THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 
 OF CALIFORNIA 
 
 LOS ANGELES 
 
 GIFT OF 
 
 John S.Prell
 
 J^^Knielit 
 
 • /
 
 tA/'Grea.t Theodolite, *y Ramsden 
 
 Used Iftf the late Gm ^. Roil iv- in the great Eiiglisli Tr,\/a7wnuttna1 OveroUon^
 
 GEOMETRICAL 
 
 AND 
 
 GRAPHICAL ESSAYS, 
 
 CONTAI NIXG, 
 
 A GENERAL DESCRIPTION 
 
 OF THE 
 
 MATHEMATICAL INSTRUMENTS 
 
 USED IN 
 
 GEOMETRY, CIVIL AND MILITARY SURVEYING, 
 LEVELLING, AND PERSPECTIVE; 
 
 WITH MANY NEW 
 
 PRACTICAL PROBLEMS. 
 
 ILLUSTRATED BY THIRTY-FOUR COPPER PLATE;, 
 
 BY THE LATE 
 
 GEORGE ADAMS, 
 
 MATHEMATICAL INSTRUMENT MAKER. TO HIS MAJESTV, &C. 
 
 THE SECOND EDITION, 
 CORRECTED AND ENLARGED BY 
 
 WILLIAM JONES, 
 
 MATHEMATICAL INSTRUMENT MAKER, 
 
 LONDON: 
 
 PRINTED BY J. DILLON, AND CO. 
 
 AND SOLD BY W. AND S. JONES, OPTICIANS, 
 HOLBORN, LONDON. 
 
 JOH^I S. Pf^ELL 
 
 Civil & Mcchardcal En-rineer. 
 
 SAN FkAMnTon/^ .. ^
 
 
 TO 
 
 THE MOST NOBLE 
 
 CHARLES, DUKE of RICHMOND, 
 AND LENNOX, 
 
 Master General of the Ordnance, &c. 
 THESE ESSAYS 
 
 ARE 
 
 V, 
 
 WITH GREAT RESPECT, 
 
 JUSTLY INSCRIBED, 
 BY HIS grace's 
 
 Most obedient. 
 
 Humble Servant, 
 GEORGE ADAMS. 
 
 to 

 
 PREFACE. 
 
 1 HOSE who have had much occasion to use the 
 mathematical ijnstrumcnts constructed to facilitate 
 the arts of drawing, surveying, &c. have Iqng 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 
 respective purposes. 
 
 This complaint has been the more general, as 
 there are few active stations in life whose profes- 
 sors are not often obliged to have recourse to ma- 
 thematical instruments. To the civil, the military, 
 and the naval architect, their use must be familiar; 
 and they are of equal, if not of more importance 
 to the engineer, and the surveyor; they are the 
 means by which the abstract parts of the mathe- 
 matics are rendered useful in life, they connect 
 theory with practice, and reduce speculation to 
 use. 
 
 Monsieur Bions treatise on tlic construction of 
 mathematical instruments, which was translated 
 into English by Mr. Stone, and published in 172^3, 
 is the only regular treatise* we have upon this 
 subject; the numerous improvements that have 
 been made in instruments since that time, have 
 
 * I do not speak of Mr. Robertson's work, as it Is confined 
 ■wholly to the instruments contained in a case of drawing instru- 
 ments.
 
 11 PREFACE. 
 
 rendered this work but of little use. It has been 
 my endeavour by the following Essays to do away 
 this complaint; and I have spared no pains to 
 render them intelligible, and make them useful. 
 Though the materials, of which they are com- 
 posed, lie in common, yet it is presumed, that 
 essential improvements will be found in almost 
 evciy part. 
 
 These Essays begin by defining the necessary 
 terms, and stating a few of those first principles 
 on which the v/hole of the work is founded: they 
 then proceed to describe the mathematical draw- 
 ing 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 micro- 
 meter screw, four new parallel rules, and other ar- 
 ticles not hitherio described : these are followed by 
 a large collection of useful geometrical problems; 
 I flatter mvsclf, that the practitioner will find many 
 that are new, and which are well adapted to lessen 
 labour and promote accuracy. In describing the 
 manner of cHviding large quadrants, I have first 
 given the methods used by instrument makers, 
 previous to the publication of that of Mr. Bird^ 
 subjoining his mode thereto, and endeavouring to 
 render it more plain to the artist by a different ar- 
 rangement. This is succeeded by geomel^rical and 
 mechanical methods of describing circles of every 
 possible magnitude; for the greater part of which 
 I am indebted to Joseph Priestley, Esq. of Bradford, 
 Yorkshire, whose merit has been already noticed 
 by an abler pen than mine.*' From this, I pro- 
 ceed to give a short view of elliptic and other com- 
 passes, and a description of Suanli's geometric pen, 
 an instrument not known in this country, and 
 
 * Pr -tlcy's Perspecti\e,
 
 PREFACE. Ill 
 
 whose curious properties will exercise the inge- 
 nuity of mechanics and mathematicians. 
 
 Trigonometry is the next subject; but as this 
 work was not designed to teach the elements of 
 this art, I have contented myself with stating the 
 general principles, and giving the canons for cal- 
 culation, subjoining some useful and curious pro- 
 blems, which, though absolutely necessary in many 
 cases that occur in county and marine surveying, 
 have been neglected by every practical writer oa 
 this subject, except Mr. Mackenzie,* and B.Donn,\ 
 Some will also be found, that are 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 
 evidem, from a view of those of the best construc- 
 tion, that large estates maybe surveyed and plot- 
 ted with greaier 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 various im- 
 provements which the instruments of science have 
 received from Mr. Ramsden, we are to reckon 
 those of the theodolite here described; the sur- 
 veyor will find also the description of a small qua- 
 drant that should be constantly used with the 
 
 * Treati?e on Maritime Suneyin^. 
 f Donn's Geometrician,
 
 IV PREFACE. 
 
 chain, improvements in the circumferentor, plain-t 
 table, protractor, &c. In treating of surveying, 
 I thought to have met with no ditHculty ; having 
 had however no opportunity of practiee 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 success must be left 
 to the reader's judgment) to remove their obscu- 
 rities, to rectify their errors, and supply their defi- 
 ficiencies; but whatever opinion he may form of 
 my endeavours, I can venture to say, he will be 
 highly gratified with the valuable communications 
 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 
 moment to give him some account of Mr. Gale's 
 improvements; they consist, first, in a new me- 
 thod of plotting, which is performed by scales of 
 equal parts, without a protractor, from the north- 
 ings and southings, eastings and westings, taken 
 out of the table which forms the appendix to this 
 work;-)- this method is much more accurate than 
 that in common use, because any small inaccuracy 
 that might happen in laying down one line is mj- 
 turally corrected in the next; whereas, in the 
 common method of plotting by scale and protrac- 
 tor, any inaccuracy in a former line is tuiturally 
 communicated to all the succeeding lines. The 
 next improvement consists in a new method of de- 
 termining the area, with superior accuracy, from 
 
 •'■' A gentleman well known for his ingenious publications on 
 finance. 
 
 -f The table is printed separate, that it may be purchaccd^ oi 
 not; as the surveyor sees convenient.
 
 PREFACE. V 
 
 the northings, southings, castings, and westings, 
 without any regard to the plot or draught, by an 
 easy computation. 
 
 As the measuring a strait hnc 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 multiplied 
 without a possibility of their bein^ discovered, or 
 corrected. I must not forget to mention here, 
 that I have inserted in this part Mr. Break's me- 
 thod of surveying and planning by the plain table, 
 the bearings being taken and protracted at the 
 same instant in the held upon one sheet of paper; 
 thus avoiding the trouble and inconvenience of 
 shifting the paper: this is followed by a small 
 sketch of maritime surveying; the use of the pan- 
 tographer, or pantagraph ; the art of levelling, and 
 a few astronomical problems, with the manner of 
 using Hadley's quadrant and sextant; even here 
 some suggestions will be found that are new and 
 useful. 
 
 I 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 thanks to Mr. La?i(iman, 
 Professor of fortification and artillery to the Royal 
 Academy at Woolwich, for 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 co-operate for their advance- 
 ment, the progress thereof would be rapid and ex- 
 tensive. This course will be found useful not only 
 to the military officer, but would make a useful 
 and entertaining part of every gentleman's educa- 
 tion. I found it necessary to abridge the papers 
 JNIr. Landnian lent me_, and leave out the calcula-
 
 VI PREFACE, 
 
 tions, as the work had already swelled to a larger 
 size than was originally intended, though printed 
 on a page unusually full. 
 
 The woi'k iinishes with a small tract on perspec- 
 tive, and a description of two instruments designed 
 to promote and facilitate the practice of that use- 
 ful art. It is hoped, that the publication of these 
 will prevent the public froiTft being imposed upon 
 by men, who, under the pretence of secresy, en- 
 hance the value of their contrivances. I knev,- an 
 instance where 40 1. was paid for an instrument 
 inferior to the most ordinary of the kind that are 
 sold in the shops. Some pains have been taken, 
 and no small expense incurred, to offer something 
 to ihe public superior in construction, and easier 
 to use, than any instrument of the kind that has 
 been hiihcrto exhibited. 
 
 I have been anxious and solicitous not to neg- 
 lect any thing that might be useful to the practi- 
 tioner, or acceptable to the intelligent. In a work 
 which embraces so many subjects, notwithstand- 
 ing all the care that has been taken, many defects 
 may still remain; I shall therefore be obliged to 
 any one who will favour me with such hints or 
 observations, as may tend towards its improve- 
 ment. 
 
 A list of the authors I have seen is subjoined to 
 this Preface. I beg leave to return my thanks to 
 the following gentlemen for their hints and va- 
 luable communications, the Rev. Mr. HawkhiSy 
 J. Piieslley, Esq. Mr. Gak, Mr. Milne, Dr. Rother^ 
 ham, Mr. Heywood, Mr. Landman, and Mr. Becky 
 a very ingenious artist.
 
 ADVERTISEMENT 
 
 BT 
 
 THE EDITOR. 
 
 JL HE first edil'mi of these Essays havings like the 
 rest of the late ingenious Author" s ijuorks^ received 
 viuch share of public approbation and encourage- 
 ment \ and being myself a joint proprietor with wy 
 hrother, S. Jones, of the copyright of all his publi- 
 cations, I conceive, that I ca?inot employ the few lei- 
 sure moments, after the business of the day, better, 
 than by revising, correcting, and ■ improving those 
 works that require reprintn/g. The present is the first 
 of my editing: considerable errors in the former edi- 
 tion have been corrected; more complete explanations 
 of instruments given, a)id many particulars of new and 
 useful articles, not noticed by the Author, with noteSy 
 ^c. are inserted in their proper places. The additions 
 and amendments are, upon the vohole, such, as I pre- 
 sume, without any pretension to superior abilities on 
 my part, will again render the work deserving of the 
 notice of all students and practitioners in the aiffercnt 
 professional branches of practical geometry. The 
 principal additions I have made are the following: 
 
 Description of a new pair of pocket Compasses, 
 containing the ink and pencil points in its two 
 legs — Improved Pei-ambulator — Way Wiser — Im- 
 proved Surveying Cross — Improved Circumferen- 
 tor-^Complete portable Theodolite — Great Theo- 
 dolite, by Ramsden — Pocket box Sextant — Artifi- 
 cial Horizon — Pocket Spirit Levels — A Pair of 
 Perspective Compasses — Keiilis improved Parallel 
 Scale — New Method of Surveying and keeping a 
 Field Book — Gunner's Callipers — Gunner's Qua- 
 drant — Gunner's Level, he. 
 
 March 30, JV. JONES. 
 
 1797-
 
 LIST 
 
 OF AUTHORS, 
 
 CONSULTED FOR THIS WORK. 
 
 Pion Construction, &c. of Mathematical 
 
 Instruments, London, 1/23 
 
 Break Systeni of Land Surveying, London, 1/73 
 
 Bonnycastle . . Introduction to Mensuration London, 1/87 
 
 Cunn Treatise on the Sector, London, 1729 
 
 Clavius Astrolabium Tribus Libris Expli- 
 
 catum, Moguntioe, 161 1 
 
 Cagnoli Traite de Trigonometric, Paris, 1/85 
 
 De la Grive . . INIanuel de Trigonometric Paris, 1 /j-i 
 
 Donn Geometrician London, \']'J5 
 
 Daudet Introduction a la Geometric, Paris, 178O 
 
 Dalrymple . . . Essay on Nautical Surveying, .... London, 1786 
 Eckhardt .... Description d'unGraphometre,. . Ala Haye, 17/8 
 
 Gardner Practical Surveying, London, 1737 
 
 Gibson Treatise of Surveying, Dublin, 17O3 
 
 Hutton Treatise on Mensuration, . London, 1788 
 
 Hume Art of Surveying, London, 1763 
 
 Hammond . . . Practical Surveyor, London, 1765 
 
 LeFebvre. . . . Oeuvres Complettes, . . . .• Maestricht, 1778 
 
 liOve Art of Surveying, London, 178(5 
 
 Mackenzie . . . Treatise of Maritime Surveying, . . London, 17/4 
 
 Mandey Marrow of Measuring, London, 1717 
 
 Nicholson . . . Navigator's Assistant, London, 1784 
 
 Noble Essay on Practical Surveying, .... London, 1774 
 
 Payne Elements of Geometry, London, I'i^'J 
 
 Trigonometry, London, 1773 
 
 Picard Traite du Nivelment, Paris, 1784 
 
 Robertson . . . Treatise of Mathematical Instal- 
 ments, London, 1775 
 
 Spiedell Geometrical Extraction, London, i6j7 
 
 Talbot Complete Art of Land Measuring, London, 1784 
 
 Hutton's Mathematical and Philosophical Dicticnary, 4to. I'J^^.
 
 A TABLE 
 
 CONTENTS. 
 
 Page 
 
 Necessary Definitions and First Principles 1 
 
 Of Mathematical Drawing Instruments 10 
 
 Of Drawing Compasses • 14 
 
 Of Parallel Rules 21 
 
 Of the Protractor 30 
 
 Of the Plain Scale 32 
 
 Of the Sector 40 
 
 Select Geometrical Problems 52 
 
 Of the Division of Strait Lines 59 
 
 Of Proportional Lines oO 
 
 Of the Transformation and Reduction of Figures 81 
 
 Curious Problems on the Division of Lines and Circles . . 93 
 
 Mr. Bird's Method of Dividing 108 
 
 Methods of describing Arcs of Circles of large Magnitude 134 
 
 To Describe an Ellipse, &:c 153 
 
 Suardi's Geometric Pen 157 
 
 Of the Division of Land l60 
 
 Of Plain Trigonometry 171 
 
 Curious Trigonometrical Problems 17^ 
 
 Of Suri'eying 194 
 
 Of the Instruments used in Surveying 199 
 
 Of the Chain 202 
 
 Of King's Surveying Quadrant 205 
 
 Of the Perambulator 208 
 
 Of the Levelling Cross 214 
 
 Of the Optical Square '^• 
 
 Of the Circumferentor 210 
 
 Of the Plain Table 2^6 
 
 Of Improved Theodolites 231 
 
 The complete portable Theodolite 243
 
 TABLE OF CONTENTS. 
 
 #agc» 
 
 llamsden's great Theodolite 243 
 
 i)i Hadley's Quadrant and Sextant 25/ 
 
 Of the Artificial Horizon 285 
 
 To Survey with the Chain 28(5 
 
 by the Plain Table -. 28S 
 
 by the Common Circumferentor 307 
 
 Of Mr. Gale's Improved Metliod of Surveying 310 
 
 jNIr. ]\1 line's JNIethod of Surveying with the Best Theodolite 319 
 
 Of Plotting 328 
 
 Mr. Gale on Plotting 333 
 
 jNIr. Milne on Plotting 340 
 
 Of Determining the Area of Land 346' 
 
 Of Maritime Survwing 36G 
 
 To Transfer one Plan from another 3/8 
 
 Description of the Pantagraph 379 
 
 Of Levelling and the best Spirit Level 383 
 
 Astronomical Problems 405 
 
 jNIilitary Geometry 442 
 
 Essays on Perspective 46j 
 
 Of instruments for Drav/ing in Perspective 48S 
 
 ADDENDA. 
 
 Emerson's Method of Surveying a large Estate 4gi 
 
 Jlodbam's new Method of Surveying and keeping a Field 
 
 Book 41)7 
 
 Keith's Improved Parallel Scale 500 
 
 Gunner's Callipers 505 
 
 Quadrant 515 
 
 • Perpendiculars il?. 
 
 Shot Gauges ^ iL 
 
 A List of the Principal Instruments and their Prices, as 
 
 made and sold by W. and S. Jones, Holborn, London , . 5lG
 
 Uoil & Mechanical Engineer 
 
 SAN FRANCHSCO, OAL. ' 
 
 GEOMETRICAL 
 
 AND 
 
 GRAPHICAL 
 
 ESSAYS. 
 
 NECESSARY DEFINITIONS, and FIRST 
 PRINCIPLES. 
 
 vXEOMETRY Originally signified, according to 
 the etymology of the 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 objects of geometry. It is a science, in which 
 human 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 practical. 
 
 Theoretical geometry considers and treats of 
 first principles abstractedly. Practical geometry 
 applies these considerations to the purposes of life. 
 By practical geometry many operations are per- 
 formed of the utmost importance to society and 
 the arts. " The effects thereof are extended 
 through the principal operations of human skill: 
 it conducts the soldier in the field, the seaman on 
 
 B
 
 2 NECESSARY DEFINITIONS, 
 
 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 science, attributed 
 to the Egyptians; and, that to the frequent inun- 
 dations of the river Nile upon the country, we owe 
 the rise of this sublime branch of human know- 
 ledge; the land-marks and boundaries being in 
 this Wily destroyed, the previous knowledge of the 
 figure and dimensions was the only method of as- 
 certaining individual property again. But, surelv, 
 it is not necessary to gratify learned curiosity 
 b^ such accounts as these; for geometry is an art 
 that must have grown with man; it is, in a great 
 measure, natural to the human mind; we were 
 born spectators of the universe, which is the king- 
 dom Tf geometry, and are continually obliged to 
 judge of heights, measure distances, ascertain the 
 iigure, and estimate the bulk of bodies. 
 
 The fu'st definition in geometry is a point, which 
 is considered by geometricians, as that v/hjeh ha3 
 no parts or magnitude. 
 
 A line is length v/ithout breadth. 
 
 A strait line is that which lies evenly between 
 its extreme points or ends. 
 
 A superficies is that which has only length and 
 breadth. 
 
 K plane angle \^ an opening, or corner, made by 
 two strait liilies meeting one another. 
 
 When a strait line AB,^^. 1, plate A, standing 
 upon another C D, m.akes angles A B C, A B D, 
 on each side equal to one another; each of these 
 angles is called a right angle; and the line A B is 
 said to ho. perpetidieular to the line C D. 
 
 It is usual to express an angle by three letters, 
 that placed at the angular point being ahvays in 
 the middle; as B is the angle of AB C.
 
 AND FIRST PRINCIPLES. S 
 
 An ohtnse angle is that which is greater than a 
 right angle. , 
 
 . An acute an^le is that which is less than a rio-ht 
 angle. 
 
 A line A B,^o-. 2, plate A, cutting another line 
 CD in E, will make the opposite angles equal, 
 namely, the angle AEC equal to B E D, and 
 AEDequaltoBEC. 
 
 A line A B, Jig. 3, plate 4, standing any-way 
 upon another CD, makes two angles CBA, AB'D, 
 which, taken together, are equal to two right 
 angles. 
 
 A plane tna??gle is a figure bounded by three 
 right lines. - 
 
 An equilateral tr'uingle is that which 
 has three equal sides. 
 
 An isosceles triangle is that which has 
 only two equal sides. 
 
 A scalene triangle is that which has 
 all its sides unequal. 
 
 A right-angled triangle is that which 
 has one rirht ano-le. 
 
 In a riglit-angled triangle, the side opposite to 
 the right angle is called the liyprAlienuse. 
 
 An chlique-angled triangle is that / 
 
 which has no rit>ht ano-le. / 
 
 In the same triangle, opposite to the greater 
 side is the greater angle; and opposite to the 
 greater angle is the greater side. 
 
 B 2
 
 4 NECESSARY DEFINITIOXS^ 
 
 If any side of a plane triangle be produced, thtf 
 outward angle will be equal to both the inward 
 remote angles. 
 
 The three angles of any plane triangle takers 
 together, 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 quadrangles, or quadrilaterals. 
 
 A square is a quadrangle, whose 
 sides are all equals and its angles all 
 right angles. 
 
 A rhomhus is 3 quadrangle, whose 
 sides are all equal, but its angles not 
 right angles. 
 
 A parallelogram is a quadrangle,- 1 | 
 whose opposite sides are parallel. | j 
 
 A rectangle is a parallelogram, whose angles are" 
 all right angles. 
 
 A rhomboid is a parallelogram, whose / 7 
 
 angles arc not right angles. / / 
 
 All other four-sided figures besides 
 these, are called trapeziums, j 
 
 A right line joining any two opposite angles of 
 a four-sided figure, is called the dwgoiial. 
 
 All plane figures contained under more than 
 four sides, are iz^Vnt^ polygons. 
 
 Polygons having five sides, are q,^\^^ pentagans% 
 those having six sides, hexagons \ with seven sides^ 
 heptagons-^ and so on.
 
 AND FIRST PRIXCIPLES. 5, 
 
 A regular polygon is that whose angles and sides 
 .are all equal. 
 
 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 be- 
 tween the same parallels, are equal. 
 
 Parallelograms having the- same base, and equal 
 altitudes, are equal. 
 
 Parallelograms having equal bases, and equal 
 altitudes, arc equal. 
 
 If a triangle and parallelogram have equal bases 
 and equal altitudes, tKe triangle is half the paral- 
 lelogram. 
 
 A circle is a plane figure, bounded by a curve 
 line called the circumference, every part, whereof is 
 equally distant from a point within the same figure, 
 called the center. 
 
 Any part of the circumference of a circle is 
 called an arch. 
 
 Any right line drawn from the center 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 J^?^r^^j ; 
 each degree into 6o equal parts, called minutes, &c. 
 
 A quadrant of a circle will therefore contain 90 
 degrees, being a fourth part of 360. 
 
 Equal angles at the centers of all circles, will 
 intercept equal numbers of degrees, minutes, &c, 
 in their circumferences. 
 
 The measure of every plane angle is an arch of 
 a circle, whose center is the angular point, and is 
 said to be of so many degrees, minutes, &c. as are 
 contained in its measurinaiarch.
 
 O NECESSAE.Y DEFINITIONS, 
 
 All right angles, therefore, are of go degrees, 
 or contain QO degrees, because their measure is a 
 quadrant. 
 
 The three angles of every plane triangle taken 
 together, contain 180 degrees, being equal to two 
 right angles. 
 
 In a right-angled plane triangle, the sum of its 
 two acute angles is 90 degrees. 
 
 The cGmplcmcnt of" an arch, or of an angle, is 
 its difi'erence 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 strait lines, appertaining to a 
 circle, called chords, sines, tangents, &c. 
 
 The cliord of an arch is a strait line, joining its 
 extreme noinls. 
 
 A diameter is a chord passincr throuo:h the center. 
 
 A segment 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, perpeiKlicular to a diameter meeting the 
 other end. 
 
 The iierscd slue of an arch is that part of the di- 
 ameter intercepted between the sine and the end 
 ofthe 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 
 center, through the other end. 
 
 The secant of an arch is the line proceeding 
 from the center, and limiting the tangent of the 
 same arch.
 
 AND FIRST PRINCIPLES. 7 
 
 The co-sine and co-tangent, &c. of any arch is 
 the sine and tangent, 8cc. of its complement. 
 
 Thus '\x\fg. A, plate 4, FO is the chord of the 
 arch FVO, and FR is the sine of the arches 
 F V, FAD; K V, II D arc the versed sines of the 
 arches FV, FAD. 
 
 V T is the tangent of the arcli F V/ and its 
 supplement. 
 
 C T is the secant of the arch FV. 
 
 A I is the co-tangent, ai'^1 C I the co-sccant of 
 the arch F V. '. 
 
 The chord of 6o°, the sine of ()0°, the versed 
 sine of 90°, the tangent of 45, and the secant of 
 0.0, arc all equal to the radius. 
 
 It is obvious, that in making use of these lines, 
 we must alwnys use the same radius, otherwise 
 there would be no settled proportions between 
 them. 
 
 Whosoever considers tlie whole extent and 
 depth of geometry, will find that the main design 
 of all its speculations is mensuration. ' To this the 
 Elements of Euclid arc 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 
 reduced to the measure' of triairgle'?j which are 
 alwavs the half of a reetan2:le of the sarnie Base 
 and ahirude, and, consequently, theii' area is ob- 
 tained by taking the half of the product of the 
 base multiplied by the altitude. 
 
 By dividing a polygon into triangles, and tak- 
 ing the value: of t'feese, that 6{ the polygon is ob-f 
 taincd; by considering the circle as' a\])blygon^ 
 with an infinitb' number of sideS^ we o-bt;»in the 
 measure thereof to a sufficient degree of accu- 
 nicv. ^
 
 8 NECESSARY DEFINITIONS, 
 
 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. 
 
 The angles and the sides determine both the 
 relative and absolute .size, not only of triangles, 
 but of all things. 
 
 Strictly speaking, angles only determine the 
 relative size ; equiangular triangles may be of very 
 unequal 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 equian-> 
 gled, but are equal in every respect, 
 
 The angles, therefore, determine the relative 
 species of the triangle; the sides, its absolute size, 
 and, consequently, that of every other figure, as 
 all are resolvible into triangles. 
 
 Yet tlie essence of a triangle seems to consist 
 much more in the angles than the sides; for the 
 angles are the true, precise, and determined boun- 
 daries 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 isoperimetrical figures* 
 that we feel how efficacious angles are, and how 
 inefficacious lines, to determine not only the kind, 
 |)ut the size of the triangles, and all kinds of figures. 
 
 * Isoperimetrical figures are such as have equal circumferencea.
 
 AND FIRST PRINCIPLES. g 
 
 For, the lines stili subsisting the same, we see 
 how a square decreases, in proportion as it is 
 changed into a more obHque rhomboid; and thus 
 acquires more acute angles. The same observa- 
 tion 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- 
 cious; and, amongst these, those have the least 
 capacity, whose angles arc most acute. 
 
 But curved surfaces, 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, tor it treats only 
 of the boundaries of things, and by angles the ul- 
 timate 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 signi-? 
 fication of the term, arc the source of all geome- 
 trical 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, 
 astronomy, &e. and the instruments generally 
 used for this purpose, are quadrants, sextants, the- 
 odolites, cireumf'erentors, &g. as described in the 
 following pages. It is necessary for the learner 
 first to be acquainted with the names and uses 
 pf the drawing instruments; which are as follow.
 
 10 DRAWING INSTRUMENTS. 
 
 OF MATHEMATICAL 
 
 DRAWING INSTRUMENTS. 
 
 Couimon Names of the frhicipal Insinuneiils, as 
 represented in Plates \, 2, and 3. 
 
 Plate \fjig. Aj is a pair of proportional com- 
 passes, without an adjusting screw. 
 
 B, a pair of best drawing compasses; /', the 
 plain point with a joint; c, the ink point; r/, the 
 dotting point; e, the pencil or crayon point; 
 PQ, additional pieces fitting into the place of 
 the moveable point /', and to which the other parts 
 are fitted. 
 
 Fj 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; ///a screw to one 
 of the legs thereof, which acts like the spring leg 
 of the hair compasses. 
 
 L, the hair compasses ; ;/, the screw that acts 
 upon the spring leg. 
 
 I K, the drawing-pen; I, the upper part; /', the 
 protracting pin thereof; K, the lov\'er, or pen part. 
 
 N, a pair of triangular compasses. 
 ■ V, a pair of portable compasses which contains 
 the ink and pencil points within its two legs. 
 
 O, the feeder and tracing point. 
 
 R, a pair of bisecting compasses, called wholes 
 and halves. 
 
 S, a small protracting pin. 
 
 T, a knife, screw-driver, and key, in one piece. 
 
 Plate 1,fig. A;, the common parallel rule. 
 
 -B_, the double barred ditto.
 
 DRAWING INSTRUMENTS. 11 
 
 C, the improved double barred parallel rule. > 
 
 D, the cross barred parallel rule. Of these' 
 rules, that figured at C is the most pcrfeeL 
 
 E, Eekhardts, or the rolling parallel rule. 
 F G H, the reetangular parallel rule. 
 
 I K L, the protraeting parallel rule. 
 
 M N O, Haywood's parallel rule. 
 
 Plate :i,fg. 1, the German parallel rule. 
 
 Fig. 2, a somieireular protraetor; fg. 2, a reet- 
 angular ditto. 
 
 Fig. '1 and 5, the two f:iccs of a sector. 
 
 Fig. (J, Jackson's parallel rule. 
 
 Fig. 7 and 8, two views of a pair of proporti- 
 onable compasses, with an adjusting screw. ■ 
 
 Fig. 9, a pair bf sectoral compasses. In this in- 
 strument are combined the sector, beam elliptical, 
 and calliper compasses; /z]^. Q, a, the sc[uare for 
 ellipses; /r, the points to work therein; d c, the 
 calliper points. 
 
 Fig. 10, a pair of beam c6mpa?sc.-. 
 
 Fig. 1 1, Sisson's protracting- scale. 
 
 Fig. 12, improved triangular compa.-.-c>. 
 
 Fig. 13, a pair of small compasses with a beam 
 and micrometer. 
 
 The strictness of geometrical dciiK/i.-ii-iiuii 
 admits 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 instruments became necessary, some ,to an- 
 swer peculiar purposes, some to facilitate opera- 
 tion, and others to promote accuracy. 
 
 It is the business of this work to describe these 
 instruments, and explain their various uses. In 
 performing this. task, a dii^ieulty arose relative to 
 the arrangement of the subject, whether- each 
 instrument, with its application,- should be de-
 
 11 DRAWING INSTRUMENTS. 
 
 scribed separately, or whether the description 
 should be iiitrodvieed under those problems, for 
 whose performance they were peculiarly designed. 
 After some consideration, I determined to adopt 
 neither rigidly, but to use either the one or the 
 other, as they appeared to answer best the pur- 
 poses of science. 
 
 As almost every artist, whose operations are 
 •connected with mathematical designing, furnishes 
 himself with a case of drawing instruments suited 
 to his peculiar purposes, they are fitted up in va- 
 rious 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, aud two spare points, which 
 may be applied accasionally to the compasses; one 
 fef 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 
 following 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 compasses, 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 cojnpasses, 
 
 A sector. 
 "■' 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
 
 DRAWING IXSTllUMEXTS. 13 
 
 fccommenclcd, 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 arc tho.-e of six inches; instru- 
 ments are seldom made longer^ but often smaller. 
 Those of SIX inches are, however, to be preferred, 
 in general, before any other size; they will effect 
 all that can be performed with the shorter ones, 
 while, at the same time, they are better adapted to 
 large work. 
 
 Large collections ate called, magazine cases of 
 instruments; these generally contain 
 
 A pair of six inch compasses with a moveable 
 leg, an ink point, a dotting point, the crayon 
 point, so contrived as to hold a whole pencil, two 
 additional pieces to lengthen occasionally one leg 
 of the compasses, 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 pointril. 
 
 A pair of small hair compasses, with a head si- 
 milar to those of the bow compasses. 
 
 A knife, a file, key, and screw-driver for the 
 compasses, in one piece. 
 
 A small set of fine water colours. 
 
 To these some of the following instruments arc 
 often added.
 
 14 DRAWING COMTAsSES. 
 
 ' A pair of beam compasses. 
 
 A pair of gunners callipers. 
 
 A pair of elliptical c,ompasscs. 
 
 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 center at 
 a great distance. 
 
 A protractor and parp.Ilel rule, such as is repre- 
 sented v.tj/o-, I K L, fic/te 1. 
 
 One or more of the parallel rules represented, 
 plate 2. 
 
 A pantographer. 
 
 A pair of sectoral compasses, forming, at the 
 same time, a pair of beam and calliper compasses. 
 
 OP DRAWING COMPASSES. 
 
 Compasses are made cither of silver or brass, 
 but with steel points. The joints should always 
 be framed of different substances; thus, one side, 
 cr part, should be of silver or brass, and the other 
 of steel. The diifcrenee in the texture and pores 
 of the two metnls causes the parts to adhere less 
 together, diminishes the v/ear, and promotes uni- 
 formity 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 imperfection. Tlie points sliould 
 l^e of steel, so tempered, as neither to be easily 
 beat or blunted; not too fine and tapering, and yet 
 meeting closely when the compasses arc shut. 
 
 As an instrument of art, compasses arc 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 Q-ivcn 
 spaces, and describe arches and circles.
 
 DRAWING COMPASSES. 15 
 
 If the arch or circle is to be described obscurely, 
 the steel points arc best adapted to the purpose; 
 if it is to be in ink or black lead., either the cLraw- 
 ing pen, or crayon points are to be used. 
 
 To use a pair of con) fusses. Place the thumb and 
 middle fmger of the right hand in ^thc opposite . ^ 
 hollows ill the shanks of the. compasses, theii press .'"li2X<^ 
 the compasses, arid'thc legs will c;pen a little way; 
 this being done, push the inncrniobt leg with the 
 third finger, elevating, at the' same tirn(^, the fur- 
 thermost, with the nail of the middtc finger^ till 
 the compasses are suflicicntly opened to receive 
 the middle and third finger; they may then be ex- 
 tended at pleasure, by pushing the furthermost leg 
 out^\'ards with the middle, or pres.'^ing it inwards 
 with the fore finger. In describing circles, or 
 arches, set one foot of the com})asics on the cen- 
 ter, and then roll the head of the compasses be- 
 tween the middle and fore finger, the other point 
 pressing at tlie same time upon the paper. They 
 shoidd be held, as upright as possible, and care 
 should be taken not to press forcibly upcn them, 
 bat rather to let them act by their own weight; 
 the legs should never be so far extended, as to tbrm 
 an obtuse angle with the paper or plane, on which 
 they are used. 
 
 I'hc ink and crayon points liave a joint just un- 
 der that part which tits into tlic compasses, by this 
 tlicy may be always ?o placed, as to be set nearly 
 perpendicular to the paper; the end of the shank, 
 of the best compasses is fram'6d so as to form a 
 strong spring, to bind firmly the moveable points, 
 and prevent them from shaking, Thi^ is found 
 to be a more effectual method than that by a scrc^w 
 
 Fig. B, plate 1, reprcsbnts a pair of t]:!e best 
 compasses, with the plain point, t:, the ink, d, the 
 dotting, c^ the crayon point.
 
 id Drawing compassEI. 
 
 In srhall cases, the crayon and ink points are 
 joined by a middle piece, with a socket at each 
 end to receive the points, which, by this means, 
 only occupy one place in the case. 
 
