca<^iired on tlie line of chords, will give the number of degrees contained in the angle. If the number of dco:rces are more tlinn QO, they must be measured ujjon the chords at twice : thus, if 120 degrees were to be practised, 6o may be taken from the chords, and those degrees be laid off twice upon the arch. Degrees taken from the chords arc always to be counted from the beginning of the scale. Of the rhanh line. This is, in fact, a line of chords constructed to a quadrant divided into eight parts or points of the compass, in order to fa- cilitate the work of the navigator in laying down a ship's course. Thus, supposing the ship's course to be N N E, and it be required to lay down that angle. Draw the line A B, fig. 14, flate 3, to represent the meri- dian, and about the point A sweep an arch with the chord of 60 degrees; then take the extent to the se- cond rhumb, from the line of rhumbs, and set it off on the arch from B to e, and draw the line A e, and the angle B A e will represent the ship's course. The second rhumb was taken, because NNE is the se- cond point from the north. Of the line of long'itiules. The line of longitudes is a line divided into sixty unequal parts, and so ap- plied to the line of chords, as to shew by inspection, the number of equatorial miles contained in a de- gree on any parallel of latitude. The graduated line of chords is necessary, in order to shew the la- titudes ; the line of longitude shews the quantity of a degree on each parallel in sixtieth j)arts of an equa- torial degree, that is, miles. The use of this line will be evident from the following example. A ship in latitude 44° 12' N. sails E. 79 miles, required her difference of longitude. Opposite to 44 ^ nearest equal to the latitude on the line of chords, stands 43 on the line of longitude, which is therefore the nuni- D3 38 THE PROTRACTING SCALES. ber of miles in a degree of longitude in that latitude. Whence as 43 : 6o : : 7Q : 1 10 miles the difference of longitude. The lines of tangents^ sejiiitangents, and secants serve to find the centres and poles of projected circles in the stereographical projection of the sphere. The line of sines is principally used for the ortho- graphic projection of the sphere ; but as the appli- cation of these lines is the same as that of similar lines on the sector, we shall refer the reader to the ex- planation of those lines in our description of that in- strument. The Vines oflahiudesand hours are used conjoint'v, and serve very readily to mark the hour lines in the construction of dials; they are generally on the most complete sorts of scales and sectors; for the uses of which see treatises on dialling.* * I think it however best to give the young . to _/" for the forenoon. In the same manner the quarters or minutes may belaid down if rcipured. Ihc edge of a ruler being placed on the point c, d'-aw the lirst five after- noon hours from th;it point thrcugh the marks 1, 2, 3, 4, 5, on the Ymtde, and continue the lines of IV and V through the centre c to the other side of the dial for the like hours of the morn- ing : lay a ruler on the point a and draw the last five forenoon huursj through the marks 5, 4. 3, 2, I, as the line/^ continuing ( 39 ) OP THE SECTOR. Amidst the variety of matliematical instruments that have been contrived to facilitate the art of drawing, there is none so extensive in its use, or of such general application as the sector. It is an universal scale, uniting, as it were, angles and pa- rallel lines, the rule and the compass, which are the only means that geometry makes use of for measuring, whether in speculation or practice. The real inventor of this valuable instrument is un- known ; yet of so much merit has the invention ap- peared, that it was claimed by Galileo^ and disputed by nations. This instrument derives its name from the tenth definition of the third book of £w//V/, where he de- fines the sector of a circle. It is formed of two equal rales, (fig. 4. and 5,plale3,) A B, D B, called legs; the hour lines of VII and VIII through the centre 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 al- ways placed on the same dialling scale take the extent of (J0°, and describe from the centre a the arch g e, then with the extent of the latitude of the place, suppose London 51^° taken from the line of chords, set one foot in a and cross the arch with the other atn. From the centre at (7 draw the linea^ for the axis of the gnomon a g /.and from^ let fall the perpendicular g i upon the meridian line///' and there will be formed a triangle a g i for a plate or triangular frame similar to this triangle, of the thickness of the interval of the j)arallel lines (7 c (I must be made and set iipright between them touch- ing at a and /' its hypothenuse or axis a g will be parallel to the axis of the earth when the dial is fixed truly, and of course cast its shadow on the hour of the day. To nuike an erect wulh dial, take the complement of the latitude of the j)lace, which from London, is yO" less 51^ = 38^ from the scale of latitudes, and ]>rorecd in all respect for the hour lines, as above for the horizonial dial, only re- v<;rsing the hours, and limiting them to the VI, and i'wc the gno- mon making the angle of the stile's height equal to the co-lalitudc 3«^, see fig .^20. For the easiest and best methods of fixing dials, see my Mdhoil of Juuliiig a Maidian Line to set Sini Diah, Cloeh, Watches, kc. 8vo. Edit D 1 40 THE SECTOR, these legs are moveable about tlic centre C of a joint clef, and will, consequcmlyj by their dif- ferent openings, represent every possible variety of plane angles. The distance of the extremities of these rules are the subtenses or chords, or the arches they describe. Sectors are made of different sizes, but their length is usually denominated from the length of the legs when the sector is shut. Thus a sector of six inches, when the legs are close together, forms a rule of 12 inches when opened ; and a foot sector is two feet long, when opened to its greatest extent. In describing the lines usually placed on this instrument, I refer to those commonly laid down on the best six-inch brass sectors. But as the principles are the same in all, and the differences little more than in the number of subdivisions, it is to be presumed that no difiiculty.will occur in the application of what is here said to sectors of a larger radius. Of this instrument. Dr. Priestley thus speaks in his Treatise on Perspective. '' Besides the small sector in the common pocket cases of instruments, I would advise a person who proposes to learn to draw, to get another of one foot radius. Two sectors are in many cases exceedingly useful, if not absolutely necessary ; and I would not advise a person to be sparing of expence in procuring a very good instrument, the uses of which are so various and important." The scales, or lines graduated upon the faces of the instrument, and vviiich are to be used as sec- toral lines, proceed liom the centre ; and are, 1. 'J'wo scales of equal parts, one on each leg, marked lin. or l. each of these scjilcs. from the great extensiveness of its use, is called the line of lines. 2. Two lines of chords^ marked cho. or c. 3. Two lines of secants, marked sec. or s. A line oi fohgovs^ marked roi. Upon the other faccj THE SECTOR. 41 the sectoral lines arc, l. Two lines of sines, marked sin. or s. '2. Two lines of tangents, marked fan. 3. Between the lines of tangents and sines, there is another line of taijg-cnts to a lesser radius to supply the defect of the former, and extending from 45^^ to 75°. Each pair of these lines (except the line of po- lygons) is so adjusted as to make equal angles at the centre, and consequently at whatever dist:mcc the sector be opened, the angles will be always respectively equal. That is, the distance between 10 and 10 on the line of lines, will be equal to 6o and 6o on the line of chords, QO and QO on the line of sines, and 45 and 45 on the line of tan- gents. Besides the sectoral scales, there are others on each face, placed parallel to the outward edges, and used as those of the common plain scale. There are on the face, ^g. 5. 1. A line of inches. 2. A line of latitudes. 3. A line of hours. 4. A line of inclination of meridians. 5. A line of chords. On the face, ^g. 4, three logarithmic scales, nanicly, one of numbers, one of sines, and one of tangents ; these are used when the sector is fully opened, the legs forming one line ; their use will be explained when we treat of trigono- metry. To read and estimate the dii'isions on the sectoral lines. Ihc value of the divisions on most of the lines nrc determined by the figures adjacent to them ; these proceed by tens, which constitute the divisions of the first order, and are numbered ac- cordingly ; but the value of the divisions on the line of lines, that are distinguished by figures, is entirely arbitrary, and may represent any value that is given to them ; hence the ligurcs 1, 2, 3,4, &;c. may denote cither 10, 20, 30, 40; or 100, 200, 300, 400, and so on. The line of lines is divided mto ten equal parts, 42 THE FOUNDATION, &C. numbered ], 2, 3, to 10; these may be called cii visions of the first order ; each of these are ag-ain subdivided into 10 other equal parts, which may be called divisions of the second order ; and each of these is divided into two equal parts, forming divi- sions of the third order. The divisions on all the scales are contained between four parallel lines ; those of the first order extend to the most distant ; those of the third, to the least; those of the second to the intermediate parallel. When the whole line of lines represents 100, the divisions of the first order, or those to which the figures are annexed, represent tens ; those of the second order, units ; those of the third order, the halves of these units. If the whole line repre- sents ten, then the divisions of the first order are units ; those of the second, tenths, and the thirds, twentieths. hi the line of tangents, the divisions to which the numbers are afl^xed, are the degrees expressed by those numbers. Every fifth degree is denoted by a line somewhat longer than the rest ; between every number and each fifth degree, there are four divi- sions, longer than the intermediate adjacent ones, these are whole degrees ; the shorter ones, or those of the third order, are 30 minutes. From the centre, to 6o degrees, the hue of shies is divided like the line of tangents; from (JO to 70, it is divided only to every degree ; trom 70 to 80, to every two degrees; Irom SO to QO, the division must be estimated by the eye. The divisions on the Vine of chords are to be esti- mated in the Siime manner as the tangents. The lesser line of tangents \s ^^YadwdtCQl every two degrees from 45 to 50; but from 50 to ()0, to every degree; from 60 to the end, to half degrees. The line of secants from to 10, is to be estimated by the eye; from 20 to 50 it is divided to every iwo OF SECTORAL LINES. 43 degrees; from 50 to 60, to every degree ; and from 60 to the end, to every half degree. OF THE GENERAL LAW OR FOUNDATION OF SECTORAL LINES. Let C A, C B, /^. \7, plate 3, represent a pair of sectoral lines, (ex. gr. those of the line of lines,) forming the angle A C B ; divide eaeh of these lines into four equal parts, in the points H, D. F, A ; I, E, G, 1] ; draw the lines ill, DE, FG, A B. Then because C A, C B, are equal, their sections are also equal, the triangles are equiangular, having a common angle at C, and equal angles at the base ; and therefore, the sides about the equal angles will be proportional ; for as C H to C A, so is H I to A B, and, therefore, as C A to C H, so is A B to H I, and, consequently, as C H to H I, so is C A to A B ; and thence if C H be one-fourth of C A, HI will be one-fourth of AB, and so of all other sections. Hence, as in all operations on the sectoral lines, there are two triangles, both isosceles and equi- angled ; isosceles, because the pairs of sectoral lines are equal by construction ; equiangled, be- cause of the common angle at the centre ; the sides encompassing the equal angles are, therefore, pro- portional. Hence also, if the lines C A, C B, represent the line of chords, sines, or tangents ; that is, if C A, A B be the radius, and the line C F the chord, sine, or tangent of any proposed number of degrees, then the line F G will be the chord, sine, or tan- gent, of the same number of dc2:rees, to the radius AB. ( 44 ) rIi A and B draw a right line, and it will be the required perpendicular. Problem 2. To raise a perpendicidar from the middle, or any other point G, of a given line A ^,Jig. 8, plate 4. 1. On G, with any convenient distance within the limits of the line, mark or set off the points n and m. 2. From n and m with any radius greater than G A, describe two arcs intersecting at C. 3. Join C G by a linc^ and it will be the perpendi- cular required. Problem 3. From a given point C, fg. 8, plate 4, out of a given line A B, to let fall a perpen- dicidar. When the point is nearly opposite to the middle, of the line, this problem is the converse of the pre- ceding one. Therefore, 1. From C, with any ra- dius, describe the arc n m. 2. From m n, with the same, or any other radius, describe two arcs in- ter'iecting each other at S. 3. Through the points C S draw the line C b, which will be the required hne. When the point is nearly opposite to the end of the line, it is the converse of Method 5, Problem 1, fig.T, plated. 1. Draw a faint line through B, and any conve- nient point E, of the line AC. 2. Bisect this line at D. 3. From D with the radius DE describe an arc cutting A C at A. 4. Through A and B E4 66 SELECT PROBLEMS, &C. draw the line A B, and it will be the perpendicular required. Another metJiod. ]. From K^ fig. (^, f laic A, or any other point in A B, with the radius A C, de- scribe the arc CD. 2. From any other point n, with the radius n C, describe another arc cutting the former in D. 3. Join ihe point CD by a line C G D, and C G will be the perpendicular re- quired. Problem 4. Through a given point C, to ch-aiv a line parallel to a given straight line A B, fig. 1 0, ^late 4. 1. On any point D, (within the given line, or without itj and at a convenient distance from C,) de- scribe an arc passing through C, and cutting the given line in A. 2. With the same radius describe another arc cutting A B at B. 3. Make B E equal to A C. 4. Draw a line C E through the point C and E, and it will be the required parallel. This problem answers whether the required line is to be near to, or far from the given line;, or wheiher the point D is situated on A B, or any where be- tween it and the required line. Problem 5. At the given point D, to make an angle equal to a given angle A B C, fig. 1 2, plate 4. 1. From B, with any radius, describe the arc n m, cutting the legs BA, B C, in the points n and m. 2. Draw the line D r, and from the point D, with the sam.e radius as before, describe the arc r s. 3. Take the distance m n, and apply it to the arc r s, from r to s. 4. Through the points D and s draw the line D s. and the angle r D s will be equal to the angle m B n,-nr A B C as j-equired. Problem 6. To extend ivith acrvracy a short straight line to any assignable length ; or, through iivo g^iven points at a small distance from each otlier to drain (X straight line. It frequently happens that a line as short as that DIVISION" OF STRAIGHT LINES. 5? between A and B, f'^. 1 1, plate 4, is required to be extended to a considerable length, which is scarce attainable by the help of a rule alone ; but may be performed by means of this problem, without error. Let the given line be A B, or the two points A and B; then from A as -jl centre, describe an arch C B I); and from the point B, lay otf B C equal to B D; and frt)m C and D as centres, with any radius, describe two arcs intersecting at E. From the point A de- scribe the arc F E G, making E F equal to E G ; then from F and G as centres, describe two arcs intersecting at H, and so on ; then a straight line from B drawn through E will pass in continuation through H, and in a similar manner the line may be extended to any assignable length. OP THE DIVISION OF STRAIGHT LINES. Proi'lem 7. To h'lsect or (]hide a given sti'aight line A B into tivo equal parts, jig. 13, plate A. ~ 1. On A and B as centres, with any radius greater than h:'lf A B, describe arcs intersecting each other at C and D. 2. Draw the line C D, and the point F, where it cuts A B, will be the middle of the line. If the line to be bisected be near the extreme edge of any plane, describe two pair of arcs of different radii above the given line, as at C and E ; then a line C produced, will bisect A B in F. By the line of lines on the sector. ] . Take A B in the compasses. 2. Open the sector till this extent is a transverse distance between ]Oand 10. 3. The extent from 5 to 5 on the same line, set off from A or B, gives the half required : by this means a given line may be readily divided into 2,4, 8, l6, 32, 64, 128, &c. equal parts. Problem 8. To divide a given straight line A B into anj member of equal farts, for instance, Jive. 58 DIVISION OF STflAIGHT LINES. Method \-> fig' 14, flate 4. I. Through A, one extremity of the line A B, draw A C, making any angle therewith. 2. Set off on this line from A to H as many equal parts of any length as A B is to be divided into. 3. Join li B. 4. Parallel to H B draw lines through the points D, E, F, G, and these will divide the line A B into the parts required. Second method, fig. 1 5, plate 4. 1 . 1 hrough B draw L D, forming any angle with A B. 2. Take any point D either above or below A B, and through D, draw D K parallel to A B. 3. On D set off five equal parts D F, F G, G H, H I, I K. 4. Through A and K draw A K, catting B D in L. 5. Lines drawn through L, and the points F, G, H, I, K, will divide the line AB into the required number of parts. Third method, fig. \'J, plate A. 1. From the ends of the line A B, draw two lines A C, B D, parallel to each other. 2. In each of these lines, beginning at A and B, set off as many equal parts less one, as A B is to be divided into, in the present instance four equal parts, A I, I K, K L, L INI, on \ C ; and four, B E, E F, F G, G H, on B D. 3. Draw lines from M to E, from L to F, K to G, I to H, and \ B will be divided into five equal parts. Fourth method, fig. X^^flate A. \. Draw any two lines C E, D F, parallel to each other. 2. Set off on each of these lines, beginning at C and D, any number of equal parts. 3. Join each point in C E with its opposite point in D F. 4. Take the extent of the given line in your compasses. 5. Set one foot of the compasses opened to this extent in D, and move the other about till it crosses N G in I. 6. Join D I, which being equal to A B, transfer the di- visions of D I to A B, and it will be divided as re- quired. II M is a line of a different length to be divided in the same number of parts. The foregoing methods are introduced on account not only of their own peculiar advantages, but be- DIVISION' OP STRAIGHT LINES. 5() cause they also are the foiiiulation of several mecha- nical methods of division. Pro»lem 9. To cut off from a given I'me AB iiuy odd part, as \d, \th, ;///, ]th, &i:. of that line, Ji^. 18, plate 4. 1. Draw through either end A, aline AC, form- ing any angle with A B. 2. Make AC equal to A B. 3. Through C and B draw the line C D. 4. Make BD equal to C B. 5. Bisect A C in a. 6. A rule on a and D will cut ofF a B equal 'd of A B. If it be required to divide A B into five equal parts, 1. Add unity to the given number, and halve it, 5+1=6, -^ = 3. 2. Divide A C into three parts ; or, as A B is equal to A C, set off Ab equal Aa. 3. A rule on D, and b will cut off b B ! th p:irt of A B. 4. Divide Ab into four equal parts by two bisections, and A B will be divided into five equal parts. To divide A B into seven equal parts, 7-f-i=3^ -!-=4. 1. Now divide AC into four parts, or bi- sect a C in c, and c C will be the ^th of A C. 2. A rule on c and D cuts oti'c B fth of A B. 3. Bisect A c, and the extent c B will divide each half into three equal parts, and consequently the whole line into seven equal parts. To divide AB into nine equal parts, 9-f-i — in -r=z5. Here, 1. Make A d equal to A b, and d C will be \xh of A C. 2. A rule on D and d cuts off d B ^ of AC. 3. Bisect Ad. 4. Halve each of these bisections, and A d is divided into tour equal parts. 5. The extent dB will bisect each of these, and thus divide AB into nine equal parts. If any odd namber can be subdivided, as 9 by 3, then first divide the given line into three parts, and take the third as anew line, and find the third there- of as before, which gives the ninth part required. Method 2. Let ti^^fig. I9, plate A, be the given line. 1. Make two equilateral triangles A D B, C D Bj one on each side of the line D B. 2. Bisect 60 DIVISION OP STRAIGHT LINES. A B in G. 3. Draw C G, which will cut off H B equal ^d of D B. 4. Draw D F, and make G F equal to D G. 5. Draw H F which cuts off B h equal ~ of A B or D B. 6. C h cuts D B in i one fifth part. 7. F i cuts A B in k equal ;th of D B. 8. C k cuts D b in 1 equal fth of D B. C). F 1 cuts A B in m equal fth of D B. 10. C ni cuts D B at n equal 7^th part thereof. Method 3. Let A B, fg, T2, phle 5, be the given line to be divided into its aliquot parts i, f , i:. 1. On AB erect the square A B C D. 2. Draw the two diagonals A C, D B, which will cross each other at E. 3. Through E draw F E G parallel to A D, cutting A B in G. 4. Join D G, and the line will cut the diagonal A C at H. 5. Through 11 draw I H K parallel to A D. 6. Draw DK crossing AC in L. 7. And through L draw M L N parallel to A Dj and so proceed as far as necessary. A G is i-, AKf, ANt of AB.* Method A. Let A B, fig. 13, flate 5, be the given line to be subdivided. 1. Through A and B draw C D, F E parallel to each other. 2. Make C A, A D, F B, F E, equal to each other. 3. Draw C E, which shall divide A B into two equal parts at G. 4. Draw A E, D B, intersecting each other at H. 5. Draw C H intersecting A B at I, making A I -fd of A B. 6. Draw D F cutting AE in K. 7. Join CK, which will cut A B in L, making A L equal -^ of AB. 8. Then draw D g, cutting AE in M, and proceed as before. Corollary. Hence a given line may be accu- rately divided into any prime number whatsoever, by first cutting off the odd part, then dividing the re- mainder by continual bisections. Problem 10. Ayi easy, simple, and very useful method of laying down a scale for di-viding lines i/ito * Hooke's Posthumous WorV», DIVISION OP STRAIGHT LIXES. 6l any munher of equal farts, or for reducing platis to arty size less than the original. \( the scale is for dividing lines into two equal parts, constitute a triangle, so that the liypothcnusc mav be twice the icngih of the perpendicular. Let it be three times for dividing them into three equal parts ; four, for four parts, and so on : fig. \ i, plate 5, represents a set of triangles so constituted. To find the third of the line by this scale. 1. Take any line in your compasses, and set off' this ex- tent from A towards !-, on the line marked one-third; then close the compasses so as to strike an arc that shall touch the base A C, and this distance will be the 7 of the given line. Similar to this is what is termed the angle of reduction, or proportion, de- scribed by some foreign writers, and which we shall introduce in its proper place. PiioBLEM ]1. To divide by the sector a given straight line into any nwnber of equal parts. Case I. Where the given line is to be divided into a number of equal parts that may be obtained by a continual bisection. In this case the operations are best performed by con- tinual bisection : let it be required to divide AB, fg.Gy plate 5, into J (3 c(|ual parts. 1. Make AB a trans- verse distance between 10 and 10 on the line of lines. 2. Takeout from thence the distance between 5 and 5, and set it from A or B to 8, and A B will be di- vided into two equal j)arts. 3. Make A 8 a trans- verse distance between 10 and 10, and 4 the trans- verse distance between 5 and 5, will bisect 8 A, and 8 B at 4 and 12 ; and thus A B is divided into four equal parts in the points 4, 8, and 12. 4'. The ex- tent A 4, put between 10 and 10, and then the dis- tance between 3 and 5 applied from A to 2, froiu 4 to 6, from 8 to 10, and from 12 to 14, will bisect each of those parts, and divide the whole line into eight equal parts. 5. To bisect each of these, we might lake the extent of A 2, and place it between 6^ DIVISION OF STRAIGHT LINES. 10 and 10 as before ; but as the spaces are toe small for that purpose, take three of them in the compasses, and open the sector at 10 and 10, so as to accord with this measure. 6. Take out the transverse measure between 5 and 5, and one foot of the com- passes in A will give the point 3, in 4 will flill on 7 and 1, on 8 will give 5 and 1 1, on 12 gives 9 and 15, and on B will give 13. Thus we have, in a correct and easy manner divided A B into 16 equal parts by a continual bisection. If it were required to bisect each of the foregoing divisions, it would be best to open the sector at 10 and 10, vfiih the extent of five of the divisions already obtained ; then take out the transverse distance be- tween 5 and 5, and set it off from the other divisions, and they will thereby be bisected, and the line divided into 32 equal parts. Case 2. When the given line cannot be divided by bisection. Let the given line be A B, Jig. 7, picite 5, to be divided into 14 equal parts, a number which is not a multiple of 2. 1 . Take the extent A B, and open the sector to it on the terms 10 and 10, and the transverse dis- tance of 5 and 5, set from A or B to 7, will divide A B into two equal parts, each of which are to be subdivided into 7, which may be done by dividing A 7 into 6 and 1, or 4 and 3, which last is preferable to the first as by it the operation may be finished with only two bisections. 2. Therefore open the sector in the terms 7 and 7, with the extent A 7 ; then take out the transverse distance between 4 and 4, this laid off from A, gives the point 4, from 7 gives 3 and 11, and from B gives 10. 3. Make A 4 a transverse distance at 10 and 10, then the transverse distance between 5 and 5 bisects A b, and lO B in 2 and 12, and gives the point 6 and 8 ; then one foot in 3 gives 1 and 5, and from 11,13 DIVISION OP STRAIGHT LINES. 63 and 9: lastly, from 4 it gives 6, and from 10, 8; and thus the line A B is divided into 14 equal parts. Problem 12. To make a scale of eqiuil 'parts con- taifiifio liny j^ii'tn nmtther in ari inch. Example. 1 o construct a scale ot" feet and inches in such a manner, that 25 ot the smallest parts shall be equal to one incli, and 12 of thein represent one fool. By the line of lines on the sector. 1. Multiply the given numbers by 4, the products will be lOo, and 48. 2. Take one inch between your compasses, and make it a transverse distance between 100 and 100, and the distance between 48 and 48 will be 12 of these 25 parts in an inch ; this extent set otf from A to I, fg. 3, pldte 5, from 1 to 2, 8ic. to I'i at B di- vides A B into a scale of 12 feet. 3. Set off one of these parts from A to a, to be subdivided into 12 parts to represent inches. 4. To this end divide this into three parts ; thus take the extent A 2 of two of these parts, and make it a transverse distance between 9 and g. 5. Set the distance between 6 and 6 from b to c, the same extent from 1 gives c, and trom e gives n,thus dividing A a into three equal parts in the points n and g. 6. By two bisections each of these may be subdivided into four equal parts, and thus the whole space into 12 equal parts. When a small number of divisions arc required, as 1, 2, or 3, instead of taking the transverse distance near the centre of the sector, the division v/ill be more accurately performed by using the follovi'ing method. Thii>. if three parts arc required from A, of which the v/hoic line contains gO, make A t), /t^'. 4, plate b, a transverse distance between go and go ; then take the distance between 87 and 87, which set oflYrom D to E backwards, and A E will contain the three desired parts. Example 2. Supposing a scale of six inches to contain 140 poles, to open the sector so that it may 64 PROPORTIONAL LINES. answer for such a scale; divide 140 by 5, which gives 70, the half of 6" equal to 3 ; because 140 was too large to be set off on the line of lines. Make three inches a transverse distance between 70 and 70, and your sector becomes the required scale. Exani'ple 3. To make a scale of seven inches that shall contain 180 fathom; -^^-=9.1 = 37, therefore make 3r a transverse between 9 and 9, and you have the required scale. OF PPvOPORTIONAL LINES. Problem 13. To cut a given line A D, fig. 14, flate 5, into two unequal parts that shall have any given proportion^ ex. gr. of C to D. 1. Draw A G forming any angle with AD. 2. From A on A G set off AC equal to C, and CE equal to D. 3. Draw E D, and parallel to it C B, which-will cut AD at B in the required proportion. To divide by the sector the line A B,//^. i. opiate 5, in the proportion of 3 to 2. Now as 3 and 2 would fall near the centre, multiply them by 2, thereby forming 6 and 4, which use instead of 3 and 2. As the parts are to be as 6 to -i--, the whole line will be 10; therefore make AB a transverse distance be- tween 10 and 10, and then the transverse distance between 6 and 6, set off from B to e, is iths of A B; or the distance between 4 and 4 will give A e iths of A B; therefore A B is cut in the proportion of 3 to L\ Example 2. To cut A B, fg. 2, plate 5, in the proportion of 4 to 5 ; here we may use the numbers themselves ; therefore with A B open the sector at 9 and 9, the sum of the two numbers ; then the dis- tance between 5 and 5 set off from B to c, or be- tween 4 and 4, set oft' A to c, and it divides AB in the required proportion. PROPOUTION'AL LINES. ()3 Note. If the numbers be too small, use their equi- multiplis ; if too large, subfllvidc them. CornlLin. From this problem \vc obtain another mode ot* dividing a straight line into any number of equal parts. PuoBLEM M. To cstniiu! I the proportion between tivo or more given %nes, cis A B, CD, E F, Jig. y, plate 5. Make i^iB a transverse distance between 10 and 10, then take the extents severally of C D and E F, and carry them along the line of lines, till both points rest exactly upon the same number ; in the first it will be found to be 85, in the second 67. Therefore A B is to C D as lOO to 85, to E F as lOO to 67, and of C I) to E F as 85 to 67. Pn o B L E M 15. To ffui a third proportional to two given right lines A and B, fig. 15, plate 5. 1. From the point D draw two right lines D E, DF, making any angle whatever. 2. In these lines take D G equal to the first term A, and D C, D H, each eijunl to the second term B. 3. Join G H, and draw C F parallel thereto ; then D F will be the third proportional required, that is, D G (A) to D C, (B.) so isDlUB) to DF. By the sector. 1. Make A B, fig. 5, plate 5,' a transverse distance between 100 and 100. 2. Find the transverse distance ot E F, which suppose 50. J. Make E F a transverse distance between 100 and 100. 4. 'lake the extent between 50 and 50, and it will be the third proportional C D required. pRORi.KM 1(3. To find a fourth proportional to three given right lines A, B, C,fg. lO, plate 5. 1. From liic point f a circle capable of containing a given an-- gle, fig. W, plate Q. Draw A C and B C, making the angles ABC, B A C, each equal to the given angle. Draw A D F 4 72 PROPORTIONAL LINES. perpendicular to A C, and B D perpendicular to BC ; with the centre D, and radius D A, or D B, describe the segment A E B, and any angle made in this segment will be equal to the given angle. A moreeasy solution of this problem will be given when we come to apply it to practice. Problem 33. To describe ati arc of a circle that shall contain any number of degrees, without com- f asses, or without finding the centre of the circle, fg. 1 2, -plate 6. Geometrically by finding points through which the arc is to pass, let A B be the given chord. 1. Draw AF, making any angie with B A. 2. At any point F, in A F, make the angle EFG, equal to the given angle. 3. Through B draw BE parallel to F G, and the intersection gives the point E, in the same manner as many points D, C, Src. may be found, as will be necessary to complete the arc. This subject will be found fully investigated here- after. Problem 34*. To inscribe a circle in a given tri- angle ABC,fig. 13, plate 6. 1. Bisect the angles A and B with the lines AO and B O. 2. From the point of intersection O, let fall the perpendicular O N, and it will be the radius of the required circle. Problem 35. To inscribe a pentagon, a hexagoriy or a decagon, in a given circle, fig. 15, plate 6. 1. Draw the diameters A B and C E at right an- gles to each other. '1. Bisect D B at G. 3. On G, with the radius G C, describe the arc C F. 4. Join C and F, and the line C F will be one side of the re- quired pentagon. The two sides DC, FDof the triangle F D C, enable us to inscribe a hexagon or a decagon in the same circle; for D C is the side of the hexagon, D F that of the decagon. FOR THE OCTAGON, &C. 73 Problem 36. To inscribe a square or an octagon in a given circle, Jig. 1(3, plate Q. 1. Draw the diameters A C, B I), at r\^\\{. angles to each other. 1. Draw the lines AD, B A, B C, C D, and you obtain the required square. FOR THE OCTAGON. Bisect the arc A B of the square in the point F, and the line A E being carried eight times round, will form the octagon required. Prob r.EM 37. ifi a given circle to inscribe an equi' lateral triatigle, an hexagon, or a dodecagon, Jig. 17, plate 6. FOR THE EQUILATERAL TRIANGLE. 1. From any point A as a centre, with a distance ccjual to the radius A O, describe the arc FOB. '2. Draw the line B F, and make B D equal to B F. 3. Join D F, and DBF will be the equilateral tri- angle required. FOR THE HEXAGON. Carry the radius A O six times round the circum- ference, and you obtain a hexagon. FOR THE DODECAGON. Bisect the arc A B of the hexagon in the point n, and the line A n being carried twelve times round the circumference, will form the required dode- cagon. Problem 38. Another method to inscribe a dode- cagon in a circle, or to divide the circnmjereiice oj a given circle into 12 equal farts, each oJ ZQ degrees, Jig. 18, flate G. 74 FOR POLYGONS. 1. Draw the two diameters I, 7, 4, 10, perpen- dicular to each other. 2. With the radius of the circle and on the four extremities ], 4, 7t iO, as centres, describe arcs through the centre of the cir- cle ; these arcs will cut the circumference in the points required, dividing it into lU equal parts, at the points marked with the numbers. Problem 39. To ^nd /he angles at the centre, and the sides of a regidar polygon. Divide 360 by the number of sides in the pro- posed polygon, thus ^f^ gives 72» ^or the angle at the centre of a pentagon. To find the angle formed by the sides, subtract the angle at the centre from 180, and the remainder is the angle required ; thus 72° from 180°, gives 108° for the angle of a pentagon. A TABLE, stciL'hig the angks at the. centres and c'lrcunifcreiices of regular polygons, from three to twelve sides inelusi-ve. Names. a." Angles at Centre. Angles at Cir. Trigon 3 120" 00' 60" 00' Square Pentagon 4 5 90 00 72 00 90 108 00 00 Hexagon (5 60 00 120 00 Heptagon Octagon >7 / 8 51 25} 45 00 128 135 34f 00 Nonagon 9 40 00 140 00 Decagon 10 36 00 144 00 Endecagon Dodecagon 11 12 32 43 -/t 30 00 1-17 150 lOVx 00 This table is constructed by dividing 360, the de- grees in a circumference, by the number of sides in each })olygon, and the quotients are the angles at the centres; the angle at the centre subtracted from 180 degreet=, leaves the anele at the circumference. FOR POLYGONS. 75 t Problf.m 40. In a given circle to inscribe any regular polygon, Jig. \4, plafeO. J. At the center C make an angle equal to the center of the polygon, as contained in the preceding tabic, an 1 join the angular points A B. 2. The distance AB will be one side of the po- lygon, which being carried round the circumfer- ence, the proper number of times will complete the figure. uinothcr method^ 'which approximates 'very nearly the truth, fig. ig, plate 0. 1. Divide the diameter A B into as many equal parts as the figure has sides. 2. From the center O raise the perpendicular O m. 3. Make m n equal to three fourths of O m. 4. Fro;n n draw n C, through the second division of the diameter. 5. Join the points A C, and the line A C will be the side of the required polygon, in this instance a pen- tagon. Problem 41. About any given triangle ABC, to circumscribe a circle, fig 2 1 , plate (3. 1. Bisect any two sides A B, BC, by the perpen- diculars mo, no. 2. From the point of intersection o, with the distance O A or O B, describe the re- quired circle. Pkoelem 42. About a given circle to circuynscribe a pentagon, fig. 20, plate (J. 1. Inscribe a pentagon within the circle, 2. Through the middle of each side draw the lines O A, OB, OC, OD, andOE. S. Through the point n draw the tangent A B, meeting O A and O B in A and B. 4 Through the points A and m, draw the line A m C, mectiuG; OC in C. 5. In like manner draw the lines C D,"D E, E B, and A U C D E will be the pentagon required. In the same manner you may about a given circle circumscribe any polygon. Problem 43. About a given square to circum- scribe a circle, fig. 22, plate ^. }. Draw the two diagonals AD, BC, intersecting 76 FOR POLYGOXS. each other at O. 2. From O, with the distance O A or O B, describe the circle A B C D, which will circumscribe the square,* Problem 4-4. On a given line A^ 1o make a re- gular hexag07i, Jig. 24, plate 6. 1. On A B make the equilateral triangle AO B. 1. From the point O, with the distance O A or O B, describe the circle A B C D E F. 3. Carry A B six times round the circumference, and it will form the required hexagon. P R o B L E M 4 5 . On a given line A B /o form a re- gular polygon of any proposed number of sides, fig. 14, plate 0. 1. Make the angles CAB, CBA each equal to half the angles at the circumference ; see the pre- ceding table. 2. From the point of intersection c, with the distance C A, describe a circle. 3. Apply the chord A B to the circumference the proposed number of times, and it will form the required polygon. Problem 46 On a given line A B, ^g. 25, plate 6j to form a regular octagon. 1. On the extremities of the given line A B erect the indefinite perpendiculars A F and B E. 2. Pro- duce A B both ways to s and w, and bisect the angle II As and oBw by the lines A H, BC. 3. Make A H and B C each equal to A B, and draw the line H C. 4. Make ov equal to on, and through v draw G D parallel to H C. 5. Draw H G and C D parallel to A F and B E, and make c E, equal to c D. 6. Through E draw E F, parallel to AB, and join the points G F and 1) E, and A B C 1) E F G H will be the octap-on rc(]uired. Problem 47. On a given right Tme A B, fig. 26, plate 6, to describe a regular pentagon, 1. Make Bm perpendicular, and equal to A B. 2. Bisect A Bin n. 3. On n, with distance n m, cross A B produced in O. 4. On A and B, with * To inscribe a circle, fig. 23, from the intersection O of two diameters and the distance O n as a radius, describe the circle as jreq^uired. RKDUCTIOX OV FIGURES. 7^ radius A O, describe arcs intersecting at 1). 5. On D, with radius A 1?, describe the arc EC, and on A and B, with the same extent, intersect iliis arc at E and C. 6. Join A E, ED, DC, C B, and you com- plete the figure. Pko B L EM 48. ITpofi a given right line A B, Jig 27, f>Ja/e (), to describe a triangle similar to the triangle CDE. 1. At the end A of the given Hne A B, make an angle F A 13, equal to the angle E C D. 1. At B make the angle A B F equal to the angle C D E. 3. Draw the two sides A F, B F, and A B F will be the required triangle. FK.OBLEM49. To describe a polygon similar to a given polygon A B C D E F, ont of its sides a b being given, f^. 28, plate 6. ] Draw A C, A D, A F. 2. Set off a b on A B, from a to r. 3. Draw r g parallel to B E, meeting A F in g. 4. Through the point gdraw g h, parallel to F C, meeting AC in h. 5. Through the point h draw the parallel h i. 6. Through i draw i k parallel to E D, and the figure A r g h i k will be similar to thefigurcABCDEF. Prob LEM 50. To reduce a figure by a scale, fig. 28 andl^, plated. 1. Measure each side of the figure .V B C D E with the scale G H. 2. Make ab as many parts of a smaller scale K L, as \ B was of the larger. 3. b c as many of K L, as B C of G II, and a c of K L, A C, &:c. by which means the figure will be reduced to a smaller one. OF THE TRANSFORMATION AND REDUCTION OP FrCURPlS. Problem 51. To change a triangle into another of equal extent, but different height, fig. 1, 2, 3, 4, Plate 7. Let A B C be the given triangle, D a point at the given height. 78 TRANSFOBMATION AXD Ca'e 1. Where the point D, fg. 1 and 2, phfe 7, is either in one of the sides, 'or in the prolonga- tion of a side. 1. Draw a line from 1) to the oppo- site angle C. 2. Draw a line A K parallel thereto from A, the summit of the given triangle. 3. Join D E, and B D E is the required triangle. Case 2. When the point D, fg. 3 and 4, pJateJ, is neither in one of the sides, no? in the prolongation thereof 1. Draw an indefinite line B D a, from B through the point D. 2. Draw from A, the summit of the given triangle, a line A a, parallel to the base B C, and cutting the line B D in a. 3. Join a C, and the triangle B a C is equal to the triangle BAG; and the point D being in the same line with B a. 4. By the preceding case, find a triangle- from D, equal to B a C ; i. e. join D C, draw a E parallel thereto, then joinDE, and BDE is the required triangle. Corollary. If it be required to change the triangle BAG into an equal triangle, of which the height and angle B D E are given : 1. Draw the indefinite jine h i) A, making the required angle with B C. 2. Take on B D a a point D at the given height; and, 3. Gonstruct the triangle by the foregoing rules. Problem 52. To make an isosceles triangle A E B, jig. 5, plate 7, equal to the scalene triangle A C B. 1. Bisect the base in D. 2. Erect the perpen- dicular D E. 3. Draw CE parallel to A B. 4. Draw A E, E B, and A E B is the required triangle. Pkoblem 53. To make an equilateral triangle equal to a given scalene triangle ABC, fg. 7 , plate 7 . 1. On the base A B make an equilateral triangle A B D. 2. Prolong B D towards E. 3. Draw C E parallel to A B. 4. Bisect DEatl, on D I describe the semicircle D F E. 5. Draw B F, the mean proportional between ]^i E, B D, 6. With B 1' from B, describe the arc FG H ; with the same radius from G, intersect this arc at H, draw B H, G H, and B G II is the triangle required^ REDUCTION^ OF FIGURES. 79 Corollary. If you want an equilateral triangle equal to a rectangle, or to an isosceles triangle ; find a scalene triangle respectively equal to each, and then work by the foregoing problem. Problem 54. To reduce a rectilinear figure A I) C D I", fg. 8 and 9, tlate 7, to aiyither equal to it, hut ivith oue suic less. 1. Join the extremities E, C, of two sides D J*', D C, of the same angle D. 2. From D draw a line D F parallel to E C. 3. Draw E, F, and you obtain a new polygon A B F F, equal to A B C D E, but with one side less. Corollary. Hence every rectilinear figure may be reduced to a triangle, by reducing it" successively to a figure with one side less, until it is brought to one with only three sides. For example ; let it be required to reduce the polygon A B C D E F, fg. 10 and 11, plate 7, into a triangle 1 A H, with its summit at A, in the cir- cumference of the polygon, and its base on the base thereof prolonged. 1. Draw the diagonal D F. 2. Draw EG pa- rallel to D F. 3. Draw FG, which gives us a new polygon, A B C G l', with one side less. 4. To re- duce ABCGF, draw AG, and parallel thereto FH; then join AH, and you obtain a j)olygon A B C II, equal to the preceding one A B C G F. 5. The polygon A B C II having a side A H, which may serve for a side of the triangle, you have only to reduce the part A B C, by drawing A C, and paralhl thereto BI ; join AI, and you obtain the required triangle I A H. N. JL In figure 10 the point A is taken at one of the angular points of the given polygon ; in figure 1 1 it is in one of thesi'ie^, in which ca^e tliere is one reduction more to be made, than when it is at the angular point. Corollary. As a triangle may be changed into anothpr of any given height, and with the angle at so MULTIPLICATION, &C. OP FIGURES. the base equal to a given angle ; if it be required to reduce a polygon to a triangle of a given height, and the angle at the base also given, you must first reduce it into a triangle by this problem, and then change that triangle into one, with the data, as given by the problem 51. Corollary. If the given figure is a parallelogram^ fig. 12, plate 7, draw the diagonal E C, and D F pa- rallel thereto ; join E F, and the triangle E B F is equal to the parallelogram E B C D. OF THE ADDITION OP FIGURES. 1 . If the figures to be added are triangles of the same height as A M B, B N C, C O D, D P E, fig. 14, plate 7, make a line A E equal to the sum of their bases, and constitute a triangle A M E thereon, whose height is equal to the given height, and A M E will be equal to the given triangles. 2. If the given figures are triangles of different heights, or different polygons, they must first be reduced to triangles of the same height, and then these may be added together. 3. If the triangle, into which they are to be summed up, is to be of a given height, and with a given angle at the base, they must first be reduced into one triangle, and then that changed into an- other by the preceding rules. 4. The ti-iangle obtained may be changed into a parallelogram by the last corollary. MULTIPLICATION OF FIGURES. ]. To multiply A MB, fig. 13, plate 7, by a given number, for example, by 4 ; or more accu- rately, to find a triangle that shall be quadruple the triangle A M B. Lengthen the base A E, so that it may be four times AB; join ME, and the triangle AME will be quadruple the triangle A M B, 2. SUBTRACTION OF FIGURES. 81 Bv reducing; any figure to a triangle, wc may obtain a triangle which may be nuihiplied in the same manner. SIBTHACTION OF FIGURES. 1, If the two triangles B AC, d:\c, fg. 15, plai& 7, are of the same height, take from the base B C of the first a part DC, eqnal to the base d c of the other, ar.d join A D ; then will the triangle AB D be the dificrenre between the two triangles. If the two triangles be not of the same height, they must be reduced to it by the preceding rules, and then the difference may be found as above ; or it a polygon is to be taken from another, and a triangle found equal to the remainder, it may be easily effected, by reducing them to triangles of the same height. 2. A triangle may be taken fi'om a polygon by drawing a line within the polygon, from a given point F on one of its sides. To eflect this, let us Knpj)ose" the triangle, to be taken from the polygon ABC DE, fig ]5, plate J, has been changed into a triangle IsiOV, fg. 2(i, whose height above its base MP is equal to that of the given point F, above A B oifig, \(S : \.\\\^ done, on AB (prolonged if necessary) lay off AG equal to OP. join F G, and the triangle AFGis equal to the triangle MOP. There are, however, three cases in the solution of this problem, which we shall therefore notice by themselves. If the base MP does not exceed AB, jig l6, fhite. 7, the point G will fall thereon, and the problem will be solved. Bat if the base M P exceeds the base k B, G will be found upon AB prolonged, fig 17 ^nd 18, plate 7;JoinFB, and draw GH parallel thereto; from the situation of this point arise the other two eases Case 1. When the point H, fig. 17, phue 7, 'S found on the side BC, contiguous to the side A B, G 82 DIVISIOK OF join F H, and the quadrilateral figure F A BH is equal to the triangle MOP. Case 2, When H, Jig. 18, plate 7, meets BC pro- longed, tVoin F draw F C and H I parallel thereto ; then join FI, and the pentagon F ABCI is equal to the triangle MOP. DIVISION OF RECTILINEAR FIGURES. 1. To divide the triangle A ME, 7?^. i:^, plate J, into four equal parts ; divide the base into four equal parts by the points B, C, D; draw M D, M C, MB, and the triangle will be divided into four equal parts. 2. If the triangle A ME, /^. \g, plate "J, is to be divided into four equal parts from a point m, in one of its sides, change it Into another A m e, with its summit at m, and then divide it into four equal parts as before; if the lines of division are contained in the triangle A INI E, the problem is solved; but if some of the lines, as m D, terminate without the triangle, join m E, and draw d D parallel thereto; join m d, and the quadrilateral m C E d is equal to C m 1) -th of A M E, and the triangle is divided into four equal parts, 3. To divide the polygon A B C D E F, fg. 20, flate"]., into a given number of equal parts, ex. gr. four, from a point G, situated in the side AF; 1. Change the polygon into a triangle A G M, whose summit is at G. 2. Divide this triangle into as many equal triangles A G H, H G I, IGK, K G M, as the polygon is required to be divided into. 3. !5ub- tract from the polygon a part equal to the triangle A G FI; then a part equal to the triangle A G I, and afterwards a part equal to the triangle A G K, and the lines G H, G Pv, G O, drawn from the point G, to make these subtractions, will divide the polygon into four equal parts, all which will be sufficiently evident from consulting the figure. RF.CTIIIXEAR FIGURES. 83 Problem 53. Three points^ Jig. 21 to 25, pla/e 7, N, O, A, l^eing given, arranged in any tnanner on a stniii^ht line, to Jind tivo other points, B, b, ;// t/ie same line so situated, that as N O : AB :: AB : N B NO : Ab :: Ab : NB A/ri Kv> NO ,pT NO,. Make A P r= , and P L z= , placing NO them one after the other, so that A L = . 2 Observing, 1. That in the two figures 21 and 22, A is placed between N and O; and that A P is taken on A O, prolonged if necelHiry. 2. In fg. 23, where the point O is situated be- tween N and A, A P should be put on the side opposite to A O. 3. In Jig. 2-1, where the point N is situated be- tween A. and O, if A P be smaller than A N, it must be taken on the side opposite to A N. 4. \ufg. 25, where the point N is also placed between A and O, if A P be greater than A N, it must be placed on A N prolonged. Now by problem \^ make M N in all the five figures a mean proportional between N O^ N P, and carry this line from L to B, and from L to b. and N O will be : A B :: A B : N B NO : Ab :: Ab : N B. Problem oQt. Two lines E F, G H, fg. 27, 28, 29, plate 7, intersecting each other at A, heijig given, to draw from C a third line W D, which shall form with the other two a triangle D A B equal to a given triangle X. 1. Froin C draw C N parallel to EF. 2. Change the triangle X into another C N O, whose summit as at the point C. ;3. Find on G H a point B, so that NO: A^ :: AB : N B, and from this point G 2 84 DIVISION OP B'IGURES. B, draw the line C B, and DAB shall be the re- quired triangle. Scholium. This problem may be used to cut off one rectilinear figure from another, by drawing a line from a given point. 'J bus, if from a point C, without or within the triangle E A H, j^^'. 30, a right line is required to be drawn, that shall cutoff a part DAB, equal to the triangle X, fg. 30 ; it is evident this may be effected by the preceding problem. If it be required to draw a line B D from a point c,. which shall cut off from the quadrilateral figure E F G H, a portion D F G B, fg. 3 ] , equal to the triangle Z ; if you are sure, that the riglit line BD will cut the two opposite sides EF, G H, prolong E F, G H, till they meet ; then form a triangle Z, equal to the two triangles Z, and FAG; and then take Z from A E II by a line B D from the point C, which is effected by the preceding problem. If it be required to take from a polygon Y, a part D F I H B equal to a triangle X ; aiid that the line B D is to cut the two sides E F, G H ; prolong these sides till they meet in A ; then make a triangle Z, equal to the triangle X, and the figure A F I H ; and then retrench from the triangle E A G the tri- angle DAB equal to Z, by a line B D, from a given point C. As all rectilinear figures may be reduced to tri- angles, we may, by this problem, take one rec- tilinear figure from another by a straight line drawn from a given point. PrxOBLEM 57. To viahc a triangle equal to any g'lVQU quadrilateral figure A W C D, fig. ?>'6^ flate 7- 1. Draw the diagonal B D. !J. Draw C E parallel thereto, intersecting A D produced in E. 3. Join A C, and A C E is the required triangle. Problem 58. To make a rectangle^ or paralle- logram equal to a given triangle A C E, fig. 33, plate 7. ADDITION OF FIGURES. 85 1. Bisect the base A E in 1). 2. Through C draw C B ijarallel to A 1). 3. Draw CI), B A, parallel to each other, and either per|>endicular to A E, or making any angle with it. And the reetangle or pa- rallelogram A B C D will be equal to the given tri- angle. Prohlem 59. 7'y ttuike a triatigh equal to a given circle, fig. a-l. ■pLite'J* Draw the radius O B, and tangent A J} perpen- dicular thereto ; make A B equal to three times the diameter of the circle, and 7 more • join A O, and the triangle A O B will be nearly equal the given circle. Problem 6o. To make a square eqiuil to a gii-en rectangle, ABC ^ fig. 35, flate 7. Produce one side A B, till \\ L be equal to the other BC. 2. Bisect AL in O. 3. With the distance A O, describe the semicircle LEA. A. Produce B C to F. 5. On B F, make the square B F Gil, which is equal to the rectangle A BCD. ADDITION AND SUBTRACTION OF SIMILAR FIGURES. Problem 61. To make a square equal to the sum of any number of squares taken together, ex. gr. equal to three given squares, 'whose sides are equal to the lines A l^Cfig- 36, plate 7. 1. Draw the infinite lines E D, D F, at rijrht angles to each other. 2. Make D G equal to A, and D H equal to B. 3. Join G and H, and Gil will be the sidcofa square, equal the two squares whose sides are A and B. 4. Make D F equal G H, D K equal C, and join K F; then will K F be the side of a square equal the three given squares. Or, after * Strictly speaking, this can only be solved but by an appro- ximation ; the area or squaring of the circle is yet a desideratum in mathematics. See Huttons Mathematical and Pbilosnphkal Dictionary ^ 2 Vols.4to, 1790". Edit. G 3 86 ADDITION AND SUBTRACTION the same manner may a square be conslructed equal to any number of given squares. Problem 62, To describe a figure equal to the sum of any given number of similar figures, fig. 36, flate 7. This problem is similar to the foregoing : 1 . Form a right angle. 2. Set off thereon two homologous sides of the given figures, as from D to G, and fi-om D to H. 3. Draw G H, and thereon describe a figure similar to one or' the given ones, and it will be equal to their sum. In the same manner you misy go on, adding a greater number of similar figures together. If the similar figures be eircles, take the radii or diameters for the homologous lines. PROBLEM 63. To make a square equal to the dif- ference of t-wo given squares, whose sides are \ B, C D, fig. ZT, plate 7. 1. Orx one end B of the shortest line raise a per- pendicular B F. 2. With the extent CD from A, cut B F in F, and B F will be the side of the re- quired square. In the same manner the difference between any two similar figures may be found. Problem 64. To inake a figure which shall be si- milar to, and contain a given figure, a certain number of times. Let M N be an homologous side of the given figure y fig. ?>^,plale 7. 1. Draw the indefinite line BZ. 2. x\t any point D, raise D A perpendicular to BZ. 3. Make B D equal to M N, and B C as many times a multiple of B D, as the required figure is to be of the given one. 4. Bisect B C, and describe the semicircle BAG. 5. Draw AC B C. 6. Make A E equal M N. 7. Draw E F parallel to B C, and E F will be the hon^o- logous side of the required figure. Problem 65. To reduce a complex figure from one scale to another, mechanically , by means of squares^ Fig 3g, fate 7. OP SIMILAR FIGURES. 87 1. Divide the given figure by cross lines into as many sqiuires as may be thought necessary. 2. Di- vide another paper into the same number of squares, either greater, equal, or less, as required. 3. Draw in every square what is eontained iu the eorrcspon- deiit square of the given figure, and you will obtain a copy tolerably exact. Problem (3d. To enlarge a nuip or p/iin, and make it fzvtce, threr, four, or Jive, &c. times larger than the original, Jig. 12, plate 8. I. Draw the indefinite line a b. 2. Raise a per- pendicular at a. 3. Divide the original plan into squares by the preceding problem. 4. Take the side of one of the squares, which set off from a to d, and on the perpendicular from ti to c, finish the square aefd, which is equal to one of the squares of the proposed plan. 5. Take the diagonal d e, set it off from a to g. and from a to 1 ; complete the square a 1 n g, and it will be double the square a e [(\. To find one three times greater, take d g, and with that extent form the square a m o h, which will be the square required. With d h you may form a square that will contain the given one aefd four times. The line d i gives a square five times larger than the original square. Problem QJ . To reduce a map ^, ^d, ~th, \th^ &c. of the original, fg. 5, plate 8. 1. Divide the given plan into squares by problem (35. 2. Draw a line, on which set off from A to B one side of one of these squares. 3. Divide this line into two equal parts at F, and on F as a centre, with F A or F B, describe the semicircle A II b. 4. To obtain 4- the given square, i\t ¥ erect the perpendicular F"!!, Jind draw the right line AH, which will be the side of the required square. 5. For ~ rd, divide A B into three parts take one of these parts, set it ofl* from A to C, at C raise the perpen- dicular C I, through I draw A I, and it will be the side of the required square. (3. For-ith, divide A B G 4 88 CURIOUS PROBLEMS ON THE into four equal parts, set off one of these from B to E, at E make E G perpendicular to A B, join B G, and it will be the side of a square ;^;th of the given one. Problem 68. To make a map or plan in propor- tion to a given one, e. gr. as three to Jive, fg. 6, plate 8. The original plan being divided into squares, 1. draw A M equal to the side of one of these squares. 2. Divide A M into five equal j)arts. 3. At the third divison raise the perpendicular C D, and draw A D, which will be the side of the required square. Problem 6g. To reduce figures by the angle of reduction. Let nh he the given side on which it is re- quired to construct a figure similar /o A C D E, fig. J , 1, 3, plate 8. 1. Form an angle L M N at pleasure, and set off the side A B from M to I, 2, PVom I, with a b, cut M L in K. 3. Draw the line I K, and several lines parallel to, and on both sides of it. The angle L M N is called the angle of proportion or reduction. 4. Draw the diagonal lines B C, AD, A E, B D. 5. Take the distance B C, and set it off from M to- wards L on M L. 6. Measure its corresporiding line K I. 7- From b describe the arc n o. 8. Now take A C, set it off (jn M L, ar.d find its corrcspon- ilent line f g. (>. From a, with the radius ( g, cut the former arc n o in c, and thus proceed till vou have completed the figure. Problem 70. To enlarge a figure /\r the angle of reductiori. Let a b c d c be the given figure, and A B- the given side, fig. ?^^ 1, and A ^ phi It ^. 1. j'crm, as in the prececHng problem, the angle LMN, by setting ofF ab from M to H, and from H with A B, cutting i\I L in I. 2. Draw H I, and parallels to, and on both sides of it. 3. lake the diagonal be, set it ofl'from M towards L, and take off its corresponding line q r. A. With q r as a radius t>n B describe the arc m n. 5. Take tjic same cor- DIVISION OF LINES AND CIRCLES. QS respondent line to ac, and on A cut m n in c, and so on for the other sides. CURIOUS PROBLEMS ON THK DIVISION" OP LINES AND CIRCLES. PROBLEM 71. To cut off from any given arc of a circle a third, a fifths a sci-enth, C5f . odd parts, and thence to divide that arc into any number of aiual farts, fig,1, plated. Example 1. To divide the arc A K B into three equal parts, C A being the radius, and C the centre of the arc. Bisect A B in K, draw the two radii C K, C B, and the chord A B ; produce A B at pleasure, and make 1^ L eq-'al A B ; bisect A C at G ; then a rule on G and L will cut C B in E, and BE will be 'd, andCE4ds of the radius C B ; on C B with CE of the radius C B, with radius CI describe the arc In M, which will be bisected in n, by the line C K ; then take the extent I n, or its equal M n, and set it off twice from A to B ; that iS; first; from A to d, and from d to o, ancj QO CURIOUS PROBLEMS ON THE oBwIll betth of the arc A B. Again, set off the same extent from B to m, and from m lo c, and the arc A B will be accurately divided into five equal parts. Examples. To divide the given arc AB into seven equal parts. A B being bisected as before, and the radii C A, C K, C B, drawn, find by problem 9 the seventh part of P B of the radius C B, and with the radius C P, describe the arc P rN ; then set off the extent P r twice from A to 3, and from 3 to 6, and 6 B will be the seventh part of the given arc AB ; the compasses being kept to the same opening P r, set it from B to 4, from 4 to 1 ; then the extent A 1 will bisect 1, 3 into 2, and 4,6 into 5 ; and thus divide the given arc into seven equal parts. Problem 72. To inscribe a regulavheptagon in a circle, fig. 8, plate 8. In^^. 8, let the arc B D be fth part of the given circle, and A B the radius of the circle. Divide A B into eight equal parts, then on centre A, with radius A C, describe the arc C E, bisect the arc B D in a, and set off this arc B a from C to b, and from b to c ; then through A and c draw A c e, cutting B D in e, and B e will be fth part of the given circle. Corollary 1. Hence we have a method of finding the seventh part of any given angle ; for, if from the extremities of the given arc radii be drawn to the centre, and one of these be divided into eight equal parts, and seven of these parts be taken, and another arc described therewith, the greater arc will be to the lesser, as 8 to 7, and so of any other proportion. Corollaiy 2. If an arc be described with the ra- dius, it will be equal to the ^th part of a circle whose radius is A B, and to the seventh part of a circle whose radius is AC, and to the sixth part of a circle whose radius is A L, &c. Corollary 3. Hence also a pentagon may be de- rived from a hexagon, fig. 9? plate 8. Let the given circle be A B C D E F, in which a pentagon is to be DIVISION OF LINES AND CIRCLES. Ql inscribed ; with the radius A C set off A P equal -th of the circle, divide A C into six ecjual parts ; then c G will be five of these parts ; with radius C G describe the arc G H, bisect A F in (j, and make G P and A II each equal to A q ; then through C and 1 1 draw the semi-diamctcr C w, cutting ilio given circle in w, join Aw, and it will be one side of the required pentagon. Cordlary 1. Hence as radius divides a circle into six equal parts, each equal (io degrees, twiee radius gives 120 degrees, or the third jnu't of the circum- ference. Once radius gives 6o degrees, and that arc bisected gives 30 degrees, which, added to 6o, divides the circumference into four equal parts ; whence we divide it into two, three, four, five, six equal parts ; the preceding corollary divides into five equal parts, the arc of a quadrant bisected divides it into eight equal parts. By problem 7 1 ^vc obtain the seventh part of a circle, and by this method divide it into any number of equal parts, even a prime number; for the odd unit may be cut off by the preceding problem, and the remaining part be subdivided by continual bisection, till another prime number arises to be cut off in the same manner. Problem 73. To divide a given right line, or an arc of a circle, into any mimbe^' of equal parts by the help of a pair of beam, or other compasses^ the distance of whose points shall not be nearer to each other than the given line, fig. 1 1 , plate 8. ^ From this problem, published by Clavius, the Jesuit, in l()ll, in a treatise on the construction of a dialling instrument, it is presumed that he was the original inventor of that species of division, called Nonius, and which by many modern mathematicians has been called the scale of Vernier. Let A B be the given line, or circular arch, to be divided into a number of equal parts. Produce them at pleasure; then take the extent AB^ and set it 92 CURIOUS PROBLEMS ON THE off on the prolonged line, as many times as the given line is to be divided into smaller parts, B C, CD, DE, E F, F G. Then divide the whole line A G into as many equal parts as are required in A B, as GH, HI, IK, K L, LA, each of which contains the given line, and one of those parts into which the given line is to be divided. For A G is to A L as A F to A B ; in other words, A L is contained f.ve times in A G, as A B in A F ; therefore, since A G, contains A F, and ^thofAB; therefore BL is the fth of A B. Then as G H contains A B plus, F ti, which is -^ th of A B, E I will be 4 ths of A B, D K ■^ths, C L -t-ths. Therefore, if we set off the interval G H from F and H, we obtain two parts at L and I, set off from the points near E, gives three parts be- tween D K, from the lour points at D and K, gives four parts at C L, and the next setting off one more of these parts ; so that lastly, the extent G H set off from the points between C and L, divides the given line as required. To divide a given line A C, or arc of any circle, into any number of equal parts, suppose ^O, Jig. 11, flate 8. 1. Divide it into any number of equal parts less than 30, yet so that they may be aliquot parts of 30 ; as for example, A G is divided into six equal parts, AB, BC, CD, DE, E F, F G, each of which are to subdivided into five equal parts. 2. Divide the first part A B into five parts, by means of the in- terval A L orJGrH, as taught in this problem. If, now, one foot of the compasses be put into the point A, (theexrent AL remaining between them unal- tered) and then into the point next to A, and so on TO the next succeeding point, the whole line A. G will he divided by the other foot of the compasses into 30 equal parts. Or it' the right line, or arc, be first divided into five equal parts, each of these must be subdivided inr.o six parts, which may be effected by bisecting DIVISION OF LINES AND CIRCLES. ^3 •ach part, and then dividing the; halves into three parts. Or it may be still better to bi gi^'^s 17, &c. so that the measure required is 50° i-?' 29'. Kxitmple 3. 'J'ake a semicircle three inches radius ■ind let the angle be 2 in 1 ; then the steps will be -1, 2, 1, 2, 3, 2', the answer 41" 10' 30' 9'". The whole circle, continued and stepped with the same opening, gave 8. 1.3. 2. 3. lor tlic same, yet the aiL-iwcrs agreed to the tenth of a second. By the same method any given line may be mea- sured, and the proportion it bears to any other straight line found. Or it will give the exact value of any straight line, ex. gr, the opening of a pair of com- passes by ste])ping any known given line with it, and this })i!ich nearer than ihe eye can discern, by comparing it ivilh any other line, as a foot, a yard, (ffc. This method will be found more accurate than by scales, or even ta"bles of sines, tangents, &c. because the measLU'e of a chord cannot be so nicely deter- mined by the eye with extreme exactness. There may be some apparent difficulty attending the rule when put in practice, it being impossible to assign any example which another person can repeat with perfect accuracy, on account of the inequality in the scales, by which the same steps, or line, will be measured by diffi^rent persons. There will, there- fore, be always some small variation in the answer ; it is however, demonstrably true, that the answer given by the problem is most accurately the measure of the given angle, although you can never delineate another angle, or line, exactly equal to the given one, tlrst measured by way of example, and this arising from the inc(]uality of our various scales, our inatten- tion in measm* ng, and the imj)erfection of our eyes. Hence, though to all appearance two angles may ap- pear perfectly e(jual to each other, this method will 11 2 lOO CUUIOUS PROBLEMS ON THB give the true measure of eachj and assign the minutest difi-'erence between them. Figure \, plate 9, will illustrate clearly this method ; thus, to measure the angle A C B, take A B between your compasses, and step B a, a b, b c, there will be c D over. Take c D, and w ith it step A e, e f , and you will have f B over ; with this opening step A g, g h, and you will have h e over, and so on. Peoblem 78. To divide a large quadrant or circle. We shall here give the principal methods used by instrument-makers, before the publication of Mr. Bird's method by the Board of Longitude, leaving it to artists to judge of their respective merits, and to use them separately, or combine them together, as occasion may require ; avoiding a minute detail of particulars, as that will be found when we come to describe Mr. Bird's method. It will be necessary, however, previously to mention a few circumstances, W'hich, though in common use, had not been de- scribed until Mr. Bird\ and Mr. Ludlams comment thereon were published. " In all mathematical instruments, divided by hand, and not by an engine, or pattern, the circles, or lines, which bound the divisions are not those which are actually divided by the compasses." '' A faint circle is drawn very near the bounding circle ; it is this that is originally divided. It has been termed the priniiti-ve circle''' '• The divisions made upon this circle are faint arcs, struck with the beam compasses ; fine points, or conical holes, are made by the prick punch, or pointing tool, at the points where these arcs cross {he primitive circle ; these are called origiiial poinls.'' " The I'lslble divisions are transferred^ from the original fx>ints to the space between the bounding circles, and are cut by the beam compasses; they Dirisiox OP Li:>-r> an'o circles. lOi are therefore always arcs of a circle, tlioucrh so short, as not to be distineruishcd from strainrht lines. Method 1 . Tlio flint or |)riinitivc, arc is tlrst struck ; the exact measure of the radius thereof is theiv ob- tained upon a standard scale with a nt^nius division of 1000 parts of an inch, which if the radius exceed ]0 inches, mny be obtained to five places of figures. This measure is liie chord of ()U. The ot'uei chords necessary to be laid off are computed by the sid)- joined jjroportion,* and then taken off fro ni the stan- dard scale to be laid down on the quadrant. Set off the chord of Oo', then add to it the choid of 30, and you obtain the 90th degree, Mr. B'lrd^ to obtain '70°, bisects ihe chord of 60"^, and then sets off the same chord from 30 10 90°, and not of 30 from 60'' to 90"". Some of the advantages that arise from this method are these ; for whether the chord of 30 be taken accurately or not from the scale of equal parts, yet the arc of 60 will be truly bi- sected, (see remarks on bisection hereafter) and if the radius unaltered be set off from the point of bi- section, it will give 90 true ; but if the chord 30, as taken from the scale, be laid off from 60 to 90, then an error in that chord will make an equal error in the place of 90''. Sixty degrees is divided into three parts by setting off the computed chord of 20 degrees, and the whole quadrant is divided to every JO degrees, by setting off the same extent from the other points. Thirty degrees, bisected by the computed chord of 15, gives 15^, which stepped from the points already found, divides the quadrant to every fifth degree. The computed chord of 6'^ being laid off, divides 30 degrees into five parts ; and set oft' from the other divisions, subdivides the quadrant into single degrees. * As the radius is to the given angle, so is the measure of the radius to halt" the required chord. H 3 102 CURIOUS PROBLEMS, &C. Thus with five extents of the beam compar.scs, and none of them less than six degrees, the quadrant is divided into gO degrees. Fifteen degrees bisected, gives 7° 30', which set off from the other divisions, divides the quadrant into half degrees. The chord of Cf 40' divides 20' into three parts, and set off from the rest of the divisions, divides the whole instrument to every ten minutes. The chord of lo'^ 5' divides the degrees into 1^ parts, each equal to five minutes cf a degree. Method 1. The chords are here su})posed to be computed as betbre, and taken off from the nonius scale. 1. Radius bisected divides the quadrant into three parts, each equal to 30 degrees. 2. The chord of 10° gives nine parts, each equal to 10 degrees. 3. Thirty degrees bisected and set off, gives 18 parts, each equal to five degrees. 4. Thirty degrees into five, by the chord of 6°; then set off as before gives 90 parts, each equal to 1 degree. 5. The chord of 6° 40' gives 2/0 parts, each equal to 20 minutes. 