 Two additional pieces, Jig. P, Q, plate 1, are 
 often applied to these compasses; these, by length- 
 ening the le^ Aj enable them tq strike larger cir- 
 cles, or measure greater extents, than they would 
 otherwise perform, and that without the inconve- 
 niences attending longer compasses. When com- 
 passes are furnished with this additional piece, the 
 moveable leg has a joint, as at h, that it may be 
 placed perpendicular to the paper. 
 
 Of the hah- compasses, Jig. L, plate 1 . They are / 
 so named, on account of a contrivance in the shank ^ 
 to set them with greater accuracy than can be ef- 
 fected by the motion of the joint alone. One of 
 the steel points is fastened near the top of the com- 
 passes, and may be moved very gradually by turn- 
 ing the screw n^ either backwards or forwards. 
 
 To use these crmipasses. 1. Place the leg, to 
 which thcscfew is annexed, outermost; 2. 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 turn- 
 ing the screw. 
 
 Of the bow compasses, fig. 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 onlv on account of their size, but also from the 
 shape of the head, which rolls with great ease be- * 
 tvvecn the fingers. It is, for this reason, custo- 
 mary to put into magazine cases of instruments, 
 a small pair of hair compasses, fig. Yi, plate 1, with a 
 head similar to the bows; these are principally
 
 DRAWING COMPASSES. ~ 17 
 
 used for repeating divisions of a small but equal 
 extent, a practice that has acquired the name of 
 stepping. 
 
 Of the drawing p 671 and protracting pin, fg. I K, 
 plate 1. The pcn^art of this instrument is used 
 to draw strait Hues; 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 sufficient cavity for the ink; the blades may 
 be opened more or less b}'- a screw, and, being 
 properly set, will draw a line of any assigned thick- 
 ness. 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 feed it 
 with ink, then regulate it to the tliickness o{ the 
 required line by the screw. In drawing lines, in- 
 cline the pen a small degree, taking care, however, 
 that the edges of both the blades touch the paper, 
 keeping the pen close to the rule and in the same 
 direction during the v/hole operation; the blades 
 should always be wiped very clean, before the pen 
 is put away. 
 
 These directions are equally applicable 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 ihe blades of the pen may be nearly perpen- 
 dicular to the paper, and both of them touch it at 
 the same time. 
 
 The protracting pin k, is onlv a short piece ({ 
 5teel wire, with a very fine point, fixed at one end 
 
 c
 
 18 DRAWING COMPASSES. 
 
 of the upper part of the handle of the drawing 
 pen. It is used to mark the intersection of Hues, 
 or to set off divisions from the plotting, scale^ and 
 protractor. 
 
 ' The feeder, fig. O, flate 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 tra- 
 cing point, as in the figure. It serves to place the 
 ink between the blades of the drawing pens, or to 
 pass between them when the ink does not flow 
 freely. The tracing point, or pointrel, is a pointed 
 piece of steel fitted to a brass handle; it is used 
 to draw obscure lines, 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 triangular compasses. A pair of these are re- 
 presented 'dXfig. 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. 
 
 These compasses, though exceedingly useful, 
 are but little known; they are very serviceable in 
 copying all kinds of drawings, as from two fixed 
 points they will always ascertain the exact position 
 of a third point. 
 
 Fig. \2, plate 3, represents another kind, which 
 has some advantages over the preceding. 1 . That 
 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 arid halves, fig. R, plate 1 . A name 
 given to these compasses, because that when 
 the longer legs are opened to any given line, the
 
 DRAWING COMPASSES. 1^ 
 
 d'iOrter ones will be opened to the half of thut 
 line; being always 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. 
 
 Proportional compasses. These compasses are of 
 two kinds, one plain, represented Jig. A, plate 1 ; 
 the other with an adjusting screw, of which there 
 are two views, one edgeways, 7?^. 8, plate 3, the 
 other in the front, 7?^. 7j plate '6-. 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 com- 
 passes, and the center is moveable, being con- 
 structed to slide with regularity in these grooves, 
 and when properly placed, is fixed by a nut ancj 
 screw; on one side of these grooves arc 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 scale for circles, a regular polygon may be in- 
 scribed in a circle, provided the sides do not ex- 
 ceed the numbers on the scale. Sometimes arc 
 added a scale for superficies and a scale for solids. 
 
 To divide a given line into a proposed number 
 (11) of equal parts. 1. Shut the compasses. 2. 
 Unscrew the milled nut, and move the slider un- 
 til the line across it coincide with the 11th divi- 
 sion on the scale. 3. Tighten the scre\v, that the 
 slider may be immoveable. 4. Open the com- 
 passesj so that the longer points may take in cx- 
 
 c 2
 
 90 DRAWING COMPASSES. 
 
 actly the given line, and the shorter will give you 
 Trth of that line. 
 
 To inscrlhe in a circle a regular 'polygon of \1 
 sides. 1. Shut the compasses. 2. Unscrew the 
 milled nut, and set the division on the slider to 
 coincide with the l^th division on the scale of 
 circles. 3. Fasten the milled nut. 4. Open the 
 compasses, so that the longer legs may take the ra- 
 dius, and the distance between the shorter legs 
 •will be the side of the required polygon. 
 
 To use the proportional compasses with an adjust- 
 ing screw. The application being exactly the same 
 as the simple one, we have nothing more to de- 
 scribe than the use and advantage of the adjusting 
 screw. 1 . Shut the legs close, slacken the screws 
 of the nuts g 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 adjuster h, the mark on the slider k 
 may be brought exactly to the division : screw fast 
 the nut^. 2. Open the compasses; gently lift 
 the end e of the screw of the nut /out of the hole 
 in the bottom of the nut ^; move the beam round 
 its pillar a, and slip the point e into the hole in 
 the pin ?/, which is fixed to the under leg ; slacken 
 the screw of the nut/; take the given line between 
 the longer points of the compasses, and screw fast 
 the nut/; then may the shorter points of the com- 
 passes be used, without any danger of the legs 
 changing their position; this being one of the in- 
 conveniences that attended the proportional com-< 
 passes, before this ingenious contrivance. 
 
 Fig. 10, plate 3, represents a pair of beam com-^ 
 passes ; they are used for taking off and transfer- 
 ring divisions from a diagonal or nonius scale, de- 
 scribing large arches, and bisecting lines or arches. 
 It is the instrument upon which Sir. Bird princi*
 
 PARALLEL RULES. 21 
 
 pally depended, in dividing those instruments, 
 whose accuraey 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 ad- 
 justing 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 extreme 
 regularity and exactness, even less than the thou- 
 sandth part of an inch. 
 
 Fig. 13, plate 3, is a small pair of beam com-* 
 passes, with a micrometer and adjusting screw, for 
 accurately ascertaining and laying down small dis- 
 tances. 
 
 Fig. 11, plate 3, represents a scale of equal parts ^ 
 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 index can'ies 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 mea- 
 sured on land. 
 
 OP PARALLEL RULES. 
 
 Parallel lines occur so. continually in every spe- 
 cies of mathematical drawing, that it is no wonder 
 so many instruments have been contrived to deli- 
 neate them w^ith more expedition than could be 
 effected by the general geometrical methods ; of
 
 ti'i PARALLEt RULES", 
 
 the various contrivances for this purpose, the {ot" 
 lowing are those most approved. 
 
 1 . T/ie common parallel nil e^ fig. A, flate 2. This 
 consists of two strait rules, which are connected 
 together, and always maintained in a parallel posi- 
 tion by the two equal and parallel bars, which 
 move very freely on the center, or rivets, by which 
 they are fastened to the strait rules. 
 
 2. The double parallel rule, fig. B, plate 2. This? 
 instrument is constructed exactly upon the same 
 principles as the foregoing, but with this advan- 
 tage, 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 former kind, they are always shift- 
 ing away from the ends of the fixed rule. 
 
 This instrument consists of two equal flat rules, 
 and a middle piece; they are connected together 
 by four brass bars, the ends of two bars are ri- 
 vetted on the middle line of one of the strait rules ; 
 the ends of the other two bars are rivetted on the 
 middle line of the other strait rule; the other ends: 
 of the brass bars are taken two and two, and rivet- 
 ted on the middle piece, as is evident from the 
 figure; it would be needless to observe, that the 
 brass bars itiove freely on their rivets, as so many 
 centers. 
 
 3. Of I lie improved doidde pai-allel rule, fig. Q, 
 plate 2. The motions of this rule are more regular 
 than those of the preceding one, but with some- 
 what more friction; its construction is evident 
 from the figure ; it was contrived by the ingenious 
 mechanic, Mr. Haywood. 
 
 4. The cross barred parallel rule, fig. D, plate 2« 
 In this, two strait rules are joined by two brass bars^ 
 which cross each other, and turn on their inter-
 
 t»ARALLEL RULES. 23 
 
 sfection as on a center; one end of each bar moves 
 on a center, the other sHdes in a groove, as the 
 lules recede from each other. 
 
 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 given to 
 draw a line parallel thereto by either of the fore- 
 going instruments : 
 
 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 gi- 
 ven point; and a line drawn along the edge 
 through the point, is the line required. 
 
 5. Of the rolling parallel rule. This 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 
 practice and attention to use it with success. 
 
 Fig. E. plate 2, represents this rule; it is a rec- 
 tangvdar parallelogram of black ebony, with slips 
 of ivory laid on the edges of the nde, and divided 
 into inches and tenths. The nde 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 cylinders are sometimes 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
 
 14 PARALLEL RULES. 
 
 the middle of the rule, that one end may not have? 
 a tendency to 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 fiat. 
 
 In using the rule with the rolling scales, to draw 
 a line parallel to a given line at any determined dis- 
 tance, adjust the edge of the ride to the given 
 line, and pressing the edge down, raise the wheels 
 a little fi'om the piiper, and you may turn the cy- 
 linders round, to bring the first division to the in- 
 dex ; then, if you move the rule towards you, look 
 at the ivory cylinder on the left hnnd, 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 numbers on the right 
 hand cylinder. 
 
 To raise a perpendicular from a given point on a 
 given line. Adjust the edge of the rule to the 
 line, placing any one of the divisions on the edge 
 of the rule to the given point; then roll the rule 
 to any distance, and make a dot or point on the 
 paper, at the same division 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 Vme. 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 perpendicular is to be draMai. 
 
 By this method of drawinsr perpendiculars, 
 squares and parallelograms may be easily drawn of 
 any dimensions. 
 
 To divide any given line into a numher of equal- 
 parts. Draw a right line from either of the ex-
 
 PARALLEL RULES. 25 
 
 treme 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 
 giv^en line is to be divided. Join the last point of 
 division, and the extreme point of the given line: 
 to that line draw parallel lines through the other 
 points of division, and they will divide the given 
 line into equal parts. 
 
 6. Of the square parallel rule. The evident ad- 
 vantages of the T square and drawing board over 
 every other kind of parallel rule, gave rise to a va- 
 riety 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 peculiarly applicable to 
 the mode of plotting recommended by Mr. Gale.* 
 Its use, as a rule for drawing parallel lines at given 
 distances from each other, for raising perpendicu- 
 lars, 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 evi- 
 dent, that it will plot with as much accuracy as 
 the beam, fig. 1 1 , plate o. 
 
 Fig. FGH, plate 2, represents this instrument; 
 the two ends F G are lov/er thou the rest of the 
 rule, that weights may be laid on them to steady 
 the rule, when both h;^nds are w;vnted. The twa 
 rules are fastened together .by the brass ends, the 
 frame is made to slide regularly between the two- 
 rules, carr\ ing at the same cime, the rule 11 'u a po- 
 sition at right angles to the edges of the rule E G,
 
 26 t-ARALLEL tlULE§. 
 
 There are slits in the frame a, b, with marks iiy 
 coincide with the respective scales on the rules, 
 while the frame is moving up or down: c is a 
 point for pricking off with certainty divisions 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 instru- 
 ment has this further advantage, that if the dis- 
 tanceSj 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 strait rule^ 
 by applying the perpendicular part against it. 
 
 7. -Fig. \}L1j, plate 1, represents the, protract- 
 ing parallel rule ; the uses of this in drawing pa- 
 rallel lines in different directions, are so evident 
 from an inspection of the figure, as to render a 
 particular description unnecessary. It answers all 
 the purposes of the T square and bevil, and is pe- 
 culiarly useful to surveyors for plotting and pro- 
 tracting, which will be seen, when we come to 
 treat of those branches. 
 
 M N O is a parallel rule upon the same princi- 
 ple 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 convenience of the purchaser. 
 Each of the rules MN turns upon a center; its 
 use as a parallel rule is evident 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; scrved- 
 for laying down divisions and angles with peculiar 
 accuracy; answers as a square, or bevil; indeed, 
 gcarce any artist can use it, without reaping con- 
 siderable advantage from it, and finding uses pe- 
 culiar to his own line of business.
 
 PARALLEL RULES. TJ 
 
 8. Of the German parallel rule, fig. 1, flate 3^ 
 This was, probably, one of the first instruments in- 
 vented to tacihtate the drawing of parallel lines. 
 It has, however, only been introduced within these 
 few years among the English artists; and, as this 
 introduction probably came from some German 
 work, it has thence acquired its name. It con- 
 sists of a square and a strait rule, the edge of the 
 square is moved in use by one hand against the 
 rule, which is kept steady by the other, the edge 
 having been previously set to the given line: its 
 use and construction is obvious {xoxnfig. 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 Tottenham; the other 
 by Mr. Marquois, of London, 
 
 Mr. Jackson s, fig. 6, plate 3, consists of two 
 equal triangular pieces of brass, ivor\ , or wood, 
 AB C, D E F, right angled at B and E; the edges 
 AB and AC are divided into any convenient num- 
 ber of equal parts, the divisions in each equal; 
 B C and D E are divided into the same number of 
 equal parts as AC, one side of DEE may be di- 
 vided as a protractor. 
 
 To draw a line G H, jfo-. V, plate 2, parallel to a 
 given line, through a point P, or at a given distance.. 
 
 Place the edge DE upon the given line IK, and 
 let the instrument form a rectangle, then slide the 
 upper piece till it come to the given point or dis- 
 tance, keeping the other steady with the left hand, 
 and draw the line G H. By moving the piece 
 equal distances by the scale on B C, any number 
 of equidistant parallels may be obtained. 
 
 If the distance, T^V. ^, plate 2, between the gi- 
 ven and required lines be considerable, place A B 
 upon I K, and E F against A C, then slide the 
 pieces alternately till D E comes to the required
 
 2$ * PARALLEL RULES. 
 
 point; in this manner it is easy also to construct 
 any square or rectangle^ he. 
 
 From any grceti point or angle V, Jig. S, plate 2, 
 to let fall or raise a perpendicular oti a givm li?ie» 
 Place either of the edges AC upon GP, and slide 
 AB upon it, till it comes to the point P, and 
 draw PH. 
 
 To divide a line into any proposed number of 
 equal pnrts, fig. T, plate 2. Find the proposed 
 number in the scale BC, 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 B C, and make a dot at I, 
 where DE cuts the line for the first part; then 
 move one or more divisions as before, make a se- 
 cond dot, and thus proceed till the whole be com- 
 pleted. 
 
 Of Marquois^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 hypothenusc that these scales derive 
 their peculiar advantages, and it is this alone that 
 renders them different from the common German 
 parallel rule: for this we are much indebted to 
 Mr. Marquois. 
 
 A¥hat 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 excel- 
 lence. On each edge of the rectangular rule are 
 placed two scales, one close to the a^gc, the other 
 within this. The outer scale, Mr. Marquois terms 
 the artificial s cale, the inner one, the natural scale: 
 the divisions "on the outer are always^rce times 
 longer than those on the inner scale, as, to derive
 
 TAPvALLEL RULES. 
 
 29 
 
 any advantage from this invention, they must al- 
 ways bear the same proportion to each other, that 
 the shortest side of the right angled triangle does 
 to the longest. The triangle 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 coin- 
 cide with the O division of the scales; the num- 
 bers 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: l. That 
 the sight is greatly assisted, as the divisions on the 
 outer scale are so much larger than those of the 
 inner one, and yet answer the same pui'pose, for 
 the edge of the right angled triangle only moves 
 through one third of the space passed over by the 
 index. 2. That it promotes accuracy, for all error 
 in the setting of the index, or triangle, is dimi- 
 nished one third. 
 
 Mr. Marquois recommends the young student 
 to procure two rules of about two feet long, having 
 one of the edges divided into inches and tenths, 
 and several triangles with their hypothenuse in dif- 
 ferent proportions to their respective perpendicu- 
 lars. Thus, if you would make it answer for a 
 scale of twenty to an inch, the hypothenuse must 
 be twice the length of the perpendicular; if a scale 
 of 30 be required, three times; of 40, four times; 
 of 50, five times, and so on. Thus also for inter- 
 mediate proportions; if a scale of 25 is wanted, 
 the hypothenuse must be in the proportion of 25 
 to 2 ; if 35, of 7 to 42, &c. Or a triangle may be 
 formed, in which the hypothenuse may be so set 
 as to bear any required proportion with the per- 
 pendicular.
 
 30 THE PROTRACTOR. 
 
 OF THE PROTRACTOR. 
 
 This Is an instrument used to protract, or lay 
 down an angle containing any number of degrees, 
 or to find how many degrees are contained in any 
 given angle. There are two kinds put into cases 
 of mathematical drawing instruments; one in the 
 form of a semicircle, the other in the form of a pa- 
 rallelogram. The circle is undoubtedly the only 
 natural measure of angles; when a strait line is 
 therefore used, the divisions thereon are derived 
 from a circle, or its properties, and the strait line 
 is made use of for some relative convenience: it is 
 thus the parallelogram is often used as a protrac- 
 tor, instead of the semicircle, because it is in some 
 cases more convenient, and that other scales, &c. 
 may be placed upon it. 
 
 The semicircular 'protractor, fig. 1, plate 3, is di- 
 vided into 180 equal parts or degrees, which are 
 numbered at every tenth degree each way, for the 
 conveniency of reckoning either from the right 
 towards the left, or from the left towards the right; 
 or the more easily to lay down an angle from 
 cither end of the line, beginning at each end with 
 10, 20, &c. and proceeding to 180 degi-ees. The 
 edge is the diameter of the semicircle, and the 
 mark in the middle points out the center. Fig. 3, 
 plate 3, is a protractor in the form of a parallelo- 
 gram: the divisions arc here, as in the semicircular 
 one, numbered both ways; the blank side repre- 
 sents the diameter of a circle. The side of the 
 protractor to be applied to the paper is made flat, 
 and that whereon the degrees are marked, is cham- 
 fered or sloped away to the edge, that an angle 
 may be more easily measured, and the divisions set 
 off with greater exactness.
 
 THE PROTRACTOR. 31 
 
 Profraclors of horn arc, from their transparency, 
 very convenient in measuring angles, and raising 
 perpendiculars. When they are out of use, they 
 should be kept in a book to prevent their warping. 
 
 Upon some protractors the numbers denoting 
 the angle for regular polygons are laid down, to 
 avoid the trouble of a reference to a table, or the 
 operation of dividing; thus, the number 5, for a 
 pentagon is set against 72°; the number 6, for a 
 hexagon, against ()0°; the number 7:, for a hepta- 
 gon, against 51i°; and so on. 
 
 Protractors for the purposes of surveying will be 
 described in their proper place. 
 
 Application of the protractor to use. 1 . A number 
 of degrees being given, to protract, or lay doivn an 
 angle, whose measure shall be'fqual thereto. 
 
 Thus, to lay down an angle of 6o degrees from 
 the point \ of the line AB,Jjg. 14, plate 3, apply 
 .the diameter of the protractor to the line AB, so 
 that the center thereof may coincide exactly with 
 the point A; then with a protracting pin make 
 a fine dot at C against Oo upon the limb of the 
 protractor; now remove the protractor, and draw 
 a line from A through the point C, and the angle 
 CAB contains the given number of degrees. 
 
 2. To fml the number of degrees contained in a 
 o-iven anole BAC. 
 
 o o 
 
 Place the center of the protractor upon the an- 
 gular point A, and the fiducial edge, or diameter, 
 exactly upon the line AB; then the degree upon 
 the limb that is cut by the line C A will be the 
 measure of the given angle, which, in the present 
 instance, is found to be do degrees. 
 
 3. From a given point A, /// (he line AB, to erect 
 a perpendicular to that line. 
 
 Apply the protractor to the line AB, so that the 
 center may coincide with the point A, and the di-
 
 32 THE PLAIN SCALE. 
 
 vision marked go maybe cut by the line A B, then 
 a line DA drawn against the diameter of the pro- 
 tractor will be perpendicular to A B. 
 
 Further uses of this instrument will occur in 
 most parts of this work, particularly its use in con- 
 structing 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 un- 
 necessarily swelled by a tedious detail and conti- 
 nual 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 merely concerned with dividing strait 
 lines. 
 
 It has been already observed, that though arches 
 of a circle are the most natural measures of an an- 
 gle, 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 o^ s'mes 
 marked sin. of tangents marked tan. q{ semltan^ 
 tangents 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 SCALEi 33t 
 
 There are two other scales, namely, the rhumhs, 
 marked ru. and long, marked lon. Scales of* 
 latitude and hours are sometimes put upon the 
 plain scale; but, as dialling is now but very little 
 studied, they are only made to order. 
 
 The divisions used for measuring strait lines are 
 called scales of equal 'parts, and are of various 
 lengths for the convenience of delineating any fi- 
 gure of a large or smaller size, according to the 
 fancy or purposes of the draughts-man. They arc, 
 indeed, nothing more than a measure in miniature 
 for laying down upon paper, &c. any known mea- 
 sure, as chains, yards, feet, &c. each pajl 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 pro- 
 portion 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 diiFerent openings of that 'in- 
 strument, a line of any length may be divided into 
 as many equal parts as any person chuscs. 
 
 Scales of equal parts are divided into two kinds, 
 the one simple, the other diagonally divided. 
 
 Six of the simply divided scales arc 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 sometimes shortened for the sake of in- 
 troducing another, called the line of chords. 
 
 The first of the larger, or primary divisions, on 
 every scale is subdivided into 10 equal parts, which 
 small parts are those which give a name to th«
 
 M TUB PLAIN SCALE, 
 
 scale; thus it is called a scale of 20, when 10 of 
 these divisions are equal to one inch. If, there- 
 fore, these lesser divisions be taken as units, and 
 each represents 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 41-, 
 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 -fJ, 36, or 36o miles or leagues, &c. and 
 bear the same proportion in the plan as the line 
 measured docs to the thing represented. 
 
 To adapt these scales to feet and inches, the 
 first primary division is often duodeeimally divided 
 by an upper line; therefore, to lay down any num- 
 ber 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 accu- 
 rately divided than by the former; for any one of 
 the larger divisions may by this be subdivided into 
 300 equal parts; and, therefore^ if the scale con- 
 tains 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-
 
 THE PLAIN SCALE. 35 
 
 equal parts, and at the opposite end there Is usu- 
 ally half an inch divided into 100 equal parts. If 
 the scale be six inches long, one end has com- 
 monly 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 
 purpose : 
 
 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 to contain; from each of these divi- 
 sions draw perpendicular lines through the eleven 
 parallels. 
 
 Secondly. Subdivide the first of these divisions in- 
 to ten equal parts, both in the upper and lower lines. 
 
 Thirdly. Subdivide again each of these subdi- 
 visions, by drawing diagonal lines from the 10th 
 below to the gth above; from the 8th 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, conse- 
 quently, 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 ca^h parallel 
 above it is one tenth of such subdivision, 'and, con- 
 sequently, one hundreth part of the whole first 
 space ; and if there be ten of the larger divisions, 
 one thousandth part of the whole space. 
 
 li] therefore, the larger divisions be accounted 
 as units, the first subdivisions will be tenth parts 
 of an unit, and the second, marked by the diago- 
 nal upon the parallels, hundredth parts of the 
 unit. But, if we suppose the larger divisions to 
 be tens, the first subdivisions will be units, and the 
 second tenths. If the larger are hundreds, then 
 \Aill the first be tens, and the second units. 
 
 D 2
 
 36 THE PLAIN SCALE. 
 
 The numbers, therefore, 576, 57,6, 5,76, nre 
 all expressible by the same extent of the compasses : 
 thus, setting one foot in the number five of the 
 larger divisions, extend the other along the sixth 
 parallel 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 of 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 I he scales of equal parts. Though 
 what I have already said on this head may be 
 deemed sufficient, I shall not scruple to introduce 
 a few more examples, in order to render the young- 
 practitioner more perfect in the management of 
 an instrument, that will be continually occurring 
 to him in practical geometry. He will have al- 
 ready observed, that by scales of equal parts lines 
 may be laid down, or geometrical figures con- 
 fitructed, whose right-lined sides shall be in the 
 same proportion as any given numbers. 
 
 Example 1. To take off the number 4,JQ 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 2. To take off the nnmher 76,4. Ob- 
 serve the points where the sixth horizontal cuts
 
 THE PLAIN SCALE. 37 
 
 tlie seventh vertical and fourth diagonal line, the 
 extent between these points will represent the 
 number 76,4. 
 
 In the first example each primary division is 
 taken for one, in the second it is taken for ten. 
 
 Exa tuple 3. To lay dovjn a line of 7,85 chains 
 hy the dlagojial scale. Set one point of your 
 compasses where the eighth parallel, counting up- 
 wards, cuts the seventh vertical line; and extend 
 the other point to the intersection of the same 
 eighth parallel with the fifth diagonal. Set off the 
 iixtent 7j85 thus found on the line. 
 
 Rxaniple 4. To measure hy the diagonal scale 
 ^ line that is already laid doivn. Take the extent 
 of the line in your compasses, 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. Thus, 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 hav^e 12 chains, 48 links, 
 or 12,48 chains. 
 
 Example 5. Three adjacent parts of any right- 
 lined triangle being given, to form the plan thereof. 
 Thus, suppose the base of a triangle, fig. \b, plate 3^ 
 40 chains, the angle ABC equal 36 deg. and angle 
 BAG equal 4 1 deg. 
 
 Draw the line AB, and from any of the 
 scales of equal parts take off 40, and set it 
 on the same line from A to B for the base of 
 the triangle; at the points A, B, make the angle 
 ABC equal to 36 degrees, and BAC to 41, and 
 the triangle will be formed; then take in your
 
 3S THE PROTRACTING SCALES, 
 
 compasses the length of the side AC, 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 pro- 
 tractor will shew that the angle ACB contains 
 103 degrees. 
 
 Example Q. Given the hase AB, Jig. l6,pJa/e3, 
 of a triangle 2,1"] yards, the angle CAB 44°, 30', 
 and the side AC 20d, yards, to constitute the said tri- 
 angle, and find the length of the other sides. 
 
 Draw the line AB at pleasure, then take 327 
 parts fi'om the scale, and lay it fi-om A to B ; set 
 the center of the protractor to the point A, lay 
 ofF44°30', and by that mark draw AC; 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 PROTRACTING 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 KC,fig. 14, plate 3, that 
 shall make with the line AB an angle, contain- 
 ing a given number of degrees, suppose 40 de- 
 grees. 
 
 Open your compasses to the extent of 6o de- 
 grees upon the line of chords_, (which is always
 
 THE PROTRACTING SCALES. 3g 
 
 equal to the radius of the circle of projection,) 
 and setting one foot in the angular point, with 
 that extent describe an arch; then taking the ex- 
 tent of 40 degrees from the said chord hne, 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, arc found nearly in the same manner; for 
 instance, to measure the angle CAB. From the 
 center describe an arch with the chord of 6o de- 
 grees, and the distance C B, measured on the 
 chords, will give the number of degrees contained 
 in the angle. 
 
 If the number of degrees arc more than 90, 
 they must be measured upon the chords at twice: 
 thus, if 120 degrees were to be practised, 60 may 
 be taken from the chords, and those degrees be 
 laid off twice upon the arch. Degrees taken from 
 the chords are always to be counted from the be- 
 ginning of the scale. 
 
 Of the rhumb line. This is, in. fact, a line of 
 chords construeted to a quadrant divided into 
 eight parts or points of the compass, in order to 
 facilitate the work of the navigator in laying down 
 a sliip's course. 
 
 Tlius, supposing the ship's course to be N N E, 
 and it be required to lay down that angle. Draw- 
 the lineAB,jf«-. \A^ plate 3, to represent the me- 
 ridian, and about the point A sweep an arch with 
 the chord of 60 degrees; then take the extent to 
 the second 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 BAe will represent the 
 chip's course. The second rhumb was taken, be ■ 
 cause NNE is the second point from the north.
 
 40 THE SECTOR.' 
 
 Of the Vine of longitudes. The line of longi- 
 tudes is a line divided into sixty unequal parts, 
 and so applied to the line of chords, as to shew 
 by. inspection, the number of equatorial miles 
 contained in a degree on any parallel of latitude. 
 The graduated line of chords is necessary, in or- 
 der to shew the latitudes; the line of longitude 
 shews the quantity of a degree on each parallel in 
 sixtieth parts of an equatorial 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 44i, nearest equal to the 
 latitude on the line of chords, stands 43 on the line 
 of longitude, which is therefore the number of miles 
 in a degree of longitude in that latitude. Whence 
 as 43 : 6o : : 79 : 110 miles the difference of Ion-, 
 gitude. 
 
 The lines of tangents, semitangents, and secants 
 serve to find the centers and poles of projected 
 circles in the stereographical projection of the 
 sphere. 
 
 The Vine 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 
 explanation of those lines in our description of 
 that instrument. 
 
 The lines of latitudes and hours are used con-r 
 jointly, and serve very readily to mark the hour 
 lines in the construction of dials; they are gene- 
 rally on the most complete sorts of scales and see- 
 tors; for the uses of which see treatises on dialling, 
 
 OP THE SECTOR. 
 
 Amidst the variety of mathematical instruments 
 that have been contrived to facilitate the art of
 
 THE SECTOR. 41 
 
 (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 
 unknown ; yet of so much merit has the invention 
 appeared, that it was claimed by Galileo, and dis- 
 puted by nations. 
 
 This instrument derives its name from the tentli 
 definition of the third book of Euclid, where he 
 defines the sector of a circle. It is formed of two 
 equal rules, (Jig. 4 and 5, plate 3,) AB, DB, 
 called legs; these legs are moveable about the 
 center C of a joint d c f, and will, consequently, 
 by their different openings, represent every possi- 
 ble 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 diffe- 
 rences little more than in the number of subdi- 
 visions, it is to be presumed that no difficulty 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 
 ^mall sector in the common pocket cases of instru-
 
 4''3 THE SIECTOR, 
 
 ments, 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 ad- 
 vise a person to be sparing of expense in procur- 
 ing a very good instrument, the uses of which are 
 so various and important." 
 
 The scales, or lines graduated upon the/aces of 
 the instrument, and which are to be used as sec- 
 toral lines, proceed from the center; and are, 
 
 1. Two scales of equal parts, one on each leg, 
 marked lin. or l. each of these scales, 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. of s. A line 
 of polygons, marked pol. Upon the other face, 
 the sectoral lines are, 1 . Two lines of sines, marked 
 sin. or s. 2. Two lines of tangents, marked tan. 
 3. Between the lines of tangents and sines, there is 
 another line of tangents to a lesser radius, to sup- 
 ply 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 center, and consequently at whatever distance 
 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 Qo 
 and 6o on the line of chords, ()0 and 90 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, j^^. 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
 
 THE SECTOR. 43 
 
 chords. On the face, fg. A, three logarlthmici 
 scales, namely, one of numbers, one of sines, and 
 one of tangents; these arc used when the sector is 
 fully opened, the legs forming one line; their 
 use will be explained when wc treat of trigono- 
 metry. 
 
 To rend and estlmale the drc'isions on the sectoral 
 lines. The value of the divisions on most of the 
 lines are dctennined by the figures adjacent to 
 them; these proceed by tens, which constitute the 
 div^isions of the first order, and are numbered ac- 
 cordingly; but the value of the div^isions 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 figures 1, 2, 3,4, &c. 
 may denote cither 10, 20, 30, 40; or 100, 200, 
 300, 400, and so on. 
 
 The line of Tines is divided into ten equal parts, 
 numbered 1, 2, 3, to 10; these may be called di- 
 visions of the first order; each of these are again 
 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 
 divisions 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 in- 
 termediate parallel. 
 
 When the whole line of lines repre?'::nts 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- 
 sent ten, then the divisions of the first order are 
 units; those of the second, tenths, and the thirds, 
 twentieths.
 
 44 ^HE FOUNDATION 
 
 Jn the Tine of tangents, the divisions to which the 
 numbers are affixed, are the degrees expressed by 
 those numbers. Evxry fifth degree is denoted by 
 a hne somewhat longer than the rest; between 
 every number and each fifth degree, there are four 
 divisions, longer than the intermediate adjacent 
 ones, these are whole degrees ; the shorter ones^ 
 or those of the third order, are 30 minutes. 
 
 From the center, to 60 degrees, the line of sines 
 is divided like the line of tangents; from 60 to 70, 
 it is divided only to every degree; from 70 to 80, 
 to every two degrees ; from 80 to 90, the division 
 jnust be estimated by the eye. 
 
 The divisions on the line of chords are to be esti- 
 mated in the same manner as the tangents. 
 
 The lesser line of tangents is graduated every two 
 degrees from 45 to 50; but from 50 to 60, to 
 every degree; from 60 to the end, to half degrees. 
 
 The line of secants from O to 10, is to be esti- 
 mated by the eye; from 20 to 50 it is divided to 
 every two degrees; from 50 to 60, to every degree ; 
 and from 60 to the end, to every half degree. 
 
 OF THE GENERAL LAW OR FOUNDATION OP 
 SECTORAL LINES. 
 
 Let C A, C B, fig. 17, plate 3, represent a pair 
 of sectoral lines, (ex. gr. those of the line of lines,) 
 forming the angle AC B; divide each of these lines 
 into four equal parts, in the points H, D, F ; 
 I, E, G; draw the lines H I, D E, F G, AB. Then 
 because C A, C B, are equal, their sections are also 
 equal, the triangles are equiangular, having a com- 
 mon angle at C, and equal angles at the base; and 
 therefore, the sides about the equal angles will be 
 jM'oportional ; for as CH to CA, so is H I to AB» 
 and, therefore, as C A to C H, so is AB to H I, and,
 
 OP SECTORAL LINES. 45 
 
 consequently, as C H to H I, so is C A to AB ; and 
 thence if CH be one fourth of CA, 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 aijid equi- 
 angled; isosceles, because the pairs of sectoral 
 lines are equal by construction; equiangled, be- 
 cause of the common angle at the center; the 
 sides encompassing the equal angles are, there- 
 fore, proportional. 
 
 Hence also, if the lines CA, C B, represent the 
 line of chords, sines, or tangents; that is, if CA, 
 AB be the radius, and the Hue CF 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 degrees, to the ra- 
 dius AB. 
 
 OP THE GENERAL MODE OP USING- 
 SECTORAL LINES. 
 