6. The chord of 7° 30' gives 540 parts, each equal to 10 minutes. 7. The chord of 7° 45' gives 1080 parts, each equal to five minutes. Or Method 3. The computed chords supposed, 1. 60 into- 4: gives 3 parts, equal 30'^ 2. 30 ^ equal 15 6 3 5*^ 3. 15 3 equal 5 18 o" 4. 20 5 equal 6 90 1° 5. 20 3 equal. 6'' 40' 270 20' t). 15 -i cqur.l 7° 30' 540 \0' 7. 15, 30 t equal 7° 45' 1080 5' Thus may the practitioner vary his numbers for MR. BIRDS METHOD. ]03 any division whatsoever, and yet j)rescrvc a sufTicient extent between the points of his compasses. If the quadrant be divided as above, to every 15 degrees, and then the eomputed arc of J() degrees set off, this arc may be divided by eomimial l)i.seetion into single degrees. If from the arc of 45°, 2° '20' be taken, or 1^ lo' from 22° 30', wc may obtain evcrv fifth minute by continual bisection. If to the arc of 7" lo' be added the arc of 62 minutes, the arc of every single minute m:iy be had by bisection. Of Mk. Bird's method of dividing. Fig. C, jpliUe Q. • In 1 7(^7 the Commissioners of Longitude proposed an handsome reward to Mr. BirJ, on condition, among other things, that he should publish an ac- count of his method of dividing astronomical instru- ments ; which was accordingly done : and a tract, describing his method of dividing, was written by him, and published by order of the Commissioners of Longitude in the same year ; some defects in this publication were supplied by the Rev. Mr. Ludlatii, one of the gentlemen who attended Mr. Bird to be instructed by him in his method of dividing, in con- sequence of the Board's agreement with him. Mr. Licdlums tract was published in 1787, in 4to. I shall use my endeavours to render this method still clearer to the practitioner, by combining and arranging the subject of both tracts. Mr. Bird's method. 1. One of the first requisites is a scale of inches, each inch being subdivided into 10 equal parts. 2. Conti,c;dousto this line of inches, there must be a nonius, in which 10.1 inches is divided into 100 equal parts, thus shewing the lOOOndth part of an H 4 104 MK. BIRD S METHOD. inch. By the assistance of a magnifying glass of one inch fbCLis, the SOOOndth part may be estimated. 3. Six beam compasses are necessary, furnished with magnifying glasses of not more than one inch focus. The longest beam is to measure the radius or chord of 6o ; the second for the chord of 4*2.40 ; the third for the chord of 30 ; the fourth for 10.20 ; the fifth for 4.40; the sixth for the chord of 15 degrees. 4. Compute the chords by the rules given, and take their computed length from the scale in the dif- ferent beam compasses. 5. Let these operations be performed in the even- ing, and let the scale and the different beam com- passes be laid upon the instrument to be divided, and remain there till the next morning. 6. The next morning, before sunrise, examine the compasses by thcseaie, and rectify them, if they are either lengthened or shortened by any change in the temperature of the air. 7. The quadrant and scale being of the same tem- perature^ describe the faint arc b ^, or primitive cir- cle ; then with the compasses that are set to the ra- dius and vv'ith a fine prick punch, make a point at ^i, which is to be the point of the quadrant ; sec Jig. 2, ^/^/^ 9. 8. With the sam.e beam compasses unaltered lay off from a to e the chord of t>0°, making a tine point at c. 9. Bisect the arc a c with the chord of 30°. 10. Then from the point c, with the beam com- passes containing 60, mark the yo'nit r, which is that of 90 degrees. ] 1. Next, with the beam compasses containing 15^, bisect the arc e r in n, which gives 75°- 12. Lay olf from n towards r the chord of 10° 20^, and from r towards n the chord of 4° 4(»' ; these two ought to meet exactly at the point g of 85° 20'. J 3. Nuv.' as in large instruments each degree is MR. BIRD S METHOD. 105 ^generally subdivided into 1*2 equal parts, of five mi- nutes eaeh, we shall iind that 85° 20' contains 10.24 such parts, because 20' e(jual 4 of these parts, and SoXi'-i makes lOiO; now 1'J24 is a number divisi- ble by continual bisection. The last computed chord was 42" 40', with which a g was bisected into o, and a o, o g, were bisected by trials. Though Mr. /i/V^ seems to have used this method himself, still he thinks it more advise. .blc to take the computed chord of 21^ 20', and by it find the pomt g ; then proceed by continual bisections till you have 102 1 parts. Thus (he arc 8 s'' 20', by ten bisections, will give us the arcs 42^40', 21° 80', lO"" 40', 5° 20', 2° 40', 1° 20', 40', 20', 10', 5'. 14. To fill up the space between g and r, 85° 20', and 90^, which is 4° 40', or 4X124-8 equal to 5() divisions ; the chord of (34 divisions was laid off from g towards d, and divided like the rest by continual bisections, as was also from a towards b. If the work, is well performed, you will again find the points 30, 45, 60, 75, and 90, without any sensible difference. It is evident that these arcs, as well as those of 15°, are multiples of the arc of 5'; for one degree con- tains 12 arcs of 5' each, of which 15° contains 180 ; the arc of 30" contains 3OO ; the arc of 6o°, 720 ; that of 75°, 900 ; and, therefore, 90° contains 1080. Mr. Gi-aham, in 1725 applied to the quadrant di- vided into 90"', or rather into 1080 parts of five mi- nutes each, another quadrant, which he divided into 96 equal parts, subdividing each of these into 16 equal parts, forining in all 1536. This arc is a severe check upon the divisions of the other ; but />//r/ says, that if his instructions be strictly followed, the coincidence between them will be surprising, and their difference from the truth exceedingly small. The arc of 96° is to be divided first into three equal parts, in the same manner as the arc of 90^^; each third contains oVl divisions, which number is divi- i05 FOR THE NONIUS. sible continually by 2, and gives l6 in each 96th part of the whole. The next step is to cut the linear divisions from the points obtained by the foregoing rules. For this purpose a pair of beam compasses is to be used, both of whose points are conical and very sharp. Draw a tangent to the arc b d, suppose at e, it will intersect the arc x y in q, this will be the distance between the points of the beam compasses to cut the divisions nearly at right angles to the arc. The point of the beam compasses next the right hand is to be placed in the point r, the other point to fall freely into the arc X y ; then pressing gently upon the screw head which fastens the socket, cut the divisions with the point towards the right hand, proceeding thus till you have finished all the divisions of the limb. FOR THE NONIUS. 1. Chuse any part of the arc where there is a co- incidence of the 90 and 96 arcs ; for example, at e, the point of 6o°. Draw the faint arc s t and i k, which may be continued to any length towards A ; "upon these the nonius divisions are to be divided in points. The original points for the nonius of the 90th arc are to be made upon the arc s t ; the origin nal points for the nonius of the 90ih arc are to be mrde upon the arc i k. Because 90 is to 96, as 15 to 16", there will be a coincidence at 15° and l6pts. 30° and 32pts, 45° and 48pts ; 60® and 64pts, 90° and 96pts. 2. Draw a tangent line to the primitive circle as before, intersecting the arc d, which gives the dis- tance of the points of the beam compasses, with which the nonius of the 90th arc must be cut. 3. Let us suppose then that the nonius is to sub- divide the divisions of the limb to half a minute^ which is effected by making 10 divisions of the no- FOR THE NONIUS. lO^ nins equal to 1 1 divisions of the limb ; measure the radius of the arc, and compute the chord of l(j, or rather 3'2 of the nonius division, which may easily be obtained by the following proportion ; if 10 divi- sions of the nonius plate make 55 minutes of a de- gree, what will 32 of those divisions make ? the an- swer is '2° 5()', the chord of which must be computed and taken from the scale of equal parts ; but as dif- ferent subdivisions by the nonius may be required, let n be the number of nonius divisions, m the num- ber of minutes taken in by the nonius b, 16, 32 or ()4-, and X the arc sought ; then as n : m :: b : X. •1. Lav off with the beam compasses, having the length of the tangent QO between the points, the point q from e, q bcjing a point in the arc s t, and e an original point in the primitive circle, and the chord of 3'J from q towards the left hand, (the chord of 32 being the chord which subtends 32 divisions on the nonius plate, or the chord of 2° 56' ; this chord to be computed from the radius with which the faint arc s t was struck, and taken off the scale of equal parts,) and divide by continual bisections ; ten of those divisions, counting from q to the left, will be the required points. The nonius belonging to the 96th arc is subject to no difficulty, as the number should always be 16, 3L% &c. that the extremes may be laid off from the divi- sions of the limb without computation. To be more particular, the length of the tangent line, or radius, with which the divisions of the nonius of the 96 arc are to be cut, must be found in the way before direc- ted for the nonius of the 90 are ; the arc i k standing instead of the arc s t. Having the tangental distance between the points of the compasses trom one of the original points in the primitive u6, lay off a point on the faint arc i k towards the left hand ; count from that point on the primitive 96 circle 17 points to the left hand, and lay off from thence another point on the faint arc i k ; the distance between those two 108 FOR THE NONIUS. points in the faint arc i k is to be subdivided by bi- sections into 16 parts, and those parts pointed ; from these points the visible divisions of the nonius are to be cut. Mr. Ludlam thinks, that instead of laying the ex- tent of the nonius single, it would be better to lay it off double or quadruple ; thus, instead of 17 points to the left hand, count 34 or 6S, and that both ways, to the right and to the left, and lay off a point from each extreme on the faint arc i k ; subdivide the whole between these extreme points by continual bi- sections, till you get 16 points together on the left band side of the middle, answering to the extent of the nonius. No more of the subdivisions are to be completed than are necessary to obtain the middle portion of lO points as before. The nonius points obtained, the next process is to transfer them on the nonius plate, which plate is chamfered on both edges ; on the inner edge is the nonius for the go arc. on the outer edge is the no- nius for the gO arc, in the middle between the two chamfers is a flat part parallel to the under surface of the nonius plate ; upon the fiat part the faint line of the -next operation, is to be drawn. To find the flace where the nonius is to begin upon the chamfered edge of the nonius 'plate, measure the distance of the centre of the quadrant from the axis of the telescope ; this distance from the axis of the telescope at the eye end, will be the place for the first division of the nonius ; then draw a faint line from the centre on fhe flat part of the nonius plate. Fasten the nonius plate to the arc with two pair of hand vices ; then with one point of the beam com- passes in the centre of the quadrant, and the other at the middle of the nonius plate, draw a faint arc from end to end ; where this arc cuts the faint line before- mentioned, make a fine point ; from this point lay off on each side another point, which may be at any distance in the arC;, only care must be taken that they MR. bird's method, &C. J 0(.) be equally distant from the middle point ; from the two last make a faint intersection as near as poissiblc to cither of the chamfered edges of the nonius plate ; through this intersection the iirst division of the no- nius nuist be cut. Mr. Bird's method of dividing his scale OF EQUAL parts. Let us suppose that we have QO inches to divide into ()00 equal parts, take the third of this number, or 300 ; now, the first power of 2 above this is 512 ; therefore, take Vo'^bs of an inch in one pair of beam compasses, W- in another, -j-^'* in a third, and ■^;% in a fourth ; then lay the scale from which these measures were taken, the scale to be divided, and the beam compasses near together, in a room facing the north ;, let them lie there the whole night ; the next morning correct your compasses,and lay off VV three times, then with the compasses -^J', VV? t-o> bisect: these three spaces as expeditiously as possible ; the space 64- is so small that there is no danger from any- partial or unequal expansion, therefore the remainder may be finished by continual bisections.- The linear divisions are to be cut from the points with the beam compasses as before described. The nonius of this scale is -.Vths of an inch long, which is to be divided into 100 equal parts, as 100 is to 101, so is 256 : 258,56 tenths of an inch, the in- teger being -,'v. Suppose the scale to be numbered at every inch from left to right ; then make a fine point exactly against -,'o, to the left of o, from this lay oft' 258,56 to the right hand, which divide after the conuDon method. If you are not furnished with a scale long enough to la) off 258,50, then set oft' \\'-, and add 8.56, from a diae-oHal scale. [ 110 ] Of Mil. Bird's pointing tool, and method OF POINTING. The pointing tool consisted of a steel wire V, inch diameter, inserted into a brass wire i inch diameter, the brass part 24- long, the steel part stood out ^, whole length Sf inches. The angle of the conical point about 20 or 25 degrees, somewhat above a steel temper ; the top of the brass part was rounded off, to receive the pressure of the finger ; the steel point should be first turned, hardened, and tempered, and then whetted on the oil stone, by turning the pointril round, and at the same time drawing it along the oil- stone, not against, but from the point ; this will make a sharp point, and alsoakiad of very fine teeth along the slant side of the cone, and give it the nature of a very fine countersink. In striking the primitive circle by the beam com- passes, the cutting point raises up the metal a little on each side the arc ; the metal so thrown up forms what is called the bur. When an arc is struck across the primitive circle, this bur will be in some measure thrown down ; but if that circle be struck again over so lightly, the bur will be raised up again ; the arcs struck jicross the primitive circle have also their bur. Two such rasures or trenches across each other, will of course have four salient, or prominent angles with- in ; and as the sides of the trenches slope, so do also the lines which terminate the four solid angles. You may therefore, feel what you cannot see ; when the conical points bear against all four solid angles, they will guide, and keep the point of the tool in the centre of decussation, while keeping the tool upright, pres- sing it gently with one hand, and turning it round with the other, you make a conical hole, into which you can at any time put the point of the beam com- passes, and feel, as well as see^ when it is lodged there. [ 1" ] Rules or maxims laid dowv by Mr. Bird. 1. The points of the beam compasses should never be broiii^ht nearer together than two or three inches, except near the end of the line or arc to be tlivided ; and there spring dividers with round moveable points had best be used. 2. The prick-punch, used to mark the points, should be very sharp and round, the conical point being formed to a very acute angle ; the point to be made by it ought not to exceed the one thousandth of an inch. When lines, or divisions, are to be traced from these points, a magnifying glass of t an inch focus must be used, which will render the im- pression or scratch made by the beam compasses suf- ficiently visible ; and if the impression be not too faint, feeling will contribute, as well as seeing, to- wards making the points properly. 3. The method of finding the principal points by computing the chords is preferable to other methods; as by taking up much less time, there is much less risk of any error from the expansion of the instru- ment, or beam compasses. 4. To avoid all possible error from expansion, Mr. Bird never admitted more than one person, and him only as an assistant ; nor suti'ercd any fire in the room, till the principal points were laid down. Mr. Bird guards, by this method, against any in- equality that might possibly happen among the ori- ginal points, by first setting out a few capital points, distributed ecjually througlj the are, leaving the in- tervals to be filled uj) afterwards ; he could by this method check tiie distant divisions with rcsj)c*ct to each other, and shorten the time of the most essential operations. 5. Great care is to be observed in pointing inter- sections, which is more difficult than in pointing from a single line, made by one point ot the coujpasses. 112 ROLES OR MAXIMS, For in bisections, the place to be pointed is laid off from the right to the left, and from the left to the right. If any error arises from an alteration of the compasses, it will be shewn double ; even if the chord be taken a little too long, or too short, it will not occasion any inequality, provided the point be made in the middle, between the two short Imes traced by the compasses. iSow, as Mr. Liuilam observes, if the bisecting chord be taken exactly, the two fore-mentioned faint arcs will intersect each other in the primitive circle, otherwise the intersection will fall above or below it. In either case, the eye, assisted by a magnifier, can accurately distinguish on the primitive circle the mid- dle between these two arcs, and a point may be made by the pointing tool. In small portions of the primitive circles, the two faint arcs will intersect in so acute an angle, that they will run into one another, and form as it were a single line; yet even here, though the bisecting chord be not exact, if the intersection be pointed as before, the point will fall in the middle of the portion to be bisected. If, in the course of bisecting, you meet with a hole already made with the pointril, the point of the com- passes should fall exactly into that hole, botb from the right and left hand, and you may readily feci what you cannot see^ whether it fit or no ; if it fits, the point of the compasses will have a firm bearing against the bottom of the conical hole, and strike a solid blov/ against it ; if it does not exactly coincide with the centre of the hole, the slant part of the point will slide down tlic slant side of the hole, drawing, or pushing the other point of the compasses trom its place. Mr. BlrcV^ method of transferring the divisions by the beam compasses from the original points, is founded on this maxim, that a right hue duinot be cut u^on brass f so as accurately to pass ihroitgh iivo given LAID DOWN BY MR. BIRD. 113 foifi/s ; but that a circle may be described from any centre to pass with accuracv through a given point. It is exceeding difficult, in the first place, to fix the rule accurately, and keep it firmly to the two points; and, seconilly, supposing it could be held properly, yet, as the very point of the knife which enters the metal and ploughs it out, cannot bear against the rule, but some other part above that point will bear iigainst it ; it follows, that if the knife be held in a different situation to or from the rule, it will throw the cutting point oat or in ; besides, any hardness or inequality of the metal will turn the knit'c out of its course, for the rule does not oppose the knife in departing from it, and the force of the hand cannot hold it to it. For these reason-, it is almost impossi- ble to draw a knife a second time against the rule, and cut within the same line as before. On the other hand, an arc of a circle may always be described by the beam compasses so as to pass through a given point, provided both points of the compasses be conical. Let one point of the compas- ses be set in the given point or conical hole in the bra^s plain ; make the other point, whatever be its distance, the central, or still point ; with the former point cut the arc, and it will be sure to pass through the given point in the brass plain, and the operation may be repeated safely, and the stroke be strength- ened by degrees, as the moving point is not likely to be shifted out of its direction, nor the cutting point to be broken. The visible divisions on a large quadrant are always the arcs of a circle, though so short as not to be dis- tinguished Irom straight lines ; they should be per- pendicular to the are that bounds them, and there- fore the still or central point of the beam compasses must be somewhere in the tangent to that arc; the bounding circle of theivisible divisions, and the pri- mitive circle should be very near each other, that the are forming the visible divisions may be as to sense I 1.14 DESCRIPTION OF MR. BIRD*S perpendicular to both circles, and each visible division shew the original point from which it was cut. Another maxim of Mr. Bird's, attributed to Mr. Graham, That it is possible practically to bisect an arc, or right line, but not to trisect, qimiquisect, &V. The advantages to be obtained by bisection have been al- ready seen ; we have now to shew the objections against trisecting, &c. 1. That as the points of trisection in the primitive circle must be made by pressing the point of the beam compasses down into the metal, the least cxtuberance, or hard particle, will cause a deviation in the first impression of a taper point, and force the point of the compasses out of its place ; when a point is made by the pointing tool, the tool is kept turning round while it is pressed down, and therefore drills a coni- cal hole. 'J. Much less force is necessary to make a scratch or faint arc, than a hole by a pressure downwards of the point of the compasses. 3. So much time must be spent in trials, that a par- tial expansion would probably take place ; and per- haps, many false marks, or holes made, which might occasion considerable error. Another maxim of Mr. Bird's was this, that step- ping was liable to great uncertainties, and not to be trusted ; that is, if the chord of l6° was assumed, and laid down five times in succession, by turning the compasses over upon the primitive circle, yet the arcs so marked would not, in his opinion be equal. DESCRIPTION OF Mr. BiRD's SCALE OF EQUAL PARTS. It consists of a scale of inches, each divided into tenths, and numbered at every inch from the left to the right, thus, 0, 1,2, 3, 8cc. in the order of the natural numbers. The nonius scale is below this, but contiguous to it, so that one common line termi- \ SCALE OF EaUAL PARTS. 115 nates the bottoms of the divisions on the scale of inches, and the tops of the divisions on the nonius ; this nonius scale contains in Icnirth 101 tenths of an inch, this length is divided into loo equal parts, or visible divisions; the left hand division of this scale is set oft' from a point ,\ of an inch to the left of i), on the scale of inches ; therefore, the right hand end of the scale reaches to, and coincides with the lOih inch on the scale of inches. FAcry tenth divi- sion on this nonius scale is figured from the right to the left, thus, 100. 90. 80. 70. ()0. 50. 40. 30. 20. 10. O. and thus on the nonius coincides with 10 on the inches; and 100 on the nonius falls against the first subdivision (of tenths) to the left hand ofo on the inches ; and these two, viz. the first and last, are the onlv two strokes that do coincide in the two scales. To take off^ any given number of inches, decimals, and millesimals of an inch ; for example, 42,764, ob- serve, that one point of the beam compasses must stand m a (pointed) division on the nonius, and the other point of the compasses in a pointed division on the scale of inches. 'i'he left hand point of the compasses must stand in that division on the nonius which expresses the number of millesimal parts ; this, in our example, is (34. To find where the other point must stand in the scale of inches and tenths, add 10 to the given num- ber of inches and tenths (exclusive of the two mille- simal figures ;) from this sum subtract the two mille- simal figures, considered now as units and tenths, and the remainder will shew in what division, on the scale of inches, the other point of the compasses must stand ; thus, in our example, add 10 to 42,7, and the sum is 52,7 ; f''Otri this subtract 6,4 and the re- mainder IS 46,3. Set then one point of the compas- ses in the 64th division on the nonius, and the other point in 40,3 on the scale of inches, and the two points will comprehend between them 42,761 inches. I2 n6 DESCRIPTION OF MR. BIRD's This will be plain, if we consider that the junction of the two scales is at the 10th inch on the scale of inches ; therefore, the compasses will comprehend 36,3 inches on the scale of inches ; but it will like- wise comprehend 64 divisions on the nonius scale. Each of these divisions is one tenth and one millesi- mal part of an inch ; therefore, 64 divisions is 64 tenths, and 64 millesimal parts, or 6,464 inches ; to 6,464 inches taken on the nonius, add 36,3 inches taken on the scale of inches, and the whole length is 42,764 inches ; and thus the. whole is taken from two scales, viz. inches and the nonius ; each subdi- vision in the former is -j^^ of an inch, each subdivi- sion in the latter is -"^-j-— ^i^-^^th of an inch. By taking a proper number of each sort of subdivisions (the lesser and the greater) the length sought is obtained. The business of taking a given length will be ex- pedited, and carried on with far less danger of injur- ing the scale, if the proposed length be first of all taken, ?iearly, on the scale of inches only, guessing the millesimal. parts ; thus, in our case, we ought to take off from the scale of inches 42,7, and above half a tenth more ; for then if one point be set in the pro- per division on the nonius, the other point will fall so near the proper division on the scale of inches, as to point it out ; and the point of the compasses may be brought, by the regulating screw, to fall exactly into the true division. If, when the points of the compasses are set, they do not comprehend an integral number of millesi- mal parts, they will not precisely fall into any two di- visions, but will either exceed, or fall short ; let the exact distance of the points of the compasses be 42,7645 ; if, as before, the left hand point be set in the 64th division of the nonius, then the right hand point will exceed 46,3 among the inches ; if the left hand point be carried one division more to the left, and stand in 05 of the nonius, then the right hand SCALF OF EQUAL PARTS. II7 point will fall short of 46,'l in the scale of Inches ; the excess in the former case being equal to the de- fect in the latter, liy observing whether the differ- ence be equal, or as great again in one case as the other, we may estimate to 'd part of a millesimal. See Mr. Binfs Tract, p. 2. It may be asked, why should the nonius scale com- mence at the 10th inch ; why not at (), and so the nonius scale lay wholly on the left hand of the scale of inches } and, in this case, both scales might be in one right line, and not one under the other ; but, in such a case, a less distance than 10 inches could not always be found upon the scale, as appears from the rule before given. The number 10 must not, in this case, be added to the inches and tenths, and then the subtraction before directed would not always be possible. Yet, upon this principle, a scale in one continued line may be constructed f§r laying off inches, tenths, and hundredths of an inch, for any length above one inch ; at the head of the scale of inches, to the left hand of O, and in the same line, set off eleven-tenths of an inch (or the multiple,) which subdivide in Mr. BinPs way, into ten equal parts. Such a compound scale would be far more exact than the common dia- gonal scale ; for the divisions being pointed, you may feel far more nicely than you can see, when the points of the beam compasses are set to the exact distance. But to return to Mr. Bifil's Tract. The nature of Mr. i^/V^^'s scale being known, there will be no difficulty in understanding his directions how to divide it. A scale of this kind is far prefer- able to any diagonal scale ; not only on account of the extreme difficulty of drawing the diagonals ex- actly, but also because there is no check upon the errors in that scale ; here the uniform manner in which the strokes of one scale separate from those of the other, is some evidence of the truth of both ; but Mr. Bird's method of assuming a much longer line 1 3 118 GRADUATION, BY SMEATOKT. than what is absolutely necessary for the scale, subdi- viding the whole by a continual bisection, and point- ing the divisions as before explained, and guarding against partial expansions of the metal, is sure to ren- der the divisions perfectly equal. The want of such a scale of equal parts (owing, perhaps, to their ignor- ance of constructing it) is one reason why Mr. Bh-d's method of dividing is not in so great estimation among mathematical instrument makers, as it justly deserves. AN OESERVATION, OR METHOD OF GRADUA- TION, OP Mr. Smeaton's. As it is my Intention to collect in this place what- ever is valuable on this subject, I cannot refrain from inserting the following remark of Mr. Smealons, though it militates strongly against one of Mr. Bird's maxims. He advises us t# compute from the mea- sured radius the chord of l6 degrees only, and to take it from an excellent plain scale, and lay it off five times in succession from the primary point of O given, this would give 80 degrees ; then to bisect each of these arcs, and to lay off one of them beyond the 80th, which would give the 88th degree ; then proceed by bisection, till you come to an arc of two degrees, which laid off" from the 88th degree, will give the QO degrees ; then proceed again by bisection, till you have reduced the degrees into quarters, or every fifteen minutes. Here Mr. Sjiieatoji would stop, be- ing apprehensive that divisions, when over close, can- not be accurately obtained even by bisection. If It were necessary to have subdivisions upon the limb equivalent to five minutes, be advises us to com- pute the chord of 21° '20' only, and to lay it off^four times from the primary point ; the last would give 85° 20', and then to supply the remainder from the bisected divisions as they rise, iiot from other commuted chords. GRADUATION, BY SMEATON. IIQ Mr. Bird asserts, that after he had proceeded by the bisections from thearcof85°'20', the several points of 30. C)0. 75.60. fell in ivilhout stfisihle incquuHly, and so indeed they might, though tliey were not equ- ally true in their places; for whatever error was in them would be communicated to all connected with, or taking their departure from them. Every hetero- geneous mixture should be avoided. It is not the same thing whether you tivicc take a measure as nearly as you can, and lay it off separate- ly, or lay off two openings of the compasses in suc- cession unaltered; for though the same openintr, carefully taken off from the same scale a second time, will doubtles fall into the holes made by the first, without sensible error ; yet as the sloping sides of the conical cavities made by the first points, will conduct the points themselves to the centre, there may be an error, which, though insensible to the sight, would have been avoided by the more simple process of lay- ing off the opening twice, without altering the com- passes. As the whole of the 90 arc may now be divided by bisection, it is equally unexceptionable with the 96 arc ; and, consequently, if an other arc of 90, upon a different radius, was laid down, they would be real checks upon each other. Mr. liamsden, in laying down the original divisions on his dividing engine, divided his circle first into five parts, and each of these into three ; these parts were then bisected four times ; but being apprehen- sive some error might arise from quinquisection, and trisection, in order to examine the accuracy of the divisions, he described another circle -,4 ^'^'^^^ within the former, by continual bisections, but found no sensible difference between the two sets of divisions. It appears also, that Mr. Bird, notwithstanding all his objections to, and declamations against the prac- tice of stepping, sometimes used it himself. \i [ 120] OP THE NONIUS DIVISIONS. It will be necessary to give the young practitioner some account of the nature and use of that admirable contrivance commonly called a nonius, by which the divisions on the limbs of instruments are subdivided. The nonius depends on this simple circumstance, that if any line be divided into equal parts, the length of each part will be greater, the fewer divisions there are in the original ; on the contrary, the length of each division will be less in proportion, as the divi- sions are more numerous. Thus, let us suppose the limb of Hadley's quad- rant divided to every 20 minutes, which are the smal- lest divisions on the quadrant ; the two extreme strokes on the nonius contain seven degrees, or 21 of the aforementioned small divisions, but that it is divided only into 20 parts ; each of these parts will be longer than those on the arc, in the proportion of 21 to 20 ; that is to say, they will be one-twentieth part, or one minute longer than the divisions on the arc; consequently, if the first, or index division of the nonius, be set precisely opposite to any degree, the relative position of the nonius and the arc must be altered one minute before the next division on the nonius will coincide with the next division on the arc, the second division will require a change of two minutes; the third, of three minutes, and so on, till the 20th stroke on the nonius arrive at the next 20 minutes on the arc ; the index division will then have moved exactly 20 minutes from the division whence it set out, and the intermediate divisions of each mi- nute have been regularly pointed out by the divisi- ons of the nonius. To render this still plainer, we must observe that the index, or counting division of the nonius, is dis- tinguished by the mark 0, which is placed on the ex- OF THE NONIUS DIVISIONS. 121 trcmc right hand division ; the numbers running regularly on thus, 20, 15, 10, 5, O. The index division points out the entire degrees and odd 20 minutes, subtended by the objects ob- served ; but the intermediate divisions are shewn by the other strokes of the nonius ; thus, look, among the strokes of the nonius for one that stands direetly opposite to, or perfectly coincident with some one division on the limb ; this division reckoned on the nonius, shews the number of minutes to be added to what is pointed out by the index division. To illustrate this subject, let us suppose two cases. The first, when the index division perfectly coincides with a division on the limb of the quadrant : here there is no difficulty, for at whatsoever division it is, that division indicates the required angle. If the in- dex divisions stand at 40 degrees, 40 degrees is the measure of the required angle. If it coincide with the next division beyond 40 on the right hand, 40 degrees 20 minutes is the angle. If with the second division beyond 40, then 40 degrees 40 minutes is the angle, and so in every other instance. The second case is, when the index line docs not coincide with any division on the limb. We are, in this instance, to look for a division on the nonius that shall stand direetly opposite to.one on the limb, and that division gives us the odd minutes, to be added to those pointed out by the index division : thus, suppose the index division does not coincide with 40 degrees, but that the next division to it is the first co- incident division, then is the required angle 40 de- grees 1 minute. If it had been the second division, the angle would have been 40 degrees 2 minutes, and so on to 20 minutes, when the index division co- incides with the first 20 minutes from 40 degrees. Again, let us suppose the index division to stand be* tween 30 degrees, and 30 degrees 20 minutes, and that the iSth division on the nonius coincides exact- 122 OF INSTRUMENTS FOR ly with a division on the limb, then the angles is 30 degrees l6 minutes. Further, let the index division stand between 35 degrees 20 minutes, and 35 de- grees 40 minutes, and at the same time the 12th di- vision on the nonius stands directly opposite to a di- vision on the arc, then the angle will be 35 degrees 32 minutes. A GENERAL RULE FOR KNOWING THE VALUE OP EACH DIVISION, ON ANY NONIUS WHATSOEVER. 1. Find the value of each of the divisions, or sub- divisions, of the limb to which the nonius is applied. 2. Divide the quantity of minutes or seconds thus found, by the number of divisions on the nonius, and the quotient will give the value of the nonius division. Thus, suppose each subdivision of the Hmb be 30 minutes, and that the nonius has 15 divisions, then i._i gives 2 minutes for the value of the nonius. If the nonius has 10 divisions, it would give three mi- nutes; if the limb be divided to every 12 minutes, and the nonius to 24 parts, then 12 minutes, or 720 seconds divided by 24, gives 30 seconds for the required value. OF INSTRUMENTS FOR DESCRIBING CIRCLES OF EVERY POSSIBLE MAGNITUDE. As there are many cases where arcs are required to be drawn of a radius too large for any ordinary com- passes, Mr. Heywood and myself contrived several instruments for this purpose ; the most perfect of these is dehneated at Jig. 5, plate 11. It is an in- strument that must give great satisfaction to every one who uses it, as it is so extensive in its nature, being capable of describing arcs from an infinite ra- dius, or a straight line, to those of two or three inches diameter. When it was lirst contrived, both Mr. DESCRIBING LARGE CIRCLES. 123 Ili'yivooii i\ud mysclt' were ignorant of what had been clone by that ever to be cflcbrated mechanician, Dr. Ilooi-r. ^ince the invention thereof, I have received some very valuable communications from different gentle- men, who saw and admired the simplicity of its con- struction ; among others, from Mr. Xic/io/sofi, au- thor of several very valuable works; Dr. Rot/ieram, Earl Stivihope, and /. Pricsthy, E?q. of Bradford, Yorkshire; the last gentleman has favoured me with so complete an investigation of the subject, and a description of so many admirable contrivances to an- swer the purpose of the artist, that any thing I could say would be altogether superfluous; I shall, there- fore, be very brief in my description of the instru- ment, represented Ji^. 5, plutc 11, that I may not keep the reader from Mr. Pries f ley's valuable essay, subjoining Dr. Hooke's account of his own contri- vance to that of ours. Much is always to be gained from an attention to this great man ; and I am sure my reader will think his time well employed in peru- sing the short extract I shall here insert. 'J'he branches A and B, fy. 5, plate 1 1, carry two independent equal wheels C, D. The pencil, or point E, is in a line drawn between the centre of the axis of the branches, and equidistant from each ; a weight is to be placed over the pencil when in use. When all the wheels have their axis in one line, and the instrument is moved in rotation, it will describe an infinitely small circle ; in this case the instrument will overset. When the two wheels C, D, have their horizontal axes parallel to each other, a right line or infinitely large circle will be described ; when these axes are inclined to each other, a circle of finite mag- nitude will be described. The distance between one axis and the centre, (or pencil,) being taken as unity, or the common radius, the numbers 1, 2, .'3, 4, &c. being sought for in the 124 OF INSTRUMENTS FOR natural tangents, will give arcs of inclination for set- ting the nonii, and at which circles of the radii of the said numbers, multiplied into the common radius, will be described. The com- mon radius multiplied by 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 (3.0 7.0 «.o 9-0 10.0 20.0 30.0 40.0 1 is the radius of a circle made by the r rollers when in- clined at these an- gles: r 5.43 1 1.19 16.42 21.48 26.34 30.58 35. 38.40 41.59 45. 63.26 71.34 75.58 78.43 80.32 81.52 82.53 83.40 84.17 87. 8 88. 5 88.34 Extracts from Dr. Hooke, on the ^Difficulty, ^c. of Drawing Arcs of Great Circles. *' This thing, says he, is so difficult, that it is almost impossible, espe- cially where exactness is required, as I was suffi- ciently satisfied by the difficulties that occurred in striking a part of the arc of a circle of 60 feet for the radius, for the gage of a tool for grinding telescope glasses of that length ; whereby it was found, that the beam compasses made with all care and cn-cum- spection imaginable, and used with as great care, DESCRIBING LARGE CIRCLES. 123 would not perform the operation ; nor by the way, an angular compass, such as described by Guldo UbaUus, by C/ai'ins, and by Bhigrave, &c. " The Royal Society met ; I discoursed of my in- strument to draw a great circle, and produced an in- strument I had provided for that j)urpose ; and there- with, by the direction of a wire about 100 feet long, I shewed how to draw a circle of that radius, which gave great satisfaction, &c. Again, at the last meet- ing I endeavoured to explain the difficulties there are in making considerable discoveries either in nature or art ; and yet, when they are discovered, they often seem so obvious and plain, that it seems more diffi- cult to give a reason why they were not sooner disco- vered, than how they came to be detected now: how easy it was, we now think, to find out a method of printing letters, and yet, except what may have hap- pened in China, there is no specimen or history of any thing of that kind done in this part of the world. How obvious was the vibration of pendulous bo- dies ? and yet, we do not find that it was made use of to divide the spaces of time, till Galileo discover- ed its isochronous motion, and thought of that pro- per motion for it, Sec. And though it may be dif- ficult enough to find away before it be shewn, every one will be ready enough to say when done, that it is easy to do, and was obvious to be thought of and invented." To illustrate this, the Doctor produced an instru- met somewhat similar to that described. Jig. 5, plate II , as appears from the journal of the Royal Society, where it is said, that Dr. Hooke produced an instru- ment capable of describing very large circles, by the help of two rolling circles, or truckles in the two ends of a rule, made so as to be turned in their sockets to any assigned angle. In another place he had extended his views relative to this instrument, that he had contrived it to draw the arc of a circle to a centre though at a considerable distance, where 126 Priestley's method the centre cannot be approached, as from ihc toj) of a pole set up in the midst of a wood, or trom ilic spindle of a vane at the top of a tower, or from a point on the other side of a river ; in all which cases the centre cannot be conveniently approached, otherwise than by the sight. This he performed by two telescopes, so placed at the truckles, as thereby to see through both of them the given centre, and by thus directing them to the centre, to set the truckles to their true inclination, so as to describe by their motion, any part of such a circle as shall be desired. METHODS OF DESCRIBING ARCS OF CIRCLES OF LARGE MAGNITUDE. By J. PrIESTLEY, EsQ. OF Bradford, Yorkshire. In the projection of the sphere, perspective and architecture, as well as in many other branches of practical mathematics, it is often required to draw arcs of circles, whose radii are too great to admit the use of common, or even beam compasses ; and to draw lines tending to a given point, whose si- tuation is too distant to be brought upon the plan. The following essay is intended to furnish some methods, and describe a ftw instruments that may assist the artist in the performance of both these pro- blems. OP finding points in, and describing arcs OP LARGE CIRCLES. The methods and instruments I shall propose for this purpose, will chiefly depend on the following propositions, which I shall premise as principles. Principle 1. The angles in the same segment of a circle, arc equal one to another. Let A C D B, fg. 1, plaie 10, be the segment of a circles the angles formed bylines drawn from the OP DESCRIBING LARGE CIRCLES. 117 extremities A and B, of the base of the segment, to any points C and D oi its arc, as the angles A C 13, A D B, are equal. '1 his is the 3 1st proposition of Eud'urs third book of the Elements of Geometry. Prmc'iple 1. If upon the ctkIs A H, fig.1, plate 10, of a right line A B as an axis, two circles or rollers C D and E F be firmly lixcd, so that the said line shall priss through the centres, and at right angles to the plains of the circles ; and the whole be sufiered to roll upon a plain without sliding. 1 . If the rollers C I) and E F be equal in diameter, the lines, described upon the plain by their circum- ferences, will be parallel right lines; and the axis A B, and every line D F, drawn between contempo- rary points of contact of the rollers and plain, will be parallel among themselves. 2. If the rollers C D and E F be unequal, then lines formed by their circumferences upon the plain will be concentric circles ; and the axis A B, and alfo the lines 1) F, will, in every situation, ten/1 to the common centre of those circles. Principle 3. If there be two equal circles or rol- lers A and B, fg. 3, plate 10, each separately fixed to its own axis, moveable on pivots ; and these axes placed in a proper frame, so as to be in the same plain, and to maintain the situation given them with respect to each other; and if the apparatus be rolled upon a plain without sliding : 1. If the axes C 1) and EF, be j)laced in a parallel situation, the circumferences of the rollers A and B will trace upon the plain straight lines; which will be at right angles to the axes C I) and E F. 2. If the axes C D and E F, continuing as before in the same plain, be inclined to each other, so as if produced to meet in soi^e point G, the rollers A and B will describe in their motions upon tiie plain arcs of the same, or of concentric circles, whose centre 128 OP THE SIMPLE BEVEL. is a point Hj in that plain perpendicularly under the point of intersection G of the two axes. I shall not stop to demonstrate the truth of the two last principles, it will easily appear on seeing the operations performed. OP THE SIMPLE BEVEL. In the performance of some of the following pro- blems, an instrument not unlike^^, 4, pJate 10, will be found useful. It consists of two rulers, moveable on a common centre, like a carpenter's rule, with a contrivance to keep them fixed at any required angle. The centre C must move on a very fine axis, so as to lie in a line with the fiducial edges C B, C D of the rulers, and project as litde as possible before them. The fiducial edges of the legs represent the sides of any given angle, and their intersection or centre C its angular point. A more complete instrument of this kind, adapted to various uses, will be described hereafter. N. B. A pin fixed in the lower rule, passes through a semicircular groove in the upper, and has a nut A which screws upon it, in order to fix the rulers or legs, when placed at the desired angle. Problem 1. Given the three points 7\, B ajid C, supposed to he in the circumference of a circle too large to be described by a pair of compasses; to find any num- ber of other points in that circumference. This may be performed various ways. As for example, fig. 5, plate 1 0. 1. Join A C, which bisect with the line FM G at right angles; from T), draw BD parallel to A C, cutting F G in E ; and making E Dr=E B, D will be a point in the same circumference, in which are A, B and C. By joining A B, and bisecting it at right angle? with I K ; and from C drawing C a parallel to A B, THE SIMPLE BEVEL. 129 cutting 1 K in L, and making L a=LC, a will be another of the required points. Continuing to dniw from the point, last found, lines alternately parallel to A C and A B ; those lines will be cut at right angles by F G and I K re- spectively ; and bv mailing the parts equal on each side ot' F G and I Iv, they become chords of the cir- cle, in which are the original points A, B, and C, and, of eon>^equencc, determine a series of points on each side of the circumference. It is plain from the construction, (which is too evi- dent to require a formal demonstration,") that the ares AD, A a, Cc, &c. intercepted between the points A and D, A and a, C and c, &c. are equal to the arc B C, and to one another. In like manniT, joining B C, and bisecting it at right angles with P Q; drawing A c' parallel to B C, and making R c'=A R, (c') is another of the re- quired points ; and, from (c') the point last found, drawing c a parallel to C A, and making a N=:N c', (a') is another point in the same circumference ; and the arcs comprehended between C c' and A a' are c(jual to that between A B. Hence, by means of the perpendiculars P Q and F G, any number of ])oints in the circumference of the circle, passing through the given ones, A, B and C may be found, whose distance is equal to A B, in the same manner, as points at the distance of B C were found by the help of the perpendiculars I K and F G. Again, if A, C and (c) or A, C and (c') be taken ns the three given points, multiples of the arc A C may be found in the same manner as those of the arc . A B were found as above described. 2. Another method of performing this problem, is as follows, Jig. 6, plate 10. Produce C B and C A ; and with a convenient radius on C, describe the arc D E ; on which setoff the parts F G, G E, &c. each equal to D F ; draw CG, C E, &c. continued out bevond G and E if necessary ; take the distance A B, K 130 PROBLEMS EY and with one foot of the compasses in A, strike an arc to cut C G produced in H ; and H is a point in the circumference of the circle that passes through the given points ABC; with the same opening A B, and centre H, strike an arc to cut C E produced in I, which will be another of the required points, and the process may be continued as far as is necessary. The reason of this construction is obvious ; for since the angles, B C A, A C H, II C I, &c, are equal, they must intercept equal arcs B A, A H, H I of the circumference. If it were required to find a number of points K, L, &c. on the other side, whose distances were equal to B C, lay down a number of angles C A K, K AL, &c. each equal to B A C, and make the distances C K, K L, &c. each equal to B C. BY THE BEVEL. This problem is much easier solved by the help of the bevel above described, as follows. SeeJ?^. 5. Bring the Centre of the bevel to the middle B, of the three given points A, B and C, and holding it there, open or shut the instrument till the fiducial edges of the legs lie upon the other two points, and fix them there, by means of the screw A, (Jig. A) j this is called setlhig the bevel to the given points. Then removing the centre of the bevel, to any part, between B and A or C, the legs of it being at the same time kept upon A and C, that centre will de- scribe (or be always found in) the arc which passes through the given points, and will, by that means, ascertain as many others as may be required within the limits of A and C, \x\ order to find i)oints without those limits, pro- ceed thus : the bevel being set above described,, bring the centre to C, and mark the distance C B upon the left leg ; remove the centre to B, and mark the distance B A on the same leg j then placing the THE KEVEL. 131 Tcntre on A, bring the right leg lipon B, and the lirst mark will fall upon (a) a point in the circum- ference of the circle, passing through A, B and C, whose distance from A is etjual to the distance B C. Removing the centre of the bevel to the point fa) last found, and bringing the right leg to A, the se- cond mark will find anotlier point (a") in the same circumference, wh(jsc distance a a" is equal A 13. Proccetling in this manner, any number of points may be found, whose distances on the circumference are alternately B C and B A. In the same manner, making similar marks on the r'i^ht leg, points on the other side, as at (c' ) and (e') are found, whose distances C c', c' c\ are equal to B A, B C respectively. It is almost unnecessary to add, that intermediate points between any of the above are given by the bevel, in the same manner as between the original points. Problem 2. Fig. 7, plate 10. Three points, A, B and C, being given, as in the last problem, to jind a fourth point D, situated in the circumference of the cir- cle passing through A, B andC, and at a given yiumber of degrees distance from any of these points \ h.for instance. Make the angles A B D, and A C D, each equal to one half of the angle, which contains the given number of degrees, and the intersection of the lines B I), CD gives the pomt D required. For, an angle at the circumference being equal to half that at the centre, the arc A D will contain twice the number of decrrecs contained by either of the angles A B 1) or AC D. Problem 3. Fig. S^ plate 10. Given three points. tf V ni the former problems, to dra'v a line from afiy of them, tending to the centre of the circle, ivhich passes through them all. Let A, B and C be the given points, and let it be required to draw A D, so as, if continued, it would K 'I 132 PROBLEMS BY pass through the centre of the circle containing A, B and C. Make the angle BAD equal to the complement of the angle B C A, and A D is the line required. For, supposing A E a tangent to the point A, then is E A D a right angle, and E A B=:B C A ; whence, B AD=:right-angle, less the Z. B C A, or the complement of B C A. Corollary 1. AD being drawn, lines from B and C. or any other points in the same circle, are easily found; thus, make ABG=BAD, which gives BG; then make BCF=::CBG, which gives CF; or C F may be had without the intervention of B G, by making A C F=:C A D. QoroUary 1. A tangent to the circle, at any of the points (A for instance), is thus found. Make B A E=B C A, and the line A E will touch the circle at A. By the hevel. Set the bevel to the three given points A, B and C, (fg. 8,J lay the centre on A, and the right leg to the point C ; and the other leg will give the tangent A G'. Draw AD perpendi- cular to A G' for the line required. For B A E being = B C A, the /. E A C is the supplement to /_ A B C, or that to which the bevel is set ; hence, when one leg is applied to C, and the centre brought to A, the direction of the other leg must be in that of the tangent GP2. Problem 4. Fig. 9, plate 10. T hree points be- ing given, as in ihs jormer problems ^ to draiv from a ^iven fourth point a line tetidi?ig to the centre of a circle passing through the first three points. Let the three points, through which the circle is supposed to pass, be A, B and C, and the given fourth point D; it is required to draw through D a line D d tending to the centre of the said circle. I'^roiM j\ and B, the two points nearest D, draw by the last problem, the lines A a, B b, tending to the said centre ; join A B.. and from any point E, taking THE BEVEL. 133 in B b, (the farther from B the hotter) draw E F pa- rallel to A a, cutting AB in F; join A D and R D, and draw FG parallel to A D, catting DB in Cr; join G E, and through D parallel thereto, draw D d for the line required. For, (continuing- l)d and B b till tl,ey meet in O,) since Aa and B b, it produced, would meet in tlie centre, and FE is parallel to Aa , wc have B F : B A :: BE : radius; also, sin(;e A D inid F G are pmallcl, li F : BA :: BG : BD; therefore BG : BD :: BE : radius, but from the parallel lines Dd andGE, we have BG : BD :: BE : BO; hence B O is the radius of the circle passing through A, B and C. By the bevel. On D with radius D A describe an arc AK; set the bevel to the three given points A, B and C, and bring its centre (always keeping the legs on A and C ) to fall on the arc AK, as at U; on A and H severally, with any convenient radius, strike two arcs crossing each other at I ; and the required line D d will pass through the points I and D. For a line drawn from A to II will be a common chord to the circles A UK and ABC ; and the line I D bisecting it at right angles, must pass through both their centres. Problem 5. Fig. g, plate 10. Three pfAiits he- ing given, as hefore^ together vcith ii fourth point, tojitul two other points, sncli, that a circle passing throvgli them and the fourth point, shall be concoilnc to that passing through three gi-ven points. Let A, Band C be the three given points, and D the fourth j)oint; it is required to find two other points, as N and P, such, that a circle passing through N, D and P shall have the same centre with that y)assing through A, B and C. The geometrical construction being y)erformed as directed by the last problem, continue E G to L, making E L =: E B ; and through B and L draw KS 3 34 PROBLEMS BY B LM, cutting D d produced in M; make A N and B P severaUy equal to M D, and N and P are ihe points required. For, since L E is parallel to M O, we have BE : LE :: B O : M O; but BE = LE by the con- struction ; therefore, M () = B O = radius of the circle passing through A, B and C, and M is in the circumference of that circle. Also, N, D and P be- ing points of the radii, equally distant from A, Aland B respectively, they will be in the circumference of a circle concentric to that passing through A, M and B, or A, Band C. By the level. Draw A a and C c tending to the centre, by problems ; set the bevel to the three given points a, B and C; bring the centre of the bevel to D, and move it upon that ])oint till its legs cut oft* equal parts AN, CQ of the lines A a and C c ; and N and Q. will be the points required. For, supposing lines drawn from A to C, and from N to Q, the segments ABC and NDQ will be simi- lar ones; and consequently, the angles contained in them will be equal. Problem 6. Fig. 10, 'plate 10. Three points ^ A, B and C, lyhig in the circumference of a circle, he- ing given as before : and a fourth pointT)^ to fnd another point F, such, that a circle passing through F and D shall touch ihe other passing through A, B and C, at any of these points; as for instance^ B. Draw 1) E a tangent to the arc A E C, by pro- blem 3, corollary 2 ; and join \\ D; draw B F, ma- king the angle D B F = K J)D, with the distance BD; on D strike an arc to cut Bp in F; and F is the point sought. Since D F =^- D B, the Z D F B = D B F; but DBF = DBE by construction ; therefore, DF B r= D P E, and E B is a tangent to the arc B D F at B ; but E B is also a tangent to the arc AFC (by con- struction, at the same point ; hence, the arc B D F touches AFC as required. THE BKVF.r. 135 Problem 7. Fig, \2, flute \0. Tijo linex tend- inis /'> o distant point being given, and also a point tn one of them \ to Jifui tivo other points, (one of ijoliich must be in the other given line J such, that a circle pass- in(T through those three points, mav have its centre at f he point cf intersection of the given Imes. LcL the given lines be A B and C I), and E the given point in one ot" them; it is required to find two otlier points, as I and H, one of which (I) shall be in the other line, such, that a circle HIE passing through the three points, shall have its centre at O, where the giv6n lines, if produced, would meet. From E, the given point, Draw E H, crossing A B at right angles in F ; make F H := F E, and 1 1 is one of the required points. From any point D in CD, the farther from E the better, draw G D paral- lel to A B, and make the angle HEI equal to half the angle GDE; and EI will cut AB in I, the other required point. For, since E H crosses A B at right angles, and H F is equal to FE, I II will be equal to I E; and the ZH EI=:^EHI; also, since GD is parallel toAB, theZGDE = Z FOE = doubleZIHE =:double ZHEI; but H EI = half Z. G 1;E by construction ; hence, the points E, I and 11 are in the circle whose centre is O. By the bevel. Draw E H at right angles to A B, and make FII r= FE as before; set the bevel to the angle G D O, and keeping its legs on the points H and K, bring its centre to the line A B, which will give the point I. Problem 8. Fig. \3, plate \0. 7'zvo lines fen.h ing to a distant point bei)ig given, to find the distance of that point. Let A B and CD be the two given lines, tending to a distant point O ; and let it be required to find the distance of that point, from any point (E for in- stance) in cither of the given lines. K 1 136 AN IMPROVED BEVEL, From E draw EF perpendicular to AB ; and from D (a point taken any where in C D, the far- ther from E the better) draw DG parallel to A B. On a scale of inches and parts measure the lengths of GE, El) and EF separately; then say, as the length of G E is to ED, so is E F to E O, the dis- tance sought. For the triangles EGl) and EFO are similar, and from thence the rule is manifest. OF INSTRUMENTS FOR DRAWING ARCS OP LARGE CIRCLES, AND LINES TENDING TO A DISTANT POINT. I shall now proceed to give some idea of a few in- struments for these purposes, whose rcu'ionah de- pends on the principles laid down in the beginning of this essay. 1. AN IMPROVED BEVEL, Fig. 12, 'plate 10, is a sketch of an instrument grounded upon principle 1, p. 126, by which the arcs of circles of any radius, without the limits at- tainable by a common pair of compasses, may be de- scribed. It consists of a ruler A B, composed of two pieces rivettcd together near C, the centre, or axis, and of a triangular part C F E D.- '\\\q. axis is a hollow socket, fixed to the triangular pnrt, about which another socket, fixed to the arm C B of the ruler A B, turns. These sockets are open in the front, for part of their length upwards, as represented in thesection at I, in order that the point of a tracer or pen, fitted to slide in the sockeC, may be more easily seen. The triangular part is furnished with a gra- duated arc D E, by which, and the vernier at i , AND ITS USE. 137 the angle D C B may be determined to a minute. A groove is made in this arc, by which, and by the nut and screw at B, or some similar contri- vance, the ruler A B may be fixed in any required position. A scale of radii is put on the arm C B, by which the instrument may be set to describe arcs of given circles, not less than '20 inches in diameter. In order to set the instrument to any given radius, the number expressing it in inches on C B is brought to cut a fine line drawn on C D, parallel, and near to the liducial edge of it, and the arms fastened in that po- sition by the screw at B. Two heavy pieces of lead or brass, G, G, made in form of the sector of a circle, the angular parts being of steel and wrought to a true upright edge, as shewn at H, are used with this instrument, whose arms are made to bear against those edges when the arcs are drawn. The under sides of these sectors are furnished with fine short points, to pre- vent them from sliding. The fiducial edges of the arms C A and C D are each divided from the centre C into 200 equal parts. The instrument might be furnished with small castors, like the pcntagraph ; but little buttons fixed on its underside, near A, V. and D, will enable it to slide with sufficient ease. SOME INSTANCES OP ITS USE. 1. Ta describe an arc, which shall pass through three given points. Place the sectors G, G, with their angular edges over the two extreme jjoints; apply the arms of the 'bevel to them, and bring at the same time its centre C (that is, the [)oint of the tracer, or pen, put into the socket) to the third poiutj and there fix the arm CB; 138 USE OF THE BEVEL. then, bringing the tracer to the left hand sector, shde the bevel, keeping the arms constantly bearing against the two sectors, till it comes to the right hand sector, by which the required arc will be described by the motion of its centre C. If the arc be wanted in some part of the drawing wilhouiihe given points, find, by problem 1, p. 128, other points in those parts where the arc is required. By this means a given arc may be lengthened as far as is requisite. 2. To describe an arc of a given rad'ms, not Jess than 1 inches. Fix the arm C B so that the part of its edge, cor- responding to the given radius, always reckoned in inches, may lie ov^er the fine line drawn on C D for that purpose : bring the centre to the point through which the arc is required to pass, and dis- pose the bevel in the direction it is intended to be drawn; place the sectors G, G, exactly to the divi- sions 100 on each arm, and strike the arc as above described. ?}. The hevel being set to strike arcs of a given ra- dius, as directed in the last paragraph, to draw other arcs ivhose radii shall have a given proportion to that of the first arc. Suppose the bevel to be set for describing arcs of 50 inches radius, and it be required to draw arcs of 6o inches radius, with the bevel so set. Say, as 50, the radius to which the bevel is set, is to 6o, the radius of the arcs required ; so is the constant number 100 to 120, tJie number on the arms C A and C 1), to which the sectors must be placed, in order to describe arcs of 6o inches ra- dius. N. B. When it is said that the bevel is set to draw arcs of a particular radius, it is always un- derstood that the sectors G, G, are to be placed at No. aoo on C A and C D, when those arcs art drawn. USE OF THE BEVEL. 139 4. ^n arc ACB (Jig. 11, plate \0) being given, to Jr.iiv other arcs concentric thereto^ ivhich shall J>ass throngh given points, as P' for instance. Through the cstrcmities A and B of the given arc draw lilies A P, B P tending to its centre, by pro- blem 3, p. 131. Take the nearest distance of the givL-n point P from the arc, and set it from A to P, and from B to P. Hold the centre of the bevel on C, (any point near the middle of the given arc) and bring its arms to pass through A and B at the same time, and there fix them. Place the sectors to the I points P and P, and with the bevel, set as before directed, draw an arc, which will pass through P', the given point, and be concentric to the given arc A C B. n 5. Through a point A, (fig. 14, flate lo) ;// the given line A B, to strike an arc of a given radius, aiui whose centre shall he in that line, produced if ne- cessary. Set the bevel to the given radius, as above de- scribed, (Method '1.) Through A, at right angles to A B, draw C D ; lay the centre of the bevel, set\xs above, on A, and the arm C A, on the line A C, and draw a line A E along the edge CD of the other arm. Divide the angle DAE into two equal parts by the line A F, place the bevel so, that its centre being at A, the arm CD shall lie on AF; while in this situation, place the sectors at No. 100 on each arm, [ind then, strike the arc. d. An arc being given, to find the length of its radius. Place the centre of the bevel on the middle of the arc, and open or shut the arms, till No. 100 on C A and C D fall upon the arc on cnch side the centre ; the radius will be found on CB (in inches) at that point of it, where it is cut by the line drawn on C D. It the extent of the arc be net eounl to that be- 140 THE OBLIQUE RULER. tween the two Numb. 100, make use of the Numb. 50, in which case the radius found on C B will be double of that sought; or the arc may be lengthened, by problem 1 , till it be of an extent sufficient to ad- mit the two Numbers 100. Many more instances of the use of this instru- ment might be given ; but from what has been al- ready done, and an attentive perusal of the foregoing problems, the principle of them may be easily con- ceived. 2. THE OBLIGUE RULER. An instrument for drawing lines that are paral- lel, is called a ^parallel ruler ; one for drawing lines tending to a point, as such lines are oblique to each other, may, by analogy, be called an oblique ruler. Fig. I'Jjplah 10, represents a simple contrivance for this purpose ; it consists of a cylindrical or pris- matical tube A B, to one end of which is fixed the roller A; into this tube there slides another CB of six or eight flat sides. The tubes slide stiffly, so as to remain in the position in which they are placed. Upon the end C, screw different rollers, all of them something smaller than A. In order to describe arcs, a drawing pen E, and a tracer may be put on the pin D, and arc retained there by a screw G ; the pen is furnished with a moveable arm E F, having a small ball of brass F at the end, whose use is to cause the pen to press with due force upon the paper, the degree of which can be regulated by placing the arm in diflcrent po- sitions. '^Ilie ruler A B being set to any given line, by rolling it along other lines may be drawn, all of which will tend to some one point in the given line, or a continuation of it, whose distance will be greater, as the distance between the jolkrs A ANOTHER OBLIQUB RULER. Ml and C is increased ; and as the diameter of C aj)*- proaches that of A ; all which is evident from prin- ciple 2, page 127. ft also apj)ears from the said principle, that du- ring the motion of the ruler, any point in its axis will accurately describe the arc of a circle, having the said point of intersection for its centre ; and, consequently, the pen or tracer, put on the pin D, will describe such arcs. The rollers, as C, which screw upon the end of the inner tube are numbered, 1, 2, 3, &e. and as many scales are drawn on that tube as there are rollers, one belonging to each, and numbered accor- dingly. These scales shew the distance in inches, of the centre or point of intersection, reckoned from the middle of the pin D, (agreeing to the point of the pen or tracer;) thus; No. 1, will describe circles, or serve for drawing lines tending to a point, whose radius or distance from D, is from 1200 inches to OOO inches, according as the tube is drawn out. No. 2 - from (300 inches to 300 inches 3 - 300 - 150 4 - 150 - 80 5 - 80-40 6 - 40-20 7 - 20 - 10 If it should be required to extend the radius or distance farther than 1200 inches, by using ano- ther ruler, it might be carried to 2400 inches ; but lines in any common sized drawing, which tend to a point above 100 feet distance, may be esteemed as parallel. 3. ANOTHER RULER OF THE SAME KIND. i%. IS, pLite \0. This is nothing more than the 142 THE CYCLOGRAPH. last instrument applied to a flat ruler, in the manner the rolling parallel rulers are made. CD is an hexagonal axis, moveable on pivot? in the heads A and F fixed upon a flat ruler : on this axis the smaller roller B, is made to slide through one half of its length ; the larger roller A, is screwed on the other end of the axis, and can be changed occasionally for others of different dia- meters. Scales adapted to each of the rollers at A, are either put on the flat sides of the axis from C to E, or drawn on the corresponding part of the flat ruler ; and the scales and rulers distinguished by the same number : at F is a screw to raise or lower the end C of the axis, till the ruler goes parallel to the paper on which the drawing is made; and at G there is a socket, to which a drawing pen and tracer is adapted for describing arcs. In using these instruments, the fingers should be placed about the middle part between the rollers ; and the ruler drawn, or pushed at right angles to its length. The tube A B, j'fg. 14, and one, or both of the edges of the flat ruler, fig. 18, are divided into inches and tenths. 4. THE CYCLOGTvAPH. This instrument is constructed upon the third principle mentioned in page 135 of this Essay. Fig, X'O, flate\0^ is composed of five rulers ; four of them D E, D F, G E and G F, forming a trape- zium, are moveable on the joints D, E, F and G ; the fifth ruler D I, passes under the joint D, and through a socket carrying the opposite joint G- The distances from the centre of the joint D, to that of the joints E and F, are exactly equal, as are the distances from G to the same joints. The rulers D E and D F pass beyond the joints E and F^, where a roller is fixed to each ; the rollers are fixed upon A DIFFERENT COXSTRUCTIOK, kc. I43 their axes, which move freely, but steadily on pivots, so as to admit of no shake by which the inclination of flic axes can be varied. The ruler I 1) passing bcvond the joint 1^, carries a third roller A, like the others, whose axis lies precisely in the direction of that ndcr ; the axes of B and C extend to K and L. A scale is put on the rnler D I, from II to G, shewing, by the position of the socket G thereon, the ]en2;th of the radius of the arc in inches, that would be "described by the end I, in that position of the irapezinm. When the socket G is brought to the end of the scale near I, the axes of the two rollers B and C, the ruler 1) I, and the axis of the roller A, are precisely parallel; and in this position, the end I, or any other point in 1) I, will decribe straight lines at right angles to D I ; but on sliding the socket G towards H, an inclination is given to the axes of B and C, so as to tend to some point in the line I D, continued beyond D, whose distance from I is shewn by the scale. A proper socket, for holding a pen or tracer, is made to put on the end I, for the purpose of de- scribing ares ; and another is made for fixing 011 any part of the ruler D I, for the more convenient description of concentric arcs, where a number are wanted. It is plain from this description, that the middle ruler D I in this instrument, is a true ohiique ruler, bv which lines may be drawn tending to a point, whose distance from I is shewn by the position of the socket G on the scale ; and the instrument is made sufBcieiuly large, so as to answer this purpose as well as the other. 5. A DIFFERE^'T CONSTRUCTION OF THE SAME INSTRUMENT. In Fi^. M), plate 10, the part intended to be used in drawing lines, lines within the trapezium 144 METHODS OP DESCRIBIXG which is made large on that account ; but this 15 not necessary; and j?^. 15, plate 10, will give an idea of a like instrument, where the trapezium may be made much smaller, and consequently less cum- bersome. D B E C represents such a trapezium, rollers, socket, and scale as above described, but much smaller. Here the ruler E D is continued a sufficent length beyond D, as to A, where the third roller is fixed ; a pen or tracer may be fitted to the end E, or made to slide between D and A, for the purpose of drawing arcs. METHODS OP DESCRIBING AN ELLIPSE, AND SOME OTHER CURVES. To describe an ellipse, the transverse and conjugate axes being given. Let A B be the given transverse, and C D the conjugate axis, ^g. 13, plate 13. Method 1. By the line of sines on the sector, open the sector with the extent A G of the semi-transverse axis in the terms of 90 and 90 ; take out the trans- verse distance of 70 and 70, 60 and 60, and so for every tenth sine, and set them off from G to A, and from G to B ; then draw lines through these points perpendicular to A B. Make G C a transverse distance between 90 and 90, and set off each tenth sine from G towards C, and from G towards D, and through these points draw lines parallel to AB, which will intersect the perpendiculars to A B in the points A, a, b, c, d, e, f, g, h, i, k, 1, m, n, o, p, q, B, for half the ellipse, through which points and the intersections of the other half, a curve being draw with a steady hand, will complete the ellipse. Method 1. With the elliptical compasses, fg. 3, plate 11, apply the transverse axis of the elliptical compasses to the line A B, and discharge the screws AN ELLIPSE. 1-15 of both the sliders ; set the beam over the transverse «'ix.is A B, atid sUdc it baokwards and forwards untd the peneil or ink point coincide with the point A, and lii^hten the serew of that slider wliich moves on the conjut^alc axis ; now turn the beam, so as to lay over the conjugate axis C D, and make the pencil or ink point coincide with the point C, and then fix the screw, which is over the slider of the transverse axis of the compasses; the compasses being thus adjusted, move the ink point gently from A, through C to B, and it will describe the semi-clHj)se ACB; reverse the elliptical compasses, and describe the other semi- ellipse B D A. These compasses were contrived by my Father in 1/48 ; they are superior to the trammel which describes the whole ellipse, as these will de- scribe an ellipse of any excentricity, which the others will not. Through any given point F to describe an ellipse , the transverse axis A B being given. Apply the transverse axis of the elliptical com- passes to the given line A B, and adjust it to the point A ; fix the conjugate screw, and turn the beam to F, sliding it till it coincide therewith, and proceed as in the preceding problem. Fig. 2, plate 11, represents another kind of ellip- tical apparatus, acting upon the principle of the oval lathes ; the paper is fixed upon the board A B, the pencil C is set to the transverse diameter by sliding it on the bar 1) E, and is adjusted to the conjugate diameter by the serew G; by turning the board A B, an ellipse will be described by the pencil. Fig. 2, A, plate II, is the trammel, in which the pins on the under side of the board A B, move for tht; de- scription of the ellipse. Ellipses are described in a very pleasing manner by the geometf ic pen, fig. 1, plate 11; this part oi that instrument is frequently made separate. To describe a parabola, •vahose parameter shall b» fqual to a given Inie. Fi^. 17, plate 13. 146 A^ HYPERBOLA. Draw a line to represent the axis, in which make A B equal to half the given parameter. Open the sector^j so that A B may be the transverse distance between QO and 90 on the line of sines, and set off every tenth sine from A towards B ; and through the points thus found, draw lines at right angles to the axis A B. Make the lines A a, 10 b, 20 c, 30 d, 40 e, &c. respectively equal to the chords of 00°, 80°, 70°, 6o°, 50°, &c. to the radius A B, and the points abcde, Sec. will be in the parabolic curve : ibr greater exactness, interme- diate points may be obtained from the intermediate degrees ; and a curve drawn through these points and the Vertex B, will be the parabola required : if the whole curve be wanted, the same operation must be pcrfwrmcd on the other side of the axis. As the chords on the sector run no further than 60°, those of 70, 80, and gO, may be found by taking the transverse distance of the sines of 35°, 40°, 45°, to the radius A B, and applying those distances twice along the lines, 20 c, 10 b, &c. 7^',^, 4, flale 1 1, is an instrument for describing a parabola; the figure will render its use sufficiently evident to every geometrician. A BC D is a wooden frame, whose sides A C, B D are parallel to each other ; E F G H is a square frame of brass or wood, sliding against the sides AC, BD of the exterior frame ; H a socket sliding on the bar E F of the interior frame, and carrying the pencil I; K a fixed point in the board, (the situation of which may be varied occasionally); EaK is a thread equal in length to E F, one end thereof is fixed at E, the other to the piece K, going over the pencil at a. Bring the frame, so that the pencil may be jn a line with the point K ; then slide it in the ex- terior frame, and the pencil will describe one part of a parabola. If the trame E F G H be turned about, so that E F may be on the other side of the- An hyperbola. 147 point K, the remaining part of the parabola may be completed. To c/e.uri/'t' an h\pcrhohi, the "ccrtex A, and asymp- totes B H, BI being green, tig. 18, pi. 13. Draw A I, AC, parallel to the asymptotes. Make A C a transverse distaneeto 45, and 43, on the up- per tangents of the sector, and apply from Bas many of these tangents taken transversely as may be thought convenient; as B D SO"", I) E 55°, and so on; and through these points dr^uv D d, E c, Sec. parallel to A C. Make A C a transverse distance between 45 and 45 of the lower tangents, and take the transverse distance of the cotangents before used, and lay them on those parallel lines ; thus making 1) d equal 40°, E C to 35°, EF to 80°, &c. and these points will be in the hyperbolic curve, and a line drawn through them will be the hyperbola required. To assist the hand in drawhiF curves thronoh a num.- her of points^ artists make use of what is termed the boiVy consisting of a spring of hard wood, or steel, so adapted to a lirm straight rule, that it may be bent more or less by three screws passing through the straight rule. A set of spirals cut out in brass, are extremely con- venient for the same purpose; for there are few curve lines of a short extent, to which some part of these will not apply. Fig. 6, plate 11, represents an instrument for drawing spirals ; A the foot by which it is affixed to the paper, B the pencil, a, h, c, d, a running line going over the cone G and cylinder H, the ends being fastened to the pin e. (3n turning the frame K N O, the thread carries the pencil progres- sively from the cone to the cyluider,and thusdescribes a spiral. The size of the spiral may be varied, by placing the thread in ditferent grooves, by putting it on the furthermost cone, or by putting on a larger cone. L2 f 148 ] OP THE GEOMETRIC PEN. The geometric pen Is an instrument in which, by a circular motion, a right line, a circle, an ellipse, and a great variety of geometrical figures, may be described. . This curious instrument was invented and described by John Baptist Suardi, in a work entitled Nuovo Istromenti per la Descrlzz'ione d'l diverse Curve An- tichi e Modeine, &c. Though several writers have taken notice of the curves arising from the compound motion of two circles, one moving round the other, yet no one seems to have realized the principle, and reduced it to practice, before /. B. Suardi. It has lately been happily introduced into the steam engine by Messieurs Watt and Bolton ; a proof, among many others, not only of the use of these speculations, but of the advantages to be derived from the higher parts of the mathematics, in the hands of an in- genious mechanic. There never was, perhaps, any instrument which delineates so many curves as the geometric pen ; the author enumerates H173, as possible to be described by it in the simple form, and with the few wheels appropriated to it for the present work. Fig. ], plate 11, represents the geometric pew, A, B, C, the stand by which it is supported ; the legs A, B, C, are contrived to fold one within the other, for the convenience of packing. A strong axis D is fitted to the top of the frame ; to the lower part of the axis any of the wheels (as i) may be adapted ; when screwed to it they are immoveable. EG is an arm contrived to turn round upon the main axis D ; two sliding boxes arc fitted to this arm ; to these boxes any of the wheels belonging to the geometric pen may be fixed, and dien moved so that GEOMETRIC PEN. liQ the wheels may take inlo each other, and ihc im- moveable wheel i ; ii is evident, that by making the arm EG revolve round the axis D, these wheels will be made to revolve also, and that the number of their revolutions will depend on the proportion between the teeth, {g is an arm carrying the pencil ; this arm slides backwards and forwards in the box c d, in order that the distance of the pencil from the centre of the wheel h may be easily varied ; the box c d is fitted to the axis of the wheel h, and turns round with it, carrying the arm (g along with it; it is evident, therefore that the revolutions will be fewer or greater, in proportion to the difference between the numbers of the teeth in the wheels h and i ; this bar and socket are easily removed for changing the wheels. When two wheels only are used, the bar { g moves in the same direction with the barEG; but if another wheel is introduced between them, they move in contrary directions. The number of teeth in the wheels, and conse- quently, the relative velocity of the epicycle, or arm {g, may be varied ad infinitum. The numbers we have used are, 8, l6, 21, 32, 40, 48, 56, 64, 72, 80, 88, 96. The construction and application of this instru- ment is so evident from the figure, that nothing more need be pointed out than the combinations bv which the figures here delineated may be produced. To render the description as concise as possible, I shall in future describe the arm E G by the letter A, and fg by the letter B. To describe Jij^ \, plate 12. The radius of A must be to that of B, as 10 to 5 nearly, tlieir velocities, or the numbers of teeth in the wheels, 10 be equal, the motion to be in the same direction. If the length of B be varied, the looped figure, delineated '^i fig. 12, will be produced. L3 150 GEOMETRIC PEN. A circle may be described by equal wheels, and any radius, but the bars, must move in contrary direction. To describe the two level figures, sec^g. 1 1, plate 12. Let the radius of A to B be as 10 to 3^-, the ve- locities as 1 to 2, the motion in the same direction. To if escribe by this cikcular motion^ a STRAIGHT LINE AND AN ELLIPSE. For a Straight line, equal radii, the velocity is 1 to 2, the motion in a contrary direction ; the same data will give a variety of ellipses, only the radii must be unequal ; the el- lipses may be described in any direction ; s,ecjig. 10, p/ate 13. Fig. 13, plate 1'2, with seven leaves, is to be for-, med when the radii are as 7 to 2, velocity as 2 to 3, motion in contrary directions. The six triangular figures, seen atjig. 2,4, 6,8, 9, 10, are all produced by the same wheels, by only varying the length of the arm B, the velocity should be as 1 to 3, the arms are to move in contrary di- rections. Fig. 3, plate 12, with eight leaves, is formed by equal radii, velocities as 5 to 8, A and B to move the same vi^ay ; if an intermediate wheel is added, and thus a motion produced in a contrary direction, the pencil will delineate Jig. l6, plaU 12. The ten-leaved figure, j?o-. 15, plate 12, is pro- duced by equal radii, velocity as 3 to 10, directions of the motions contrary to each other. Hitherto the velocity of the epicycle has been the greatest ; in the three following figures the curves are produced when the velocity of the epicycle is lesS'than that of the primum mobile. Yovjig. 7, the radius of A to B to be as 2 to 1, the velocity as 3 to 2 ; to be moved the siime way. For ^^. 14, the radius of A, somewhat less than the diameter given to B, the velocity as 3 to 1 j to be moved in a contrary direction. For_^^. 5., equal radii, velocity as 3 to I ; moved DIVISION OP LAND. 151 the same way. Tliese instances are sufficient to shew how much mav be performed by tliis instru- ment ; with a few additional pieces, it may be made to describe a cycloid, with a circular b.'jse, spirals, and particularly the spiral of ArchiiDcde-i, &c. OF THE DIVISION' OF LAND. To know how to divide land into any number of equal, or unequal parts, according to any assigned proportion, antl to make proper allowances for the difFercnt qualities of the land to be divided, form a material and useful branch of surveying. In dividing of land, numerous cases arise ; in some it is to be divided by lines parallel to each other, and to a given fence, or road ; sometimes, they are to intersect a given line ; the division is often to be made according to the particular directions of the parties concerned. In a subject which has been treated on so often, novelty is, perhaps, not to be de- sired, and scarcely expected. No considerable im- provement has been made in this branch of surveying since the time of SpeuJell. Mr. Talbot, whom we shall chiefly follow, has arranged the subject better than those who preceded him, and added thereto two or three problems ; his work is well worth the surveyor's perusal. Some problems also in the fore- going part of this work should be considered in this place. Pr o b l e m 1 . To divide a triafi^le in a oiven ratio l?y right lines drawn from atiy angle to the opposite side thereof. 1. Divide the opposite side in the proposed ratio. 2. Draw lines from the several points of division to the given angle, and then divide the triangle as re- (juired. ThuSj to divide the triangle A B C, Jig. 13, plate La 152 DIVISION OF LAXD. 8, containing 267 acres, into three parts, in propor- tion to the numbers 40, 20, 10, the lines of division to proceed from the angle C to A B, whose length is 28 chains ; now, as the ratio of 40, 20, 10, is the same as 4, 2, 1, whose sum is 7, divide A B into seven equal parts ; draw C a at four of these parts, C b at six of them, and the triangle is divided as re- quired. Ar'iihmet'ically. As 7, the sum of the ratios, is to A B 28 chains ; so is 4, 2, 1 , to 1 6, 8, and 4 chains respectively ; therefore Aa = 16, A b = 8, and b B = 4 chains. To know how many acres in each part, say, as the sum of the ratios is to the whole quantity of land, so is each ratio to the quantity of acres ; {7 : 26,5 :: 4 : 15,142857 -^ = triangle A C a 7 : 2d,5 :: 2 r 7,5/1428 > = triangle a C b 7 : 26,5 :: 1 : 3,785714 J = triangle b C B.* Problem 2. To divide a triangular field into any number of parts, and in any given proportion, from a given 'point in one of the sides. 1. Divide the triangle into the given proportion, from the angle opposite the given point. 2. Reduce this triangle by problem 51, so as to pass through tlie given point. Thus, to divide the field A B C, fig. 14, plate 8, of seven acres, into two parts, in the proportion of 2 to 5, for two different tenants, from a pond b, in B C, but so that both may have the beneut of the pond. 1. Divide BC into seven equal parts, make B a = 5, then C ar=:2 draw A a ; and the field is divided in the given ratio. 2. To reduce this to the point b, draw A b and a c parallel thereto, join c b, and it will be the required dividing line. Operation in the field. Divide B C in the ratio re- quired, and set up a mark at the point a, and also at * Talbot's Complete Art gf Land M(;asurlng. DIVISION' OP LAND. 153 the pond ^; at A, witli the theodolite, or other in- strument, measure the angloh Aa; at <; lay off the same angle A a c, which will give the point c in the side A c, trom whence the fence must go to the pond. '2. To divide A B C, /j;. 15, plated, into three equal parts from the pond c. 1 . Divide A B into three equal parts, A a, a b, b B, and C a C b will divide it, as required, from the angle A ; reduce these as above directed to c d and c e, and c e, and they will be the true dividing lines. Problem 3. To di-vule a triangular Ji eld in any re- quired ratio, hy lines drawn parallel to one side, and cutting the others. Let ABC, fg. ]6,p/ate 8, be the given triangle to be divided into three equal parts, by lines parallel to A B, and cutting AC, BC. Rule 1 . Divide one of the sides that is to be cut by the parallel lines, into the given ratio. 2. Find a mean proportional between this side, and the first division next the parallel side. 3. Draw a line pa- rallel to the given side through the mean propor- tional. 4. Proceed in the same manner with the re- maining triangle. Example 1 . Divide B C into three equal parts B D, DP, PC. 2. Find a mean proportional between B C and D C. 3. Make C G equal to this mean pro- portional, and draw GH parallel to A B. Proceed in the same manner with the remaining triangle C H G, dividmu;- G C into two equal parts at I, find- ing a mean proportional between C Gand C I ; and then making C L equal to this mean proportional, and drawing LM parallel to A B, the triangle will be divided as required. A square^ or rectangle, a rhombus, or rhomhoideSy may he divided into any given ratio, by lines cutting two opposite parallel sides, by dividing the sides into the proposed ratio, and joining the points of division. Problem 4. To divide a right-lined figure into any proposed ratio, by lines proceeding from one angle. 154 DIVISION OP LAND. Problem 5. To divide a right-lined figure into any proposed ratio, hy lines proceeding from a given point in one of the sides. Problem 6. To divide a right-lined figure itito any proposed ratio y hy right lines proceeding Jrom a given poiyit within the said figure or field. It would be needless to enter into a detail of the mode of performing the three foregoing problems, as the subject has been already sufficiently treated of in pages 84, 85, &c. Problem 7«* " J^ i^ required to divide any given quantity of ground into any given number of parts, and in proportion as any given numbers.'''' Rule. " Divide the given piece after the rule of Fellowship, by dividing the whole content by the sum of the numbers expressing the proportions of the several shares, and multiplying the quotient severally by the said proportional numbers for the respective shares required." Example. It is required to divide 300 acres of land among A, B, C and D, whose claims upon it are re- spectively in proportion as the numbers 1, 3, 6, 10, or whose estates may be supposed lOOl. 300L 6ool, and lOOOl. per annum. The sum of these proportional numbers is 20, by which dividing 300 acres, the quotient is 15 acres, which being multiplied by each of the numbers 1, 3-, 6, 10, we obtain for the several shares as follows : a. r. p. A's share = 15 : : 00 B's share = 45 : O : 00 C's share = QO : O : 00 D's sh'dre = 150 : : 00 Sum = 300 : O : 00 the proof. "* Probiein *7, an erroneous rule given by a late writer, introduced nere to prevent the practitioner being led into error. — From Xalbot's C<)mplete Art of Land Measuring, Problem 1, page 205 itnd appendix^ page 4^0. DIVISIO>J OP LAND. 155 F)iit this is upon supposition that the land is all of an ecjual value. \ow let us suppose the land, to beliiid off for each person's share, is of the following different values per acre, viz. A's=5s. B's=8s. C's=r2s. D's=l5s. an acre; whose sum, 40s. divided by 4, their number, quotes 10s. for the mean value per acre. And, according to a late author, \vc must augment or diminish each share as follows : as { s. s. a. a. 5 ; 10 : . 15 : 30 A's share 8 10 : : 45 : 56',3 B's share o . 10 : : go : 7^ Cs share 5 10 : : 150 : 100 D's share \rL } sum of shares=2Glj3 acres. which is less than 300, by more than 38 acres, and shews the rule to be absolutely false; for when each person's share is laid out as above, there remains 38 acres unapplied ; I suppose for the use of the sur- veyor. But suppose we change the value of each person's land, and call A's 15s. B's 12s. C's 8s. and D's 5s. then we shall have as r 15 : 10 J 12 : 10 J 8 : 10 I 5 ; 10 15 : 10 A's share 45 : 37,5 B's share go : 112,5 C's share 150 : 300 D's share } sum of the shares 400 acres: how must the surveyor manage here, as he must make 46o acres of 300 ; for here D's share only takes the whole 300, where is he to find the l60 acres for the other three shares ? But enough of this; see it truely and methodically performed in the pext problem. 156 DIVISION OP LAND. Problems. // is required to divide any given quantify of land among any given number of persons , in froportion to their several estates, and the value of the land that falls to each person s share. Ride. Divide the yearly value of each person's estate by the value per acre of the land that is allotted for his share, and take the sum of the quotients, by which divide the whole given quantity of land, and this quotient will be a common multiplier, by which multiply each particular quotient, and the product will be each particular share of the land. Or say, as the sum of all the quotients is to the whole quantity of land, so is each particular quotient to its proportional share of land. Example. Let 300 acres of land be divided among A, B, C, D, whose estates are lOOl. 300l. 600I. and lOOOl. respectively per annum ; and the value of the land allotted to each is 5, 8, 12, and 15 shillings an acre, as in the example to the last problem. 600 Then 1000 100 =20, 5.09=37,5 12 = 50, and 15 174,166-. plier ■=-^Qi^^^, and the sum of these quotients is 1,72248, the common multi- 300 174,16 then 1,72248 1,72248 1,72248 1,72248 X X X X 20 = 37,5 - 50 = m,m^ = acres. 34,45 64,593 8 b", 124 114,832 A's share B's share C's share D's share sum of the shares =: 299,999 ^cres, or 300 very nearly, and is a proof of the whole. Let us now change the values of land as in the last problem, and see what each will have for his share. Suppose A's 15, B's 12, C's 8, and D's 5 DIVISION OP LAND. 157 shillings an acre; then 11^— 6, 666 ??^=25,—^ ^ 15 ' 12 ' 8 -- ^^t figures, the remaining figure towards the left band gives the content in acres, and consequentlv the number of acres at first sight ; the remaining decimal fraction, multiplied by 4, gives the roods, and the de- cimal part of this last product multiplied by 40, gives the poles or perches. Thus, if a field contains 16,54321 square links, we see immediately that it contains 16 acres, 54321 multiplied by 4, gives 2,17284 or 2 roods and 17284 parts ; these, multiplied by 40, produce (3,91360, or 6 poles, 91360 parts. Dlrecliotis for using the chain. Marks are first to be set up at the places whose distances arc to be ob- tained ; the place where you begin may be called your first station ; and the station to which you measure, the second station. Two persons arc to hold the chain, one at each end ; the foremost, or chain leader, must be provided with nine arrows, one of which is to be put down perpendicularly at the end of the chain when stretched out, and to be afterwards taken up by the follower, by way of keeping an account of the number of chains. When the arrows have all been put down, the leader must wait till the follower brings him the arrows, then proceeding onwards as before, but without leaving 1Q2 INSTRUMENTS FOR STRAIGHT LINES. an arrow at the tenth extention of the chain. Jn order to keep an account of the number of times which the arrows are thus exchanged, they should each tie a knot on a string, carried for that purpose, and which may be fastened to the button, or button- hole of the coat ; they should also call out the num- ber of those exchanges, that the surveyor may have a check on them. It is very necessary that the chain bearers should proceed in a straight line ; to this end, the second, and all the succeeding arrows, should always be so placed, that the next foregoing one may be in a line vi^ith it, the place measured from, and that to which you are advancing; it is a very good method to set up a staff at every ten chains, as well for the purpose of a guide to preserve the rectilinear direction, as to prevent mistakes. All distances of offsets from the chain Hne to any boundary which are less than a chain, are most con- veniently measured by the offset staff; the measure must always be obtained in a direction perpendicular to the chain.* The several problems that may be solved by the chain alone, will be found in that part of the work, which treats of practical geometry on the ground.'l' ^ Various tables of english and foreign measures, useful to sur- veyors, &c. are annexed to this work. f The best method of surveying by the chain, and now generally used by the more skilful surveyors, I judge, a sketch of here will be acceptable to many readers. It consists of forming the estates into triangles, and applying lines within them parallel and contiguous to every fence and line to be laid down, with offsets from these lines when necessary. The peculiar advantage of this method is, that, after three lines are measured and laid ^ down, every other line proves itself upon a^jplication. Thus, if the triangle a be he laid down, and the points d and e given in the sides, when the line d c has been mea- sured for the purpose of taking a fence con- tiguous to it, it will prove itself when laid down, from the two extremities being given. " This method cannot be used in woods, where the principal lines could not be observed, or in surveying roads or very detached parts of estates ; in such cases recourse must be had to the theodolite^ or other angular instrument. [ 193 ] DRSCRIPTION AKD USE OP THE SURVEYING" ttL'ADRANT, FOR ADJUSTING AND REGULAT- ING THE MEASURES OBTAINED BY THE CHAIN WHEN USED ON HILLY GROLXD, INVENTED BY R.King, sl kveyor. There are two circumstances to be considered in the measuring ot lines in an inclined situation : the Jirst regards the plotting, or laying down the mea- sures on paper ; the second, the area, or superficial content of the land. With respect to the first, it is evident that the oblique lines will be longer than the horizontal ones, or base ; if, therefore, the plan be laid down according to such measures, all the other parts thereof would be thereby pushed out of their true situations ; hence it becomes neces- sary to reduce the hypothcnusal lines to horizontal, which is easily effected by Mr. King's quadrant. With respect to the area, there is a difference among surveyors ; some contending that it should be made according to the hypothcnusal ; others, ac- cording to the horizontal lines : but, as all have agreed to the necessity of the reduction for the first purpose, we need not enter minutely into their reasons here ; for, even if we admit that in some cases more may be grown on the hypothcnusal plain than the horizontal, even then the area should be given according to both suppositions as the hilly and uneven ground requires more labour in the working. The quadrant A B, fg. 1, plate l4, is fitted to a wooden square, which slides upon an offset staff, and may be fixed at any height by means of the screw C, which draws in the diagonal of the staff, thus embracing the four sides, and keeping the limb of the square perpendicular to the staff ; the staff should be pointed with iron to prevent wear; when the staff is fixed in the ground on the station line^ the O 194 kixg's srnvEYixG quadrant. square answers the purpose of across staff, and may, if, desired, have sights fitted to it. The quadrant is three inches radius, of brass, is furnished with a spirit level, and is fastened to the limb DE of thesqurtre, by the screw G. When the several lines on the limb of the qua- drant have their first division coincident with their respective index divisions, the axis of the level is pa- rallel to the staf}'. The first line next the edge of the quadrant is numbered from right to left, and is divided into 100 parts, which shews the number of links in the horizontal line, which are completed in 100 links on the hypothenusal line, and in proportion for any lesser number. The second, or middlemost line, shews the number of links the chain is to be drawn forward, to render the hypothenusal measure the same as the horizontal. The third, or uppermost line, gives the perpendi- cular height, when the horizontal line is equal to 100. To use the quadrant. Lay the staff along the chain line on the ground, so that the plain of the quadrant may be upright, then move the quadrant till the bubble stands in the middle, and on the several lines you will have, 1. The horizontal length gone for- ward in that chain. 1. 1 he links to be drawn for- ward to complete the horizontal chain, 3. The per- pendicular height or descent made in going forward one horizontal chain. The two first lines are of the utmost importance in surveying land, which cannot possibly be planned with any degree of accuracy without having the ho- rizontal line, and this is not to be obtained by any instrument in use, without much loss of time to the surveyor. For with this, he has only to lay his staff on the ground, and set the quadrant till the bubble is in the middle of the space, which is very soon per- formed, and he saves by it more time in plotting his survey, than he can lose in the field ; for as he IMPROVED MEASURING WHEEL. IQS completes the horizontal chain as he goes forwards, the njfsets are always In their rijr/it places, and the licld-boiik being kt^pt by horizontal measure, his lines are .liways sure to close. if thi; superficial content by the hypothenusal measure be required for any particular ])urpose, he has that likewise by entering in the margin of his field-book the links drawn forward in each chain, having thus the hypothenusal and horizontal length of every line. The third line, which is the perpendicular height, mav be used with success in finding the height of timber ; thus measure with a tape of 100 feet, the surface of the ground from the root of the tree, and find, by the second line, how much the tape is to be drawn forward to complete the distance of 100 horizontal feet ; and the line of perpen- diculars shews how many feet the foot of the tree is above or below the place where the 100 feet distance is completed. Then inverting the quadrant by means of sights fixed on the staff, place the staff in such a position, as to point to that part of the tree whose height you want ; and slide the quadrant till the bubble stands level, you will have on the line of perpendiculars on the quadrant, the height of that part of the tree above the level of the place where you are ; to which add or subtract the perpendicular height of the place tVom the foot of the tree, and you obtain the height required. OF THE PERAMBULATOR, OR IMPROVED MEA- SURING WHEEL ; the WAY-WISER, AND THE PEDOMETEK. Fig. 6, pJaU ]/, represents the perambulator, which consists of a wheel of wood A, shod or lined with iron to prevent the wear ; a short axis is fixed to this wheel, which communicates, by a long pinion 02 I96 THE WAY-WISER AND PEDOMETER. fixed in one of the sides of the carriage B, motion to the wheel-work C, included in the box part of the instrument. In this instrument, the circumference of the wheel A, is eight feet three inches, or half a pole; one revolution of this wheel turns a single- threaded worm once round ; the worm takes into a wheel of 80 teeth, and turns it once round in 80 revolutions ; on the socket of this wheel is fixed an index, which makes one revolution in 40 poles, or one furlong : on the axis of this worm is fixed an- other worm with a single thread, that takes into a wheel of 40 teeth ; on the axis of this wheel is another worm with a single thread, turning about a wheel of 160 teeth, whose socket carries an index that makes one revolution in 80 furlones, or 10 miles. On the dial plate, see fig. 7» there are three graduated circles, the outermost is divided into 220 parts, or the yards in a furlong ; the next into 40 parts, the number of poles in a furlong ; the third into 80 parts, the number of furlongs in 10 miles, every mile being distinguished by its proper Roman figure. This wheel is much superior to those hitherto made, 1 . Because the worms and wheels act without shake, and, as they have only very light indices to carry, move with little or no friction, and are, therefore, not liable to wear or be soon out of or- der ; which is not the case with the general num- ber of those that are made, in which there is a long train of wheels and pinions, and consequently much shake and friction. 2. The divisions on the graduated circles are at a much greater distance, and may therefore be subdivided into feet, if re- quired. 3. The measure shewn by the indices is far more accurate, as there is no shake nor any loss of time in the action of one part or another. T'he instrument is sometimes made with a double wheel for steadiness when using, and also with a bell THE WAY-WISER AND PEDOMETER. 197 connected to tlic wheel-work, to strike the number of miles gone over. This instrument is very useful for measuririg roads, commons, and every thing where expedition is required ; one objection is however made to it, namely, that it gives a measure somewhat too long by entering into hollows, and going over small hills. This is certainly the case ; the measuring wheel is nol an infallible mode of ascertaining the horizontal distance between any two places ; but then it may with proprictv be asked whether any other method is less fallible ? whether upon the whole, and in the circumstances to which the measuring wheel is usually appropriated, the chain is not equally uncertain, and the measure obtained from it is as liable to error, as that from the wheel. The ivay-wlser is a similar kind of instrument, but generally applied to carriages, for measuring the roads or distance travelled. The best method for constructing such a one is represented '\v\ Jig. 8, plate 17. A piece of plate iron A is screwed to the inside nave of the wheel ; this being of a curvi- lineal shape, in every revolution of the coach-wheel B it pushes against the sliding bar C, which, at the other end, withinside of the brass box of wheel- work D, is cut with teeth, and thereby communi- cates motion to the wheel-work in the box. The bar is re-acted upon by a spring in the box, so as to drive it out again for the fresh impulse from the iron piece on the nave, at every revolution. As the wheels of carriages differ in size, the wheel-work is calculated to register the number of revolutions, and shew by three indices on the dial plate to the amount of 20,000. In any distance, or journey performed, the length of feet and inches in the circumference of the wheel must be first accu- rately measured, and that multiplied by the number shewn on the dial of the way-wiser gives the dis- tance run. 03 IQS THE SURVEYING CROSS. By means of rods, universal joint, &c. it is often made to act within the carriage, so that the person may, at any moment, without the trouble of getting out, see the number of the revolutions of the wheel. If the instrument is to be always applied to one wheel, a table may easily be constructed' to shew the dis- tance in miles and its parts by inspection only. The. pedomeler IS exactly the same kind of instru- ment as the way-wiser. The box containing the wheels is made of the size of a watch-case, and goes into the fob, or breeches pocket ; and, by means of a string and hook fastened to the waistband or at the knee, the number of steps a man takes in his regular paces are registered, from tlie action of the string ■ upon the internal wheel-work, at every step, to the amount of 30,000. It is necessary to ascertain the distance walked, that the average length of one pace be previously known, and thatmtdtiplied bythenum- ber of steps registered on the dial-plate. This instru- inent requires a person to be very uniform in his paces, and the acting distance invariable, or a false step or pull will be made. OP THE SURVEYING CROSS, ^^. 2, plate 14. The cross consists of two pair of sights, placed at right angles to each other: these sights are some- times pierced out in the circumference of a thick tube of brass about 2 Mnches diameter, sc.cji^. 3^ plate 14. Sometimes it consists of four sights strongly fixed upon a brass cross ; this is, when in use, screwed on a staff having a sharp point at the bottom to stick in the ground ; one of this kind is represented at fg.2,flate\A:. The four sights screw off to make the instrument convenient for the pocket, and the Staff w hich is about A~ or five feet in length (for both the crosses) unscrews into three parts to go in a port- manteau, &c. The surveying cross is a very useful instrument for IMPROVED BY JONES. IQQ placing ofoffsets, or even for measuring small pieces of gnnind ; its accuracy depends on ihc sights being exactly at right angles to each other. It may be proved by looking at one object through two of the sights, and observing at the same time, without moving the instrument, another object through the other two sights ; then turning the cross upon the staff, look at the same objects through the opposite sights ; if they are accurately in the direction of the sights, the instrument is correct.* It is usual, in order to ascertain a crooked line by offsets, f^r^t to measure a base or station line in the longest direction of the piece of ground, and while measuring, to find by the cross the places where per- pendiculars would fall from the several corners and bends of the boundary; this is done by trials, fixing the instrument so, that by one pair of sights both ends of the line may be seen ; and by the other pair, the corresponding bend or corner ; then measuring the length of the said perpendicular. To be more particular, let A, h, i, k, 1, TO,fg. 35, ^late Q, be a crooked hedge or river ; measure a straight line, as ABf along the side of the foregoing line, and while measuring, observe when you are opposite to any bend or corner of the hedge, as at c, d, e; from thence measure the perpendicular offsets, as at c h, d i, &c. with the offset staff, if they are not too long; if so, with tl>e chain. The situation of the offsets are readily found, as above directed, by the cross, or Kings surveying quadrant ; they are to be registered in the field-book. Of surveying iiilh the chain and cross. What has * I have made snme additions to the box cro?s staff, which have been found useful and convenient for the pocket, where great accuracy is not required. See Jig. 6. A compa>^s and needle at the top A to give the hearings, and a moveable graduated base at B, by rack-work and pinion C, to give an angle to 5' of a degree by the nonius divided on the box above. Thus the surveyor may have a small theodolite, circumferentor, and cross staff all in one in- Btrymcnt, Edit. O 4 200 OF THE OPTICAL SaUARE. been denominated by many writers, surveying by the chain onIy,\s, in fact surveying by the cross and chain ; for it is necessary to use the cross, or optical square, for determining their perpendicular lines, so that all that has been said, even by these men, in favour of the c/iam alone, is founded in fallacy. To survey the triangular field AB C, fg. -zl, plate p, by the chiiin and cross : 1. Set up marks at the corners of the field. 2. Beginning, suppose at A, measure on in a right line till you are arrived near the point D, vi'here a perpendicular will fall from the angle, let the chain lie in the direction or line AB. 3. Fix the cross over AB, so as to see through one pair of sights the mark at A or B, and through the other, the mark at C; if it does not coincide at C with ^:he mark, the Gross must be moved backwards or forwards, till by trials one pair of the sights exactly coincide with tltc mark at C, and the other with A or B. 4. Observe how many chains and links the point D is from A, suppose 3.03. which must be entered in the field- book. 5. Measure the perpendicular DC, 643. 7. finish the measure of the base line, and the work is done. This mode is used at present hy many surveyors , probably because there is no check whereby to discover their errors, which must be very great, if the survey is of any extent. To plot this, make A B equal 11. 41. A D equal to 3.03. on the point D erect the perpendicular D C, and make it equal 6. 43. then draw A C, B C, and the triangle is formed. OP THE OPTICAL SQUARE, fg. 4, plate J4, This instrument has the two principal glasses of Hadleys quadrant, and was contrived by my hither ; it is in most, if not in all respect superior to the com- mon surveying cross, because it requires no staff, may be used in the hand, and is of course of great use to a military officer. It consists of two plain n:)irrprs, so disposed, that an object seen by reflexion OP THE OPTICAL SQUARE. CiOl from both, will appear to coincide with another ob- ject seen by Jircct I'ision whenever the two ob- jects subtend a right angle from the centre of the instrnment. and serves therefore to raise or let fall pcrpeiulicnlars on the gromul, as a square does on p.'iper, of which we shall give some examples. Its ap- plication to the purposes of surveying will be evident from these, and what has been already said con- cerning the cross. Fig. 4, plate U, is a representation of the instru- ment witiiout its cover, in order to render the con- struction more evident. There is a cover with a slit or sight for viewing the objects ; the object seen di- rectly, always concidcs with the object seen by re- flexion, when they are at right angle to each other*. From a given point in a given line^ to raise a perpen- dicular. 1. The observer is to stand with this in- strument over the given point, causing a person to stand with a mark or fixing one at some convenient distance on the given line. 2. An assistant must be placed at a convenient distance, with a mark some- where near the line in which it is supposed the per- pendicular will fall ; then if on looking at one of the objects, the other be seen in a line with it, the place where the mark of the assistant is fixed is the re- quired point. From a given point over a given linl to let fall a per- peruUcidar. Every straight line is limited and deter- mined by two points, through which it is supposed to pass; so in a fieUl the line is determined by two fixed objects, as steeples, trees, marks erected for the purpose, &c. For the present operation, the two objects that determine must be on one side the point where the perpendicular falls ; or, in other words, the observer must not be between the objects, he must place himself over the line, in which he will al- ways be when the two objects coincide; he must move himself backwards on this line, till the mark, * Sec the description of a considerable improvement upon it, after the description of the Hadley's sextant //^z/t', 19. Edit. 202 OF THE CIRCUMFERENTOR. from whence the perpendicular is to be let fall, seen by direct vision, coincides wilh one of the objects which determine the given line seen by reflexion, and the instrument will be over the required point. To measure inaccessible distances by the optical square. Required the distances from the steeple A, jig. 20, flate 9, to B. Let the observer stand with his instrument at B, and direct an assistant to move about C with a stafFas a mark, until he sees it coin- cide by direct vision with the object at A ; let him fix the staff there ; then let the observer walk along the line until A and B coincide in the instrument and BD will be perpendicular to x^ C ; measure the three lines B C, D C, B D, and then the following proportion will give the required distance, for as DCistoDB, soisBC to A B. Second method. 1 . Make B C, fg. 2 1 , flate p, perpendicular to B A. 2. Divide B C into four equal parts. 3. Make C D perpendicular to C B. 4. Bring F E A into one line, and the distance from G to F will be 7 of the distance from B to A. OP THE CIRCUMFERENTOR, fig. 1, fhte 15. This instrument consists of a compass box, a mag- netic needle, and two plain sights, perpendicular to the meridian line in the box, bv which the bearin2:s of objects are taken from one station to another. It is not much used in England where land is valuable ; but in America where land is not so dear, and where it is necessary to survey large tracts of ground, over- stocked with wood, in a little time, and where the surveyor must take a multitude of angles, in which the sight of the two lines forming the angle may be hindered by underwood, thecircumfcrentoris chiefly used. The circumferentor, sec fig. 1, plate 15, consists of a brass arm, about 14 or 15 inches long, with THE CIRCUMFERENTOR. 203 sights at each end, and in the middle thereof a cir- cular box, with a glas^ cover, of about 5'r inches diameter; within the box 'S a brass graduated circle, the upper surface divided into 3(iO degrees, and num- bered 10. 20. 30. to 306 ; every tenth degrees, is cat down on the inner edge of the circle. The bot- tom of the box is divided into four parts or quadrants, each of which is subdivided into 90 degrees num- bered from the meridian, or north and south points, each way to the east and west points ; in the middle of the box is placed a steel pin finely pointed, called the centre pin, on which is placed a magnetic needle, the quality of which is such, that, allowing for the difference between the astronomic and magnetic meridians, however the instrument may be moved about, the bearing or angle, which any line makes with the magnetic meridian, is at once shewn by the needle. At each end of the brass bar, and perpendicular thereto, sights are fixed ; in each sight there is a large and small aperture, or slit, one over the other, these are alternate ; that is, if the aperture be upper- most in one sight, it will be lowest in the other, and so of the small ones ; a fine piece of sewing silk, or a horse hair runs through the middle of the large slit. Under the compass box is a socket to fit on the pin of the staff; the instrument may be turned round on this pin, or fixed in any situation by the milled screw ; it may also be readily fixed in an horizontal direction by the ball and socket of the staff, moving for this pur- pose the box, till the ends of the needle are equi- distant from the bottom, and traverse or play with freedom. Occasional variatiojis in the construction of this instrument, are 1. In the sights, which are some- times made to turn down upon an hinge, in order to lessen the bulk of the instrument, and render it mare convenient for carriage ; sometimes they arc made to slide on and off with a devetail, sometimes 204 THE CIRCUMFERENTOR. to fit on with a screw and two steady pins. 2. In the box, which in some instruments has a brass cover, and very often a spring is placed within the box to throw the needle off the centre pin, and press the cap close against the glass, to preserve the point of the centre pin from being blunted by the conti- nual friction of the cap of the needle. 3. In the needle itself, which is made of different forms. 4. A further variation, and for the best, will be no- ticed under the account of the improved circum- fercntor. The surveying compass represented ^^fig. 3, plate 15, is a species of circumferentor, which has hitherto been only applied to military purposes ; it consists of a square box, within which there is a brass circle divided into 36o degrees ; in the centre of the box is a pin to support a magnetic needle ; a telescope is fixed to one side of the box, in snch a manner as to be parallel to the north and south line; the telescope has a vertical motion for viewing objects in an in- clined plain ,; at the bottom of the box is a socket to receive a stick or staff" for supporting the instrument. To use the circumferentor. Let A B C, Fig. 19, plate g, be the angle to be measurerd. 1. The in- strument being fixed on the staff, place its centre over the point B. 2. Set it horizontal, by moving the ball in its socket till the needle is parallel to the bottom of the compass box. 3. Turn that end of the compnss box, on which the N. or fleur de lis is engraved next the eye. 4, Look along B A, and observe at what degree the needle stands, suppose 30. 5. Turn the instrument round upon the pin of the ball and socket till you can see the object C, and suppose the needle now to stand at r25. 6. Take the former number of observed degrees from the latter, and the remainder 95 is the required angle. If, in two observations to find the measure of an angle, the needle points in the first on one side 360°, and m the second on the other_, add what one wants THE CIRCLMPEREXTOR. 205 of 360°, to what the otlier is past 360°, and the sum is the required angle. This irencral idea of the use ot the circunilcrcntor, it is presumed, will be sufFieient for the present; it will be more partieularly treated of hereafter. When in flic use of the cireumfercntor, you look through ihe upper sigliis, from the ending of the staliuu to the beginning, it is ealled a bjck sights but when you look through the lower slit from the be- ginning of the station towards the end, it is termed the Jure sioht. A theodolite, or any instrument which is not set by the needle, must be fixed in its place, by taking back and foresights at every station, for it is by the foregoing station that it is set parallel ; but as the needle preserves its parallelism throughout the whole survey, whosoever works by the eircum- ferentor, need take no more than one sight at every station. There is, indeed, a difference between the mag- netic and astronomic or true meridian, which is called the variation of the needle. This variation is diti^erent at different places, and is also different at different times ; this difference in the variation is called the variation of the variation ; but the increase and decrease thereof, both with respect to time and place, proceeds by such very small increments or de- crements, as to be altogether insignificant and in- sensible, within the small limits of an ordinary sur- vey, and the short time required for the performance thereof.* * The present variation, or, more properly, the dcdinaUon of the needle, is near 23" W. of the north, at I/^jndon ; or two point* in general may be allowed on an instrument to the E. to fix th^ just meridian. ]\s inclbuiio?!, or dip, was about 72^ of the north pole below the horizon in the year ^77^- Th^ inclination, as well as the declination of the needle, is found to be continually varying ; and, from the observations and hypotheses hitherto made, not to developc any law by which its positi(,n can be determined, for any future time. Kuir. . t [ 206 ] OF THE IMPROVED CIECUMFEEENTOR. The excellency and defects of the preceding in- strument both originate in the needle ; from the re- gular direction thereof, arise all its advantages ; the unsteadiness of the needle, the difficulty of ascer- taining with exactness the point at which it settles, are some of its principal defects. In this improved construction these are obviated, as will be evident from the following description. One of these instru- ments is represented ntjig. 2, plate 15. A pin of about three quarters of an inch diameter goes through the middle of the box, and forms as it were a vertical axis, on which the instrument may be turned round horizontally ; on this axis an index A B is fastened, moving in the inside of the box, having a nonius on the outer end to cut, and subdi- vide the degrees on the graduated circle. By the help of this index, angles may be taken with much greater accuracy than by the needle alone ; and, as an angle may be ascertained by the index with or without the needle, it of course removes the difficul- ties, which would otherwise arise if the needle should at any time happen to be acted upon, or drawn out of its ordinary position by extraneous matter ; there is a pin beneath, whereby the index may be fastened temporarily to the bottom of the box, and a screw, as usual, to fix the whole occasionally to the pin of the ball and socket, so that the body of the instru- ment, and the index, may be either turned round together, or the one turned round, and the other re- main fixed, as occasion shall require. A further im- provement is that o^ preventing all horizontal motion of the hall in the socket; the ball has a motion in the socket, every possible motion is necessary, except L3ES OF THE C IRC UMFEUENTOR, &C. lOJ th: horizontal one, which is here totally destroyed, and every other possible motion left perfectly free* GE^;ERAL IDEA OP THE USE OF THESE INSTRL MEXTS. For this purpose, let fg. 1, flute 18, represent a field to be surveyed, 1. Set up the eircuinfcrcntor at any corner, as at A, and therewith take the course or bearing, or the angle that such a line makes with the mao^netic meridian ^hewn by the needle, of the side AB, and measure the length thereof with the chain. If the circumferentor be a common one, having no index in the box, the course or bearing is taken by simply turning the sights in a direct line from A to B, and when the needle settles, it will point out on the graduated limb the coarse or number of de- grees which the line bears from the magnetic me- ridian. But if the circumferentor has an index in the box, it is thus used. 1. Bring the index to the north point on the graduated limb, and fix it there, by fastening the body of the instrument and the under part together by the pin for that purpose, and turn the instrument about so that the needle shall settle at the same point; then fasten the under part of the instrument to the ball and socket, and taking the pin out, turn the sights in a direct line from A towards B, so will the course and bearing be pointed out on the graduated circle, both by the needle and by the * The instrument is made to turn into a vertical position, and by the addition of a spirit level to take altitudes and depressions. The index A B has been found to intcrt'ere too much with the free play of the needle. In the year l/pl 1 contrived an external nonius piece ii,_fig. a, to move against and round the graduated circled, either with or without rack-work or pinion. The circle and com- pass plate are fixed, and the nonius piece and outside rim and sight carried round together when in use. This has been generally ap- proved of. Edit. 208 USES OF THE CIRCUMFERENTOR, ScC. index. This done, fasten the body of the instru- ment to the under part again, and having set the in- strument up at B, turn the sights in a direct line back from B to A, and there fasten the under part of the instrument to the ball and socket ; then take out the pin which fastens the body of the instrument to the under part, and turn the sights in a direct line towards C, and proceed in the same manner all round the survey ; so will the courses or bearings of the several lines be pointed out both by the needle and the index, unless the needle should happen to be drawn out of its course by extraneous matter; but, in this case, the index will not only shew the course or bear- ing, but will likewise shew how much the needle is so drawn aside. After this long digression to explain more minutely the use of the instruments, we may proceed. 2. Set the circumferentor up at B, take the course and bearing of B C, and measure the length thereof, and so proceed with the sides C D, D E, E F, F G, G A, all the way round to the place of beginning, noting the several courses or bearings, and the lengths of the several sides in a field-book, which let us suppose to be as the following : o / 1. A B North 7 West 21. 00. 2. BC North 55 15 East 18. 20. 3. CD South 62 30 East 14. 40. 4. D E South 40 West 11. 5. E F South 4 15 East 14. 6. F G North 73 43 West 12. 40. 7. GA South 52 West p. 17. N. B. By north 7° west, is meant seven degrees to the westward, or left hand, of the north, as shewn by the needle ; by north .55° 15' east, fifty-five de- grees fifteen minutes to the eastward, or right hand of the north, as shewn by the needle. In like manner by south 62° 30' east, is meant sixty-two degrees and thirty minutes to the cast-ward. THF. COMMOM THEODOLITE. !20() or left hand of the south ; and by south 40° west, ibrty degrees lo the westward, or right hand of the south. . The '21 chains, 18 chains 20 links, &c. arc the lengths or distances of the respective sides, as mea- sured by the chain. FifT. 4, plate 15, represents a small circumfercntor, or theodolite ; it is a kind that was much used by General /^-yy, for delineating t!)c smaller parts of a survev. The diameter is 4 inches. It is better to have thesiglit pieces double, as shewn iuji^. 2. OF THE COMMON THEODOLITE, The error to wliich an instrument is liable, where the whole dependence is placed on the needle, soon rendered some other invention necessary to measure angles with accuracy ; among these, the common theodolite, with four plain sights, took the lead, be- ing simple in construction, and easy in use. The common theodolite is represented Ji^. 5, phte\A\ it consists of a brass graduated circle, a moveable index A B ; on the top of the index is a compass with a magnetic needle, the compass box is covered with a glass, two sights, C, D, are fixed to the index, one at each end, perpendicular to the jjlain of the instrument. • There are two more sights E F, which arc fitted to the graduated circle at the points of 360° and ISO'^; they all take on and off for tiui convcnicncy of packing. In each sight thcrJ is, as in the circumfercntor, a large and a small aj;crturc placed alternately, the large aper" ture in one sight being always opposed to the nar- row aperture in the other ; underneath the brass circle, and in the centre thereof, is a sprang to fit on the pin of the ball and socket, which fixes on a three- legged staff. The I ircle is divided into degrees, which are all P 210 THE COMMON" THEODOLITE. numbered one way to 36o^ usually from the left to the rio-ht, supposing yourself at the centre of the instrnnient ; on the end of the index is a nonius di- vision, by which the degrees on the limb are subdi- vided to five minutes ; the divisions on the ring of the compass box. are numbered in a contrary direction to those of the limb. As much of geometrical mensuration depends oa the accuracy of the instrument, it behoves every surveyor to examine them carefully ; different me- thods will be pointed out in this work, according to the nature of the respective instruments. In that under consideration, the index should move regu- larly when in use ; the theodolite should always be placed truly horizontal, otherwise the angles mea- sured by it will not be true ; of this position you may judge with sufficient accuracy by the needle, for if this be originally well balanced, it will not be parallel to the compass plate, unless the instrument be horizontal ; two bubbles, or spirit levels, are sometimes placed in a compass box at right angles to each other, in order to level the instrument, but it appears to me much better to depend on tho needle : 1. Because the bubbles, from their smallness, are seldom accurate. 2. Because the operator can- not readily adjust them, or ascertain when they indi- cate a true level. To examine the instrument; on an extensive plain set three miarks to form a triangle; with your theodo- lite take the three angles of this triangle, and if these, when added together, make 180°, you may be certain of the justness of your instrument. To examine the needle ; observe accurately where the needle settles, and then remove it from that situation, by placing a piece of steel or magnet near it ; if it afterwards settles at the same point, it is so far right, and you may judge it to be perfectly so, if it settles properly in all situations of the box. If THE COMMON Tli KODOLITE. 211 in any situation of the box a deviation is observed, tlic error is most probably occasioned by some parti- cles of steel in the metal, of which the compass box is made. To examine the graduations ; set the index di- vision of the nonius to the beginning of each degree of the theodolite, and if the hist division of the no- nius always terminates precisely, at each application, with its respective degree, then the divisions are ac- curate, Ciiuliofis in the use of the tnstnuneyit, 1 . Spread the legs that support the theodolite rather wide, and thru>t them lirmly into the ground, that they may neither yield, nor give unequally during the observation. 2. Set the instrument horizontal. 3. Screw the ball iirmly in its socket, that, in turn- ing the index, the theodolite may not vary from the objects to which it is directed. 4. Where ac- curacy is required, the angles should always betaken twice over, oftcncr where great accuracy is material, and the mean of the observation must be taken for the true angle. To measure an an6o°, which are numbered in a direction contrary to those on the horizontal limb. The bot- tom of the box is divided iiito four parts, or quad- rants, each of which is subdivided to every 10 de- grees, numbered from the mcridan, or north and south points each way to the east and west points. In the middle of the box is a steel pin finely pointed, on which is placed the magnetic needle ; there is a wire trigger for throwing the needle off its point when not in use. Of the horizontal limh A A. This limb consists of two plates, one moveable on the other ; the outs'de edge of the upper plate is chamfered, to serve as an index to the degrees on the lower. The upper plate, together with the compass, vertical arc, and tele- scope, are easily turned round by a piniou fixed to the screw c ; d is a nut for fixing the index to any part of the limb, and thereby making it so secure, that there is no danger of its being moved out of its place, while the instrument is removed Irom one sta- tion to another. The horizontal limb is divided to half degrees, and numbered from the right hand to- wards the left, 10, 20, 30, ^vC. to 3(io ; the divi- sions are subdivided by the nonius scale to every Uiinute of a desfree. On the upper plate, opposite to the nonius, are a IMPROVF.D THP.ODOLITES. 2 21 few dlvl'sions similar to those on the vertical arr, giving the 100th ports for measuring the diameter of trcTs, buililines, &c. The whole instrument fits on the conical ferril of a strong brass headed staff, with three substantial wooden logs ; the top, or head of the staff, consists of two brass plates E, parallel to exich other ; four •screws pass through the upper plate, and rest on the lower plate ; by the action of these screws the situa- tion of the plate may be varied, so as to set the hori- zontal limb truly level, or in a plain parallel to the horizon -, for this purpose, a strong pin is fixed to the underside of the plate, this pin is connected with a ball that fits into a socket in the lower plate ; the axis of the j)in and ball are so framed, as to be al- ways perix^ndicuh'.r to the plate, and, consequently, to the horizontal limb. To (iJjnst the theodolite. As so much of surveying depends on the accuracy of the instruments, it is ab- solutely necessary that the surveyor should be very- expert in their adjustments, without which he cannot expect the instruments will properly answer the pur- poses they were designed for, or that his surveys will have the requisite exactness. The necessary adjustments to the theodolite, which \Ve have just described, arc, 1. That the line of sight, or collimation, be exactly in the centre of the cvliiidric rings round the telescope, and which lie in the Y's. 2. That the level be parallel to this line, or the axis of the above-mentioned rings. 3. The hori- zontal limb must be so set, that when the vertical arc is at zero, and the upper part moved roimd, the bubble of the level will remain in the middle of the open space. Previous to tiic adjustments, place the instrument upon the staff, and set the legs thereof firmly upon the ground, and at about three feet frotn each otiier, so that the telescope may be at a proper height for the eye, and that two of the screws on the staff tliat 222 IMPROVED THEODOLITES, are opposite to each other may be nearly in the di- rection of some conspicuous and distant object. To adjust the line of coUimation. Having set up the theodolite agreeable to the foregoing direction, di- rect the telescope to some distant object, placing it so that the horizontal hair^ or wire, may exactly coincide with some well defined part of the object; turn the telescope, that is, so that the tube of the spirit level D may be uppermost, and observe whe- ther the horizontal hair still coincides with the ob- ject ; if it does, the hair is in its right position if; not, correct half the difference by moving the hair, or wire, which motion is effected by easing one of the screws in the eye tube, and tightening the other ; then turn the telescope round to its former position, with the tube of the spirit level lowermost, and make the hai/ coincide with the object, by moving the ver- tical arc ; reverse the telescope again, and if the hair does not coincide with the same part of the object, you must repeat the foregoing operation, till in both positions it perfectly coincides with the same part of the object. The precise situation of the horizontal hair being thus ascertained, adjust the vertical hair in the same manner, laying it for this purpose in an horizontal position : the spirit tube will, during the adjustment of the vertical hair, be at right angles to its former position. When the two wires are thus adjusted, their intersection will coincide exactly with the same point of the object., while the telescope is turned quite round ; and the hairs arc not properly adjusted, till this is effected. Adjustment of the level. To render the level pa- rallel to the lineof collimation, place the vertical arc over one pair of the staff screws, then raise one of the screws, and depress the other, till the bubble of the level is stationary in the middle of the glass ; now take the telescope oat ofihcY's, and turn it end IMPROVED THEODOriTES. 023 for end, that is, let the eye end lay where the object etui was placed ; and if, when in this situation, the bubble remains in the middle as before, the level is well adjll^tcd ; if it does not, that end to which the bubble runs is too high; the position thereof must be corrected by tinning with a screw-driver one or both of the screws which pass through the end of the tube, till the bubble has moved half the distance it ought to come to reach the middle, and cause it to move the other half by turning the staff screws. Re- turn the telescope to its former position, and if the adjustments have been well made, the Nibble will remain in the middle ; if otherwise, the process of altering the level and the staff screws, with the re- versing, must be repeated, till it bears this proof of its accuracy. In some instruments there is s provi- sion for raising or lowering the Y's a small degree, in order more conveniently to make the bubble con- tinue in its place when the vertical arc is at o, and the horizontal limb turn round. To adjust the level of the horizontal limb. Place the level so that it may be in a line with two of the staff screws, then adjust it, or cause the bubble to become stationary in the middle of the open space by means of these screws. Turn the horizontal limb half round, and if the bubble remains in the middle as before, the level is well adjusted; if not, correct half the error by the screws at the end of the level, and the other half by the staff screws. Now return the horizontal limb to its former position, and if it remains in the middle, the errors are corrected ; if not, the process of altering must be pursued till the error is annihilated, bee this adjustment in the description of Ramsdens theodolite. When the bubble is adjusted, the horizontal limb may always be levelled by means of the staff screws. [ 224 J OP THE THEODOLITE, AS IMPROVED BY KAMSDEX, jig. 12, plate l6. Among the improvements the instruments of science have received from Mr. Ramsden^ and the perfection with which he has constructed them, we - are to rank those of the theodoUte ; in the present instance, he has happily combined elegance and neatness of form, with accuracy of construction ; and the surveyor will contemplate with pleasure this in- strument, and the various methods by which the parts concur to give the most accurate result. The principal parts of this instrument are however so similar to the foregoing, that a description thereof must, in some degree, be a repetition of what has been already described, and requires less detail here. F F represents the horizontal limb, of six, seven, or eight inches in diameter, but generally of seven inches, so called, because when in use, it ought al- ways to be y^laced parallel to the horizon. It con- sists, like the former, of two plates, the edges of these two are chamfered, so that the divisions and the nonius are in the same plain, which is oblique to the plain of the instrument. The limb is divided into half degrees, and subdivided by the nonius to every minute ; it is numbered to 36o° from the north towards the east : besides these, the tangents to 100 of the radius are laid down thereon. The upper plate is moved by turning the pinion G : on this plate are placed, at right angles to each other, two spirit levels for adjusting more accurately the horizontal limb. N O P is a solid piece fitted on the upper horizon- tal plate, by means of three capstan head screws, passing through three similar screws. Fy the action i)f these, the vertical arc may be set perpendicular to AN IMPROVED THEODOLITE. 225 tiie horizontal limb, or be made to move in a vertical plain. On tliis solid piece, are fixed two stout supports, to carry the axis of* the vertical arc, which arc is moveable by the pinion E. On the upper part of the vertical arc, are the Y's and loops to support and confine the telescope; the Y's arc tangents to the cyhndric rings of the telescope, which rings are turned, and then ground as true as possible, and are prevented from moving backwards or forwards, by means of two shoulders. The telescope is achro- matic, and about twelve inches in length, and may be adjusted to the eye of the observer, or the distance of the object, by turning the milled nutB. The wires are adjusted by the screws in the eye tube at A. Under the telescope is fixed a spirit level C, tire distance of whose ends from the telescope may be re- gulated by the screws c, c. Heneath the horizontal limb there is a second or auxiliary telescope, which has both an horizontal and vertical motion : it is moved horizontally by the jnilled screw G, and when directed to any object, is fixed in its situation by another milled screw H ; it moves vertically on the axis ; there is an adjustment to this axis, to make the line of coUimation move in a vertical plain. By the horizontal motion, this telescope is easily set to what is called the backset stations ; the under telescope keeping in view the back object, while the upper one is directed to the fore object. Underneath the lower telescope is a clip to fasten occasionally the main axis ; this clip is tightened by the finger screw L, and when tightened, a small motion of the adjusting screw K will move the telescoj)e a few degrees, in order to set it with great accuracy. I'eneath these is the staflT, the nature of which will be sufficicnily evident from what was said thereon in the descripti(jn of the last theodolite, or by inspection of the figure. To adjust the levels of the hon%ontal plate, \. Place the instrument on its staff, with the legs thereof ac Q 226 AN IMPROVED THEODOLITE, &C. such a distance from each other, as will give the in- strument a firm footing on the ground. 2. Set the nonius to 36o, and move the instrument round, till one of the levels is either in a right line with two of the screws of the parallel plates, or else parallel to such a line. 3. By means of the two last mentioned screws, cause the bubble in the level to become sta- tionary in the middle of the glass, 4. Turn the ho- rizontal limb by the milled nut half round, or till the nonius is at 180, and if the bubble remains in the middle as before, the level is adjusted ; if it does rot, correct the position of the level, by turning one or both the screws which pass through its ends, till the ))ubb}e has moved half the distance it ought to come, %o reach the middle, and cause it to move the other half by turning the screws of the parallel plates. 