 It is necessary to explain, in this place, a few 
 terms, either used by other writers in their de- 
 scription of the sector, or such as we may occasi- 
 onally use ourselves. 
 
 The solution of questions on the sector is said to 
 Ixj simple^ when the work is begun and ended on 
 the same line; compound^ when the operation be- 
 gins on one line, and is finished on the other. 
 
 The operation varies also by the manner in 
 which the compasses are applied to the sector. 
 If a measure be taken on any of the sectoral lines, 
 beginning at the center, it is called a lateral dis- 
 tance. But if the measure be taken from any point 
 in one line, to its corresponding point on the line 
 of the same denomination, on the other leg, it is 
 called a transverse or parallel distance*
 
 46 THE MODE OP USING 
 
 The divisions of each sectoral line are bounded 
 by three parallel lines; the innermost of these if5 
 that on which the points of the compasses are to 
 be placed, because this alone is the line which 
 goes to the center, and is alone^ therefore, the sec- 
 toral line. 
 
 We shall now proceed to give a few general in- 
 stances of the manner of operating with the sector, 
 and then proceed to practical geometry, exempli- 
 fying its use further in the progress of the work, 
 as occasion offers. 
 
 Alult'ipli cation by the line of lines. Make the 
 lateral distance of one of the factors the parallel 
 distance of 10; then the parallel distance of the 
 other factor is the product. 
 
 Exaniple. Multiply 5 by 6, extend the com- 
 passes from the center of the sector to 5 on the 
 primary divisions, and open the sector till this dis- 
 tance become the parallel distance from 10 to 10 
 on the same divisions; then the parallel distance 
 from 6 to 6, extended from the center of the see- 
 tor, shall reach to 3, which is now to be reckoned 
 30. At the same opening of the sector, the pa- 
 rallel distance of 7 shall reach from the center to 
 35, that of 8 shall reach from the center to 40, &c. 
 
 Division by the line of lines. Make the lateral 
 distance of the dividend the parallel distance of 
 the divisor, the parallel distance of 10 is the quo- 
 tient. Thus, to divide 30 by 5, make the lateral 
 distance of 30, viz. 3 on the primary divisions, the 
 parallel distance of 5 of the same divisions; then 
 the parallel distance of 10, extended from the 
 center, shall reach to 6. 
 
 Proportion by the line of Vines. Make the lateral 
 distance of the second term the parallel distance 
 of the first term ; the parallel distance of the third 
 term is the fourth proportional. Example. To
 
 SECTORAL LINES. 47 
 
 find a fourth proportional to 8, 4, and 6, take tlic 
 lateral distance of 4, and make it the parallel dis- 
 tance of 8; then the parallel distance of 6, ex- 
 tended from the center, shall reach to the fourth 
 proportional 3. 
 
 In the same manner a third proportional Is 
 found to two numbers. Thus, to find a third 
 proportional to 8 and 4, the sector remaining as 
 in the former example, the parallel distance of 4, 
 extended from the center, shall reach to the third 
 proportional 2. In all these cases, if the number 
 to be made a parallel distance be too gi'cat for the 
 sector, some aliquot part of it is to be taken, and 
 the answer multiplied by the number by which 
 the first number was divided. Thus, if it were 
 required to find a fourth proportional to 4, 8, 
 I'.nd 6; because the lateral distance of the second 
 term 8 cannot be made the parallel distance of the 
 first term 4, take the lateral distance of 4, viz. the 
 half of 8, and make it the parallel distance of the 
 tirst term 4 ; then the parallel distance of the third 
 term 6, shall reach from the center to 6, viz. the 
 half of 12. Any other aliquot part of a number 
 may be used in the same way. In like manner, if 
 the number proposed be too small to be made the 
 parallel distance, it may be multiplied by some 
 number, and the answer is to be divided by the 
 same number. 
 
 To protract angles hy the line of chords. Case 1 . 
 When the given degrees are under 6o. 1 . Wiih 
 any radius \^,fig. 14, plate 3, on A as a center, 
 describe the arcJi B G. 2. Make the same radius 
 a transverse distance between 6o and 60 on the 
 line of chords., 3. Take out the transverse dis- 
 tance of the given degrees, and lay this on the 
 arch from B towards G, which will mark out the 
 angular distance required.
 
 43 USES OP SECTORAL SCALED 
 
 Case 2. When the given degrees are more ihatt 
 60. 1 . Open the sector, and describe the arch as 
 before. 2. Take i or i of the given degrees, and 
 take the transverse distance of this h or I, and lay 
 it off from B towards G, twice, if the degrees were 
 halved, three times if the third was used as a 
 transverse distance. 
 
 Case 3. When the required angle is less than 
 6 degrees; supposes. 1. Open the sector to the 
 given radius, and describe the arch as before. 
 2. Set off the radius from B to C. 3. Set off the 
 chord of 57 degrees backward from C to f, which 
 will give the arc f b of three degrees. 
 
 SOME USES OF THE SECTORAL S(*ALES OF SINES, 
 TANGENTS, AND SECANTS.* 
 
 Given the radius of a circle^ (suppose equal to 
 two inches^) required the sine and tangent ofl'^ 30 
 to that radius. 
 
 Solution. Open the sector so that the trans- 
 verse distance of 90 and 90 on the sines, or of 45 
 and 45 on the tangents, may be equal to the given 
 radius, viz. two inches; then will the transverse 
 distance of 28° 30', taken from the sines, be the 
 length of that sine to the given radius; or if taken 
 from the tangents, will be the length of that tan- 
 gent to the given radius. 
 
 But if the secant oflS° 30' was required? 
 
 Make the given radius, two inches, a transverse 
 distance to O and 0, at the beginning of the line 
 of secants; and then take the transverse distance 
 of the degrees wanted, viz. 28° 30'. 
 
 * Robertson on Mathematical Instrumenta,
 
 OF SINES, TANGENTS, &C. 
 
 49 
 
 A tangent greater than 45° (suppose 6o°J is 
 found thus. 
 
 Make the given radius, suppose two inches, a 
 transverse distance to 45 and 45 at the beginning 
 of the scale of upper tangents; and then the re- 
 quired number 60^ 00' may be taken from this 
 scale. 
 
 The scales of upper tangents and secants do not 
 run quite to 76 degrees; and as the tangent and 
 secant may be sometimes wanted to a greater 
 number of degrees than can be introduced on the 
 sector, they may be readily found by the help of 
 the annexed table of the natural tangents and se- 
 cants of the degrees above 75; the radius of the 
 circle being unity. 
 
 Degrees. 
 
 Nat. Tangent. 
 
 Nat. Secant. 
 
 76 
 
 4,011 
 
 4,133 
 
 77 
 
 4,331 
 
 4,445 
 
 78 
 
 4,701 
 
 4,810 
 
 79 
 
 5,144 
 
 5,241 
 
 80 
 
 5,671 
 
 5,759 
 
 81 
 
 6,314 
 
 6,392 
 
 82 
 
 7,115 
 
 7,185 
 
 83 
 
 8,144 
 
 8,205 
 
 84 
 
 9,514 
 
 9,567 
 
 85 
 
 11,430 
 
 1 1 ,474 
 
 86 
 
 . 14,301 
 
 14,335 
 
 87 
 
 19,081 
 
 19.107 
 
 88 
 
 28,636 
 
 28,654 
 
 89 
 
 57,290 
 
 57,300 
 
 Measure the radius of the circle used upon any 
 scale of equal parts. Multiply the tabular num- 
 ber by the parts in the radius, and the product 
 will give the length of the tangent or secant 
 sought, to be taken from the same scale of equal 
 parts.
 
 50 USES OP SECTORAL SCALES 
 
 Example. Required the length of the tangent 
 and secant of 80 degrees to a circle, whose radius, 
 measured on a scale of lb parts to an inch, is 47 5 of 
 those parts? 
 
 tangent. secant. 
 
 Against 80 degrees stands 5jd7l 5,759 
 
 The radius is 47,5 47,5 
 
 28355 28795 
 
 39697 40313 
 
 22Ci84 23036 
 
 269,3725 273,5525 
 
 So the length of the tangent on the twenty- 
 fifth scale will be 269-3 nearly. And that of the 
 secant about 273^. 
 
 Or thus. The tangent of any number of de- 
 grees may be taken from the sector at once; if the 
 radius of the circle can be made a transverse dis- 
 tance to the complement of those degrees on the 
 lower tangent. 
 
 Example. To find the tangent of "JS degrees to 
 a radius of two inches. 
 
 Make two inches a transverse distance to 12 de- 
 grees on the lower tangents; then the transverse 
 distance of 45 degrees will be the tangent of 78 
 degrees. 
 
 In like manner the secant of any number of de- 
 grees may be taken from the sines, if the radius of 
 the circle can be made a transverse distance to the 
 co-sine of those degrees. Thus making two 
 inches a transverse distance to the sine of twelve 
 degrees; then the transverse distance of 90 and 90 
 will be the secant of 78 degrees. 
 
 From hence it will be easy to find the degrees 
 answering to a given lincj expressing the length
 
 OP SINES, TANGENTS, &C. 51 
 
 of a tangent or secant, which is too long to be 
 measured on those scales, when the sector is set to 
 the, given radius. 
 
 Thus, for a tangent, make the given line a 
 transverse distance to 45 and 45 on the lower tan- 
 gents; then take the given radius, and apply it to 
 the lower tangents ; and the degrees where it be- 
 comes a transverse distance is the co-tangent of 
 the degrees answering to the given line. 
 
 And for a secant; make the given line a trans- 
 verse distance to QO and 90 on the sines. Then 
 the degrees answering to the given radius, applied 
 as a transverse distance on the sines, will be the 
 co-sine of the degrees answering to the given se- 
 cant line. 
 
 Given the length of the sine, tangent, or secant, of 
 any degrees-, to find the letigth of the radius to that 
 sine, tangent, or secant. 
 
 Make the given length a transverse distance to 
 its given degrees on its respective scale: then, 
 
 In the sines. The transverse distance of 90 and 
 90 will be the radius sought. 
 
 In the lower ta?igents. The transverse distance 
 of 45 and 45, near the end of the sector, will be 
 the radius sought. 
 
 In the upper tangents. The transverse distance 
 of 45 and 45, taken towards the center of the 
 sector on the line of upper tangents, will be the 
 center sought. 
 
 In the secant. The transverse distance of O and O, 
 or the beginning of the secants, near the center of 
 the sector, will be the radius sought. 
 
 Given the radius and any line representing a sine, 
 tangent, or secant; to find the degrees corresponding 
 to that line. 
 
 Solution. Set the sector to the given radius, 
 according as a sine^ or tangent^ or secant is con- 
 cerned. 
 
 .. E 2
 
 5'Z SELECT PROBLEMS 
 
 Take the given line between the compasses; 
 apply the two feet transversely to the scale con- 
 cerned, and slide the feet along till they both rest 
 on like divisions on both legs; then will those di- 
 visions shew the degrees and parts corresponding 
 to the given line. 
 
 To find the length of a versed sine to a given 
 number of degrees, and a given radius. 
 
 Make the transverse distance of gO and 90 on 
 the sines, equal to the given radius. 
 
 Take the transverse distance of the sine com- 
 plement of the given degrees. 
 
 If the given degrees are less than 90, the diffe- 
 rence between the sine complement and the ra- 
 dius gives the versed sine. 
 
 If the given degrees are more than 90, the sum 
 of the sine complement and the radius gives the 
 versed sine. 
 
 To open the legs of the sector, so that the corres- 
 ponding double scales of lines, chords, sines, and tan- 
 gents^ may malie each a right angle. 
 
 On the lines, make" the lateral distance ]^, a 
 distance between eight on one leg, and six on the 
 other leg. 
 
 On the sines, make the lateral distance QO, a 
 transverse distance from 45 to 45; or from 40 to 
 50; or from 30 to 60; or from the sine of any de- 
 grees to their complement. 
 
 Or on the sines, make the lateral distance of 45 
 a transverse distance between 30 and 30. 
 
 SELECT GEOMETRICAL PROBLEMS. 
 
 Science may suppose, and the mind conceive 
 things as possible, and easy to be effected, in 
 which art and practice often find insuperable dif-»
 
 IN GEOMETRY. 53 
 
 jfrculties. " Pure science has to do only with 
 ideas; but in the application of its laws to the use 
 of life, we are constrained to submit to the imper- 
 fections of matter and the influence of accident.'* 
 Thus practical geometry shews how to perform 
 what theory supposes; in the theory, however, it 
 is sufficient to have only a right conception of the 
 objects on which the demonstrations are founded; 
 drawing or delineations being of no further use 
 than to assist the imagination, and lessen the exer- 
 tions of the mind. But in practical geometry^, we 
 not only consider the things as possible to be ef- 
 fected, but are to teach the ways, the instruments, 
 and the operations by which they may be actually 
 performed. It is not sufficient here to shew, that 
 a right line may be drawn between two points, or 
 a circle described about a center, and then de- 
 monstrate their properties; but we must actually 
 delineate them, and exhibit the figures to the 
 senses: and it will be found, that the drawing of 
 a strait line, which occurs in all geometrical ope- 
 rations, and which in theory is conceived as easy 
 to be effected, is in practice attended with consi- 
 derable difficulties. 
 
 To draw a strait line between two points upon 
 a plane, we lay a rule so that the strait edge thereof 
 may just pass by the two points; then moving 
 a fine pointed needle, or drawing pen, along this 
 edge, we draw a line from one point to the other, 
 which for common purposes is sufficiently exact; 
 but where great accuracy is required, it will be 
 found extremely difficult to lay the rule equally, 
 with respect to both the points, so as not to be 
 nearer to one point than the other. It is difficult 
 also so to carry the needle or pen, that it shall nei- 
 ther incline more to one side than the other of the
 
 54 SELECT PROBLEMS 
 
 rule; and thirdly, it is very difficult to find aniTe 
 that shall be perfectly strait. If the two points be 
 very far distant, it is almost impossible to draw 
 the line with accuracy and exactness; a circular 
 line may be described more easily, and more ex- 
 actly, than a strait, or any other line, though even 
 then many difficulties occur, when the circle is 
 required to be of a large radius. 
 
 It is from a thorough consideration of these dif- 
 ficulties, that geometricians will not allow those 
 lines to be geometrical, which in their description 
 require the sliding of a point along the edge of a 
 rule, as in the ^ellipse, and several other curve 
 lines, whose properties have been as fully inves- 
 tigated, and as clearly demonstrated, as those of 
 the circle. 
 
 From hence also we may deduce some of those 
 •maxims which have been introduced into practice 
 by Bird and Smeaion, which will be seen in their 
 proper place. And let no one consider these re- 
 flections as the effect of too scrupulous exactness, 
 or as an unnecessary aim at precision; for as the 
 foundation of all our knowledge in geography, 
 navigation and astronomy, is built on observation, 
 and all observations are made with instruments, it 
 follows, that the truth of the observations, and the 
 accuracy of the deductions therefrom, will princi- 
 pally depend on the exactness with which the in- 
 struments are made and divided; and that these 
 sciences will advance in proportion as those are 
 less difficult in their use, and more perfect in the 
 performance of their respective operations. 
 
 There is scarce any thing which proves more 
 clearly the distinction between mind and body, 
 and the superiority of the one over the other, thart 
 a reflection on the rigid exactness of speculative
 
 IN GEOMETRY. 55 
 
 geometn'', and the inaccuracy of practice, that is 
 not directed by theory on one hand, and its ap- 
 proximation to perfection on the other, when 
 guided by a just theory. 
 
 In theory, most tigurcs may be measured to an 
 ahnost infinite exactness, yet nothing can be more 
 inaccurate and gross, than the ordinal r) methods of 
 mensuration; but an intelligent practice finds a 
 medium, and corrects the imperfection of our me- 
 chanical organs, by the resources of the mind. If 
 we were more perfect, there would be less room 
 for the exertions of our mind, and our knowledge 
 would be less. 
 
 If it had been easy to measure all things with 
 exactness, we should have been ignorant of many 
 curious properties in numbers, and been dej)rived 
 of the advantages we derive from logarithms, sines, 
 tangents, &c. If practice were perfect, it is doubt- 
 ful whether we should have ever been in possession 
 of theory. 
 
 We sometimes consider with a kind of envy, the 
 mechanical perfection and exactness that is to be 
 found in the works of some animals; but this per- 
 fection, which does honor to the Creator, does 
 little to them; they are so perfect, only because 
 they are beasts. 
 
 The imperfection of our organs is abundantly 
 made up by the perfection of the mind, of which 
 we are ourselves to be the artificers. 
 
 If any wish to see the difRcullies of rendering 
 practice as perfect as theory, and the wonderful re- 
 sources of the mind, in order to attain this degree 
 of perfection, let him consider the operations of 
 General Roy, at Hounslow-hcath; operations that 
 cannot be too much considered, nor too much 
 praised by every practitioner in the art of geo- 
 metry. See Fhilosoplh Trans, vol. 80^ et seq.
 
 56 SELECT PROBLEMS 
 
 Pr b l e m 1 . To erect a perpendicular at or near 
 the end of a given right line, C D, Jig. 5, plate 4. 
 
 Method 1 . On C, with the radius C D, describe 
 a faint arc ef; on D , with the same radius, cross 
 ef'At G, on G as a center; with the same radius, de- 
 scribe the arc D E F; set off the extent D G twice, 
 that is from D to E, and from E to F. Join the 
 points D and F by a right Hne, and it will be the 
 perpendicular required. 
 
 Method '2, fg. 5. On any point G, with the 
 radius D G, describe an arc FED; then a rule 
 laid on C and G, will cut this arc in F, a line join- 
 ing the points F and D will be the required per- 
 pendicular. 
 
 Method 3. 1 . From the point C, fig. 6, plate 4, 
 with any radius describe the arc r n in, cutting the 
 line AC in r. 2. From the point r, with the same 
 radius, cross the arc in n, and from the point n, 
 cross it in m. 3. From the points n and m, with 
 the srime, or any other radius, describe two arcs 
 cutting each other in S. 4. Through the points 
 S and C, draw the line S C, and it will be the per- 
 pendicular required. 
 
 Method 4. By the line of lines on the sector, fig. 7, 
 plate A. 1 . Take the extent of the given line A C. 
 2. Open the sector, till this extent is a transverse 
 distance between 8 and 8. 3. Take out the trans- 
 verse distance between 6 and 6, and from A with 
 that extent sweep a faint arc at B. 4. Take out 
 the distance between 10 and 10, and with it from 
 C, cross the former arc at B. 5. A line drawn 
 through A and B, will be the perpendicular re- 
 quired; the numbers 6, 8, 10, are used as multi- 
 ples of 3, 4, 5. 
 
 By this method, a perpendicular may he easily and 
 accurately erected on the ground. 
 
 Method 5. Let AC, fig. 7, plate 4, be the givea
 
 IN GEOMETRY. 5/ 
 
 line, and A the given point. 1 . At any point D, 
 with the radius DA, describe the arc EAB. 
 
 2. With a rule on E and D, cross this arc at B. 
 
 3. Through A and B draw a right Hne, and it will 
 be the required perpendicular. 
 
 Problem 2. To raise a perpendicular from the 
 middle, or any other point G, of a given line AB, 
 fg. 8, plate 4. 
 
 1 . On G, with any convenient distance within the 
 limits of the line, mark or set oiFthe points n and m. 
 2. From n and m with any radius greater than GA, 
 describe two arcs intersecting at C. 3. Join CG by 
 a line, and it will be the perpendicular required. 
 
 Problem 3. Froin a given point C,fig. Q, plate 
 
 4. out of a given line K^i tolet fall a perpendicular. 
 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 n m, with 
 the same, or any other radius, describe two arcs in- 
 tersecting each other at S. 3. Through the points 
 C S draw the line C S, which will be the required 
 line. 
 
 When the point is nearly opposite to the end of 
 the line, it is the converse of Method 5, Problem 1, 
 fig' 1, plate A. 
 
 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 AC at A. 4. Through A and B 
 draw the line AB, and it will be the perpendicular 
 required. 
 
 Another method. 1 . From A, fig. Q, plate 4, or 
 any other point in AB, with the radius AC, 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 the point C D by a line
 
 58 SELECT l^ROBtEMS, 8cC. 
 
 C G D, and C G will be the perpendicular re- 
 quired. 
 
 Problem 4. Through a given point C, to draw 
 a line parallel to a given strait line AB, fg. 10, 
 plate A. 
 
 1. On any point D, (within the given line, or 
 without it, and at a convenient distance from C,) 
 , describe an arc passing: through C, and cutting the 
 given line in A. 1. With ihe same radius describe 
 another arc cutting AB at B. 3. Make BE equal 
 to AC. 4. Draw a line CE 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 whether the point D be situated on AB, or any 
 where between it and the required line. 
 
 Problem 5. At the given point D, to make an 
 angle equal to a given angle AB C,Jig, \'2, plate 4. 
 
 1. From Bj with any radius, describe the arc 
 n m, cutting the legs B A B C, in the points 
 n and m. 2. Draw the line D r, and from the 
 point D, with the same radius as before, describe 
 the arc rs. 3. Take the distance m n, and apply 
 it to the arc rs, 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, or AB C, 
 as required. 
 
 Problem 6. To extend with accuracy a short 
 strait line to any assignable length ; or, through two 
 •given points at a small distance from each other to 
 draw a strait line. 
 
 It frequently happens that a line as short as that 
 between A and B, j^>. 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 AB, or the two points A
 
 DIVISION OF* STRAIT LINES. 6^ 
 
 and B; then from A as a center, describe an arch 
 CBD; and from the point B, lay otF BC equal 
 to B D ; and from C and D as centers, with any 
 radius, describe two arcs intersecting at E. From 
 the point A describe the arc F E G, making E F 
 equal to E G ; then from F and G as centers, de- 
 scribe two arcs intersecting at H, and so on ; then 
 a strait 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 STRAIT LINES. 
 
 Problem 7. To bisect or divide a given strait 
 line AB into two equal partSyfg. 13, plate 4. 
 
 1, On A and B as centers, with any radius 
 greater than half AB, describe arcs intersecting 
 each other at C and D. 1. 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 dif- 
 ferent radii above the given line, as at C and E; 
 then a line C produced, will bisect AB in F. 
 
 By the line of lines on the sector. 1. Take AB in 
 the compasses. 2. Open the sector till this extent 
 is a transverse distance between 10 and 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, 16, 
 32, 64, 128, &c. equal parts. 
 
 Problem S. To divide a given strait line AB 
 into any number of equal parts, for instance, five. 
 
 Method I, fig. 14, plate 4. 1. Through A, one 
 extremity of the line AB, draw AC, making any 
 angle therewith. 2. Set off on this line from A to 
 H as many equal parts of any length as AB is to
 
 60 DIVISION OF STRAIT LINE3. 
 
 be divided into. 3. Join H 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, j^g. 15, plate 4. 1. Through B 
 draw LD, forming any angle with AB."" 2. Take 
 any point D either above or below AB, and 
 through D, draw D K parallel to AB. 3. On D 
 set otF five equal parts D F, F G, G H, HI, IK. 
 4. Through A and K draw A K, cutting BD 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, jig. 1/, plate 4. 1. From the 
 ends of the line AB, draw two lines AC, 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 AB is to be divided into, in the present 
 instance four equal parts, AI, IK, K L, LM, on 
 AC; and four, BE, E F, FG, GH, on B D. 
 3. Draw lines from M to E, from L to F, K to G, 
 I to H, and AB will be divided into five equal 
 parts. 
 
 Fourth method, fg. l6, plate A. 1. 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 NG in I. 
 6. Join DI, which being equal to AB, transfer 
 the divisions of D I to AB, and it will be divided 
 as required. H M is a line of a different length 
 to be divided in the same number of parts. 
 
 The foregoing methods are introduced on ac- 
 count not only of their own peculiar advantages.
 
 DIVISION OF STRAIT LINES. 6l 
 
 but because they also are the foundation of several 
 mechanical methods of divi'-ion. 
 
 Problem 9- To cut; off from a given line AB 
 any odd part, as yd, \th, ^th, yth, &c. of that line, 
 fig. \Q, plate 4. 
 
 1. Draw through either end A, a line AC, 
 forming any angle with AB. 2. Make AC equal 
 to AB. 3. Through C and B draw the line CD. 
 4. Make B D equal to C B. 5. Bisect AC in a. 
 6. A rule on a and D will cut off a B equal id of 
 AB. 
 
 If it be required to divide AB into five equal 
 parts, 1 . Add unity to the given number, and 
 halve it, 5 + 1=6, ^=3. 2. Divide AC into 
 three parts; or, as AB is equal to AC, set off Ab 
 equal A a. 3. A rule on D, and b will cut off 
 bB rth part of AB. 4. Divide Ab into four 
 equal parts by two bisections, and AB will be di- 
 vided into five equal parts. 
 
 To divide AB into seven equal parts, 7-f-i=8, 
 T=4. 1. Now divide AC into four parts, or bi- 
 sect a C in c, and c C will be the ^th of AC. 2. A 
 rule on c and D cuts off' c B fth of AB. 3. Bisect 
 Ac, and the extent cB 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-|-l = 10 
 •^=5. Here, 1. Make Ad equal to Ab, and d C 
 will be Ith of AC. 2. A rule on D and d cuts 
 off dB i of AC. 3. Bisect Ad. 4. Halve each 
 of these bisections, and Ad is divided into four 
 equal parts. 5. The extent d B will bisect each 
 of these, and thus divide AB into nine equal 
 parts. 
 
 If any odd number can be subdivided, as Q by 3, 
 tben first divide the given line into three parts, 
 ar>d take the third as a new line, and find the third
 
 62 DIVISION OF STRAIT LINES. 
 
 thereof as before, which gives the ninth part re- 
 quired. 
 
 Method 2. Let DB, fg. 19, plate A, be the 
 given Hne. 1. Make two equilateral triangles 
 ADB, CDB, one on each side of the line D B. 
 2. Bisect AB in G. 3. Draw C G, which will 
 cut off H B equal id of D B. 4. Draw D F, and 
 make GF equal to D G. 5. Draw H F which 
 cuts off B h equal k of AB or D B. 6. C h cuts 
 D B in i one fifth part. 7- F i cuts AB in k equal 
 ^th of DB. 8. Ck cuts Db in 1 equal |th of 
 DB. 9. Fl cuts AB in m equal ^th of DB. 
 10. Cm cuts D B at n equal ^th part thereof. 
 
 Method 3. Let KB, Jig. 12, flate 5, be the gi- 
 ven line to be divided into its aliquot parts h, t, i. 
 1. On AB erect the square ABCD. 2. Draw 
 the two diagonals AC, D B, which will cross each 
 other at E. 3. Through E draw FEG parallel 
 to AD, cutting AB in G. 4. Join D G, and the 
 line will cut the diagonal AC at H. 5. Through 
 H draw IHK parallel to AD. 6. Draw DK 
 crossing AC in L. 7- And through L draw 
 MLN parallel to AD, and so proceed as far as 
 necessary. AG is h, AK |, AN \ of AB.* 
 
 Method 4. Let AB,/^. 13, plate 5, be the given 
 line to be subdivided. 1 . Through A and B draw 
 C D, F E parallel to each other. 2. Make CA, 
 AD, FB, FE equal to each other. 3. Draw 
 CE, which shall divide AB into two equal parts 
 at G. 4. Draw A E, DB, intersecting each other 
 at H. 5. Draw CH intersecting AB at I, mak- 
 ing AI id of AB. 6. Draw DF cutting AE in 
 K. 7. Join CK, which will cut AB in L, mak- 
 ing AL equal \ of AB. 8. Then draw Dg, cut- 
 ting AE in M, and proceed as before. 
 
 * Hooke's Posthumous Works.
 
 DIVISION OF STRAIT LINES. 63 
 
 Corollary. Hence a given line may be accu- 
 rately divitied into any prime number whatsoever, 
 by lirst cutting oti the odd part, then dividing the 
 remainder by continual bisections. 
 
 Problem 10. An eusy\ sii/ij)Ie, and ve7y useful 
 method of laying down a scale for di-v'id'wg lines Into 
 any number of equal 'parts, or for reducing -plans to 
 any size less than the original. 
 
 If the scale is for dividing lines into two equal 
 parts, constitute a triangle, so that the h) pothe- 
 nuse may be twice the length of the perpendicular. 
 Let it be three times for dividing them into three 
 equal parts; four, for four parts, and so on : Jig. 11, 
 plate 5, represents a set of triangles so consti- 
 tuted. 
 
 To find the third of the line by this scale. 
 1. Take any line in your compasses, and set off 
 this extent from A towards I, on the line marked 
 one third; then close the compasses so as to strike 
 an arc that shall touch the base AC, and this 
 distance will be the t of the given line. Similar 
 to this is w^hat is termed the angle of reduction, 
 or proportion, described by some foreign writers, 
 and which we shall introduce in its proper place. 
 
 Problem 11. To divide by the sector a given 
 strait line into any nundjer of equal parts. 
 
 Case 1. Where the given line is to be divided 
 into a number of equal jiarts that may be ob- 
 tained by a continual bisection. 
 
 In this case the operations arc best performed by 
 continual bisection; let it be required to divide 
 A^, fig. Q, plate o, into l6 equal parts. 1. Make 
 AB a transverse distance betw^een 10 and 10 on 
 the line of lines. 1. Take out from thence the 
 distance between 5 and 5, and set it from A or B 
 to 8, and AB will be divided into two equal parts. 
 3. Make A 8 a transverse distance between 10 and
 
 64 DIVISION OP STRAIT L1NES» 
 
 10, and 4 the transverse distance between 5 and 5, 
 will bisect 8 A, and 8 B at 4 and 12; and thus AB 
 is divided into four equal parts in the points 4, 8, 
 and 12. 4. The extent A 4, put between 10 and 
 10, and then the distance between 5 and 5 applied 
 from A to 2, from 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 bi- 
 sect each of these, we might take the extent of 
 A 2, and place it between 10 and 10 as before; 
 but as the spaces are too 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 mea- 
 sure. 6. Take out the transverse measure between 
 5 and 5, and one foot of the compasses in A will 
 give the point 3, in 4 will fall on 7 and 1, on 8 
 will give 5 and 11, on 12 gives 9 and 15, and on 
 B will give 13. Thus we have, in a correct and 
 easy manner divided AB into 16 equal parts by a 
 conthiual .bisection. 
 
 If it were required to bisect each of the forego- 
 ing divisions, it would be best to open the sector 
 at 10 and 10, with the extent of five of the divi- 
 sions already obtained ; then take out the trans- 
 verse distance between 5 and 5, and set it off from 
 the other divisions, and they will thereby be bi-- 
 sected, and the line divided into 32 equal parts. 
 
 Case 2. When the given line cannot be di- 
 vided by bisection. 
 
 Let the given line be AB, fy. 7, pJate 4, to be 
 divided into 14 equal parts, a number which is not 
 a multiple of 2. 
 
 1. Take the extent AB, 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 
 AB into two equal parts, each of which are to be 
 subdivided into 7, which maybe done by dividing
 
 - blVISIO^r OP StRAlT LINES. 65 
 
 A 7 into 6 and 1, or 4 and 3, which last is pre- 
 ferable 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 trans- 
 verse 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 A4 a transverse distance at 10 and 
 10, then the transverse distance between 5 and 5 
 bisects A b, and lOB in 2 and 12, and gives the 
 point 6 and 8; then one foot in 3 gives 1 and 5, 
 and from 11, 13 and Q: lastly, from 4 it gives 6, 
 and from 10, 8; and thus the line AB is divided 
 into 14 equal parts. 
 
 Problem 12. To make a scale of equal j>arts 
 containing any given number in an inch. 
 
 Example. To construct a scale of feet and inches 
 in such a manner, that 25 of the smallest parts shall 
 be equal to one inch, and 12 of them represent 
 one foot. 
 
 By the line of lines on the sectoi . 1 , Multiply the 
 given numbers by 4, the products will be 100, and 
 48. 2. Take one inch between your compasses, 
 and make it a transverse distance between 100 and 
 100, and the distance between 48 rnd 48 will 
 be 12 of these 25 parts in an inch; this extent set 
 off from A to l,fg. 3, plate 5, from 1 to 2, &c. to 
 12 at B divides AB into a scale of 12 feet. 3, Set 
 off one of these parts from A to a, to be subdi- 
 vided into 12 parts to represent inches. 4. To 
 this end divide this into three parts; thus tr.ke the 
 extent A 2 of two of these parts, and make it a 
 transverse distance between 9 and 9. 5. Set the 
 distance between 6 and 6 from b to e, the same 
 extent from 1 gives g, and from e gives n, thu^ 
 dividing A a into three equal parts in the points n
 
 06 PROPORTIONAL LINES'. 
 
 -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 are required, 
 as I, 2, or 3, instead of taking the transverse dis- 
 tance near the center of the sector, the division 
 will be more accuYately performed by using the 
 following method. 
 
 Thus, if three parts are required from A, of 
 which the whole line contains QO, make AT),Jjg. 4, 
 plate 5, a transverse distance between 90 and c}0; 
 then take the distance between 87 and 87, which 
 set off from D to E backwards,^ and AE will con- 
 tain the three desired parts. 
 
 Example 2. Supposing a scale of six inches to 
 contain 140 poles, to open the sector so that it 
 may answer for such a scale; divide 140 by 2, 
 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. 
 
 Example 3. To make a scale of seven inches 
 tliat shall contain 180 fathom; •^=9.-^=3^, 
 therefore make 3§ a transverse between 9 and 9, 
 and you have the required scale.. 
 
 OF PROPORTIONAL LINES. 
 
 Problem 13. To cut a given line A D, fig. 1 4 , 
 jilale 5, Into tivo iinequal parts that shall have any 
 given proportion, ex. gr. of C to D. 
 
 1, Draw AG forming any angle with AD. 
 2. From A dn AG set off AC equal to C, and 
 CE equal to D. 3. I>aw ED, and parallel to it 
 C B, which will cut AD at B in the required pro- 
 portion.
 
 PROPORTIONAL LINESi 6} 
 
 To divide by the sector the line AB, fg. 1, 
 if^hite 5, in the proportion of 3 to 2. Now as 3 and 
 "2 would fall near the center, 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 4, the whole 
 line will be 10; therefore make AB a transverse 
 distance between 10 and 10, and then the trans* 
 verse distance between 6 and 6, set off from B to 
 e, is Iths of AB; or the distance between 4 and 4 
 will give Ae iths of AB; therefore AB is cut in 
 the proportion of 3 to 2. 
 
 Example 2. To cut AB, Jig. 2, plate 5, in the 
 proportion of 4 to 5 ; here we may use the num-^ 
 b^rs themselves ; therefore with A B open the sec- 
 tor at 9 and Q, the sum of the two numbers; then 
 the distance between 5 and 5 set off from B to c, 
 or between 4 and 4, set off A to c, and it divides 
 AB in the required proportion. 
 