5. Return the horizontal limb to its former position, and if the adjustments have been well made, the bubble will remain in the middle; if otherwise, the process of altering must be repeated till it bears this proof of accuracy. 6. Now regulate the screv^s of the staff head, so that the bubble remain in the middle while the limb is turned quite round. 7- Adjust the other level by its own proper screws, to agree with that already adjusted. To adjust the level under the telescope. 1. The horizontal plate being levelled, set the index of the nonius of the vertical arc to o, pull out the two pins, and open the loops which confine the telescope. 2. Adjust the bubble by its own screws. 3. Reverse the level, so that its right hand end may now be placed to the left ; if the bubble continues to occupy the middle of the glass it is in its right position ; if not, correct one half of the error by the capstan screws under the plate, and the other half by the screws under the level. 4. Reverse the level, and correct, if there is any occasion, continuing the operation till the error vanishes, and the bubble stands in the mid- die in both positions. VARIATION IN' THEODOLITES. 227 To adjust the line of coUimation. 1 . Direct the te- lescope, so that the horizontal wire may coincide with some well defined part of a remote object. 2. Turn the telescope so that the buhhle may be uppermost; if the wire does not coincide with the same part of the obiect as before, correct halt the difference by mov- ing the vertical circle, and the other half by moving the wire, which is efFceted by the screws in the eye tube of the telescope ; and so on repeatedly, till the difference wholly disappears. Lastly, adjust the vertical wire in the same manner ; when the two wires are properly adjusted, their intersection will coincide exactlv with the same point of an object, while the telescope is turned quite round. VARIATIONS IN THE CONSTRUCTION OF THEO- DOLITES WITH TELESCOPIC SIGHTS. To accommodate those who may not wish to go to the price of the foregoing instruments, others have been made with less work, in order to be afforded at a lower price ; one of these is represented ^ifg> 5, plate 15. It is clear from the figure, that the differ- ence consists principally in the solidity of the parts, and in their being no rack-work to give motion to the vertical arc and horizontal limb. The mode of using it is the same with the other, and the adjust- ment for the line of collimation, and the level under the telescope, is ])erfectly similar to the same adjust- ments in the instruments already described, a further description is therefore unnecessary. A larger kind is also made with or without rack- work, similar to Jig. 1, plate l6. A small one, about four inches in diameter, was invented by the late Mr. Benj. Martin, in which the telescope, about six inches in length, with a level, has no vertical motion, but the horizontal motion is given bv a pinit>n ; and it may be turned into a vertical po- Q 2 228 DESCRIPTION OP THEODOLITES, sition to take angles of altitudes or depressions. The divisions by the nonius are to five minutes. THE FOLLOWING ACCOUNT OF THEODOLITES IS ADDED BY THE EDITOR. It generally happens, that the observer has occa- sion to take the vertical and horizontal angles at the same time, by portable as well as by larger theodoli- tes ; the following is, therefore, recommended as the most complete and portable instrument hitherto made, and is in truth almost the best theodolite in minia- ture. Its construction renders it somewhat more expensive than those before described. It is the one that the late author alluded to in a note, page 315 of the former edition of this book, but had not time to describe it. Fig. 7, plate 14, is a representation of the instru- ment. The graduated limb and index plate A, A, are about four inches in diameter, and move by rack and pinion B; it reads off by means of the nonius to three minutes of a degree ; If the observer should not object to very fine divisions, it may be to two mi- nutes of a degree. The achromatic telescope C is about six inches in length, and contains a small spirit bubble at C, partly sunk into the tube ; it turns upon a long axis, and is moved very accurately by rack and pinion on the arc atD. This arc is necessarily of a short length, but admits about 30 degrees motion on each side of O, for altitudes or depressions. The staff"^ which from one piece opens into a tripod, is about five feet in length, and has the parallel plates of ad- justment at the top. A small screw from these screws into a socket under the limb A, and by an external rim of metal, the horizontal motion only of the theo- dolite is produced, when the plates are properly set by the screws. The telescope rests upon a cradle, and by opening the two semicircles tom to the instrument there is another conical brass socket, 1 inches in diameter, that turns easily ou that in the cenire of the octagon underneath. In the cover of this socket is an hole concentric to the instrument, to admit the thread or wire to pass, which suspends the plummet at L. There is a small box with a winch handle at M, that serves occasion- ally to raise or lower the plummet. To the three feet there are screws, such as at N, for levelling tbo CY RAMSDEN. 233 instrument ; and also three blocks of box wood, and three brass conical rollers under the feet screws, tixed to the lower surface of the mahoc^any, to give the whole a perfectly easy motion ; O, O, O, are three of the four screws attached to tiic octagonal plain, for accurately centering the instrument by the plummet. P and Q represent two positions of screws to give -a circular motion to the entire ma- chine, but these having been found to act by jerks from the great weight, another apparatus or clamp, see jig. 2, was adjusted, attached to the curbs, con- sisting of a brass cock fixed to, and projecting from the curb of the instrument ; the cock being acted upon by two screws working in opposite directions, and which are clamped to the curb of the octagon. The curb upon which the feet of the instrument rest, carries the mahogany balustrade R R, fitted to receive a mahogany cover, that guards the whole in- strument. In this cover are four small openings, one for each of the vertical microscopes S, S, one for the clamp of the circle, and one for the socket of the //boy^'s joint. This cover secures the circles and its cones from dirt, and serves conveniently for laying any thing upon, that may be wanted near at hand ; and particularly lanthorns used at night, for reading off the divisions on the limb of the instrument. There is a lower telescope T, lying exactly under the centre of the instrument, and directed through one of the openings on the balustrade, and used only for terrestrial objects, requires but a small elevation, and has an axis of ] 7 inches in length, supported by the braces attached to the feet. It is moved by rack- work, by turning the pinion at V. There is a small horizontal motion that can be given to the right hand of the axis of the end of this telescope. The whole instrument being nicely levelled, the upper telescope at zero, aiul likewise on its object, the lower telescope by help of this adjustment is brought accurately to the same object, from the point of commencements fron) which the angles are to be measured. 234 THE GREAT THEODOLITE, There are three flat arms, one of which is repre- sented at U, fixed by screws to the edge of the bell- metal plate. These arms are also braced to the feet of the instrument, rising as they project outwards to- wards the circumference of the circle, going beyond it about 1:^ inches. One arm, lying directly over one of the feet, is that to which is attached wheels and screws moved by Hook'ii joint, not seen in the figure, and also a clamp to the circle. It is this that produces the accurate motion of the circle. The other two arms, one of which also lies over a foot, and the other directly opposite to it, become the dia- meter of the circle, having their extremities termina- ted on a kind of blunted triangular figure, forming the bases of pedestals, whereon stand the vertical microscopes S, S. The arms, braces, base, &c. are every where pierced, in order to lessen weight with- out diminishing strength. The angles are not read off in this instrument by a nonius as common to others, but with microscopes, and which form the most essential part of the in- strument. But a short account of them can be given here, an adequate idea cnn only be obtained by a re- ference to the FhilosopMcal' Transactions y page 145 and 14g. That horizontal microscope for the ver- tical angles has been already mentioned. The two vertical ones S, S, are used for reading off the divi- sions on the opposite sides of the circle immediately under them. Each microscope contains two slides, one over the other, their contiguous surfaces in the foci of the eye glasses. The upper one is a very thin brass plate, at its lower surface is attached a fixed wire, having no other motion than what is ne- cessary for adjustment, by the left hand screw to its proper dot, as hereafter to be explained. The other slide is of steel of one entire piece, directly under the former, of sufficient thickness to permit a.microme- Xcv screw of about 'J^l threads in an inch to be formed of it. To its upper surface is fixed the immoveable wire, which changes its place by the motion of the BY RAMSDEV. 235 micrometer head. Tliis head is divided into 6o equal part*;, each of which represents one second or angular motion of the telescope. This steel slide is attached by a chain to the spring of a watch coiled up within a IE\ S QUADRANT. 241 nonius scale lie close to ihe divisions ; it is also fur- bished wirli ;i scrrw fn lix ilic index in any desired isiiion. The best insUiinients have an adjnsiini^ screw fit- led to tlie index, that it may be moved more slowly, and with j^reater regularity and accuracy than by the hand. It is pro[)cr however to observe, that the in- dex must be previously fixed near its right j)osition by the above-mentioned screw, before the adjusting tjcrew is put in motion. See B C, fi^. 4. The circular arcs on the arc of the quadrant are drawn from the centre on which the index turns : the smallest excontricity in the axis of the index would be productive of considerable errors. The position of the index on the arc, after an ob- servation, points out the number of degrees and mi- nutes contained in tlie observed angle. Of the index-glass E. Upon the index, and near its axis, is fixed a plain speculum, or mirror of glass, quicksilvered. It is set in a brass frame, and is placed so that the face of it is perpendicular to the plain of the instrument ; this mirror being fixed to the index, moves along with it, and has its direction changed by the motion thereof. This glass is designed to receive the image of the sun, or any other object, and reflect it upon cither of the two horizon glasses F and G, according to the nature of the observation. The brass frame with the glass is fixed to the in- dex by the screw c ; the other screw serves to replace it in a perpendicular position, if by any accident it has been deranged, as will be seen hereafter. The index glass is often divided into two parts, the one silvered, the other black, with a small screen in front. A single black surface has indeed some ad- vantages ; but if the glasses be well selected, there is little danger to be apprehended of error, from a want of parallelism ; more is to be feared from the sur- faces not being flat. R 24^ DESCRIPTION OF Of fhe horizon glasses F, G, On the radius AB of the octant, arc two small speculums. '1 he sur- face of the upper one is parallel to the index glass, when the counting division cf the index is at o on the arc ; but the surface of the lower one is perpen- dicular to the index glass, when the index is at o de- grees on the arc : these mirrors receive the reflected ■ rays from the object, and transmit them to the ob- server. The horizon glasses are not entirely quicksilvered; the upper one I, is only silvered on its lower part, or that half next the quadrant, the other half being transparent, and the back part of the frame is cut away, that nothing may impede the sight through the vnsilvered part of the glass. The edge of the toil of this glass is nearly parallel to the plane of the instru- ment, and ought to be very sharp, and without a flaw. The other horizon glass G is silvered at both ends; in the middle there is a transparent slit, through which the horizon, or other object^ may be seen. Each of these glasses is set in a br^ss frame, to which there is an axis ; this axis passes through the wood-work, and is fitted to a lever on the under side of the quadrant ; by this lever the glass may be turn- ed a few degrees on its axis, in order to set it paral- lel to the index glass. The lever has a contrivance to turn it slowly, and a button to flx it. To set the glasses perpendicular to the plane of the quadrant, there are two sunk screws, one before and one be- hind each glass ; these screws pass through the plate, on which the frame is fixed, into another plate, so that by loosening one, and tightening the other of these screws, the direction of the frame, with its mir- ror, may be altered, and thus be set perpendicular to the plane of the instrument. Of the shadesy or dark glasses, K. There are two red or dark glasses, and one green one ; they ar* HADLBY*S ftUADRANT. ^43 t^SRd to prevent the bright rnys of the sun, or the glare of the moon, from hiiriinv the eye at the time of oh-^ci vation. 1 hoy are each o\ them set in a brass frame, whieh turns on a centre, so that they may he used separately, or togetlier, as the liright- ness of ilie sun n)ay require. The green gla.ss may be used also alone, if the sun be very faint ; it is also used for taking the altitude of the moon, and in ascertaining her distance troni a fixed .star. When these glasses are used for the fore observa* tion, they are fixed as at K in Ji^. 1 ; when used for the back observation, thcv are removed to N. Of fie livo sight vanes H, I. Each of these vanes is a perforated piece of brass, designed to direct the sight parallel to the plane of the quadrant. That whieh is fixed at I is used for the fore, the other for the back observation. The vane I has two holes, one exactly at the height of the quicksilvered edge of the horizon glass, the other somewhat higher, to direct the sight to the mid- dle of the transparent piirt of the mirror, for those objects which are bright enough to be reflected from the unsilvered part of the mirror. Of ihc divisions on the limb of the quadrant^ and of the nonius on the index. For a description of these divisions, see page 120. Directions to hold the instrument. It is recom- mended to support the weight of the instrument by tiie right hand, and reserve the left to govern the in- dex. Place the thumb of the right hand against the edge of the quadrant, under the swelling part on whieh the fore sight I stands, extending the fingers across the back of the quadrant, so as to lay hold on the opposite <^^^i:i^ placing the fore finger above, and the other fingers below the swelling part, or near the fore horizon glass ; thus you may support the instru- ment conveniently, in a vertical position, by the right hand oulv ; by resting the thumb of the left R 2 -244'" ADJUSTMENTS OF hand against the side, or the fingers against the mid- dle bar, you may move the index gradually either way. In the back observation, the instrument should be supported by the left hand, and the index be go- verned by the right. Of the axis of 'vision, or live of sight. Of the two objects which are marie to coincide by this instru- ment, the one is seen directly by a ray passing through, the other, by a ray reflected from, the same point of the hcjrizon glass to the eye. This ray is called the visual ray ; but when it is considered merely as a line drawn from the middle of the horizon glass to the eye-hole of the sight vane, it is called the axis of vision. The axis of a tube, or telescope, used to direct the sight, is also called the axis of vision. The quadrant, if it be held as before directed, may- be easily turned round between the fingers and thumb, and thus nearly on a line parallel to the axis of vision ; thus the plane of the quadrant will pass through the two objects when an observation is made, a circumstance absolutely necessary, and which is more readily effected when the instrument is furnished with a telescope : within the telescope are two parallel wires, which by turning the eyeglass tube may be brought parallel to the plane of the quadrant, so that by bringing the object to the mid- dle between them, you are certain of having the axis of vision parallel to the plane of the quadrant. OP THE NECESSARY ADJUSTMENTS. It is a^pcculiar excellence of IJadley's quadrant, that the errors to which it is liable are easily detected, and scon rectified ; the observer may, therefore, if he will be attentive, always put his instrument in a state iit for accurate observation. The importance of this instrument to navigation is self-evident ; yet HADLEY*S QUADRANT. 245 much of this importance depends on the accuracy with which it is made, and the necessary attention of the observer ; and one would hardly think it possible that any obscr\er would, to save the trilling sum of one or two guineas, prefer an in)[)erfect instrument to one that was rightly constructed and accurately made; or that any consideration should induce him to neglect the adjustments of an instrument, on whose truth he is so highly interested. But such is the nature of man ! he is too apt to be lavish on bau- bles, and penurious in matters of consequence; ac- tive about trifles, indolent where his weliare and hap- ])iness are concerned. The adjustments for the fore observation arc, I . To set the fore horizon glass pa- rallel to the index glass; this adjustment is of the utmost importance, and should always be made pre- vious to actual observation. The second is, to see that the plane of this glass is perpendicular to the plane of the quadrant. 3. To see that the index glass is perpendicular to the plane of the instrument. To adjust th-e fore horizon glass. This rectifica- tion is deemed of such importance, that it is usual to speak of it as if it included all the rest, and to call it ADJUSTING THE INSTRUMEXT. It is SO tO plaCC the horizon glass, that the index may shew upon the arc the true angle between the objects : for this pur- pose, set the index line of the nonius exactlv against o on the limb, and fix it there by the screw at the un- der side. Now look through the sight I at the edge of the sea, or some very distant well-defined small object. The edge of the sea will be seen directly through the unsilvered part of the glass, but by re- flection in the silvered part. If the horizon in the silvered part exactly meets, and forms one continued line with that seen through the unsilvered part, then is the instrument said to beadju'^^ted, and the horizon glass to he parallel to the index glass, hut if the ho- rizons do not coincide, then loosen the milled nut on the under side of the quadrant, and turn the ho- 245 ADJUSTMENTS OF rizon glass on its axis, by means of the adjusting le- ver, till you have made them perfectly coincide ; then fix the lever firmly in the situation thus ob- tained, by tightening the milled nut. This adjust- ment ought to be repeated before and after every as- tronomical observation. So important is this rectification, that experienced observers, and those who are desirous of being very accurate, will not be content with the preceding uiode of adjustment, but adopt another method, which is usually called finding the index error ; a method pre- ferable to the foregoing both for ease and accuracy. To find the index error. Instead of fixing the in- dex at o, and moving the horizon glass, till the image ofa distant object coincides with the same object seen directly ; let the horizon glass remain fixed, and move the index till the. image and object coincide ; then ob- serve whether the index division on the nonius agrees, with the o line on the arc ; if it does not, the num- ber by which they difi^er is a quantity to be added to the observation, if the index line is beyond the o on the limb ; but if the index line of the nonius stands between o and QO degrees, then this error is to be subtracted from the observation.* We have already observed, that the part of the arc beyond o, towards the right hand, is called the arc of excess ; and that the nonius, when at that part, must be read the contrary wav. or, which is the same thing, you may read them off in the usual way, and then their complement to 20 min. will be the real number of degrees and minutes to be added to the observation. To make the index glass and fore horizon glass per- * This adjustment, may be made more accurately, or the error bet- ter found, by using the sun instead of the horizon ; but this method requires another set of dark glasses, to darken the direct rays of the sun ; such a set is applied to the best instruments, and this method of adjustment is explained in the following description of the sextuat. hadley's quadrai*t. Q.47 petif/irnhir to the plane of the instrument. Though these adjustments are neither so necessary nor im- portant a'^ the prccc(hng one ; vet, as after being once performed, they do not require to be repeated for a considerable time, and as they add to the accu- racy of ob:5ervation, thcyousfht not t<>bc neglected ; and further, a knowledge of them enables the ma- riner to examine and form a proper judgment of his instrument. To adjust the index glasR. This" adjustment con- sists in setting the plane of the index glass perpendi- cular to that of the instrument. Method 1 . By means of the two adjusting tool?, represented at fig. 2 and 3, which are two wooden frames, with two lines on each, exactly at the same distance from the bottom. Place the quadrant in an horizontal position on a table, put the index about the middle of the arc, turn back the dark glasses, place one of the above-men- tioned tools near one end of the arc, the other at the opposite end, the side with the lines towards the in- dex glass ; then look down the index glass, directing the sight parallel to the plane of the instrument, you \vill see one of the tools by direct vision, the other by reflection in the mirror ; by moving the index a little, they may be brought exactly together. If the lines coincide, the mirror is riu;htly fixed ; if not, it must be restored to its proper situation by loosening the screw c, and tightening the screw d ; or, ince versa, by tightening the screw c, and releasing the screw d. Method 2. Hold the quadrant in an horizontal position, with the index glass close to the eye ; look nearly in a right line down the glass, and in such a manner ; that you may see the arc of the quadrant by direct view, and by reflection at the same time. If they join in one direct line, and the arc seen by reflection forms an exact plane with the arc seen bv direct view, the glass is perpendicular to the plane o^ R 4 24S ADJUSTMENTS OP the quadrant ; if not, the error must be rectified by altering the position of the screws behind the frame, as directed above. To ascertain zvhether the fore horizon glass he per- pendicular to the plane of the itistrument. Having ad- justed the index and horizon glasses agreeable to the foregoing directions, set the index division of the nonius exactly against o on the limb ; hold the plane of the quadrant parallel to the horizon, and observe the image of any distant object at land, or at sea the horizon itself; if the im ige of the horizon at the edge of the silvered part coincide with the object seen directly, the glass is perpendicular to the plane of the instrument. If it fall above or below, it must be ad- justed. If the image seen by reflection be higher than the object itself seen directly, release the fore screw and tighten the back screw ; and, vice versa, if the image seen by reflection be lower, release the back screw and screw up the fore one ; and thus pro- ceed till both are of an equal height, and that by mov- ing the index you can make the image and the ob- ject appear as one. Or, adjust the fore horizon glass as directed in page 262 ; then incline the quadrant on one side as much as possible, provided the horizon continues to be seen in both parts of ihe glass. If, when the in- strument is thus inclined, the edge of the sea con- tinues to form one unbroken line, the quadrant is perfectly adjusted ; but if the reflected horizon be separated from that seen by direct vision, the specu- lum is not perpendicular to the plane of the quad- rant. And if the observer is inclined to the right, with the face of. the quadrant upwards, and the re- flected sea appears higher than the real sea, you must slacken the screw before the horizon glass, and tighten that which is behind it ; but if the reflected sea a})pears lower, the contrary must be performed. Care must be always taken in these adjustments to loosen one screw before the other is screwed up, and HADLBY S aUADRANT. QIQ to leave the adjusting screws tight, or so as to draw with a moderate force against each other. This adjustment may be also made by the sun, moon, or star ; in this case the (|uach-ant may be held in a vetrical position ; if the image seen by reflection appciirs to the right or left of the object seen directly, then the glass must be adjusted as before by the two screws. OF THE ADJLSTMEXTS FOR THE BACK OUSER- VATIOX. The back observation is so called, because the back is turned upon one of the two objects whose angular distance is to be measured. The adjustment consists in making the reflected image of the object behind the observer coincide with another seen directly before him, at the same time that the index division of the nonius is directly against the o division of the arc. The method, therefore, of adjusting it consists in measuring the distance of two objecis nearly 180 de- grees apart from each other ; the arc passing through each object must be measured in both its parts, and if the sum of the parts be 36o degrees, the speculums are adjusted ; but, if not, the axis of the horizon glass must be moved till this sum is obtained. Set the index as far behind o as twice the dip* of the horizon amounts to ; then look at the horizon through the slit near G, and at the same time the op- posite edge of the sea will appear by reflection in- verted, or upside down. Ly moving the lever of the axis, if necessarv, the edges may be made to coincide, and the quadrant is adjusted. There is but one position in which the quadrant * This is according to the height of the observer's eye above the ^a. See Hobcrtsons Nafi^atwn, 250 DEFECTS 1^ BACK OBSERVATION'S. can be held with the limb downwards, without caus- ing the reflected horizon to cross the part seen by direct vision. If, on trial, this position be found to be that in which the plane of the quadrant is perpendicular to the horizon, no farther adjustment is necessary than the fore-mentioned one; but if the horizons cross each other when the quadrant is held uprio-ht, ob- serve which part of the reflected horizon is fowest. If the right-hand part be lowest, the sunk screw which is before the horizon glass must be tightened after slackening that which is behind the glass ; but if the right hand is highest, the contrary must be performed : this adjustment is, however, of much less importance than the preceding, as it does not so much affect the angle measured. INCONVENIEXCIES AND INACCURACIES OF THE BACK OBSERVATION. The occasions on which the back observation is to be used are, when the altitude of the sun or a star is to be taken, and the fore horizon is broken by ad- jacent shores ; or when the angular distance between the moon and sun, or a star, exceeds go degrees, and is required to be measured for obtaining the longi- tude at sea : but there are objections to its use in both cases ; for if a known land lie a few miles to the north or southward of a ship, the latitude may be known from its bearing and distance, without havino- recourse to observation : and again, if the distance of the land in miles exceed the number of minutes in the dip, as is almost always the case in coasting along an open shore, the horizon will not be broken, and the fore observation may be used ; and lastly, if the land be too near to use the fore observation^ its ex- treme points will in general be so far asunder, as to prevent the adjustment, by taking away the back ho- rizon. In the case of measurino: the anefular dis- HADLEY S SEXTANT. 251 tanccs of the heavenly bodies, the very great accu- racy required in these observations, makes it a matter of importance that the adjustments should be well made, and frccjuently examined into. iJut the quan- tity of he dip is varied by the pitching and rolling of the ship ; and this variation, which is })crccptible in the measuring altitude by the fore observation, is doubled in the adjustment for the back observation, and amounts to several minutes. It is likewise ck- ceeding difficult, in a ship under way, to hold the quadrant for any length of time, so that the two ho- rizons do not cross each other, and in the night the edge of the sea cannot be accurately distinguished. All these circumstances concur to render the adjust- ment uncertain ; the fore observation is subject to none of these inconveniencies.* DESCRIPTION, USE, AND METHOD OF ADJUSTING HADLEY's SEXTANT. As the taking the angular distances of the moon and the sun, or a star, is one of the best and most accurate methods of discovering the longitude, it was necessary to enlarge the are of the octant to the sixth part of a circle ; but as the observations for deter- mining the longitude must be made with the utmost accuracy, the framing of the instrument was also al- tered, that it might be rendered more adequate to the solution of this important problem, llcncc arose the present construction of the sextant, in the description of which, it is presumed that the foregoing pages have * Nicholson's Navigator's Assistant. 252 DESCRIPTION OF been read, as otherwise we should be obh'ged to re- peat the same observations. Sextants are mostl}' executed (some trifling varia- tions excepted) on two plans ; in the one, all the ad- justments are left to the observer : he has it in his power to examine and rectify every part of his in- strument. This mode is founded on this general principle, that the parts of no instrument can be so iixed as to remain accurately in the same position they had when first put out of the maker's hands ; and that therefore the principal parts should be made moveable, that their positions may be examined and rectified from time to time by the observer. In the second construction, the principal adjust- ment, or that of the horizon glass for the index error, is rejected ; and this rejection is grounded upon two reasons : 1. That from the nature of the adjustment it fre- quently happens that a sextant will alter even during the time of an observation, without any apparent cause whatever, or without being sensible at what pe- riod of the observation such alteration took place, and consequently the observer is unable to allow for the error of adjustment. 2. That this adjustment is not in itself sufficiently exact, it being impossible to adjust a sextant with the same accuracy by the coincidence of two images of an object, as by. the contact of the limbs thereof; and hence experienced and accurate observers have always directed the index error to be found and al- lowed for, which renders the adjustment of the hori- zon glass in this direction useless ; for it is easy to place it nearly parallel to the index glass, when the instrument is made, and then to fix it firmly in that position by screws. The utility of this method is confirmed by experience : many sextants, whose in- dices had been determined previous to their being carried out to India, have been found to remain the same at their return. HADLEY S SEXTANT. 253 Notwitlistanding the probable certainty of the ho- rizon glass remaining permanently in its situation, the observer ought from time to time to examine the index error of his instrument, to see if it remains the same; or make the pro[)cr allowances for it, if any alteration should have taken place. One material point in the formation of a sextant is so to construct it, that it may support its own weight, and not be liable to bend by any inclination of the plane of the instrument, as every flexure would alter the relative position of the mirrors, on which the de- termination of an angle depends. Besides the errors now mentioned, to which the sextant or quadrant, &:c. is liable, there is another which seems inseparable from the construction and materials of the instrument, and have been noticed by several skilful observers. It arises from the bending and elasticity of the index, and the resistance it meets with in turning round its centre. To obviate this error, let the observer be careful always to finish his observations, by moving the index in the same direction which was used in setting it to o for adjusting, or in finding the index error. The direction of the motion is indeed indiHbrent ; but as the common practice in observing is to finish the observations by a motion of the index in that di- rection which increases the angle, that is, in the fore observation from o towards 90, it would be well if the observer would adopt it as a general rule, to finish the motion of the index, by pushing it from him, or turning the screw in that direction which carries it farther from him. By finishing the motion of the index, we mean that the last motion of the index should be for some minutes of a dcj-ree at least in the required direction. [ 254 ] DESCRIPTION OF THE SEXTANT. Fig. 4, plate IQ, represents the best sextant of brass, so framed, as not to be liable to bend. The arc A A, is divided into ]20°, each degree is divided into three parts, or 20 minutes,which are again subdi- vided by the nonius into every half minute, or 30 se- conds : see the nature of the nonius, and the gene- ral rule for estimating the value thereof, in the pre- ceding part of these Essays. Every second division or minute on the nonius, is cut longer than the in- termediate ones. The nonius is numbered at every fifth of these longer divisions, from the right towards, the left, with 5, 10, 15, and 20, the first division to- wards the right hand being to be considered as the index division. I A the radius of the sextant is now generally made about 8 or 10 inches, and the arc A A is best of silver. In order to observe with accuracy, and make the images come precisely in contact, an adjusting screw B is added to the index, which may be moved with greater accuracy than it can by hand ; but this screw- does not act until the index is fixed by the finger *crew C. Care should be taken not to force the ad- justing screw when it arrives at either extremity of its- adjustment. When the index is to be moved any considerable quantity, the screw Cat the back of the sextant must be loosened ; but when the index is brought nearly to the division required, this back screw should be tightened, and then the index may be moved gradually by the adjusting screw. A small shade is sometimes fixed to that part of the index where the nonius is divided, this being covered witb white paper, reflects the light strongly upon the divisions. There are four tinged glasses at D, each of whicb* is set in a separate frame turning on a centre : they are used Lo screen and save the eye from the bright- hadley's sextant. 255 ncss of the solar image, and the glare of the moon, and may be used separately, or together, as occasion requires. There are three more such glasses placed behind » the horizon glass at V., to weaken the rays of the sun or moon, when they arc viewed directly through the horizon glass. The paler glass is sometimes used in observing altitudes at sea, to take oH' iho strong glare of the horizon. The trame of the index glass I, is firmly fixed by a strong cock to the centre plate of the index. The horizon glass F, is fixed on a frame that turns on the axes or pivots, which move in an exterior frame : the holes in which the pivots move may be tightened by four screws in the exterior frcime ; G is a screw by which the horizon glass may be set perpendicular to the plane of the instrument ; should this screw be- come loose, or move too easy, it may be easily tight- ened by turning the capstan headed screw H, which is on one side the socket, through which the stem of the finger screw passes. The sextant is furnished with a plain tube. Jig, 7, without any glasses ; and to render the objects still more distinct, it has also two achromatic telescopes, one, Jig. 5, shewing the objects erect, or in their na- tural position ; the longer one, Jig. 6, shews them in- verted. It has a large field of view, and other ad- vantages ; and a little use will soon accustom the ob- server to the inverted position, and the instrument will be as readily managed by it as by the plain tube only. By a telescope, the contact of the images if more perfectly distinguished ; and by the place of the images in the field of the telescope, it is easy to per- ceive whether the sextant is held in the proper plane for observation. By sliding the tube that contains the eye glasses in the inside of the other tube, the image of the object is suited to different eyes, and made to appear perfectly distinct and well defined. q,d6 description of The telescopes are to be screwed into a circular ring at K ; this ring rests on two points against an exterior ring, and is held thereto by two screws : by ^ turning one, and tightening the other, the axis of the telescope may be set parallel to the plane of the sex- tant. The exterior ring is fixed on a triangular brass stem that slides in a socket, and by means of a screw at the back of the sextant, may be raised or lowered so as to move the centre of the telescope to point to that part of the horizon glass which shall be judged the most fit for observation. Fi^. 8, is a circular head, with tinged glasses to screw on the eye end of either of the telescopes, or the plain tube. The glas- ses are contained in a circular plate, which has four holes ; three of these are fitted with tinged glasses, the fourth is open. By pressing the finger against the projecting edge of this circular plate, and turning it round, the open hole, cr any of the tinged glasses, may be brought between the eye glass of the teles- cope and the eye. Fig. 9, a small screw driver. Fig. 10, a magnify- ing glass, to read off the divisions by. Tofaidthe index error of the sextant. To findthe index error, is, in other words, to shew what number- of degrees and minutes is indicated by the nonius, when the direct and reflected images of an object co- incide with each other. The most general and most certain method of as- certaining this error, is to measure the diameter of the sun, by bringing the limb of its image to coin- cide Vk'ith the limb of the sun itself seen directly, both on the quadrantal arc, and on the arc of excess. If the diameter takc/U by moving the index forward on the quadrantal arc be greater than that taken oa the tire of excess, then half the diflercncc is to be subtracted ; but if the diameter taken on the arc of excess be greater than that by the quadrantal arc, half the dilierence is to be added. If the numbers hadley's sextant. 157 shewn on the arc be tlic same in both cases, the glas- ses arc truly pnrallel, aiul there is no index error ; but it" the numbers bo dilFcrcnt, then half the diti'er- cnee is the index error.* It is however to be observed, that when the index is on the arc of excess, or to the ri^^ht of o, the com- plement of the numbers shewn on the nonius to 20 ought to be set down. Several observations of the sun's diameter should be made, and a mean taken as the result, which will give the index error to very great exactness. KxuinpL'. Let the numbers of minutes shewn by the index to the right of zero, when the limbs of the two images are in contact, be 20 minutes, and the odd number shewn by the nonius be 5, the comple- menc of this to 20 is 1.5, which, added to 20, gives 35 minutes ; and, secondly, that the number shewn by the index, when on the left of zero, and the oppo- site limbs are in contact, be 20 minutes, and by the nonius g' 30", which makes together 20f 30" ; tht« subtracted from 35' gives 5' 30", which divided by 2, gives 2 45" for the index error ; and because the greatest of the two numbers thus found, was, when the index was to the right of the o, this index error must be added to the number of degrees shewn on the arc at the time of an observation. To set the horizon glass perpendicular to the plane of the sextant. Direct the telescope to the sun, a star, or any other well-dt fined object, and bring the direct object and reflected image to coincide nearly with each other, by moving the index ; then set the two images parallel to the plane of the sextant, by turn- ing the screw, and the images will pass exactly over * 111 other words the difTerence between the degree and minute shewn by the index : first, when the lower reflected limb of the sua is exactly in contact with tlie.^uppe^limb ot the sun ; and secondly, ■when the upjier edge of the im:ige is in contact with the lower edge of th^ object, divided by 2, will bo the index error, s 258 DESCRIPTION OF each other, and the mirror will then be adjusted in this direction. I'o set the axis of the telescope parallel to the plane of the sextant. We hnve already observed, that in measuring angular distances, the line of sight, or plane of observation, should be parallel to the plane of the instrument, as a deviation in that respect will occasion great errors in the observation, and this is most sensible in large angles : to avoid these, a teles- cope is made use of, in whose field there are placed two wires parallel to each other, and equidistant from tlie centre. Ihese wires may be placed parallel to the plane of the sextant, by turning the eye glass tube, and, consequently, by bringing the object to the middle between them, the observer may be cer- tain of having the axis of vision parallel to the plane of the quadrant. To adjust the telescope. Screw 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 instrument ; then seek two objects, as the sun and moon, or the moon and a star, whose distance should, for this purpose, exceed gO degrees ; the dis- tance of the sun and moon is to be taken great, be- cause the same deviation of the axis will cause a greater error, and will consequently be more easily discovered. Move the index, so as to bring the limbs of the sun and moon, if they are made use of, exact- ly in contact with that vi'ire which is nearest to the plane of the sextant ; fix the index there ; then^ by altering a little the position of your instrument, make the images appear on the wire furthest from the sex- tant. If the nearest limbs be now precisely in con- tact, as they were before, then the axis of the teles- cope is in its right situation. But if the limbs of the two objects appear to separate at the wire that is furthest from the plane of the instrument, it shews that the object end of the telescope inclines towards IIADLEYS SEXTANT. 25^ the pl;ine of the instrument, and must be rectified by tiajhtcning the screw nearest i!»c sextant, having pre- viously unturned the screw furthest from it. If the images overlap each other at the wire furthest from the sextant, the olijcct endof the telescope is inclined from the plane of the sextant, and the highest screw must be turned towards the right, and the lowest screw towards the loft : by repeating this operation a few times, the contact will be precisely the same at both wires, and consequently the axis of the telescope will be parallel to the plane of the instrument. To exaini/ie the glasses of a sextant, or quadrant. 1. To find whether the two surfaces of any one of the reflecting glasses be parallel, apply your eye at one end of it, and observe the image of some object reflected very obliquely from it ; if that image ap- pears singly, and well defined about the edges, it is a proof that the surfaces are parallel ; on the contrary, if the edge of the reflected image appears misted, as if it threw a shadow from it, or separated like two edges, it is a proof that the two surfaces of the glass are inclined to each other: if the image in the spe- culum, particularly if that image be the sun, be view- ed through a small telescope, the examination will be more perfect. 2. To find whether the surface of a reflecting glass be plane. Chuse two distant objects, nearly on a level with each other ; hold the instrument in an ho- rizontal position, view the left hand object directly through the transparent part of the horizon glass, and move the index till the reflected image of the other is seen below it in the silvered part ; make the two images unite just at the line of separation, then turn the instrument round slowly on its own p)ane, so as to make the united images move along the line of se- paration of the horizon glass. If the images con- tinue united without receding from each other, or va- rying their respective position, the reflecting surface S 2 260 OF HAD ley's sextant, is a good plane. The observer must be careful that he does not give the instrument a motion about the axis of vision, as that will cause a separation if the planes be perfect. 3. To find if the two surfaces of a red or darken- ing glass are parallel and perfectly plane. It is diffi- cult, nay, almost impossible, to procure the shades perfectly parallel and good ; they will therefore, ac- cording to their different combinations, give differ- ent altitudes or measures of the sun and moon. The best way to discover the error of the shades, is to take the sun's diameter with a piece of smoaked glass before your telescope, all the vanes being re- moved ; then take away the smoaked glass, and view the sun through each shade and the several combi- nations thereof. If the two images still remain in contact, the glasses are good ; but if they separate, the error is to be attributed to the dark glasses, which must either be changed, or the error found in each combination must be allowed for in the observations. If you use the. same dark glasses in the observation as in the adjustment, there will be no error in the ob* served angle.* OP hadley's sextant, as used in survey- ing. No instrument can be so conveniently used for taking angles in maritime surveying as Hadley's sex- tant. It is used with equal facility at the mast head, as upon deck, by which its sphere of observation is much extended : for, supposing many islands to be visible from the mast head, and only one from deck, no useful observation can be made by any other in- ■* The dark glasses are generally left free to turn in their cells : so that, after one observation has been made, turning either of them half round, another observation may be had^ and iialf.thc difference thus given is the error of the glass. Edit, AS USED IN SURVEYING. 26l strumcnt. But by this, angles may be taken at the mast head from the one visible ol)iect with great ex- actness ; and further, taking angles from heights, as hills, or a ship's mast head, is almost the only way of exaetly describing the ligurc and extent of the shoals. It has been objected to the use of Hadley's sextant for surveying, that it docs not measure the horizon- tal angles, by which alone a plan can be laid flown. This observation, however true in theory, may be ob- viated in practice by a little caution. If an angle be measured between an object on an elevation, and another near to it in a hollow, the dif- ference between the base, which is the horizontal an- gle, and the hypothenusc, which is the angle ob- served, may be very great ; but if these objects are measured, not from each other, but from some very distant object, the difierence between the angles of each from the very distant object, will be very near the same as the horizontal angle. This may be still further corrected, by measuring the angle not be- tween an object on a plane and an object on an ele- vation, but between the object on a plane and some object in the same direction as the elevated object, of which the eye is sufficiently able to judge. How to observe the horizontal angle, or angular dis- tance, helzveen tzvo objects. First adjust the sextant, and if the objects are not small, fix on a sharp top, or corner, or some small distinct part in each to observe; then, having set the index to o dog. hold the sextant horizontally, as above directed, and as nearly in a plane passing through the two objects as you can ; direct the sight through the tube to the left hand ob- ject, till it is seen directly through the transparent part of the horizon glass ; keeping that object still in sight, then move the index till the other object is seen by reflection in the silvered part of the horizon glass ; then bring both objects together by the index, S 3 262 OF HADLEY's SEXTAXTj and by the inclination of the plane of the sextant when necessary, till they unite as one, or appear to joia in one vertical line in the middle of the line which divides the transparent and reflecting parts of the horizon glass; the two objects thus coinciding, or one appearing directly below the other, the index then shews on the limb the angle which the two ob- jects subtend at the naked eye. This angle is always double the inclination of the planes of the two re- flecting glasses to one another ; and, therefore, every degree and minute the index is actually moved from o, to bring the two objects together, the angle sub- tended by them at the eye will be twice that number of degrees and minutes, and is accordingly numbered so on the arc of the sextant ; which is really an arc of 60 degrees only, but graduated into ICO degrees, as before observed. The angle found in this manner between two ob- jects that are near the observer, is not precise ; and may be reckoned exact only when the objects are above half a mile off. For, to get the angle tridy ex- act, the objects should be viewed from the centre of the index glass, and not where the sight vane is placed ; therefore, except the objects are ?o remote, that the distance between the index glass and sight vane vanishes, or is as nothing compared to it, the angle wrll not be quite exact. This inaccuracy in the angle between near objects is called the parallax of the instrument, and is the angle which the distance between the index glass and sight vane subtends at any near object. It is so small, that a surveyor will seldom have occa-ion to regard it; but if it shall happen that great accuracy is required, let him choose a distant object exactly in a line with each of the near ones, and take the angles between them, and that ■will be the true angle between the near objects. Or, observe the angle between near objects, when the sextant has been first properly adjusted by a distant AS USED IN SURVEYING. ^63 object ; then adjust it by the left hand object, which will bring the index on the arc of excess beyond o degrees ; add that excess to the angle found between the objects, and the sum will he the true angle be- tween thcni. If one of the objects is near, and the other distant, and no remote object to be found in a line with the near one, adjust the sextant to the near object, and then take the angle between them and the parallax will be ibund. Exiiiiiplt'. To measure the horizontal angle AB C, Jig. 19, pliile (), with the sextant. 1. bet up such marks at A and C, as may be seen when you are standing at B. 2. Set o on the index to coincide with o on the quadrantal arc. 3. Hold the sextant in an horizontal position, look through the sight and horizon glass at the mark A, and ob- serve whether the image is directly under the object ; if not, move the index till they coincide. 4. If the index division of the nonius is on the arc of excess, the indicated quantity is to be added* to the observed angle ; but if it be on the quadrantal arc, the quan- tity indicated is to be subtracted : let us suppose five to be added. 5. Now direct the sight through the transparent part of the horizon glass to A, keep that object in view, move the index till the object C is seen by reflection in the silvered part of the horizon glass. 6, The objects being now both in view, move the in- dex till thev unite as one, or appear in one vertical line, and the index will shew the angle subtended at the eye by the two objects : suppose 75.20, to which add 5 for the index error, and you obtain 75.25, the angle required; if the angle be greater than 120*', which seldom happens in practice, it may be subdi- vided by marks, and then measured. No instrument can be more convenient or expedi - tious than the sextant, for se ting of offsets. Adjust the instrument, and set the index to ()0 degrees ; walk along the station line with the octant in your S 4 164 NEW POCKET BOX SEXTANT. &C. hand, always directing the sight to the farther station staff ; let the assistant walk along the boundary line ; then, if you wish to make an ofl^set from a given point in the station line, stop at that place, and wait till you see your assistant by reflection, he is then at the jDoint in the boundary through which that offset pas- ses ; on the other hand, if you wish an offset from a given point or bend in the boundary, let the assistant Stop at that place, and do you walk on in the station line till you see the assistant by reflection in the oc- tant, and that will be the point where an oftset from the proposed point or bend will fall. The manner of using this instrument for the solu- tion of those astronomical problems that are necessary in surveying, will be shewn in its proper place. «- DESCRIPTION OF A NEW POCKET BOX SEX- TANT, AND AN ARTIFICIAL HORIZON, BY THE EDITOR. Fig. 1 1, f late \Q, is a representation of a very con- venient pocket sextant, and contains a material im- provement on the reflecting cross staff before de- scribed, see^g. 4, plate 14. In military operations, as well as trigonometrical ones, it has been found of very essential service. AB a round brass box three inches in diameter, and one inch deep. A C is the index turning an index glass within the box, a, a, are the two outside ends of the screws that confine an horizon glass also within the box. An angle is observed by the sight being directed throng.h an hole in the side of the box about D, upon and through the horizon glass and the second opening at E, and the angle is read off to one minute by the divided arc and nonius F, G, H. By sliding a pin projecting on the side of the box, a dark glass is brought i;efore the sight hole, not shewn in the figuie ; by j)ushing the pin at b, a dark screen for the sun is interposed b<;^- XEW POCKF.T BOX SF.XTAXT, &('. 265 twcen the index and horizon glasses. I is an endless «crc\v, sometimes applied to give a very accurate mo- tion, like the tangent screw to the index of a sextant. Or a racked arc and pinion may be applied at about r, d, wliich I think in some respects better. 'J he following table is sometimes engraved upon the cover that goes over the box when shut up. liy the sextant being set to any of the angles contained in this table, an height or distance of accessible or inaccessible objects is obtained in a very simple and expeditious manner. iMul Angle. Angle. Div. 1 45° 00' 45' 00 1 2 63 2(5 26 34 2 3 71 34 18 26 3 4 75 58 14 02 4 5 78 41 11 19 5 6 80 32 9 28 6 8 82 52 7 08 8 10 84 17 5 43 10 Make a mark upon the object, if accessible, equal to the height of your eye from the ground. Set the index to any of the angles from this table, and walk from the object, till the top is brought by the glas- ses to coincide with the mark ; then, if the angle be greater than 45^, multiply the distance by the corre- sponding figure to the angle in the table; if it be less, divide, and the product, or quotient, will be the height of the object above the mark. If the object be inaccessible, set the index to the greatest angle in the table that the least distance from the object will admit of, when by moving back^vards and torwnrds, till the top of the object is brought to a level with the eye, and at this place set up a mark equal to the height of the eye. Then set the intlcx to any of the lesser angles^ and go backwards in a line from the ob- '266 THE ARTIFICIAL HORIZON. ject, till the top is made to appear on the level with the eye, or mark before set; set here another mark, measure the distance between the two marks, and this divided by the difference of the figures in the last column, against the angle made use of, the quotient will give the height of the object above the height of the eye, or mark. For the distance, multij^ly the height of the object by the numbers against either of the angles made use of, and the product will be the distance of the object from the place where such an- jrle was used. If the index is set at 45 the distance is the height of the object, and vice versa. The index set to 90° becomes a reflecting cross staff, and is used accord- ing to the directions in page 263. The sextants, as before described by the author, of the best kind, are made of brass, or other metal. The radii now most approved of are from six to ten inches, their arcs accurately divided by an engine, and the nonii shewing the angles to 30, 15, or even 10 seconds ; but the fine divisions of the latter are liable to be obliterated by the frequent cleaning of the instrument. THE ARTIFICIAL HORIZON. In many cases it happens that altitudes are to be taken on land by the sextant ; which, for want of a natural hoiizon, can only be obtained by an artificial one. There have been a variety of these sort of in- struments made, but the kind now to be described is allowed to be the only one that can be depended upon. Fig. 12, plute ig, represent the horizon fixed up for use. A is a wood or metal framed roof con- taining tvv'o true parallel glasses of about 5 by 34- inchcs, fixed not too tight in the frames of the roof. This serves to shelter from the air a wooden trough filled with quicksilver. In making an observation by it with the sextain, the relicctcd image of the TO SURVEY WITH THE CHAIN ONLY. 2^7 sun, moon, or other object, is brought to coincide with the same object reflected from the glas'^es of the sextant; half the angle sliewn upon the Hmb is the altitude above the horizon or level required. It is neces>ary in a set of observations that the roof be al- ways placed the same way. When done with, the roof folds UP flat-ways, and, with the quicksilver in a bottle, &c. is packed into a portable flat case. TO SURVEY WITH THE CHAIN ONLY. The diflicuhics that occur in measuring with ac- curacy a straight line, render this method of survey- ing altogether insufficient for measuring a piece of ground of any extent ; it would i)e not only extreme- ly tedious, but liable to many errors that could not be detected ; indeed there are very few situations where it could be used without King's surveying quadranty or some substitute for it. The method is indeed in itself so essentially defective, that those who have praised it most, have been forced to call in some in- strument^ as the surveying cross and optical square, to their aid. Little more need be said, as it is evi- dent, as well from the nature of the subject, as from the practice of the most eminent surveyors, that the measuring of fields by the chain can only be proper for level ground and svmJl enclosures ; and that even then, it is better to go round the field and measure the angles thereof, taking ofl^sets from the station lines to the fences. That this work may not be deemed imperfect, we shall introduce an example or two se- lected from some of the best writers on the subject ; observing, however, that fields that are plotted from Pleasured lines, are always plotted nearest to the truths when those lines form at their junction angles that ap' proach neatly to a right angle. Example 1. To survey the triangular field ABC, fg. 22, plate Q, by the chnin and cross. Set up marks at the corners^ then begin at one of them, and 268 TO SURVEY WITH THE CHAIN ONLY. measure from A to B, till you imagine that you are near the point D, where a perpendicular would fall from the angle C, then letting the chain lie in the line A B, fix the cross at D, so as to see through one pair of the sights the marks at A and B ; then look through the other pair towards C, and if you see the marks there, the cross is at its right place ; if not, you must move it backwards and forwards on the line A B, till you see the marks at C, and thus find the point D; place a mark at D, set down in your field book the distance x\ D, and complete the measure of A B, by measuring from D to B, 11.41. Set down this measure, then return to D, and measure the perpen- dicular D C, 6.43. Having obtained the base and perpendicular, the area is readily found : it is on this principle that irregular fields may be surveyed by the chain and cross ; the theodolite, or Hadley s sextant^ may even here be used to advantage for ascertaining perpendicular lines. Some authors have given the method of raising perpendiculars by the chain only; the principle is good, but the practice is too operose, tedious, and even inaccurate to be used in surveying ; for the method, see Geometry on the Ground. Example 'I. To measure the four-sided figures, A B C D, fig. 34, plate Q. AE 214 DE 210 AF 362 BF 306 AC 592 Measure either of the diagonals, as AC, and the two perpendiculars D E, B F, as in the last problem, which gives you the above data for completing the- figure. Example 3. To survey the irregular 'nc^A,fig. 22, ■plate 13. Having set up marks or station staves wherever it may be necessary, walk over the ground, and consider how it can be most conveniently divi- ded into triangles and trapeziums, and then measure them by the last two problems. QF SURVEYING BY, &C. 2G(J ' It is best to subdivide the field into as few separate triant^lcs as possible, but rather into trapeziums, by I drawing diagonals from corner to corner, so that the I perpendicular may fall within the figure; thus the I figure is divided into two trapeziums AHC(i, G D E F, and the triangle G C D. Measure the diagonal A C, and the two perpendiculars G M, B N, then the base GC, and the perpendicular Dq ; lastly, the diagonal D h\ and the two perpendiculars, p E, O G, and you have obtained sufHeient for your purpose. OF SL'RVEYIXG BY THE PLAIN TABLE. We have already given our opinion of this instru- ment, and shewn how far only it can be depended upon where accuracy is required ; that there are many cases where it may be used to advantage, there is no doubt ; that it is an expeditious mode of survey- ing, is allowed by all. I shall, therefore, here lay down the general modes of surveying with it, leaving it to the practitioner to select those best adapted to his peculiar circumstances, recommending him to use the modes laid down in example 3, in preference to others, where they may be readily applied. He will also be a better judge than I can be, of the ad- vantages of Mr. Breaks method of using the plain table. Excimple 1. To take by the plain table the plot of a piece of land A B C 1) E, fg. 3(), plale g, at ouc station near the middle, from whence all the corners may be seen. Let RTSV, ^^. 37 , plate g, represent the plain table covered with a sheet of paper, on which the plan of the field, fg. 3(j, is to be drawn ; go round the field and set up objects at all the corners there- of, then put up and level your plain table, turning it about till the south point of the needle points to the 270 OP SURVEYING BY N. point, or 360" in the compass box ; screw the table fast in that position, and then draw a line P p parallel to one of the sides for a meridian line. Now choose some point on the paper for your station line, and make there a fine hole with a small circle of black lead round it ; this is to represent the station point on the land, and to this the edge of the index is to be applied when directed to an object. Thus, apply the edge of the index to the point 0, and direct the sight to the object at A, when this is cut by the hair, draw a blank line along the cham- fered edge of the index from towards A, after this move the index round the point as a centre, till you have successively observed through the sights, the several marks at A, B, C, D, E ; and when these marks coincide with the sights, draw blank or ob- scure lines by the edge of the index to 0. Now measure the distance from the station point on the ground to each of the objects, and set off by your scale, which should be as large as your paper will ad- mit of, these measures on their respective lines ; join the points A B, B C, CD, D E, E A, by lines for the boundaries of the field, which, if the work b« properly executed, will be truly represented on the paper. N. B. It is necessary, before the lipes are mea- sured, to find by a plumb-line the place on the ground under the mark on the paper, and to place an arrow at that point. Example 2. Let fg. 33, plate 9, represent the piece of ground to be surveyed from one station point, whence all the angles may be seen, but not so near the middle as in the foregoing instance ; go round the field, and set up your objects at all the cor- ners, then plant the table where they may all be con- veniently seen ; and if in any place a near object and one more remote are in the same line, that situation is to be preferred. Thus, in the present case, as at THE PLAIN TABLE. 27 1 O, ^^ coincides with/;, and c with If; the tabic is planted thereon, making the lengihway of the tabic correspond to that of the field. Make your point- bole and circle to represent the place of the table on the land, and apply the edge of the index thereto, so as to see through the slit the mark at a cut by the hair ; then with your pointrel draw a blank line from O towards ii, do the same by viewing through the sights the several marks c, J, e, f, g, keeping the edge of the index always close to 0, and drawing blank lines from towards each of these marks. Find by a plumb-line the place on the ground un- der on the paper, and from this point measure the distances first to g, and proceed on in the same line to /;, writing down their lengths as you come to each ; then go to a, and measure from it to 0, then set off from your scale the respective distance of each on its proper line : after this, measure to c, and continue on the line to /', and set off their distances; then mea- sure from to d, and from ^V. to the angle h. As there is no diffieuhy in taking tlie oftVets from the other station lines, we sliall not proceed farther in plotting it on the out- side; for a sight of the figure is sufficient. .Some surveyors would [)lant their table at a place between 3t)4 and 6q8 in the first station line, and take the two angles, which are here plotted by the intersection of lines, as the by-angles a and h were taken at O 1 within the field ; but if the boundary should not be a straight line from one angle to tlie other, then their distance should be measured, and offsets taken to the several bends in it. You plot an inaccessible distance in the same man- ner as the tree in 7?^. 3'2 ; for if you could come no nearer to it th;ni the station line, yet you might with a scale measure its distance from I, or 2, or any part of the line between them in the plan, the same as if you measured it with the chain on the land ; observing to make the stations at such a dis- tance from one another, that the lines drawn towards the tree may intersect each other as near as possible to right angles, drawing a line from each to 1 ;* writing down the degree the north end of the needle points to, as it should point to the same degree at each station ; remove the table, and set up a mark at © i. The imperfections in all the common methods of using the plain table, are so various, so tedious, and liable to such inaccuracies, that this instrument, so much esteemed at one time, is now disregarded by all those \\ho aim at correctness in their work. Mr. Brt'cik iias endeavoured to remedy the evils to which * In general the mark © always denotasa station or place where the instrument is planted. The dotted lines leading from one sta- tion to ano'her, are the station lines ; flie black lines, the bounda- ries ; the dotted lines from the boundary to thj station line are ot^sets. r T 3 278 OF SURVEY IXG BY this instrument is liable, by adopting another method of using it ; a method which I think does him consi- derable honour, and which I shall therefore extract from his complete " System of Land Siirveying,''' for the information of the practitioner. Example 1 . To take the plot of a field A B C D E F, fig. 1 1, plate 13, f'oni one station therein. Choose a station from whence you can see every corner of the field, and place a mark at each, number- ing these with the figures 1, 2, 3, 4, &c. at this sta- tion erect your plain table covered with paper, and bring the south point o{ the needle to the flower-de- luce in the box ; then draw a circle O P Q H upon the paper, and as large as the paper will hold. Through the centre of this circle draw the line N S parallel to that side of the table which is paral- lel to the meridian line in the box, and this will be the meridian of the plan. Move the chamfered edge of the index on O, till you observe through the sights the several marks A, B, C, and the edge thereof will cut the circle in the points 1, 2, 3, &c. Then having taken and pro- tracted the several bearings, the distances must be measured as shewn before. To draw the flan. Through, the centre and the points 1, 2, 3, &c. draw lines A, B, C, D. E, F, and make each of them equal to its re- spective measure in the field ; join the points A, B, C, &c. and the j)lan is finished. Example 2. To take the plot of a field A B C D, ^c. from several stations., fig. 14, plate 13. Having chogen the necessary stations in the field, and drawn the circle P QR, which you must ever observ^e to do in every case, set up your instrument at the first station, and bring the needle to the meri- dian, which is called adjusting the instrument ; move the index on the centre 0, and take an observation at Ay B, C, 2, //, and the fiducial edge thereof THE PLAIN' TABLK. 279 will intersect the circle in 1, 2, 3, &c. Then remove vonr instrument to the second station in the field, and applyinp: the edge oC the index to the centre and the mark O '2 in the circle, take a back sight to the first station, and fasten the ta])le in this position ; then move the index on the centre 0, and direct the sights to the remaining angular marks, so will the fiducial edge thereof cut the circle in the points 4, J, (j, &c. The several distances being measured with a chain, the work in the field is finished ; and rniered in the book thus : THE PIEf.D BOOK EXPLAINED. No. ! Dist, No. Dist. I 2 3 02 b 520 344 360 730 386 3 5 6 I 0'-i 370 470 550 550 ' To draiv the Plan, Having chosen 1 upon paper to represent the fjibt station in the field, lay the edge of a parallel ru- ler to O and the mark 1, and extend the other edge till it touch or lay upon © 1, and close by its edge draw a line 1^1=520. Then lay the ruler as before to and the mark 2, and, extending the other edge to 1, draw thereby the line l,2r=344, which gives the corner B, as the line 1 ,1 does the corner A. Af- ter the same manner project 2, together with the corners C, H. Again^ apply the edge of the ruler to and the point 4, and extend the other edge till it touch 2, and draw the line 2,4 = :370, which will T 4 280 OF SURVEYING BY give the point or corner D. Thus project the re- maining corners E, F, G, and the plan is ready for closing. Exaj7iple 3. To take the f Jot of several f elds AB CD, B E C F, D I H K, ^;7^ I F G H, fro7n sta- tions chosen at or near the middle of each, Jig. 15, pJate 13. Adjust your plain table at the first station in A B C D, and draw the protracting circles and me- ridians A'^ S ; then by proposition I, project the an- gles or corners of J B C D, B E FC, and also the second station, into the points 1, 2, 2, 3, 4, 5, 6. Again, erect your instrument at the first station, and lay the index on the centre and the mark 2, and direct the sights by turning the table to the se- cond station, then move the index till you observe the third station, and the edge thereof will cut the circle in 3. Then remove the instrument to the third station, lay the index on the centre and the mark 3, and take a back observation to the first station ; after which by the last proposition find the points 7, S, 4, 9, in the circle P Q R. As to the measuring of distances, both in this and the two succeeding propositions they shall be passed over in silence, having sufficiently displayed the same here- tofore ; what I intend to treat of hereafter, is the method of taking and protracting the bearings in the field, with the manner of deducing a plan there- from. To dra-vv the Flan, Choose any point 1 for your first station ; ap- ply the edge of a parallel ruler to the centre and the point 1, and having extended the other edge to 1, draw the line 1,1=370, which will give the corner A. In like manner find the other corners B, Cj Di together with 2 ; which being. joined. T!fE PLAIK TABLE. 