 No/€. If the numbers be too small, use their 
 equimultiples; if too large, subdivide them. 
 
 Corollary. From this problem we obtain ano- 
 ther mode of dividing a strait line into any number 
 of equal parts. 
 
 Problem 14. To estimate the proportion he' 
 tween two or more given Vines, as AB, CD, EF, 
 fig' 9^ Plated. 
 
 Make AB a transverse distance between 10 and 
 10, then take the extents severally of CD and EF^ 
 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 Qt ^ 
 Therefore AB is to CD as 100 to 85, to EF as 
 100 to idj, and of C D to E F as 85 to Q7 . 
 
 Problem 15. To find a third proportional ta 
 two given right lines A and B,fig. 15, plate 5. 
 
 1 . From the point D draw two right lines D E, 
 D F, making any angle whatever. 2. In these 
 
 p 2
 
 08 PROPORTIONAL LINES, 
 
 lines take DG equal to the first term A, and DC, 
 DH, each equal to the second term B. 3. Join 
 GH, and draw CF parallel thereto; then DF 
 will be the third proportional required, that is^ 
 DG (A) to D C, (B,) so is D H (B) to D F. 
 
 By the sector. 1. Make AB, fg. 5, flate 5, a 
 transverse distance between 100 and 100. 2. Find 
 the transverse distance of E F, which suppose 50. 
 3. Make EF a transverse distance betw^een 100 and 
 100. 4. Take the. extent between 50 and 50, and 
 it will be the third proportional C D required. 
 
 Pr o b l e m 1 6. To find a fourth ■proportional to 
 three given right lines A, B, C,fig. l6, flate 5. 
 
 1. From the point a draw two right lines, mak* 
 ing any angle whatever. 2. In these lines make 
 a b equal to the first term A3 a c equal to the se- 
 cond B, and a d equal to the third C. 3. Join be, 
 and draw de parallel thereto, and ae will be the 
 fourth proportional required ; that is, a b (A) is ta 
 a c (B,) so is a d (C) to a e. 
 
 By the sector, 1 . Make the line A a transverse*- 
 distance between 100 and 100. 2. Find the trans- 
 verse measure of B, which is 60. 3. Make c the 
 third line a transverse measure between 100 and 
 100. 4. The measure between 60 and 60 will be? 
 the fourth proportional. 
 
 Problem 17. To find a mean proportional he"- 
 tween t-zvo given strait lines A and B,fig. 1 7 , plate 5. 
 
 1 . Draw any right line, in wliich take C E equal 
 to A, and E A equal to B. 2. Bisect AC in B, and 
 with BA, or B C as a radius, describe the semi- 
 circle ADC. 3. From the point E draw EG 
 perpendicular to AC, and it will be the mean pro- 
 portional required. 
 
 By the sector. Join the lines together, (sup- 
 pose them 40 and f)0) and get the sum of them,- 
 130? then find the half of this sum 05, and half
 
 PROPORTIONAL LINES. 6g 
 
 the difference 25. Open the line of lines, so that 
 they may be at right angles' to each other; then 
 take with the compasses the lateral distance 65, 
 and apply one foot to the half difference 25, and 
 the other foot will reach to (io, the mean propor- 
 tional required; for 40 to 6o, so is 6o to QO. 
 
 Problem 18. T'o cut a g'l-ven line AB into ex- 
 treme and mean froport'ion^jig. 18, flate 5. 
 
 1. Extend AB to C. 2. At A erect a perpen- 
 dicular AD, and make it equal to AB. 3. Set the 
 half of AD or AB from A to F. 4. With the 
 radius FD describe the arc D G, and AB will be 
 divided into extreme and mean proportion. A G 
 is the greater segment. 
 
 By the sector. Make AB a transverse distance 
 between 6o and 6o of the line of chords. 2. Take 
 x)ut the transverse distance between the chord of 
 36, which set from A to G, gives the greatest 
 segment. 
 
 Or make AB, a transverse distance between 54 
 and 54 of the line of sines, then is the distance be- 
 tween 30 and 30 the greater segment, and 18 and 
 18 the lesser segment. 
 
 Problem IQ. To divide a given strait line In 
 the same proportion as anotlur given strait line is di- 
 vided, jig. 10, plate 5. 
 
 Let AB, or GD be two given strait lines, the 
 first divided into 100, the second into (X) equal 
 parts; it is required to divide EF into 100, and 
 GH into 60 equal parts. 
 
 Make EF a transverse distance in the terms lOO 
 and 100, then the transverse measure between p<) 
 and go set from E to 90, and from F to 10; the 
 measure between 80 and 80 set from E to 80, and 
 from F to 20, and the measure between 70 and 70 
 set from E gives 70, from F 30. The distance 
 between 60 and 60, gives 60 and 40 i and 
 lastly, the transverse measure between 50 ajid 5Q
 
 70 PROPORTIONAL LINES. 
 
 bisects the given line in the point 50, and we shall, 
 by five transverse extents, have divided the line 
 EF into 10 equal parts, each of which are to be 
 subdivided into 10 smaller divisions by problems 
 11 and 12. 
 
 To divide G H into 6o parts, as we have sup- 
 posed CD to be divided, make GH a transverse 
 distance in the terms of 6o, then work as before. 
 
 Problem 20. To find the angular poiJit of two 
 ^iven lines AK, c P, fig. 22, plate 5, which incline 
 to each other without producing either of them. 
 
 Through A draw at pleasure AN, yet so as not 
 to cut c P too obliquely, 
 
 1. Draw the parallel lines A-, Gf, K^. 2. Take 
 any number of times the extents, AN, GO, K P, 
 and set them on their respective lines, as from N to 
 y and I/, from o to (3 and £, from P to a and (J, and 
 a line through ^, £, u, and another through «, P, y, 
 tv'ill tend to the same point as the lines AB, c P. 
 
 Method 2. 1. Through AB and CD, /^. 21, 
 plate 5, draw any two parallel lines, as GH and 
 FE. 2. Set off the extent B D twice, from B to 
 G, and D to H; and the extent AC twice, from 
 A to F, and from C to E. 3. A line passing 
 through F and G will intersect another line pass- 
 ing through E and H in I, the angular point re- 
 quired. 
 
 The extent FA, G B, may be multiplied or di - 
 vided, so as to suit peculiar circumstances. 
 
 Corollary. Hence, if any two lines be given that 
 tend to the same angular point, a third, or more 
 lines may be drawn that shall tend to the same 
 point, and yet pass through a given point. 
 
 Solution by the sector. Case 1. When the pro- 
 posed point e is between the two given lines VL 
 and a h^fig. 23, plate 5. 
 
 Through e dniw a line a v, cutting ab at a, and 
 VL at V^, then from any other point b, in ab, the
 
 PROPORTIONAL LINES. 71 
 
 ftirtlier the better, draw b x parallel to a V, and 
 cutting VL in x, make av a transverse distance 
 between 100 and 100, on the line of Hnes; take 
 the extent ev, and find its transverse measure, 
 which suppose to be 60; now make xb a transverse 
 distance between 100 and 100, and take out the 
 transverse distance of the terms 6o, which set off 
 from X to f, then a line drawn through the points 
 e arid f, shall tend to the same inaccessible point q 
 with the given lines ab, VL. 
 
 Case 1. When the proposed point e is without 
 the given lines ab and VL, fg. 24. 
 
 Through e draw any line ev, cutting ab in a, and 
 VL in v; and from any other point b in ab draw- 
 ee F parallel to ev, and cutting VL in x, mak:e 
 ve a transverse distance in the terms of 100; find 
 the transverse measure of av, which suppose 72; 
 make xb a transverse measure of 7'2, and take out 
 the distance between the terms of 100, which set 
 off from X to f, and a line ef drawn through e and 
 f, will tend to the same point with the line ab. 
 
 If the given or required lines fall so near each 
 other, that neither of them can be measured on the 
 terms of 100, then use any other number, as 80, 
 70, 60, &c. as a transverse measure, and work with 
 that as you did with 100. 
 
 This problem is of considerable use in many 
 geometrical operations, but particularly so in per- 
 spective; for we may consider VL as a vanishing 
 line, and the other two lines as images on the pic- 
 -ture; hence having any image given on the picture 
 that tends to an inaccessible vanishing [)oint, as 
 many more images of lines tending to the same 
 point as may be required, are readily drawn. This 
 problem is more fully illustrated, and all the va- 
 rious cases investigated;, in another part of this 
 work.
 
 72 PROPORTIONAL LINES, 
 
 Problem 21. Upon a given right line AB, Jig, 
 19, plate 5, to make an equilateral triangle. 
 
 1. From A and B, with a radius equal to AB, 
 describe arcs cutting in C. 2. Draw AC and 
 BC, and the figure ACB is the triangle re- 
 quired. 
 
 An isosceles triangle may be formed in the same 
 manner. 
 
 Problem 22. To make a triangle., whose three 
 sides shall he respectively equal to three given lines, 
 
 A, B, C, fig. 20, plate 5, provided any two of them 
 ]?e greater than the third. 
 
 1 . Draw a line B C equal to the line B. 2. On 
 
 B, with a radius equal to C, describe an arc at A, 
 3. On C, with a radius equal to A, describe ano- 
 ther arc, cutting the former at A. 4. Draw the 
 lines AC and AB, and the figure AB C will be the 
 triangle required. 
 
 Problem 23. Upoti a given line AB, fig. I, 
 plate 6, to describe a square. 
 
 1 . From the point B draw B D perpendicular, 
 and equal to AB. 2. On A and D, with the ra- 
 dius AB, describe arcs cutting in C. 3. Draw 
 AC and CD, and the %ure ABCD is the re- 
 quired square. 
 
 Problem 24. To describe a rectangle or paral- 
 lelogram, ivhose length and breadth shall be equal to 
 two given lines A and B, fig. 2, plate 6. 
 
 1 . Draw C D equal to A, and make D E per- 
 pendicular thereto, and equal to B. 2. On the 
 ])oints E and C, with the radii A and B, describe 
 arcs cutting in F. 3. Join C F and EF; thea 
 C D E F will be the rectangle required. 
 
 Problem 25, Upon a given line AB, to construct 
 a rhombus., fig. 3, plate 6. 
 
 1. On B, with the radius AB, describe an arc 
 at D. 2. On A, with the same radius, describe
 
 ?ROPORTIONAi. LINES. 73, 
 
 an arc at C. 3. On C, but still with the same 
 radius, make the intersection D. 4. Draw the 
 lines AC, DC, DB, and you have the required 
 figure. 
 
 Having two given lines AB, AD, and a given, 
 anghy to construct a rhomhoides. 
 
 Make the angle AC D equal to the given angle, 
 and set off CD equal to AB, and AC equal to 
 AD; then from A, with the distance AB, describe 
 an arc atB; intersect this arc with tlie extent AD, 
 set off from D; join AB, B D, and the figure is 
 completed. 
 
 Problem 2(5. Having the diagonal AD, and 
 four sides AB, B D, DC, AC, to construct a trape- 
 zium^fig. b^ plate 0. 
 
 Draw an occult line AD, and make it equal to 
 the given dicigonal. Take AB in the compasses, 
 and from A strike an arc at B; intersect this arc 
 from D with the extent DB, and draw AB, DB; 
 now with the other two lines AC, CD, and from 
 A and D make an intersection at C ; join DC, AC, 
 and the figure is completed. 
 
 Pr o B L E M 27 • Having the four sides and one an- 
 gle, to construct a trapezium, fg. 5, plate d. 
 
 M ike the fine AB equal to its given side, and 
 at A make the angle CAB equal to the given an- 
 gle, and iVC equal to the given side AC; then 
 with the extent B D from B, describe an arc at D, 
 intersect this from C with the extent C D; join the 
 several lines, and the figure is obtained. 
 
 Problem 28. To find the center of a circle^ 
 fg. Q, plate b. 
 
 1. Draw any chord AB, and bisect it with the 
 chord CD. 2. Bisect CD by the chord EF„ 
 and their intersection o will be the center of the 
 circle.
 
 74 PROPORTIONAL LINES. 
 
 Problem 29. To describe the circumference of 
 a circle through any three given fo'ints AB 0,fg. 7, 
 ^late 6, provided they are not in a strait line. 
 
 1. From the middle point. B draw the chords' 
 EA and B C. 2. Bisect these chords with the 
 perpendicuhir lines n O, n O. 3. From the point 
 of intersection O^ and radius OA, OB, and O C, 
 you may describe the required circle ABC. By 
 this problem a portion of the circumference of 
 a circle may be finished, by assuming three 
 points. 
 
 Problem 30. To draw a tangent to a given 
 circle^ that shall pass through a given point A, fg. 8 
 and Q, plate 6. 
 
 Case 1. When the point A is in the circumfe- 
 rence of the Q\vc\c,fg. 8. plate 6. 
 
 1. From the center O, draw the radius OA. 
 1. Through the point A, draw C D perpendicular 
 to O A, and it will be the required tangent, 
 
 Case 2. When the point A is without the cir- 
 cle, ^j^. 0,//<r//^ 6. 
 
 1. From the center O draw OA, and bisect it" 
 in n. 2. From the point n, with the radius n A, 
 or n Oj describe the semicircle AD O, cutting the 
 given circle in D. 3. Through the points A and 
 D, draw AB, the tangent required. 
 
 Problem 31. To cut off from a circle, fg. 10, 
 plate 0, a segment containing any proposed angle, 
 ex. gr. 120°. 
 
 Let F be the point from whence it is required (o 
 draw a chord which shall contain an angle of 120^'. 
 
 1. Through F draw FR a tangent to the circle. 
 
 2. From F draw FA, making an angle of 60 de- 
 grees, with the tangent FR, and FCA is the 
 segment required.
 
 FOR THE PENTAGON, ScC. 75 
 
 Problem 32. On a given line AB to describe 
 the segment of a circle caj^able of cotitaining a given 
 Single, fjg. 1 \, plate 6. 
 
 Draw AC and B C, making the angles ABC, 
 BAC, each equal to the given angle. Draw AD 
 perpendicular to AC, and BD perpendicular to 
 B C; with the center D, and radius DA, or DB, 
 describe the segment AEB, and any angle made 
 in this segment will be equal to the given angle. 
 A more easy solution of thi? problem will be given 
 when we come to apply it to practice. 
 
 Problem 33. I^j describe an arc of a circle 
 that shall contain any number of degrees^ voithout com- 
 pulses, or vjithout finding the center of the circle^ 
 fg. \2, plate 6. 
 
 Geometrically by finding points through which 
 the arc is to pass, let AB be the given chord. 
 
 1. Draw AF, making any angle with BA. 
 2. At any point F, in AF, make the angle E F G, 
 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, i&c. may be found, as will be necessary to com- 
 plete the arc. * 
 
 This subject will be found fully investigated 
 hereafter. 
 
 Problem 34. To inscribe a circle in a given 
 triangle, ABC,fg 13, plate 0. 
 
 1 . Bisect the angles A and B with the lines A O 
 and B O. 2. From the point of intersection O, 
 let fall the perpendicular ON, and it will be the. 
 radius of the required circle. 
 
 Pr o B L e M 3 5 . To inscribe a pentagon, a hexagon y 
 or a decagon, in a given circle, fig. 1 5, plate 6, 
 
 1. Draw the diameters AB and CE at right 
 angles to each other. 2. Bisect DB at G. 3. On 
 (j, with the radius GC, describe the arc CF,
 
 75 FOR THE OCTAGON, kc, 
 
 4. Join C and F, and the line C F will be one side 
 of the required pentagon. 
 
 The two sides DC, FDof the triangle F DC, 
 enable us to inscribe a hexagon or decagon in the 
 same circle; for D^ is the side of the hexagon, 
 D F that of the decagon. 
 
 Problem 36. To inscribe a square or an octagon 
 in a given circle, fig. l6, plate 6. 
 
 1. Draw the diameters AC, B D, at right angles 
 to each other. 2. Draw the lines AD, BA, BC, 
 C D, and you obtain the required square. 
 
 FOR THE OCTAGON. 
 
 Bisect the arc AB of the square in the point F, 
 and the line AE being carried eight tirties round, 
 •will form the octagon required. 
 
 Problem 37. In a given circle to inscribe an 
 equilateral triangle, an hexagon, or a dodecagon^ 
 Jig. \7, plated. 
 
 FOR THE EQUILATERAL TRIANGLE. 
 
 1 . From any point A as a center, with a distance 
 equal to the radius AO, describe the arc FOB. 
 
 2. Draw the line B F, and make B D equal to B F. 
 
 3. Join DF, and DBF will be the equilateral triT. 
 angle required. 
 
 FOR THE HEXAGON. 
 
 Cany the radius AO six times round the cir- 
 cumference, and you obtain a hexagon. 
 
 FOR THE DODECAGON. 
 
 Bisect the arc A B of the hexagon in the point n, 
 and the line An being carried twelve times round
 
 IfOR THE DODECAGON, ScC. 
 
 If 
 
 the circumference, will form the required dode-* 
 cagon. 
 
 Problem 38. Another method to inscrihe a do" 
 decagon in a circle, or to divide the circumference of 
 a given circle into 1 2 equal farts , each of 30 degrees^ 
 fig. 18, plated. 
 
 1. Draw the two diameters 1, 7^ 4, 10, perpen- 
 dicular to each other. 2. With the radius of the 
 circle and on the four extremities 1, 4, 7^ 10, as 
 centers, describe arcs through the center of the 
 circle; these arcs will cut the circumference in the 
 points required, dividing it into 12 equal parts, at 
 the points marked with the numbers. 
 
 Problem 39. To find the angles at the center^ 
 and the sides of a regular -polygon. 
 
 Divide 360 by the number of sides in the pro- 
 posed polygon, thus ^ gives 7*25 for the angle at 
 the center of a pentagon. To find the angle 
 formed by the sides, subtract the angle at the cen- 
 ter from 180, and the remainder is the angle re- 
 quired; thus 72° from 180'^, gives 108*^ for the an • 
 gle of a pentagon. 
 
 A TABLE, sbtiv'nig the angles at the centers and circumferences of 
 regular polygons, from three to tivelvc sides inclusive ^ 
 
 
 ^ 
 
 
 
 
 
 Names. 
 
 
 Angles at Center. 
 
 Angles 
 
 at Cir, 
 
 Trigon 
 
 3 
 
 120" 
 
 00' 
 
 60° 
 
 OO' 
 
 Square 
 
 4 
 
 90 
 
 00 
 
 90 
 
 00 
 
 Pentagon 
 
 ,5 
 
 72 
 
 00 
 
 108 
 
 GO 
 
 Hexagon 
 
 (i 
 
 00 
 
 00 
 
 120 
 
 00 
 
 Heptagon 
 
 7 
 
 51 
 
 25f 
 
 128 
 
 34|. 
 
 Octagon 
 
 8 
 
 45 
 
 00 
 
 135 
 
 00 
 
 Nonagon 
 
 .0 
 
 40 
 
 00 
 
 140 
 
 00 
 
 Decagon 
 
 10 
 
 36 
 
 00 
 
 144 
 
 GO 
 
 Endecagon 
 
 11 
 
 32 
 
 43tZ_ 
 
 147 
 
 16^ 
 
 Dodecagon 
 
 J2 
 
 30 
 
 00 
 
 150 
 
 00
 
 fS FOR POLYGONS^ 
 
 This table is constructed by dividing 36o, tfi« 
 clcgrecs in a circumference, by the number of sides 
 in each polygon, and the quotients are the angles 
 at the centers ; the angle at the center subtracted 
 from 1 80 degrees, leaves the angle at the circum- 
 ference. 
 
 Problem 40. In a given circle to inscribe a,ny 
 regular polygon^ fig. 14, plate 6. 
 
 1 . At the center c make an angle equal to the 
 center of the polygon, as contained in the pre- 
 ceding table, and join the angular points AB. 
 2. The distance AB will be one side of the polygon, 
 which being carried round the circumference, the 
 proper number of times will complete the figure. 
 
 Another method, which approximates very nearly 
 the truth, fig. IQ, plate 6. 
 
 li Divide the diameter AB into as many equal 
 parts as the figure has sides. 2. From the center 
 O raise the . perpendicular Om. 3. Make m ii 
 equal to tinec fDurths of Om, 4. From n draw nC^ 
 through the second division of the diameter. 5. 
 Join the points AC, and the line AC will be the 
 side of the required polygon, in this instance a 
 pentagon. 
 
 Problem 41. Ahont any given triangle ABC, 
 io circamscrihe a circle, fig. 21, plate 6. 
 
 1. Bisect any two sides AB, BC, by the per- 
 pendiculars mo, no. 2. From the point of inter- 
 section o, with the distance OA or OB, describe 
 the required circle. 
 
 Problem 42. About a given circle to circum- 
 scribe a pentagon, fig. 20, plate (). 
 
 1. Inscribe a pentagon within the circle, 2. 
 Through the middle of each side draw the lines 
 O A, O B, O C, O D, and O E. 3. Through the 
 point n draw the tangent AB, meeting OA and 
 OB in A and B. 4. Through the points A and m.
 
 FOR POLYGONS-. ^^ 
 
 eli'aw the line AmC, meeting OC in C. 5. In 
 like manner draw the Hnes C D, D E, E B, and 
 AB C D E will be the pentagon required. 
 
 In the same manner you may about a given cir- 
 cle circumscribe any polygon. 
 
 Problem 43. About a given square to circum^ 
 scribe a circle, f.g. 22, flate £). 
 
 1. Draw the two diagonals AD, B C, intersect- 
 ing each other atO. 2. From O, with the distance 
 OA or OB, describe the circle ABCD, which 
 will circumscribe the square. 
 
 Problem 44. 0;/ a givefi line A.^ to make it 
 regular hexagon, fig. 1A, flute 6. 
 
 1. On AB make the equilateral triangle AOB, 
 2. From the point O, with the distance OA or 
 OB, describe the circle ABCDEF. 3. Carry 
 AB six times round the circumference, and it will 
 form the required hexagon. 
 
 Pr OBLEM45. On a given line A B to form a re-- 
 gular polygon of any proposed nwnber of sides, fig. l-4» 
 plate (). 
 
 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 CA, describe a circle. 3. Apply 
 the chord AB to the circumference the proposed 
 number of times, and it will form the required po- 
 lygon. 
 
 Problem 46. On a giveji line AB, fig. 25, 
 plate 6, to form a regular octagon. 
 
 1. On the extremities of the given line AB 
 erect the indefinite perpendiculars AF and BE,- 
 
 2. Produce AB both ways to s and w, and bisect 
 the angle nAs and oBw by the lines AH, B C. 
 
 3. Make AH and BC each equal to AB, and 
 draw the line H C. 4. Make ov equal to on, and 
 through V draw GD parallel to HC. 5. Draw
 
 60 **0R POLYGONS. 
 
 H G and C D parallel to AF and B E, and make 
 cE equal to cD. 6. Through E draw E F, pa-^ 
 rallel to AB, and join the points GF and DE, 
 and ABCDEFGH will be the octagon required. 
 
 Problem 47. On a given right line KE,Jig. 26^ 
 ^Jate 6, to describe a regular pentagon. 
 
 1. MakeBm perpendicular, and equal to AB. 
 1. Bisect AB in n. 3. On n, with distance n m, 
 cross AB produced in O. 4. On A and B, with 
 radius AO, describe arcs intersecting at D. 5. On 
 D, with radius A B, describe the arc E C, and on A 
 and B, with the same extent, intersect this arc at 
 E and C. 6. Join AE, E D, D C, CB, and you 
 complete the figure. 
 
 Problem 48. Upon a given right line AB, 
 jig. 27, plate 6, to describe a triangle similar to the 
 triangle C D E. 
 
 1. At the end A of the given line AB, make 
 an angle FAB, equal to the angle E C D. 2. At 
 B make the angle AB Y equal to the angle C D E, 
 3. Draw the two sides AF, BF, and ABF will 
 be the required triangle. 
 
 Pro b l e m 49. To describe a polygon similar to a 
 given polygon ABCDEF, one of its sides ab being 
 given^ Jig. 1Q, plate 6. 
 
 1. Draw AC, AD, AF. 2. Set off ab on AB, 
 from a to r. 3. Draw rg parallel to BE, meeting 
 AF in g. 4. Through the point g draw gh, pa- 
 rallel to FC, meeting AC in h. 5. Through the 
 point h draw the parallel hi. 6. Through i draw 
 i k parallel to E D, and the figure A r g h i k will 
 be similar to the figure ABCDIeF. 
 
 Problem 50. To reduce a figure by a scale ^ 
 fig. 28 and 29, plate 6. 
 
 1. Measure each side of the figure AB C D E 
 with the scale G H. 2. Make ab as many parts 
 of a smaller scale K L, as AB was of the larger.
 
 DEDUCTION OF FIGURES. 81 
 
 3. be as many of K L, as B C of G H, and ac of 
 K L, AC, &c. by which means the figure will be 
 rxiduced to a smaller one. 
 
 ■OF THE TRANSFORMATION AND REDUCTION 
 OF FIGURES. 
 
 Pr o b l e m 5 1 . To change a tnaugle into another 
 of equal extent, hut different height, fg. 1, 2, 3, 4, 
 jilate 7. 
 
 Let AB C be the given triangle, D a point at 
 the given height. 
 
 Case 1. Where the point T),fig. l and 2, plate 
 7, is either in one of the sides, or in the prolonga- 
 tion of a side. 1 . Draw a line from D to the oppo- 
 site angle C. 2. Draw a line AE parallel thereto 
 from x\, the summit of the given triangle. 3. Join 
 D E, and B D E is the required triangle. 
 
 Case 2. When the point Y>,fg. 3 and 4, plate 7, 
 is neither in one of the sides, nor in the prolonga- 
 tion thereof. 1. Draw an indefinite line BDa, 
 from B through the point D. 2. Draw from A, 
 the summit of the given triangle, a line A a, pa- 
 rallel to the base BC, and eutting the line BD in a. 
 3. Join aC, and the triangle B a C is equal to the 
 triangle B AC; and the point D being in the same 
 line with B a. 4. By the preceding case, find a 
 triangle from D, equal to BaC; i.e. join DC, 
 draw a E parallel thereto, then join D E, and 
 B D E is the required triangle. 
 
 Corollary. If it be required to change the tri- 
 angle BAC into an equal triangle, of which the 
 height and angle BDE are given: 1. Draw the 
 indefinite line B DA, niaking the required angle 
 with B C. 2. Take on B 13 a a point D at the 
 given height; and, 3. Construct the triangle by 
 the foregoing rules. 
 
 G
 
 82 TRANSFORMATION AND 
 
 Prob LEM 52. To make an isosceles triangle AEB^ 
 Jig. 5, plate 7, equal to the scaletie triangle AC B. 
 
 1. Bisect the base in D. 2. Erect the perpen- 
 dicular DE. 3. Draw CE parallel to AB. 4. 
 Draw AE;, EB, and AEB is the required tri- 
 angle. 
 
 Problem 53. To make an equilateral triangle 
 equal to a given scalene triangle AB C, fg. 7 , plate 7 . 
 
 1. On the base AB make an equilateral triangle 
 ABD. 2. Prolong BD towards E. 3. Draw 
 CE parallel to AB. 4. Bisect DE at I, on D I 
 describe the semicircle D F E. 5. Draw B F^ the 
 mean proportional between BE, B D. 6. With 
 B F from B, describe the arc F G H ; with the 
 same radius from G, intersect this arc at H, draw 
 B H, G H, and B G H is the triangle required. 
 
 Corollary. If you want an equilateral triangle 
 equal to a rectangle, or to an ispsceles triangle ; 
 find a scalene triangle respectively equal to each, 
 and then work by the foregoing problem. 
 
 Problem 54. To reduce a rectilinear figure 
 ABCDE,j^^-. 8 and g, plate J, to another equal to 
 ity hut with one side less. 
 
 1 . Join the extremities E, C, of two sides D E, 
 D C, of the same angle D. 2. From D draw a 
 line DF parallel to E"C. 3. Draw EF, and you 
 obtain a new polygon ABFE, equal to ABCDE, 
 but with one side less. 
 
 Corollary. Hence every rectilinear figure may 
 be reduced to a triangle, by reducing it succes- 
 sively 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 ABCDEF, fig. lO and ll, plate 7, into 
 a triangle I AH, with its summit at A, in the cir- 
 cumference of the polygon, and its base on the 
 base thereof prolonged.
 
 REDUCTION OP FIGURES, 83 
 
 1. Draw the diagonal DF. 2. Draw EG pa- 
 rallel to DF. 3. Draw F G, which gives us a new 
 polygon, ABCGF, with one side less. 4. To re- 
 duee ABCGF, draw AG, and parallel thereto 
 FH; then join AH, and you obtain a polygon 
 ABCH, equal to the preceding one ABCGF. 
 5. The polygon ABCH having a side AH, which 
 may serve for a side of the triangle, you have only 
 to reduce the part ABC, by drawing AC, and 
 parallel thereto BI; join A I, and you obtain the 
 required triangle I AH. 
 
 N. B. In figure 10 the point A is taken at one 
 of the angular points of the given polygon; in 
 figure 11 it is in one of the sides, in which case 
 there is one reduction more to be made, than when 
 it is at the angular point. 
 
 Corollary. As a triangle may be changed into 
 another of any given height, and with the angle 
 at the base equal to a given angle; if it be rc- 
 fjuired 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, 
 Jig. 12, plate 7, draw the diagonal EC, and DF 
 parallel thereto; join EF, and the triangle EBF 
 js equal to the parallelogram EBCD- 
 
 OP THE ADDITION OP FIGURES. 
 
 1 . If the figures to be added are triangles ot the 
 same height as AMB, BNC, COD, DPE, 
 Jig. 14, plate 7, make a line AE equal to the sum 
 of their bases, and constitute a triangle A ME 
 thereon, whose height is equal to the given height, 
 and AM E will be equal to the given triangles. 
 
 G 2
 
 84 MULTIPLICATION, &C. OP FIGURES. 
 
 2. If the given figures are triangles of different 
 heights, or difTerent 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 triangle obtained may be changed into 
 a parallelogram by the last corollary. 
 
 MULTIPLICATION OF FIGURES. 
 
 1. To multiply AMB, /^. 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 AMB. Lengthen the base AE, so that 
 it may be four times AB ; join M E, and the trian- 
 gle AM E will be quadruple the triangle AMB, 
 2. By reducing any figure to a triangle, we may 
 obtain a triangle which may be multiplied in the 
 same manner. 
 
 SUBTRACTION OF FIGURES, 
 
 1. If the two triangles BAG, dac, fg. 15, 
 plate 7, are of the same height, take from the base 
 B C of the first a part D C, equal to the base d c 
 of the other, and join AD; then will the triangle 
 ABD be the difference 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 if 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.
 
 iJIVISiON OP FIGURES. 85 
 
 ■2. A triangle may be taken from a polygon by 
 drawing a line within the polygon, from a given 
 point F on one of its sides. To effect this, let us 
 suppose the triangle, to be taken from the polygon 
 ABCDE, ^^g. 16, plate 7, has been changed into 
 a triangle MOP, Jig. 26, whose height above its 
 base M P is equal to that of the given point F, 
 above AB offg. l6; this done, on AB (prolonged 
 if necessary) lay off AG equal to O P, join F G, 
 and the triangle AFG is 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, 
 flate 7, the point G will fall thereon, and the pro- 
 blem will be solved. 
 
 But if the base M P exceeds the base AB, G 
 will be found upon AB prolonged, ^^. 17 and 18, 
 f/a/e 7 ; join F B, and draw G H parallel thereto ; 
 from the situation of this point arise the other two 
 cases. 
 
 G/se 1. When the point H, Jig. 17, pliite 7, is 
 found on the side BC, contiguous to the side AB, 
 joinFH, and the quadrilateral figure FAB His 
 equal to the triangle MOP. 
 
 Case 2. When H, fg. 18, plate 7, meets BC 
 prolonged, from F draw FC and HI parallel 
 thereto; then join FI, and the pentagon FAB 
 C I is equal to the triangle MOP. 
 
 DIVISION OF RECTILINEAR FIGURES. 
 
 1 . To divide the triangle AM E,^g. 13, plate 7, 
 into four equal parts; divide the base into four 
 equal parts by the points B, C, D; draw M D, 
 M C, M B, and the triangle will be divided into 
 four equal parts.
 
 S6 DIVISION OP 
 
 2. If the triangle AME, fig. \g, plate 7, is to 
 be divided into four equal parts from a point m, 
 in one of its sides, change it into another Ame, 
 •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 AME, the problem is 
 solved; but if some of the lines, as mD, terminate 
 without the triangle, join mE, and draw dD pa- 
 rallel thereto; join md, and the quadrilateral 
 m C E d is equal to CmD ith of AM E, and the 
 triangle is divided into four equal parts. 
 
 3. To divide the polygon ABCDEF, fig. 10, 
 plate 7, 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 GM, 
 whose summit is at G. 2. Divide this triangle 
 into as many equal triangles AGH, HGI, IGK, 
 K G M, as the polygon is required to be divided 
 into. 3. Subtract from the polygon a part equal 
 *to the triangle AGH; then a part equal to the 
 triangle AG I, and afterwards a part equal to the 
 triangle AGK, and the lines GH, GR, GO, 
 drawn from the point G, to make these subtrac- 
 tions, will divide the polygon into four equal 
 parts, all which will be sufficiently evident from 
 consulting the figure. 
 
 Problem 55. J^hree points, fig. 21 to 15,pJate7, 
 N, O, A, being given, arranged in any manner on a 
 strait line, to find two other poitits, B_, b, in the sam^ 
 line so situated, that as 
 
 NO : AB :: AB : NB 
 
 NO : Ab :: Ab : N 1> 
 
 Make AP= , and PL= , placing 
 
 4. 4 
 
 them one after the other, so that A L= .♦
 
 RECTILINEAR FIGURES. 87 
 
 Observing, 1. That in the two figures 21 and 
 22, A is placed between N and O; and that AP 
 is taken on A O, prolonged if necessary. 
 
 2. In Jig. '23, where the point O is situated be- 
 tween N and A, AP should be put on the side 
 opposite to AO. 
 
 3. In^g. 24, where the point N is situated be- 
 tween A and O, if AP be smaller than AN, it 
 must be taken on the side opposite to AN. 
 
 4. In Jig. 25, where the point N is also placed 
 between A and O, if AP be greater than A N, it 
 must be placed on AN prolonged. 
 
 Now by problem 17 make M N in all the five 
 figures a mean proportional between NO, N P, 
 and carry this line from L to B, and from L to b. 
 and NO will be : AB : : AB : NB 
 NO : Ab :: Ab : NL. 
 
 Problem 56. Two lines EF, GH, ^^. 27, 
 28, 29, plale 7, intersecting each other at A, being 
 given, to draw from C a third line B D, which shall 
 form with the other two a triangle DAB equal to a 
 given triangle X. 
 
 1 . From C draw CN parallel to EF. 2. Change 
 the triangle X into another C N O, whose summit 
 is at the point C. 3. Find on GH a point B, so 
 that N O : AB : : AB : N B, and from this point 
 B, draw the line CB, 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. 
 