2Sl finishes the field j4 B C D AOer the same method constrm t the other tields /^ EFC, D f H K, IFGfl, and }0U have done. THE FIELD BOOK. REMARKS. No. DiST. REMARKS. The open Sir Will. Jones's 1 2 ©2 3 4 2 5 6 7 3 ©3 7 ©4 8 <) 1 a 10 8 ©1 3;o 380 580 403 440 InABC D. Field. B EFL D I HK. Ground. ©2 In BLFI. John Simpson's jB£F/ closes at Return to 4(X) 4(50 Ground. / /' G H. ©1 (>00 In ABC D. South John Spencer's D I tl K closes at ©3 630 432 400 In DI H K. I FG H. Field. Ground. Ed Johnstone s ©4 In I FGH. Ground South / /" G H closes at 380 Field. Example A. To take the plot of a field A BCD E F, hy going ro7(ml the same, fig. \2, plate 13. Set up your plain tabic at the first station in the field ; move the fiducial edge of the index on the centre 0, and take an observation at the mark placed at the second station, then will the same iiducial edge cut the circle P Q R in the point 1. Then re- 282 OF SUKVEYIXG BY move your instrument to the second station, and placing the edge of the index on O and the point ], take a back sight to the first, or last station ; then directing the index on the centre to the third, or next station ; the edge thereof will cross the circle in the point 2. In like manner the instrument being planted at every station, a back sight taken to the last preceding one, and the index directed forward to the next succeeding station, will give the protracted points 3, 4j 5, 6". THE FIELD BOOK. ' REMARKS. 0. QL. 0. REMARKS. 70 50 85 ©1 250 550 In JBCDEF 02 In Ditto. 84 ■ Corner 65 440 03 In Ditto. 60 465 04 In Ditto. 72 58 365 80 750 05 In Ditto. 40 68 302 60 680 06 In Ditto. 58 50 355 67 653 I'coOl Close Ditto. rUE I'LAIN' TAr.LK, 283 To Jraiv the Plufi. CIioosc any point ], to dcnoLc the first station. Lay the edge ot"a parallel nilcr on the centre Q 'i" ^; the instrument being planted at c, and the sights directed to d, the bearing of r- d, will be 331^ 45'. In the same man- ner proceed to take the bearinn; of other lines round the wood, observing carefully the following general rule : Keep the flower-de-luce from you, and take the bearing of each line from the north end of the needle. lines. bearings. links. ab ^60. 30 1 2 12 be 2()2. 12 1015 cd 331. 45 1050 dc 59. 00 1428 c i 1 12. 15 645 fa 151. ^0 IBOG * Wyid's Practical Surveyor;, p. /■/. COMMON CIRCUMFERENTOn. 289 Instead of planting the circunirerentor at every station in the field, the bearings of the several lines may be taken if it be planted only at every other station. So if the instrument had been planted at /», and the fiower-de-luce in the box kept towards you when YOU look back to the station a, and from you when you look forwards to the station c, the bearings of the iines (/ /', and /' r, would be the same as before ob- served ; also the bearings of the lines c d, and d e, might be observed at d^ and e f^ and fa, at /; so that instead of planting the instrument six times, you need in this case plant it but three times, which saves some labour. But, since you must go along every station line, io measure it, or see it measured, the trouble of setting down the instrument is not very great, and then also you may examine the bearing of each line as you go along ; and, if you suspect an error in the work, by the needles being acted upon by some hidden magne- tic power, or from your own mistake, in observing the degrees that the needle points to, you may correct such error at the next station before you proceed. As when the instrument was planted at a, and the sights directed at /', the flower-de-luce from you, the north end of the needle pointed to 26o° 30' ; now beiti^ come to /', direct the sights back to a mark at 25 the held. O A corner. 10 17 o o 60 O Oithc field ,To the boundary of To the place of beginning. To survey a tract of land, consisting of any numher (jf fields lying together. 1. Take the outside boundaries of the whole tract as before directed, noting in your field book where the particular fields intersect the outside boundaries • and then take the internal boundaries of the several fields fronn the place where they so butt on the out- side hounds. Lcifg. 4, plate 18, represent a tract of land to be surveyed, consisting of three fields. First, begin at any convenient corner, as at A, and proceed taking the courses, distances, offsets, and re- rnarks, as in the following field book. U3 294 OP MK. GALES METHOD BY FIELD BOOK. REMARKS. Jff- 5ets. Station Oft- lines. sets. REMARKS. 1st. At a corner against Winterton farm, and H. Smith's land. Lands of Henry Smith. 00 S. 87° W. 00 00 East field. 5 50 1 10 To the corner of east 11 30 00 and west field. 15 00 |o 60 Corner of west field. Hunterdon farm. 2d. N. 18° 30. W 58 16 9 60 \0 00 The above corner. 00 3d. N. 12°15.E. 00 4 10 8 90 00 60 00 West corner. 4th. N. 37° E. 5 40 17 00 40 00 To the corner of west field and north field. Lands of Jacob Williams. 70 00 5th. s. ye"" 45. E. 6 50 l6 00 00 North field. Winterton farm. 6th. S.5° 15 W. 9 60 14 30 ig 00 25 00 56 10 70 00 To the corner of north andeast fields. N.B. A gate into field 20 links from the corner. 7th. S. 74° 30. W. 2 30 7 65 86 00 East field. r roth s place of begi nning r >• THE IMPROV€D C IRC UMFERENTOR. 295 FIELD BOOK. REMARKS. W«st field. North field. Oif- Station I Off- sets, lines. I set.s. 8th. At the corner ot cast and west field, viz. The first otiVct on the first station line. REMARKS. North. 8 30 16 61 75 00 00 West field. Comer of west and north fields against Huntendon farm. gth. East. (» 5 40 13 77 East field. lp:ast field. iTo the corner of ^ ^ north and cast fields "0|aoraInst Winterton lOlh. Back to the last station. N. 62° 03 .W| OO^Torth field. 3 40 1 001 o 70 00 7 p2 lO 00| To take a survey of an estate, manor y and lordship. An estate, manors or lordship, is in reality a tract of land, consisting of a number of fields; it ditfcrs in no respect from the last article, excepting in the ninnher of fields it may contain, and the roads, lanes, or waters that may riui through it, and is of course surveyed in the same manner. It is the best in the first place to take the whole of the outside boundaries, noting as above directed the several offsets, the several places where the boun- daries are intersected by roads, lanes, or waters, the places where the boundaries of the respective fields butt on the outside bounds, and where the gates lead into the respective fields, and whatever other objects, as windmills, houses, &c. that may happen to be worthy of being taken notice of. If, however, there should be a large stream of water running through the estate, thereby dividing it into two U4 igd TO SURVEY AN ESTATE, parts, and no bridge near the boundary, then it will be best to survey that part which lies on one side of the stream first, and afterwards that part which lies on the other side thereof: if the stream be of an irregular breadth, both its banks forming boundaries to several iickls sliould be surveyed, and its breadth, where it enters, and where it leav^es the estate, be determined by the rules of tri- gonometry. In the next place, take the lanes or roads, (if any such there be) that go through the estate, noting in the same manner as before, where the divisions be- tween the several fields butt on those lanes or roads, and where the gates enter into those fields, and what other objects there may be worth noticing. Where a lane runs through an estate, it is best to survey in the lane, because in so doing, you can take the ofi^sets and remarks both on the right hand and the left, and thereby carry on the boundaries on each side at once. If a large stream run through and separate the estate, it should be surveyed as above-mentioned ; but small brooks running through a meadow, require only a few offsets to be taken from the nearest station line, to the principal bends or turning in the brook. In the last place take the internal divisions or boundaries between the several fields, beginning at any convenient place, before noted in the field book, where the internal divisions butt on the outside of the grounds, or on the lanes, 8cc. noting always every remarkable object in the field book. Exmnple. Ltt^g. 5, picife 18, represent an estate to be surveyed ; begin at any convenient place as at A, where the two lanes meet, proceed noting the courses, distances, &c. as before directed from A to B, from B to C, and so on to D, E, F, G, H, I, K, L, Mand A, quite round the estate. Then proceed along the lane from A to N, O and MANOB, AND LORDSHIP. 297 I, setting the courses, distances, oflscts,and remarks, as before. Tliis done, proceed to the internal divisions, be- ginning at any convenient place, as at O, and pro- ceed, always taking your notes as before directed, from O to P, Q and R, so will you have with the notes previously taken, the dimensions of the north iield ; go back to Q, and proceed from Q to H, and you obtain the dimensions of the cojjsc. Take ES and SP, and you will have the dimensions of the home field; go back to S, and take ST, and you will have the dimensions of the land, applied to do- mestic purposes of buildings, yards, gardens, and orchards, the particulars and separate divisions of which being small, had better be taken last of all ; go down to N, and take N D, noting the offsets as well to the brook, as to the fences, which divide the meadow from the south and west fields, so will you have the dimensions of the long meadow, together with the minutes for laying down the brook therein ; go back to U, and take U B, and you obtain the dimensions of the west field, and also of the south field ; go back again to N, and take N W, noting the offsets as well to the brook as to the fence which divides the meadow from the east field, thus will you have the dimensions of the east field, and the minutes for laying down the brooks in the meadow ; then go to L, and take LX, which gives you the dimensions of the east meadow, and of the great field; and lastly, take the internal divisions of the land, appropriated to the domestic purposes of buildings, yards, orchards, gardens, &c. The method of taking the field notes is so intircly similar to the examples already given, that they would be altogether unnecessary to repeat here. To the surveyor there can need no apology for in- troducing, in this place, the method used by Mr. Milne, one of the most able and expert surveyors of the present day ; and I think he will consider 298 MR. MILNE S METHOD himself obliged to Mr. Milne, for communicating, with so much liberality, his deviations from the com- mon practice, as those who have hitherto made any improvements in the practical part of surveying, have kept them as profound secrets, to the detriment of science and the young practitioner. As every man can describe his own methods, in the clearest and most intelligible manner, I have left Mr. Milne s in his own language. Mr. Milxe's method of surveying. " The method 1 take to keep my field notes in surveying land, differing materially from those yet published ; if, upon examination, you shall think it may be useful or worthy of adoption, you may, if you please, give it a place in your treatise on sur- veying. What I, as well as all surveyors aim at, in going about a survey, is accuracy and dispatch ; the first is only to be acquired by care and good instruments, the latter by diligence and long practice. From twenty years experience, in the course of which I have tried various methods, the following is what I at last adopted, as the most eligible for carry- ing on an extensive survey, either in England, Scot- land, or any other cleared country. Having taken a cursory view of the ground that I am first to proceed upon, and observed the plainest and clearest tract, upon which I can measure a cir- cuit of three or four miles, 1 begin at a point con- venient for placing the theodolite upon, and making a small hole in the ground, as at A, flaU 20, one assistant leading the chain, the other having a spade for making marks in the ground, 1 measure in a di- rect line from the bole at A to B, noting down, as I go along, on the field sketch the several distances from place to place, and sketching in the figure of the road, as 75 links to the east side of the avenue. OP SURVEYING. 2()C) 400 links, toucliing the south side of the rond, with an otiset ot" 40 links to the otJier side, 5CK) links to the corner of a wood, and 817 to the corner of another wood ; which last not allowini^ nie to carry the line farther, I make a mark at B, where 1 mean to plant the instrument, and beginning a new line, measure along the side of the road 883 links to C, where I make a mark in the ground, writing the same in my field sketch, taking care always to stop at a })lace from which the last station can be seen, when a pole is placed at it. Beginning a new line, and measuring in a direct line towards D, I have 3'IJ links to a line of trees, SQO links opposite a corner of paled inclosures, where I make a mark to have recourse to, and 8()3 links to the end of the line, where I make a mark in the ground at D ; from thence measuring in a direct line to E, I here have 600 links of steep ground, therefore make a mark in the ground at the bottom of the steepness, so that when I come to take the angles with the theodolite, I may take the depression thereof; con- tinuing out the line, I have 1050 links to a bridge, with an offset of 20 links to the bridge, and 80 links to the paling ; at 1247 there is a line of trees on the let't, and here also the line comes to the north side of the road, and 1388 links to the end of the line at £, and so proceed in like manner round the circuit to F, G, H, I, K, L, M, N ; and from thence to A where I began. The holes, or marks made in the ground, are represented by round dots on the field sketch ; steepness of ground is expressed on the sketch by faint strokes of the pen, as at 1), F, &c. Figures ending a station line are written larger than the intermediate ones ; offsets are written down op- posite the places they were taken at, and are marked cither to the right, or left of the line on the field sketch, just as they happen to be on the ground. If the field sketch here given was enlarged, so as to fill a sheet of paper, there would then be room for in- 300 MR. MILNe's method serting the figures of all the offsets, which the smallness of this does not admit of. In measuring these circuits, or station lines, too much care cannot be taken by the surveyor to measure exact ; I therefore, in doing them, always choose to hold the hindermost end of the chain my- self. The next thing to be done is to take the angles or bearings of the above described circuit, and also the altitude or depression of the different declivities that have been measured up or down. Having previously prepared a sheet of Dutch paper with meridian lines drawn upon it, as in j>lale 21, also a horn protractor with a scale of chains upon the edge thereof, and a small ruler about a foot in length ; and having two assistants provided with a pole each, to which are attached plumb lines for keeping them perpendicular, and a third assistant for carrying the theodolite; I pro- ceed to plant the instrument where I began to measure, or at any other angular point in the cir- cuit; if the wind blows high, I choose a point to begin at that is sheltered from it, so that the needle may settle steady at the magnetic north, vi'hich is in- dispensably necessary at first setting off', at the same time taking care that no iron is so near the place as to attract the needle. The best theodolite for this purpose is the large one, sec Jig. 1, plate l6, the manner of using of which I shall here describe. The spirit level C, having been previously ad- justed to the telescope A, and the two telescopes pointing to the same object, I begin by levelling the instrument, by means of the four screws M acting between the two parallel plates N, first in a line with the magnetic north, and then at right angles thereto. This accomplished, I turn the moveable index, by means of the screw G, till it Cf^incides with 180° and 360*^, and with the mag- OF SURVEYING. 301 nctic north nearly, screwing it fast by means of the nut H, and also the head of the instrument by means of the pin L ; I make the north point iti the compass box coincide with the needle very ex- actly, by turning the screw K ; both telescopes being then in the magnetic meridian, I look, through the lower one I, and notice what distant object it points to, Tlicn unscrewing the moveable index by the aforesaid nut H. the first assistant having been pre- viously sent to the station point marked N, I turn about the telescope A, or moveable index, by means of the nut G, till it takes up the pole now placed at N, raising or depressing the telescope by means of the nut E, till the cross hairs or wires cut the pole near the ground. This done, I look through the lower telescope to see that it points to the same object it did at first ; if so, the bearing or angle is truly taken, and reading it upon the limb of the in- strument F, find it to be 40° 55' S. W. I then take my sheet of paper, and placing the horn pro- tractor upon the point A, plate 21, along the meri- dian line, passing through it, I prick off the same angle, and with the first ruler draw a faint line with the pen, and by the scale set oft' the length of the line, which I find to be, by my first sketch, plate 20, 7690 links ; and writing both bearing and dis- tance down, as in plate 21 y and again reading off the angle to compare it with what I have wrote down, I then make a signal to the first assistant to come forward with his pole ; in the mean time I turn the moveable index about, till the hair or wire cut the pole which the second assistant holds up at B, and looking through the lower telescope, to sec that it points on the same object as at first, I read the angle upon the limb of the theodolite, 48° 55' S. E. and plotting it oft' ui)on the sketch with the horn protractor, draw a straight line, and prick oft' by the scale the length to B, 817 links ; writing the 302 MR. MILNE S METHOD same down on the field sketch, and again reading the angle, to see that I have written it down right, 1 screw fast the moveable index to the limb, by means ot the screw iJ, and making a signal to the second assistant to proceed with his pole, and plant it at the third station C, while I with the theodolite proceed to 1), leaving the first assistant with his pole where the instrument stood at A. Planting the instrument by means of a plumb- line over the hole, which the pole made in the ground at the second station, and holding the moveable index at 48° 55' as before, and the lim.b levelled, 1 turn the limb till the vertical wire in the telescope cuts the pole at A nearly, and there screw it fast. I then make the vertical wire cut the pole very nicely by means of the screw K ; then looking through the lower telescope to see and remark what distant objects it points to, I loosen the index screw, and turn the moveable index till the vertical hair in the telescope cuts the pole at C nearly, and, making the limb fast, I make it do so very nicely by means of the screw K, and then fix the index. Reading the degrees and minutes which the index now points to on the limb, I plot it in my field sketch as before, the bearing or angle being 48° 3 O', and the length 885 links. Sending the theodolite to the next sta- tion C, the same operation is repeated ; coming to D, besides taking the bearing of the line D E, I take the depression from D, to the foot of the decli- vity 6oo links below it, which I perform thus : Having the instrument and the double quadrant level, 1 try what part of the body of the assistant, who accompanies me for carrying the instrument, the telescope is against, and then send him to stand at the mark at the bottom of the declivity, and making the cross wires of the telescope cut the same part of his body equal to the height of the in- strument, I find the depression pointed out upon the quadrant to be } of a link upon each chain, or SURVEYING. 303 , hich upon six chains is four links to be subtracted from 1383 links, the measured length of the line, leaving 1381 links for the horizontal length, which I mark down in my Held sketch, plate 2, in the manner there written. Also from the high ground at D, seeing the temple O, I take a bearing to it and plotting otT the same in my field sketch, draw a straight line; tlicsanie temple being seen from sta- tion L, and station M, I take bearings to it, from each ; and tVom the extension of these three bear- ings intersecting each other in the point (), is a proof that the lines have been trulv measured, and the an- gles right taken. Again, from station E to station F, the ground rises considerably, therefore, besides the bearing, I take the altitude in the same manner I did the depression ; and do the same with every considerable rise and fall round the circuit. Coming to station N, and having the moveable index at 6j^ 15' N. W. taking up the pole at M, and making fast the limb, I loosen the index screw, and turn it till I take up the pole at the first sta- tion A. This bearing ought to be 40° 55' N. E. being the ^niuj number of degrees and minutes it bore from A south-west, and if it turn out so, or within a minute or two, we call it a good closure of the circuit, and is a proof that the angles have been accurately taken. But if a greater error appears, I rectify the move- able index to 40° 55', and taking up the pole at A, make fast the limb, and take the bearing to M, and so return back upon the circuit again, till 1 find out the error. But an error will seldom or never happen with such an instrument as here de- scribed, if attention is paid to the lower telescope ; and besides, the needle will settle at the same degree (minutes by it cannot be countcdj in the bo.K, as the moveable index will point out on the limb, provided it is a calm day, and no extraneous matter to attract 304 MR. MILNE S METHOD it. If it does not, I then suspect some error has been committed, and return to the last station to prove it, before I go farther. With regard to plotting off the angles with the horn protractor in the field, much accuracy is not necessary; the use of it being only to keep the field sketch regular, and to preserve the figure of the ground nearly in its just proportion. Coming home, I transfer from the first field sketch, plate 20, all the intermediate distances, offsets, and objects, into the second, plate 21, nearly to their just proportions ; and then I am ready to ])roceed upon surveying the interior parts of the circuit. For doing this, the little light theodolite represented ^t fg. A, plate 15, or the more complete one, jig. 7} plate 14, will be sufii- ciently accurate for common surveying. It is un- necessary for me to describe farther the taking the angles and mensurations of the interior subdivisions of the circuit, but in doing of which, the young surveyor had best make use of the horn protractor and scale ; because, if he mistakes a chain, or takes an angle wrong, he will be soon sensible thereof by the lines not closing as he goes along ; the more experienced will be able to fill in the work sufficiently clear and distinct without them. The field sketch will then be such as is given in plate 22, which indeed appears rather confused, owing to the smallness of the scale made use of to bring it within compass ; but if this figure was enlarged four times, so as to fill a sheet of paper, there would then be room for entering all the figures and lines very distinctly. Having finished this first circuit ; before I begin to measure another, I examine the chain I have been using, by another that has not been in use, and fmd that it has lengthened more or less, as the ground it has gone through was rugged or smooth, or as the wire of which it is made is thick or small ; OP SURVEYING. 305 the thick or great wire drawing or lengthening more than the small. The careful surveyor, if he has more than two or three clays chain work to do, will take care to have a spare chain, so that he may every now and then correct the one by the other. The offset staff is inadequate for this purpose, being too short. For field surveying, when no uncommon accuracy is required, cutting a bit olF any link, on one side, of 50 links, and as much on the other side, answers the purpose better than taking away the rings. The above method is what I reckon the best for taking surveys to the extent of 100,000 acres ; be- yond this, an error from small beginnings in the mensuration becomes very sensible, notwithstanding the utmost care; therefore, surveying larger tracts of country, as counties or kingdoms, requires a different process. Surveys of this kind are made from a judi- cious series of triangles, proceeding from a base line, in length not less than three miles, measured upon an horizontal plane with the greatest possible accu- racv. Thomas Milne.'''' OP PLOTTING, OR MAKING A DRAUGHT OF THE LAND FROM THE FIELD NOTES. OP PLOTTING, AND OP THE INSTRUMENTS USED IN PLOTTING. By plotting, we mean the making a draught of the land frouj the field notes. As the instruments X sod OF PLOTTING. necessnry to bs used by the surveyor in taking the dimensions of land, arc buch whcrevviih he may mea- sure the length of a side, and the quantity of an angle in the field; so the instruments commonly used in making a plot or draught thereof, are such wherewith he may lay down the length of a side, and the quantity of an angle on paper. They therefore consist in scales of equal part. s for laying down the lengths or distances, and. p'/olractors for laying down the angles. Scales of equal parts are of different lengths and differently divided ; the scales commonly used by surveyors, are called feather-edge scales; these are made of brass, ivory, or box ; in length about 10 or 12 inches, but may be made longer or shorter at pleasure. Each scale is decimally divided, the whole length, close by the edges, which are made sloping in order to lay close to the paper, and num- bered 0, 1 , 2, 3, 4, &c. "which are called chains, and every one of the intermediate divisions is ten links, the numbers are so placed as to reckon back- wards and forwards; the commencement of the scale is about two or three of the larger divisions from the fore end of the scale ; these are numbered backwards from o towards the left hand with the numbers or figures 1, 2, 3, &c. these scales are often sold in sets of 6, of various chains to the inch. See fg.l.pIateTl. The application and use of this scale is easy and expeditious, for to lay down any number of chains from a given point in a given line; place the edge of the scale in such a manner that the o of the scale may coincide with the given point, and the edge of^ the scale with the given line ; then with the protract- ing pin, point off from the scale the given distance in chains, or chains and links. Fig. F G rl, fkUe 2, represents a new scale or rather scales of equal parts, as several may be laid , the work is right, otherwise not.* Then in an additional column put the whole quantity of northing or of southing made at the * The truth of this observation cannot bnt appear self evident to the reader. For the meridians within the limits of an ordinarysur- vey having no sensible difference from parallelism, it must necessa- rily follow, that if a j»erson tnivel any way soever wiih such small limits, and at length come round to the place where he set out, he must have travelled as far to the northward as to the southward,^ and to the eastward as to the westward, though the practical sur- veyor will always find it difficult to make his work close with this perfect degree of exactness. X4 312 MK. gale's method termination of each of the several lines of the sur- vey ; which will be determined by adding or sub- tracting the northing or southing made on each par- ticular line, to or from the northing or southing made on the preceding line or lines. And in an- other additional column, put the whole quantity of easting or westing made at the termination of each of the several lines of the survey, which will be de- termined in like manner by adding or subtracting the easting or westing made on each particular line, to ox from the easting or westing made on the prece- ding line or lines. The whole quantity of the northings or south- ings, and of the eastings or westings, made at the terminations of each of the several lines, being thus contained in these two additional columns, the plot may be easily laid down from thence, by a scale of equal parts, without the help of a pro- tractor. It is best, howev^er, to use a pair of scales with a clasp,* whereby the one may at pleasure be so fast- ened to the other, as to move along the t^xgo, there- of at right angles, so that the one scale may repre- sent the meridian, or north and south line, the other, an east and west line. It may also be ob- served, that in order to avoid taking the scales apart during the work, it will be necessary that the whole of the plot lay on one side of that scale which repre- sents the meridian ; or in other words, the stationary scale should represent an assumed meridian^ laying \vholly on one side of the survey. On ibis account it will be necessary to note in a third additional co- lumn of the preparatory table, the distances of each of the corners or terminations of the lines of the survey, east or west from such assumed meridian. In practice, it is rather more convenient that this as-. * Pr one of those at fg. F G H, or I K I.^ plate 2 OP PLOTTING. 313 sumed meridian should lay on the west side, than on the cast side of the siincy. The above-mentioned additional column, as has been already observed, contains the distance of each of the corners or terminations of the lines of the survey, eastward or westward, from that magnetic meridian which passes through the place of begin- ning the survey. In order, therefore, that the as- sumed meridian should Iny entirely without the sur- vey, and on the west side thereof, it will only be ne- cessary that its distance from that meridian, which passes through the place of beginning, should be somewhat greater than the greatest quantity of west- ing contained in the said second additional column. Let then an assumed number somewhat greater than the greatest quantity of westing contained in the second additional column, be placed at the top of the third additional column, to represent the dis- tance of the place of beginning of the survey from the assumed meridian. Let the several eastings, con- tained in the second additional column, be added to this assumed number, and the several westings sub- tracted from it ; and these sums and remainders be- ing respectively placed in the third column, will shew the distance of the several corners or termina- tions of the lines of the survey from the assumed me- ridian. The preparatory table being thus made, take a sheet of paper of a convenient size, or two or more sheets pasted together with a little paste or mouth glue, in case a single sheet should not be large enough ; and on the left hand of the intended plot draw a pencil line to represent the assumed meri- dian, on which lay the stationary scale. Place the moveable scale to any convenient point on the edge of the former, and point off by the edge of the lat- ter, according to any desired number of chains to an inch, the assumed distance or number ofcliains contained at the top of the above-mentioned third 314 MR. GALES METHOD column of the preparatory table. Move the move- able scale along the edge of the stationary one, to the several north or south distances contained in the first of the above-mentioned additional columns ; so will the points thus marked off represent the several corners or terminations ot the lines of the survey ; and lines being drawn from one point to the other, will of course represent the several lines of the survey. Example. Let it be required to make a plot or draught of the tield notes, p. 208, according to Mr. Gale^ method. The preparatory table will be as follows : 1 N. and .S. E and W. Distances 1 f!islances distances oftheends Courses and c tanccs as taken the field. 1 is- in Northings, southings, from the eastings, and westings, place of made on those respective besinning from the place of beginning of each line from the as- courses and distances, to the end to the end sumed of each of each meridian. line. line. N. S. E. W. o th. E. 3 00 I.N. 7 W.ljl 00 20 84 2 56 N. 20 8C W. 2 56 E. 44 2.N.55 15E.18 20 10 37 14 96 N. 31 81 E. 12 o9 15 39 ;i.S. 62 S0E.14 40 6 65 12 77 N. 34 .56,E. 25 16 28 161 4.S.40 W.ll 8 45 7 07 xV. 16 ISIE. 18 09 21 09| 5.S. 4 l.'iE.H 13 96 1 04 N. 2 17)E. 19 13 22 13 6.N.73 45W.12 40 3 47 11 90 N. 5 64 E. 7 23 10 93 7.S.52 W. 9 17 5 64 ? 23 iM. 00 00 3 00 .')4 68 34 68 28 76 28 76 The northings and southings, eastings and west- ings in the above table, are taken from the first table in the Appendix ; thus, first find the course 7 degrees in the tabic, and over against 21 chains, in the column marked dist. you have 20.843 in the column marked N. S. which, rejecting the right hand figure 3 for its insignificancy, is 20 chains 84 links for the quantity of northing, and in the co- kimn marked E. W. you have 2.559, ^'^'^.V ^""car 2 chains and 56 links for the westing made on that course and distance. 2. Find the next course OP PLOTTING. 315 55.15 in the tabic, and over against 18 chains in the column dist. you have lo chains 2f) links in the coluiiin marked N. S. and 14 chains 7p links in the column marked E. W. and over against 20 links in the column dist. you have 1 1 links in the column N. S. and l() links in the column marked E, W. which, put together, make 10 chains 37 links for the northing, and 14 chains 05 links for the easting made on the second course • and so of the rest. The north and south distances made from the place of beginning to the end of caeh line, contained in the next column, are determined thus. On the first course, 20 chains and 84 links of northing was made; on the second course, 10 chains 37 links of northing, which, added to the preceding, makes 31 chains 21 links of northing ; on the third course was made 6 chains 65 links of southing, which subtracted from the preceding 31 chains 21 links of northing, makes 24 chains 56 links of northing ; and so of the rest. The east and west distances made from the place of beginning to the end of each line, contained in the next right hand column, are determined in the same manner ; thus, on the first course was made 2 chains 56 links of westing ; on the second course 14 chains 25 links of easting, from which subtracting the preceding 2 chains 56 links of westing, there re- mains 12 chains SQ links of easting ; on the third course was made 12 chains 77 links of casting, which added to the preceding 12 chains 39 links, makes 2.5 chains 16 links of casting; and so of the rest. The distances of the end of each line from the assumed meridian, contained in the next right hand column, are thus determined. The first assumed number may be taken at pleasure, provided only that it exceeds the greatest quantity of westing con- tained in tlie preceding column, whereby the as- 3l6 ^iR. gale's method Slimed meridian shall be entirely out of the survey. In the foregoing example, the greatest quantity of westing contained in the preceding column is Q, chains 06 links ; the nearest whole number greater than this is 3 chains, which is accordingly taken and placed at the top, to represent the distance between the assumed meridian and the place of beginning the survey ; from this 3 chains subtract 2 chains 06 Jinks of westing, there remains 44 links for the dis- tance between the terminations of the first line and the assumed meridian. The 12 chains 39 links of easting in the next step, is added to the assumed 3 chains, which make 15 chains 39 links for the dis- tance of the termination, by the second line from the assumed meridian. The 25 chains 16 links of eastiiig in the next step, being added in like manner to the assumed 3 chains make 28 chains ]6 links for the distance of the termination. of the third line from the assumed meridian ; and so on, always adding the eastings and subtracting the westings from the first assumed number. The preparatory table being completed, take a sheet of paper or more, joined together if necessary, and near the left hand edge thereof rule a line as N. S. Jig. 1, plate IS, to represent the assumed me- ridian ; on this line lay the stationary scale, and as- suming ^ as a convenient point therein to represent the point directly west from the place of beginning, bring the moveable scale to the point a, and lay off the first number contained in the last-mentioned co- lumn of the preparatory table, viz. 3 chains, from a to A, and A will represent the place of beginning of the survey. Move the moveable scale along the stationarv one to the first north and south distance from the place of beginning, viz. 20.84 tVom a to b, and lay off the corresponding distance froni the as- sumed meridian, 0.44. from b to B, and draw A B, sp will A B represent the first line of the survey ; OP PLOTTING. 317 again, move the moveable scale along the stationary one, to the third north and south distance from the place of beginning, viz, N. 31.21 from a to c ; and lay olf the corresponding distance from the assunK:d meridian, viz. 15.3p, from c to C, and draw B C, $o will B C represent the second line of the survey. Again move the moveable scale along the stationary one to the third north and south distance from tiic place of beginning, viz. 24-. 5(), from a to d, and lay off the corresponding distance from the assumed me- ridian, viz. 2S.l6, and draw C D, which will be the third line of the survey. Proceed in the same man- ner till the whole be laid down, and A B C D E F G will be the required plot. This method of plotting is by far the most perfect, and the least liable to error of any that has been con- trived. "It may appear to some to require more la- bour than the common method, on account of the computations required to be made for the pre{)ara- tory table. These computations arc however made with so much ease and expedition, by the help of the table in the appendix, that this objection would vanish, even if the computation were of no other use. hut merely for plotting ; but it must be observed that these compulations are of much further use in deter- mining the area or quantity of land contained in the; survey, which cannot be asccrtaiiied with equal ac- curacy in any other way. When this is considered, it will be found that this method is not only prclcr- ablc on account of its superior accuracy, but is at- tended with less labour on the whole than the com- mon method. If a pair of scales, such as are above recommen- ded, be not at hand, the work may be laid down from a single scale, by first marking off the N. and S. distance on the line N. S. and afterwards launcr off the corresponding cast distances at ri^ht angles thereto. 31S MR. gale's method To plot the field notes, p. 313. 1. Lay down all the station lines, viz. 7" W. 21 chains, N. 55. 15, E. 18. 20, &c, contained in the middle column of the field book, as before directed, without paying any regard to the offsets, until all the station lines, represeted in fig. 3, f>L 18, by dotted lines, be laid down ; then lay off the respective offsets at right angles from the station line, at the respective dis- tances at which they were taken, as is done at j^^. 3, a bare inspection of which will make the work per- fectly plain. 7c» plot the field notes ^ p. 294. 1. Lay down the whole of the outside boundaries, the station lines first, the offsets afterwards, as directed in the preced- ing articles ; and then lay down the internal divisions or boundaries of the respective fields, in the same manner as is done in fig. 4. OBSERVATIONS ON PLOTTING, BY MR. MILNE. The protractor, whether a whole or a semicircle, ought not to be less in diameter than seven or eight inches, to insure the necessary degree of accuracy in plotting the angles of a survey. The degrees on the limb are numbered various ways, but most commonly from 10° 20°, &c. to 36o°. It would be right to repeat the numbers the contrary way, when the breadth of the limb will admit of it. Others are numbered 10° 20°, &c. to 180°, and contrary. Sur- veyors will suit themselves with that kind best adap- ted to their mode of taking angles in the field ; so that in which ever way the limb of the instrument they survey with is graduated, the protractor had best be the same. It is true, a surveyor of much practice will read oft' an angle mentally very readily, in which ever way the instrument happens to be graduated. For instance 1 have in the field taken the angles OP PLOTTING. 319 made with the magnetic meridian, in which case I count no angle above 89° 5[)' ; if the angle comes to g(/o', I call it due E. or due W. if it is 8g'' 5t/, I write to it N. E. or S. E. or else N. W. or S. W. as it happens to turn out. Now 8y" j()' N. E. upon a protractor numbered to 3(3o° O', and placing 3(3o to the north, reads the same ; but Scf og' S. E. reads 90° T, and 89' 09' S. W. reads '2(39^ jq' ; and lastly, 89'' 59' N. W. reads 270° T. Again, if the angle or bearing is due south, it reads on this protractor 180'' ; if 0° l' S. E. it reads 179'^ 59 ; and if o'' l' S. W. it is 180° l'; and so of any other intermediate angle or bearing. The great inducement to surveyors for talking the angles by the bearings the lines make with the mag- netic meridian, is the having the needle as a check in the course of the survey ; and when the circuit comes to be plotted, having it in his power to point off all the angles, or bearings of the circuit, at once planting of the protractor. This mode of surveying, or taking angles, is also more expeditious, and more accurate than any other, provided the index of the instrument is furnished with a clamp for making it fast to the limb, while it is carrying from one station to another. The protractor for plotting this way of surveying had best be graduated 10, 20, &c. to 180, on the right hand, and the same repeated on the left, and again repeated contrary to the former ; but as pro- tractors for general use are graduated 10, 20, &:c. to 360, I shall here describe a very useful and conve- nient one of this latter kind, and then proceed to give an example of plotting all the angles of a cir- cuit by it from one station. This protractor is represented at 7?^. bypJ'ute I7. I'.s diameter, from pointer to pointer, is (j[ inches ; the centre point is formed by two lines crossing each other at right angles, which are cut on a piece ol' 320 MR. GALfe's METftOD glass. The limb is divided into degrees and half degrees, having an index with a nonius graduated to count to a single minute, and is furnished with teeth and pinion, by means of which the index is moved round, by turning a small nut. It has two pointers, one at each end of the index, furnished with springs for keeping them suspended while they are bringing to any angle ; and being brought, applying a finger to the top of the pointer, and pressing it down, pricks off the angle. There is this advantage in having two pointers, that all the bearings round a circuit may be laid, or pricked off, although the index tra- verses but one half of the protractor.* Letjf^. 5, flate 13, be the circuit to be plotted. I draw the magnetic meridian N. S. Jig. 6, plate 13, and assigning a point therein for station 1, place the centre of the above described protractor upon it, with 360° exactly to the north, and 180° to the south of the magnetic meridian line N. S. For the con- veniency of more easily reading off degrees and mi- nutes, I bring the nonius to the south side of the pro- tractor, or side next to me ; and seeing by the field sketch, or eye draught, fg, 5, that the bearing from station 1 to station 1 is 87° 30' S. E. I readily con- ceive it wanting ^"^30' of being due east, or Q0° 0' ; therefore, adding mentally 2° 30' to 90° o', 1 make * The protractor, represented fg. 7, tlate 11, is circular, as before partly described at page 30Q; the centre is also formed by the intersection of two lines at right angles to each other, which are cut on glass, that all parallax may thereby be avoided ; the index is moved round by teeth and pinion. The limb is divided into de- grees and half degrees, and subdivided ti> every minute by the no- nius ; the pointer may be set at any convenient distance from the centre, as the socket which carries it moves upon the bar B C, and is fixed thereto by the nut D. At right angles to the bar B C, and moveable ultli it, there Is another bar E F; on this bar different scales of equal parts are placed, so that by moving a square against the inner edge thereof, angles may be transferred to any distance from the centre, containing the same number of degrees marked out by the index ; the use of which will be evident from Mr. Milne's obiervatlons. OP PLOTTING. 3*21 g'r 30' on this protractor, to wliich I bring the no- nius, and then make a dot by jjressinc: down the pointer of (he protractor, and with my lead pencil mark it 1. The bearinc; from station 2 to station 3 being 1U~ 13' N. E. sec Jig. 5, I bring the nonius, for the sake of expedition and case of rcacHng off, to 10° 15' S. W. which is upon this protractor lyo'' 15', and pressing down the north pointer, it will mark (;ft' 10° 15' N. K. which I mark with my pencil, 2. The bearing from station 3 to station 4 being 85° 13' N. E. I readily see this is wanting 4° 47' of being due cast; therefore I bring the s nonius to within that of due west, which upon this protractor is 265° 13', and pressing down the N. E. pointer, it will prick off 85° 13' X. E. which I mark 3. The bearing from station 4 to station 5 being 11° 5' N. W. I bring the nonius to l()8"55', and with the opposite, or north-west pointer, dot off 1 1° 5' N. W. which I mark 4. In like manner I dot off the bearings from sta- tion 5 to 0, from 6' to /, and from 7 to I, which closes the circuit ; marking them severally with my pencil 5,(3, 7- Then laying aside the protractor, and casting my eve about the tract traced by the pointer of the pro- tractor (or the bearing from station 1, marked 1, I apply the ruler to it and station I, and drawing a line, mark off" its length by compasses and scale, eight chiiins, which tixes station 2. Again, casting my eye about the tract traced by the pointer of the protractor, for the 2 bearinf^^, marked 2, I apj)ly the parallel ruler to it and sta- tion 1, and moving the nilcr jjarallcl eastward, till its edge touches station 2, I from thence draw a line northward, and by scale and compasses mark its length, five chains, to station M. Thus the bearing lo'' 15' N. K. at station 1, is transferred to station 2, aS will easily bi; <"onceived by supposing the points 322 Mr. gale's method. a, b, moved eastward, preserving their parallelism tilt they fall into the points of station 2 and 3, forming in^^. 6, the parallelogram a, b, 3, 2. It is evident this bearing of 10° 15' N. E. is by this means as truly plotted as if a second meridian line had been drawn parallel to the first from sta- tion 2, and the bearing 10° 15' N. E. laid off from it, as was the common method of plotting previous to the ingenious Mr. WilTiam Gardiner communica- ting the more expeditious and more accurate mode to the public. However, its accuracy depends entire- ly upon using a parallel ruler that moves truly paral- lel, which the artist will do well to look to, before he proceeds to this way of plotting. Station 3 being thus fixed, to find station A, I cast my eye about the tract traced by the pointer of the protractor for the bearing marked 3, and applying the parallel ruler to it and station 1, move the ruler parallel thereto northward, till it touch station 3 ; and from thence draw a line north-eastward, and by scale and compasses marking its length, three chains, station 4 is found. Now conceive the points c, d, moved northward, preserving their parallelism till they fall into the points of stations 3 and 4 ; and thus station 4 has the same bearing from station 3, as c or d has from station 1. In the sam.e manner I proceed to the next bearing 4 ; and placing the ruler to the point made by the pointer of the protractor at 4 and station 1, and mov- mg the ruler parallel till its edge touch station 4 ; from thence draw a line, and mark off its length, 5 chains, to station 5. Here e, f, is transferred to 4th and 5th stations, and the 5th station makes the same angle with the meridian from the 4th station, as c, or f, does with the meridian from the first station. In like manner, applying the parallel ruler to sta- TO DETERMINE THE AREA, &C. 323 tlon 1, and the several other bearings of the circuit 5, 6, and 7 ; the points g, h, will be transferred to 5th and 6th, and i, k, to (ith and 7ih ; and lastly, the bearing from the 7 th station will fall exactly into the first, which closes the plot of the circuit. It is almost unnecessary to observe, that the dotted lines marked a, b, c, he. Jig. 6, are drawn there only for illustrating the oi)eraiion : all that is necessary to mark in practice are the figures 1, '2, 3, &c. round the tract traced by the pointer of the protractor. T. Milne: OP DETERMINING THE AREA OR CONTENT OF LAND, 1. The area or content of land is denominated by acres, roods, and perches. Forty square perches make a rood. Four roods, or l6o square perches, make an English acre. And hence 10 square chains, of four perches each, make an English acre. 2. The most simple surface for admensuration is the rectangular parallelogram, which is a plane four- sided figure, having its opposite sides equal and pa- rallel, and tlic adjacent sides rectangular to one an- other ; if the length and breadth be equal, it is cal- led- a j-^m//r ; but, if it be longer one way than the «ther, it is commonly called an ol>lorig. y 2 324 TO DETERMINE THE AREA TO DETERMINE THE AREA OF A RECTANGULAR PARALLELOGRAM. Rule. Multiply the length by the breadth, and that product will be the area.* It has been already observed, that the best method of taking the lengths is in chains and links ; hence, the above-mentioned product will give the area in chains and decimal parts of a chain. And as 10 square chains make an acre, the area in acres will be shewn by pointing off one place of the decimals more in the product than there are both in the multtpl'ica?id znd nwl/ipl/er ; which answers the same purpose as dividing the area into chains by 10; and the roods and perches will be determined by multiplying the decimal parts by 4 and by 40. Example. Suppose the side of the foregoing square be 6 chains and J5 links ; what is the area ? Multiply G.ys bv 6.75 3376 4/25 4050 Acres 4.55625 4 Roods in an Acre. Roods 2.22500 40 Perches in a Rood. Terches 9.OOOOO Acres. Roods. Perches. Ans. 429. Suppose the length of the foregoing oblong be 9 chains and 45 links, and the breadth 5 chains and 15 links ; how many acres doth it contain ? * Euclid, Book i. Prop, 35. Scholium. TO DETERMINE THE AREA, &C. 325 Multiply 0.15 by 5.15 4725 4/25 Acres 4.86675 4 Roods 3.46700 40 Perches 18.68000 A. R. P. Ans. 4 3 IS. [The decimal parts, 68 are not worth regarding.] If the parallelogram be not rectangular, but ob- lique angled, the length must be multiplied by the perpendicular breadth, in order to give ihe area.* 3. After the parallelogram, the next most simple surface for admensuration is the triangle. TO DETERMIXE THE AREA OF A TRIANGLE. Ri^k. Multiply the base, or length, by half the perpendicular, let fall thereon from the opposite angle, and that product will be the area of the tri- angle. -j- Exampks. In the triangle C ABC, right angled at B, let the base A B be 8 chains and 25 links, and the perpendicu- lar B C (3 chains and 40 links ; how many acres doth it con- a tain ? ^' * Sec Euclid. Book i. Prop. 35 and 36. t Euclid, Book i. Prop. 41. Scholium. Every triangle is equal to the one half of a parallelogram, having tb« same base and altitude. Y 3 326 TO DETERMINE THE AREA Multiply 8.25 by (the half of 6.40, viz.) 3.20 10500 2475 Acres 2.6400O 4 Roods 2.5(5000 40 Perches 22.40(XX) A. R. P. Ans. 2 2 22. In the oblique angled triangle D E F, let the base D E be 12 chains and 25 links, and the perpendicular F G, let fall on the base D E from the opposite angle D F, be 7 chains and .30 links ; how many acres doth it contain ? Multiply 12.25 by (the half of 7.30, viz.) 3.65 6125 7350 3675 Acres 4.47125 4 Roods 1.8850(1 40 A. An?. 4 Perches 35.40000 R. P. 1 35. OF A TRIANGLE. 327 It would be the same thing, if the perpendicular were to be multiplied by half the base ; or, if the base and perpendicular were to be multiplied together, and half the product taken for the area. 4. The next most simple surface for admensuration is a four-sided figure, broader at one end than at the other, but having its ends parallel to one another, and rectangular to the base. Its area is determined by the following Rule. Add the breadths at each end together, and multiply the base by half their sum, and that product will be the area. Or otherwise, multiply the base by the whole sum of the breadths at each end, and tliat product will be double the area, the half of which will be the area required. Example. Let the base be 10 chains and go links, the breadth at the one end 4 chains and 75 links, and the breadth at the other end 7 chains and 35 links ; what is the area ? A.75X 7-35/ 2)12.10 sum 6.05 half sum 10.90 54450 6050 Acres d.59450 4 Roods 2.3/800 40 Perches 15.12000 K. ns. 6 R. P. 2 15. Y 4 32g TO DETERMINE THE AREA 5. Irregular four-sided figures are called trapezia. The area of a trapezium may be determined by the following Rule. Take the diagonal length from one ex- treme corner to the other as a base, and multiply it by half the sum of the perpendiculars, falling there- on from the other two corners, and that product will be the area. Example. In the trape- zium A B C D, let the di- agonal AC be 11 chains and 90 links ; the perpen- dicular B F 4 chains andAi 40 links, and the perpendicular D E 3 chains and go links i what is thie area ? B F 4.40 DE 3.90 2)8.30 sum 4. J 5 half sum AC 11.90 37350 415 415 Acres 4.93850 4 Roods 3.75400 40 Perches 30.l6000 A. K. P, Ans. 4 3 30. If the diagonal AC were multiplied by the whole sum of the perpendiculars, that product would be OP A TRIAVGIE. 329 iiouble the area ; the half of which would be the area required. 6. The area of all the other figures, whether re- gular * or irregular, of how many sides soever the figure may cousist, uiay be tlctermincd eitlier by di- viding the given figure into triangles, or trapezia, and measuring those triangles, or trapezia, separately, the sum total of which will be the area required; or ot/iirii'ise, the area of any figure may be determined by a computation made from the courses and dis- tances of the boundary lines, according to an univer- sal theorem, which will be mentioned presently. 7. The method of determining the area by divid- ing the given figure into trapezia and triangles, and measuring those trapezia' and triangles separately, as in article 5 and article 3 of this part, is generally practised by those laud measurers, who are employed to ascertain the number of acres in any piece of land, when a regular land surveyor is not at hand ; and for this purpose, they measure with the chain the * The regular ^gures, polygons and circles, do not occur in prac- tical surveying. (.)n this head, therefore, I sliall only obsei-ve, that the method of determining the area of a regular polygon, which is a figure containing any number of equal sides and equal angles, is, to multiply the length of one of the sides by the number of sides which the polygon contains, and then to multiply the product by half the perpendicular, let fall from the centre to any one of the sides, and this last product will be the area. The length of the perpendicular, if not given, may be determines by trigonometry, thus : Divide 360 degrees by double the number of sides contained in the polygon, and that quotient will be half the angle at the centre ; then say, as the tangent, or half the angle at tlie centre, is to half the length of one of the sides j so is radius, to the perpendicular sought. The area of a circle is thus determined ; if the diameter be given, say, as 1 : is to 3.1415p2 : : so is the diameter : to the cir- cumference ; or, if the circumference be given, say, as 3.J415p2 is to 1 : : so is the circumference : to the diameter. Then multiply the circumference by i of the diameter, and the product will be the area. 330 OF AN IRREGULAR FIGURE. bases and perpendiculars of the several trapezia and triangles in the field. Thus, for example, suppose the annexed figure be a field to be measured according to this method, the land measurer would mea- sure with the chain the base B G, and the per- pendiculars H C, and I A, of the trapezium p ABCG; the base GD, and the perpendiculars K F, and L C, of the trapezium G C D F ; and the base D F, and perpendicular M E, of the triangle D E F. In order to determine where on the base lines the perpendiculars shall fall, the land measurer com- monly makes use of a square, consisting of a piece of wood about three inches square, with apertures there- in, made with a fine saw, from corner to corner, crossing each other at right angles in the centre ; which, when in use, is fixed on a staff, or the cross ; see jigs. 2, 3, and 6, plale 14. The bases and perpendiculars beingthus measured, the area of each trapezium is determined as in arti cic 5, and the area of the triangle as in article 3 of this part ; which, being added together, gives the area of the whole figure required. This method of determining the area, where the whole of each trapezium and triangle can be seen at one view, is equal in point of accuracy to any method whatever. It may likewise be observed, that a plot or map may be very accurately laid down from the bases and perpendiculars thus measured : this me- tliod, however, is practicable only in open or cleared OF AX IRREGULAR FIGURE. 331 countries ; in countries covered with wood, it would be altogether iinpracncablc, because it would be im- possible to sec from one part of the tract to another, so as to divide it into the necessary trapezia and tri- angles. Ikitin open countries, the only objection to this method, is the great labour and tediousness of the field work ; in a large survey, the number of the trapezia and triangles would be so very great, that the measuring of all the several bases and perpendi- culars would be a very tedious business, 'ihe same ends are as eftcctually answered, with infinitely less labour, by taking the courses and distances of the boundary lines, in the manner mentioned p. 290. 8. Many surveyors who take their field notes in courses and distances, make a plot or maj) thereof, and then determine the area from the plot or map so made, by dividing such plot or map into trapezia and triangles, as in the above figure, and measuring the several bases and perpendiculars by the scale of equal parts, from which the plot or map was laid down. By this method, the area will be determined some- thing nearer the truth, but it falls short of that ac- curacy which might be wished, because there is no determining the lengths of the bases and perpendi- culars by the scale, w'thin many links ; and the smaller the sciile shall be by which the plot is laid down, the greater, of coarse, must be the inaccuracy of this method of determining the area. This method, however, notwithstanding its inac- curacy, will always be made use of where tables of the northing, southing, easting and westing, are not at hand. 9. Proper tables of the northing, southing, east, ing and westing, fitted purposely for the surveyor's use, such as are contained in the Af>pend'ix to this 7ir^^/, ought to be in the hands of every surveyor. By these tables the area of any survey is determined with )32 AN UNIVERSAL THEOREM. great accuracy and expedition, by an easy computa- tion from the following UNIVERSAL THEOREM. If the sum of the distances, on an east and west Ihie, of the two ends of each line of the survey, from any assumed meridian laying entirely out of the survey, le multiplied hy the respective northing 6;south- 1 N G 77uide on each respective line ; /^^ d i f f e r e x c e between the sum of the north vrotivcis, and the sum of the sovTH products, will he double the area of the survey.* d * Demonstration. Let ABC ^ D E F be any survey ;, and let ^ the east and west lines A a, B b, C c, D d, E e, and F f, be drawn from the ends of each of ^ the lines of the survey to any assumed meridian, as N. S. lay- ig entirely out of the survey. / CO ^.J^ Then. [ say, the difTcrcnce between the £um of the products A a-4-B bx a b, B b+C ex b c, C c+D dx c d 5 and the sum of the products Dd+EeXde, Ee-f-Ffxef, Ff+Aaxfa3 will be double the area of A B C D E F A. For, by article 4,page327,the former productswill be, respectively, double the areas of the spaces A E b a, B C c b, and C D dc, and their sum will of course be double the area ot the space A a d D C B A. In like m.anner, the latter products will be, respectively, double the areas of the spaces D E e d, E F f e, and F A a f j and their sum will of course be double the area of the space A ad DEFA. But it is evident, that the difi'erence between the areas of the spaces A a d U E F A, and A a d D C B A, must be the area of the space A B C D E F A ; consequently, the ditterence between their doubles must be double the area of A B C D E F A, AN UNIVERSAL THEOREM, 333 For example. Let it be required to determine the area of the survey mentioned in p. 200, from the field notes there given. The northing;s, southings, castings and westings, as also the north anJ smith Jistanccs^ and the east utul ivcit JisljHces made from the place of beginning to the end of each line ; and likewise the distances of the ends of each line from the assumed meridian, have been alrcaily placed in their respective columns, and explained in the preparatory to the explanation of the method of plotting there recommended ; but it will not be improper to give them a place here again, together with the further columns for deter- mining the area ; in order that the whole, together with the respective uses of the several parts, may ap- pear under one view, as follows. Again, the differen ce between t he sum of the products M m-i-N nX m n, N n+O o x n o, Q q+R rX q r ; and the sum of the products p Q o-»-P p Xo p, P p+Q qXp q, Rr-)-Mnixrm; will be double -y the area of M X (3 P a R M. For, reasoning as before, the sum of the first mentioned products will be double the area of Q q o O N M R QA^yi 11 r m Is/l; and the stim of the l ast mentioned products will be double the area of U q o O P U-j-iM R r m TVl. But it is cvidtnt, that Qqo O P Q— Q q o ,•£ "^ 5 G rt ti l-s «f i"^ 5i a J rj E u o WW WWW WW O 0) 1 ■c -= OJ c *^ — o U: 2- &!.•; ?^ Z: Z Z ;z; ^ C r-t 13 ■- C t« R^ 8 ^ ^ r< 3 C _a) •9 "• Ff -3 r! cr ,/ re &l; J '^. 1) o ■o r^ o c^ (O > -o o O 0^ c? p» •^ t^ CO CM lo t^ ^ ..- o t- o t^ ■~ 'f (>> rl or) r- r- o< 1^ o o to CO CO ;i o ii rt 3 -s _ ■;: ^ ■5 E *" ^ = " £ 3 ;r; t/T-^ ►: *■ e ^ So-j rt ^ g c c g » ^ ^ 5^ ■" t« ST- "-' !^ C/ •— bc r- *-> •^ t^ •^ C TT" --^ ^ - C S 5f 2 ji V rz 0) .- ■>-' ^- '^ '♦^ *:i ^ '^c: s s ?i s u 2 ^WK^W^!^ '-'CO ^ ■*• oo "^ c» c -* ^ "-^ 'O ^o ^ t^ o 2 2 -■/ w: x' Z ji •— »« ci •* lO (O C^ EXPLANATION, &C. 335 The three last columns relating to the area, are the only ones that remain to bi- explained. The first of these three columns contains the sums of the distances of the two ends of each line of the survey from the assumed meridian ; which, according lo the foregoing universal theorem, are multipliers for determining the area. 'J hese are formed by adding each two of the successive num- bers in the preceding left hand column togctlicr. Thus 3.00 is the distaiice of the j)lace of beginning from the assumed meridian, and 0.44 is the dis- tance of the end of the first line from the assumed meridian ; and their sum is 3.44 for the first mul- tiplier. Again 0.44 added to 15.39 makes 15.83 for the second multiplier ; and 15.39 added to 28.1{> makes 45.15 for the third multiplier ; and SO of the rest. The north products contained in the next co- lumn, are the products of the muUiplication of the several northings contained in the columns marked N, by their corresponding multipliers in the last mentioned column. Thus 7.1t)89() is the pro. duct of the multiplication of 20.84 by 3.44; an ad- ditional place of decimals being cut off, to give the product in acres. The south products contained in the next ri^ht hand column are, in like manner, the products of the multiplication of the several yoz/////,'/^j- in the column marked b, by their corresponding multipHcrs. 'j hus 28.96075 is the product of the multiplication of ().0'5 by 43.55; an additional place of decimals beino- cut off, as before, to give the product in acres; and so of the rest. The north products being then added toc^etiier into one sum, and the south products into another sum, and the lesser of these sums being subtracted from the greater, the rettuiineler, by the above uni- versal theorem, is double the area of the survey: which being divided by 2, gives the area required ; vi/.. 336 THE FIELD BOOK. 51.73087, equal, when reduced, to 51 acres, 2 roodj?, and 36 perches 9-IO.* Nothing can be more simple or easy in the ope- ration, or more accurate in answering the desired ends, both with respect to the proof of the work, the plotting, and the computation of the area, than this process. The northings, southings, castings and westings, which give the most decisive proof of the accuracy or inaccuracy of the field work, are shewn by inspection in table 1. From these the numbers or distances for plotting are formed by a simple addition or subtraction. And the multipliers for determining the area, are formed from thence, by a simple addition, with much less labour, than by the common method of dividing the plot into trapezia, and taking the multipliers by the scale of equal parts. The superior accuracy and ease with which every part of the process is performed, cannot, it is ima- 2;ined, fail to recommend it to every practitioner, in- to whose hands this tract shall fall. 10. If the boundary lines of the survey shall have crooks and bends in them, which is generally the case in old settled countries, those crooks and bends are taken by making offsets from the station lines ; as explained in page 29 1. In these cases the area comprehended within the station lines is determined as in the last example ; and the areas comprehended between the station lines and the boundaries are de- termined separately, as in the third and fourth ar- ticles of this part, from the offsets and base lines, noted in the field book ; and are added or subtracted * It must be observed, that, in order to determine the area by this method of computation, it is indispenaibly necessary that the several lines of the survey be arranged regularly, one after another, all the way round the survey, in case they should not have been so^ taken in the ^leld, because, withoiit such arrangement, the north- ings, or southings made on each line, and the distances Irum, the as- suijied meridian, would not correspond \yith one another. UNIVERSAL THEOREM. 337 rcspcctivclv, accnnline: as the station lines sliall happen to be will/in or ivilhout the field surveyed ; wliieh. o'conrse, , (Ijcing the difference between 15. Go and 21.00 on the station line.) and a perpendicidar off- set 0.(J5 ; which, according to the 3d article of this part, contains an area of 0.17550. These added together make 0.63<}87 for the area of the offi^ets on the first station line ; and, as this station line is z-jithin the survey, this area must, of course, be aJJtJ to the above-mentioned area comprehended wilhin the station lines. Next, for the area comprehended between the se- cond station line and the boundary of the field, we * A [dot urthi^ .survev is given in f^. 3, pht^ 13. z ^ 338 EXAMPLE TO THE have a base line 18.20, and a perpendicular or ofiTset 0.60; which, according to article 3 of this part, con- tains an area of 0.54600. And, as this station line is without the survey, this area must of course be suhtracted ^xom the above-mentioned area. The areas of all the other offsets are determined in the same manner, and are respectively as follows; viz. On ihtjirst station line On the second On the third On ihe fourth On \\\e. fifth On the sixth On the seventh Totals The area comprehended as above Offsets to be added . . . Areas of the Offsets. j To be added. A. 0.63687 To be sub- tracted. 0.54600 0.64800 0.36250 1.08875 0.68750 0.27510 2.08812 2.15660 within the station lines. Sum Offsets to be subtracted Remains the area of the A.51.7.3O87 2.08812 53.S18.Q(} 2.15660 survey required A. 51.66239 4 Roods 2.64956 40 Perches 25.98240 11. When the survey consists of a number of fields lying together, it is best to determine the area of the 'whole first ; and afterwards, the area of each of the fields separately ; the sum of which, if the work be true, will of course agree with the area of the whole ; which forms a check on the truth of the computations. I UNIVERSAL THEOREM. 339 But here it must be observed, that the boun- daries of eaeh field must be arranged for the com- putation, so as to proeccd reguhirly one after an- other all the way round the field ; because, as was observed in the note to the gth article of this part, without such arrangement, the northing or southing made on each line, and the distances from the assumed meridian, would not correspond with one another. For example. Suppose it were required to ascer- tain the area of the survey mentioned in page 294, of the several fields therein contained, from the field book thcie given.* 'i he bearings and lengths of the station lines on the outside boundaries are as follow; viz. • ' ch. 1. 1st. S. 87 W 15.00 2(1. N. 18 30 W 9.60 3d. N. 12 15 E 8..gO 4W\. N. 37 E 17.00 5th. S. 76 45 E 16.00 dth. S. 5 15 W 25.00 7th. S. 74 30 W 7.Q5 The area comprehended within these station line.<;, comjmttd according to the above-mentioned universal theorem, is A. 66.90493 The area of the offsets to be added 0.56/15 (computed as in the last example) ■ — • Sum 67.47203 The area of the offsets to be subtracted 2.56715 I'hc area of the whole survey Acres 64.90493 t Roods 3.61972 40 Perchei .... 24.78880 Se^ the plot of this sur\cy injfp^. 4, plate 18 Z2 340 EXAMPLE TO THE Then, for determining the area of the east field, we have the following lines, viz. 1st. Part of the first ?tation line in the ch. 1. field book, S. 8/° W. 5.50 2d. The offset to the corner of the field, N. 30"" W. 1.10 3d. The eighth .station line in the field book, North — 16.61 4th, The ninth station line in the field book, East — IS.// 5th. The offset from the 'corner of the field to the sixth station line in the field book, S. 8-4° 45' E. 0.56 6th. Part of the sixth station line in the field book, S. 5° 15' W. 15.40 7th. The seventh station line in the field book, S. 74° 3 / W. 7.65 The area comprehended within these lines, com- puted, according to the foregoing universal theo- rem, is ;•••.••: A.22.91936 There are no offsets to be added in this field, the areas of the offsets to be subtracted, computed as in the last example, are 2.22052 Remains the area of the east field. Acres, 20.6£)884 4 Roods 2.79536 40 Perches 31.81440 For ascertaining the area of the west field^ we have the following lines, viz. 1st. The offset from the south-easterly corner to the first station line in the ch. 1. field book, S. 3° E. 1.10 2d. Part of the first station line in the field book, S. 5/° W. 9.5O 3J. The second station line in the field book, N. 18° 30'W, 9.60 4th. The third station line in the field book, N. 12° 15' E. 8.90 5th. Part of the fourth station line in the fieM book, N. 37° E. 5.40 ()th. The offset to the corner,, S. 53° E. 04O UNIVERSAL THEOREM. 341 7th. The tenth station line in the ficKl ch. 1. bonk, reversed, . S. 0'2° 30' E. y .[)2 8th. The eiqhth statiim line in the ticld book, reversed Soutli — lO.Oi The area comprehended within those lines, com- puted according to the foregoing universal theo- rem, is A.22. 1 Sai8 The areas of the otVsets to be added, computed as in the last example, 0.6228/ Slim 22.S0;}35 The areas of the offsets to be subtracted I.159OO Remains the area of the west field, Acres 21 .64435 4 Roods 2., J 740 40 Perches 23.09600 For ascertaining the area of the north field, wc have the tollowing lines, viz. 1st. The offset from the south-westerly corner to the fourth station line in the ch. I field book, N. 53° \V. 0.40 2d. Part of the fourth station line in the field book, N. 37° E. 11.60 3d. The fifth station line in the field book, S. 76° 45' E. 16.00 4th. Part of the sixth station line in field book, S. 5^ 15' W. 9.60 5th. The olfset to the corner, N. 84" 45' W. 0,56 6th. The ninth station liti« in the field book, reversed West — 13-77 /th. The tenth station line in the field book, N. 62° 30' W. 7.92 The area comprehended within these lines, com- puted according to the following universal theo- rem, is A.21. 