 Thus, if from a point C, without or within the 
 triangle E A ll,j7g. 30, a right line is required ta 
 be drawn, that shall cut off a part DAB, equal to 
 the triangle yL,Jig. 30 ; it is evident this may bft 
 effected by the preceding problem.
 
 SS DIVISION OF FIGURES, 
 
 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,^^^. 31, equal 
 to the triangle Z; if you are sure, that the right 
 line BD will cut the two opposite sides EF, GH, 
 proloni>; E F, GH, till they meet; then form a 
 triangle Z, equal to the two triangles Z and FAG; 
 and then take Z from AEH by a line BD from 
 the point C, which is etfected by the preceding 
 problem. 
 
 If it be required to take from a polygon Y, a 
 part DFIHB equal to a triangle X; and that 
 the line BD is to cut the two sides EF, GH; 
 prolong these sides till they meet in A; then make 
 a triangle Z, equal to the triangle X, and the fi- 
 gure AF I H ; and then retrench from the triangle 
 E A G the triangle 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 strait line drawn 
 from a given point. 
 
 Problem 57. To f?iake a triangle equal to any 
 given quadrilateral figure ABCD,_;?^. 2>2>, plate"] , 
 
 1. Draw the diagonal B D. 2. Draw CE pa- 
 rallel thereto, intersecting AD produced in E. 
 3. Join AC, and ACE is the required triangle. 
 
 Problem 58. To maize a rectangle^ or paralle- 
 logram equal to a given triangle ACE, j^^. 33, 
 plate y. 
 
 1. Bisect the base A E in D. 1. Throus:h C 
 draw CB parallel to AD. 3. Draw CD, "BA, 
 parallel to each other, and either perpendicular to 
 A E, or making any angle with it. And the rect- 
 angle or parallelogram ABCD v/ill be equal to X\\e 
 given triangle.
 
 ADDITION OF FIGUKES. SQ 
 
 "Problem 59. To make a triangle equal to a 
 ghefi circle, ^g. ^4, plate 7.* 
 
 Draw the rudius OB, and tangent AB perpen- 
 dicular thereto; make AB equal to three times 
 the diameter of the cirele, and f more; join AO, 
 and the triangte^OB will be nearly equal the 
 given eirele. 
 
 Problem 6o. To make a square equal to a 
 given rectangle, AB C D, fig. 35, plate 7. 
 
 Produce one side AB, till B L be equal to the 
 other BC. 2. Bisect A.L in O. 3. With the 
 distance AO, describe the semicircle LFA. 4. 
 Produce B C to P\ 5. On BP" make the square 
 BFGH, which is equal to the rectangle ABCD. 
 
 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 gl-veii squares, whose sides are equal to 
 the lines AB C,fig. 36, plate 7. 
 
 1. Draw the indefinite lines ED, DF, at right 
 angles to each other. 2. Make D G equal to A, 
 and D H equal to B. 3. Join G and H, and G PI 
 will be the side of a square, equal the two squares 
 whose sides are A and B. 4. Make DL equal GPI, 
 DK equal C, and join K L; then will KL be the 
 side of a square equal the three given squares. Or, 
 after the same manner may a square be constructed 
 equal to any number of given squares. 
 
 Pr o b l e m 62. To describe a figure equal to the sum 
 •f any given number of similar figures, fig. 3Q, plate 7 , 
 
 This problem is similar to the foregoing : 1 . P'orm 
 a right angle. 2. Set off thereon two homologous 
 sides of the given figures, as from D to G, and from 
 
 * Strictly, thii can only be solved but by an approximation ; the area, oj 
 sfjuaiiiig of the circle is yet a desiderata in mathematics. See Huttun' s l^/(^• 
 thcmuii.al and I'kiks-jphktil J)ictionarj-, > Vols. 4to. 1 796. Edit.
 
 00 ADDITION AND Sl/BTRACTION 
 
 D to H. 3. Draw GH, and thereon describe a 
 ligure similar to one of the giv^en ones, and it will 
 be equal to their sum. In the same manner you 
 may go on, adding a greater number of similar 
 figures together. 
 
 If the similar figures be circles, take the radii 
 or diameters for the homologous lines. 
 
 Problem 63. To 7?iake a square equal to the 
 difference of two given squares ^ whose sides are AB, 
 CD, f g.^T, plate T. 
 
 1 . On one end B of the shortest line raise a per- 
 pendicular B F. 2. With the extent C D 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 make a figure which shall he 
 similar to^ and contain a given figure^ a certain mimher 
 of times. Let ^^ he an homologous side of the. 
 givefi figure, fig. ^iS, plate J. 
 
 1. Draw the indefinite line BZ. 2. At any 
 point D, raise DA perpendicular to BZ. 3. Make 
 B D equal to M N, and BC as many times a mul- 
 tiple of B D, as the required figure is to be of the 
 given one. 4. Bisect BC, and describe the semi- 
 circle BAC. 5. Draw AC, BC. 6. Make AE 
 equal M N. 7. Draw EF parallel to BC, and 
 E F will be the homologous side of the required 
 figure. 
 
 Problem 65. To reduce a complex figure from 
 one scale to another, mechanically, by means of squares. 
 Fig. 3C), plate 7. 
 
 1 . Divide the given figure by cross lines into as 
 many squares 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 contained in th«
 
 OP SIMILAR FIGURES. Ql 
 
 correspondent square of the given figure, and you 
 will obtain a copy tolerably exact. 
 
 Problem 66. To enlarge a map or plan, and 
 make it twice, three, four, or five, &c. times larger 
 than the original, fig. 12, plate 8. 
 
 1 . Draw the indefinite line ab. 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 a to e, 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 I ; complete 
 the square al n g, and it will be double the square 
 a e f d. To find one three times greater, take d g, 
 and with that extent form the square amoh, 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 1 gives a square five times 
 larger than the original square. 
 
 Problem 6^ . To reduce a map ^, \d, \th, \th, 
 "^c. of the original, fig. b, plate 8. 
 
 1 . Divide the given plan into squares by pro- 
 blem 65. 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 
 center, with FA or FB, describe the semicircle 
 AHB. 
 
 4. To obtain \ the given square, at F erect thc 
 perpendicular F H, and draw the right line AH^ 
 which will be the side of the required square, 
 5. For id, divide AB into three parts; take one 
 of these parts, set it off from A to C, at C raise the 
 perpendicular C I, through I draw AI, and it will 
 be the side of the required square. 6. For vth, di- 
 vide AB into four equal parts, set off one of these 
 from B to E, at E make E G perpendicular to AB^.
 
 §2 CURIOUS PKOBLEMS ON THE 
 
 join B G, and it will be the side of a square |th of 
 the given one. 
 
 Problem 68- To malie a map or plan hi propor- 
 tion lo a given one^ e. gr. as three to froe, Jig. 6^ 
 plate 8. 
 
 The original plan being divided into squares, 
 1. draw AM equal to the side of one of these 
 squares. 2. Divide AM into five equal parts, 
 8. At the third divison raise the perpendicular CD, 
 and draw AD, which will be the side of the re- 
 quired square. 
 
 Problem 69. To reduce figures hy the angle of 
 reduction. Let a b he the given side on •which it is 
 required to construct a figure similar to ACDE, 
 fig. \,rL,'i, plate 8. 
 
 1 . Form an angle L M N at pleasure, and set 
 off the side AB from M to I. 2. From I, with ab, 
 cut ML in K. 3. Draw the line IK, and se=- 
 veral lines parallel to, and on both sides of it* 
 The angle Lr^l Nis called the angle of proportion 
 or reduction. 4. Draw the diagonal lines B C, 
 AD, AE, BD. 5. Take the distance B C, and 
 set it off from M towards L on ML. 6. Measure 
 its corresponding line KL 7- From b describe 
 the arc no. 8. Now take AC, set it off on M L, 
 and find its correspondent line ig. 9. From a, 
 with the radius ( g, cut the former arc no in c, and 
 thus proceed till you have completed the figure. - 
 
 Problem 70- To enlarge a figure by the angle 
 of reduction. Let abcde he the given figure, and 
 AB the given side, fig. 3,1, and A, plate 8. 
 
 1. Form, as in the preceding problem, the an- 
 gle LMN, by setting off ab from M to H, and 
 from H with AB, cutting M L in L 2. Draw 
 H I, and parallels to, and on both sides of it. 
 3. Take the diagonal be, set it ofi'from M towards 
 L, and take off" its corresponding line qr. 4. With
 
 ^ DIVISION OF LINES AND CIRCLflS. QS 
 
 qr as a radius on B describe the arc mn. 5. Take 
 the same correspondent line to ac, and on A cut 
 mn in c, and so on for the other sides. 
 
 CURIOUS PROBLEMS ON THE DIVISION OP 
 LINES AND CIRCLES. 
 
 Problem 71. To cut off from a?iji given arc of 
 a circle a third., a fifth., a se-vetith, &c. add parts j 
 (ind thence to divide that arc i?ito anjy number of equal 
 parts J fig. 7 J plate 8. 
 
 Exaniple 1. To divide the arc AK B into three 
 equal parts, CA being the radius, tind C the cen- 
 ter of the arc. 
 
 Bisect AB in K, draw the two radii CK, C B, 
 and the -chord AB; produce AB at pleasure, and 
 make BL equal AB; bisect AC at G; then a 
 rule on G and L will cut CB in E, and B E will 
 be id, and C E ^ Js of the radius C B; on CB with 
 C E describe the arc Eed; lastly, set off the extent 
 E e or D e from B to a, and from a to b, and the 
 arc AK B will be divided into three equal parts. 
 
 Corollary. Hence having a sextant, quadrant, 
 &c. accurately divided, 2 the chord of any arc set 
 off upon any other arc of h that radius will cut off 
 an arc similar to the first, and containing the same 
 number of degrees. 
 
 Also id, ith, tth, &c. of a larger chord will 
 constantly cut similar arcs on a circle whose radius 
 is yd, ;^th, fth, &c. of the radius of the first arc. 
 
 Example 2. Let it be required to divide the arc 
 AKB into live equal parts, or to find the ^t\\ 
 part of the arc AB. 
 
 Having bisected the given arc AB in K, and 
 drawn the three radii CA, CK, CB, and having 
 found the fifth part of I B of the radius C B, with 
 radius C I describe the arc I n M, which will be 
 
 ■V
 
 P4 CURIOUS PROBLEM^ ON THR 
 
 bisected in n, by the line CK; then take the ex- 
 tent 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,, 
 and oB will be ith of the arc AB. Again, set off 
 the same extent from B to m, and from m to c, and 
 the arc AB will be accurately divided into five 
 equal parts. ' 
 
 Example 3. To divide the given arc AB into 
 seven equal parts. AB being bisected as before, 
 and the radii CA, C K, C B, drawn, find by pro- 
 blem 9 the seventh part of PB of the radius CB, 
 and with the radius CP describe the arc PrN; 
 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 Al will bisect 1, 3 into 2, 
 and 4, 6 into 5 ; and thus divide the given arc into 
 seven equal parts. 
 
 Problem 7-- T^o inscribe a regular heptagon 
 in a circle, jig. 8, 'plate 8. 
 
 In j?ir. 8, let the arc B D be ?th part of the gi- 
 ven circle, and AB the radius of the circle. Di- 
 vide AB into eight equal parts, then on center A, 
 with radius AC, describe the arc CE^ bisect the 
 arc B D in a, and set off' this are Ba from C to b, 
 and from b to c; then through A and c draw Ace, 
 cutting BD in e, and Be will be |th part of the 
 given circle. 
 
 Corollary 1. Hence we have a method of find- 
 ing the seventh part of any given angle; for, if 
 from the extremities of the given arc radii be 
 drawn to the center, 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 7j and sa 
 of any other proportion.
 
 BIYISION OF LINES AND CIRCLES. 05 
 
 Corollary 2. If an arc be described with the 
 radius, it will be equal to the ^th part of a circle 
 whose radius is AB, and to the seventh part of a 
 circle whose radius is AC, and to the sixth part 
 of a circle whose radius is AL, &c. 
 
 Corollary 3. Hence also a pentagon may be 
 derived from a hexagon, fig. 9, plale 8. Let the 
 given circle be ABCDEF, in which a pentagon 
 is to be inscribed ; with the radius AC set off AF 
 equal ith of the circle, divide AC into six equal 
 parts; then cG will be five of these parts; with 
 radius CG describe the arc GH, bisect AF in q, 
 and make GP and PH each equal to A q; then 
 through C and H draw the semi-diameter C w, 
 cutting the given circle in w, join Aw, and it will 
 be one side of the required pentagon. 
 
 Corollary 4. PIcnce as radius divides a circle 
 into six equal parts, each equal Go degrees, twice 
 radius gives 120 degrees, or the third part of the 
 circumference. . 
 
 Once radius gives 60 degrees, and that arc bi- 
 sected gives 30 degrees, which, added to 60, di- 
 vides 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 di- 
 vides it into eight equal parts. By problem 71 we 
 obtain the seventh part of a circle, and by this me- 
 thod 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 number of equal farts hy 
 the help of a fair of heani, or other compasses, the
 
 p6 CURIOUS PROBLEMS ON THE 
 
 distance of whose pohits shall not he nearer to each 
 other than the given line, fig. \\, plate 8. 
 
 From this problem, published by Clavius, the 
 Jesuit, in l6l 1, in a treatise on the construction of 
 a dialling instrument, it is'presunicd that he was the 
 original inventor of that species of division, called 
 No7/ms, and which by many modern mathemati- 
 cians has been called the scale of Fernier. 
 
 Let AB be the given line, or circular arch, to be 
 divided into a number of equal parts. Produce 
 them at pleasure; then take ^c extent AB, and 
 set it off on the prolonged line, as many times as 
 the given line is to be divided into smaller parts, 
 BC, CD, DE, EF, FG. Then divide the whole 
 line AG into as many equal parts as are required in 
 AB, asGH, HI, IK, KL, LA, each of which 
 contains the given line, and one of those parts 
 into which the given line is to be divided. For 
 AG is to A L as A F to AB; in other words, AL 
 is contained live times in AG, as AB in AF; 
 therefore, since AG contains AF, and ith of AF, 
 AL .will contain AB, and \\h of AB; therefore 
 BL is the fth of AB. Then as GH contains 
 AB plus, FH, which is fths of AB, EI will be 
 Iths of AB, D K Ith, C L l^ths. Therefore, if we 
 set ofi'the interval GH from F and H, wc obtain 
 two parts at L and I, set off from the points near 
 E, gives three parts between D K, from the four 
 points at D and K, gives four parts at CL, 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 AC, or arc of any circle, 
 into any number of equal parts, suppose 30, fig. 1 1 , 
 plate 8. 
 
 1. Divide it into any number of equal parts 
 less than 30, yet so that they may be aliquot parts
 
 'DIVISION OP LIXES AND CIRCLES. Q7 
 
 of 30; as for example, AG is divided into six equal 
 parts, A B, BC, CD, DE, EF, FG, each of which 
 are to be subdivided into five equal parts. 2. Di- 
 vide the first part AB into five parts, by means of 
 the interval AL or GH, as taught in this problem. 
 If, now, one foot of the compasses be put into the 
 point A, (the extent AL remaining between them 
 unaltered) and then into the point next to A, and 
 so on to the next succeeding point, the whole line 
 AG will be divided by the other foot of the com- 
 passes into 30 equal parts. 
 
 Or if the right line, or arc, be first divided into 
 five equal parts, each of these must be subdivided 
 into six parts, which may be effected by bisecting 
 each part, and then dividing the halves into three 
 parts. 
 
 Or it may be still better to bisect three of the 
 first five parts, and th-en to divide four of these 
 into three, which being set off from every point, 
 will complete the division required. 
 
 Corollary. It frequently happens that so many 
 small divisions are required, that, notwithstanding 
 their limited number, they can be hardly taken 
 between the points of the compasses without error; 
 in this case use the following method. 
 
 If the whole number of smaller parts can be 
 subdivided, take so many of the small parts in the 
 given line, as each is to be subdivided into, yet so 
 that they may together make up the whole of the 
 given line. For if the first of these parts be cut 
 into as many smaller parts as the proposed number 
 requires every one of them to contain, and the 
 same is also done in the remaining parts, we shall 
 obtain the given number of smaller parts. 
 
 If 84 parts are to be taken in the proposed line, 
 first bisect it, and each half will contain 42 ; bisect 
 these again, and you have four parts^ each of which 
 
 K
 
 Q8 CL'RIOUS FROBtEMS OX THE 
 
 is to contain 21 ; and these, divided into thrce^ 
 give 12 parts, each of which is to contain seven 
 parts; subdivide these into seven each. 
 
 But if the proposed number of small parts can- 
 not be thus subdivided, it will be necessary to take 
 a number a little less or greater, that will be capa- 
 ble of subdivision ; for if the superfluous parts are 
 rejected, or those wanting, added, we shall obtain 
 the proposed number of ])art3. Thus if 74 parts 
 are to be cut from any given line of 80 parts; 1. 
 Bisect the given line, and each ^ will contain 40. 
 2. Bisect these again, and you have four parts to 
 contain 20 each. 3. Each of these bisected, you 
 have eight parts to contain 10 each. 4. Bisect 
 these, and you obtain l6 parts, each to be divided 
 into five parts. 6. llejcct six of the parts, and 
 the remainder is the 74 parts proposed. 
 
 Or if 72- parts be proposed; divide the line into 
 24 equal parts, and each of these into three parts, 
 and you obtain 72; to which adding two, you 
 will obtain the number of 74. 
 
 Problem 74. To cut off from the c'lrcumfer-eiice 
 of any given arc of a circle unj number of degrees anci 
 minutes, jig. 3, plate Q. 
 
 1. Let it be proposed to cut off from any arc 57 
 degrees; with the radius of the given arc, or circle, 
 describe a separate arc as AB, and having set off 
 the radius from A to C, bisect AC in E, then AE 
 and EC will be each of them an arc of 30 deo-rees. 
 Make CB equal to AE, and AB will be a qua- 
 drant, or QO degrees, and will also be divided into 
 three equal parts; next, divide each of these into 
 three by the preceding rules, and the quadrant 
 will be divided into nine equal parts, each contain- 
 ing 10 degrees. Lastly, divide the first of these 
 into 10 degrees, then set one loot of the compasses 
 into the seventh single degree, and extend the
 
 DIVISION OF LINES ANb CIRCLES. QQ 
 
 Other to the 50th, and the distance between the 
 points of the compasses will contain 57 degrees, 
 which transfer to the given arc. Or at two ope- 
 rations; first, take 50 degrees, and then from the 
 first 10 take seven, and you have 57 degrees. 
 
 2. Let it be required to cut off from any given 
 arc of a circle 45 degrees, 53 minutes, f^. 3, 
 plafe Q. 
 
 Divide the arc of 53 degrees of the quadrant 
 AB, whose radius is AC (or rather its equal arc 
 FG) into 6o equal parts, first into 5, and then 
 one of these into three; or first into three, and 
 one of these into five. Again, one of these bi- 
 sected, and this bisection again bisected, gives the 
 6oth part of an arc of 53 degrees. 
 
 For the 5th part of the arc F G is F H, con- 
 taining 12 parts; its third part is FI, containing 
 four parts; the 2 of FI is N, which contains two 
 parts; and FN again bisected in K, leaves FK the 
 60th part of the arc FG; consequently, FK com- 
 prehends 53 minutes; therefore, add the arc FK 
 to 45 degrees, and the arc AF will contain 45 de- 
 grees 53 minutes. 
 
 Corollary. If w^e describe a separate arc LM 
 with the radius AC, and set off thereon the extent 
 L M of Ol degrees of the given arc AB, and divide 
 LM into 60 equal parts; thus, first into two, then 
 both of these into three, and then the first of these 
 three into 10 by the former rules, which gives the 
 60th part of the arc L M. And as one division 
 of the arc LM contains by construction one de- 
 gree of the quadrant AB, and one sixtieth part of a 
 degree more, that is one minute, therefore two di- 
 visions of L M contain two degrees, and two mi- 
 nutes over; three divisions exceed three degrees by 
 three minutes, and so of all the rest. 
 
 H 2
 
 100 CURIOUS PROBLEMS ON THE 
 
 Whence if one division of LM be set off froiil- 
 any degree on AB, it will add one minute to that 
 degree; two adds two minutes; three, three mi- 
 nutes, and so on. 
 
 When the division is so small that the compasses 
 will hardly take it in without error, take two, 
 t-hree, four, or more of the parts on L, set them off 
 from as many degrees back from the degree in- 
 tended, and you will obtain the degree and minute 
 required. 
 
 Problem 75. To diviile a circle into any uneven 
 number of equal 'parts. 
 
 Example 1 . Let it be required to divide a circle 
 into 34dT equal parts. 
 
 Reduce the whole into 3d parts, which gives us 
 1040; find the greatest multiple of 3 less than 1040, 
 which may be bisected; this number will be found 
 in a double geometrical progression, whose first 
 term is 3, as in the margin ; 7G8 the ninth immber, 
 is the number sought, as in the margin. Subtract 
 768 from 1040, the remainder is 27 '2, then 
 find how many degrees and minutes this 
 remainder contains by the rule of three. 
 As 1040 is to 3(i0 degrees, so is 272 to 
 94° 9' 23". Now set off 94° 9' 23" ujjon 
 the circle to be divided, and divide, the 
 remaining part of that circle by continual 
 bisections, till you come to the mimbcr 3, 
 which will be one of the required divisions 768 
 of the 34(3 equal parts, by which dividing the. arc 
 of 94° 9' 23" you will have the whc^lc circle di- 
 vided into 346| equal parts; for there will be 25(i 
 divisions in the greatest arc, and 9OT in the other. 
 
 Exainph 2. Let it be rc(}uired to divide a circle 
 into 179 e(|unl jjarts. Find the grciitest nun:iber 
 not excec'ding 179> which may be continually bi- 
 
 3 
 
 2 
 
 6 
 
 2 
 
 12 
 
 2 
 
 24 
 
 2 
 
 48 
 
 2 
 
 9(3 
 
 2 
 
 192 
 
 2 
 
 384 
 
 2 
 
 7fis 
 
 o.
 
 DIVISION OP LINES AND CIRCLES. 101 
 
 sectcd to unity, which you will find to be 128. 
 Subtract 128 from 179, the remainder is 51 ; then 
 find what part of the circle this remainder will 
 occupy by the following proportion, as 179 is to 
 360, so is 51 to 10'2°34' ll''; set off' from the 
 circle an arcof 102" 34' ll", and divide the re- 
 maining part of the circle by continual bisections, 
 seven of which will be unity in this example; by 
 which means this part of the circle will be divided 
 into 128 equal parts, and the remaining 51 may be 
 obtained by using as many of the former bisections 
 as the space will contain, so that the whole cir- 
 cumference will be divided into 1 79 equal parts. 
 
 ExiDiiple 3 . Let it be required to divide a circle 
 into 29^ equal parts, to rej)resent the days of the 
 moon's age. 
 
 Reduce the given number into halves, which 
 gives 59 parts; seek tlie greatest lunnbcr, not ex- 
 ceeding 59, which may be continually bisected to 
 unity, which you will find to be 32. Subtract this 
 from 59, the remainder is 27 ; and find, as before, 
 the angle equal to the remainder by this propor- 
 tion, as 59 is to 3()0, so is 27 to l64° 44' 44"; set 
 off 164° 44' 44", divide the remaining part of the 
 circle by continual bisections, \\hich will divide 
 this portion into 32 parts, and from that, the rest 
 into 27^, making 29^ parts, as required. 
 
 Example 4. Let it be required to divide a circle 
 into 365" o 49" e(|ual parts, the length of a tropi- 
 cal yeal\ 
 
 Reduce the whole into minutes, \\hich will be 
 525949; then seek the greatest multiple of 1440, 
 the minutes in a solar day, that may be halved, and 
 is at the same time less than 525949; this you will 
 firftl to be 368640, which subtracted from 525949, 
 leaves 157309. To find the number of degrees
 
 102 CURIOUS PROBLEMS ON THfi 
 
 that is to contain this number, use the following; 
 proportion; as 525949 is to 157309 multiplied 
 by 360, so is 157309 to 107° 40' if A^" . 
 
 Now set off an angle of 107" 40' 27" 49''' upon 
 the circle to be divided, and divide the remaining 
 part of that circle by continual bisections, till you 
 come to the number 1440, which in this case is 
 unity, or one natural solar day; by which, dividing 
 the arc of 107° 40' 27'' 49'", the whole circle will 
 be divided into 365° b' 49"; for there will be 256 
 divisions, or days, in the greater arc, and 109° b' 
 49" in the lesser arc. 
 
 Example 5. Let it be required to divide a circle 
 into 365i equal parts; which is the quantity of a 
 Julian year. 
 
 Reduce the whole into four parts, which gives 
 tis 1461 ; 1024 is the greatest multiple of 2, less 
 than 1461 ; when subtracted from 1461, we have 
 for a remainder 437- Then by the following pro- 
 portion, as 1461 is to 437 X360, so is 407 to 
 107° 40' 46" 49'", the degrees to be occupied by 
 this remainder. 
 
 Set off an angle of 107° 40' 47" upon the circle to 
 be divided, and divide the remainder by continual 
 bisections, until you arrive at unity, by which di- 
 viding the arc, you will have the whole circle 
 divided into 365^ parts. 
 
 Problem 76. To divide a quadrant y or circle ^ 
 into degrees, fig. 10, flaie 8. 
 
 Let ABbe the quadrant, C the center thereof. 
 With the radius AC describe the two arcs AD, 
 B E, and the quadrant will be divided into three 
 equal parts, each equal to 30 degrees; then divide 
 each of these into five parts, by the preceding rules, 
 and the quadrant is divided into 15 equal parts; 
 bi^iect these parts^ and then subdivide as already
 
 tUVISlO^ OP LINES AXD CIRCLES. 105 
 
 directed, and the quadrant will be divided into 
 90 degrees. Other methods will be soon ex- 
 plained more at large. 
 
 Problem 77- To find what part any smaller line 
 or arc, is of a greater, as for exa)nple, any angle is of 
 a semicircle. 
 
 Take the smaller with a pair of compasses, and 
 with this opening step the greater. With the re- 
 mainder, or surplus, step one of the former steps; 
 with the remainder of this, step one the last 
 steps, setting down the number of steps each time. 
 About five times will measure angles to five se- 
 conds. 
 
 Then to find the fraction, expressing what part 
 of the whole the smaller part is, 
 
 Suppose the number of steps each time to be 
 e. d. c. b. a. 
 
 9. 7. 8. 1. 5. Then 5 X 2 -|- 1 = 11, and 11 X 
 8 4- 5 = 93, and 93 X 7 + 1 1 = 6(52, and 662 
 X 94-93=6051; so that Si-ix is the fraction re- 
 <]uired. 
 
 If we call the number of steps ab c d e begin- 
 ning at the last, the rule may run thus : multiply 
 a by b, and add 1 ; multiply that sum by c, and 
 add a; multiply this sum by d, and add the pre- 
 ceding sum ; multiply this sum by e, and add the 
 preceding sum; then the two last sums arc the 
 terms of the fraction., 
 
 d, c. b. a. 
 
 Example 1 . Suppose thc'Stcps are 3. 5. 1. 9. then 
 ^ X 1 H- 1 = 10, 10 X 5 -}- 9 = 59, and 59 X 
 3 4- 10 =: 187; hence the terms are -frr. Now 
 180^ X 59 = 10620, this divided by 187 gives 
 56°, with a remainder of 148, 148 X 60 = 8880, 
 *wT, gives 47, &c. so that the measure required 
 is 56° 47' 2j^".
 
 104 CURIOUS PROBLEMS ON THE 
 
 Example 3. Take a semicircle three inches ra- 
 dius and let the angle be 2 in 1 ; then the steps 
 will be 4, 2, 1, 1, 3, 2."the answer 41° lo' 50" g"\ 
 
 The whole circle, continued and stepped with 
 the same opening, gave 8. 1. 3. 2. 3. for the same, 
 yet the answers agreed to the tenth of a second. 
 
 By the same method any given line may be 
 measured, and the proportion it bears to any other 
 strait line found. Or it will give the exact value 
 of any strait line, ex. gr. the opening of a pair of 
 compasses by stepping any known given line with 
 it, a?id this much riearer than the eye can discern, hy 
 comparing it with any other line, as afoot, a yard, &c. 
 
 This method will be found more accurate than 
 by scales, or even tables of sines, tangents, &c. be- 
 cause the measure of a chord cannot be so nicely 
 determined 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 different persons. 
 There will, therefore, be always some small varia- 
 tion 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, ex- 
 actly equal to the given one, first measured by 
 way of example, and this arising from the inequa- 
 lity of our various scales, our inattention in mea- 
 suring, and the imperfection of our eyes. Hence, 
 though to all appearance two angles may apj^ear 
 perfectly equal to each other, this method will give 
 the true measure of each, and assign the minutest 
 difference between them.
 
 DIVISION OF LINES AND CIRCLES. 103 
 
 Figure I, plate g, will illustrate clearly this me- 
 thod; thus, to measure the angle AG B, take AB 
 between your compasses, and step Bii, a b, be, 
 there will be c D over. 
 
 Take c D, and with it step Ae, ef, and you will 
 have'fB over; with this opening step Ag, gh, 
 and you will have h e over, and so on. 
 
 Problem 78. To divide a large quadrant or 
 circle. 
 
 We shall here give the principal methods used 
 by instrument-makers, before the pu})]ication of 
 Mr. Bird's method by the Board of Longitude, 
 leaving it to artists to judge of their respcclive 
 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, pre\iousIv to men- 
 tion a few circumstances, \\'hich, though in com- 
 mon use, had not been described until Mr. /ii/v/'s 
 and Mr. Ludlam\ comment thereon were pub- 
 iished. 
 
 " 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 boundino; 
 circle; it is this that is originally divided. It has 
 been termed the priinili-ve circled 
 
 *' The divisions made upon this circle arc faint 
 arcs, struck with the beam compasses; fine 
 points, or conical holes, are made by the prick 
 punch, or pointing tool, at the points wliere these 
 arcs cross the primitive circle; these arc called 
 original points ^ 
 
 " The visible divisions are transferred from the 
 original points to the space between the bounding
 
 106 CURIOUS PROBLEMS ON THE 
 
 circles, and are cut by the beam compasses ; they 
 are therefore always arcs of a circle, though so 
 short, as not to be distinguished from strait lines." 
 
 Method 1. The faint or primitive arc is first 
 fetruck; the exact measure of the radius thereof 
 is then obtained upon a standard scale with a no- 
 nius division of 1000 parts of an inch, which if 
 the radius exceed 10 inches, may be obtained to 
 live places of figures. This measure is the chord 
 of 6o. The other chords necessary to be laid off 
 are computed by the subjoined proportion,* and 
 then taken off from the standard scale to be laid 
 down on the quadrant. 
 
 Set off the chord of 6o°, then add to it the chord 
 of 30, and you obtain the goth degree. 
 
 Mr. Bird, to obtain 90°, bisects the chord of 
 60°, and then sets off the same chord from 30 to 
 90°, and not of 30 from 6o° 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 bisected, (see remarks on bisection 
 hereafter) and if the radius unaltered be set off 
 from the point of bisection, 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 ^(f. 
 
 Sixty degrees is divided into three parts by set' 
 ting off the computed chord of 20 degrees, and 
 the whole quadrant is divided to every 10 de- 
 grees, 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 
 
 * As the radius Is to the given angle, so is the measure of 
 the radius to half the required chord.
 
 DIVISION OP LINES AND CIRCLES. 107 
 
 already fouiKl, divides the quadrant to every fifth 
 degree. 
 
 The computed chord of 6" being laid off, di- 
 vides 30 degrees into five parts; and set otr'froni 
 the other divisions, subdivides the quadrant into 
 single degrees. 
 
 Thus with five extents of the beam compasses, 
 and none of them less than six degrees, the cpia- 
 drant is divided into 90 degrees. 
 
 Fifteen degrees bisected, gives 7° 30', \\Jiieh set 
 off from the other divisions, divides the quadrant 
 into half degrees. 
 
 The chord of d° 40' divides 20'^ into tln-ce })ans, 
 and set off from the rest of the divisions, divides 
 the whole instrument to every ten minutes. 
 
 The chord of 10° 5' divides the degrees into 
 12 parts, each equal to five minutes of a degree. 
 
 Method 1. The chords are here supposed to l^e 
 computed as before, and taken ofi' from the noniuis 
 scale. 
 
 1. Radius bisected divides the quadrant inio 
 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 oft'' as before gives QO parts, each cf[ual to 
 1 degree. 
 
 5. The chord of 6° 40' gives 270 parts, each 
 equal to 20 minutes. 
 
 (1 The chord of 7° 30' gives 540 parL^, cacll 
 equal to 10 minutes. 
 
 7. The chord of 7'' 45' gives 1080 parts, cacll 
 equal to five minutes. 
 
 Qx M<ilhod?j. The computed chords supposed.
 
 108 MR. BIRD S METHOD 
 
 1 . 60 into i gives 3 parts, equal 30*^ 
 
 2. 30 i equal J3 6 15" 
 
 3. 15 3 equal 5 18 5"^ 
 
 4. 20 5 equal () 90 l'' 
 
 5. 20 3 equal 6"" 40' 270 20' 
 
 6. 15 i equal 7° 30' 540 lO' 
 
 7. 15, 30 5 equal 7"" 45' 1080 5' 
 
 • Thus may the praetitioncr vary his numbers for 
 
 any division whatsoever, and yet preserve a suffi- 
 cient extent between the points of his com- 
 passes. 
 
 If the quadrant be divided as above, to every 1 5 
 degrees, and then the computed arc of 1 6 degrees 
 set off, this arc may be divided by continual bisec- 
 tion into single degrees. If from the arc of 45°, 
 2° 20' be taken, or 1° lo' from 22° 30', we may 
 obtain every fifth minute by continual bisection. 
 If to the arc of 7° lO' be added the arc of 62 mi- 
 nutes, the arc of every single minute may be had 
 by bisection. 
 
 Op Mr. Bird's method of dividing. 
 Fig. 2, ■p/ate 9. 
 
 In 1767 the Commissioners of Longitude pro- 
 posed an handsome reward to Mr. Bin/, on con- 
 dition, among other things, that he should publish 
 an account of his method of dividing astronomicai 
 instruments; which was accordingly done: and a 
 tract, describing his method of dividing, was writ- 
 ten by him, and published by order of the Com- 
 missioners of Longitude in the same year; some 
 defects in this publication were supplied by the 
 Rev. Mr. Ludiam, one of the gentlemen who at- 
 tended Mr. Bird to be instructed by him in his 
 method of dividing, in consequence of the Board's
 
 OP DIVIDI]^G. 109 
 
 agreement with him. Mr. Ludlams tract was 
 published in 1787, in 4to, 
 
 I shall use my endeavours to render t]iis 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. 
 
 1. Contiguous to 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 inch. By the assistance of a magnifying 
 glass of one inch focus, the SOOOndth part may be 
 estimated. 
 
 3. Six beam compasses are necessary, furnished 
 witli magnifying glasses of not more than one inch 
 focus. The longest beam is to measure the radms 
 or chord of 60 ; the second for the chord of 42.40; 
 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 
 different beam compasses. 
 
 5. Let these operations be performed in the 
 evening, and let the scale and the different beam 
 compasses be laid upon the instrument to be di- 
 vided, and remain there till the next morning. 
 
 6. The next morning, before sun-rise, examine 
 the compasses by the scale, 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 
 temperature, describe the faint arc bd, or primi- 
 tive circle; then with the compasses that are set
 
 no ani. bird's Method. 
 
 to the rftclius and nitli a line prick punch, make a 
 point at a, which is to be the O point of the qua- 
 drant; SCQ Jjg.'l, plate g. 
 
 8. With the same beam compasses unalterecf 
 lay off from a to cthc chord of 60°^ making a fin« 
 point at c. 
 
 9. Bisect the arc ac with the chord of 30°. 
 
 10. Then from the point c, \vilh the beam com- 
 passes containing 60, mark the point v, which is 
 that of 90 degrees. 
 
 1 1 . Next, with the beam compasses containing 
 15°, bisect the arc er in n, which gives Jo^. 
 
 12. Lay off from n towards r the chord of 
 10° 10', and from r towards n the chord of 4° 40'; 
 these two ought to meet exactly at the point g of 
 85° 20'. 
 
 13. Now as in larp-e instruments each dcQ-ree is 
 generally subdivided into 12 equal parts, of five 
 minutes each, we shall find that 85° 20' contains 
 10.24 such parts, because 20' equal 4 of these 
 parts, and 85X12 makes 1020; now 1024 is a 
 number divisible by continual bisection. 
 
 The last computed chord was 42° 40', with 
 which a g was bisected in o, and a o, o g, were 
 bisected bv trials. Though Mr. Bird seems to 
 have used this method himself, still he thinks it 
 more adviseable to take the computed chord of 
 21° 20', and by it find the point g; then proceed 
 by continual bisections till you have 1024 parts. 
 Thus the arc 85° 20', by ten bisections, will give 
 vs the arcs 42° 40', 21° 80', 10° 40', 5° 20', 2° 40', 
 1°20', 40', 20', 10', 5'. 
 
 14. To fill up the space between g and r, 85* 
 •20', and 00°, which is 4° 40', or 4X12 + 8 equal to 
 60 divisions; the chord of ()4 divisions was laid oft" 
 from g towards d, and divided like the rest by con- 
 tinual bisections,, as was also from a towards- b;
 
 MR. BIRD S METHOfi. Ill 
 
 If the work is well performed, you will again find 
 the points 30, 45, ()0, 75, and gO, 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 contains 12 arcs of 5' each, 
 of which 15° contains 180; the arc of 30° containa 
 360; the arc of 6o°, 720; that of 75°, QOO; and, 
 therefore, 90° contains 1080. 
 
 Mr. Graham, in 1725 applied to the quadrant 
 divided into 90°, or rather into 1080 parts of fivo 
 minutes each, another quadrant, which he divided 
 into 96 equal parts, subdividing each of these into 
 16 equal parts, forming in all 1536. This arc is a 
 severe check upon the divisions of the other; but 
 Bird says, that if his instructions be strictly fol- 
 lowed, the coincidence betwech them will be sur- 
 prising, and their difference from the truth exceed- 
 ingly 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 512 divisions, which number is 
 divisible continually by 2, and gives 16 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 
 is^harp. Draw a tangent to the arc b d, suppose at 
 e, it will intersect the arc xy 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.
 
 113 FOU THE NONIUS. 
 
 proceeding thus till you have finished all the dm* 
 sions of the limb. 
 
 POR THE NONIUS.- 
 
 1. Chuse any part of the arc where there h a 
 coincidence of the 90 and Qb arcs; for example, 
 at e, the point of 6()°. Draw the faint arc s t and 
 ik, which may be continued to any length towards 
 A; upon these the nonius divisions are to be di- 
 vided in points. The original points for the no- 
 nius of the goth arc are to be made upon the arc 
 St; the original points for the nonius of the g()th 
 arc are to be made upon the are i k. 
 
 Because go is to g6, as 15 to 16, there will be a. 
 coincidence at 15° and l6pts, 30° and 32pts, 45° 
 and 4Spts; 6o° and 64pts, go° and g()pts. 
 
 2. Draw a tangent line to the primitive circle as 
 before, intersecting the are d, which gives the dis- 
 tance of the points of the beam compasses, with 
 which the nonius of the goth arc must be cut. 
 
 3. Let us suppose then that the nonius is to 
 
 subdivide the divisions of the limb to half a minute, 
 
 which is eifected bv makino; 10 divisions of the 
 
 ^ . . . 
 nonius equal to 11 chvisions of the limb; measure 
 
 the radius of the arc, and compute the chord of 16, 
 or rather 32 of the nonius division, which may ea- 
 sily be obtained by the following proportion; if 
 10 divisions of the nonius plate make 55 minutes 
 of a degree, what will 32 of those divisions make ? 
 the answer is 2*^ 56', the chord of which must be 
 computed and taken from the scale of equal parts; 
 but as different subdivisions bvthe nonius mav be 
 required, let ;/ be the number of nonius divisions, 
 h the number of minutes taken in bv the nonius b, 
 id, 32 or (34, and X the arc sought; then as u : 
 m :; b : X.
 
 FOR THE NONIUS. 113 
 
 4. Lay off with the beam compasscSj having the 
 length of the tangent ()() between the points, the 
 point q from c, q being a point in the arc s t, and 
 G an original point in the primitive circle, and the 
 chord of 32 from q towards the left hand, (the 
 chord of 32 being the chord which subtends 32 di- 
 visions 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, 32, &c. that the extremes may be laid off from 
 the divisions 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 arc to be cut, must be found 
 in the way before directed for the nonius of the 
 90 arc; the ark ik standing instead of the arc st. 
 Having the tangental distance between the points 
 of the compasses from one of the original points 
 in the primitive 96, lay off a point on the faint arc 
 i k towards the left hand ; count from that point 
 on the ])rimitive 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 points 
 in the faint arc i k is to be subdivided by bisec- 
 tions into 16 parts, and those parts pointed; from 
 these points the visible divisions of the nonius are 
 to be cut. 
 
 Mr. LiuTlam thinks, that instead of laying the 
 extent of the nonius single, it would be better to 
 lay it off double or quadmple; thus, instead of 17 
 points to the left hand, count 34 or 68, and that 
 both ways, to the riglit and to the left^ and lay off 
 
 1
 
 114 FOR THE NONIUS. 
 
 a point from each extreme on the faint arc ik; 
 subdivide the whole between these extreme points 
 by continual bisections, till you get l6 points toge- 
 ther on the left hand side of the middle, answering^ 
 to the extent of the nonius. No more of the sub- 
 divisions are to be completed than are necessary to 
 obtain the middle portion of l6 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 90 arc, on the outer edge is the 
 nonius for the9t)arc,in the middle between the two 
 chamfers is a flat part parallel to the under sm-face 
 of the nonius plate; upon the flat part the faint 
 line of the next operation, is to be drawn. 
 To find the place 'where the ?ionius is to begin upon 
 the chamfered edge of the nonius plate, measure the 
 distance of the center 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 center on the flat part of the 
 nonius plate. 
 
 Fasten the nonius plate to the arc with two 
 pair of hand voices; then with one point of the beam 
 compasses in the center 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 be equally distant 
 from the middle point; from the two last make a 
 faint intersection as near as possible to either of 
 the chamfa'ed edges of the nonius plate; through 
 this intersection the first division of tlic nonius 
 must be cut.
 
 [115] 
 
 Mr. Bird's method op dividing his scale 
 op equal parts. 
 
 Let us suppose that we have 90 inches to divide 
 into 900 equal parts, take the third of this number, 
 or 300; now, the first power of 2 above this is 
 512; therefore, take Wths of an inch in one pair 
 of beam compasses, r-i in another, W in a third, 
 and T^ 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 W three times ; then with the com- 
 passes Vs , W, Tff, bisect these three spaces as ex- 
 peditiously 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 Wths 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 integer being t^. Suppose the scale to be 
 numbered at every inch from left to right; then 
 make a fine point exactly against -r^, to the left of o, 
 from this lay off 258,56 to the right hand, which 
 divide after the common method. 
 
 If you are not furnished with a scale long enough 
 to lay off 258,50, then set off ¥0°, and add 8,56, 
 from a diagonal scale. 
 
 Op Mr. Bird's pointing tool, and method 
 of pointing. 
 
 The pointing tool consisted of a steel wire ^o 
 inch diameter, inserted into a brass wire i inch 
 
 I 2
 
 Il6 Mil. bird's pointing tool, &c. 
 
 diameter, the brass part 1^ long, the steel part 
 stood out f, whole length 3^ inches. The angle 
 of the conical point about 20 or 25 degrees, some- 
 what above a steel temper; the top of the brass 
 part was rounded oft', to receive the pressure of 
 the fmger; 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 also a kind 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 
 compasses, the cutting point raises up the metal a 
 little on each side the arc ; the metal so throw^n 
 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 cir- 
 cle be struck again ever so lightly, the bur will be 
 raised up again ; the arcs struck across the primi- 
 tive circle have also their bur. Two such rasurcs 
 or trenches across each other, will of course have 
 four salient, or prominent angles within ; and as 
 the sides of the trenches slope, so do also the lines 
 which terminate the four solid angles. You may 
 therefore, /^^/ what you cannot see; when the co- 
 nical points bear against all four solid angles, they 
 will guide, and keep the point of the tool in the 
 center of decussation, while keeping the tool up- 
 right, pressing it gently with one hand, and turn- 
 ing it round with the other, you make a conical 
 hole, into which you can at any time put the point 
 of the beam compasses, and feel, as well as see, 
 when it is lodged there.
 
 [ H7] 
 
 RULES OR MAXIMS LAID DOWX BY Mr. BiRD. 
 
 1. The points of the beam compasses should 
 never be brought nearer together than two or three 
 inches, except near the end of the line or arc to be 
 divided ; 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 thou- 
 sandth of an inch. When lines, or divisions, are 
 to be traced from these points, a magnifying glass 
 of ^ an inch focus must be used, which will render 
 the impression or scratch made by the beam com- 
 passes sufficiently visible; and if the impression be 
 not too faint, feeling will contribute, as well as 
 seeing, towards 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 instrument, 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 suftbred any fire 
 in the room, till the principal points were laid 
 down. 
 
 Mr. 5/V<^ guards, by this method, against any 
 inequality that might possibly happen among the 
 original points, by first setting out a few capital 
 points, distributed equally through the arc, leaving 
 the intervals to be filled up afterwards; he could 
 by this method check the distant divisions with 
 respect to each other, and shorten the time of thtj 
 most ^sential operations.
 
 118 KULE3 OR MAXIMS, 
 
 5. Great care is to be observed in pointing in- 
 tersections, which is more difficult than in points 
 ing from a single line, made by one point of the 
 compasses. 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 toa 
 long, or too short, it will not occasion any ine^ 
 quality, provided the point be made in the middle, 
 between the two short lines traced by the com- 
 passes. 
 
 NoAv, as Mr. Ludlam observes, if the bisecting 
 chord be taken exactly, the two fore-mentioned 
 faint arcs will intersect each other in the primi- 
 tive 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 middle between these two 
 arcs, and a point may be made by the pointing tool. 
 
 In small portions of the prmiitive 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 Vv^ill 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 compasses should fall exactly into that hole, 
 both from the right and left hand, and you may 
 readily feel 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 blow against it; if 
 it does not exactly coincide with the center of the 
 hole, the slant part of the point will slide down tb?
 
 LAID DOWN BY MR. BIRD. 1 ig 
 
 slant side of the hole, drawing, or pushing the 
 other point of the compasses from its place. 
 
 Mr. Bird's method of transferring the divisions 
 by the beam compasses from the original points, is 
 founded on this maxim, that a right tins cannot be 
 cut upon hrasSy so as accurately to pass through two 
 given points; but that a circle may be described 
 from any center to pass with accuracy througli a 
 given point. It is exceeding difficult, in the nrst 
 place, to fix the rule accurately, and keep it firmlv 
 to the two points; and, secondly, 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 against it; it fol- 
 lows, that if the knife be held in a different situa- 
 tion to or from the rule, it will throw the cutting 
 point out or in; besides, any hardness or ine- 
 quality of the mxtal will turn the knife out of its 
 course, for the rule does not oppose the knife in 
 departing from it, and the force of the hand can- 
 not hold it to it. For these reasons it is almost 
 impossible 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 al- 
 ways 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 compasses be set in the given point or con.ical 
 hole in the brass 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 strengthened by degrees, as the 
 moving point is not likely to be shifted out of i\s> 
 4clir.ection, nor the cutting point to be broken.
 
 120 !IULE3 OR MAXIM5, &:C. 
 
 The visible divisions on a large quadrant are 
 always the arcs of a circle, though so short as not 
 to be distinguished from strait lines; they should 
 be perpendicular to the arc that bounds them, and 
 therefore the still or central point of the beam 
 compasses must be somewhere in the tangent to 
 that arc; the bounding circle of the visible divi- 
 sions, and the primitive circle should be very near 
 each other, that the arc forming the visible divi- 
 sions may be as to sense 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 Vuie, but not to trisect, quinquisect, 
 ^c. The advantages to be obtained by bisection 
 have been already seen; we have now to shew the 
 objections against trisecting, &c. 
 
 1. That as the points of triscction in the primi- 
 tive circle must be made by pressing the point of 
 the beam compasses down into the metal, the least 
 extuberance, 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 conical hole. 
 
 1. 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 
 partial expansion would probably tiikc place; and, 
 perhaps, many false marks, or holes made, which 
 might occasion considerable error. 
 
 Another maxim of Mr. Bird's was this, that 
 stepping was liable to great uncertainties, and not 
 to be trusted; that iS;, if the chord of l6° W9S aS"
 
 BIRD S SCALE OF EQUAL PART?. 121 
 
 snnied, and laid down five times in siicecssion, bj 
 turning the compasses over upon the primitive cir- 
 cle, yet the ares so marked would not,, in his opi- 
 nion be ecjual. 
 
 DE^CRIPTIOX OF Mr. BiRD^S SCALE OF 
 EQUAL PARTS. 
 
 It consists of a scale of inches, each divided into 
 tenths, and numbered at every incli from the lelt 
 to the right, thus, O, 1,2, 3, &c. in the order of the 
 natural numbers. The nonius scale is below this, 
 but contiguous to it, so that one common line ter- 
 minates the bottoms of the divisions on the scale 
 of inches, and the tops of the divisions on the 
 nonius; this nonius scale contains in length 101 
 tenths of an inch, this length is divided into 100 
 equal parts, or visible divisions; the left hand di- 
 vision of this scale is set off from a point -rs of an 
 inch to the left of 0, on the scale of inches; there- 
 fore, the right hand end of the scale reaches to, 
 and coincides with the 10th inch on the scale of 
 inches. Every tenth division on this nonius scale 
 is figured from the right to the left, thus, 100. C)0. 
 80. 70. (JO. 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 ofoon the inches; and 
 these two, viz. the first and last, are the only two 
 strokes that do coincide in the two scales. 
 
 To take off any given number of inches, deci- 
 mals, and millesimals of an inch; for example, 
 42,7()4, observe, that one point of the beam com- 
 pass must stand in a (j)ointed) division on the no- 
 nius, and the other point of the compasses in a 
 pointed division on the scale of inches;.
 
 122 DESCRIPTION OF MR. BIRD 5 
 
 The left hand point of the compasses mii5t 
 stand in that division on the nonius \vhich ex- 
 presses the number of millesimal parts j this, in our 
 example, is (i4. 
 
 To find where the other point must stand in the 
 scale of inches and tenths, add 10 to the given 
 number of inches and tenths (exclusive of the two 
 millesimal figures;) from this sum subtract the 
 two millesimal 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; from this 
 subtract 6,4 and the remainder is 46,3. Set then 
 one point of the compasses in the 64th division on 
 the nonius, and the other point in 46,3 on the scale 
 of inches, and the two points will comprehend be- 
 tween them 42,764 inches. 
 
 This will be plain, if we consider that the junc- 
 tion 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 likewise comprehend 64 divisions on 
 the nonius scale. Each of these divisions is one 
 tenth and one millesimal part of an inch ; there- 
 fore, 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 subdivision in the former is 
 tV of an inch, each subdivision in the latter is -r^-j- 
 WcTsth 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 
 expedited, and carried on with far less danger of
 
 SCALE OF EQUAL PARTS. 123 
 
 injaring the scale, if the proposed length be first 
 of all taken, nearly^ on the scale of inches onlv, 
 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 proper 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 di- 
 vision. 
 
 If, when the points of the compasses are set, they 
 do not comprehend an integral number of mille- 
 simal parts, they will not precisely fall into any 
 two divisions, but will cither 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 divi- 
 sion more to the left, and stand in 65 of the nonius, 
 then the right hand point will fall short of 46,2 in 
 the scale of inches; the excess in the former case 
 being equal to the defect in the latter. By ob- 
 serving whether the difference be equal, or as 
 great again in one case as the other, we may esti- 
 mate to 4d part of a millesimal. See Mr. Binfs 
 Tract, p. 2. 
 
 It may be asked, why should the nonius scale 
 cxDmmence at the lOth inch; why not at 0, 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 ap- 
 pears from the rule before given. The number 10 
 must not, in this case, be added to the inches and
 
 124 DE^CRIPIOX OF MR. BIBd's, &C. 
 
 tenths, and then the subtraction before directed 
 would not always be possible. 
 
 Yet, upon this principle, a scale in one conti* 
 nued line may be constructed for laying ofl' 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 0, and in the same line, set off 
 eleven tenths of an inch (or the multiple,) which 
 subdivide in Mr. Birds way, into ten equal parts. 
 Such a compound scale would be far more exact 
 than the common diagonal scale; for the division* 
 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. Bird's Tract. 
 
 The nature of Mr. Bird's scale being known, 
 there will be no difficulty in understanding his 
 directions how to divide it. A scale of this kind 
 is far preferable to any diagonal scale; not only 
 on account of the extreme difficulty of drawing the 
 diagonals exactl)', but also because there is no 
 check upon the errors in that scale; here the uni- 
 form manner in which the strokes of one scale se- 
 parate from those of the other, is some evidence of 
 the truth of both ; but Mr. Bird's method of as-^ 
 suming a much longer line than what is absolutely 
 necessary for the scale, subdividing the whole by 
 a continual bisection, and pointing the divisions as 
 before explained, and guarding against partial ex-r 
 pansions of the metal, is sure to render the divi- 
 sions perfectly equal. The want of such a scale 
 of equal parts (owing, perhaps, to their ignorance 
 of constructing it) is one reason why Mr. Bird's 
 method of dividing is not in so great estimation 
 among mathematical instrument makers, as it 
 justly deserves.
 
 I 125 J 
 
 An OBSERVATION", OR METHOD OF GRADUA- 
 TION, OF Mr. Smeaton's. 
 
 As it is my intention to collect in this place 
 whatever is valuable on this subject, I cannot 
 refrain from inserting the following remark of 
 Mr. Smeaton^, though it militates strongly against 
 one of Mr. Blnfs maxims. He advises us to com- 
 pute from the measured radius the chord of 16 de- 
 grees only, and to take it from an excellent plain 
 scale, and lay it off five times in succession from 
 the primary point of given, this would give 80 
 degrees; then to bisect each of these arcs, and to 
 lay off one of them beyond the SOth, 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 90 degrees; 
 then proceed again by bisection, till you have re- 
 duced the degrees into quarters, or every fifteen 
 minutes. Here Mr. Smeaton would stop, being 
 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, he advises us 
 to compute the chord of 21° 10' only, and to lay it 
 oft' four times from the primary point; the last 
 would give 85° 20', and then to supply the re- 
 mainder from the bisected divisions as they rise, 
 not from other computed cliords. 
 
 Mr. 5i/v/ asserts, that after he had proceeded by 
 the bisections from the arc of 85° 20', the several 
 points of 30. 60. 75. ^Q. fell in without sensible ine- 
 quality, and so indeed they might, though they 
 were not equally true in their places; for whatever 
 error was in them would be communicated to all 
 connected with, or taking their departure from
 
 126 GRADUATIOXj B Y* SMEATO^'■. 
 
 them. Every heterogeneous mixture should be 
 avoided. 
 
 It is not the same thing whether you t-ivlce take 
 a measure as nearly as you can, and lay it oft' 
 separately, or lay oft^ two openings of the com- 
 passes in succession unaltered^ for though the 
 same opening, carefully taken oft" from the same 
 scale a second time, will doubtless 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 center, there may be an error, 
 which, though insensible to the sight, would have 
 been avoided by the more simple process of laying 
 oft^ the opening twice, without altering the com- 
 passes. 
 
 As the whole of the QO arc may now be divided 
 by bisection, it is equally unexceptionable with 
 the ()6 arc; and, consequently, if another arc of 
 90, upon a different radius, was laid down, they 
 would be real checks upon each other. 
 
 Mr. Ramsde7i, in laying down the original divi- 
 sions 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 apprehensive some error might arise from 
 quinquiscction, and triscction, in order to examine 
 the accuracy of the divisions, he described another 
 circle tV inch within the former, by continual bi- 
 sections, but found no sensible difiTerence between 
 the two sets of divisions. It appears also, that 
 Mr. Bird, notwithstanding all his objections to, 
 and cteclamations against the practice of stepping, 
 sometimes used it himself.
 
 [ 1^7 J 
 
 OF THE NONIUS DIVISIONS. 
 
 It will be necessary to give the young practiti- 
 oner 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 circum- 
 stance, 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 propor- 
 tion, as the divisions are more numerous. 
 
 Thus, let us suppose the limb of Hadley's quad- 
 rant divided to every 20 minutes, which are the 
 smallest divisions on the quadrant; the two ex- 
 treme strokes on the nonius contain seven degrees, 
 or 21 of the afore-mentioned 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 are; consequently, if the first, or 
 index division of the nonius, be set precisely oppo- 
 site to any degree, the relative position of the no- 
 nius and the are must be altered one minute be- 
 fore the next division on the nonius will coincide 
 with the next division on the arc, the second di- 
 vision vv'ill 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 minute 
 have been regularly pointed out by the divisions of 
 the nonius.
 
 128 OF THE XOXIUS DIVISIOXs. 
 
 To render this still plainer, we must observe 
 that the index, or counting division of the nonius, 
 is distinguished by the mark 0, uhich is placed on 
 the extreme right hand division; the numbers 
 limning 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 arc shewn 
 by the other strokes of the nonius; thus, look 
 umong the strokes of the nonius for one that stands 
 directly opposite to, or perfectly coincident with 
 some one division on the limb; this division reck- 
 oned 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 quad- 
 rant : here there is no difficulty, for at whatsoever 
 division it is, that division indicates the required 
 angle. If the index 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 de- 
 grees 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 
 tionius that shall stand directly opposite to one on 
 the limb, and that division gives us the odd mi- 
 nutes, to be added to those pointed out by tlic 
 index division: thus, suppose the index division 
 does not coincide with 40 dcsrrecs, but that the 
 next division to it is the first coincident division, 
 then is the required an^lc 40 degrees 1 minute.
 
 VALUE OF ANY NONIUS. 12Q 
 
 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 coincides with 
 the first 20 minutes from 40 degrees. Again, let 
 us suppose the index division to stand between 
 30 degrees, and 30 degrees 20 minutes, and that 
 the Itith division on the nonius coincides exactly 
 with a division on the limb, then the angle is 
 30 degrees \6 minutes. Further, let the index 
 division stand between 35 degrees 20 minutes, 
 and 35 degrees 40 minutes, and at the same time 
 the 1 2th division on the nonius stand directly op- 
 posite to a division on the arc, then the angle will 
 be 35 degrees 32 minutes. 
 
 A GENERAL RULE FOR KNOWING THE VALUE 
 OF EACH DIVISION, ON ANY NONIUS WHAT- 
 SOEVER. 
 
 1. Find the value of each of the divisions, or 
 subdivisions, 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 va- 
 lue of the nonius division. 
 
 Thus, suppose each subdivision of the limb be 
 30 minutes, and that the nonius has 1 5 divisions, 
 then 4f gives two minutes for the value of the 
 nonius. If the nonius has 10 divisions, it would 
 give three minutes; 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 tor the required value.
 
 130 OP INSTRUMENTS FOR 
 
 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 
 compasses, Mr. lieywood and myself contrived 
 several instruments for this purpose; the most 
 perfect of these is delineated iat Jig. 5, plate 11. 
 It is an instrumenjt that must give great satisfac- 
 tion to every one who uses it, as it is so extensive 
 in its nature, being capable of describing arcs from 
 an infinite radius, or a strait line, to those of two 
 or three inches diameter. When it was first 
 contrived, both Mr. Heywood and myself were 
 ignorant of what had been done by that ever to 
 be celebrated mechanician. Dr. Hooke. 
 
 Since the invention thereof, I have received 
 some very valuable communications from different 
 gentlemen, who saw and admired the simplicity 
 of its construction; among others, from Mr. Ni- 
 cholsofi, author of several very valuable w^orks; 
 Dr. Rotherhim, Earl Stanhope, and /. Priestley^ 
 Esq. of Bradford, Yorkshire; the last gentleman 
 has favoured me with so complete an investigation 
 of the subject, and a description of so many admi- 
 rable contrivances to answer the purpose of the 
 artist, that any thing I could say ^vould be alto- 
 gether superfluous; I shall, therefore, bo v^ery brief 
 in my description of the instrument, represented 
 Jig. 5, plate 11, that I may not keep the reader 
 ti'om Mr. Priestley''^ valuable essay, subjoining 
 Dr. Plooke'n account of his own contrivance 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 pe- 
 rusing the short extract I shall here insert.
 
 DESCRIBING LARGE CIRCLES. 
 
 IS! 
 
 The branches A and ^, Jig. 5, plate \\, carry- 
 two independent equal wheels C, D. The pencil^ 
 or point E, is in a line drawn between the center 
 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 
 axes 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 magni- 
 tude will be described. 
 
 The distance between one axis and the center, 
 (or pencil,) being taken as unity, or the common 
 radius, the numbers 1, 2, 3, 4, &c. being sought 
 for in the natural tangents, will give arcs of incli- 
 nation for setting the nonii, and at which cir- 
 cles of the radii of the said numbers^ multiplied 
 into the common radius, will be described. 
 
 The com- 
 mon radius 
 multiplied 
 by 
 
 0.1 
 
 
 
 5.43 
 
 0.2 
 
 
 
 11.19 
 
 0.3 
 
 
 
 16.42 
 
 0.4 
 
 
 
 21.48 
 
 0.5 
 
 
 
 26. 34 
 
 0.6 
 
 is the radius 
 
 of a 
 
 30.58 
 
 0.7 
 
 circle made 
 
 by 
 
 35. 
 
 0.8 
 
 > the rollers when < 
 
 38.40 
 
 0.9 
 
 inclined at these 
 
 41. 59 
 
 1.0 
 
 angles : 
 
 
 45. 
 
 2.0 
 
 
 
 63.26 
 
 3.0 
 
 
 
 71.34 
 
 4.0 
 
 
 
 75.58 
 
 5.0 
 
 
 
 78.42 
 
 6.0 J 
 
 
 
 L. 80.32 
 
 K 2
 
 132 
 
 OF INSTRUMENTS FOR 
 
 The com- 
 mon radius 
 multiplied 
 by 
 
 " 7.0- 
 
 8.0 
 
 9.0 
 
 < 10.0 > 
 
 20.0 
 
 30.0 
 
 - 40.0 _ 
 
 is the radius of a 
 circle made by 
 the rollers when 
 inclined at these 
 angles : 
 
 81.52 
 82.53 
 
 83. 40 
 
 84. 17 
 
 87. S 
 
 88. 5 
 88.34 
 
 Extracts from Dr. Hooke, on the Difficulty, &c. 
 of Drawhig ^rcs of Great Circles. " This thing, 
 says he, is so difficult, that it is almost impossible, 
 especially where exactness is required, as I was 
 sufficiently satisfied by the difficulties that oc- 
 curred in striking a part of the arc of a circle of 
 6o 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 circumspection imaginable, and used 
 with as great care, would not perform the opera- 
 tion; nor by the way, an angular compass, such as 
 described by Guido UbalJus, by Cla-vius, and by 
 Blagrave, &c. 
 
 " The Royal Society met; I discoursed of my 
 instrument to draw a great circle, and produced 
 an instrument I had provided for that purpose; 
 and therewith, 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 meeting I endeavoured to explain 
 the difficulties there are in makino- considerable 
 discoveries either in nature or art; and yet, when 
 they are discovered, they often seem so obvious and 
 plam, that it seems more difficult to give a reason 
 why they were not sooner discovered, than ho\V 
 they came to be detected now: how easy it was, 
 wc now think, to find out a method of printing
 
 DESCRIBING LARGE CIRCLES. 133 
 
 letters, and yet, except what may have happened 
 in China, there is no specimen or history of any 
 thing of that kind done in this part of the Avorld. 
 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 dis- 
 covered its isochronous motion, and thought of 
 that proper motion for it, &c. And though it 
 may be difficult enough to find a way 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 in- 
 strument somewhat similar to that described,^?^-. 5, 
 flate 11, as appears from the journal of the Royal 
 Society, where it is said, that Dr. Hooke produced 
 an instrument capable of describing very large cir- 
 cles, by the help of two rolling circles, or truckles 
 in the two ends of a rule, made so as to be turned 
 ill their sockets to any assigned angle. In another 
 place he had extended his views relative to this in- 
 strument, that he had contrived it to draw the arc 
 of a circle to a center at a considerable distance, 
 where the center cannot be approached, as from 
 the top of a pole set up in the midst of a wood, or 
 from the 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 center cannot be conveniently ap- 
 proached, otherwise than by the sight. This he 
 performed hy two telescopes, so placed at the 
 truckles, as thereby to see through both of them 
 the given center, and by thus directing them to 
 the center, to set the truckles to their true incli- 
 nation, so as to describe by their motion, any part 
 of such a circle as shall be desired.
 
 134 Priestley's method 
 
 Methods of describing arcs of circles 
 
 OF LARGE MAGNITUDE. By J. PrIESTLEY, 
 
 Esti. 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 
 situation is too distant to be brought upon the plan. 
 The following essay is intended to furnish some 
 methods, and describe a few instruments that may 
 assist the artist in the performance of both these 
 problem.s. 
 
 OF finding points in, and describing arcs 
 
 OF 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, are equal one to another. 
 
 Let AC T>^,fig. \, plate 10, be the segment of 
 a circle; the angles formed by lines drawn from 
 the extremities A and B, of the base of the segment, 
 to any points C and D in its arc, as the angles 
 ACB, ADB, are equal. 
 
 This is the 31st proposition of EuclicVs third 
 book of the Elements of Geometry. 
 
 Principle 1. If upon the ends AB, Jig.^, plate 10, 
 of a right line AB as an axis, two circles or rollers 
 CD and EF be firmly fixed, so that the said 
 line shall pass throiigh the centers, and at right 
 angles to the plains of the circles ; and the whole 
 be suffered to roll upon a plain without sliding ;
 
 OP DESCRIBING LARGE CIRCLES. 135 
 
 1. If the rollers CD and EF be equal in dia- 
 meter, the lines, described upon the plain by their 
 circumferences, will be parallel right lines; and 
 the axis AB, and every line DF, drawn between, 
 contemporary points of contact of the rollers and 
 plain, will be parallel among themselves. 
 
 2. If the rollers CD and EF be unequal, then 
 lines formed by their circumferences upon the 
 plain will be concentric circles; and the axis AB, 
 and also the Hues DF, will, in every situation, 
 tend to the common center of those circles. 
 
 Principle 3. If there be two equal circles or rol- 
 lers A and ^,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 appa- 
 ratus be rolled upon a plain without sliding: 
 
 1 . If the axes C D and E F, be placed in a pa- 
 rallel situation, the circumferences of the rollers 
 A and B will trace upon the plain strait lines; 
 which will be at right angles to the axes C D 
 and EF. 
 
 2. If the axes CD and EF, continuing as be- 
 fore in the same plain, be inclined to each other, 
 so as if produced to meet in some point G, the 
 rollers A and B will describe in their motions upon 
 the plain arcs of the same, or of concentric cir- 
 cles, whose center is a point H, in that plain per- 
 pendicularly 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.
 
 r 136. ] 
 
 OF THE SIMPLE BEVEL. 
 
 In the performance of some of the following 
 problems, an instrument not vmlike jig, 4, plate 
 10, will be found useful. It consists of two rulers, 
 moveable on a common center, like a carpenter's 
 rule, with a contrivance to keep them fixed at any 
 required angle. The center C must move on a 
 very fine axis, so as to lie in a line with the 
 fiducial edges CB, CD of the rulers, and project 
 as little as possible before them. The fiducial 
 edges of the legs represent the sides of any given 
 angle, and their intersection or center C its an- 
 gular point. 
 
 A more complete instrument of this kind, 
 adapted to various uses, will be described here- 
 after. 
 
 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 po'mts A, B and C, 
 sup-posed to be in the circumference of a circle too 
 Jarge to he described by a fair of compasses ; to find 
 any number of other points in that circumference. 
 
 This may be performed various ways. As for 
 example, fig. 5, plate 10. ^ 
 
 1. Join AC, which bisect with the line FM G 
 at right angles; from B, draw BD parallel to 
 AC, cutting FG in E; and making ED =:EB 
 D will be a point in the same circumference, in 
 which are A, B and C. 
 
 By joining AB, and bisecting it at right angles 
 with IK; and from C drawing C a parallel to AB, 
 cutting IK in L, and making La =:LCj a will 
 be another of the required points,
 
 THE SIMPLE BEVEL. 137 
 
 Continuing to draw from the point, last found, 
 lines alternately parallel to AC and AB; those 
 lines will be cut at right angles by F G and I K 
 respectively; and by making the parts equal on 
 each side of F G and I K, they become chords of 
 the circle, in which are the original points A, B, 
 and C, and, of consequence^ determine a series 
 of points on each side of the circumference. 
 
 It is plain from the construction, (which is too 
 evident to require a formal demonstration,) that 
 the arcs AD, A a, C c, &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 manner, joining BC, and bisecting it 
 at right angles Avith PQ; drawing A c' parallel to 
 B C, and making R c'=AR, (c/) is another of the 
 required points; and, from (c') the point last 
 found, drawing c'a' parallel to C A, and making 
 a'N^Nc', (a') is another point in the same cir- 
 cumference; and the arcs comprehended between 
 C c' and A a' are equal to that between AB. 
 Hence, by means of the perpendiculars P Q. and 
 FG, any number of points in the circumference 
 of the circle, passing through the given ones, A, 
 B and C may be found, whose distance is equal to 
 AB, in the same manner, as points at the distance 
 of B C were found by the help of the perpendi- 
 culars I K and F G. 
 