81422 The areas of the offsets to be added, computed as in the last examj)le, 1 .25030 Sum.' 23.00452 Z3 342 MARITIME SURVEYING. The areas of the offsets to be subtracted 0.50060 Remains the area of the north f eld. Acres 22.56372 4 Roods 2.2548S 40 Perches 10.K|520 A. 11. P. The area of the east field, 20 2 31 West field, 21 2 23 North field, 22 2 10 Total A. S4 3 34 The agreement of this total with the area of the whole, as first ascertained, shews the work to be right. We must not, however, in actual practice expect to find so perfect an agreement as the above. A difference of a few perches, in actual practice, is no impeachment either of the field work, or of the com- putations. For other methods of surveying, and a new plan of a field book, see the Addenda to this work. OF MARITIME SURVEYING. It was not my intention at first to say any thing concerning iniiritimc surveying, as that subject had been very well digested by Mr. Murdoch Machenz'ie^ whose treatise on Marilime Surveying ought to be in every person's hands who is engaged in this MARITIME SURVEYING. 343 branch of siirvcyinf^, a brancli which has been hi- therto too much neglected. A few general j)rin- ciples only can he laid down in this work ; these, however, it is presumed, will be found sufficient for most purposes ; when the practice is seen to be easy, and the knowledge thereof readily attained, it is to be hoped, that it will constitute a part of every sea- man's education, and the more so, as it is a subject in which the safety of shipping and sailors is very much concerned. 1. Make a rough sketch of the coast or harbour, and mark every point of land, or particular varia- tion of the coast, with some letter of the alphabet ; either walk or sail round the coast, and fix a staff with a white rag at the top at each of the places marked with the letters of the alphabet. If there be a tree, house, white cliff, or other remarkable object at any of these places, it may serve instead of a sta- tion staff.* 2. Choose some level spot of ground upon which a right line, called a fundamental base, may be mea- sured either by a chain, a measuring pole, or a piece of log-line marked into feet ; generally speaking, the longer this line is, the better ; its situation must be such, that the whole, or most part of the sta- tion-staves may be seen from both ends thereof; and its length and direction must, if possible, be such, that the bearing of any station-staff, taken from one of its ends, may differ at least ten degrees from the bearing of the same staff taken from the other end; station-staves must be set at each < nd of the fundamental base. If a convenient right line cannot be had, two lines and the interjacent angles may be measured, and the distance of their extremes found by construction, may be taken as the fundamental base. If the sand measured has a sensible and gradual * See Nicholson's Navigator's Assiatant. Z4 344 MARITIME SURVEYING. declivity, as from high-water mark to lovv-vcater. then the length measured may be reduced to the horizontal distance, which is the proper distance, by making the perpendicular rise of the tide one side of a right-angled triangle; the distance measured along the sand, the hypothenusc ; and from thence finding the other side trigonometrically, or by pro- traction, on paper ; which will be the true length of the base lina. If the plane measured be on the dry land, and there is a sensible declivity there, the height of the descent must be taken by a spirit-level, or by a quadrant, and that made the perpendicular side of the triangle. If in a bay one straight line of a sufficient length cannot be measured, let two or three lines, forming angles with each other, like the sides of a polygon, be measured on the sand along the circuit of the bay ; these angles carefully taken with a theodolite, and exactly protracted and calculated, will give the straight distance between the two farthest extremi- ties of the first and last line. 3. Find the bearing of the fundamental base by the compasses, as accurately as possible, with Had- ley's quadrant, or any other instrument equally ex- act ; take the angles formed at one end of the base, between the base line and lines drawn to each of the station-staves ; take likewise the angles formed be- tween the base line and lines drawn to every re- markable object near the shore, as houses, trees, windmills, churches. &c. which may be supposed useful as pilot marks ; from the other end of the base, take the angles formed between the base and lines drawn to every one of the station staves and ob- jects; if any angle be greater than the arc of the qua- drant, measure it at twice, by taking the angular distance of some intermediate object from each ex- treme object ; enter all these angles in a book as they are taken. 4. Draw the fandamental base upon paper from a MARrTIML SURVERIXG. 345 scale of equal parts, and from its ends respectiveJy draw unlimitted lines, forming with it the angles taken in the snncy, and mark the extreme of each line with the letter of the station to which its angle corresponds. The intersection of every two lines, whose extremes are marked with the same letter, will denote the situation of the station or object to which, in the rongh dranght, that letter belongs; through, or near all the points of intersection which represent station-staves draw a waving line with a pencil to re- j)resent the coast, 5. At low water sail about the harbour, and take the soundings, observing whether the ground be rocky, sandy, shelly, &c. These soundings may. be entered by small numeral figures in the chart, by taking at the same time the bearings of two remark- able objects ; in this excursion, be particular in cx- aminmg the ground oft' points of land which project out into the sea, or where the water is remarkably smooth, without a visible cause, or in the vicinity of small island, Sec. observe the set and velocity of the tide of flood, by heaving the log while at an- chor, and denote the same in the chart by small darts. The time of high water is denoted by Roman numeral letters ; rocks are denoted by small crosses ; sands, by dotted shading, the figures upon which usually shew the depth at low water in feet ; good anchoring places arc marked by a small anchor. Upon coming near the shore, care must be taken to examine ami correct the outline of the chart, by observing the inflections, creeks, &e. more mi- nutelv. 6. In a small sailing vessel go out to sea, and take drawings of the a[)pearanee of the land, with its bearings ; sail into the harbour, observe the ap- ])earancc of its entrance, and particularly whether there be any false resemblance of an entrance by which ships may be deceived into danger ; remark 340 MARITIME SURVEYING. the signs or objects, by attending to which, the har- bour may be entered in safety; more especially where it can be done, let the ship steer to the anchoring place, keeping two remarkable objects in one, or on a line. 7. Coasts are shaded off on the land side ; houses, churches, trees, S:c. on sliore, are drawn small in their proper figures ; in a proper place of the chart, draw a mariner's compass, to denote the situation of the rhumbs, &c. ; and on one side of the flower-de-luce, draw a faint half flower-de-luce at the point of north by compass ; draw also a divided scale of miles, or Iciagues, which must be taken from the same original scale as the fundamental base. Charts or plans may be neatly drawn with Indian ink, or the pen and common ink, and are as useful as any others, but they are frequently done in water colours; for which purpose the beginner will derive more advantage from viewing a proper drawing, or from overlooking a proficient at work, than from a multitude of written instructions. ^n example to illiistrate the foregoing precepts. Let ABC D E F G H, fig. 1 1, plate 23, represent a coast to be surveyed, and I K L, an adjacent island; it is required to make an accurate chart of the same. To 7nake the actual ohservat'wn on shore. B3' sailing or walking along the shore, a rough sketch is made, and station staves set up at the points of land, A, 1;, C, D, E, F, G, H, and al'^o at I, K, L, on the island, precept 1. During this operation, it is observed, that no proper place offers itself at which a funda- mental base can be drawn, so as to command at once a view of all the stations ; it will therefore, be necessary to survey the coast in two separate parts, by making use of two. base lines ; if the coast had been more extended, or irrea;ular, a oreatcr niimbcr MARITIME SURVEYING. 34/ of base lines might have been necessary. Between the points B ancl C, the ground is level, and a eon- sidcrahle nuinbcr of the soiith'-rnmost points may be seen from l)olh points, conformable to precept 2 ; make B C, therefore, the first fundamental base, its length by admeasurement is found to be 812 fa- th(Mns, and its hearing by compass from B to C, is N. 11° 14' E. from each end of this base, measure with Hdciltys cjuadrant or sextant the angles formed between it, and lines drawn to each station in sight, precept 3, enter, or, tabulate them as follows : First fundamental base B 0=812 fathoms, bear- ing of C from B,N. ll°U'E. Angle A B C = 1 ?>5 07 Ai igle A C B — 22 54 L B C = 85 00 L C B — 7 1 02 KBC= 561:8 KCB— 91 40 E B C = 43 32 E C B -77. 1 1 3 50 DBC= 13 30 DC B— 145 20 After having made these observations, it will be re- quisite to proceed to the northern part of the coast ; in all cases where a coast is surveyed in separate parts, it is best to measure a new fundamental base for each part, when it can be conveniently done ; a line from the station E drawn towards F, is well adapted to our purpose ; let E P, therefore, be the second base line, its length by admeasure- ment is found to be 77«S fathoms, and its bearing by compass from E to P, N. :)8° 20' E. Measure the angles formed by lines drawn from each end of tljis base, as before directed, and tabulate them as follows : Second fundamental base E P = 778 fathoms, bearing of P, from E. N. 38° 20' E. 348 GEOMETRICAL CONSTRUCTION. Angle H E P = 88 30 Angle H P E = 73 00 I EP = 7l 3^^ I P E = 28 54 G E P = 43 b9 G P E = 1 1 1 00 FEP = 27 51 FPE = 127 18 It is Sufficiently apparent, that the connexion be- tween the two parts of this survey is preserved by the second fundannental base being drawn from the point E, whose situation was before determined by obser- vations from the first base line ; if this particular po- sition of the second base had not been convenient, and it had been taken at a distance from ev^ery point determined in situation tVom the first base, the con- nexion would have required an observation of the bearing of one of the said points from each end of the second base : thus, suppose the line I P to be the second base line, instead of E P ; the position of I P with respect to the given point E, may be known by taking the bearings of K, from I and P. All the observations which are required to be made on shore, befng now completed, it will be advisable to construct the chart before we proceed to the other observations on the water, for, by this method, an op- portunity is offered of drawing the waving line of the coast with more correctness than could otherwise be obtained. GEOMETRICAL C0X6TR U CTION, ^^. \2, pldie 1^. On the point B as a centre, decribe the circle n e s w, the diameters n s and w e being at right an- gles to each other, and representing the meridiai^ and parallel ; from n set off the arc n N, equal to the quantity of the variation of the compass ; draw the diameter N S, and the diameter W E at right an- gles to the same ; these lines will represent the mag- netic meridian, and its parallel of latitude. GEOMETRICAL CONSTRUCTION'. 34g From the centre B, draw the first base line B C =:= 8r2 fathoms, N. 1 1'' 1 1' E. by compass : tioin one extremity B, draw the unlimitcfl lines B a, B I. B k, B e, B d, forming, respectively with the base, the anirlcs observed fmm thence ; fmm the other extremity C, on this the whole instrument moves when in use. To the longer instruments are sometimes placed twcf moveable rollers, to support and facilitate the mo- tions of the pantographer; their situation may b6 varied as occasion requires. The graduations are placed on two of the rules, B and D, with the proportions of -i, |, ^, &c. to V-z* marked on tliem. The pencil holder, tracer, and fulcrum, must in all, rases be in a right line, so that when they are set to any number, if a string be stretched over them, and they do not coincide with it, their is an error either in the setting or graduations. 1 he long tube which carries the pencil, or crayon, moves easily up or down in another tube ; there is a string affixed to the long, or inner tube, passing af- terwards through the holes in the three small knobs to the tracing point, where it may, if necessary, be fastened. By pulling^ this string, the pencil is lifted OF THE PANTOGRAPHER. 357 «p occasionnlly, and thus prevented from making false or improper marks upon die copy. To reduce in any of the proportions l\ ■\, t, f, ^c as marked on the livo hiirs B, ttud D. Suppose, for ex- ample, V i^ rCijuired. I'lacc the two sockets at t on the bars B and D. l*lace the fulcrum, or lead weight at B, the pencil socket, wiili pencil at D, and the tracing point at C. Fasten down upon a smooth board, or rablc, a sheet of white paper under the pen- cil I), and the original map, &c. under the tracing point C, allowing yourself room enough for the va- rious openings of the instrument. Then with a steady hand carefully move the tracing point C over the outlines of the map, and the pencil at I) will de- scribe exactly the same figure as the original, but -^ the size. In the same manner for any other pro- portion, by only setting the two sockets to the num- ber of the required proportion. The pencil-holder moves easily in the socket, to give way to any irregularity in the paper. There is a cup at the top for receiving an additional weight, either to keep down the pencil to the paper, or to in- crease the strength of its mark. '1 here is a silken string fastened to the pencil-hol- der, in order that the pencil may be drawn up off the paper, to prevent false marks when crossing the original in the operation. If the original should be so large, that the instru- ment will not extend over it at any one operation, two or three points must be marked on the original, and the same to correspond upon the copy. The fulcrum and copy may then be removed into such situations as to admit the copying of the remaining part of the original ; first observing, that when the tracing point is applied to the three points marked oi; the original, the pencil falls on the three correspond- ing points upon the copy. In this manner, by re- peated shiftings, a pentagraph may be made to copy an original of ever so large dimensions. A a 3 358 DESCRIPTION OP THE To enlarge in any of the propGrtio7is f, f, 7, ^c. Suppose Y, You set the two sockets at f, as before, and have only to change places between the pencil and tracing point, viz. to place the tracing point at D, and the pencil at C. To copy of the same size, but reversed. Place the two sockets at 4:? the fulcrum at D, and the pencil atB. There are sometimes divisions of 100 unequal parts laid down on the bars B and D, to give any in- termediate proportion, not shewn by the fractional numbers commonly placed, Pentagraphs of a greater length than two feet are best made of hard wood, mounted in brass, with steel centres, upon the truth of which depends entirely the equable action of this ivseful instrument. o p LEVELLING. The necessity of finding a proper chanipel for conveying water occurs so often to the surveyor, thiiL finy work on that subject, which neglected to treat on the art of levelling, would be manifestly imper- fect ; I shall therefore endeavour to give the reader a satisfactory account of the instruments used, and the mode of using them. A DESCRIPTION OP THE BEST SPIRIT LEVEL. Fig. 3, plale IJ , represents one of the best con- structed Spirit levels, mounted on the most com- BEST SPIRIT LEVEL. 359 plctc Staves, similar to those aflixctl to a best theo- dolite. The achromatic telescope, A, B, C, is moveable, cither in the plane of the horizon, or with a small in- clination thereto, so as to cut any object whose ele- vation, or depression, from that plane, does not ex- ceed 12 degrees; the telescope is about two feet long, is furnished with fine cross wires, the screws to adjust which are shewn at^, for determining the axis of the tube, and forming a just line of sight. By turning tiic milled screw 1>, on the side of the tele- scope, the object glass is moved outwards, and thus the telescope suited to dinbrcnt eyes. The tube c c with the spuit bubble 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 the ad- justinent. The two supporters D, E, on which the telescope is placed, are nearly in the shape of the letter Y, the inner sides of the Y's are tangents to the cylindric ring of the telescope. The lower ends of these supporters are let perpendicularly into a strong bar ; to the lower end of the support E, is a miiled nut F, to bring the instrument accurately to a level ; and at the other end of the bar at H, is a screw for tightening the suj)port 1) at any height. Between the two supports is a compass box G, divi- ded into four quarters of 90° each, and also into 3(3o°, "with a mngnelical needle, and a contrivance to throw the needle off the centre when it is not used ; and thus constituting a perfect circumferentor. The compass is in one piece with the bar, or is sometimes made to take on and off by two screws. To the under part of the compass is fixed a conical brass ferril K, which is fitted to the bell-metal frus- tum of a cone at the top of the brass head of the staves, having at its bottom a ball, moving in a socket, in the plate fixed at the top of the three metal joints for the legs. L, L, are two strong brass parallel A a 4 360 DESCKIPTION OF THE plates, with four adjusting screws, h, h, h, which are used for adjusting the horizontal motion. The screw at M is for regulating this motion, and the screw N for making fast the ferril, or whole instrument, when necessary. By these two screws the instrument is either moved through a small space, or fixed in any position with the utmost accuracy. The staves being exactly the same as those applied to the best theodo- lites, render any further description of them here unnecessary. It is evident from the nature of this instrument that three adjustments are necessary. 1. To place the intersection of the wires in the telescope, so that it shall coincide Vv'ith the axis of the cylindrical rings on. which the telescope turns. 2. To render the level parallel to this axis. 3. To adjust for the hori- zontal motion quite round upon the staves. To adjust the cross wires. If* while looking through the telescope at any object, the intersection of the wires does not cut precisely the same part during a revolution of the telescope on its axis, their cidjust- ment is necessary, and is easily obtained by turning the little screws tf, <3, a. The two horizontal screws &re to move the vertical wire, and act in opposite di- rections to er.eh other ; one of v.diich is to he tight- ened as the opi^osiie is slackened, (this not being at- tended to, v.'ili endanger the breaking of the wire) till the wire has been moved sufficiently. The up- per screw to the horizontal wire is generally made with a capstan head, so that by simply turning it to the right or left hand, the requisite motion of the wire is produced, and thus the intersection brought exactly in the axis of the telescope. To adjust the spirit level at ordy one station. When the spirit level is adjusted to the telescope, the bub- ble of air will settle in the middle, or nearly so, whe- ther the telescope be reversed or not on its supports D, E, which in this case are not to be moved. The BEST sriRlT LEVFL. 36l w hole level being placed lirinly on its staves, the bub- ble ot'air brought lo the inidiiic by turning the serew F, the rims /", /, of the Y's open, and when the tele- scope is taken otl' and laid the contrary way on its supports, should the bubble ot' air not come to rest in the middle, it then proves that the spirit level is not true to the axis of the tube, and requires adjust- ment. The end to which the bubble of air goes must be noticed, and the distance of the bubble case, and heiglit of one support, so altered by turning tim screws at c and F, till by trial the bubble comes to the middle in both positions of the telescope. This very ficiic mode of adjustment of the level is one great improvement in the instrument ; for, in all the old and common constructed levels, which did not admit of a reversion of the telescope, the spirit level could only be well adjusted by carrying the instru- ment into the field, and at two distant stations ob- serving both forwards and backwards the deviation upon the station staves, and correcting accordingly ; which occasioned no small degree of time and trouble not necessary by the improved level. To aJJusf for ihc horizo7ital motion. The level is said to be com [)lctely* adjusted, when, after the two previous ones, it may be moved entirely round upoa its stafF-head, without the bubble changing materi- ally its place. To perform which, bring the telescope over two of the parallel plate screws z^*, ^, and make it level by unscrewing one of these ser-ews while you are screwing up the 0{)posite one, till the air bubble is in the middle, and the screws up firm. Then turn the instrument a quarter round on its staft^j till tho telescope is directly over the other two screws ^, h, unscrewing one screw, and screwing up at the same time the opposite one, as before, till the bubble sct^ tics again in the middle. The adjustment of the staff plates arc thus made for the horizontal motion of the instrument, and the telescope may be moved 362 DESCRIPTION OFj &C. round on Its staff without any material change of thci place of the bubble, and the observer enabled to take a range of level points. The level tube is also made to adjust in the horizontal direction by two opposite screws at its joint e, so that its axis may be brought into perfect parallelism to that of the telescope. The telescope is generally glassed to shew objects erect. In the common sort of levels with a shorter telescope from 12 to "20 inches in length, and fiilh- out a circumfcrentor, they are glassed to shew the ob- jects inverted. To an expert observer this will make no difference; and, there being but two eye-glaases instead of four, as in the other, about three or tour inches are saved in the length of the telescope. A short brass tube, to screen the sun's rays from the object glass, is sometimes made to go on the ob- ject end c of the telescope, and a screw-driver, and istee: pin for the capstan- headed screws are packed in the same case with the instrument.* • Fig. 10, is a representation of a brass mounted pocket spirit level, about six inches in lengthy with sights, having a ground bot torn, intended tor ordinary purposes cither in levelling a plane or de- termining level points. When about 12 inches in length, with double sights and adapted to a staff", it may serve for conducting small parcels of water, or draining a field, &c. Fig. II, is one that 1 have constructed with some improvement. The brass perpendicular piece A, is made to slide en cccabionally by a dovetail. Ey this piece, and the bottom of the level together, any standing square pillar, or other object^ may be set to the level, and perpendiciilar at the same time. The case of the spirit level B, swings on two pivots; the horizc'nial position, therefore, of a ceil- ing or under surface of a plane may be as readily obtained as any inferior plane in the common way ; fhe spirit level thereby always remaining the same, and the position of the base only changing to admit of a contact with the plane to be levelled. In which case also the perpendicular side A, becomes equally useful. The adjustment of these pocket levels is very easily proved, or made, by bringing the bubble in the middle upon any table or base ; if, upon reversion afterwards on the same place precisely, the bubble keeps to the middle, it is adjusted ; if not, by means of a screw-driver turn one of the screws at the proper end, till it he fo jraised or depressed as to cause the bubble to stand the rcAtrsing, at OP LEVELLTSG, 363 Two m;ihogany station staves often accompany this instrument ; they consist of two parts, each part is about five feet ten inches long, so that whcfi tbry arc pulled out to their greatest extent, they form a ten feet rod; and every ioot is divided into 100 equal jjarts. To each oi' these staves there is a sliding vane A, y?^. 9, pla^e 17, with a brass wire across a square ho;c made in the vane ; when this wire coinc-idcs with the horizontal wire of the telescope, it shews the height of the apparent level above the ground at that place. OF LEVELLING. Levelling is the art which instructs us in finding how much higher or lower any given point on the surface of the earth is, than another given point on the same surface ; or in other words, the difference in their distance from the centre of the earth. Those points arc said to be le-velj which are cqui- distantfrom the centre of the. earth. The art of level- ling consists, therefore, 1st. In finding and marking two, or more level points that shall be in the circum- ference of a circle, whose centre is that of the earth. 2dly. In comparing the points thus found, with other points, in order to ascertain the difference in their distances from the earth's centre. heljig. 1, plute 23, represent the earth ; A, its centre ; the points 1^, C, i), E, F, upon the circum- ference thereof are level, because they are equally distant from the centre ; such arc the waters of the sea, lakes, 6om B, of the radius B A, two equal lines B C, B C, Jig. (3, flale 23, be drawn, making equal angles, C B A, DBA, then will Q and D be equally distant from the centre; thouglr the level mav be obtained by these oblique lines, yet it is far easier to obtain it by a line perpendicular to the radius. When the level line is perpendicular to the radius, and touches it at o;ie of its extremes, the other ex- tremity will mark the apparent level, and the true level is found by knowing how much the apparent one exceeds it in height. 'Jo find the height of the apparent above the true level for a certain distance, square that distance, and then divide the product by the diameter of the earth, and the quotient will be the required difference ; it follows clearly, that the heights of the apparent level at ditfcrent distances areas the squares of those dis- tances ; and, consequently, that the difrerence is f^reater, or smaller, in proportion to the extent of the ine ; for the extremity of this line separates more from the circumference of the circle, in proportion as it recedes from the point of contact. 1'iius, let 366 DIFFERENT METHODS OP A, Jig. 7, plale 23, be the centre of the earth, BC the arc which marks the true level, and B, E, D, the tangent that marks the apparent level ; it is evident, that the secant A D exceeds the radius A B, by C D, which is the difference between the apparent and true level ; and it is equally evident, that if the line extended no flirther than E, this difference would not be so great as when it is extended to D, and that increases as the line is lengthened. When the distance does not exceed 25 yards, the difference between the two levels may be neglected ; but if ii be 50, iOO, &c. yards, then the error result- ing from the difference will become sensible and re- quire to be noticed. A TABLE, Which shews the quantity of curvature below the apparent level in inches, for every chain up to J 00. tr Inches 5' nrl Inches ! ^ 5' Inches Inches ] 0,(X)125 14lo,24 27 0,91 40 2,00 2 0,005 15 0,23 2B 0,98 45 2,28 3 0,01125 1 6 0,32 \W 1 ,05 50 3,12 4 0,02 17|0,3t) 130 1,12 55 3,78 5 0,03 18,0,40 31 1,19 t)0 65 4,50 6 0.D4 19-0,45 32 1 ,27 5,31 / 0,06" 20j0,50 33 1,35 70 6", 12 8 0,08 2i:o,55 34 j J, 44 1 75 7,03 9 0,10 '22:0,60 3511,53 80 8,00 10 0,12 i 23 0,67 30 l,fJ2 85 0,03 11 0,15 '240,72 87 1.71 1 90 10,12 12 0,18 25,0,78 0\j l.bO i 95 11,28 1310,21 j 26 0,84 »9 1,91 100 12,50 The jfirst 7nethod of finding and marking two level points^ is by a tangent whose point of contact is ex- MARKING LEVEL POINTS. 367 actly in the middle of the level line ; this method may be practised without regarding the diftcrcnco between the apparent and true level ; but, to he used with success, it would be well to place your instru- ment as often as possible at an equal distance from the stations; for it is clear, that it' from one and the same station, the instrument remaining at the same height, and u'^cd in the same manner, two or more points of sight be observed, equally distant from the eye of the observer, they will also be equi-distant from the centre of the earth. Thus, let the instrument, ^j^. 8, plate23, be placed at equal distances trom C, 1), E, F, the two points of sight marked upon the station-staves C G, D H, will be the level points, and the difference \\\ the height of liie points E, F, will shew how much one place is higlier than the other. Second method. This consists in levelling from one point immediately to another, placing the instrument at the srations where the staves were fixed. This may also be performed without noticing the diifer- ence between the true and apparent level ; but then it requires a double levelling, made from the first to the second station, and reciprocally from the second to the first. Thus, let the two stations be BE, fig. p, plate 23, the station-staves C B, D E, which in the practice of levelling may be considered as parts of the radii A B, A E, though they be really two perpendiculars parallel to each other, without the risk of any error. In order to level by this method, plant the instru- ment at B, let the height of the eye, at the first ob- servation, be at F, and the point of sight found be G ; then remove the instrument to E, and fix it so that the eye may be at G ; then, if the line of sight cuts F, the points F and (j are level, being equally dis- tantfrom the centre of the oarth^ as is evident from the figure. 368 OF THE PRACTICE But if the situation of the two stations be such, that the height of the eye at the second station could not be made to coincide with G, but only with H, fig. \0, plate 23, yet if the line of sight gives I as far distant from F as G is from H, the two lines F G, H Ij will be parallel, and their extremities level points ; but if that is not the case, the lines of sight are not parallel, and do not give level points, which however might still be obtaine<:l by farther observa- tions ; but, as this mode is not practised, we need not dwell further upon it. The velocity of running water is proportional to the fall ; where the fall is only three or four inches in a mile, the velocity is very small ; some cuts have been made with a fall from four to six inches ; a four inch fall in a straight line is said to answer as well as one of six inches with many windings. The distance from the telescope to the staff should not, if possible, exceed 100 yards, or five chains ; bu^ 50 or 6o yards arc to be preferred. OF THE PRACTICE OP LEVELLING, 1st. Of simple Levelling, We term that simple levelling, when level points are determined from one station, whether the level be fixed at one of the points, or between them. Thus, let A V>,fig. l^ flale C4, be the station points of the level, C. D, the two points ascertained, and let the height Feet, Inches. from A to C be 6 O from B to D be 9 O their difference is 3 shewing that B is three feet lower than A. OP LEVELLING. 36Q In this example, the station points of the level are l)elo\v the line of sight and level points, as is 2;ene- rally the case ; but if they were above, as in Jig. 2, plate 24, and the distance of A to C he six feet, and from 13 to 1) nine feet, the difference will be still three feet, that B is higher than A. 2d. Of compound LeveJl'mg. Compound levelling is nothing more than a col- lection of many single or simple operations compared together. To render this subject clearer, we shall suppose, that for a jiarticular purpose it were neces- sary to know the difference in the level of the two points A and N, Jig. 3, plate 24 ; A on the river Zome, N on the river Belam. As I could find no satisfactory examples in any English writer that I was acquainted with, and not being myself in the habits of making actual surveys, or conducting water from one place to another, I was under the necessity of using those given by Mr. Lc Febvre. Stakes should be driven down at A and N, exact- ly level with the surface of the water, and these should be so fixed, that they may not be changed until the whole operation is finished ; the ground between the two rivers should then be surveyed, and a plan thereof made, which will greatly assist the ope- rator in the conducting of his business ; by this he will discover that the shortest way from A to N, is by the dotted line AC, G H, H N, and he will from hence also determine the necessary number of stations, and distribute them properly, some longer than the others, according to ihe nature of the case, and the situation of the ground. In this instance ] 2 stations are used; stakes should then be driven at the limits of each station, as at h., B, C, D, E, F, &c. they should be driven about 18 inches into tha Bb 370 OF THE PRACTICE, &C. ground, and be about two or three inches above its surface ; stakes should also be driven at each station of the instrument, as at 1/2, 3, 4, Sec. Things being thus prepared, he may begin his work ; the first station will be at 1, equi-distant from the two limJts A, B; the distance from A to A, l66 yards ; and, consequently, the distance on each side of the instrument, or from the station stake, will be S3 yards. Write dov.'n in the first column, the first limit A ; in the second, the number of feet, inches, and tenths, the points of sight, indicated on the station staff at A, namely, 7.6.0. In the third column, the second limit B; in the fourth, the height indicated at the station staff B, namely, 6.0.0. Lastly, in the fifth column, the distance from one station staff to the other, in this instance 1 66 yards. Now remove the level to the point marked 2, which is in the middle between B and C, the two places where the station staves are to be held, ob- serving that B, which was the second limit in the former operation, is the first in this; then write dovv'n the observed heights as before, in the first column B; in the second 4.6.0; in the third C ; in the fourth, 5.6.2; in the fifth, 56o, the distance between B andC. As, from the inequality of the ground at the third station, it is not possible to place the instrument in the middle between the two station staves, find the most convenient point for your station, as at 3 ; then measure exactly how far this is from each station statr, and you will find from 3 to C, l(iO yards; from 3 to D, 80 yards : the remainder of the operations will be as in the preceding station. In the fourth operation it will be necessary to fix the station, so as to compensate for any error that might arise from the inequality of the last ; there- fore mark out 80 yards from the station staflf" D, to OK THE LEVEL, &C. 371 the point 4 ; and 160 yards from l to R; and this must Ixi carctully attended to, as by siieh compensa- tions the work, may be much raciliiatcd. Proeeed in the same manner with the eioht re- o njaining stations, as in tlic lour former, observing to enter every thing in its respective column ; when the whole is finished, add the sums of each colum to- gether, and then subtract the less from the greater, thus from 82 : 2 : 5, take 76 :Q : 7, and the remain- der 5 : 4 : 8, is what the ground at N is lower than ihe ground at A. f. in. f. In. yards. A 7 6 B 6 160 ii 4 6 C 5 6 2 250 C 12 8 6 D 8 4 240 D E 4 1 250 E 6 JO F 2 Jl 250 F 7 4 G 4 8 6 300 G 7 7 H JO 250 H 4 6 4 I 8 10 110 J 6 3 K 10 130 K 6 4 3 L 5 8 6 250 I- 7 M 8 4 3 250 M 6 5 N 7 10 250 7() 9 7 82 2 5 2680 82 2 5 76 9 7 5 4 G OF THE PROFILE OP TIjLE LEVjEL, The next thing to be obtained is a section of this level ; for this purpc^se draw a dotted line, as o o. Jig. A,plale 24, either above or below the plan, which line may be taken for the level or horizontal line : then let fidl perpendiculars upon this line from all the station points and places, where the station slaves were fixed. 372 OP THE PEOPILE Beginning at A, set off seven feet six Inches upon this line from A to a; for the height of the level point determined on the staff at this place, draw a line through scription and use of Hadley's Quadrant, t Nicholsoii's Navigator's Assistant. [ 3S7 ] MERIDIONAL ALTITLDES. The meridian altiuulc of the sun* is found by attending a few minutes before noon, and taking his altitude from time to time ; when the sun's al- titude remains for some time without any consider- able inrrcasc, tlie observer must be attentive to mark, the eoincidence of the limb of the sun with the horizon, till it perceptibly dips below the edge of the sea. The quantity thus observed is the meri- dional altitude. TO TAKE THE ALTITUDE OF A STAR. Before an observer attempts to take the altitude of a star, it will be proper for him to exercise him- self by viewing a star with the quadrant, and learn- ing to follow the motion of the reflected image without losing it, lest he should take the image of some other star, instead of that whose altitude he is dcsii'ous to obtain. His qundrant being properly ad- justed, let him turn the dark glasses out of the way, and then, 1 . Set the index of the nonius to. the o line of the limb. 2. Hold the quadrant in a vertical position, agrce- a1)!e to the foregoing directions. 3. Look, through the sight vane and the trans- parent part of the horizon glass, straight up to the star, which will coincide with the iuKigc seen in the silvered part, and form one star ; but, as soon as yoL; move the index forward, the reflected image will descend below the real star ; you must follow this image, by moving the whole body of the qua- drant downwards, so as to keep it in the silvered part * At [)laccs where it rises and sets. Cc 2 ^ 388 FOP. FINDING THE ALTITUDE, &C. of the horizon glass, as the motion of the index de presses it, until it comes down exactly to the edge oi the horizon. It is reckoned better to observe close than open ; that is, to be well assured that the objects touch each other; and this opinion is well founded, as many persons are near-sighted without knowing it, and see distant objects a little enlarged, by the addition of a kind of penumbra, or indistinct shading off into the adjacent air. There are but two corrections to be made to the observed altitude of a star, the one for the dip of the horizon, the'other for refraction. EULES FOR FINDING THE LATITUDE; THE SUn's ZENITH, DISTANCE, AND DECLINATION AT NOON BEING GIVEN. The first subject for consideration is, whether the sun's declination be north or south ; and, secondly, whether the required latitude be north or south. If the latitude, or declination, be both north, or both south, they are said to be of the same denomina- tion ; but if one be north, and the other south, they are said to be of different denominations. Thirdly, take the given altitude from QO*^ to obtain the zenith distance. Rule 1. If the zenith distance and declination be of the same name, then their difference will give the latitude, whose denomination is the same with the declination, if it be greater than the zenith dis- tance ; but the latitude is of a contrary denomi- nation, if the zenith distance be less than the decli- nation. Rule 2. If the zenith distance and declination have contrary names, their sum shews the required latitude, whose name will be the same as the decli- jiation. THE LATITUDE. 38^ Or, by the two following rules ynu may find the latitude of the pk'.ru tVoin the altitude and declination given. R'de 1. It' the altitude and declination are of different names, that is, the one north, the other south, add QO degrees to the declination ; from that sum subtract the meridian altitude, the remain- der is the latitude, and is of the same name as the de- clination. Rule 2. If the meridian altitude and declina- tion arc of the same denomination, that is, both north or both souih, then add the declination and altitude together, and subtract that sum from 90 degrees, if it be less, and the remainder will be the latitude, but of a contrary name. But if the sum exceeds 90 degrees, the excess will be the latitude, of the same name with the declination and alti- tude. EXAMPLES FOR FINDING THE LATITUDE BY MERIDIONAL OBSERVATION.* Example 1. Being at sea, July 29, 1779, the me- ridian ahitude of the sun's lower limb was observed to be 34° 10' N. the eye of the observer being 25 feet above the sea. The latitude of the place is re- quired. The sun's declination for the third year-j- after leap * Nicholson's Navigator's Assistant. \ The annua! course of the seasons, or the natural year, conslstins^ of nearly 365 days six hours, and the current year being rockoned 3fi5davs, it is evident that one whole day would be lost in four years if the six hours were constantly rejected. To avoid this in- convenience, which, if not attended to, would cause the seasons to shift in process of time through all the months of the year, an ad- ditional day is added to the month of February every fcrurth year ; this fourth year is termed leap year, and is found by dividing the year of our Lord by 4 ; leap year leaves no remainder ; other years are called the first, second, or third year after leaji year, according z» the remainder is 1, 2, or 3. The Nautical Almanack, and Rohcrtson's Treatise on Navigation, contain the best tables of the sun's declinatioaj &c. Sec. Cc3 SQO TO FIND THE LATITUDE BY year on July 2Q, is found in Table III. to be 18' 46' N. the dip for 25 feet elevation is five minutes, the refraction for 34° is one minute ; therefore. Altitude of sun's lower limb 3-i° 10' N. Add semi-diameters l6 3-1 26 Subtract dip and refraction 6 Correct altitude 34 20 Zenith distance or co-alt. 55 40 Subtract declination 18 46 N. Remains latitude 36 54 S, or of tiie contrary name to the declin. Example 2. October 26, 17 SO, sun's meridional altitude, lower limb 62° Og' S. required the latitude. Height of the eye 30 feet. 1780 is leap year, and the sun's declination in October 26, is 12° 45' S. The dip for 30 feet eleva- tion is six minutes, the refraction for 62° is ^ minute. Therefore, . Sun's apparent alt. 1. 1. 62° 00' S. -j- semi-diam.— -dip and refrac. 0^ Correct alt. 62 18v Zenith dis. 2/ 41^ S Sun's decHnation 12 45 S. DifFerence is lat. 14 56^ N. or of the contrary name. Example 3. Jan. 7, 1776, altitude of sun's lower limb at noon 87° lO' S. height of the eye 30 feet, required the latitude. MERIDIONAL OLS F.R V ATIOX. 391 Sun's apparent a!t. I. 1. 8"° 10' ■+■ scmi-diam.— !(> 8 57 41 8 5y 27 12 58 6 9 7 6" 5 31 3 b ^9 3 4 3 2 45^ 12 2 15 12 2 ^H 12 2 12J * Thii gives the whole interval between the obiervations, to which, if your clock is known ttjvary much in that time, you must ad'l the clock's loss, or subtract lis gain, during that intcr\al, and half this Corrected (if necessary) inteival uiust be added to the time of forenoon, Sec. 400 MERIDIONAL PROBLEMS. As the seconds differ add them together, i-5 and divide the sum by 3, by which you obtain Jf | a mean. h. Therefore, 12 m. 2 s. 14 3)42(14 the mean. 8 equation for six hours-. 12 2 1 14 55 8 1 equation of time sub. 12 19 7 clock near 20' too fast for — equal time. For accurate Tables of Equations to Equal Altittides, and the. Method ofjinding the Longitude at Sea by Time-keepers, I refer the reader to Mr. Wales's pamphlet. 8vo. 1794. MERIDIONAL PROBLEMS BY THE STARS, AND VARIATION OF THE MAGNETIC NEEDLE BY THE SUN.* "%. To fix a meridian line hy a star, when if can he seen at its greatest elongation on each side of the ■pole. Provide two plummets. Let one hang from a fixed point ; let the other hang over a rod, sup- ported horizontally about six or eight feet above the ground, or floor in a house, and so as to slide occa- sionally along the rod : let the moveable plummet hano- four or five feet northward of the fixed one. Sometime before the star is at its greatest elongation, follow it in its motion with the moveable plummet, so as a person a little behind the fixed plummet may aUvays see both in one, and just touching the star. When the star becomes stationary, or moves not be- yond the moveable plummet, set up a light on a staff, by signals, about half a mile off, precisely in the direction of both the plummets. Near 12 hours * Mackenzie's Maritime Surveying, p. 47- MEltlDIOXAL PROBLEMS. 401 i. ft er wards, when the star comes towards its greatest '>!oiigatioii on the otlior side of the |)()le, with your eye a tool or two hchiiid the fixed jjluinmet, follow it with the moveable pliunmct, till you perceive it sta- tionary as before ; mark the (iirection of the plum- mets then by a light put up precisely in that line : take the angle between the places of the two lights with //.a/Av's (|uadrant, or a good theodolite ; the centre of the instrument at the fixed plummet, and a pole set up in day-light, bisecting that angle, will be exactly in the meridian seen from the fixed plummet. This observation will be more accurate if the eye steadied by viewing the plumb-lines and star through a small slit in a plate of brass stuck upright oi» a stool, or on the top of a chair-back. This problem depends on no former observations whatever ; and^ as there is nothing in the operation, or instruments, to aifect its accuracy, but what any one may easily guard against, it may be reckoned the surest foundation for all subsequent celestial observa- tions that require an exact meridian line. The only disadvantage is, that it cannot be performed but in winter, and when the stars may be seen for 12 hours together ; which requires the night to be about 15 hours long. To fx a meridian line hy two circumpolar stars that have the same right ascensioriy or dijjer precisely 1 SO decrees. P'ix on two stars that do n.ot set, and whose right ascensions are the same, or exactly 180 degrees dif- fereat : take them in the same vertical circle by a plumb-line, and at the same time let a light be set up in that direction, half a mile, or a mile off, and the light and plummet will be exactly in the me- ridian. In order lo place a distant light exactly in the di- rection of the plumb-line and stars^ proceed in the D d 402 MERIDIONAL PROBLEMS. following manner. Any night before the observa- tion is to be made, when the two stars appear to the . eye to be in a vertical position, set up a staif in the place near which you would have the plumm.et to hang ; and placing your eye at that staff, direct an assistant to set another staff upright in the ground, 30 or 40 yards off", as near in a line with the stars as you can. Next day set the two statfs in the same places ; and in their direction at the distance of about a mile, or half a mile northward, cut a small hole in the ground, for the place where the light is to stand at night; and mark the hole so, that a person sent there at night may find it. Then choose a calm night, if the observation is to be made without doors ; if it is moon-light, so much ihe better, and half an hour before the stars appear near the same vertical, have a lighted lanthorn ready lied to the top of a pole, and set upright in the distant hole marked for it the night before ; and the light will then be very near the meridian, seen from the place marked for the plummet. At the same time, let another pole, or rod, six or eight feet long, be supported horizontally where the plummet is to hang, six or seven feet from the ground, and hang the plumb-line over it, so as to slip easily along it, either to one hand or the other, as there may be occasion, or tie a staff firmly across the top of a pole, six or seven feet long; fix the pole in the LTOund, and make the plumb-line hang over the cross staff. Let the weight at the end of the line be pretty heavy, and swing in a tub of water, so that it may ijot shake by a small motion of the air. Then shift the plumb-line to one hand or the other, lill one side of the line, when at rest, cut the star which is nearest the pole of the world, and the middle of the light together : as that star moves, continue moving the plumb-line along the rod, so as to keep it always on the light and star, till the other star comes to the MERIDIONAL PR0BLBM3. 403 same side of the plumb line also; and then the plummet and light will be exactly in the meridian. There arc not two remarkable stars near the north pole, with the same right ascension precisely, or just a semi-circle's ditFerencc : but there arc three stars that arc very nearly so, viz. the pole star, e in Ursa- Major, and "/ in Casiopeia. If either of the two last, particularly f, arc taken in the same vertical with the pole star, they will then be so very near the meridian, that no greater exactness need be desired for any purpose in surveying. At London y is about V west of the meridian then; and e much nearer it, on the west side likewise. Stars towards the S. pole proper for this observation arc, y, in the head of the cross; a, in the foot of the cross ; and u^ in the head of the phenix. 7 Jincl a meriJuin luic by a clrcwnpolar star, when it is at its greatest azimuth from the pole. By a cir- cumpolar star is here meant, a star whose distance froiu the pole is less than either the latitude, or co- latitude of the place of observation. 1. Find the latitude of the place in which you are to observe the star. 2. Fix on a star whose declination is known, and calculate its greatest azimuth from the north, or ele- vated pole. 3. Find at what time it will be on the meridian the afternoon you are to observe, so as to judge about what time to begin the observation. 4. Prepare two plummets, and follow the star with one of them, as directed in page 400, until the star is stationary ; then set up a light half a mile, or a, mile oil', in the direction of the plumb-line and star, and mark the place of the light in the ground, and also of the plumb-line. 5. Next day set up h pole where the light stood, with a flag flying at it : then with a theodolite, or Hadlejs ciuadrant, set the index to the degree and Dd 2 404 MERIDIONAL PEOBLEMS. minute of the star's azimuth from the north, found before ; direct, by waving to one hand or the other, an assistant to set up a staff on the same side of the pole with a flag, as the pole of the world was from it, so as at the plumb-line these two lines may make an angle equal to the azimuth of the star, and the plumb-line and staft" will then be in the meridian. When any star is descending, it is on the west side of the pole of the world ; while it ascends, it is on the cast-side of it. The pole of the world is always between the pole star and Ursa-Major : so that when Ursa-Major is W. or E. of the pole star, the pole of the world is W. or E. of it likewise. To find the greatest azimuth of a circumpolar star from the meridian, use the following propor- tion. As co-sine of the lat. : R : : S. of the star's dis- tance from the pole : S. of its greatest azimuth. The greatest azimuth is when the star is above the horizontal diameter of its diurnal circle. On the N. side of the equator, the pole star is the most convenient for this observation ; for the time when it is at its greatest elongation from the pole, may be known sufficiently near by the eye, by ob- serving when 5, in Ursa-Major, and 7, in Casiopeia, appear to be in a horizontal line, or parallel to the horizon ; for that is nearly the time. Or, the time may be found more precisely by making fast a small piece of wood along the plumb line when extended, with a cross piece at right angles to the top of the up- right piece, like the letter T ; when the plummet is at rest, and both stars are seen touching the upper edge of the cross piece, then they arc both horizontal. It is another conveniency in making use of this star, that it changes its azimuth much slower than other stars, and therefore a^brds more time to take its di- rection exact. MERIDIONAL PROLLEMS. 405 On the soutli side of the equator, the head of the cross is the most convenient for this observation, being nearest to the S. pole; and the time of its coming to the horizontal diameter of the diurnal circle, when it appears in a horizontal line with the foot of the cross, or the head of the phenix ; and at the eqnaior its greatest azimuth is one hour 1() mi- nates before or after that, as it is E. or W. of the meridian. The pole of the world is between the head of the cross and the last of these stars. To find ivhen any star will come to the meridian either in the south or north. Find the star's right ascension, in time, from the most correct tables, also the sun's right ascension for the day and place pro- posed ; their difference will shew the difTcrencc be- tween their times of coming on the meridian in the south, or between the pole and zenith which will be after noon if the sun's right ascension is least, but before noon if greatest. Eleven hours 58 minutes after the star has been on the meridian above the pole, it will come to the meridian north of it, or be- low the pole. Given the latitude of the place, and the declinations and right ascensions of tivo stars in the same vertical line, to find the horizontal distance of that vertical from the meridian, the time one of the stars will take to come from the vertical to the meridian, and the pre- cise time of the observation. Case 1. JVhen the two stars are northward of the zenith, plate l6, fig. 4. Let Z P O be a meridian, Z the place of obser- vation, HO its horizon, and Z P a vertical circle passing through tv.o known stars, s and S ; P the pole, E Q the equator, P s D and P S d circles of declination, or right ascension, passing through these stars : then is Z P the co-latitude of the place, P s and PS the co-declination of the two stars respec- tively, and the angle s P S the nearest distance of l)d 3 406 MERIDIONAL PROBLEMS. their circles of right ascension ; V O the arc of the ^ horizon between the vertical circle and the meridian ; and d Q the arc of the equator between the star S and the meridian. When, in the triangle P Z s, the angles P Z s and s P Z, measured by the arcs V O and D E, are found, the problem is solved. Solution I. Begin with the oblique-angled tri- angle P s S, in which are given two sides P s and PS, the co-declinations of the two stars respectively, and the included angle S P s ; which angle is the dif- ference of their right ascensions when it is less than 180 degrees ; but if their difference is more thnn 1 80 degrees, then the angle S P s is equal to the l(?sser right ascension added to what the greater wants of 36o degrees. From hence find the angle S s P by the following proportions : As radius is to the co-sine of the given angle S Ps, so is the tan. of the side opposite the required angle P s, to the tangent of an angle, which call M. M is like the side opposite the angle sought, if the given angle is acute ; but unlike that side, if the given angle is obtuse.* Take the difference between the side PS, adjacent to the required angle, and M ; call it N. Then, Sine N : sine M : : tan. of the given angle S P s : tan. of the required angle S s P, which is like the given angle, if M is less than the side P S adjacent to the required angle ; but unlike the given angle, if M is greater that P S, 2. Next in the oblique-angled triangle Ps Z there are given two sides, PZ, the co-latitude of the place of observation ; PS, the co-declination of the stars; * Two arcs or angles are said to be like, or of the same kind, when both are less than gO°, or both more than C)0°; but are said to be unlike, when one is greater, and the other less than 00° ; and are made like, or unlike to another, by taking the supplement to 180* degrees of the arc, or angle, produced iu like pruportion, in place of what the proportion brings out. MERIDIONAL PROBLEMS. 407 and ihc angle PsZ opposite lo one of them, which is the supplement of P s S, last found to ISo"^ ; from thence find the angle P Z s opposite the otiicr side^ by the follouitig proportion : Sine of P Z : sine P s Z : : sine P s : P Z s, cither acute or obtuse. This angle P Zs, measured by the are of the hori- zon V O, is therefore equal to the horizontal dis- tance of the vertical of the two stars from the meri- dian. Let the direction of the vertical be taken by a plumb-line and distant light, as before directed, or by two plumb-lines, and marked on the ground. Next (lay, the degrees and minutes in the arc V O, may be added to it by Hadleys quadrant, and a pole set up there, which will be in the direction of the meri- dian from the plumb-line. 3. Last, in the same triangle P s Z find the angle s P Z between the given sides, by the following pro- portions : As radius is to the tan. of the given angle PsZ, so is the co-sine of the adjacent side P S, to the co- tan, of M. M is acute, if the given angle and its adjacent sides are like ; but obtuse, if the given angle and the adjacent sides are unlike. As the co-t. of the side adjacent to the given angle P s, is to the co-t. of the other side P Z ; so is the co-s. M, to the eo-s. of an angle, which call N. N is like the side opposite the given angle, if that angle is acute ; but unlike the side opposite the given angle, if that angle is obtuse. Then the required angle s P Z is either equal to the sum or difference of M and N, as the given sides are like or unlike. 'I he angle s PZ, thus found, added to s P S, and their sum subtracted from 180° will leave the ande o d P Q, or the arc d Q that measures it, which is the Dd 4 408 MERIDIONAL PKOBLEMS. arc of the equator the star S must pass over in coming from the vertical, Z V, to the meridian. Which converted into time, and measured by a clock or watch, beginning to reckon the precise moment that a plumb-line cuts both stars, will shew the hour, minute, and second, that S is on the meridian. Find, by the right ascension of the star a«d sun, at what time that star should come to the meridian jn the north, the night of the observation ; subtract from it the time the star takes from the vertical to the meridian, and the remainder, corrected by the sun's equation, will be the time when the two stars were in the same vertical. The lowest of the two stars comes soonest to the meridian below the pole, the highest of them comes soonest to that part of the meridian which is above the pole. The nearer in time one of the stars is to the me- ridian when the observation is made, it is the better ; for then an ordinary watch will serve to measure the tirne sufficiently exact. It is still more advantage- ous if one of them is above the pole, when the other is below it. The nearer one of the stars is to the pole, and the farther the other is from it, the more exact will this observation be, because the change of the ver- tical will be the sooner perceived. For this reason, in north latitudes, stars northward of the zenith arc preferable to those that are southward of it. If the two stars are past the meridian when they are observed in the same vertical, then the arc d C^ gives the time S took to come tVom the meridian to the vertical, and must be added to the time when that star was on the meridian, to give the tirnc of the observation ; and the arc of the horizon Y O must be marked on the ground on the side of the vertical, contrary to v/hat it would have been in the forego- ing supposition ; that is^ eastward bclo'u: the pole, and Vi'cstvvard nhove it, MERIDIONAL PROBLEMS. 409 When two stars come to the vertical line near the meridian, it may be ilifficuU to judge on which side ()( it they are at that time ; lor determining this, the following rules may serve. The right ascension ot two stars may be either each less, or each 7Hore than 180 degrees, or one more and the other less. When the right ascension of each of the stars is either less, or more, than ISO degrees, they will come to the same vertical on the east-side of the meridian when the star, w ith the greatest right ascension is the lo'ucest ; but on the ivest-side of the meridian when it is highest. IVhen the right ascension of one of the stars is more than 1 80 degrees, and that of the other less. If the right ascension of the highest is less than 180°. but greater than the excess of the other's right ascension above 180°, then they come to the vertical on the east-side of the meridian. But if the riirht ascension of the higher star is less than that excess, they come to the vertical on the iccsi-side of the meridian. If the right ascension of the higher star is more than 180 degrees, and that excess is less than the right ascension of the lower star, then they come to the vertical on the ^vest-side of the meridian ; but if the excess of the higher star's right ascension above 1 80° is ;«c/r^ than the right ascension of the lower star, then they come to the same vertical on the cast- side of the meridian. Case 2. JVhen the two stars are in the same I'crti- cal^southiiard of the zenith, plate ]6,Jig. 3. When two stars are observed in the same ver- tical line sonthward of the zenith, or toward the depressed pole, the operations and solutions arc nearly the same as in Case 1. For let PZO be a meridian, Z the place of observation, O H its ho- rizon, ZV a vertical circle passing through the 410 TO FIND THE VABIATI6N two Stars S and s^ P the pole, Q E, the equator, PD and Pel two circles of declination, or right ascension, passing through the two stars respec- tively ; then is Z P the co-latitude of the place, P S and P s the co-declination of the two stars re- spectively, the angles S P s the difference of their right ascensions, the angle S Z Q, or O V the arc of the horizon which measures it, the distance of the vertical from the meridian ; the angle s P Z, or d Q which measures it, the arc of the equator, which s must pass over from the vertical to the meridian, which converted into time, and mea- sured by a watch, will shew when s is on the me- ridian ; and subtracted from the calculated time that the star should come to the meridian, will, when corrected by the sun's equation, give the true time of the observation. SPs is the triangle to begin the solution with ; and in the triangle s P Z, the angles s P Z and s Z P, when found, will give the solution of the problem, as in Case 1. For the supplement s Z P to 180 degrees is the angle V Z O, or its measure O V, the horizontal distance of the vertical from the meridian ; the angle sPZ, or its measure d Q, is the equatorial distance of the vertical from the meridian ; for all which the solution in Case 1, properly applied, will serve. On the south-side of the zenith, when the highesp of the two stars has the /f<7j-/ right ascension, they come to the same vertical on the east-side of the me- ridian ; but when the highest star has the greatest. right ascension, they come to the same vertical on the west-side of the meridian. Problem. To find the suns amplitude at risings or setting, and from thence the 'variation of the magnetic needle, with an azimuth compass. Make the needle level with the graduated circle in the box. Then, when the sun's lower edge is a OP THE MAGNETIC NEEDLE. 411 scmi-diamcter above the horizon, take the bearing of if^ centre, from the X. or S. whichever is nearer, t!iroLigh the sights making the thread bisect the sun's disk, and that subtracted from 90°, will be the Sim's magnetic amplitude, or distance, from the E. or W. points by the needle. Next, calculate the sun's true amplitude for that day, by the following proportion: As the cosine of the latitude is to radius, so is the sine of the sun's declination at setting or rising to the sine of his amplitude from the W. orK. Which will be X. or S. as the sun's declination is N. or S. and the distance in degrees and minutes be- tween the true E. or W. and the magnetic, is the va- riation of the needle. An easy and sure way to prevent mistakes, which the unexperienced arc liable to in this cal- culation, is, to draw a circle by hand, representing the visible boundary of the horizon, and on it to mark the several data by guess ; then, by inspect- ing the figure, it will easily appear how the varia- tion is to be found, whether by addition or subtrac- tion, and on which side of the north it lies. For ex- ample : Plate 16. fig. 5. Suppose the variation was sought at sun-setting. Draw by hand a circle N W S E, to represent your visible horizon ; in the middle of it mark the point C, for your station ; from C, draw the line C W, to represent the true west ; then on the north or south side of that line, according as the sun sets northward or southward of the true west, draw the line C ©, representing the direction of the sun's centre at setting, and another line C w, for the magnetic west, cither on the north or south side of 0, as it was observed to be, and at its judged distance. Then by observ- ing the situation of these lines, it will easily occui whether the magnetic aniplitudc and true ampli- 412 TO FIND THE VARIATION tude are to be added, or subtracted, to give the variation ; and on which side of the true north the vjiriation Hes. In the present supposition, w © is the maj[;netic amphtude, and O W the true ampli- ' tude; W therefore must be subtracted from O \v, to give w W the distance of the one from the other ; and n, the magnetic N. 90*^ from vv, must be west- ward of N, the true north. This method is sufficiently exact for finding the variation, but it is not exact enough for fixing a pre- cise meridian line; because of the uncertainty of the refraction, and of the sun's centre: but if the sun ascends, or descends, with little obliquity, the error then will be very little. To J/nd the sim's azhnut/i, and from thence to find the variation of the needle. I'irst, let the latitude of the place be exactly f6und. Next, let the quadrant be carefully adjusted for observation. Then, two or three hours before or after mid-day, take the altitude of the sun's centre as exactly as possible, making the vertical wire of the telescope biset the sun's disk : and, without al- tering the plane of the quadrant in the least, move the telescope vertically till you see some distant sharp object on the land, exactly at the vertical wire; and that object will be in the direction of the sun's azimuth when the altitude of its centre Was laken. If no such object is to be seen, let a pole be set up in that direction, about half a mile off', or as far as can be seen easily. Next calculate the sun's azimuth by the following rule. Add the complement of the latitude, the com- plement of the altitude, and the complement of the sun's declination to C)0° together, and take the half of that sum, and note it down ; subtract the complement of the declination from the half snm, nnd take the remainder ; then take the comple- OF THE MAGNETIC NEEDLE. 41. J incnt arithmetical* of the sines of the comple- ment of theahitiulc, and of the complement of tlic latitude, and add them together, and to them aild the sines of the four-mentioned half sum and re- mainder : half the sum ot these four logarithms is the eo-sineof half the azimuth required. Therefore, find by the tables what angle that co-sine answers to, double that angle, and that will be the sun's azitnuth from the north. If the sun's declination is S. in north latitude, or N. south Liiitudc, in place of taking the comjdlenicnt of the declination to (JO^, add 90** to it, and proceed as before. In south latitudes the azimuth is found in the same manner ; only, the sun's azimuth is found from the S. Then, to find the variation, place your needle below the 'centre of the quadrant, set it level, and find how many degrees the pole, or object, in the sun's azimuth, bears from the north by thq needle ; and the difference between that and the azimuth found by calculation, is the variation of the needle sought. If the sun ascends, or descends with little obli- quity, a meridian line may be fixed pretty exactly this way, because a small inaccuracy in the altitude of the sun's centre, will not be sensible in the azi- muth. But, when the sun docs not rise liigh on the meridian, this method is not to be relied on, w hen great exactness is necessary ; for tlicn every inaccuracy in latitude, altitude, and refraction, occasions severally a greater error in the azimuth. To mark the meridian line on the ground, place * The complement arithmetical of :i logarithm is found thus : begin at the left band of the logaritlim, and subtract each figure from 9, and the last tigure from 10, setting down the several re- mainders in a line j and that number will be the arithmetical conx- {jlement required. 414 PRACTICAL GEOMETRY , the centre of a theodolite, or Hadkys qnacirantj where the centre of the quadrant was when the* sun's altitude was taken, and putting the index to' the degree and minute of the azimuth, direct, by waving your hat towards one side or the other, a pole to be set up, making an angle with the former pole placed in the azimuth, equal to the sun's azimuth found; and that last-placed pole will be in the meridian, seen from the centre of the qua- drant. N. B. The sun's declination in the tables must be corrected by the variation arising from the difference of the time between your meridian and that of the tables; and also for the variation of declination for the hours before, or after noon, at which the sun's altitude was taken. The two last problems are constructed and fully explained in Robertsons, Moore s, and other Treatises on Navigation ; it is, therefore, needless to be more particular here. A COURSE or PRACTICAL GEOISIETRY* ON THE GROUND, BY ISAAC LANDMAN, PROFESSOR or FORTIFICATION AND ARTILLERY ROYAL MILITARY ACADEMY AT WOOLWICH- It is as easy to trace geometrical figures on the ground, as to describe them on paper j there is, * I have been obliged to omit in this Course, so liberally commu- ON THE GROUND. 415 I however, some small ditFcrciicc in the mode of operation, because the inslriimeiits are difFcrcni. A rod, or chain, is here used instead of a scale ; the spade, instead of a pencil ; a cord fastened to two staves, and stretched between them, in-;tead of a rule, the same cord, by fixing one uf the staves in the ground, and keeping the other moveable, answers the purpose of a pair of compasses ; and with these few instruments every geometrical figure necessary in practice, may be easily traced 0:1 the ground. Pkob LKM 1 . To draw upon the ground a straight line through t-ujo given points A, B, fig. 1, plate 'IJ . Plant a picket, or staff, at each of the given points A IJ, then fix another, C, between them, in such manner, that when the eye is placed so as to see the edge of the staif A, it may coincide with the edges of the staves B and C. 'I'he line may be prolonged by taking out the staff A, and planting it at U, in the direction of B and C, and so on to any required length. The accuracy of this operation depends greatly upon fixing the staves u}>Tight, and not letting the eye be too near the staff", from whence the observation is made. Prob i,EM 2, To measure a straight line. We have already observed, that there is no ope- ration more difficult than that of measuring a straight line accurately ; when the line is short, it is generally measured with a ten feet rod ; for this purpose, let two men be furnished each with a ten feet rod, let the first man lay his rod down on the line, but not take it up till the second has laid his down on the line, so that the end may exactly co- nicHtcd by Mr. Lan'lman, the calculations that illustrate the exam- ples ; this, 1 hope, will not in the lea.st les.scn its use, a.s this work will fall into the hands of very few who arc iguoraiil of the nature and application of lugariihms. 'St \ it\ id I us ' 416 PRACTICAL GEOMETRY incide with that of the first rod; now let the first \ man Hft up his, and count one, and then lay down at the end of the second rod ; the sccon man is now to lift up his, and cotint two ; and th continue till the whole line is measured. Staves should be placed in a line at proper distances from each other by Problem 1, to prevent the operators from going out of the given line. When the line is very long, a chain is generally used; the manner of using the chain has been al- ready described, page 1 gO. Problems. To measure distances hy pacing, and to make a scale of faces, which shall agree with another, containing fathoms, yards, or feet. In military concerns, it is often necessary to take plans, form maps, or procure the sketch of a field of battle, and villages of cantonment, or to recon- noitre fortified places, where great accuracy is not required, or where circumstances will not allow the use of instruments ; in this case it is necessary to be well accustomed to measuring by the common pace, which is easily effected by a little practice; to this end measure on the ground 300 feet,and as the com- mon pace is 2^ feet, or 120 paces in 300 feet, walk over the measured space till you can finish it in 120 paces, within a pace or two. Example. Let us suppose that we have the map of a country containing the principal objects, as the villages, towns, and rivers, and it be necessary to linish it more minutely, by laying down the roads, single houses, hills, rocks, marches, &c. by mea- suring with the common pace ; take the scale be- longing to the map, and make another relative to it, in the following manner, whose parts are paces. Let the scale of the map be L'OO fathoms, draw :i line A B, Jig. 3, plate 27, equal to this scale, and divide it into four equal parts, AC, CD, &c. each of which will represent 50 fathoms ; bisect A C at ON THE GROUND. 