 Again, if A, C and (c) or A, C and (c') betaken 
 as the three given points, multiples of the arc 
 AC may be found in the same manner as those of 
 the arc AB were found as above described. 
 
 2. Another method of performing this problem, 
 is as follows, ^g. 6, p/afe 10. Produce CB and 
 CA; and with a convenient radius on C, describe 
 the arc D E ; on which set off the parts I^^ G, G E, 
 kc. each equal to DF; draw CG^ C E, &c. con-
 
 138 PROBLEMS BY 
 
 tinucd out. beyond G and E if necessary ; take the 
 distance AB, and wiih one foot of the compasses 
 in A, strike an arc to cut CG 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 AB, and center H, strike an 
 arc to cut C E produced in I, which will be ano- 
 ther 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, BCA, ACH, H C I, &c. are 
 equal, they must intercept equal arcs B A, AH, HI 
 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 AK, 
 KAL, &c. each equal to BiVC, and make the dis- 
 tances CK^ KL, &;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. See Jig. 5. 
 
 Bring the center 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, fjig. A) ; this is called setting the bevel to the 
 given points. Then removing the center 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 center will describe (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 witliin the limits of A and C.
 
 THE BEVEL. 139 
 
 In order to find points without those limits, 
 proceed thus : the bevel being set above described, 
 bring the center to C, and mark the distance C B 
 upon the left leg ; remove the center to B, and 
 mark the distance BA on the same leg; then pla- 
 cing the center on A, bring the right leg upon B, 
 and the first m.ark will fall upon (a) a point in the 
 circumference of the circle, passing through A, B 
 and C, whose distance from A is equal to the dis- 
 tance B C. Removing the center of the bevel to 
 the point (a) last found, and bringing the right leg 
 to A, the second mark will find another point (a") 
 in the same circumference, whose distance a a"" is 
 equal AB. Proceeding in this manner, any num-r 
 ber of points may be found, whose distances on 
 the circumference are alternately B C and BA. 
 
 In the same manner, making similar marks on 
 the right leg, points on the other side, as at (c') 
 and (c") are found, whose distances C c', c' c", are 
 equal to BA, B C respectively. 
 
 It is almost unnecessary to add, that interme- 
 diate points between any of the above are given by 
 the bevel, in the same manner as between the ori- 
 ginal points. 
 
 Problem 1. Fig. 7, plate 10. Three j)o'mis, K^ 
 B and C, being given, as In the last problem, to find 
 a fourth point D, situated In the circumference of the 
 circle passing through A, B and C, and at a green 
 7iumber of degrees distant from any of these polnts\ 
 A for instance. 
 
 Make the angles AB D, and AC D, each equal 
 to one half of the angle, which contains the given 
 number of degrees, and the intersection of the 
 lines B D, C D gives the point D required. 
 
 For, an angle at the circumference being equal 
 to half that at the center, the arc AD will con-
 
 140 PROBLEMS BY 
 
 tain twice the number of degrees contained by 
 either of the angles ABD or ACD. 
 
 Problem 3. Fig. Q, plale 10. Giveti three 
 points, as in the former problems, to draw a line from 
 any of them, tending to the center of the circle, which 
 passes through them all. 
 
 Let A, B and C be the given points, and let it 
 be required to draw AD, so as, if continued, it 
 would pass through the center of the circle con- 
 taining A, B and C. 
 
 Make the angle BAD equal to the complement 
 of the angle B CA, and AD is the line required. 
 
 For, supposing AE a tangent to the point A, 
 then is EAD a right angle, and EAB=BC A; 
 whence, BAD=: right angle, less the Z. B CA, 
 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 C F ; or C F may be had without the inter- 
 vention ofBG, by making ACF=CAD. 
 
 Corollary 1. A tangent to the circle, at any of 
 the points (A for instance), is thus found. 
 
 Make BAE=BCA, and the line AE will 
 touch the circle at A. 
 
 By the bevel. Set the bevel to the three given 
 points A, B and C, (fg. Q,) lay the center on A, 
 and the right leg to the point C; and the other 
 leg will give the tangent AG'. Draw AD per- 
 pendicular to AG'' for the line required. 
 
 For BAEbeing=BCA, the Z EA C is the 
 supplement to /_ ABC, or that to which the 
 bevel is set; hence, when one leg is applied to 
 C, and the center brought to A, the direction of 
 the other leg must be in that of the tangent G E.
 
 THE BEVEL. 141 
 
 Problem 4. Fig. g, plate 10. Three points 
 heing given, as in the former problems, to draw from 
 a given fourth point a line tending to the center of a 
 circle passing through the three first 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 Hne Dd tending to the center of the said circle. 
 
 From A and B, the two points nearest D, draw 
 by the last problem, the lines Aa, Bb, tending to 
 the said center; join AB, and from any point E, 
 taking in B b, (the farther from B the better) draw 
 EF parallel to Aa, cutting AB in F; join AD 
 and B D, and draw FG parallel to AD, cutting 
 DB in G; join GE, and through D parallel 
 thereto, draw Dd for the line required. 
 
 For, (continuing D d and B b till they meet in 
 O,) since A a and B b, if produced, would meet in 
 the center, and F E is parallel to A a, we have 
 B F : BA :: BE : radius; also, since AD and FG 
 iire parallel, B F : BA :: B G : B D; therefore 
 BG : BD :: BE : radius; but from the parallel 
 lines Dd and GE, we have BG : BD :: BE : BO; 
 hence B O is the radius of the circle passing 
 through A, B and C. 
 
 By the level . On D with radius DA describe 
 an arc AK; set the bevel to the three given points 
 A, B and C, and bring its center (always keeping 
 the legs on A and C) to fall on the arc AK, as at H ; 
 on A and H severally, with any convenient radius, 
 strike two arcs crossing each other at I; and the 
 required line Dd will pass through the points 
 I and D. 
 
 For a line drawn from A to H will be a common 
 chord to the circles AHK and ABC; and the 
 line ID bisecting it at right angles, must pass 
 through both their centers.
 
 112 I'ROBLEMS BY 
 
 Problem 5. Fig. g, plate 10. Three po'mfs 
 heing give?i, as before^ together with a fourth point, 
 to jind two other 'points ^ such, that a circle passing 
 through them and the fourth point, shall he concentric 
 to that passing through three given points. 
 
 Let A, B and C be the three given points, and 
 D the fourth point; it is required to find two 
 other points, as N and P, such, that a circle pass- 
 ing through N, D and P shall have the same cen- 
 ter with that passing through A, B and C. 
 
 The geometrical construction being performed, 
 as directed by the last problem, continue E G to 
 X<, making E L=E B; and through B and L draw 
 BLM, cutting Dd produced in M; make AN 
 and B P severally equal to M D, and N and P are 
 the points required. 
 
 For, since LE is parallel to MO, we have BE : 
 LE :: BO : MO; but BE = LE by the con- 
 struction ; therefore, M O = B O = radius of the 
 circle passing through K, B and C, and M is in the 
 circumference of that circle. Also, N, D and P 
 being points of the radii, equally distant from 
 A, M and B respectively, they will be in the cir- 
 cumference of a circle concentric to that passing 
 through A, M and B, or A, B and C. 
 
 By the bevel. Draw Aa and C c tending to the 
 center, by problem 3 ; set the bevel to the three 
 given points A, B and C; bring the center of the 
 bevel to D, and move it upon that point till its legs 
 cut off equal parts AN, CQ of the lines Aa and 
 C c; and N and Q wall be the points required. 
 
 For, supposing lines drawn from A to C, and 
 from N to Q, the segments ABC and N D Q will 
 be similar ones; and consequently, the angles 
 contained in them w^ill be equal. 
 
 Problem 6. Fig. 10, plate 10. Three points, 
 A, B and C, lying in the circumference of a circle^
 
 THE BETEL. 143 
 
 hehtg given as hefore: and a fourth point D, to Jind 
 another point F, such, that a circle passing through 
 F and D shall touch the other passing through A, B 
 a7kl C, at any of these points \ as for instance^ B. 
 
 Draw BE a tangent to the arc ABC, by pro- 
 blem 3, corollary 2; and join BD; draw B F, 
 making the angle DBF=EBD, with the dis- 
 tance BD; on D strike an arc to cut BF in F; 
 and F is the point sought. 
 
 Since DF=DB, theZDFB = DBF; but 
 D B F=r D B E by construction ; therefore, D F B 
 = D B Ej and E B is a tangent to the arc B D F 
 at B; but E B is also a tangent to the arc AB C 
 (by construction) at the same point; hence, the 
 arc B D F touches AB C as required. 
 
 Problem 7. Fig- 13, plate 10. Two lines tend- 
 ing to a distant point being given, and also a point in 
 ene of them; to jjyid tiio other foints, (one of which 
 must he in the other given line,) such, that a circle 
 passing through those three points, may have its center 
 at the point of intersection of the given lines. 
 
 Let the given lines be AB and C D, and E the 
 given point in one of them ; it is required to find 
 two other 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 
 center at O, where the given lines, if produced, 
 would meet. 
 
 From E, the given point, draw E H, crossing 
 AB at right angles in F; make FH=FE, and H 
 is one of the required points. From any point D 
 in C D, the farther from E the better, draw G D 
 parallel to AB, and make the angle HE I equal 
 to half the angle GDE; and EI will cut AB in I, 
 the other required point. 
 
 For, since EH crosses AB at right angles, and 
 H F is equal to F E, I II will be equal to I E, and
 
 144 INSTRUMENTS FOR LARGE CIRCLES. 
 
 the /_ HEI=:Z.EHI; also, since GDIs pa- 
 rallel to AB, the Z GDE = Z FOE = double 
 Z. IHEn= double Z.HEI; butHEI = half 
 /^GDE by construction; hence, the points E, 
 I and H are in the circle whose center is O. 
 
 By the bevel. Draw EH at right angles to AB, 
 and make FH=r:FE as before; set the bevel to 
 the angle G D O, and keeping its legs on the 
 points H and E, bring its center to the line AB, 
 which will give the point I. 
 
 Problem 8. Fig. 13, plaie 10. Two lines tend'- 
 hig to a (list ant point being given, to find the distatice 
 of that point. 
 
 Let AB and C D be the two given lines, tend- 
 ing to a distant point O; and let it be required 
 to tind the distance of that point, from any point 
 (E for instance) in either of the given lines. 
 
 From E draw E F perpendicular to AB ; and 
 from D (a point taken any where in C D, the far- 
 ther from E the better) draw D G parallel to AB. 
 On a scale of inches and parts measure the lengths 
 of G E, ED and E F separately ; then say, as the 
 length of GE is to ED, so is EF to EO, the 
 distance sought. 
 
 For the triangles E G D and E F O are similar, 
 and from thence the rule is manifest. 
 
 OP 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 
 instruments for these purposes, whose rationale de- 
 pends on the principles laid down in the beginning 
 of this essay.
 
 [ 1^15 I 
 
 1. AM IMPROVED BEVEL. 
 
 • 
 
 Fig. 12, p/a/e 10, is a sketch of an instrument 
 grounded upon principle 1, p. 134, by which the 
 arcs of circles of any radius, without the limits at- 
 tainable by a common pair of compasses, may be 
 described. 
 
 It consists of a ruler A B, composed of two pieces 
 rivetted together near C, the center, or axis, 
 and of a triangular part C F E D. The axis is a 
 hollow socket, fixed to the triangular part, about 
 which another socket, fixed to the arm C B of the 
 ruler AB, turns. These sockets are open in the 
 front, for part of their length upwards, as repre- 
 sented in the section at I, in order that the point 
 of a tracer or pen, fitted to slide in the socket, 
 may be more easily seen. 
 
 The triangular part is furnished with a gradu- 
 ated arc D E, by which, and the vernier at B, 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 AB may be fixed in any required 
 position, 
 
 A scale of radii is put on the arm C B, by which 
 the instrument may he 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 numbcn* expressing it in inches on C B is 
 brought to cut a fine line drawn on C D, parallel, 
 and near to the fiducial edge of it, and the arms 
 fastened in that position 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
 
 146 AN IMPROVED BEVEL, 
 
 when the arcs are drawn. The under sides of 
 these sector^ are furnished with fine short points, 
 to prevent them from sHding. 
 
 The fiducial edges of the arms C A and C D are 
 each divided from the center C into 2,00 equal 
 parts. 
 
 The instrument might be furnished with small 
 castors^ like the pentagraph; but little buttons 
 fixed on its underside, near A, E and D, will ena- 
 ble it to slide with sufficient ease. 
 
 SOME INSTANCES OP ITS USE. 
 
 I. To describe an arc, which shall pass throvgli 
 three give?i joints. 
 
 Place the sectors G, G, with their angular edges 
 over the two extreme points ; apply the arms of the 
 bevel to them, and bring at the same time its cen- 
 ter C (that is, the point of the tracer, or pen, put 
 into the socket) to the third point, and there fix the 
 arm C B ; then, bringing the tracer to the left hand 
 sector, slide 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 center C. 
 
 If the arc be wanted in some part of the drawing 
 ivithoiit the given points, find, by problem 1, p, 
 13(), other points in those parts where the arc is 
 required. By this means a giv^en arc may be 
 lengthened as far as is requisite. 
 
 '2.. To describe ati arc of a given radius, not less 
 than 10 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 over the fine line drawn on C D 
 for that purpose: bring the center to the point 
 through which the arc is required to pass^ and did-
 
 AXD ITS USE. 147 
 
 pose the bevel in the direction it is intended to be 
 drawn ; plaee the sectors G, G, exactly to the di- 
 visions 100 on each arm^ and strike the arc as 
 above described. 
 
 3. The bevel being set to strike arcs of a given 
 radius, as directed in the last paragraph, to drazv 
 other arcs whose 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 
 ares of 60 inches radius, with the bevel so set. 
 
 Say, as 50, the radius to which the bevel is set, 
 is to 60, the radius of the arcs required; so is the 
 constant number 100 to 120, the number on the 
 arms C A and CD, to which the sectors must be 
 placed, in order to describe arcs of 60 inches 
 radius. 
 
 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. 100 on CA and CD, when those arcs are 
 drawn . 
 
 4. An arc AC B (fig. 11, plate \0) being given, 
 to draxv other arcs concentric thereto, which shall 
 pass through given points, as V for ijistance. 
 
 Through the extremities A and B of the given arc 
 draw lines A P, B P tending to its center, by pro- 
 blem 3, p. 140. Take the nearest distance of the 
 given point P from the arc, and set it from A to P, 
 and from B to P. Hold the center of the level 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 
 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 ACB. 
 
 L. 2
 
 148 USE OP THE BEVEL, 
 
 5. Through a point K, (fig. 14, plate lo) in the 
 iriven line AB, to strike an arc of a given radius, 
 and whose center shall lie in that line, produced if 
 necessary. 
 
 Set the bevel to the given radius, as above dc- 
 seribed, (Method 2 J 
 
 Through A, at right angles to AB, draw CD; 
 lay the eenter of the bevel, set as above, on A, and 
 the arm CA, on the line AC, and draw a line AE 
 along the edge C D of the other arm. Divide the 
 angle DAE into two equal parts by the line AF, 
 jjlaee the bevel so, that its center being at A, the 
 arm CD shall lie on AF; while in this situation, 
 place the sectors at No. 100 on each arm, and then 
 strike the arc. 
 
 (). An arc being given, to find the length of its 
 radius. 
 
 Place the center of the bevel on the middle of 
 the arc, and open or shut the arms, till No. 100 
 on CA and CD fall upon the are on each side 
 the center; 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. 
 
 If the extent of the are be not equal to that be- 
 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 length- 
 ened, by problem 1, till it be of an extent suffici- 
 ent to admit the two Numbers 100. 
 
 Many more instances of the use of this instru- 
 ment might be given; but from what has been 
 already done, and an attentive perusal of the fore- 
 going problems, the principle of them may be 
 easily conceived.
 
 [ i^i9 ] 
 
 2. THE OBLIQUE RULER. 
 
 An instrument for drawing lines that are paral- 
 I'Cl, 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 ob- 
 lique ruler. 
 
 Fig. \T , plate 10, represents a simple contrivance 
 for this purpose; it consists of a cylindrical or pris- 
 matical tube AB, 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 are 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 dif- 
 ferent positions. 
 
 The ruler AB 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 rollers A 
 and C is increased; and as the diameter of C ap- 
 proaches that of A; all which is evident from 
 principle 2, page 134. 
 
 It also appears from the said principle, that du- 
 ring the motion of the ruler, any point in its 
 axis will accurately describe the are of a circle, 
 having the said point of intersection for its center; 
 and, consequently, the pen or tracer, put on the 
 pin D, will describe such arcs.
 
 150 ANOTHER OBLIQUE RULEE. 
 
 The rollers, as C, which screw upon the end of 
 the inner tube are numbered, 1, 2, 3, &c. and as 
 many scales are drawn on that tube as there are 
 rollers, one belonging to each, and numbered ac- 
 cordingly. These scales shew the distance in in- 
 ches, of the center or point of intersection, reckon- 
 ed 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 600 inches, ac- 
 cording as the tube is drawn out. 
 
 No. 2 - from 6oo inches to 300 inches 
 
 3 
 4 
 5 
 6 
 7 
 
 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 esteem- 
 ed as parallel. 
 
 3. ANOTHER RULER OF THE SAME KIND. 
 
 Tig. 18, pidfe 10. This is nothing more than 
 the last instrument applied to a fiat ruler, in the 
 manner the rolling parallel rulers are made. 
 
 C D is an hexagonal axis, moveable on pivots 
 in the heads A and F fixed upon a flat ruler; on 
 tills 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 
 
 300 
 
 - 
 
 150 
 
 150 
 
 - 
 
 80 
 
 80 
 
 - 
 
 40 
 
 40 
 
 - 
 
 20 
 
 20 
 
 - 
 
 10
 
 THE CYCLOGRAFH. 151 
 
 diameters. 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 correspondmg part 
 of the flat ruler; and the scales and rulers distin- 
 guished 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 rol- 
 lers; and the ruler drawn, or pushed at right an- 
 gles to its length. The tube AB,J?^-. 14, and one, 
 or both of the edges of the flat ruler, fg. 18, are 
 divided into inches and tenths. 
 
 4. THE CYCLOGRAPH. 
 
 This instrument is constructed upon the 
 third principle mentioned in page 135, of this 
 Essay. 
 
 Fig. \6, plale 10, is composed of five rulers; 
 four of them DE, D F, GE and GF, forming 
 a trapezium, 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 center of the 
 joint D, to that of the joints E and F, arc exactly 
 equal, as are the distances from G to the same 
 joints. The rulers DE and DF pass beyond 
 the joints E and F, where a roller is fixed to each; 
 the rollers are fixed upon their axes, wliich move 
 freely, but steadily on pivots, so as to admit of no 
 shake by which the inclination of the axes can be 
 varied. The ruler I D passing beyond the joint 
 D, carries a third roller A, like the others, whose
 
 152 THE CYCLOGRAPH OF 
 
 axis lies precisely in the direction of that ruler; 
 the axes of B and C extend to K and L. 
 
 A scale is put on the ruler D I, from H to G, 
 shewing, by the position of the socket G thereon, 
 the length of the radius of the arc in inches, that 
 would be described by the end I, in that position 
 of the trapezium. 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 D I, and the axis of the 
 roller A, arc precisely parallel ; and in this posi- 
 tion, the end I, or any other point in D I, will de- 
 scribe strait lines at right angles to DI; 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 tra^ 
 cer, is made to put on the end I, for the pur- 
 pose of describing arcs; and another is made for 
 lixing on any pnrt of the ruler D I, for the more 
 convenient description of concentric arcs, where a 
 number are \^'anted. 
 
 It is plain from this description, that the middle 
 ruler D I in this instrument, is a true oblique ruler, 
 by which lines may be drawn tending to a point, 
 whose distance from I is shcA^ n by the position of 
 the socket G on the scale; and the instrument is 
 made sufficiently large, so as to answer this pur- 
 pose as well as the other. 
 
 5. A DIFFERENT CONSTRUCTION OF THE SAME 
 INSTRUMENT. 
 
 In Fig. 16, phite 10, the part, intended to be 
 used in drawing lines, lies within the trapezium, 
 whi^'h is made large on that account: but this is 
 not necessary; ixndjig. 15^ f^/d/e 10, >\ill give an
 
 A DIFFERENT CONSTPvUCTIOX. 153 
 
 3(1c;i ot' a like instrument, wlicrc the trapezium 
 may be made much smaller, and consequcntlj 
 less cumbersome. 
 
 DBEC represents such a trapezium, rollers, 
 socket, and scale as above described, but much 
 smaller. Here the ruler E D is continued a suf- 
 jlicient 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 OF DESCEIBING AN ELLIPSE, AND 
 SOME OTHER CURVES. 
 
 To describe an ellipse, the transverse and conjugate 
 axes hein^r s'lven. 
 
 Let AB be the given transverse, and CD the 
 conjugate axis, 7^^. \3,plate 13. 
 
 Method I. By the line of sines on the sector, 
 open the sector with the extent AG of the semi- 
 transverse a,xis in the terms of 90 and 90 ; take 
 out the transverse distance of 70 and 70, (io 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 AB. Make 
 G C a ti'ansverse 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 intcrsebt the per- 
 pendiculars 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 drawn with a steady 
 hand, will complete the ellipse. 
 
 Method 2. V^'xxh. the elliptical compasses, /[^. 3, 
 flate 1 1, apply the transverse axis of the elliptical 
 rompasses to the lijie AB, and discharge the 
 screws of both the sliders; set the beam over the
 
 154 METHODS OP DESCRIBING 
 
 transverse axis AB, and slide it backwards and 
 forwards until the pencil or ink point coincide 
 with the point Aj and tighten the screw of that slider 
 which moves on the conjugate 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 com- 
 passes ; the compasses being thus adjusted, move 
 the ink point gently from A, through C to B, and 
 it will describe the semi-ellipse A C B ; reverse the 
 elliptical compasses, and describe the other semi- 
 ellipse B D A. These compasses were contrived 
 by my Father in 1748; they are superior to the 
 trammel which describes the whole ellipse, as 
 these will describe an ellipse of any excentricity, 
 which the others will not. 
 
 Through any given 'point F to describe an ellipse ^ 
 ihe transverse axis A B being given. 
 
 Apply the transverse axis of the elliptical com- 
 passes to the given line AB, 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 el- 
 liptical apparatus, acting upon the principle of 
 the oval lathes; the paper is fixed upon the board 
 AB, the pencil C is set to the transverse diameter 
 by sliding it on the bar D E, and is adjusted to the 
 conjugate diameter by the screw G; by turning 
 the board AB, an ellipse will be described by the 
 pencil. Fig. 2, A, plate 11, is the trammel, in 
 which the pins on the under side of the board AB, 
 move for the description of the ellipse. 
 
 Ellipses are described in a very pleasing manner 
 hy the geofnetric fen, Jig. I, plate 11; this part of 
 that instrument is frequently made separate.
 
 AX ELLIPSE. 155 
 
 To describe a parabola, ivhose parameter shall he 
 equal to a given line. Fig. 1 7 , plate 1 3 . 
 
 Draw a line to represent the axis, in which make 
 AB equal to half the given parameter. Open the 
 sector, so that AB may be the transverse distance 
 between QO and go 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 AB. Make the lines A a, 10 b, 
 20 c, 30 d, 40 e, &c. respectively equal to the 
 chords of 90°, 80°, 70°, 60°, 50°, &c. to the radius 
 AB, and the points abcde, &:c. will be in the 
 parabolic curve: for greater exactness, interme- 
 diate points may be obtained from the interme- 
 diate degrees; and a curve drawn through these 
 points and the vertex B, will be the parabola re- 
 quired : if the whole curve be wanted, the same 
 operation must be performed on the other side of 
 the axis. 
 
 As the chords on the sector run no further than 
 60°, those of 70, 80 and 90, may be found by 
 taking the transverse distance of the sines of 35 , 
 40°, 45°, to the radius AB, and applying those 
 distances twice along the lines, 20 c, 10 b, &c. 
 
 Fig. A, plate 11, is an instrument for describing 
 a parabola; the figure will render its use suffi- 
 ciently evident to every geometrician. ABCD 
 is a w^ooden- frame, whose sides AC, B D are pa- 
 rallel to each other; E F GH is a square frame of 
 brass or wood, sliding against the sides AC, B D 
 of the exterior frame; H a socket sliding on the 
 bar EFofthe interior frame, and carrying the 
 pencil I ; K a fixed point in the board, (the situa- 
 tion of which may be varied occasionally) ; E a K 
 is a thread equal in length to E F, one end thereof 
 is fixed at E, the other to the piece K, going over 
 ihc pencil at a. Bring the frame, so that the pen-
 
 156 AN HYPERBOLA. 
 
 cil may be in a line with the point K ; then slide it 
 in the exterior frame, and the pencil will describe 
 one part of a parabola. If the frame E F G H be 
 turned about, so that EF may be on the other 
 side of the point K, the remaining part of the pa- 
 rabola may be completed. 
 
 To describe an hyperhoJa^ the vertex A,. a?id 
 asymptotes BH, BI being given. Fig. 18, pi. 13. 
 
 Draw A I, AC, parallel to the asymptotes. 
 Make AC a transverse distance to 45, and 45, on 
 the upper tangents of the sector, and apply from B 
 as many of these tangents taken transversely as 
 may be thought convenient; as B D 50°, D E 55'', 
 and so on ; and through these points draw D d, 
 Ec, &c. parallel to AC. 
 
 Make AC 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 Dd 
 equal 40°, E C to 35°, E F to 80°, &e. and these 
 points will be in the hyperbolic curve, and a line 
 drawn through them will be the hyperbola re- 
 quired. 
 
 To assist the hand in drawing curves through a 
 number of points^ artists make use of what is termed 
 the bow., consisting of a spring of hard wood, or 
 steel, so adapted to a firm strait rule, that it may 
 be bent more or less by three screws passing 
 through the strait rule. 
 
 A set of spirals cut out in brass, are extremely 
 convenient 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, b, c, da running line 
 going pver the cone G and cylinder H, the ends
 
 GEOMETRIC PEN. 157 
 
 being fastened to the pin e. On turning the frame 
 K N O, the thread carries the pencil progressively 
 from the cone to the cylinder, and thus describes 
 a spiral. The size of the spiral may be varied, by 
 placing the thread in different grooves, by putting 
 it on the furthermost cone, or by putting on a 
 larger cone. 
 
 OF THE GEOMETRIC PEN. 
 
 The geometric pen is an instrument in which, 
 by a circular motion, a right line, a circle, an el- 
 lipse, and a great variety of geometrical figures, 
 may be described. 
 
 This curious instrument was invented and de- 
 fcribed by John Baptist Suanli, in a work entitled 
 ]\'uovo Istromenti per la Ddscrizzione di diverse 
 Curve Antichi e Moderne, &c. Tec (^i>- :^:-' . /'.'">'., 
 
 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 inge- 
 nious mechanic. There never was, perhaps, any 
 instrument which delineates so many curves as the 
 geometric pen; the author enumerates 1273, as 
 possible to be described by it in the simple form, 
 and with the few wheels appropriated to it for the 
 present work. 
 
 Fig 1, plate 11, represents the geometric pen% 
 A. B, C, the stand by which it is supported; the
 
 158 GEOMETRIC PEX. 
 
 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 this axis any of the wheels 
 (as i) maybe adapted; when screwed to it they 
 arc immoveable. 
 
 E G is an arm contrived to turn round upon the 
 main axis D; two sliding boxes are fitted to this 
 arm ; to these boxes any of the wheels belonging to 
 the geoilietric pen may be fixed, and then moved 
 so that the wheels may take into each other, and 
 the immoveable wheel i; it is evident, that by 
 making the arm E G 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. 
 
 fg 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 cen- 
 ter 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 fg along with it; 
 it is evident, therefore, that the revolutions will be 
 fewer or greater, in proportion to the difi:erencc 
 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 bar EG; 
 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 in infinitum. 
 
 The numbers we have used arc, 8, \6, 24, 32, 
 40, 48; 5(i, 64, 72, 80, 88, Qt).
 
 GEOMETRIC PEN. ISg 
 
 The construction and application of this instru- 
 ment is so evident from the figure, that nothinjr 
 more need be pointed out than the combinations 
 by which the ligures here dehneated may be pro- 
 duced. 
 
 To render the description as concise as pos- 
 sible, I shall in future describe the arm E G by 
 the letter A, and f g by the letter B. 
 
 To describe Jig. 1, ■plale 12. The radius of A 
 jnust be to that of B, as 10 to 5 nearly, their velo- 
 <'ities, or the numbers of teeth in the wheels, to be 
 equal, the motion to be in the same direction. 
 
 If the length of B be varied, the looped figure, 
 delineated at Jig. 12, will be produced. 
 
 A circle -may be described by equal wheels, and 
 any radius, but the bars, must move in contrary 
 directions. 
 
 To describe the two level figures, see Jig. 11, 
 ph/fe 12. Let the radius of A to B be as 10 to 3f, 
 the velocities as 1 to 2, the motion in the same 
 direction. 
 
 To describe by this ci r c u l a r m o t i o n , A s t r a i t 
 LINE AND AN ELLIPSE. For a strait line, equal 
 radii, the velocity as 1 to 2, the motion in a con- 
 trary 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; see 
 fg. 10, plate 13. 
 
 Fig. 13, plate 12, with seven leaves, is to be 
 formed when the radii are as 7 to 2, velocity as 
 2 to 3, motion in contrary directions. 
 
 The six triangular figures, seen aXjig. 2, 4, 6, 8, 
 9, 10, are all produced by the same wheels, by 
 only varying the length of the arm B, the velocity 
 $hould be as 1 to 3, the arms are to move in con- 
 trary directions.
 
 l60 DIVISION OP LAXD. 
 
 Fig, 3, f)Ja/c 12, ^^'ith eight leaves, is formed by 
 equal radii, veloeitics as 5 to S, A and B to move 
 the same wav; if an intermediate wheel is added, 
 and thus a motion produced in a contrary direc- 
 tion, the pencil will delineate j'fi'-. l6, flute 12. 
 
 The ten-leaved tigure, fig. 15, flafe 12, is pro- 
 duced by equal radii, velocity as 3 to 10, direc- 
 tions of tlie motions contrary to each other. 
 
 Hitherto the velocity of the epicycle has been 
 the greatest; in the three following figures the 
 CLirvcis arc produced when tlic velocity of the epi- 
 cycle is less than that of the pi-imimi mobile. 
 
 For jig. 7, the radius of A to B to be as 2 to 1, 
 the velocity as 3 to 2 ; to be moved the same way. 
 
 For Jig. 14, the radius of A, somewhat less than 
 the diameter given to B, the velocity as 3 to 1 ; to 
 be moved in a conti'ary direction. 
 
 For Jrg. 5, equal radii, velocity as 3 to 1 ; moved 
 the same way. These instances are sufficient to 
 shew how much may be performed by this instru- 
 ment; with a few additional pieces, it may be 
 made to describe a cycloid, with a circular base, 
 spirals, and particularly the spiral of Archime- 
 des, &c. 
 
 OP THE DIVISION OP LAXD. 
 
 To know how to divide land into any number 
 of equal, or unequal parts, according to any as- 
 signed proportion, and to make proper allowances 
 for the different qualities of the land to be divided, 
 form a material and useful branch of surveying. 
 
 In dividing of land, numerous eases 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
 
 DiVISION OP LAND. l6l 
 
 often to be made according to the particular di- 
 rections of the parties concerned. In a subject 
 which has been treated on so often, novelty is, 
 perhaps, not to be desired, and scarcely expected.- 
 No considerable improvement has been made in. 
 this branch of surveying since the time of SpeidelL 
 Mr. Talbot, whom we shall chiefly follow, has 
 arranged the subject better than those who pre- 
 ceded him, and added thereto two or three pro- 
 blems ; his work is well worth the surveyor's peru- 
 sal. Some problems also in the foregoing part of 
 this work should be considered in this place. 
 
 Problem l. To divide a triangle in a given 
 ratio, hy right lifies drawn from any angle to the op- 
 posite 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 
 required. 
 
 Thus, to divide the triangle K^C, fig. 13, 
 plate 8, containing 26^ acres, into three parts, in 
 proportion to the numbers 40, 20, 10, the lines of 
 division to proceed from the angle C to AB, 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 AB 
 into seven equal parts; draw Ca at four of these 
 parts, Cb at six of them, and the triangle is di- 
 vided as required. 
 
 Arithmetically. As 7? the sum of the ratios, is 
 to AB 28 chains; so is 4, 2, 1, to 1 6, 8, and 4 
 chains respectively; therefore A a = 1(3, 'jh = 8, 
 and b B irz: 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; 
 
 M
 
 i6'2 DIVISION OF LAND. 
 
 7 : 26,5 :: 4 : 15,142857 ^ = triangle AC 3 
 7 : 26,5 :: 2 : 7,571428 > = triangle aCb 
 7 : 2(3,5 :: 1 : 3,765714 J = triangle bCB,^' 
 
 Problem 2. To divide a triangular field into 
 any numher of parts, and in any given proportio?!, 
 froju a given point in one of the sides. 
 
 1 . Divide the triangle into the given proportion, 
 from the angle opposite the given point. 2. Re- 
 duce this triangle by problem 51, so as to pass 
 through the given point. 
 
 Thus, to divide the field ABC, fig. \4, 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 benefit of 
 the pond. 
 
 1 . Divide B C into seven equal parts, make B a 
 = 5, then Ca = 2, draw A a, and the field is di- 
 vided in the given ratio. 2. To reduce this to the 
 point b, draw Ab and ac parallel thereto, join cb, 
 and it will be the required dividing line. 
 
 Operation in the field. Divide BC in the ratio 
 required, and set up a mark at the point a, and 
 also at the pond h; at A, with the theodolite, or 
 other instrument, measure the angle bAa; at a 
 lay off the same angle A a c, which will give the 
 point e in the side A c, from whence the fence 
 must go to the pond. 
 
 2. To divide ABC,^^-. 15, plate 8, into three 
 equal parts from the pond c. 1. Divide AB 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 they 
 will be the true dividing lines. 
 
 * Talbot's Complete Art of Land Measuring.
 
 DIVISION OP LAND. l63 
 
 Problem 3. To dhide a triangular Ji eld in any 
 required ratio ^ hy lines drawn parallel to one side^ 
 and cutting the others. 
 
 Let ABC, fig, 16, fdate 8, be the given tri- 
 angle to be divided into three equal parts, by lines 
 paraUcl to AB, and cutting AC, BC. 
 