417 at E, divide A E into five equal parts, Ac, e f, f pf, &c. each of whicli will be five fathoms ; draw GH j)arallcl to AB, and at any distance therefrom; then through the points of divitiion A, e f, g h, K, C, &c. draw lines perpendicular to A l), and cut- ting G H, which will be thereby divided into as many equal parts as A B. Two hundred fathoms, at '2 J feet [)cr pace, is 480 paces ; therefore write at JI, the last division of G H, 480 paces, at I 300, at K 240, and so on. To lay down on the plan any distance measured in paces, take the number of fa- thoms from the line A B, corresponding to the num- ber of paces in G II, which will be the distance to be laid down on the plan. Wlien plans, or maps, are taken by the plain ta- ble, or surveying compass, this is an expeditious me- thod of throwing in the detail, and a little practice soon renders it easy. A TABLE, For reducing the common pace of 2J feet into feet and inches. Paces Feet I IK P. IG Feet Inc. P. Feet Inc. P. 46 Feet Inc. P. Feet Ihc. 1 2 6 40 77 G 115 152 6 2 .5 17 42 6 ■VJ 80 47 117 6 Cr2 155 S 7 6 18 45 ■yj 82 48 120 G.'3 157 6 4 10 10 47 6 J4 85 49 50 122 6 (J4 160 3 12 6 20 50 87 6 125 65 162 6 6 13 21 52 6 .:J6 90 51 127 6 66 165 7 17 6 22 55 37 92 6 52 130 67 167 6 8 20 2y 24 57 6 1 ■■38 39 95 bo 54 132 6 1 68 170 9 22 6 60 97 6 155 69 172 6 10 23 25 62 6 40 100 55 137 6 70 175 11 27 6 2G 65 41 102 6 5G 140 71 177 6 12 .'30 27 28 07 6 42 43 105 57 58 142 6 72 180 13 32 6 70 107 6 145 73 182 6 14 3.5 29 72 6 44 110 59 147 6 :a 1K5 15 I 37 6 I" 75 45 112 60 150 • ■' 187 G Ee 418 PRACTICAL GEOMETRY Distances of a certain extent mav be measured by the time employed in pacing them ; to do this, a person must accustom himself to pace a given extent in a given time, as 600 paces in five mi- nutes, or 120 in one minute; being perfect in this exercise, let it be required to know how many paces it is from one place to another, which took up in pacing an hour and a quarter, or ']b minutes. Multiply 75 by 120, and you obtain 120, the mea- sure required. This method is very useful in military opera- tions ; as for instance, when it is required to know the itinerary of a country for the march of an army^ or to find the extent of a field of battle, an encamp- ment, &c. It may be performed very well on horse- back, having first exercised the horse so as to make him pace a given space in a determinate time, and this may be effected in more or less time, accord- ing as he is trained, to walk, trot, or gallop the original space. Peoblem a. To ivaJk in a straight line from a poposed point to a given object, Jig . 2, •^late 27. Suppose the point A to be the proposed point from whence you are to set out, B the given ob- ject ; fix upon another point C, as a bush, a stone, or any mark that you can find in a line with B ; then step on in the direction of the two objects B, C ; when you arc come within 10 or 15 paces of C, find another object D, between C and B, but in a line with them ; it is always necessary to have two points constantly in view, in order to walk in a straight line. This problem is of great use in measuring dis- tances, and surveying by the pace ; particularly in rcconnoitcring cither on foot or on horseback. It •may alto be used wjieii a battalion is ordered to take a position at the distance of 3 or 400 paces, and pa- rallel to that in which it is standing; to effect this, let iwo non-commissioned officers, who are well cxcr- OM THE GROUND, 419 v^ed ill the practice of this problem, step out from iC extremities, or wings of the battalion, place .em^elves square with the front of the battalion, then fix upon two objects straight before them, and on a signal given, set out and step the rcijuired number of paces, then halt, tims becoming two guides for placing the battalion. Pkoblem 5. To pi lice a troop EF, between tuo jiveri poiN/s A, B,^g. 4, plate 2/ . To pcrtbrra this, let two persons, each with a pole in his hand, separate about 50 or do paces from each other ; and then move on till the poles arc situated in the direction A, B, /". e. so that the person at C may see the point 1) in a direction with B, at the same time that the person at D sees C in a direction with the object A. Met/ioJ 2, 7?tf. 5. To perform the same with one person. Take a staff" C F, Jjg. 5, plate 2/, three or tour feet long, slit the top thereof, and place a straight rule, six or seven inches long, in this slit ; place this instrument between the two points, and see if A and B coincide with E, then view D from B, and if A coincides therewith, all is right ; if not, move the rule forward or backward till it is in the direction of D, E. Pkoelem 6. To raise a perpendicular from a point C, to the given line A B, Jig. 6, plate 2^ . Set off two points E, U, in A B, equally distant from C; fold a chord in two equal parts, place a staiF in the middle at F, fasten the two ends to the staves E and D; then stretch the chord tight, and the point F will be the required point, and the line CF will bq perpendicular to A B. Problem 7. From a given point F, out of the line A 1^, to let fall a pirpendiciilar C V^fig- 7, plate 27. Fold your chord into two equal parts, and fix the middle thereof to the point F, stretch the two halves to 2\. B, and where they meet that line plant E e2 420 PRACTICAL GEOMETRY two Staves, as at E, D, divide the distance E D iato two equal parts at C, and the line C F will be perpen- \ diculartoAB. ' *• Problem 8. To raise a perpendicular B C at the end B of the I'me B ^,fig. 8, plate 27. From D taken at pleasure, and with the length D B, plant a stafFat the point A, in the direction FB; with the same length set off from D towardsC, plant a staff C in the direction D A, and B C vrill be per- pendicular to F B. Methods, Jig. g, p'ate 17. By the numbers 3, 4, 5, or any multiple thereof. Set off from C to D, in the direction C A, four feet or four yards, and plant a staff at D, take a length of three feet and five feet, or yards, according as the former mea- sure was feet or yards ; fatsen one end of the chord at C,and the other at D; then stretch the chord so that three of these parts may be next to the point C, and five next to D ; plant a staft'at H, and C H will be perpendicular to CA.* Problem g. To draw a line C F parallel to the line A Bj and at a given distance from it, fg. 10, plate 27. At A taken at pleasure in A B, raise a perpendi- cular A C, then from B taken also at pleasure in AB, draw BF perpendicular to A B ; make AC, B F, each equal to the given distance ; plant staves at C and F, and in the direction of these two plant a third, E, and the line C E F will be the re- quired line. Problem 10. To make an angle abc on thg ground equal to a given angle A B C, fig. 1 .1 a7id 1 2, plate 17 . Set off any number of equal parts from B to C,and from B to A, and with the same parts measure A C ; describe on the ground with these three lengths a * Much trouble and time in these alterations may be ^vcd by ..using a cross-staff, before described; sec p. 198. ^dvi. ON THE GROUND. 421 triangle a be, and the angle abc thereof will be equal to the angle A B C. PnoiJLEMll. To ^rolojig the line AB, no/ with' siamliftg the obstucle G, and to obtain the length thereof. Jig. 13, p/j/e -27. At B raise the perpendicular H C, draw C D per- pcndieular to B C, and E D to C D, make ED equal lo BC; then raise a ])cvpendicular EF to ED, and E F will be in a straight line with AB. The measures of the several lines A B, CD, E F, added together, give the measure of the line AF. This problem is particularly useful in mining. Problem 12. To draw a line BC, parallel to the inaccessible face he of the bastion a, b, c, d, e, /// order to place a battery at C, "where it ivdl produce the greatest effect, fg. \\,plaie 27. Place a staff at the point A, out of the reach of musket shot, and in a line with b e ; from A draw A G, perpendicular to b A ; set off from A to G about 375 paces, and at G raise G B perpendicular to A G, and produce this line as far as is requisite for placing the battery, /'. e. so that the direction of the fire may be nearly perpendicular to one third of the face b c. Problem 13. To make an angle upon paper egiial to the given angle ABC, fig. 15 and 16, plate IT . Set off S6 feet from B to C, and plant a staff at C; set off also 36 feet from B to A, and plant a staff at A, and measure the distance A C, which we will suppose to be equal to 52 feet. Let D G be a given line on the paper; then from G with 36 parts taken from any equal scale, describe the arc DH, then with 52 parts from the same scale, mark off the distance DH; draw G H, and the angle D G H is equal to the angle ABC. Problem 14. To measure J rom the outside an angle ABC, formed by two walls A B, C B, fig. 1 7j plate 27. Ee3 422 PRACTICAL GEOMETRY Lay off 30 feet from B toE in the direction A B, and plant a staff at E ; set off the same measure from . B to F in the direction B C, and measure F E, and you may obtain the measure of your angle, either by laying it down on paper^ as in the last problem, or by calculation. Problem 15. To ascertain the opening of an inac^ eessihle flanked angled CD of the bastion A B C D E, flg.\S,plat^17. Place a staff at F^ in the direction of the face B C, but out of the reach of musket shot ; make F G perpendicular to F B, and equal to about 250 paces ; at G make G H perpendicular to G F^, and produce it till it meets, at H, the direction of the 'face D C ; place a staff at I in a line with D H, and the angle IHK will be equal to the required angle BCD, and the value of I H K may be obtained by Problem 13. Problem 16. To ascertain the length of the line AB, accessible only at the fzuo extremities K^,fg. ig, fJate1*j» Choose a point O accessible to A and B, draw A O, O B, prolong A O to D, and make O D equal to A O ; prolong BO to C, and make O C equal to B O, and C D will be equal to A B, the required line. Problem 17- To let fall a perpendicular from the inaccessible point D, tipo7i the right line A 6,7^^.21, plate "Ti. From A and B draw A D, B D, let fall the perpen- diculars AH, B F, upon the sides A D, BD, plant a staff at their intersection C_, and another at E, in the direction of the points C, D, jand E C D will be the required perpendicular. Problem J 8. To flnd the breadth AB, of a river, fg. 10, plate 17. Plant a staff at C in a line with A B, so that B C may be about I of the length of A !^., draw ^ ON THE GROUND. 423 line C E In any direction, the longer the better. Set up a staff" at D. tlic middle of C E, find the intersection F of E B, DA; draw DG p.irallel to B C; measure B F, F K; and you will find A B by the following proportion; as FE — UF : iJ C or DG :: BF : AB. Inaccessible distances may be obtained in other modes, when circumstances and the ground will per- mit, ai in the following problem. Problem \g. Let it he required to measure the distdtice A B 0/ the produced capital of the ruvelifi, from the head of the trenches B, to the an^le A, fg. '22, plate 27. Draw B E perpendicular to A B, and set off" from B to E 100 feet, produce B E to C, and make E C equal to one eighth or tenth part of 1> E ; at C raise the perpendicular CD, plant staves at E and C, then move with another staff" on C D, till this staff" is in a line with E and A. Measure C D, and you will obtain the length of A B by the following proportion t as E C : C D : : B E : A B ; thus if BE =100 feet, E C = 10 feet, and C D = 38 feet, then as 10 : 100 : : 38 : 380 feet, the distance of A B. Problem 20. To measure the inaccessible distance A H, fg. 23, plate 27. Plant a staff" at C, a point from whence you can readily sec both A and B. By the preceding pro- blems, find the length of C A, C B, make C D as many parts of CA, as you do C E of C B, and join D E ; then as C D : D E : : C A : A B.* * yln t'ijsy victhod of finding the distance of forts or other ohjects. — If you go oft" at any angle, as yo, and continue in that direction until you bring the object and your tirit station under :'.;i angle of 03, the distance measured from the iirst station is equal to half the distance the object is from your fiist station. But should the ground not admit of going off"at an angle of 90, go oft' at an angle of 45, and whatever distance in that direction you find the object nnd the first station under aai angle of 106i, is half the distance of tUe object nnd tirit station, bee al90})age 202 of these Essays. £ C4 42^ PRACTICAL GEOMETRY Pr o B L E M 2 1 . To determine the direction of the ca~ \ fitalof a bastion produced, fig. 24, pJale'lT . Upon the produced lines B D, BE, of tiie two faces C B, A B, place the staves E and D ; find the lengths of E B and DB, by Problem IQ, and mea- sure D E. Then as the capital G B divides the angle ABC, and its opposite E B D, into two equal parts, we have as D B : B E :: DF ; E F. and as DB-|-BE:BE::DE:EF, and, consequently the point F, which will be in the direction of the ca- pital G B produced. Problem 22. Through the point C to draiv a line IK parallel to the inaccessible line A B, fig. 25, plate 1"] . Assume any point D at pleasure, find a point E in the line A D, which shall be at the same time in a line with C B ; from E draw E G parallel to D B, and through C draw G C F parallel to AD, meet- ing B D in F. Plant a staff at H in the line E G, so that H may be in a line with F A, and a line I, H, C, K, drawn through the points H C, will be pa- rallel to A B. Problem 23. To take a plan of a place A, B, C, D, E, by similar triangles^ fig. 26 and IJ , pLlT . 1. Make a sketch of the proposed place, SLsfi^. 11. 2. Measure the lines A B, B C, CD, D E, EA, and write the measure obtained upon the corresponding lines a b, be, c d, d e, e a, of the sketch. 3. Measure the diagonal lines A C, A 1), and write the length thereof on a c, a d, and the figure will be reduced to triangles whose sides arc known. 4. To obtain a plan of the buildings, measure B G, B F, G H, I K, F K, &c. and write down the measure on the sketch. 5. Proceed in the same manner with the other buildings, &c. 6. Draw the figure neatly from a scale of equal parts. Problem 24. To take a plan of a wood, or via? shy ON THE enouND. 425 ground^ by measuring round about if, fig. 28 and 29, l^lale 27. Make a rough sketch, fig. 29, of the wood, set up staves at the angular points, so as to form the circumscrihing lines A B, B C, C D, and measure these lines, and set down the measures on the corre- sponding lines in the sketch, then find the value of the angles by Problems 13 and 14. Problem 25. Tu take a plan of a river ^ fig. 30 and 3 \, plate 'I7. 1. Draw a.sketch of the river. 2. Mark out aline A B, fig. 30. 3. At C, upon A B, raise the per- pendicular C H. 4. Measure AC, C H. 5. Mea- sure from C to D, and at D make D I perpendicular to AB, and measure D I. 6, Do the same the whole length of A B, till you have obtained the principal bcndings of the river, writing down every measure when taken on its corresponding line in the sketch, aiid you will thus obtain sufficient data for drawing the river according to any propor- tion. Problem 26. To take a plan of the neck of land A, B, C, D, E, F, G,fig. 32 and^3, plate 27. Take a sketch of the proposed spot, divide the figure into triangles ABC, A CD, AD E, A E G, by staves or poles placed at the points A, B, C, D, E, F, G, measure the sides A B, B C, A C, C D, A D, A E, A F, F G, A G of the triangles, writing down these measures upon the corresponding lines a b, b c, a c, &c. then measure A H on A B, and at H raise the perpendicular H I, and measure its length ; do the same at K, M, &c. writing down the measures obtained on their corresponding lines ah, hi, i k, k 1, &c. Proceeding thus, yon will as- certain a sufficient number of points for laying down your plan by a sc.alc of equal parts. r 426 3 PRACTICAL TRIGONOMETRY. Problem 27. To ascertain the height of a build- ing, fig. 34, plate 28. Measure a line F E, from the foot of a building-, so that the angle C D A may be neither too acme nor obtuse ; thus suppose E F = 130 feet, place your theodolite at D, and measure the angle ADC = 34° 56'. 'J'hen as radius is to tangent 34^^ 56'^ so is E F =: 130 feet, to A C. Which, by working the proportion, you will find to be SQ feet ^^ parts, or 89 feet 7.92 inches, to which adding four feet for DE, or its equal C F, you obtain the whole height 93 feet 7-22 inches. Problem 28. To find the ajtgle formed hy the Vine of aim, and the axis of a piece of ordnance produced, the caliber and dimensions of the piece being knoivn, fig. 35, plate 28. Suppose the line BI to be drawn through B, the summit of the swelling of the muzzle, and parallel to C D, the axis of the piece : the angle A B I will be equal to the angle A E C, formed by the line of aim A F, and the axis CD ; then in the right-an- gled triangle A I B, we have the sides A 1 and B I to find the angle A B T, which we obtain by this pro- portion ; as IB : AI :: radius, tangent of angle A B I equal to A E C required. Problem 29. The elevation, three degrees, of a light twelve-pounder being given, to find the height to •which the line of aim rises at the distance of 1 100 yards, 'which is about the range of a tisoelve-pounder , ivith an elevation of three degrees, fig. 35, plate IS. Ihe line of aim, which we suppose to be found by the preceding problem, forming with the axis of a light twelve-pounder l° 24', will make with the ho- rizon an angle of 1° 36' ; thus the height at the ho- rizontal distance of 1200 yards will be the second Oy THE GROUND. 427 side of a right-angled triangle, where the angle ad- jacent to the side of 1200 yards is 1° 30", and may therefore be obtained by the following proportion ; as radius to the tangent of l"^ 3(5', so is 1200 to 33 yards 1 foot 6 inches = FG. PiiOiiLEM 30. The jirst embrasure K of a rl- cochee buttery beltig direct, to Jind the inclhuit'iin of the sevaith embrasure V>, i. e. the angle formed bj the Hue of direction B C, and the hreast-vjork A 1^, at the seventh embrasure A B ; ;'/ is supposed that all the pieces are directed toivards a point C, at the distance of ] 500 feet, fig. 2,^, plate 28. The line of direction A C of the first embrasure is supposed to be perpendicular to the breast-work A B ; therefore we have to find the angle A B C of the right-angled triangle B A C, in which the right angle is known AC=1500 feet, and AB is deter- mined by the size, the distance, and number of embrasures : thus, suppose the distance from the middle of one embrasure to another be 20 feet, this multiplied by 0, will be 120 feet, and equal to A B; then as AB to A C, so is radius to the tangent of angle ABC, 85° 26'. Problem 31. As the hurler DE, fig. 37, plate 28, is always perpendicular to the directing line of the gun, and as at least one end of it ought to be laid against the breast-work, it will make an angle A F D, which was found by the preceding pro- blem to be 85° 26' ; therefore, knowing the length DE of the hurler, and consequently its half D F, it will be easy to calculate the distance B F from the breast-work where the middle F of the hurler ought to be placed, upon the line of direction jof the gun. Problem 32. To ascertain the height of a build- ing from a giveii point, from whence it is unpossible to measure a base in any direction, the points A and F being supposed to he in the same horizontal line , fig. 38, fhitt 28. 428 PRACTICAI. TRIGONOMETRY Measure the angles CED, A ED, let CED be equal 43° 12', the angle AED equal 2° 26', and the height EF from the centre of the instrument to its foot five feet; then in the right-angled tri- angle ADE we have AD=:E1'=5 feet, the an- gle D E A=:2'' 26' to find 1) E, which may be ob- tained by the following proportion ; as radius : co-tangent of angle D E A, so is AD to D E 118 feet ; then in the right-angled triangle C D E we have D E 1 IS feet, angle C E D 33" 12' to find CD, which is found by the following proportion ; as ra- dius to tangent of angle CED, join D E to D C 77 feet, which added to E F 5 feet, is 82 feet, theheight of the tower. Problem 33. The distayue AC, 135 totsesj from ihe ■point Q^ to the fianked angle of the bastion being given^ and also AB, 186' toises, the exterior side of the polygon^ io find B C, fig. 3Q, plate 28. ] . Find the angle B by the following propor- tion ; as A B is to the sine of angle C, so is A C to the sine of angle B, 39° 8' ; because it is plain from the circumstances of the case that B must be acute, and therefore the angle B A C is also known. 2. B C is found by the following proportion ; as sine angle C is to AB, so is sine of angle F» A C : BC, 213 toises, three feet. Problem 34. To find the height of the building KC,fg.AO. 1. At B measure the angle FBC in the direc- tion F B. 2. Set off any distance B D as a base. 3. Measure the angle B D C, C B D is the supple- ment of F B C, and B C D is the supplement of CBD+BDC. Then as sin. Z B CD : B D : : C D B : B C, B C being found, we have in the right-angled tri- angle F B C, the side B C and angle FBC, to find F C, which is found by this proportion ; as radius to sin. angle FBC, so is B C : F C, F C added to ON TME GROrxd. 4?^ \ F, the height of tlic instiumciit, gives the height I the tower. Problem 35. To ascertain the distance between two inaccessible objects, C D, fig. 4 1 , plate 28. 1. Measure a bjse A B. from whose extremities voii can see the two objects C, D. 2. Measure the angles C A l), D A B, DBA, C B A, C B D. '3. In the trianjr'ie BAG we liave the side K B and angles ABC, B A C, to find B C, which is found by the fnllowina: proportion ; as sine of angle A C B : AB :: sine BAG : B C. 4. The angles CAB, G B A, added together and subtracted from 180, gives the angle A C B. 5. In the triangle A B D we havx: the angles D A B, A B D, and conse- quently A L) B, and the side A B to find BY), which is found by the following proportion ; as the sine of angle A I) B is to A 1j, so is the sine of angle D A B : B D. 6. In the triangle C JJ D we have the two sides B G, B D, and the angle C B D, to find the angle G D B, and the side CD; to find G D B we use this proportion ; as the sum of the two given sides is to their difiercnce, so is the tan- gent of half the sum of the two unknown angles to the tangent of half their diHerenee; the anHc C D B being found, the following proportion will give G 1^ ; as sine angle G D B to B G, so is siwc aii- gle G B D to G D. Problem 36. To draw a line through the point B, parallel to the inaccessible line CD, Jig. 41, plate 28. Find the angle BG D by the preceding problem, and then place your instrument at B, and with B G make an angle G B E equal to BCD, and E B will be parallel to G D. Pro 15 L EM 37- To ascertain several points in the same direction, though there are obstacles which pre- vent one extremity of the line being seen from the other, fg. 42, plate 2^. i. Assume a point G at pleasure, from which the 430 PRACTICAL TRIGONOMETRY two extremities of the line A B may be seen. 2, Measure the distances C B, C A, and the angler A C Bj and find the value of the angle C A B, 3. Measure the angle A CD, and then in the tri- angle A C D we have the side A C, and the two an- gles D A C, A C D, and therefore the third to find C D, and as sine angle A D C to A C, so is sine an- gle C A D to C D. 4. Set off C D, making with A C an angle equal the angle A C D, equal to the measure thus found, and the point D will be in a line with A B; and thus as many mote points, as G, may be found as you please ; m ihh manner a mortar bat- tery may be placed behind an obstacle, so as to be in the direction of the line AB. Thus also you may fix the position of a richochec battery M, fig. 43, so as to be upon the curtain A D produced. Problem 38. To measure the height of a hilly whose foot is inaccessible, fig. 44, plate 28. 1. Measure a base F G, from whose extremities the point A ie visible. 2. Measure the angles ABC, ACB, ACD. 3. In the triangle ABC we have B C, and the angles A B C, A C B, to find A C; but as sine angle B A C is to B C, so is sine angle ABC to A C. 4. In the right-angled triangle ADC we have the side A C, and the angle A CD, to find AD; but as radius is to sine angle ACD, so is A C to Al), the height required. Problem 39. To take the ma^ of a country, fig. Aib^ plate 28. First, choose two places so remote from eachother, that their distance may serve as a common base fotf the triangle to be observed, in order to form the map. Let A, B, C, D, E, F, G, PI, I, K, by several re- markable objects, whose situations are to be laid down in a map. Make a rough sketch of these objects, according to their positions in regard to each other ; on this OK THE CROLND. 431 skrtcli, the diftcrcnt iDcasures taken in the course of the observations arc to be set clown. Measure the base A B, whose length should be ])roj)ortionate to the distance of the extreme objects from A and B; Irom A, the oxtremitv of the base, measure the angles E A li, FAB, GAB, CAB, D A B, formed at A with the base \ B. I'Vom B, the other extremity of the base, observe the angles E B A, F B A, G B A, C B A, DBA. If any object cannot be seen from the points A and B,anolher point must be found, or the base changed, so that it may be seen, it being necessary for the same object to be seen at both siations, because its positions can only be ascertained by the intersection of the lines from the ends of the base, with which they form a triangle. It is evident from what has been already said, that having the base AB given, and the angles observed, it will be easy to find the sides, and from them lay down, with a scale of equal parts, the several trian- gles on your map, and thus fix with accuracy the position of the different places. In forming maps or plans, where the chief points are at a great distance from each other, trigonome- trical calculations are absolutely necessary. Bat where the distance is moderate, after having measured a base and observed the angles, instead of calculating ihc sides, the situation of the points may be found by laying down the angles with a protractor; this method though not so exact as the preceding, answers sufficiently for most military ope- rations. Problem 40. The use of the surveying compass, fig. 3, flute 15, indetermtnmg the particulars of va- rious objects to he inserted in a plan^ fig. 4(3, plate 2^. The first station being at A, plant staves at the re- quisite places, and then place tiie compass at A, di- recting the telescope to C; observe the number of 432 PRACTICAL TRIGONOMETRY degrees north AC, made by tbc needle and the tele- scope, or its parallel line drawn through the centre of the compass, and mark this angle in your sketch, see jf^. 46; observe and mark in the same manner the other angles, north AO, north A P, north AQ; then measure A C, and at the second station C, ob- serve the angles north C M, north C Q, north C O, north C P, north C D, and so on at the other sta- tions ; in observing the angles, attention must b© paid, when the degrees pass ISO, to mark them pro- perly in the sketch, in order to avoid mistakes in pro- tracting. Problem 41. To raise perpendiculars, and form angles equal to given ajigles hy the surveying compass, jig. 47, plate 28. Let it be required to trace out the field work, A D C B, at the head of a bridge. The direction of the capital F E being given, de- scribe the square A B C D in the following manner ; place your compass at F, and as A B is to be perpen- dicular to FE, direct the telescope to E, and observe^ when the needle is at rest, the number of degrees it points to; then turn the compass box upon its centre, till the needle has described an arc of 90°, and place a staff in the direction of the telescope towards A, and A F will be perpendicular to F E; continue A F towards B, and make AF, BF, each equal 30 toises at E; in the same manner raise the perpendicular DC, and make ED, EC, each equal 30 toises ; join DA, CB, and you have the square A BC D. Make C G, D K, each equal to 7 of A B, through G and K draw the line G K, set off G H, K I, each equal io 7 of A B or C D, draw the lines of defence I C, H 1), place the compass at I, and the telescope. in the direction I H ; observe where the needle points when at rest, and turn the compass till the needle has described an arc of 100, the value of the angle L I H of the Hank, place a staff at L in the LSE OP THE PLAIN TABLE, &C. 433 direction of the telescope, and at the same time in the line I fl, whicb gives the length of the face 1) L :: kI the Hank IL; tlic lace CM and flank HM ..re ascertained in the same manner. Make BQ equal to ;- of A li, and A R equal to -— r-, and draw 1) II, C Q, through Z the middle of A D, draw O Z, make R T equal to f A 13 ; at T and with T D, form an angle D T U, equal to 105°; plant a statf at U, so that it may be in the line ZO, and at the same time in the direction T U, which gives the flank T U and face O U. Make Q S equal ; A B, at S form an angle CS P equal to 120^, draw the line S P, meeting the river, and the lines O, U, T, D, L, I, H, will be the line of the tcte de pont required. THE USE OF THE PLAI.V TABLE IN MILITARY OPERATIONS. Problem 42. To take the plan of a camp^fg. 2, plate 29. Place the table at A, where you can conveniently see the greater part of the field, and having made a scale on it, fix a fine needle perpendicular to the table at the place that you fix upon to represent the point A ; the fiducial edge of the index is always to be applied against the needle. Turn the plain table so that the index may point to the object B, and be so situated as to take in the field ; then plant a staff at C, in a line with B, point the index to the windmill F, and draw the indefinite line A V : then point it to K, the right wing of the cavalry K L, and draw A K, then to L, and draw A L, afterwards point the index to the steeple I, and then to the points M, O, P, N, H, E, G, then draw a line on the table parallel to the north and isouth of your compass, to represent the magnetic meridian. F f 434 USE OF THE PLAIx\ TABLE Remove the plain table to C, {)Umting a staff at C, measure A C, and set off that measure by your scale from A upon the line AC, and fix the needle at C ; then set the index upon the Hne A C^ and turn the table till the line o{ sight coincides with A, fasten the table, point the index to F, and draw C F, in- tersecting A F in F, and determining the position of the windmill F ; from C draw the indefinite lines C M, C K, C L^ &c. which will determine the points M K L, Sic. draw K L and m n parallel thereto, to represent the line of cavalry. Remove the table from C to D, setting up a staff at C, measure C D, and set off the distance on C B from your scale, place the needle at D, the index on C D, and turn the table till C coincides with the sights, and take the remarkable objects which could not be seen from the other stations. Staves should be placed at the sinuosities of the river, and lines drawn at the stations A, C, D, B, to these staves, which will give the windings of the river. Having thus determined the main objects of the field, sketch on it the roads, hillsy &;c. Problem 43. 7o ^ah a flan of the trenches of an attach^ fig, 1, flate 29. The plain of trenches, taken with accuracy, gives a just idea of the objects, and shews how you may close more and niore upon the enemy, and be covered from their enfilade fire, and also how to proceed in the attack without multiplying useless works, which increase expense, augment the labour, and occasion a great loss of men. Measure a long line A B, parallel to the front of the attack D F, place the table at A, and set up a staff at B, point the index to B, and draw a line to represent A B, fasten the table, and fix a needle at tiie point A, direct the line of sights to the flanked angle of the ravelin C, and draw A C ; proceed in the game manner with the flanked angles D^ li, F, G, IN MILITARY OPERATIONS. 435 to draw the lines ;it the opening of the trenches, plant staves at II and K, from A draw a line on the table in the direction A II, measure A II, and set off" that n)casare from your scale upon the line AH oi) the table. Remove the plain table from A to B, set up a staff at A, lay the fiducial edge of the index against the line A B, and turn the table about till the staff at A coincides with the line of sight, then fasten the table, ilirt>ct the sights to C, and draw B C, inter- secting \ at C; in the same manner ascertain the flanked angles D, E, F, G, draw a line in the direc- tion B R, and set off the measure thereof from your scale. Remove the table from B to R, and set up a staff at B where the plain table stood, lay the index upon the line corresponding with R B, then turn the table about till the line of sight is in the direction B R, screw the table fast, direct the sights towards P, and draw on the table the line K P, and by the ^caie lay off its measure on that line. Remove the plain table from R to P, lay the index upon the line P R, and turn the table about till the line of sight coincides with R, screw the table fast, and draw a line upon it in the direction P Q ; mea- sure PQ, and take the same number of parts from your scale, and set it off on the line, and so on with the other station Q. Plaving removed the table to the station S, and duly placed it with regard to Q, from the point S, draw the lines S V, S '1\ S U, setting off (Vom your scale their lengths, corresponding to their measures on the ground. Having thus taken the zigzags RPQ, S T W, and the parts U U W Z of the parallels, you remove the plain table to 11, and proceed in like manner to take the zigzags H I K, &c. as above, which will represent on your plain table the plan of the attack required. F f ^ [ 430 J OF LEVELi.ix\G, fig. 48, plate 28. Levelling is an operation that shews the height of one place in respect to another ; one place is said to be higher than another, when it is more distant from the centre of the earth than the other ; when a line has all its points equally distant from the cen- tre, it is called the line of trne level ; whence, be- cause the earth is round, thai line must be a curve, and make a part of the earth's circumference, as the line ABED, all the points of which are equally distant from the centre C of the earth ; but the line of sight A G, which the operation of levelling gives, is a right line perpendicular to the semi-diameter of the earth CA raised above the true level, denoted by the curvature of the earth, and this in proportion as it is mere extended ; for which reason, the opera- tions which we shall give, are only of an apparent level, which must be corrected to have the true level, when the line of sight exceeds 300 feet. Suppose, for example, that A B vi'as measured upon the surface of the earth to be 6000 feet, as the diameter of the earth is near 4201 8240 feet, you will find B F by the following proportions. * 4'20 18240 : 6000 :: 60OO : BF equal to 0,85677 f which is 10,28124 in. that is to say, be- tween two objects A and F, 6000 feet distant from each other, and in the vSame horizontal line, the dif- ference B F of the true level, or that of their distance from the centre of the earth, is 10,28124 in. When the di {Terence between the true and appa- rent level, as of B F, has been calculated, it will be easy to calculate those which answer to a less dis- tance ; for we may consider the distances B F, b f, * As the arc A B = 6OOO feet is but very small, it may be con- sidered equal to the tangent A F, and in this respect, A F is a mean proportional between the whule diameter, or twice the radius B C, and the ©xterior part B F. OF LEVELLING. 437 as almo>t equal to the lines A I, A i, which arc to each otlicr as the squares of the chorils, or of the arcs A B, a b, because in this case the chords and the arcs inav be taken one for the other. Thus to find the difference f b of level, which answers to .5000 feet, make the following propor- tion ; 6000 f. : 5000 f. : : 0,85677 : fb, which will be equal to 0,7K39C) f. or 8,56788 in. The point F, which is in the same horizontal line with A, is said to be in the apparent level A. and ••the point B is the true level of F ; so that B F is the diffvirence of the true level from the apparent. Problem 44, Jig. 49, plate 28. The above no- tions being snj)posed to know the difference of level between points B and A, which arc not in the same horizontal line ; then at A make use of an instru- ment proper to take the angle BCD, and having niea>ured the distance C D, or C I, by a chain which must be kept horizontal in different parts of it on the ground ALV B, you may in the triangle C DB, considered as rectangular in D, calculate B D, to which add the height C A of the instrument, and calculate the ditference of level D I, as we have shewn above. But as this method requires great accuracy in measuring the angle BCD, and an instrument very exact, it is often better to get at the same end with a little more trouble, v/hich is shewn by the follow- ing method. Problkm 45. Use of the spirit or ii-uter level, fg. bO, plate 28. Place the level at E, at equal distances from B and G, fix or,c station staff at B, the other at G; your instrument being adjusted, look at B, and let the vane be moved till it coincides with the line of sight ; then look at the staff G, and let the vane be moved till it coincides with the line of sight 11, and the difference in height shewn by the vanes on the Ff 3 438 AN ESSAY two Staves, will be the difference in the level be- tween the two points B and G. Thus, suppose the vane at G was at 4 f. 8 in. and at B 3 f . 9 in. sub- tract one from theother,and the remainder l 1 inches, will be the difference in the level between the two points B and G ; you may proceed in the same man- ner with the other points , but more need not be said on this head, as I have already treated this sub-- ject very fully in the foregoing part of this work. AN ESSAY ON PERSPECTIVE ; AND A DESCRIPTION QT SOME INSTRUMENTS, FOR FACILITATING THE PRACTICE OP THAT USEFUL ART, DEFINITIONS. DefinitiGn \, Perspective is the art of delineating the representations of bodies upon a plane, and has two distinct branches, linear and aerial. Definition 2. Linear perspective shews the me- thod of drawing the visible boundary lines of ob- jects upon the plane of the picture, exactly where those lines would appear if the picture were trans- parent ; this drawing is called the outline of those objects it represents. Di'jinilion 3. Aerial perspective gives rules to fill up this drawing with colours, lights, and shades, such as the objects themselves appear to have,, whca ON PERSPECTIVE. 43Q viewed at that point where the eye of the spectator is phiccd. To illastralc those definitions, suppose the picture to be a plate of glass inclosid [n a frame P L N, Jjg. 1, p/afe 30, through which let the eye of the spectator, placed at E, view the ohjrct QRST; /roni the given point E, let the visual lines E Q, E R, E S, &c. be drawn, cutting the glass, or pic- ture, in q, r, s, &c. these points of intersection will be the perspective rcj)rcscntations of the original QRJ^T, &c. and if they ate joined, q rst will be the true perspective delineation of the original figure. And lastly, if this out-line be so coloured in every part, as to deceive the eye of a spectator, viewing the same at E, in such a manner, that he cannot tell whether he views the real object itself, or its repre- sentation, it may be truly called the picture of the object it is designed for. DefniticH 4. When the eye, or projecting point, is supposed at an indefinitely great distance, com- pared with the distance of the picture and the ob- ject to be represented, the projecting lines being then supposed parallel, the delineation is called a parallel one, and by the corps of engineers, i7iiU- tiiry fer sped I've. Defiuilion 5. If this system of parallel rays be perpendicular to the horizon and to the picture, the projection is called a ])lane. Definition 6. If the parallel rays be horizontal, and the picture upright, it is called an elevation. Definition 7. A right line E e, fig. 2, phite 30, from the eye, E, cutting the plane of the picture at right angles, and terminating therein, at e, is cal- led the distance of the j)icture. Definition 8. The point e, where this central ray cuts the picture, is called the centre of the picture. Definition 9. The point X, Jig. 2, plate 30, where any original line cuts the picture, is called the inter- section of that line. lii A 440 AN ESSAY Defmiioji ^0. The seat ¥, fig. 2, flute 30, of any point E upon a plane, is where a perpendicular from that point cuts the plane; thus, if a perpen- dicular E F, be drawn from any elevated point E, to the ground plane FGN at F, this point F is called the scat of the point E upon the ground plane. Definition 11. Apparent magnitude is measured by the degree of opening of two radials passing through the extremes of bodies whose apparent ir.agnitude of Q R, fig. 3, flate 30, to an eye at E, is measured by the optic angle Q E R. Hence it is evident, that all objects viewed under the same angle, have the same apparent magnitude. Definition \1. The intersection G N, fig. 2, flate 30, of the picture with the ground plane, is called the ground line. Definition 13. A plane passing through the eye, and every where parallel to the ground (or ground plane, as it is commonly called, because it is sup- posed every where flat and level) is called the ho- rizontal plane, as E h n, fig. 2, flate 30. Definition 14. The intersection of the horizontal plane with the picture is called the horizontal line ; thus hen, fig. 2, flate 30, is called the horizontal line, being the intersection of the horizontal plane E h e n, parallel to ground plane FGN. Definition 15. The centre of any line is where a perpendicular from the eye cuts it. Definition l6. A plane i E F gh^ fig. 2, no. 2, flate 30, passing through the eye E, perpendicu- lar to the ground, is called the vertical flane ; and that part of the ground plan gfN which lies to the left hand, is called amplitudes to the left, and all that lying on the other side, amplitudes to the right, their measures being taken on the base line Q F N, or its parallels, as depths are by i' g, or its parallels. rrofosiiion 1. Parallel and equal straight lines ON PERSPECTIVE. 44l appear less, as they are farther removed from the eye. Let O P, Q R, fig. 3, pJalc 30, be two equal and parallel straight lines, viewed at the point E ; Q R being the farthest off, will appear the least. For, draw E Q and E R, cutting O P in r ; then Q R and O r have the same apparent magnitude, (defimtion 1 1 ,) but O r is only a part of O P. C'jrrJlary, Similar figures parallclly situated, ap- pear less, the farther they are removed from the eye. Proposinofi 1. All original parallel straight lines, P R, fig. 3, plate 30, which cut the picture at P, appear to converge to the same point O therein ; viz. that point which is the intersection with the picture, and a line passing through the eye parallel to the original parallel lines. For, since P R is parallel to E O Q, let R Q be parallel to O P, and therefore equal to it. By the last proposition, the farther R Q is taken from O P, the nearer it (the representation of the point R) ap- proaches to the fixed point O, and the case is the same with any other line parallel to P R ; and, therefore, if all the parallels be indefinitely produced, that is, at least till the straight lines measuring their distance become invisible to the eye, they will all appear to vanish together in this point, which, in consequence thereof, is called iht vcm'ishing point of all those parallels. Corollary 1 . Hence all original parallel planes, P RS P, intersecting the picture, as in P p, seem to converge to a right line therein, viz. that line O o, which is the intersection of the picture, with a plane Q O E o q, parallel to them all passing through the eye. For, since p S, fig. 3, plate 30, is parallel to o q, therefore p S will appear to converge to its vanish- ing point o, by the foregoing proposition ; and iince both the points R, S, equally appear to tend 441 AN ESSAY to the respective points O o, therefore the line R S also appears to approach to the line O o ; and this is the case with all planes parallel to P R S p, and therefore if they are produced till they become in- visible, they will appear to meet in O o, whence this line is called the vanishing of all those parallel planes. Corollary 2. The representations of original lines pass through their intersections and vanishing points; and of original planes, through their intersections and vanishing lines. Corollary 3. Lines parallel to the picture have parallel representations. For, in this case the line E O, which should pro- duce the vanishing point, never cuts the picture. Corollary 4. Hence the representations of plane figures parallel to the picture, are similar to their originals. Let q r s t u, Jig. 4, fhte 30, be the representa- tion of the original figure Q R S T U, to an eye at E ; then all the lines q r, r s, s t, &c. being re- spectively parallel to their originals Q R, R S, ST, &c. as well as the diagonals, t q, T Q, &c. there- fore the inscribed triangle q t u, and Q T \J, are similar to each other, and so of all the other tri- ani^les ; and, therefore, the whole figure q r s t to QRST. Corollary 5. The length of any line in the re- presentation is to that of its respective original, as the distance of the picture to that of the original plane ; for all the triangles E Q U, E q u, E Q R, E q r, &c. are similar ; therefore, as E q is to q u, so is E Q to Q U ; that is, as q u is to Q U, so is the distance of the picture to the distance from the plane Q U. Problem 1. IJav'uig the centre and distance of the picture given^ to find the representation rf any ^iven point thereon, fig. 5, ■plate 30. ON PERSPECTIVE. 443 From the eye E, to the centre of the pietiirc c, draw E c, and parallel to it Qd from the given point Q, cutting the picture in d; join c d, and draw E Q, intersecting c d in q, the perspective place of the original point Q. For, e d is the representation of the original line d Q, indefinitely produced from its intersection q, till it appears to vanish in c; therefore (jnm.'st be somewhere in this line, pr. 1, cor. '1 ; it must alsp be somewhere in E Q, and therefore in the point q, where they intersect. Problem 2. The cetiirc and distance of the pic- ture being given, to find the representation of a given line Q R, fig. 6, plate 30. Produce R Q, to intersect the picture in X, draw EV parallel to RQX, cutting the picture in V, the vanishing point of the line R X ; draw E R and E Q, cutting V X the indefinite representa- tion of RX, in r and q, and rq is the represen- tation of R Q. This needs no demonstration. Problem 3. Having the centre and distance of the picture, to find the representation of aji original plane, whose positioti with respect to the picture is given. Draw any two lines, except parallel ones, upon the plane, and find their vanishing points by the last problem ; through thCvSe points a line being drawn will be the vanishing line required. This is evident from prop. 1, cor. 1. Problem 4. Having the centre and distance of the picture given, to find the vanishing point of lines perpendicular to a plane whose representation is giveuy fig. 7, plate 30. Let PL be the vanishing line of the given re- presentation, X its centre, e that of the picture, and cE its distance; join EX, and perpendicular to it draw E f, cutting X e in f. 444 AN ESSAY For, EPL is the original plane, producing the Tanishing line P X L, and E f being a visual ray, parallel to all the original lines that are perpendicu- lar to the plane EPL, or its parallels ; f is there- fore the vanishing point of all of them. Corollary 1. When the vanishing line passes through the centre of the picture ; that is, when the parallel planes arc penpendicular to that of the picture, the points X and e coinciding with Ef become perpendicular to E e, or parallel to the picture, and therefore the lines will have parallel re- presentations. Corollary. 2. If the original lines were desired to make any other angle with the original plane than a right one, it is only making x E f equal to it. Problem 5. The cenire o.^ and dhtance E e, of the picture heing given^ and the vanishing p§mi f, of a line parallel to E f, to find the vanishhig line of planes perpendicidar to that Une whose parallel ^{pro- duces the 'vanishing point f, fg. 7, plate 30. Join f e, and make EX perpendicular to, cutting f c, produced in X, draw F L perpendicular to c x, and P L will be the vanishing line required. The planes may form any angle instead of a right one, if that f EX be made equal to the same. • Problem 6. The centre and distance of the pic- ture heing given^ and the vanishing point of a Une, to find the vanishing line of planes, perpendicular to the liyie zvhose vanishing point is given, fg. "i , plate 30. From f, the given vanishing point, through e, the centre of the picture, draw feX, and through the eye E draw E f, perpendicular to which draw EX, cutting fein X; make FXL perpendicular to X ef, and PL will be the vanishing line. Problem 7- Having given the centre and dis- tance of the picture, the inclination of two planes, the vanishing line of one of them, and the vanishing point of their common interseclion, to find the vanishitig lins of the other plane. ON PERSPECTJVE. 445 Case 1. Let the inclination of the planes lie a right angle, and let P L be the vanishing line, P the vanishing point of their common intersection. Ijy Problem 5 tind f, the vanishing point of lines perpendicular to the plane, whose vanishing line is P L ; join P f, which is the vanishing line rc- tjuircd. For, since this last plane is perpendicular to the former, f will be the vanishing point of one line in it, (prop. 2. ) and P being the vanishing point of another, therefore P f is the vanishing line. Case 2. When the inclination n e m, fig. 8, plate 30, is greater or less than a right angle. Let N X be the vanishing line of one plane, x the vanish- ing point of its intersection with the other plane ; froiw the eye E draw E x, and perpendicular there- to, the plane NEM intersecting x N in N, and the picture in NfM, make the angle NKM equal to n e m the given inclination, join x M,which will be the required vanishing line. lor, the planes being parallel to the original planes by construction, M x, N x are their vanishing lines, cor. 2, pr. 2. Pboblem 8. To find the projeclloii of any solid figure. ^ Fina the representation of any one of its faces by Problem 3, and of the others by the last ; if any side be convex or concave, a number of points may be found therein by Problem 1, and curves drawn evenly through them will represent the curved su- perficies required; or squares, or other regular iigures may be inscribed or circumscribe4 about the origi- nal figures, and these plain figures being projected by the foregoing methods, together with a few points therein, by which means the curves may be similarly dravvn about or within these projected squares, &c. by this means also may tangents be drawn to the re- presentations of all kinds of curve lines, in all kinds of situations. [ 446 ] GENERAL REMARK. So far I have endeavoured to render the principles- \ of perspective obvious by a mere inspection of the figures. For the young artist^ whose mind seldom conforms to mathematical reasoning, may, in all the foregoing problems, suppose the plane of the picture placed upright upon that of the paper, which paper he may consider as the ground plane, the operator's eye E, being always in its proper situation with re- j spect to the picture, as well as to the planes of the ' original objects. The data, or things required to be known before objects can be put into perspective, are their plans and elevations, see Defin. 5 and 6, which must be actually laid down by a convenient scale adapted to the size you mean your picture shoi^ld be, which may be very easily accomplished ; for you must remember, that the true extreme visual rays^ that is, those which pass from the eye to the two op- posite borders of the picture, must not make an an- gle le-^s than two- thirds, or greater than three- fourths of a right one. With respect to the distance of the picture, it must be remembered, that objects cannot be seen distinctly nearer to a common eye than six inches, and therefore in the smallest miniature pieces the distance must exceed that quantity; the height of the eye should be about half the distance of the picture,and about the third part of the picture's whole height. For example, if the whole height of the picture be three, the distance from it should be two, and the height of.the eye one. If the plan, when laid down by its proper scale, be bounded by a quadran- p;le G N c d, fig. 9, no, 1, plate 30, then the ground line of the picture is supposed to be placed upon the shortest side of it, and upright to its plane ; there- fore this side of the figure should be made exactly equal to the breadth of the picture, and the other two adjacent sides should, if produced^ meet at the dis- GENERAL REMARK. 4if fance of the picture, as at F the foot of the observer, as is represented in Jig. g, no. 3. The problems al- ready given are sufficient for all cases that can hap- pen in putting objects in whatsoever position into perspective ; and though they arc perhaps solved in such a manner, as to give the clearest ideas of the genuine practice of perspective, yet others may pos- sibly prefer some of the following methods, vvhieh are however easily deduced from the preceding system. Example 1. Having given the centre e of the picture P L N f, fg. '2, no. 2, plate 30, its distance cE 12 inches, height of E F six inches, to find the representation of a point T, whose depth in the plan is eight inches, and amplitude to the left five inches. Method 1. Form fin f-JSf, take f n = five inches, and having draw T perpendicular to f N, make it equal to eight inches, then will T be placed in its proper situation, and n will be its seat on the pic- ture ; join T F, (F as usual being the foot of the observer's eye upon the ground plane, that is, in the present example, six inches perpendicularly below the eye E) cutting f N in t, draw t o perpendicular to f N, and join T E, intersecting t o in o, the re- presentation of the point T required. Otherwise, if the point T be situated on the ground plane, draw FT, cutting {^ in t, and ts perpen- dicular to f N, cutting the visual E T in o, the point required. If the point T has any elevation perpendicular over the same point, as at S, make T S etjual to the height it should have, and draw E S, cutting the perpendicular o t s in s, the perspective point of S required. Remark. This method is very convenient in some cases, as when there are many windows, &c. per- pendicularly over one another, &c. 448 OENEilAL REMARK. Method 1. In the horizontal line e E,Jig. 2, no. 3, fhte 30, from the centre of the picture e, take e E equal to the distance of the picture ; from the given point Q. draw Qd perpendicular to the ground line G N, make d Q' equal to d Q, and join Q."^ E and e d, intersecting each other in q, the represen- tation of the point Q. Method 3. Perpendicularly over the centre e, jig. 1, no. A, plate 30, of the picture, take E e, equal to the distance of the picture, draw Qd per- pendicular to G N, join E Q and e d, intersecting each other in q, the point as before. Remark. Both these methods are in fact the same as that in Problem 1, as may be seen by comparing the figures with each other, being marked with the same letters for that purpose. It may be farther re- marked, that both may be alternately used in the same piece, remembering to use that which you judge will give the bluntest intersection, that is, whichever makes the angle e q E the greatest. And now supposing a clear know^ledge both ot the theory and practice to be obtained, it may not be amiss to shew how naturally a practice mores elegant and simple, may be deduced, viz. by sup- posing all the planes to coincide with the paper, or plane of the picture. Thus, jff^. 5, flate 30, if the triangles q e E, q d Q revolve round d q e till they fall into the plane P LN, no change takes place in any of the com* parative distances E e, E Q, &c. and the point q preserves its situation on the picture, as in fg. 1, no. A ; and again, if in fg. 2, tio. 4, e E be moved round on the point e, till it falls into the horizontal line at n, and d Q be moved similarly round, till it becomes, as in fg. 3, no. 3, parallel to e E, the point q still remains unaltered. Method A. Let m Nf s r, fg. 2, no. 5, pJate 30, be the ground line, f e E perpendicular to it, passing GENERAL RExMARK. 449 through ihc centre of the picture e, and let fK* equal to fN. be the distance of the picture, NF being pcrpcndicuhir to N F, and the representation ot the point Q, whose distance from N r is r Q. Draw F/ Q, cutting Nr in s, taking N m equal to Q r, and draw c m, cutting N P in o, draw oq parallel to N r, and s q to No, intersecting each other in q, the point required. Remark. The lines E' Q c m need not be drawn, but a dot made at s, where the ruler crosses n r and em; a T square * may also be applied to the line N s r, and a dot made on its fiducial edge at s ; then if the square be slid up till the fiducial edge crosses the mark at o, the point s will be transferred to q, the representation of the given point obtained without drawing any lines over the picture. (This method was first discovered by Mr. Beck, an ingeni- ous artist, well known for many usefijl contrivances.) It is also more accurate than the last method in gene- ral, as the intersections are more obtuse. But nevertheless when the two points in represen- tation are projected very near together, it is the best way to work by the intersections and vanishing points of the original lines ; for then having the full extent of the representation, its true directions may be ascertained very correctly. The vanishing points by this method are thus found ; suppose for instance of the line I Q intersecting the ground line in I, in f e E' take e E equal to E^ f, that is, equal to the distance of the picture ; draw E V parallel to I Q, intersecting the horizontal line ho in V the vanishing point of I Q and all its parallels. * ^titfg. 20, plate 31. This is a very u.seful article in drawing ; a ruler a. Is tixtd as a square to /> ; there is aL-o a moveable piece r. This ruler applied clo^e to the side of a true drawing board, will ad- mit of parallel Jines being (irawn, as well as oblique ones, with more case and expedition than by the common parallel rulers. Hon. Gg 450 GENERAL REMARK. Rxample 2. To put any plane figure, as Q R S T, or M N O r, into perspective, plate 30, fig, g. This is, in fact, only a repetition of the last ; for by linding the projections of the several points, as before, nothing remains but to join them properly, and the thing is done ; thus the points m, n, o, p, projected from M, N, O, P, being joined, give in p o n, the representation of the quadrangle M J^ O N, &c. But if any line, as p o, should be very short, so that a small error in the point o would considerably alter the direction of it, find the intersection and vanishing point of its original ; or, if the vanishing point falls at too great a distance, as it very frequently does in practice, find the intersec- tion E"", and vanishing point Z of any other line passing through the point O, as the diagonal O M, and the truth of the projection may be depended upon. All the three last methods may be made use of in projecting the same figure ; thus the point p, by the second method, that is, make e E in the horizontal line equal to E* k, the distance of the picture, draw P Y perpendicular to S X, and take Yk eqnal to Y P, join Ek and eY, intersecting in p, which is a better intersection than the third method woi.ld give, viz. by drawing EP; but yet inferior to that in the fourth method. Example 3. To put any solid body into perspcQ- tive. Fi:;d the scats of all the points upon the ground plane, and project them as before ; let G be the seat of one of the points, g its projection, pro- duce e g to the ground line a E'', and make E* H perpendicular, and equal to the height of the given point, from its scat join He, and it will cut the perpendicular gg"", in g', the reprcsentatign of the equired point. Or, by the fourth method, G^ S being equal to the distance of G from S X. Make G' G' perpendicular and equal to the GENERAL REMARK. 451 s^ivcn height, draw G" c, cutting S U in e', then e* g' parallel to S X, will cut the perpendicular gg' in the point g' required; or the point g may he transferred to g" by the T square, without draw- ls? SS'' ^^ ^' ?'• Proceed in this manner, till you have obtained all the requisite points in the figure, marking or num- bering them as you proceed, or else correct them by straight or curved lines according to your original, by which means you may easily see the connexion ofvour work at all times without confusion. Reiniirks. 1. Sometimes when there is a great number of small parts in a body to be perfectly mailc out, it may not be amiss to draw squares over the ground plane of the object, and find the seats of its several elevated points ; then by turning these squares into perspective, the positions of the several points will likewise be found by inspection, and the horizontal row of squares will serve as a scale for the altitudes of bodies, whose seats lie in that row, or both plans and elevations may be used, as in fig. 10, p'ate 30. 2. Perspective may also be practised without hav- ing any recourse to ground plans; for, by taking the horizontal angles, the amplitudes of objects are to be ascertained, and by vertical ones their heights and depths in the picture ; the angles thus taken may be entered in a table of this form, Vert. Z Horizontal. Vert. Up. Left. Right. Down. Gg 1 452 CEXERAL REMARK. In protracting which angles you must make the distance of the picture radius, iind lay down the an- gles by a line of tangents adapted to that radius, which is therefore best done by means of the sector ; or more expeditiously by the fourth method, page 448; for, if a protractor be fixed at E% fig. 9, plate 30, horizontal angles will cut the ground line S X in the same places, as if the lines forming them passed over the original jjoints in the plan ; and if the protractor be fixed at e, the same may be said of S U with re- spect to elevations. Solid bodies may also be put into perspective, by drawing lines in particular directions, as from the centre of a circle, or of concentric ones, and finding the representations of the same, raise perpendiculars of the proper heights, always supposng the bodies to be transparent, fig. 10, 'plate 30. OP SHADOWS. Luminous bodies, as the sun, moon, lamps, &c. are generally considered as points ; but the artist takes the advantage of their not being perfectly so, by sof- tening the extremities of his shadows, which are so softened in nature, and for this reason, which may be thus explained. . Let R S, fig. 11, no. 2, plate 31, be the radius of a luminous body, whose seat is r s, and centre is S ; from the opaque body O, draw x r, touching the ex- tremities of both the bodies on both sides, by which means a penumbra, or semi-shade, q x n, will be formed on each side of the main shadow, which be- comes extremely tender towards the outer extremity, and from thence gradually strengthens till it blends with the uniform shadow, which will, if the diameter of the iuminous be greater than that of the opaque one, measured in the direction of their centres, con- verge to a point, as n fi, fig. 1 1, folate 3 1. N. B. The opaque body which casts the shadow is called the shading body ; and those that are im- mersed in the shadow, arc called shadowed bodies. OP SHADOWS. 453 Problem 9. 7o find the shaihw of any object upon a plane. Througli the luminous body draw planes toucbinir all the illumined planes of the object, and the inter- sections of these planes with the given plane, will give the boundary of the shadow required. Example 1. Let the lumiiKMis point be tlic sun, the plane of projection ARSB, ^-r^. I'i, phite 3\, and the object the parallelogram A B C D. Tbe sun's rays, on account of his distance, may be supposed parallel, therefore the planes OR A, O S B, are parallel ; and, therefore, since C D is parallel to B A, R S is parallel to it also, and the shadow a parallelogram, whose length is to the height of the object, supposed upright, as radius to the tangent of the sun's altitude ; and its breadth as radius to the sine of the inclination of its rays, with B A the base of the parallelogram. Therefore find the seat of one of the rays, as O R A, and make the angle ADR equal to the comple- ment of the sun's altitude, and make the parallelo- gram B R, and the thing is done. Example 2. Let the luminous body be a lamp placed at O, fig. 13, plate 31 ; the object, a paralle- logram A B C J), standing upon the plane ARS D. Having drawn O G perpendicular to the plane A R S 1), and the rays O B R, O C S ; from G, the scat of the lamp on the plane A S, draw GAR, G D S, intersecting the rays OB, O C, in R and S> and A RS D will be the shade required. Example 3. Where the shadow from the sun, &c. passes over different objects. Continue the sides 'CADB to A', A", I/, B", &c. fig. 14, plate 31, and where A G, B H, cut the body GHIK, draw K G A", I H B, making the angles A' G A", B H B', equal to the inclination of the plane G I to G B, proceed in the same manner with every new plane M N ; or if the object is cur- vilinear, tangents will always pass through the lines Gg J 454 OF MILITARY PERSPECTIVE, C A' and D B respectively, except when they are perpendicular. Example 4. Let a lamp O, fig. 15, plate 31, throw a shadow on the body R S U T, and let QTU be the central line of the shadow from the parallelo- gram, A B, upon the ground ; draw O B, G P, parallel to Q 1\ and from T to OP draw t P, touching the plane U s of the body ; from P draw P S u, P w U, cutting .the extremities of the shadow in U and u, and w U u S will be the shadow of A B upon the face US ; proceed in like manner with all the other illumined faces. N. B. If t P never meets OP, it denotes the face of the body to be parallel to A B, and the shadow on that face to be a parallelogram. The shadows, being thus ascertained, may be put into perspective by the foregoing rules. Problem 10. To find the refiectlons of objects upon polished surfaces. Let fall a perpendicular upon the reflecting plane, to which draw a radial from the eye, as much below the horizontal line, as the real object appears to be above it. Example 1. Let A B, j%-. ]&, plate 31^ be any object placed on the water ; from B draw B b, per- pendicular to the surface A b, which continue till the angles B E b and C E b are equal ; that is, (E b being a horizontal line) till b c is equal to bB, and b C will be the reflection of AB. Example?.. When objects are upright, the lines may be produced below the horizontal Iffic, as much as the real ones are above it. OF PARALLEL, OR MILITARY F ERSPKCTI VE, In this kind of projections, the eye is supposed to be j)lnccd at an indefinite distance from the ob- ject in the diagonal, and looking down upon it ia OP AERIAL PERSPECTIVE. 455 an angle of 45'^, so that the top, one sicie, and one end, are seen under the '^iime angle, and therefore appear In their true {Moportions with respect to each other ; and therefore heights, lengths, and breadths mu C. And then I go to A, and measure every field in my way thither. Next I go from A towards 1), and set down as be- fore, all my crossing of the hedges ; and the length A D, when I come at D. And in like manner I mea- sure along D" C, setting down all the crossings of the hedges as before, with whatever else is remark- able, as where a highway crosses at d. Having finished all the main stations, we mast beofin to make inner stations. Therefore I take F ' and G for two stations, being in the lines A B and BC, the hedges from F to G running almost straight ; then I measure from F towards G, and at /, I find a hedge going to the left, and going on to^, I find another hedge going to the right; and at //, I cross the burn. At i, there is an angle, to which I make an offset. Going on further, I come at a cross hedge /, going to the right; and then measure on to G, the end of the station. Now in going from FtoG, we can take all the angles that the sides of the fields make with the stationary line F G, and then measure their lengths ; by which these fields may be laid down on paper. 'i hen I take another inner station at I, and mea- suring from A to o, I come to the opposite corner of the field ; then measuring on to p, I cross a hedge ; then I proceed to my station I. 'J hen I measure from I to F, and take an offset to ?/, where the hedge crosses the brook. Then I come to the corner of the last field at m ; and then measure to the opposite corner at F, the other station. In your going from A to I, you may take the angles that the hedges make with your stationary line A I, and measure these hedges, and then they may be laid down. And the like in going from 1 to F. OF SURVEYING. 46/ All this being done, take a new station H, and measuring- from B towards II, all the hedges lie al- most in a right line. So going along we come at a cross hedge, and going further we come at a tree, in the hedge we measure along ; going further we come at two other cross hedges ; and a piece fur- ther we cross the brook. ; going on we come at a cross hedge ; going on still, we come to another cross hedge ; all these hedges are to be left. Then going on stiJl further, we have a windmill to the right ; and afterwards a cross hedge to the left, and then we measure on to the station 11. Then measuring from H towards C, we have a house on the left; and then go on to C. And the fields may be all surveyed as you go along B II and H C, and then laid down. And after this manner you must proceed through the whole, taking new stations till all be done," Mr. John Rodham's new method of survey- ing, WITH THE PLAN OP THE FIELD-BOOK, plate 34. *' The field book is ruled into three columns. la the middle one are set down the distances on the chain line at which any mark, offset, or other obser- vation is made ; and in the right and left hand co- lumns arc entered the offsets and observations made on the right and left hand respectively of the chain line. It is of great advantage, both for brevity and perspicuity, to begin at the bottorfi of the leaf and .V rite upwards ; denoting the crossing of fences, by lines drawn across the middle column, or only a part of such a line on the right and left opposite ihe figures, to avoid confusion ; and the corners of fields, and other remarkable turns in the fences where offsets arc taken to, by lines joining in the manner H h 2 468 eodham's new method the fences do, as will be best seen by comparing the book with the plan annexed to the field-book,- as shewn \uplalc 34. The marks called, a, b, c, he. are best made in the fields, by making a small hole with a spade, and a chip or small bit of wood, with the particular letter upon it^ may be put in, to prevent one mark being taken for another, on any return to it. But in general, the name of a mark is very easily had by referring in the book to the line it was made in. After the small alphabet is gone through, the ca- pitals may be next, the print letters afterwards, and soon, which answer the purpose of so many dif- ferent letters; or the marks may be numbered. The letter in the left hand corner at the beginning of every line, is the mark or place measured from ; and, that at the right hand corner at the end, is the mark measured to : but when it is not convenient to go exactly from a mark, the place measured from, is described si/ch a distance from one mark to- wards another ; and where a mark is not measured to, the exact place is ascertained by saying, turn to the right or left hand, such a distance to such a mark^ it being always understood that those distances ar© taken in the chain line. The characters used, are f for /;/;-« to the right hand, ~\ for turn to the left hand^ and A placed over an offset, to shew that it is not taken at right angles with the chain line, but in the line with some straight fence ; being chiefly used when crossing their directions, and it is a better way of ob- taining their true places than by offsets at right angles. When a line is measured whose position is de- termined either by former work (as in the case of producing a given line, or measuring from one known pla<:eormark to another) or by itself (as in the third side of a triangle) it is called a fast line, or SURVEYING. 469 and a double line across the book is rlravvn at the conclusion otit ; but if its position is not determined (as in the second side of a triangle) it is called a loose line, and a siuf^le line is drawn across the book. When aline becomes determined in po'^ilion, and is afterwards continued, a double line half through the book is drawn. When a loose line ia measured, it becomes ab- solutely necessary to measure some line that will determine its position. Thus, the first line a h, be- ing the base of a triangle, is always determined ; but the position of the second side /;/', does not be- come determined, till the third side j b is measured; then the triangle may be constructed, and the posi- tion of both is determined. At the beginning of a line, to fix a loose line to the mark or place measured trom, the sign of turning to the right or left hand must be added (as at j in the third line ;) otherwise a stranger, when laying down the work, may as easily construct the triangle hj b on the wrong side of the line a //, as on the right one : but this error cannot be fallen into, if the sign above named be carefully observed. In choosing a line to fix a loose one, care must be taken that it does not make a very acute or obtuse angle ; as in the triangle /> B r, by the angle at B being very obtuse, a small deviation from truth, even the breadth of a point at^ or r, would make the error at B, when constructed, very considerable ; but by constructing the triangle /> B ^, such a deviation is of no consequence. Where the words, leave off, arc written in the field-book, it is to signify that the taking of ofF- scts is from thence discontinued ; and of course something is wanting between that and the next offset." Hh 3 [ 470 ] MR. KEITHS IMPROVED PARALLEL SCALE. Mr. Thomas Keith, teacher of the inathematics, his considerably improved the German parallel ruler, or that of Mr. Marquois, see pages 25 and lQ. By making the hypothenuse, and the perpendicular line to it from the opposite angle, in the ratio of 4 to J, and adding several scales, &c. its uses are consider- ably extended for drawing plans of fortifications, and other branches of the mathematics. Fig. 2, 3, 4, and 5, plate 33, represent the two faces of the ruler, and the triangle of half the real dimensions. The slider is a right-angled triangle, the perpen- dicular is divided into inches and tenths, and the base has three indices, and other divisions requisite to be used with the scale. The ruler contains l6 different scales, which, by the help of the slider may be increased to 20, with- out guessing at halves and quarters. The figures on the one end shew the number of divisions to an inch, and the letters on the other end are necessary to exemplify its use. It likewise contains the names of the polygons, the angles at their centres, and a scale of chords by which all polygons may be readily constructed. ITS USE AS A COMMON PARALLEL RULER. To draw a line parallel to the slider. Lay the hy- pothenuse, or sloped edge of the triangle in the po- sition you intend to have your line, place the scale against the base of the triangle, and draw a line along the slope edge, keep ihe scale fixed, and move the slider to the left or right hand, according as you want a parallel line above or below the other. To drwjo a line faralhl to the scale. Lay the scale krith's improved, &c. 471 in the position you intend to have your lines, and draw a line alon^ the cd^^c of it. Place the base of the triimp^le again-^t this edge, tho middle index standing at O on the scale, and make a mark against any one division on the pcrpendicnlar ; turn the iri- angle the other side uppermost, and n)ake a mark against the same division ; join these marks, and this line will bo parallel to tlie former. The same may be done by sliding the triangle, without turning it, if the lines are not required to be very long. To draw a line parallel to the sluler at a ^rcat Jis' tauce. Draw a line along the slope edge of the tri- angle, and fix the scale as before, make a mark against 1 on the perpendicular ; the index being at O, turn the triangle the other side uppermost, and make a mark against the same division, take away the tri- angle, and move the scale to these marks, apply the triangle again ; proceed thus till you have got the proper distance, then draw a line along the slope edge, and it will be parallel to the former. To draw a line parallel to the scale at a great dis- tance. Draw a line along the edge of the scale, place the base of the triangle against that edge, the middle index being at O, and make a mark against 2 on the perpendicular ; turn the triangle the other side up- permost, and make a mark against the same division ; take away the triangle, and move the scale to these marks ; proceed thus till you have got the proper distance, then draw a line along the scale, and it will be parallel as required. ITS USE IN ERECTING PERPENDICULARS. If a line is wanted perpendicular to the scale, ap- ply the base of the slider to it, and draw a line from the scale along the perpendicular of the triangle ; should a longer perpendicular be wanted, apply the perpendicular of the triangle to the scale, and draw a line along the base. Hh 4 472 KEITHS IMPROVED If a perpendicular is wanted to a line, which has been drawn along the hypothenuse of the tria-ngle, keep the scale fixed, and apply the hypothenuse of the triangle to it, then draw a line along the per- pendicular of the triangle. ITS U6E IN CONSTRUCTING POLYGONS. Having made choice of any one scale, take the side of your polygon from it. Take the degrees under the name of your polygon, and subtract them from 180 ; at each end of the above side of your po- lygon, make angles equal to half the remainder, and the distance from the intersection of the lines, which form the angles, to either end of the side of your polygon, will give the radius of its circumscribing circle. Or, the three angles, and one side being given of a triangle, the radius, or remaining sides, may be found by the rules of trigonometry. JTS USE IN DRAWING PARALLEL LINES AT ANY GIVEN DISTANCE, WITHOUT THE ASSIbTANCE OP A PAIR OP COMPASSES. If the side of your 'p(Aygon was taken frojii the scale C. More the slider from O to 1, 2, 3, &c. on the scale D, and you will draw a parallel line of the width of 1,2, 3, &tc. divisions on the scale C. This scale is 20 fathoms to an inch. In a similar manner the scales G, L, and P, are to be used. If the side nj your folygon iias taken from the scale C, calling the divisions tivo each. Move the slider from O, on the scale A, to 1, 2, 3, &c. on the same scale, and you will draw a parallel line of half the width of 1, 2, 3, &c. divisions on the scale C. This scale is 40 fathoms to an inch. PARALLEL SCALE. 473 Similar instructions may be applied to the scales G, L, and P. If the side of your polygon was taken from the scale C, calling the divisions three each. When the mid- dle index stands at O, the divisions marked Con the slider will make straight lines with 5 and 6 on t!ie scale D. J5y moving the slider to the right or left till the other divisions thereoi> make straight lines successively with 5, 6, &c. you will draw a parallel line of one-third of the width of 1,2, 3, &c. di- visions on the scale C. Ihis scale is 6o fathoms to an inch. Similar directions must be observed in usinij the •scales G, L, and P. If the side of your polygon "jcas taken from the scale A. Move the slider from O, to 1, 2, 5, &c. on the scale B, and you wiil draw a parallel line of the width of 1, 2, 3, &c. divisions on the scale A. This scale is lO fathoms to an inch. Similar directions must be observed in usinir the scales E, I, and N. Jf the side of your polygon was taken from the scale D. Move the slider from O, to 2, 4, 6, &c. on the scale B, and you will draw a parallel line of the width of 1, 2. 3, &c. divisions on the scale D. This scale is five t'athoms to an inch. In making use of H, M, and Q, similar directions must be observed. Note. There is no absolute necessity for always moving the siidcr from O, and it maybe used cither side uppermost, :\ny one of the three indices may likewise be made use of, &c. 'Jo render the scale still more perfect, Mr. Keith has made the following additions. 1. The divifeions on the scale A, and the fourth scale from A, have been subdivided lor the purpose of constructing sections, &:c. '1 en of the divisions on the scale A make one inch ; if, therefore, you 474 KEITHS IMPBOVED call these divisions six each, the small divisions on the same scale, nearest to the left hand, will be one each ; if you call the large divisions on the scale A, five, four, or three each, then the sets of smaller di- visions in a successive order from the left hand di- vision above-mentioned, towards the right, will be one each. 2. Five of the divisions on the fourth scale from the edge make an inch ; if, therefore, you call these divisions 3, 5, 7j 9? or 1 1 each, then each of the sets of small divisions, from the left hand towards the right, will be subdivided into parts of one each. Thus you have a scale divided into 2^, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, and 6o equal parts to an inch, exclusive of the scales on the three other edges of the ruler. 3. The line P on the slider stands for polygons ; R, radius ; the figures in this line being the radii of the several polygons under which they stand, the side being 180 toises in each. If the side of your polygon be different from 180 toises, say, as 180 toises are to the radius under the number of sides your polygon contains, so is the side of your polygon to its radius. Ex. What is the radius of an octa- gon, the exterior sides being 120 toises? as 180 : 235 • 18 : : 120 : 156 ' 78 Ans. The remiiining lines are to be read thus — The English foot is to the French foot as 107 is to 114, or as 1 is to l.o65, a toise six French feet, a fathom six English feet. The English foot is to the Rhynland foot as p715 is to 10000 ; the Rhynland foot is to the French foot as 1033 is to lOSs ; the Rhynland rod is 12 Rhynland feet. It m.ay not be amiss to remark, that either the scale or the slider will erect a perperdicular instan- taneously, for the long divisions across the scale make right angles with the edges. [ 475 ] OF THE GUNNERS CALLIPERS, OR COMPASSES. This article is generally included in the magazine case of instruments tor the military officer, or engi- neer ; is a very usetul mathematical instrument in the artillery service, and, as its description was omit- ted by our late author, I have taken the opportunity of inserting some account of it here. The principal uses of this instrument are to take the diametersof common shot, the bore or caliber of a piece of ordnance, estimate the weight of shot, quan- tity of powder, &c. for guns of given dimensions, and other particulars in practical gunnery. Ft^. 6 and 7, arc the re{)resentations of the two faces of the callipers marked A, B, C, 1). They consist of two thin flat brass rulers moving on a joint, curved internally to admit the convex figure of a ball, whose diameter is to be taken by the points at the end A ; these points are of steel, to prevent much wear. The rulers are made from six to 12 inches in length from the centre, according to the number of lines and tables to be engraved upon them. The usual length for pocket cases is six inches, the scales upon which I shall now describe. On Fi^. 6, ruler A, is contained, 1. A scale of inches, divided into tenths, and continued to 12 inches on the ruler B. 2. A table shewing the quantity of powder neces- sary for charging the chambers of brass mortars and howitzers. 3. On the ruler B, is a line marked Inches, be- ginning from the steel point, for t^iving the diame- ters of the calibers of guns in inches. 4. A line marked Guns, contiguous to the preced- ing, shewing the nominal pounders, or weight of shot for the res|x;ctive bores of the guns in inches. 5. On ihc semi-circular head of the rule is a semi- 476 DESCRIPTION OP circle divided into degrees, figured in contrary direc- tions, to measure angles by, and give the elevation of cannon, &c. 6. Next to the preceding, is a circular scale marked Shot Diameters, which, v;ith the chamfered edge marked Index, shews the convex diameter in inches of a shot, or other object placed between the points. The quadrant part of the joint prevents this being represented in the figure. On the other face of the callipers, C and D, Jig. 7, are engraved, 7. on the ruler C, a table xn?iTk&^ Brass Guns, shew- ing the quantity of powder necessary for the proof and service charges of brass guns. 8. The line of lines, marked Lin. g. Two circular scales on the head of the ruler, marked Shot, shewing the weights of iron shot, as taken by the points of the callipers. On the ruler D, besides another corresponding line of lines, 10. A table shewing the quantity of powder ne- cessary for proof and service charges of iron guns, from 4 to 42 pounders. 11. To these are sometimes added various figures of a circle, cube, &c. with numbers. 12. A table of the weights and specific gravities of a cubic foot of various metals, ivory, wood, wa- ters, &c. EXPLANATION OF THE LINES AND TABLES. 1. The line of inches, graduated contiguous to the exterior edges of the sides A and 13, Jig, 6, when opened to a straight line, makes a measure of 12 inches and tenths for the purpose of a com- mon rule. 2. "^1 he table shewing the quantity of powder ne- cessary for mortars and howitzers. Ihis table is THE GUXNERS CALLIPERS. 4/7 adapted for both sea and land. Thus, by inspection merely is shewn that a 13 inch brass mortar at sea re- quires 30 pounds of powder; by land only lOpounds. Mortars and ho.vitzers under 10 inches have only the land quantities inserted. 3. The line of inches for concave diameters. It commences from the steel point at 2 inches, and con- tinues on to 10 inches, and is subdivided to halves and quarters. When used, the legs are placed across each other, and the steel points brought to a contact with the internal concave surface of the gun, at the diametrical, or greatest possible distance ; the calli- pers then being- taken out, and inspected where in tiie scale one external edge of the rule is upon the other scale, that division will give in inches and quar- ters the caliber of the gun required. 4. The line shewinii* the weia:ht of shot for the bores. Adjoining to the preceding line is the one marked Gum, proceeding from 1 - to 42 pounds. It is therefore evident, that at the same time the caliber in inches is given, the weight of the shot is also given by inspection on the scale, by the side of the ruler crossing both ; and, from the weight of the shot, the caliber in inches. Thus 4^- inches shews a Q-inch shot, and vice versa. 5. The semi-circular graduation on the joint- head. The degrees are figured to 180 contrary- ways, so as one set to be the complement of 180 to the other ; and they are used to lay down or mea- sure angles, find the elevation of cannon, Szc. I'irst. To lay Joivn any angle suppose 30 degrees. Open the legs till the chamfered edges cut 30, and two lines drawn against the outside c<\gQ.s> will be at the inclination, or angle of 30 degrees, tending to a proper centre. If a centre ])oint be desired, cross the legs till the other chamfered edge cuts 30 ; the outside edges will then be at the angle of 1)0 de- grees, and the place of their intersection the angu- lar point. At% DESCRIPTION OP Secondly. To measure an entering or infernal an- gle. So apply the outward edges of the rulers, that they may exactly coincide with the legs or sides of the given angle. The chamfered edge on the semi-circle on the outward set of figures will point out the degrees contained in the given angle. Thirdly. To measure a salient, or external angle. Place the legs of the callipers over each other, and make their outside edges coincide with the leiis or sides of the angle to be measured ; then will the chamfered edge cut the degree required on the inner semi-circle, equal to the angle measured. Fourthly. To determine the elevation of a cannon or mortar. A rod or stick must be placed into the bore of a cannon, so as to project beyond its mouth. To the outward end of this stick, a string with a plummet must be suspended ; then with the legs of the callipers extended, so as to touch both the string and rod, the chamfered edge will cut the degrees equal to the complement of 90° of the true angle of the elevation of the cannon, figured on the outer se- mi-circle. So that if the edge had cut 30 degrees, the true elevation would have been 6o° the comple- ment to 90. 6. The circular line of Inches ^on the head next to the preceding, marked Shot Dia?n. This scale ex- tends from to 10 inches, and is subdivided into quarters. The chamfered edge marked Inches, is the index to the divisions. 7o measure hy this scale the convex diameter of shot, &c. Place the shot between the steel points of the callipers, so as to shew the greatest possible extent ; the chamfered edge, or index, will then cut the division shewing the diameter of the shot in inches and quarters. N. B. The diameter given by this scale gives the shotVather less than by the scale No. 4, on ac- count of a customary allowance of what is called IVindage. THE GLNNKKS CALLIPERS. 47^ ' 7. The table marked Brass GnnSy to shew the quantity of powder necessary for proof and service charges of brass guns. Tliis tabic is arranged for the size of the piece, called cither Heavy, Middle, or Lights inro their respective columns. Kx. Sup- pose it is required to knou' the quantity of powder for a proof and service charge of a heavy A'l pounder. Under A'l are 3 I lb. 8 oz. for the proof charge, and 21 lb. for the service charge. 8. The line of lines, marked Lin. This is a line of equal parts, and for various occasions may be so used. For other problems, being the same as usually laid down upon the sector, I must refer the reader to the description of that instrument already given in this work, page SQ. Q. Two circular scales on the joint head for giv- ing the weight of iron shot, marked shoi. These scales are engraved upon the chamfered part of the hciid of the callipers ; they are, in fact, but one con- tinued scale, only separated for the advantage of being more conspicuous. 1'hey shew the weights of iron shot from ^ ^o 42, pounds. The diameter of the shot is to be taken by the points of the callipers, and then by an index engraved on the ruler is pointed out the proper weight. 10. The table shewing the quantity of powder necessary for proof and service charges of iron guns, marked Iron Guns. An inspection of the figure, or the callipers themselves, shew the table formed into three columns. The first, marked Iron Guns, shewing the weight of the shot ; the second. Proof, shewing the quantity of powder for the proof charge; the third. Service, the quantity requisite for a ser- vice charge ; and all these adjusted from f to 42 pounders. 11. The mathematical figures. The most vacant part of the callipers are sometimes filled up by six mathematical figures, with numbers armexcd to each 480 DESCRIPTION OP to assist the learner's memory, and indicating as follows. 1. A circle with two diametrical lines, about which are the numbers 7.21, and 113.355; they both are the proportion of the diameter to the cir- cumference, the latter being nearer the truth than the former. The following examples may be there- fore readily worked by them. Required the circumference of a circle to any given diameter. As 7 : 22 : : given diameter : the circumference required. Or 113 : 355 : : diameter : the circumference more exactly. And the converse of these. As 22 : 7 : : circumference : the diameter. As 355 : 113 :: circumference : the diameter more exactly. 2. A circle with an inscribed and circumscribed square, and inscribed circle to this smaller square. To this fio-ure are annexed the numbers 28. 22. 14. 11. denoting that the larger square is 28, the in- scribed circle is 22. The area inscribed at the square in that circle 14, and the area of the smaller in- scribed circle 1 1. Both the proportion of the squares and circles are in the proportion of 2 to 1 ; and from them the area of any circle may be found, having its diameter given. For example. Required the area of a circle whose diameter is 12. Now the square of 12 is 144. Then as 28 : 22 : : 144 : 113.1, ihe area. Or as 14 : 11 :: 144 : 113.1. 3. Represents a cube inscribed in a sphere ; the afiixcd number 8()j shews that ri cube of iron in- ecnbed in a sphere of 12 inches in diameter, weighs 89 j pounds. THE GLXXERS CALLIPEKS. 481 4, Represents a sphere inscribed in a cube, with the numbers '143 aflixed ; it is to shew the weight in pounds of an iron globe 12, inches in diameter, or a globe inscribed in a cube, whose side is 12 inches. 5, Rej)resenis a cylinder and cone, tlic diameter and heigl)t of which are one foot. To tiic cylinder is affixed the number 3tJ4,5, the weight in pounds of an iron cylinder of the above dimensions. The cone b^hewn by the number 12 1,5 that it is the weight b ?of the same base and height, and is one- third of that of the cylinder. Cones and cylinders of equal weight and bases are to one another as 1 lo 3. 6, Represents an iron cube, whose side is 12 inches, to weigh 4t)4,5 pounds. The figure of the pyramid annexed to the cube, the base and height of which, if each 12 inches, denotes the height to be one-third of the cube's weight, viz. 154^. Remark. Globes to globes are as the cubes of their diameters ; cubes to cubes, as their cubes of the length of their sides ; of similar bodies, their weights are as their solidities. Hence the dimen- sions and weight of any body being given, the weight or dimensions of any other similar body may be found, Examples. 1 . The sides of a cube of iron hei7!g tivo feel, re- quired the iveight. The sixth figure shews that a cube of iron, whose sides are each 1 foot, weighs 404,5. Therefore, As 1 (the cube of 1 foot) : 4^4,5 : : 8 (the cube of 2. feet) : 37 id, the weight of an iron cube whose sides are 2 feet. 2. The diuineler of an iron shot being six inches, required the weight. . \'.y the fourth fij2;ure a one foot iro'i ball weighs •243 pounds, and (i inches»=,5 the X ''^ '» ^*^*^t' 1 i 482 DESCRIPTION OP Therefore, as 1 (the cube of 1 foot) : 243 (the weight) : : ,125 (cube of ,5 or j^-^) ; 30,375 pounds, the weight required. Another ride. Take f the cube of the diameter in inches, and f of that eighth, and their sum will be the weight required in pounds exactly. Or, the \feight of a four inch shot being nine pounds, the proportions 64 : Q may be used with equal ex- actness. 3. The IT eight of an iron hall heing given, to find its diameter. This rule is the converse of the preceding, the same nuiDbers being used ; as 243 (the weight) : J (cube of 1 foot;) or, as Q pounds : 64 (cube of 4.) Another rule. Multiply the weight by 7? and to the product add ■ of the weight, and the cube root of the sum will be the diameter in inches. Thus, an iron ball of 12 pounds weight will be found to be 4,403 inches. The methods of this rule are too evi- dent to require worked examples. 4. The following example will make up a suffi- cient nnmber, by which the learner may know how to apply readily any number from the figures just described. A parcel of shot and camion weighing five tons, or 1 1 200 pounds is to he melted, and cast into shot of three and five inches diameter, arid weight of each sort to he the same ; required the number there will he of each. 1st. Half of 11200 is 560 pounds. 2d. Cube of 3 inches, or cube j of a foot=^^-. Then, as 1- (cube of 1 foot) : 243 (the weight of a 1 foot ball) : : -^V '• 3,79, vveight of a 3 inch iron shot. N. B. In this proportion 64 divides 243 on ac- count of the fraction. Now 5 inch 63= .V of a foot, its cube tV^ts • THE GUKNER S CALLIPERS. 483 Therefore, as l : 2J3 :: -,'-,Vt • ^7,67, weight of an iron .3 ineh sliot. Dividing 5(iOO by 3, 7.*) gives 1504, the number required of the 3 inch shot. Dividing 56()0 by ^7■,^7 gives 313, tlic required number of the 5 inch shot. A tabic of speciHc gravities and weights of bo- dies is added, or not. at the pleasure of the purcha- ser. It \% not esscmiiil to the general uses of the cal- lipers, although many curious and useful problems relative to the weights and dimensions of bodies may be obtained from it in the most accurate manner.* 'i he quantity of lines placed upon the callipers may be increased or arranged at pleasure, The fol- lowing 19 were the greatest number that I ever knew of being placed upon them. 1. The measures of convex diameters in inches. ,2. The measures of concave diameters in inches. 3. 'I'he weights of iron shot from given diameters. 4. The weight of iron shot proper to given gun bores. 5. i he degrees of a semi-circle, (j. The proportion of Troy and Averdupoisc weight. 7- The proportion of English and French feet and pounds. 8. Factors useful in circular and spherical iigures. 9, Tables of the specific gravity and weights of bodies. 10. Tables of the quantity of powder ne- cessary for proof and service of brass and iron guns. 11. Rules for computing the number of shot or shells in a finished pile. 12. Rules concerning the fall of heavy bodies. 13. Rules for the raising of water. 14. The rules for shooting with cannon or mortars. 15. A line of inches. 1(3. Logarith- mic scales of numbers, sines, versed sines, and tan- gents. 17. A sectoral line of equal parts, or the line of lines. 18. A sectoral line of plans or super- ficies. 19. A sectoral line of solids. * See Mr. Adiims's Lectures, five vols. 8vo. a new and. improvcrl edition of which is now in the press, and under my correction M: and C being on the slider, vi'hich is slid in a groove made for the purpose, and the other two on the rule. The figures or numbers on the slide being placed between the two lines serves of course for both. The three lines ABC are called double Vines, be- cause the figures from 1 to 10 are contained twice in the length of the slide, and the other lowest line Dj a single hue proceeding from 4- to 40, and is called the Girt Line, from its utility in computing the contents of trees and timber. The use of the double lines A and B is for work- ing proportions, and finding the areas of plane figures, and the use of the girt line U, and the other double line C, is for measuring solids. On the girt line are marks at 17.15, lettered W G, (ivine gallons) and at 18-95 A G (ale gallons) which are wine and ale gauge points to this instrument, to answer the principal purpose of a gauging rule. On the same side of the rule is placed a table of the amount of timber per load of fifty cubic feet at the price of 6d per foot, to 24 pence. At 6d per foot it will appear to be ll. 5s. at f)d. ll. 17s. 6d. &c. Other scales and tables are sometimes substituted for these, as a table of board measures, one of timber measure, a line shewing for what length of any breadth will make a foot sijuare, also aline sliewing what length tor any thickness will make a solid foot ; the former line serving to complete the table of board measure, and the latter the tabic of timber jneasure. THE SLIDIKG RULE. ^97 The method of notation on the rule is very sim- ple, the fignrcs in order, urc niunbcrcd from the left hand towards the right. Who ». f,/l'^^';^,S:"'^;';f is accounted I, then 1 in the muMle w,ll be 10, and the one at the end U)i) ; and when 1 at I we be- o-innini^ is accounted 10. then 1 in the mida e is 100, and tlie 10 at the end 1000. and so on. And all the smaller divisions and sp.iccs are ot course al- tered in your mind j)rop -rtionaliy. When the learner has made hmiself tainiliar with the notation, he will be surprized to hud with what dispatch and accuracy he will be able to work the tollowing problems, and much less liable to er- rors, than by computing with the pen. By a foot slide a solution to about the '200dth part of the whole may be relied on. , . ,i Problem 1 To nmhifly ''-"' mmhers logelhsr, as 16 and . q. Set 1 on B to the mtiltiplicr 1 on A, the;, ngainsi the multiplieand igon !i «'!' ^e t^^ ■•^- quircd product 304 on A. 2. To find the fro- duct of 33 and ig. Set I on B to the tm.lfphcand zTo^^. then because IQ on B runs beyond the end of the line, seek it on the other raduts or other preceding part of the Imc, wh.ch in value willbe^ Q or Ath l»rt, so jhat agamst I.9 on B Ti 1 be found 60,5 on A. Thts, therefore, tnult- nlied aeain by K', which is done by talung away ^ W t -"eeitnal point, gives the product 005 In like? manner the' ,>roduet ol 270 by 54 w,ll be found to be 14,580. , FaOBLEM 2. To divule one number hy ""olhe, ^, JRO hx 1" Set the divis:.r 12 on A, to on B, then ag^n t'ihc dividend 480 on A, will be the quo- tient 40 on B. 2. To divide 7^80 by 24. Let the div" or 24 on A be set to , on B, thenbce.use 7O8O fs no contained on A, the tenth 768O o,, A tm.st be ouMu for, and agah.st it on B w.ll be fou.ul he quotient 32, «hieh increased te,> tunes (or the 488 EXAMPLES ON above reasons will give 320 the two required queS' tions. So 396 by 27, gives 1A.6, and 741 by 42, gives 17.6. Problem 3. To square any numVer, as lb. Set 1 on B to 25 on A, then against 25 on B, stands ^25 on A, the square of 25 required. Or by the other two lines. Set 1 on C, to 10 on Dj observing that if you call the 10 on D 1, the 1 on C will be 1, but if you call the 10 on D 10^, then the 1 on C will be 100, and if you call the 10 on D 100, the 1 on C will be 1000, and so on. So against 25 stands 625 ; against 30, 9OO ; 35, 1225 ; and 40, 1600, reckoning the 10 on Das 10. Problem 4. To extract the square root of any nmiiber. Set 1 on C to 10 on D, which is the same position of the slider as in the preceding problem ; estimate the value of the two lines in the same way ; then against every number found on C stands a square root on D, thus the square root of 529 will be found to be 23, 9OO to be 30, and 300 to be 17.3,4264 to be 65.3. Pr o B L E M 5 . To find a mean froporiional letween any two nimibers, as 9 and 25. Set one number 9 on C to the same number on D, then against the other number 25 on C will be 1 5 on D the mean proportional required. For 9 : 15 : : 15 : 25. Required the mean proportional between 29 and 430. Set 29 on C to 29 on 1), and it will be found that 430 on C will be situated beyond the scale D, or it will be contained on C. Therefore take the lOOdth part of it, and seek for A.o on C, and against it on D is 11.2, which multiplied by lO, gives 112 the mean proportional required. It is therefore evident that if one of the numbers over-r runs the line, it is proper to ta]i.e the lOOdth part, and augment the answer ten times. The mean pro- portional between 29 and 320, is 9O.3, and between 71 and 274 is 17.3. THE SLIDING RULE. 489 Problem 6. To find a third proportional to any two numbers, us 1\ and \\1. Set the first 21 on B, to the second 32 on A ; then against the second 32 on B stands 48.8 on A, the third j)roportional required Then the third proportional to 1/ and 29 is 49.4, and to 73 and 14 is 2.7. Problem 7- Three mimhers being given, to find a fourth proportional , suppose 12, 28 and 57. The solving of this problem is performing the rule of three. Set the first term 12 on B, to the second 28 on A, then against the third term oj on B will be found the fourth 133 on A. If either of the middle numbers fall beyond the line, take -,', th part of that number, and augment the answer ten times. The fourth proportional to Oaa, and 29, will be. found Q7.Q, and to 27, 20, 73, the number 54.0. After the working of the preceding examples, it is presumed that the reader will find no difficulty in performing on the rule all the numerical solu- tions to the various geometrical problems treated of in page 73 and sequel, of this work. OP TIMBER MEASURE. Problem 1. To find the area, or superficial con- tent 171 feet in a hoard or plank. Multiply the length by the breadth, and the product will be the content required. Note. If the board is tapering, add the breadth of the two ends together, and take \ their sum for the mean breadth, and multiply the length by this mean breadth. By the slid' ng rule. Set 12 on B to the breadth or inches on A, then against the length in feet on B, the content on A in feet and fractional parts will be shewn. Example. A board 11 inches broad, and 13 feet Ago TIMBER MEASURING. long will, by the above rule, be found to contain 174 feet. Exa?7ipJe 1. What is the value of a plank whose length is 12 feet 6 inches, and mean breadth ll, at lid per square foot. By the slide rule. As 12 B : 11 A : : 121 B : 11-1 A the number of feet. Then a.^ 1 foot on B to 1.5 pence on A so is 1 1' feet on B to 17 pence on A, the value required. In a very irregular mahogany plank of 14 feet in length, it was found necessary to measure several breadths at equal distances from each other, viz. at every two feet. 1 he breadth of the less end was six inches, and that of the greater end one foot. The interme- diate breadths were one foot, one foot six inches, twe feet, two feet, one foot and two feet. It is required how many square feet was contained in the plank ? By dividing the sum of the breadths by their num- bers, the mean breadth will be found l^ feet, and by working by the above rules, the answer to be ipi feet. To find the solid content of squared or four-s'uUd thnher. Multiply the mean breadth by the mean thick- ness, and the product again by the length, and it will give the content required. By jhe sliding rule. As the length in feet on C, is to 12 on D, when the greater girt is in inches, or to 10 on D, when it is in tenths of feet : so is the quarter girt on D to the content on C. Remarks-, if the timber be squared, half the sum of the breadth and depth in the middle is consi- dered as the quarter girt above-mentioned. But if tijc timber be round, it is customary to girt the piece round the middle with a string one-fourth part of this girt squared, and multiplied b}' the leuL'th gives the solitiity. Pocket measuring tapes, with the quarter girt divided thereon to shew the (juarter girt by inspection, are most convenient in taking the girt. " TIMBER MEASURING. 49I The quarter girt rule for squared tiiiiber is only sufiicicntly near to tlie truth when the breadtli and height of the sides are ihe same, or the diifcrencc not more than one or two inches, in all greater dif- ferences the measure by the girt method will be too considerable, and the rules above given will be j)roper to adhere to. To shew the fallacy of taking- the quarter girt for the side of a mean square, the fal- lowing example has been given. Suppose a piece of timber to be 24 feet long, and a foot square throughout, and let it be slit into two equal parts from end to end. Then the sum of the solidity of the two parts added together by the quarter girt method, which is to multiply the square of the quarter girt in inches h\ the length in feet^ and divide by 144, will be 27 feet, but the true solidity is 24 feet, and if the two pieces were unequal the dit- ference would be still greater. Note. If the circumference or girt be taken in inches, and the length in feet, divide the product by 144 to give the contents in feet. If the timber tapers regularly from one end to the other, the breadth and thickness taken in the middle, will be the mean breadth and thickness or half the sum of the two end dimensions, will be the mean. If it does not taper regularly, but is thicker in some places, than in others, let several dimensions be taken, and their sum divided by the number of them, will give the mean dimensions, or girt it in several places equally distant from each other, and divide the sum of the circumferences by their number for a mean circumference, the square of one quarter of the mean circumference multiplied by the length will give the solidity. Example ]. If a piece of s(]uared timber be 2 feet 9 inches deep, 1 foot 7 inches broad, and l(i feet 9 inches long, required the contents in sohd feet. 492 TIMBER MEASURING. Multiplying the depth, breadth, and thickness together by the above rule, the true content is 72.93 feet, or 73 feet nearly. But by the quarter girt, the content will be found as fol- lows : — 26 inches quarter girt. 9.6 Decimals. 16.75 = 16 foot 9 in. 6/6 156 52 10050 11725 10050 676 square 144) 1132300 (78.63 feet, 1243 » 910 460 28 rem. T;y THE SLIDING RULE. As 161 on C : 12 on D : : 26 on D : to 781- on C This shews that from the inequality of the depth and breadth, the common girt method gives more than the true content. In the following ex- amples, taken from Dr. Hut ion s Treatise on Metisu- ratioji, Svo. page 499, the difference is immaterial, on account of the depth and breadth not being ma- terially different. Example 2. Required the content of a piece of T1M*BKR MEASURING. 4i)3 timber, whose length is 0.4]: ^eet, and the mean breadth aud thickness each 1.04 feet? Answer, 2Ci; feet. Example 3. Wliat is the eontcnt of a piece of timber whose length is 1\\ feet, and the mean breadth and thickness each 1.04 feet. Answer, 2(5'^ feet. Example 4. Required the con ten f of a piece of timber whose length is 37-36 feet : at the greater end the breadlh is 1-78, and thickness 1.78, and thicknes 1.25, and at the less end ihc breadth 1.04, and thickness o.Ql. Answer, 41.278 feet. The spare examples arc to be worked by both the methods. Problem 3. To find the solidity of round or un- squared thnher. RURE I, or COMMON RULE. Multiply the square of the quarter girt, or of ^- of the mean circumference, by the length for the content. BY THE SLIDING RULE. As the length upon C : 12 or lO upon D : : quar- ter girt, in I'ithsor loJis, on D : content on C. Remarks. This is the method adopted and ge- nerally used by all land surveyors and timber dealers, the buyer is allowed to take the girt of the tree any where between the greater end and middle of the tree, if it tapers. All branches or boughs, the quarter girt of whiclt is less than () inches, and also any part of the trunk, not Jess than two feet in cireuipference, is. not con- sidered as timber. An allowance is generally made to the buyer, on account of the bark, of one inch and hjlf in every foot of (juarter girt, or ^th of the girt; but, except- ing of elm, it is considered to be tgo much. In 494 TIMBER MEASURIXG. general one inch on the circumference of the tree or whole girt, or -^'-^th of the quarter girt should be considered as a fair and average allowance. When the buyer is not allowed his choice of girt in taper trees, he may take the mean dimensions, as in the former problems, either by girting it in the middle for the mean girt, or at the two ends, and take half their sum. Or if the tree be very ir- regular, the best way is to divide it into several lengths, and find the content of each part sepa- rately. This commonly used rule gives the answer about i^th less than the true quantity in the tree, or nearly the quantity it would be, after the tree was trimmed square, which circumstance, instead of being an objection, as stated by some mathemati- cian^, I think it to be a fair consideration for the loss of wood as trimmed in squaring down the tree to make useful shaped timber. Rule 2. Giving the result ve?y near to the truth. Multiply the square of 4 of the mean girt, by twice the length, and the product will be the true content very nearly. By the sliding rule. As twice the length on C : 12 or 10 on D :: 4^ of the girt on liith or I iths on D : the content on C. Example 1. A tree is in length 9 feet 6 inches, and its mean girt 14 feet, required its content ? Decimals. 2.8 2 8 224 784 19 70^6 784 Content 1489B Duo( Je-cimals, 9 9 7 2 9 7 5 7 2 1 3 1 8 1 10 1 »9 148 11 8 [ 495 ] BY THE SLIDING RULE. c D D A.S IQ : 10 : : 28 : M9 Or \]) : 12 : : 33 : J4CJ Exawple 2. If a tree is 24 feet lonjr, and the mean girt 8 feet, what is the content. Ans. 123 feet nearly. Example 3. What is the value of an oak tree, the girt of which at the lesser end is Qi. feet, and its length to the part where it is of 2 feet girt, 1'J~ feet ; it hath two boughs, the girt at the thiekcr end of the one is 4*9, and at the thicker end of the other 3*94, the length of the timber of the former where it becomes of only 2 feel girt, is 9 feet ; and the length of the latter 7 ^^ : required the value at 2I. 3s. 9d. per load of 50 cubic feet, allowing T^^rtli of the me;ui girt for tb.c bark of the bough, and — th of the same in the boughs, by both the rules. Answer. True value 2l. 18s. 6d. unjust value 2I. 5s. 9;d. Some of the two preceding problems, as well as the three following, are extracted from Dr. Hut- ton's exctlhnt Treatise 07i Mensuration, 8vo. 1802. 'J he only true rule for finding the contents o^ taper round timber, must be found in the rule for finding the contents of a frustum of a cone ; but it is to ) tedious for dispatch in business, and consi dered too nice for the ordinary materials of timber. The sceond rule above as given by Dr. Hiitton, ap- proximates nearer to the truth than the common one, and it is curious for the reader to know, that by both these rules the contents of timber eut into two parts, may be made to measure to more than the Avhole. and as arc shewn bv the three following problems given in Dr. Hulloii^ Mensuration. 496 TIMBER Measuring. Pkoblem4. To find where a piece of round ta- fer'ing t'lmher must he cut, so that the two farts, ynea- sured separately, according to the cormnon method of measuring^ shall produce a greater solidity than when cut m any other party and greater than the whole. Cut it through exactly in the middle, or at half of the length, and the two parts will measure ta the most possible by the common method. Example. Supposing a tree to girt 14 feet at the greater end, 2 feet at the less, and consequently 8 feet in the middle ; and that the length is 32 feet. Then, by the common method, the whole tree measures to only 128 feet ; but when cut through at the middle, the greater part measures to 121, and the less part to 25 feet ; whose sum is 146 feet ; which exceeds the whole by 18 feet, and is the most that it can be made to measure to, by cutting it into two parts. Problem 5.. To find where a tree should he cut, so that the part next the greater end may measure to the most fossihle. From the greater girt take three times the less ; then, as the difference of the girts is to the remainder, so is one-third of the whole length, to the length from the less end to be cut off. Or, cut it where the girt is one-third of the greatest girt. Note. If the greatest girt does not exceed three times the least, the tree cannot be cut as is required by this problem. For, when the least girt is exactly equal to one-third of the greater, the tree already measures to the greatest possible ; that is, none can l;e cut off, nor indeed added to it, continuing the same taper, that the remainder or sum may measure to as much as the who»c. And when the least girt exceeds one-third of the greater, the result by the rule shews how much in length must be added^ that the result may measure to the most possible. TIMBER MEASURING. 4(j7 Example. Tiikinpr Jicrc t!jc same example as be- fore, we shall have as \l: 8 : : V '- 7 =^ ll>c leneih to be cut off; and conscqneiiti> the lenfrfh of the remaining piirt is 21^ ; also '-/ t= 4 Ms the girt at the section. ) fence the content of the re- maining part is 135 ^' feet; wlicrcas the whole tree, by the same method, measures only to 128 feet. Pro,' lem 6. To cut a tree, so that the pvt ttcxt the greater end may meanire^ hy the commo)i nieti,oJ, to exactly the same quantify as the zvhoie measures to. Call (he sum ot the girts of the two ends f, and their difference d. 'i hen multipl)' ^ by the sum of J and /Is, and from the root of the product take the difference between d and !2 s. ; then, as 2 d is to the remainder, so is the wliole length, to the length to be cut oi; the sinall end. And if s be taken from the said root, half the re- mainder will be the girt at the section. Example. Using the same example, we have j- z= l6, ^= 12, and the length X=: 32; hence fl X C/^4i -f d) d) - Is -\- d) = ^; X a 2 (i/wO X 12) — 20) = W^7— 26^=i3-5pyii7 =t= the length to be cut off; and consequently 18-1CX)883 is the length of the remaining part. Also ,: \/{:4sx^)-^)—'s=^ 2/57 — 8 = 7.09C)6o9 is, the girt at the section. Hence the girt in the middle ot the greater part is — '—- ^ — -=:= 10.54C)S31, whose Uh part is 2"037458 ;. and con- sequently the content of the same part is 2'637458'^ X 18.40C»883 = 128, the very same as the whole tree measures to, notwithstanding above ' part is cut off the length. For an useful table to facilitate the compulation (;f the contents of timber, the reader is referred to the tables at the end of this work. K k t 498 J A COLLECTION OF TABLES OF BRITISH AND FOREIGN MEASURES, AND WEIGHTS, TAKEN FROM THE BEST AUTHORITIES. BrUish Measures, and JVeights The following Tables are inserted for the occasional use of tht young Surveyor. 1 . A Table of Eriglish Luteal Measures. 7.92 incli 1 link 12 m 1 foot 36 4A- 3 1 yard 198 25 161 5h 1 pole 792 100 Q6 22 4 1 chain 7920 1000 660 220 40 10 1 furl. 63360 8000 5280 1760 320 80 8 1 mile Kpole, perch, and rod are the same measure. 691 miles, or 060640 feet, is computed to be equal to one degree of a great circle of the earth, or so much must any person change his place on a me- ridian to alter his latitude one degree, or on the equator to alter his longitude one degree. K fathom is equal to 6 feet, a customary measure for depths. A hand is 4 inches, by which the heights of horses are measured. The following measures are usually applied to cloth : — Inches 9 36 AS 5>s> 1 Nail 4 j6 12 20 24 1 Quarter 4 3 5 6 ] Yard 1 Ell I'lemish 1 El! 1-nglish J Ell French [ 499 J 2. A TiiUicfSquar/ Mijsures. 9 leet 1 yarJ V72. 30! 1 perch 4356 494 1 6 1 chain 1 1210 40 n I rood 435<5o 4S40 1 iGO 1 10 4 1 nrrc 1 1 278784CX) 30C)7tiOO 102-100 6400 2560 640 1 mile N B. Perch the same as pole, aiid a square cluin is the tenth part of an acre. 144 Square inches are equal to 1 square foot, 62.7264 square inches, equal to 1 square link, 10.000 square links is 1 «?quare chain. B. Noble, in his G(joda:sia fiibeni'tca, 8vo. p. 9 and 10, states, that an Irish square perch is 49 yards, and that an Irish acre is 7850 yards, or 160 square perches. Ilcnce an Irish acre, is to an English acre, as 7850 to 4840, or as 49 to 30. A Table of Lineal Scotch Measure. Inches 37.^2 71424- Foot 1 Ell 3.1 1 18.6' 6 744 240 ^^•'^ {li)84$ Yards Fall. Furl. 40 1 Mile 8 1 611 ideg. liirhes Limb? of Feot Ells Kali's ?hort Chain Long Mile a Chain Roods 8.928 1 74.4 24 4 892.8 100 in 36 6 1 80 Rood 0.666 1. 53 f [ 500 ] A Table of Square Scotch Measures. Sqr. Inches Sq. Feet Sq. Ells Sq.Falk, or Sq. Rood Sq. Acr* Sq Roods 144 1 9.61 1 345.96 36 1 13838.4 1440 40 55353.6 5760 160 4 1 640 1 Mile Correspondence between English ond Scotch Measures. 24.8 English yards, are equal to one Scotch chain, 80 chains, or one Enghsh mile, are equal to 71 Scotch chains, the proportion of the English to the Scotch chain, is therefore as 80 to 7 1.* 6130 Square English yards are equal to a Scotch acre. One English acre is equal to 0.787 for a little more than ^^ of a Scotch acre. One Scotch acre is equal to 1.27 (or little more than It) of an English acre. A Table of English Solid or Cubical Measures. Inches Foot 1728 1 AQ656 27 1 yard 1 Load of timber is 50 cubical feet 1 Load of earth is 27 feet, or 1 cubical yard. Of French Measures. The old measures, or those previous to the late French Revolution, were as contained in the an- nexed tables. The new measures, or those esta- blished since the. Revolution, follow directly after. * Gray, in hi?, Art of Land Measuring, 17^7 ^ P- '/> states a Scotch chain to be 24 ells, and each link to be 8,b8 Scotch inches. [ -^01 1 -■/ Table of old French Lineal Measure. Foinl Line - 'J 1 Pouce 144- I'J 1 Picd-dcRoi ,-,iQ lit , , 1 Toisc du Clintclct 1 / -0 l-i-t 1'- 1 oil'* 1 & lie 1 Acaucimc. U):>(;8 844 7ii () 1 :3S0ib" i31<58 CO-i- iiii 3 ]^ Pcrchc Royale. An English foot is equal to 0.9383 of a French foot. A French foot is equal to 1.06.575 English feet. A French toisc is c(juul to 6.3945 English feet. Their square measures for land surveying, &c. arc as follows : The Paris or pent ^ consisted of 100 square ^^r^/z^j, of 18 Paris lineal feet each, or 324 square Paris feet, therefore making in the whole 32400 square Paris feet (or 367-0 square English feet) for the ar- fent^ which in round numbers is about ^ths of the English acre. The Mesnre Royale du France pour Varpentage des terres established the perche to be 21 feet of lineal measure, or consequently 484 square feet, which augmented 100 times gives 4S dc 403 square Paris feet (or .548.53.36 English feet for thearpent, which is somewhat above 1^ English acre. The former arpent of 32400 feet was generally about Paris. The cubic Paris foot or inch, is to the English tubic foot, nearly as 1.21 to 1 ; so that one cubic Paris foot or inch is about 1 ' English cubic foot or inch ; hence 5 Paris cubic feet make 6 English cubic leet. OF THE NEW FRENCH MKASURES. The old French measures concisely described in K k3 502 STANDARD FRENCH MEASURES. the prece:Hng pages, were not the only kind of li- n:?\ ineasurcs u=c(l, each province in Prance had partly its own peculiar measures. 'Jhe diversity of the agrarian measure v,'as infinite. In some places the arpent was considered as 100 perches, in others as 120, and in others as 128. The perch itself had also various local lengths, varying from g to 2) feet. Their itinerary measures vi'erc all locally various. The effects of the late revolution therefore necessa- rily cxtcndid to this corrupt object, and all the measures and weights of that nation are now de- duced from one simple lineal standard which they denominate a metre. And the adoption of this elementary measure by thenation, with the other ne- cessary ones constructed upon it, was oro'ercd by de- cree of the National Convention on the 5 th of April, 1796, (dii 18 Germ'mal, dul'an 3 de Vlre Repuhlicaine Francaise ) The method that the French mathematicians or academicians took to determine this invariab'C stand- ard of measure, was by taking a segment of the earth's circumference, and for this purpose several degrees of the meridian were carefully and accu- rately measured by a just rod, and from, that the whole meridian or C( mplete circumjcrence of the earth is calculated. They have taken the forty mil- lioneth part of the circumference of the earth, or which is the same thing, the ten millioneth part of the quarter of the terrestrial meridian, or the dis- tance of the equator to the pole, as the primitive measure or metre, and laid it down accordingly on scales. 'J he metre by a comparison with accurate scales of English inches, feet, &c. at the tempera- ture of 62°i ahrenhcit's thermometer, has been found to be equal to 39.371 inches. This is one of the two best and most simple me- thods of determining an original standard measure, without referring to a pattern, it also serves to ve- rify any reputed standard one, and afibrds the nieaug NEW FRENCH MEASURE. 503 of finding the lcng:h of the standard measure again at any time, supjjosing the original pattern to be lost or destroyed. The other mct'nod is rendered still more simple, and rccjuircs less trouble and expcnce in its opera- tion, it is bv means of a pendulum, and may be uiulerstood tVom the following brief explanation. The length of a pendulum that vibrates seconds has, by several philosophers, been adopted as a stan- dard of measure. It has been found that in the latitude of London, that a pendulum performing 6o vibrations in a minute, or 3600 in an hour, will in len^ith be very little more than 39,1 inehcs, and that aecording as the place is nearer to or firther from the equator, or the latitude decreasing or in- creasing, the length must be shorter or longer to perform the same number of vibrations. The quan- tity by which the pendulum must be shortened or lengthened, in order that it may vibrate seconds in any given latitude, may be readily calculated, or be seen by an inspection of a table of lengths among the tables annexed to this work. Hence a pendu- linn that vibrates the above or any other number of times in a n^inute, may be adopted as a standard of measure to any knowMi latitude. The difficultv of measuring the precise distance between the real point of suspension, and the centre of oscillation of the pendulum is the principal source of inaccuracy, to wliich this method is subject. '1 he centre of oscillation of a pendulum is that point of a pendulum, where all the matter or the gravitating power of all its parts of the body or bo- dies that constitute the pendulum, are concentrated or conceived to be concentrated. 'Jo obviate this difiiculty, the late ingenious, Mr. Jllulehiirst, con- trived a clock-work mechanism, with a moveable point of suspension to the pendulum, so as to be lengthened or shortened at pleasure, and so of course 4 K 504 A STANDARD MEASURE. adjustable to vibrate any number of times in a given interval of time. The difference of the lengths of the same pendulum Vvhen it performed a certain number of vibrations in one hoar, and when it per- formed another certain number of vibrations like- wise in one hoar, he pr >po-^cd to use as a stan- dard of measure, and by which means the above- mentioned sources of error would in a great measure be obviated. SccAIr. IF /'ife/iursfs " Attempt to obtain Measures of Length, &c. from the mensu- ration of time and true length of the pendulum." ISir George Shackburgh Evelyn, after the death of Mr. Whitehurst, upon the same machine, re- peated all the experiments that the machine was capable of, and deduced the following conclusion, " that the difference of the length of two pendu- lums, such as Mr. Whitehurst, as vibrating 42 and S4 times in a minute of mean time in the lati- tude of London, at 113 feet above the level of the sea, in the temperature of 6o°, and the barome- ter at 30 inches, is equal to 5C). §9358 inches of the parliamentary standard, from whence all the measure of superficies and capacities are deducible.'' Phi- losophical Transactions, for '\79^-> p^g^ 174. Having shewn the means by which a standard lineai measure may be obtained, we shall now in a concise way place before the reader the new 1 rench measures. The new French measures, as well as their weights, are arranged in decimal order. We shall give here only the measures, the weights are placed hereafter, with the weiohts of other nations. The numbers that are annexed to the followmg names express the number of English inches. The milimetre is the 10th part of the contimetre, the contimetre is the 10th part of the decimetre, the decimetre is the 10th part of the metre, and so on. OP FRENCH MEASURES, 505 Measures of Length. Milimctre O.039371 Eng. inches. Centimetre ().3()371 Decimetre 3.937 1 Metre 39.371 Decametre 393.7 1 Hecatometre 3937. 1 Chiliomctre 3937 1. Myriomctrc 3937 10. The operation by these measures is thus shewn to be very simple, merely to multiply on by 10, or re- moving the decimating point to the left for multi- plication, and the right for division. The following measures arc the itinerary ones, English miles, fur. yds. feet. inch. A decametre A hecatometre A chiliometre A myriometre 10 1 9.7 109 1 1 4 213 1 10.2 6 1 130 6 English. Eight chiliometrcs are nearly equal to five miles Measures of Cofacity. Cubic inches Millilitre 0.06l03 English. Contilitrc <).ni028 Decilitre 6.1028 litre 61.028 Deciilitre 610.28 Hccatolitre 6] 02,8 Chiliolitre 6 1028. Myriolitrc 6l()280. 506 OF ENGLISH WEIGHTS. A litre is nearly equal to 27 English wine pints, and 14- decilitres are nearly equal to three English wine pints. A chilioliire is equal to one tun^ and. 12.75 English wine gallons, 27ie Jgrarian Measures. The Are, a square decametre^ or a square perch, is now the principal unity oi Agrarian measures., or of French land surveying. It is equal to 100 cen- tiares^ or 100 square metres, and is equivalent to 3.95 English perches. The hecatare, or new aspent, is 100 ares, and is, equal to 2 acres, 1 rood, and 35.4 perches English. Of English Weights. A standard weight is determined from a standard lineal measure ; for a body of an uniform substance may be taken, and made very exactly of any given dimensions, and that will serve as the standard weight. In this manner it has been ascertained, that a cubic inch of pure distilled water, when the barometer is at 29.74 inches, and the thermometer at 06°, weighs 252.422 parliamentary grains, 5760 of which are equal to one pound Troy. About the year 1757? Mr. John Bird, an accurate mathematical instrument divider, made several very accurately divided brass standard linear scales, about three feet in length, one is preserved in the Exchequer, another by the Assay Master in the Mint, and an- other in the House of Commons. By fome accu- rate and useful examinations of scales by Sir Geo. Shuckburgh Evelyn, the latter scale is the same, or agrees within an insensible quantity with the antient standards of our realm. Also, according to tlie standard, weights made by Mr. Harris, under the order of the tiousc of Commons, in the year OP ENGLISH WEIGHTS. 50 7 1/58, the weight above of the cubic inch of pure dis- tilled water, was a mean result from :i g-reat iiiiui- bcr of trials of the- old weights in the Exchequer. In Great Britain three sorts of weights have been priniMpallv a(Io[)tcd. Trovx'ei'^ht, ylvoirJupois iveii^lit y and Apothecaries iva^ht^\i\\\ the Troy pound of 57t)0 grains abovemcntioncd has been adopted as the best integer for the siandaid weight. Troy Weights. Which is used for weighing gold, silver, dia- monds, costly li(jUors, and a few other articles. 2-1 grains tnake one pcnnyivelght. 20 pennyweights make one ounce. 12 ounces make one found. Apothecaries and chemists compound and vend their medicines by these weights ; but in general purchase their articles by the averdupois weights. Avoirdupois Weight. The Avoirdupois weights are used in commerce, for weighing all kinds of grocery, fruit, tobacco, butter, cheese, iron, brass, lead, tin, soap, tallow, pitch, resin, salt, wax, &c. l6 drams make one ounce. 1(3 ounces make one pound. 14 pounds make one stone. 28 pounds make a quarter of an hundred "Jiright. 4 quarters of a hundred (or 1 12 pounds) make one hundred weight.. 20 hundred weight i^or 2240 pounds) make one ton. A correspondence helTcee troy arui averdupois weight. 41 Ounces troy, nre equal to 4.3 ounces averdupois. 1 Ounce troy is ccjual to 1.09707 ounce averdu- pois. 508 ENGLISH MEASURES. 1 Ounce averdupois is equal to 0. 91152 ounce troy. 1 Pound troy is equal to 0.8227A of a pound aver- dupois. 1 Pound averdupois is equal to 1.21545 pound troy, or to 1 pound, 11 pennyweights, and 20 gains troy. 50 Pounds averdupois of old tro}', or 6o pound of new hay, make a truss, and 1800 weight of old hay make a London load. 46 Pounds averdupois of straw make a truss, and 36 trusses a load. 14 Pound of most gross commodities make a slonc. There are a few weights of other denominations ■used in other parts of England, in articles of traf- fic, for the particulars of which the reader is re- ferred to treatises relative to these articles. In Scotland, hay, wool, Scotch lint, hemp, butter, cheese, tallow, &c. are sold by the Trene, or old Scotch Measure. 20 Ounces make 1 pound. l6 Pounds make 1 stone. German JVnights, as used at Amsterdam. 29tt grains \ r 1 deep 16 deeps {^^.J) 1 ounce make l6 pounds) (. 1 stone l6 ounces ^ j 1 pound One English pound troy is equal to 0.757 of an Amsterdam pound weight. By the following table is shewn, that the capa- city of one barrel is equal to 34 gallons, or 136 quarts, or to 2/2 pints, or to 9588 cubic inches, according to the country measure; also, it shews WINE MEASURE. 509 that 35,. cubic inches arc equal to a pint, 2 pints to a quart, &c. In a similar way by the nine mea- sures, and to the Scotch hquid measure. Beer measure for London is 3(3 gallons to the bar- rel. Ale measure 32 gallons to the barrel. Beer and Ale measure for the country is 34 gal- lons to the barrel. Beer and Ale Measures. iCubicInchcs: Pints Qts (.Tall;i'>arrs Hhds, I'cer Ale Country measure. Beer Ale Country measxire. Beer Ale Country measure Beer Ale Country measure Beer Ale Country nieisure. 1522S 1353t) 14382_ Toi52~ 90-24 9J98 282 432,210; 54 384J192I 48 408i204| 51 288Jl44|"30 25(J 12s 32 272 136 34 li 4 I J-Vine Measures. ICubic Inch( ;s|PintslQuarts;(;allons|Barrels'|Hhcls| 14553 1 SO-i 1 252 1 03 ! 2 111 7276.5 1 252 1 \1Q \ 31^ 1 1 I 231 1 HI 4 1 I 1 1 ^7,75 2| 1 1 231 Cubic inches make one gallon, wine measure, by an Act of Parliament, 5th Queen Anne ; but the standard gallcn at Guildhall, contains only 224 inches. [ 510 ] Scotch Liquid Measure. Cut lie lnche?lGills|Mutchkins Chopins PintsiQuartsjGalls. Hhds.J 13235,7 12048; 512 250" 1261 64 i 16 1 1 1 827,23 i 128 32 i6 81 4 1 11 206,8 1 321 8 1 4 21 1 1 103,104 1 16| 4 I 2 I 1| 51,7 1 81 2 1 1 25,85 1 4| 1 6,-^02 1 l| The Stirling jug, containing one Scotch pint, is the original standard ot all liquid and dry measures, and of all weights in Scotland. It contains 103.464 cubic inches. When accurately filled with water at Leith, the water weighs 3 pounds and 7 ounces of Scots troy (equal to 55 ounces, or to 26 180 Eng- lish troy grains), so that one ounce weighs 476 Eng- lish troy grains. By the Act of Union, the barrel for English country measure of 34 gallons, whose capacity is 958S cubic inches, is reckoned equal to 12 Scots gallons, making 9926.7 cubic inches. Subsequent to the French Revolution, all the measures and weights of that nation are deduced from the metre, (see fage 501.) English Dry Measure of Capacity. Pints Quarts Pottles Galls. Pecks Bush. Qrs. Hay Last load. 2 1 4 2 1 8 4 2 1 16 8 4 2 1 64 32 16 8 4 1 512 256 128 64 32 8 1 2560 1280 640 320 160 40 5 1 5120 2560 1280 640 320 80 10 2 1 SCOTCH MEASURE. 511 The capacity of the staiulard Winchester bushel is equal to 2150.42 cubic inches. A striked bushel is to a heaped bushel as ?, to 4, that is, a heaped bushel is one third more than a striked bushel. A peck, is equal to 537.6 cubic inches. The avoirduj)ois weight of a bushel of wheat at a mean is 6o pounds, of barley 50 pounds, and of oats 38 pounds. The weight of a pint, or ,%th part of a peck, at a mean, ot wheat is 15 avoirdui)ois ounces, of barley ] 2 ! ounces of oats is g\- ounces. A cubic foot of Newcastle coal weighs about 50 pounds, and 6o cubic feet make 1 chaldron. A heaped bushel of coals weighs about 83 pounds avcrdupois, and 36 bushels, or the chaldron weighs about '16.67 hundred weight, or 2()6s pounds avoir- dupois. In Scotland a leppce, or feed for a horse, is equal to 200. 345 cubic inches. An English cubic foot of water weighs 62^ pound avoirdupois, or 1000 ounces. A Table to reduce byf)ot}ienusal Lines to Horizojiial, the Hv^otbcnuse being 100 Links. Hyp. ICO.l Links. ||Hyp- 100. Links. 1 deduct Hyp. 100. Liukb. Z.of elev. Deduct i ^ of elev. ^ of elev. deduct or depres. fr.c.ich or depres. fr. each or depre?. fr. each lOolk. 1 OOlks.l 1 iOOlks. 4° 03' t 4 19° 57'\ 6 ■ 35'' 54' •9 5 44 21 34 7 3G 52 20 7 01 23 04 8 37 49 2| 8 7 1 24 30 9 38 35 22 9 57 ^h 25 50 10 39 39 23 11 29 2 27 09 ii 40 32 2i 12 .'■>() 2i 28 22 12 1 41 25 25 14 0-1 3' 29 32 13 1 42 1(5 20' 15 12 H 30 41 J 4 '1.3 07 27 Hi 1(J 4 31 47 15 13 .V 28 17 15 ^h 32 52 16 1 .tl 'Id 29 18 12 5 33 54 17 1 45 M 30 19 te 5^ 34 55 18 1 ( 512 ) Parlous Standard National Measures. English Inches. The old V?iris foot 12. 792 The new Paris standard inctre 39.3/1 The Scotch foot 12.(yjl The Scotch ell fsame as English' 45.000 The Ri-ynland foot of Denmark 12.302 The Swedish/oo^ 11 .6()2 The Amsterdam foot 11.1/2 ell 26.8 The Russian ar^Z'ra? 28.35 The Vienna foot in Austria 12.44 The Spanish vara of Madrid 39.166 of Seville 33.12/ ofCastille 32-952 The Twtxxifoot 20. 17 rees 23 .5 trdbucco 12 1 .2 The Genoa palm -i ^'g canna 87-^ The Venice braccio for measuring" silk 25 3 for measuring cloth 27. 1'he Piorence braccio C 22. S I 22.92 The hraccio of Rome, for architects 30.72 for merchants 34.07 The Roman canna 78- The ancient Ronian/wfl/f 1 1.(535 'pohn 8.82 The ancient Greek foot 12.O9 Tile Naples canna 82.9 pahi 10.31 The P.ologna/oc?/ 15. The braccio of Milan 2O.7 of Bologna 2-i.50 of Parma and Flacenza 2O.9 of Lucca 23.5 of Bresica and Mantua 25.1 The Royal /w.' of China 12.6 Englhh Feet. The ancient Roman mile (by Plinius) \ 4840.5 (by Strabo) J 4903. ufadiuvi 606 The Egyptian stadium 730.8 The U of the Chinese 606 C 5i3 ) A TABLE, tor expedUiausly Measuring Timber. : Quarter Area. Quarter \ 1 Quarter Girt. Girt. Area. 1 Girt. Area. Inches. Feet. Inches. Feet. ! Ini-hcs. Feet 6 .250 12 1.000 18 2.250 ^ .2/2 l^i: 1 .ai2 16t 2.370 6? .-04 12i- 1 .085 19 2.506 ^ .317 12i 1.129 I9t 2.640 7 .340 13 1.174 20 2.717 7i- .3(J4 13^ 1-219 20t 2.917 74 .390 13i 1.265 21 3.002 7i .417 13| 1.313 2lt 3.209 8 .444 14 1.3dl 22 3.362 9i .472 14^ 1.410 22t 3.516 &i .501 14' 1 .400 23 3.6-3 H .531 \.ii 1.511 23t 3.835 9 .562 15 1,5()2 24 4.000 9i .594 1-H 1.615 24t 4.168 .0'25" 15^ 1 .do's 25 4.340 9i ■659 15i 1.722 25t 4^16 10 .694 16 1.777 26 4.6g2 lOJ .730 i6i 1.833 26t 4.876 10^ .766 i(5i 1.890 27 5.062 lOJ .803 10> 1 .94s vk 5.252 11 .840 17 2.OOO 28 5.444 lu .878 i7i 2.0G6 28 j; 5.UiO Hi .918 I7t 2.126 29 5.S40 Hi ■950 I7i 2.1S7 29t 1 30 5.044 6.250 The number of the area, against the quarter girt in the table, U be multiplied by the length of the timber in feet, and the pro- uct will be the content in feet and dccimtls of a foot. Thus, appose a piece of timber be 16 feet 9 inches long, (or 16.75 feet,) nd the quarter girt 20 inches, ly the lablj then 16.75 x 2,717 =45,50975 feet the contents. LI [ 514 ] A Tahle, shelving hoiv much a Pendulum that swings Seconds at the Equator, ivould gain every 24 hours in different Latitudes ; and hotv much the Penduhcm ivoidd need to he lengthened in these Latitudes, in ordt r to make it swincr seconds therein. o Laiuude Time Lencthen-j 1 atitude Time Lengthen- of the gained ing of the of the gained ing of the Place. 111 one pendulum Place. m one i'endulu.n Day. to swing Day. to swing Seconds. 1 econds. De,'. Seconds. Inch. Parts. Deg. Seconds. nch. Parts. 5 1 .7 .0016 50 134 .0 .1212 10 6 -9 .0002 55 153 .-J .1386 15 15 .3 ' .0138 60 171 .-. .1549 20 26 .7 .0246 65 187 .5 .1696 25 40 .8 .036p 70 201 .6 .1824 30 57 .1 .0516 75 213 .0 .1927 35 75 .1 .0679 80 221 .4 .2033 40 94 .3 .«S53 85 226 .5 .2050 45 il4 .1 .1033 90 228 .3 .2065 A pendulum that swings seconds at the equator must bc-V-V parts of an inch shorter than one that swings seconds at London ; and a pendulum that swings seconds at the poles must be -co- parts of an inch longer than one that swings seconds at Lon- don. The cause of this difference arises from the spheroidical figure of the earth, and the centrifugal force diminishing (and so acting gradually, less and less,) from the equator to the North and South poles, as the diurnal motions of the places are slower and slo'iver. The length of a pendulum that swings seconds at the equator is 3g inches ; and the length of a pendu- lum that swings seconds at the poles^ is 3g.26G inches. f 515 ) A TABLE, To reduce ary length takat in Links into F,et onA fnrhis. 1 iiik-i. Feet Fnches. 1 i-iuks. ilvct. 1 tiichftj. 1 •' lrlk^ Tea. liKllCd 1 O 7-92 1 34 22 5,2s 1 (y? 44 2,04 2 ) 3,84 1 35 23 • ,20 63 4-1 o,.-.o' 3 1 11, /ti 36 23 9, 2 6(1 45 6.48 4 2 7.68 37 24 5,01 1 70 40 2.40 o 3 3,60 38 • 25 0,90 7 46 ,0,1 6 3 11,52 ! 39 25 8,88 72 47 6,21 / 4 7.11 40 26 4,80 73 48 1 2, . !> 8 5 3,30 41 2/ 0.72 7* 48 I0,()S 9 5 11,28 42 27 8,64 75 49 6,CX) 10 6 7.20 43 28 4,56 70 50 1 ,0J 11 7 3,,2 44 29 0,48 77 50 .0,84 12 / 11,04 45 29 8,40 1 7S 5 5,76 13 8 6,f>6 4Q 30 4,32 1 79 52 1,68 14 p 2,t>8 47 3| 0,24 so 52 9,60 15 9 10,80 48 3 8,16 81 53 5,.>2 It) 10 6,72 49 32 4,08 82 54 1 ,-1-1 17 11 2,04 50 33 0,00 83 54 g,:u, 18 11 RV56 51 33 7.92 84 55 5,2 c- 19 12 6,48 52 34 3,84 85 56 i,2.:> 20 13 2,40 53 34 11,76 s6 56 9,12 21 13 10,32 5i 35 7,6S 87 57 5,01 22 14 6,24 55 36 3,60 88 56 o,ou 23 15 2.16 56 36 11,52 1 89 , 53 8,bS 24 15 10,08 1 5/ 37 7A-i 90 59 4, SO 2.3 16 6,00 1 59 38 3, .36 9> 60 O.7-! 26 17 1,92 59 38 11,28 92 60 8,61 27 17 0,84 6() 39 7,20 93 61 4,5',' 28 18 5,76 1 61 40 3,!2 9-i 02 0,48 '-i9 19 1 ,68 1 62 40 i 1 ,04 1 95 02 8,40 30 19 9,6o| 63 41 6,96 1 ()«) 63 4,32 31 1 20 5,52 1 64 42 2,88 j 97 6i 0,24 32 21 1,44 65 42 10,bO 98 64 b,i6 33 21 9,3(i 66 43 6,72 99 65 4, Ob FfN/S. Priuttd bj \V'. (jk-iiiliiuiing, .:.';; IL.Uwij G»riii;ii. A LIST Of the principal Instruments described in this Work, and their Pr':ce<:, as made and sold by W. and S. Jones, 30^ Holhcr?!, London. PLATE I. Partial, or complete magazine cases of instruments, as ^' ^' represented in the plate, from ll. I8s, to 10 10 PLATE II. Parallel rulers, Jig. A, B, C, D, and E, according to the length and mounting, from 2s. each, to 2 13 Q N. B. Oi jigs. A, or B, or C, or D, or E, one of them is included in the cases above, to order. Improved ruler, FGH 1 11 Q Ditto I K L 1 16 o Ditto represented at M, from 10s. 6d. to O l6 PLATE III. Fig. \ , Marquois's parallel scales, divided f) 2 and 3. Protractors, v/ood or ivory, from 2s. to. . , 7 4. Sectors of box-wood, ivory, or brass, &c. accord- ing to the length, from 2s. 6d. to 3 13 6 7- Brass proportional compasses, six inches 1 \5 Q Without the adjusting rod, fg. A, plate 1 Ill 6 9 and Q a. Elliptical, sector, and beam compasses . 3 13 6 10. Beam compasses from ll. 4s. to 4 4 1 1 . Sisson's beam compasses 3 3 12. Triangular compasses 1 1 O Ditto represented at Jig. N, plate 1 13 O 13. Calliper and beam, small size 1 11 6 PLATE X. Bevel, &c. rulers, fgs. 4, 12, 15, lO, 17, and 18, v.irious prices from lOs. O'd. to 3 13 6 PLA'I E XI. Fig. 1 . Suardi's geometric pen 4 4 O Ditto, with a greater variety of wheels, kc. . . 7 7 [ 517 ] C- '■ i Fr^. 2. EUipti<"al drawing board 2 12 6 — — 3. Elliptical brass compas»cs 2 12 6 .. 4 . Parabolic drawing machine 2 12 6 5. Cyclograph for drawing circles of large radii ... 4 14 (i 6. Spiral machine 4 14 6 . ■■ -/. Protractor, perpendicxJar, &c. ., 5 5 O PLATE XIV. Fig. 1. King's surveying quadrant with sights 1 18 O 2. Portable surveying brass cross and staff 118 3. Brass pocket box cross and staff 18 i. Ditto, without a staff , 10 (5 6. Improved ditto, with staff, from ll. lis. 6d. to . . 3 3 O 4. Optical square 1 4 5. Ten inch common theodolite and staves 5 5 O 7- Best complete portable theodolite 10 10 PLATE XV, Fig. 1. Common circumferentor and staves ... 3 13 6 6. Ditto, without staff, 2 12 6 2. Improved ditto, with staff, 4l. 14s. 6d. to 5 15 6 3. Surveying compass with telescope 2 2 — — »7- Miner's compass without telescope, U. 5s. and 1 15 O ■ 4 . Six inch pocket common theodolite and staves . . 3 3 . 5. Six inch theodolite by rack work and staves .... 12 12 Ditto without rack work, common 8 8 PLATE XVI. Fig. 1 . Second best 7 or 8 inch theodolite and staves, l61. lOs. and 22 1 O 2. Very best improved ditto, ditto 31 JO O N. B. For large theodolite see the frontispiece, from 801. to ^ 300 PLATE X\1I. Fig.) 1 . Plane table and staves in a box 4 4 2. Beighton's improved ditto 9 8 3. Spirit level of the best kind with telescope and staves, according to the adjustments and finishing, from &1. 8s. to 12 12 Common ditto, without compass, from 3l. 13s. Od, to 6 f> O [ 518 ] Fig. 4. Six inch circular protractors, ll, 18s. to 2 10 5. Ditto by rack work, very best 4]4 6 Common circular brass protractors, 6s. to 18 O Perambulator, or measuring wheel 6 i6 6 Way wiser for carriages 61. 6s. to 15 15 .0 9- -tatiun staves^ the pair 2 13 10. Pocket spirit levels, from 10s. 6d. each, to ... . Ill 1 1 . Improved universal ditto 1 11 6 PLA'lE XIX. Fjg. 1. Hadley's quadrant, ebony and brass, 31. to ... , 3 13 6 ——4. Best metal sextant, 8 to 10 inches radius . . , , . 13 2 6 Bass stand and counteipoise for a ditto 4 4 Second best sextants, 8l. 8s. to 10 10 11. Pocket box sextant from 2l. 2s. to 3 3 O 12. Artificial horizon ll. l6s. and complete 2 2 PLATE XX'II. Fig. 2. Feather edged 12 inch box plotting scales, each ..030 Ditto ivory, each 8 O PLATE XXX L Fg. ]g. Pantagraph, 2 feet, best 4 14 6 Common ditto, 12 to IS inches, from ll. 4s. to 3 20. T. Squares, from 6s. ()d. to PLATE XXXIT. Fig. I. Perspective machine G 2. Ditto, ditto 5 3. Perspective compass 1 IS PLATE XXXIil. jF.^. 2, 3, 4, and 5. Keith's improved parallel scales in a case .' 10 G Ditto in wood and ivorj-^, or all ivory 1 5 6 and 7, Gunners callipers, from 2l. 12s. 6d. to . . G 6 8. Gunner's quadrant 2 12 6 . Q. level and perpendicular 1 l6 For farther particulars see W. and S. Jokes's general catalogue. 13 6 10 6 6 5 IS 9 = ^^CvS^^gl IVERSirr OF CUIFORMK _ ^^^ 14 DAY USE V - K^TURN TO DESK FROM WHICH BORBOWED LOAN DEPT. WT" T.-,rfar ^ ? niQ?. ft fia-DLO JUN5 7f -gASf63 iAR . k>5^^g 8£C CIR. fEB 1 2 76 ^^^ LD21A-50W-2 '71 (P200l8l0)476— A-32 General Library Umversity of California Berkeley IIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA j"-Vr*V rr^4"v. B f LFY LIBRARIf CD5M73n31 ^m^'^' Z / m&K....J F THE UNIVERSITY OF CALIFORNU LIBRARY OF THE UNIVERSITY OF CALIFOR F THE UNIVERSITY OF CALIFORNU LIBRARY OF THE UNIVERSITY OF CALIFOR