 Ride 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 
 parallel to the given side through the mean pro- 
 portional. 4. Proceed in the same manner with 
 the remaining triangle. 
 
 Example 1 . Divide B C into three equal parts 
 B D, D P, P C. 1. Find a mean proportional be- 
 tween B C and DC. 3. Make C G equal to thi>; 
 mean proportional, and draw GH parallel to AB. 
 Proceed in the same manner with the remaining 
 triangle C H G, dividing G C into twO equal p:irt5 
 at I, finding a mean proportional between C G 
 and CI; and then making C L equal to this mean 
 proportional, and drawing LM parallel to AB, 
 the triangle will be divided as required. 
 
 A square, or rectangle^ a rhombus, or rhomho'ides^ 
 may he divided into any given ratio, hy lines cutting 
 two opposite parallel sides, hy dividing the sides into 
 the proposed ratio, a?id joining the faints of division. 
 
 Problem 4. To divide a right-lined figure into 
 any proposed ratio, by lines proceeding from one angle. 
 
 Problem 5. To divide a right-lined figure into 
 miy proposed ratio, by lines proceeding from a given 
 point in one of the sides. 
 
 Problem (3. To divide a right-lined figure into 
 aiiv proposed ratio, by right lines proceeding from a 
 given point within the said figure or field. 
 
 It would be needless to enter into a detail of 
 the mode of performing the three foregoing pro- 
 
 M 2
 
 1(54 DltlSION OF LAND. 
 
 blems, as the subject has been already sufficiently 
 treated of in pages 84, 85, &c. 
 
 Problem 7-"^ " ^^ is required to divide any given 
 qttavtity of ground into any given nujnher of farts, 
 and inprojyortio?! 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 respectively in proportion as the numbers 1 .. 
 3,6, 10, or whose estates may be supposed lOOl. 
 300l. 600l. and lOOOl. per annum. 
 
 The sum of these proportional numbers is 20, 
 by which dividing 300 acres, the quotient is 1 5 
 acres, which being multiplied by each of the num- 
 bers 1,3, 6, 10, we obtain for the several shares 
 as follows: 
 
 a. 
 A's share = 15 ; 
 
 r. 
 ; : 
 
 ; 00 
 
 B's share = 45 : 
 
 ; ; 
 
 : 00 
 
 C's share = gO : 
 
 : ; 
 
 : 00 
 
 D's share = 1 50 : 
 
 : : 
 
 : 00 
 
 Sum = 300 ; 
 
 : ; 
 
 : 00 the proof. 
 
 Bat this is upon supposition that the land is all of 
 an equal value. 
 
 Now let us suppose the land, to be laid off for 
 each person's share, is of the following different 
 
 ■* Problem /;, an erroneous rule given by a late writer, intro- 
 duced here to prevent the practitioner being led into error. — 
 From Talbot's Complete Art of Land Measuring, Problem J, 
 page 20o; and Appendix^ page 410.
 
 DIVISION OP LAND. 
 
 i65 
 
 v^-ilues per acre, viz. A's = 5s. B's=:8s. C's = l2s. 
 D's = 1 5s. an acre; whose sum, 40s. divided by 4, 
 their number, quotes 10s. for the mean value per 
 acre. And, according to a late author, we 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 
 
 12 
 
 10 : 
 
 : 90 : 
 
 75 
 
 C's share 
 
 15 
 
 10 : 
 
 : 150 : 
 
 100 
 
 D's share 
 
 sum of shares =26 1,3 acres,' 
 
 \v"hich 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 re- 
 main 38 acres unapplied; I suppose for the use of 
 the surveyor. 
 
 But suppose we change the value of each per- 
 son's land, and call A's 15s. B's 12s. C's 8s. and 
 D's 5s. then we shall have 
 
 s. 
 
 s. 
 
 a. 
 
 a. 
 
 
 15 
 
 10 : 
 
 ; 15 
 
 10 
 
 A's share 
 
 12 
 
 10 : 
 
 : 45 
 
 37,5 
 
 B's share 
 
 8 
 
 10 : 
 
 : go 
 
 112,5 
 
 C's share 
 
 5 
 
 10 : 
 
 : 150 
 
 300 
 
 D's share 
 
 as 
 
 sum of the shares 46o 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 l6o 
 acres for the other three shares ? But enough of 
 this; see it truly and methodically performed in 
 the next problem.
 
 l66 DITISIOX OP LAND. 
 
 Problem 8. // is required to diviile any gwen 
 quantity of land among any given number of fersons, 
 in proportion to their several estates^ and the value of 
 the land that falls to each persons share. 
 
 Rule. Divide the yearly value of each person's 
 estate by the value per acre of the land that is al- 
 lotted for his share, and take the sum of the quo- 
 tientSj by which divide the whole given quantity 
 of land, and this quotient will be a common mul- 
 tiplier, 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 the land. 
 
 Example, Let 300 acres of land be divided 
 
 among A, B, C, D, whose estates are lOOl, 300l. 
 
 t)00l. 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. 
 
 rp, 100 ^^ 300 ^^ „ 600 .^ , 
 
 Ihen = 20, =:=:37,D =50, and 
 
 5 8 12 
 
 — — -=^^^,^0^, and the sum of these quotients is 
 
 r^ 300 
 
 1 74, 1 6o . • . -z — rrr-^==-- 1 ,1 224 8, the common muL 
 1/4,100 
 
 tiplicr : 
 
 acres, 
 
 1,72248 X 20 zzz: 34,45 A's share 
 
 then ^ 1^72248 X 37,5 = 64,593 B's share 
 
 men's 1^7^248 X 50 = 86,124 C's share 
 
 1,72248 X 6(3,66(5 = 114,832 D's share 
 
 sum of the shares = 299,999 acres, or 
 300 very nearly, and is a proof of thf whole,
 
 DIVISION OF LAND. 1^7 
 
 Let us now change the values of land as In the 
 last problem, and see what each will have tor his 
 6hare. Suppose A's 15, B's 12, C's 8, and D's 5 
 
 100 ^ ^^^ 300 (ioo 
 
 shilhngs an acre; then-— =6, 666, — -r^Qo,— — - 
 
 1 1 Z o 
 
 ^' 1000 , , 
 
 =7o, — : — =200, and the sum of these quo- 
 tients is 306,666, therefore — ,. ,.,.,> =0,Q7826 
 
 the common multiplier : 
 
 I? 
 
 then 
 
 acres, 
 0,97826 X QfiQQ = 6,5217 A's share 
 97826 X 25, = 24,4565 B's share 
 0,97826 X 75, — 73,3695 C's share 
 .0,97826 X 200, = 195,6520 D's share 
 
 the sum of the shares = 299,9997 acres, 
 which proves the whole to be right. 
 
 Exajnph 2. Let 500 acres be divided among 
 six persons, whose estates areas follow; viz. A's 
 40l. B's 20I. C's lOl. D's lOOl. E's 4001. and F's 
 lOOOl. per annum, and the value of the land most 
 convenient for each is A's, B's and C's, each 7s. 
 D's lO's. E's 15s. and F's 12 s. an acre; now each 
 estate divided by the value of his share of the land, 
 will stand thus : 
 
 
 s. 
 
 /. 
 
 
 
 A 
 
 7 ) 
 
 40 
 
 
 5,71428- 
 
 B 
 
 7 ) 
 
 20 
 
 
 2,85714 
 
 C 
 
 7 ) 
 
 10 
 
 
 1,42857 
 
 D 
 
 10 ) 
 
 100 
 
 
 10, 
 
 E 
 
 15 ) 
 
 400 
 
 
 26,66666 
 
 F 
 
 12 ) 
 
 1000 
 
 
 83,33333- 
 
 > quotient 
 
 sumof thequotients=:l29,99998j or 130.
 
 ids DIVISION OP LAND. 
 
 Now =3.846153, the common multiplier; 
 
 130 
 
 by which, multiplying each quotient, we shall 
 
 have for each share as follows, viz. 
 
 
 
 acres. 
 
 A 
 
 = 
 
 21,9778 
 
 B 
 
 = 
 
 10,9918 
 
 C 
 
 = 
 
 5,4941 
 
 D 
 
 = 
 
 38,46l5 
 
 E 
 
 = 
 
 102,5641 
 
 F 
 
 = 
 
 320,5127 
 
 their sum is = 500,0020, and thus proves 
 the whole. 
 
 N. B. If any single share should contain land 
 of several different values, then use the means to 
 divide his estate by. 
 
 Also if there be different quantities as well as 
 values, find what each quantity is worth at its va- 
 lue, and add their sums together; then say, as the 
 sum of the quantities is to this sum; so is one acre 
 to its mean value to be made use of. 
 
 Now having found each person's share, they 
 may be laid out in any form required, by the di- 
 rections and problems given in the next section. 
 But when scv^eral shares contain land of the same 
 value, it is best to lay out their sum in the most 
 convenient form, and then subdivide it. 
 
 For dividing of coinvions, &c. Surveyors generally 
 measure the land of different value in separate par- 
 cels, and find the separate value thereof, which 
 added in different sums, gives the whole content, 
 and whole value. 
 
 By problem 1, they find each man's propor- 
 tional share of the whole value, and then lay out
 
 LAYING OUT LAND. I69 
 
 for each person, a quantity of land equal in value 
 to his share; this they eiiect by first laying out a 
 quantity by guess, and then casting it up and find- 
 ing its value; and if such value be equal to his 
 share of the whole value, the dividing line is right; 
 if otherwise, they shift the dividing line a little, 
 till by trial they find a quantity just equal in value 
 to the value of the required share. 
 
 If any single share contains land of several dif- 
 ferent values, each is measured separately, and 
 their several values found; and if the sum of them 
 be equal to the value sought, the division is right; 
 if not, it must be altered till it is so. 
 
 OF LAYING OUT ANY GIVEN QUANTITY OF 
 LAND, 
 
 As the quantity of land is generally given in 
 acres, roods, and perches, it is necessary, first, to 
 reduce them to square links, which may be per- 
 formed by the following rule. 
 
 To reduce acres, roods, and perches into square 
 links. 
 
 Rule 1 . To the acres annex five cyphers on the 
 right hand, and the whole will be links. 2. Place 
 five cyphers to the right of the roods, and divide 
 this by 4, the quotient will be links. 3. Place 
 four cyphers on the right hand of the perches, di- 
 vide this by l6, the quotient will be links. 4. 
 These sums added together, give the sum of 
 square links in the given quantity. 
 
 Problem 1. To Jay out a piece of land, contain- 
 mg any given number of acres, inform of a square. 
 
 This is no other than to determine the side of a 
 square that shall contain any desired number of 
 Acres; reduce, therefore;, the given number of
 
 ITO LAYING OUT LAND. 
 
 acres to scjiiare links, and the square root thereof 
 will be the side of the square required. 
 
 Problem 2. To lay out any desired quantity of 
 land in form of ,a parallelogram ^ having either its 
 base or altitude given. 
 
 Divide the content, or area, by the given base, 
 and the quotient is the altitude; if divided by the 
 altitude, the quotient is the base; by the same 
 rule a rectangle may be laid out. 
 
 Problem 3. To lay out a?iy desired quantity of 
 land in form of a parallelogram, whose has e shall he 
 2, 3, 4, &'c. times greater than its altitude. 
 
 Divide the ^area by the number of times the 
 base is to be greater than the altitude, and extract 
 the square root of the quotient; this square root 
 will be the required altitude, which being multi- 
 plied by the number of times that the base is to be 
 greater than the altitude, will give the length of 
 the required base. 
 
 Problem 4. To lay out a given quantity of land 
 in form of a triangle, having either the base or the 
 ferpendicidar given. 
 
 Divide the area by half the given base, if the 
 base be given ; or by half the given perpendicular, 
 if the perpendicular be given ; and the quotient 
 will be the perpendicular, or base required. 
 
 Problem 5. To lay out any given quantity of 
 land in a regular polygon. 
 
 1 . Find in the following table, the area of a 
 polygon of the same name with that required, the 
 side of which is 1. 2. Divide the proposed area 
 by that found in the table. 3. Extract the square 
 root of the quotient, and the root is the side of the 
 polygon recjuired.
 
 TLATN TRIGONOMETRY. 
 
 171 
 
 No. 
 sides 
 
 Names. 
 
 Areas. 
 
 
 
 3 
 
 Triangle 
 
 0,433 
 
 4 
 
 Square 
 
 1, 
 
 5 
 
 Pentag;on 
 
 1,72 
 
 6 
 
 Hexagon 
 
 2,598 
 
 7 
 
 Heptagon 
 
 3,634 
 
 8 
 
 Octagon 
 
 4,828 
 
 9 
 
 Nonagon 
 
 6,182 
 
 10 
 
 Decagon 
 
 7,694 
 
 11 
 
 Undecagon 
 
 9,365 
 
 12 
 
 Duodecagon 
 
 11,196 
 
 Problem 6. To lay out any quanl'ity of land In 
 a circle. 
 
 1. Divide the area by ,7854. 2. Extract the 
 square root of the quotient, for the diameter re- 
 quired. 
 
 OF PLAIN TRIGONOMETRY. 
 
 Plain trigonometry is the art of measuring and 
 computing the sides of plain triangles, or of such 
 whose sides are right lines. 
 
 As this work is not intended to teach the ele- 
 ments of the mathematics, it will be sufficient for 
 me just to point out a few of the principles, and 
 give the rules of plain trigonometry, for those 
 cases that occur In surveying. In most of those 
 cases, it is required to find lines or angles, whose 
 actual admeasurement is difficult or impracticable; 
 they are discovered by the relation they bear to 
 other given lines or angles, a calculation being in- 
 stituted for that purpose; and as the comparison 
 of one right line with another right line, is more 
 convenient and easy, than the comparison of a right 
 line to a curve; it has been found advantageous tQ
 
 172 PLAIN TRIGONOMETJRY. 
 
 measure tlic quantities of angles, not by the arc it- 
 self, which is described on the angular point, but 
 by certain lines described about that arc. 
 
 If any three parts* of a plain triangle be given, 
 any required part may be found both by construc- 
 tion and calculation. 
 
 If two angles of a plain triangle are known in 
 degrees, minutes, &c. the third angle is found by- 
 subtracting their sum from 180 degrees. 
 
 In a right-angled plain triangle, if either acute 
 angle (in degrees) be taken from go degrees, the 
 remainder will express the other acute angle. 
 
 When the sine of an obtuse angle is required, 
 subtract such obtuse angle frorn 1 80 degrees, and 
 take the sine of the remainder, or supplement. 
 
 If two sides of a triangle are equal, a line bisect- 
 ing the contained angle, will be perpendicular to 
 the remaining side, and divide it equally. 
 
 Before the required side of a triangle can be 
 found by calculation, its opposite angle must first 
 be given, or found. 
 
 The required part of a triangle must be the last 
 term of four proportionals, written in order under 
 one another, whereof the three first are given or 
 known. 
 
 In four proportional quantities, either of them 
 may be made the last term ; thus, let A, B, C, D, 
 be proportional quantities. 
 
 As first to second, so is third to fourth, A:B::C:D. 
 As second to first, so is fourth to third, B : A : : D ; C. 
 As third to fourth, so is first to second, C : D :: A : B. 
 As fourth to third, so is second to first, D : C : : B : A. 
 
 Against the three first terms of every propor- 
 tion, or stating, must be written their respective 
 values taken from the proper tables. 
 
 * This is imperfectly stated by several writers. One of the 
 given parts must be a side. A triangle consists of six parts^ via. 
 three sides and three angles. Edit.
 
 If the value of the first term be taken from the. 
 sum of the second and third, the remainder will 
 be. the value of the fourth term or thing required ; 
 because the addition and subtraction of loearithms, 
 corresponds with the multiplication and division 
 ■of natural numbers. 
 
 If to the complement of the first value, be added 
 the second and third values, the sum, rejecting the 
 borrowed index, \vill be the tabular number ex- 
 pressing the thing required : this method is gene- 
 rally used when radius is not one of the propor- 
 tionals. 
 
 The complement of any logarithm, sine, or tan- 
 gent, in the common tables, is its diflerence from 
 the radius 10.000.000 , or its double 20.000.000. 
 
 CANONS FOR TRIGONOMETRICAL 
 
 CALCULATION, 
 
 1. The following proportion is to be used when 
 two angles of a triangle, and a side opposite to one 
 of them, is giv'Cn to find the other side. 
 
 As the sine of the angle opposite the given side, 
 is to the sine of the angle opposite the required 
 side ; so is the given side to the required side. 
 
 2. When two sides and an angle opposite to 
 one of them is given, to find another angle; use 
 the following rule : 
 
 As the side opposite the given angle, is to the 
 side opposite the required angle ; so is the sine of 
 the given angle, to the sine of the required angle. 
 
 The memory will be assisted in the foregoing 
 cases, by observing that when a side is wanted, the 
 proportion must begin with an aiigle; and when an 
 arigle is wanted, it must begin with a side. 
 
 3. When two sides of a triangle and the inclu- 
 ded angle are given, to find the other angles and 
 side.
 
 174 CANOTsTS. 
 
 As the sum of the two o-ivcn sides is to tlieJf 
 difiereiiee; so is the tangent of half the sum of 
 the two unknown angles, to the tangent of half 
 their difFerenee. 
 
 Half the difference thus found, added to half 
 their sum, gives the greater of the two angles, 
 whieh is the angle opposite the greatest side. If 
 the third side is wanted, it may be found by solu- 
 tion 1. 
 
 4. The following steps and proportions are to be 
 used when the three sides of a triangle are given, 
 and the angles required. 
 
 Let ABC, plate 9, fg. 30, be the triangle; 
 make the longest side AB the base; from C the 
 angle opposite to the base, let fall the perpen- 
 dicular C D on AB, this will divide the base into 
 two segments AD, BD. 
 
 The difFerenee between the two sei^ments is 
 found by the following proportion: 
 
 As the base AB, or sum of the two segment •?;; 
 is to the sum of the other sides (AC-f-B C) ; so is 
 the difference of the sides (AC— B C), to the dif- 
 ference of the segments of the base (AD — D B). 
 
 Half the difference of the segments thus found, 
 added to the half of AB, gives the greater seg- 
 ment AD, or subtracted, leaves the less DB. 
 
 In the triangle ADC, find the angle AC D by 
 solution 2; for the two sides AD and AC arc 
 known, and the right angle at D is opposite to 
 one of them. 
 
 The complement of ACD, gives the angle A. 
 
 Then in the triangle ABC, you have two sides 
 AB, BC, and the angle at A, opposite to one of 
 them, to find the angles C and B.
 
 I" 175 ] 
 
 ©P THB LOGARITHMIC SCALES ON THE SECTOR, 
 
 There arc three of these lines usually put on the 
 sector, they arc often termed the Guntcr's lines, 
 and are made use of for readily working propor- 
 tions; when used, the sector is to be quite opened 
 like a strait rule. 
 
 If the 1 at the beginning of the scale, or at thft 
 left hand of the first interval, be taken for unity, 
 then the 1 in the middle, or that which is at the 
 end of the first interval and beginning of the se- 
 cond will express the number 10; and the ten at 
 the end of the right-hand of the second interval or 
 end of the scale, will represent the number 100. 
 if the first is 10, the middle is 100, and the last 
 1000; the primary and intermediate divisions in 
 each interval, are to be estimated according to 
 the value set on their extremities. 
 
 In working proportions with these lines, atten- 
 tion must be paid to the terms, whether arithme- 
 tical or trigonometrical, that the first and third 
 term may be of the same name, and the second 
 and fourth of the same name. To work a pro- 
 portion, take the extent on its proper line, from 
 the first term to the third in your compasses, and 
 applying one point of the compasses to the se- 
 cond, the other applied to the right or left, ac- 
 cording as the fourth term is to be more or less 
 than the second, will reach to the fourth. 
 
 Example 1. If 4 yards of cloth cost 18 shillings, 
 what will 32 yards cost ? This is solved by the line 
 of numbers; take in your compasses the distance 
 between 4 and 32, then apply one foot thereof on 
 the same line at 18, and th« other will reach 144, 
 the shillings required.
 
 17^ CURIOUS AND USEFUL. 
 
 Example 2. As radius to the hypothcnuse 120, 
 so is the sine of the angle opposite the base 30° 
 17' to the base. In this example, radius, or the 
 sine of 90, and the sine of 30° \f taken from the 
 line of sines, and one foot being then applied to 
 120 on the line of numbers, and the other foot on 
 the left Avill reach to 6o| thelength of the required 
 base. The foot was applied to the left, because the 
 legs of a right-angled triangle are less than the 
 iiypothenuse. 
 
 Example "i. As the cosine of the latitude 51° 
 oO', (cqual-thc sine of 38° 30') is to radius, so is 
 tlie sine of the sun's declination 20° 14', to the 
 sine of the sun's amplitude. Take the distance 
 between the sines of 38° 30' and 20° 14' in your 
 compasses; set one foot on the radius, or sine of 
 90°, and the other will reach to 33 ;°, the sun's 
 amplitude required. 
 
 CURIOUS AND USEFUL TRIGONOMETRICAL- 
 PROBLEMS. 
 
 The following problems, though of the greatest 
 use, and sometimes of absolute necessity to the 
 surveyor, arc not to be found in any of the com- 
 mon treatises on surveying. The maritime sur- 
 veyor can scarce proceed \vithout the kno\Adedge 
 of them ; nor can a kingdom, province, or county 
 be accurately surveyed, unless the surveyor is well 
 acquainted with the use and application of them. 
 Indeed, no man should attempt to survey a county, 
 or a sea coast, who is not master of these problems. 
 The second problem, which is peculiarly useful 
 for determining the exact situation of sands, or 
 rocks, within sight of three places upon land, 
 whose distances arc well known, was first proposed
 
 GEOMETRICAL PROBLEMS. 177 
 
 by Mr. Tozvnly, and solved by Mr. Collins^ Philo- 
 sophical Transactions, No. 69. There is " no 
 problem more useful in surveying, than that by 
 which we find a station, by observed ang-les of 
 three or more objects, whose reciprocal distances 
 are known; but distance, and bearing from the 
 place of observation are unknown, 
 
 " Previous to the resolution of these problems, 
 another problem for the easy finding the segment 
 of a circle, capable of containing a given ang^le, 
 is necessary, as will be clear from the following 
 observation. 
 
 " Two objects can only be seen under the same 
 angle, from some part of a circle passing through 
 those objects, and the place of observ^ation. 
 
 " If the angle under which those objects appear, 
 be less than 90°, the place of observation will be 
 somewhere in the greater segment, and those ob- 
 jects will be seen under the same angle from every 
 part of the segment. 
 
 " If the angle, under which those objects are 
 seen, be more than 90°, the place of observation 
 will be somewhere in the lesser segment, and those 
 objects will be seen under the same angle from 
 every part of that segment*." Hence, from the 
 situation of three known objects, we are able to 
 determine the station point with accuracy. 
 
 Problem. To describe on a given line VtQ, fig, 
 29, plate 9, a segment of a circle, capable of contain- 
 ing a given angle. 
 
 Method 1 . Bisect B C in A. 2. Through the 
 point of bivseetion, draw the indefinite right line 
 DE perpendicular to B C. 3. Upon B C, at the 
 point C, constitute the angles DCB, FCB, G C B, 
 
 * Dalrymple's Essay on Nautical Surveying, in which ano- 
 ther mode of solving this problem is given. 
 
 N
 
 178 CURIOUS AXD USEFUL 
 
 II C B, respectively equal to the difference of the 
 
 angles of the intended segments and 90 degrees: 
 the angle to be formed on the same side with the 
 segment, if the angle be less than 9O; but on the 
 opposite, if the angle is to be greater than go de- 
 grees. 4. The points D, F, G, H, where the an- 
 gular lines CD, CF, CG, CH, &e. intersect 
 the line D E, will be the centers of the intended 
 segments. 
 
 Thus, if the intended segment is to contain an 
 angle of 120°, constitute on B C, at C, (on the op- 
 posite side to which you intend the segment to be 
 tleseribed,) the angle D C B equal to 30°, the dif- 
 ference between go° and 120°; then on center D, 
 and radius DC, describe the segment C, 120, B, 
 in every part of which, the two points C and B 
 will subtend an angle of 120 degrees. 
 
 HI want the segment to contain 80 degrees at 
 O, on B C make an angle B C G, equal 10 degrees, 
 and on the same side of B C as the intended seg- 
 jnent; then on G, with radius G C, describe seg- 
 ment C 80 B, in every part of w hich C and B will 
 subtend an angle of 80 degrees. 
 
 Method 2. By the sector. Bisect B C as before, 
 and draw the indefinite line D E, make AC radilis, 
 and with that extent open the sector at 45 on the 
 line of tangents, and set off on the line DE, the 
 tangent of the difference between the observed 
 angle and po degrees, on the same side as the in- 
 tended segment, if the observed angle is less than 
 QO; on the contrary side if more than 90 degrees. 
 
 If the angle is of pO degrees, A is the center of 
 the circle, and AC the radius. 
 
 The intersection of the line D E with any cir- 
 cle, is the center of a segment, corresponding to 
 Juiif the angle of the segment in the first eircle. 
 Hence, if the difference between tlie obse;-vcd
 
 GEOMETRICAL PROBLEMS. l/C) 
 
 angle and QO'', be more than the scale of tangcntv«« 
 contains, find the center to double the angle ob- 
 served, and the point where the circle cuts the in- 
 definite line, will be the required center. 
 
 Problem 1. To determine the position of a po'inl^ 
 from whence three points or a triangle can be dis- 
 covered, whose distances are hiown. 
 
 The point is either without, or within the given 
 triangle, or in the direction of two points of the 
 triangle. 
 
 Case 1. If lien the three given objects form a trl~ 
 angle, ami the point or station whose position Is re- 
 quired, Is without the triangle. 
 
 Example. Suppose I ^\■ant to determine the 
 position of a rock T>,fg. IJ, plate Q, from the 
 shore; the distances of the three points A, C, B, 
 or rather the three sides AC, CB, iVB, of the tri- 
 angle A, B, C, being given. 
 
 In the first place, the angles AD C, C D B^ must 
 be measured by an Hadley's sextant or theodolite; 
 then the situation of the point D may be readily 
 found, cither by calculation or construction. 
 
 By construction. Method 1. On AC, fg. 28, 
 plate 9, describe by the preceding problem, a cir- 
 cle capable of containing an angle equal to the 
 angle ADC; on CB, a segment containing an 
 angle equal to the angle CDB; and the point of 
 intersection D is the place required. 
 
 Another method. Make the angle EB A, fg. 3g, 
 />/tf/«? 9, equal to the angle ADE, and the an2:le 
 B AE equal to the angle E D B. Through A,^B, 
 and the intersection E, describe a circle AEB D ; 
 through E, C, draw E C, and produce it to inter- 
 sect the circle at D; join AD. B D, and the distan- 
 ces AD, C D, B Dj will be the required distances. 
 
 Calculation: ' In the triangle ABC, are given 
 the three sides, to find the angle BAC. In the 
 
 N 2
 
 180 CUllIOUS AND USEFUL 
 
 triangle AE B, arc given the angle BAE^ the 
 angles ABE, AEB, and the side A B, to find 
 AEandBE. 
 
 In the triangle AED, we have the side AE, 
 and the angles AED, AD E, and consequently 
 D FA, to find the sides AD. 
 
 The angle ADE, added to the angle AEC, 
 and then taken from 180°, gives the angle DAE. 
 The angle CAE, taken from the angle DAE, 
 gives the angle CAD, and hence D C. Lastl}', 
 the angle AEC, taken from AE B, gives D E B^, 
 and consequently, in the triangle DEB, we have 
 E B, the angle DEB, and the angle E D B, to 
 iind B D. 
 
 In this method, when the angle B D C is less 
 than that of BAC, the point C will be above the 
 }3oint E; but the calculation is so similar to the 
 foregoing, as to require no particular explanation. 
 
 When the points E and C fall too nearly toge- 
 ther, to produce E C towards D with certainty, 
 the first method of construction is the most ac- 
 curate. 
 
 Case 2. When the gwen place or station D, fig. 
 38, plate Q, is ivithoiit the triangle made by the three 
 given objects ABC, but in a line "with one of the sides 
 produced. 
 
 Measure the angle AD B, then the problem 
 may be easily resolved, either by construction or 
 calculation. 
 
 By construction. Subtract the measured angle 
 ADB from the angle CAB, and you obtain the 
 angle A B D; then at B, on the side BA, draw the 
 angle ABD, and it will meet the produced side 
 CA at D; and DA, DC, DB, will be the re- 
 tyiired distances. 
 
 By calcidation. \\\ the triangle ABD, the angle 
 D is obtained by obsciTation, the ^ BAD is the
 
 GEOMETRICAL PROBLEMS. 181 
 
 supplement of the angle CAB to 180°: two angles 
 of the triangle being thus knovvn^ the third is also 
 known; we have, consequently, in the triangle 
 AB D, three angles and one side given to find the 
 length of the other two sides, which are readily 
 obtained by the preceding canons. 
 
 Case 3. When the station po'mt is in one of the 
 sides of the give?! triangle. Jig. \, flate 13. 
 
 By construction. 1 . Measure the angle B D C. 
 1. Make the angle BAE equal to the observed 
 angle. 3. Draw C D parallel to EA, and D is the 
 station point required. 
 
 By calculation. Find the angle B in the triangle 
 ABC, then the angles B and B D C being known, 
 we obtain D C B ; and, consequently, as sin. angle 
 B D C to B C, so is sin. angle D C B to B D. 
 
 Case 4. JVhen the three given places are in a 
 strait line, jig. 2, flate 13. 
 
 Example. Being at sea, near a strait shore, I ob- 
 served three objects, A, B, C, which were truly 
 laid down on my chart; I wished to lay down the 
 place of a sunken rock D ; for this purpose the 
 angles AD B, B D C, were observed with Hadlcy's 
 quadrant. 
 
 By comtruction. Method 1. On AB, jig. 3, 
 plate 13, describe the segment of a circle, capable 
 of containing the observed angle AD B. On BC 
 describe the segment of a circle, capable of con- 
 taining the angle BDC; the point D will be at 
 the intersection of the arcs, and by joining DA, 
 D B, DC, you obtain the required distances. 
 
 Method 2. Make the angle AC E,7%-. 4, flate 
 13, equal to ADB, and the angle EAC equal to 
 BDC; and from the point of intersection E, 
 through B, draw a line to E D, to intersect the 
 arc ADC; join A, D; and D, C, and DA, DB, 
 D C, are the required distances.
 
 182 CURIOUS AND USEFUL 
 
 By calculation. 1. In the triangle CAE, Jig. 4, 
 plate 13, we have all the angles, and the side AC, 
 to find AE. 2. In the triangle ABE, AB, AE, 
 and the included angle are given, to find the an- 
 gles AE B, ABE. 3. In the triangle B D C, the 
 angle B D C and D C B (=AB E) are given, and 
 consequently the angle D C B, and the side B C ; 
 henee it is easy to obtain D B. 
 
 Case 5. When the station falls within the tri-^ 
 angle, formed hy the three giveti objects, fg. 19, 
 plate 13. 
 
 Let ABC represent three towers, whose dis- 
 tance from each other is known; to find the dis- 
 tance fi-om the tower D, measure the angles ADC, 
 BDC, ADB. 
 
 Construction. On two of the given sides AC, AB, 
 fig. 20 plate \o, describe segments of circles capa- 
 ble of containing the given angles, and the point D 
 of their intersection will be the required place. 
 
 Another method. Or, we may proceed as in some 
 of the foregoing cases; making the angle ABE, 
 fig.21, plate 13, equal to the angle A DE, and 
 BAE equal B D E; and describe a circle through 
 the three points A, B, E, and join E, C, by the line 
 EC; and the point D, where BC intersects the 
 circle E, A, D, B, E, will be the required station. 
 
 Case 6. When the station point I),fg. 7, plate 
 13, falls without the triangle AB C, but the point C 
 falls towards D. 
 
 Thus, let AB C,fig. 7, plate 13, represent three 
 towers, whose respective distances from each other 
 are known ; required their distance from the point 
 D. Measure the angles ADC, BDC, and to prove 
 the truth of the observations, measure also ADB. 
 
 Then, bj construction, method 1. On AC, fig. 
 8, plate 13, describe a circle capable of containing 
 the angle BDC, and on AB, one capable oi conr
 
 GEOMETRICAL PROBLEMS. 183 
 
 taining the angle AD B, and the point of intersec- 
 tion will be the plaee required. 
 
 Or, it may he constructed by method 2, case 1, 
 which gives us the point Yy^Jig. g, plate 13, com- 
 pared with Jig. 3Cj, plate 0. The calculation is 
 upon principles so exactly like those in method 2, 
 case 1, that a further detail would be superfluous. 
 
 The point of station Yi found instrumentally. The 
 point D may be readily laid down on a draught, 
 by drawing on a loose transparent paper indefinite 
 right lines DA, D B, DC, at angles equal to 
 those observed ; which being placed on the draught 
 so as each line may pass over, or coincide with, it*i 
 respective object, the angular point D will then 
 coincide with the place of observation. Or, 
 
 Provide a graduated semicircle of brass, about 
 six inches in diameter, having three radii with 
 chamfered edges, each about 20 inches long, (or 
 as loug as it may be judged the distance of the 
 stations of the three given objects may require) 
 one of which radii to be a continuation of the dia- 
 meter that passes through the beginning of the de- 
 grees of the semicircle, but immoveably fixed to it, 
 the other two moveable round the center, so as to 
 be set and screwed fast to the semicircle at any 
 angle. In the center let there be a small socket, 
 or hole, to admit a pin for marking the central point 
 on the draught. When the sloped edges of the 
 two moveable radii are set and screwed fast to the 
 semicircle, at the respective degrees and minutes 
 of the two observed angles, and the whole instru- 
 ment moved on the draught uptil the edges of the 
 three radii are made to lie along the thrpe stasime- 
 trie points, each touching its resp.ectiye point, the 
 center of the semicircle will then be in the point 
 of station D; which may be marked on the draught, 
 {through the socket, with a pin. Such an instru-
 
 184 CURIOUS AND USEFUL 
 
 ment as this may be called a station-pointer; and 
 would prove convenient for finding the point of 
 station readily and accurately, except when the 
 given objects were near; when the breadth of the 
 arc, and of the radii, and of the brass about the 
 center of the semicircle, might hinder the points 
 from being seen, or the radii so placed as to com- 
 prehend a very small angle between them. 
 
 The three succeeding p'ohl ems may occur at sea, in 
 finding the distances and position of the rocks, sands, 
 &c. from the shore, In many cases of maritime sur- 
 veylng; they are also very serviceable In making a 
 Tiiap of a country, from a series of triangles derived 
 from one or more measured hases. 
 
 Problem 1. Given the distance of two oh'iects 
 KB, fig. 5, plate Q, and the angles ADB, B JD C, 
 BCA, to find the distance of the two stations D, C, 
 from the objects A, B. 
 
 By construction. Assume d c any number at plea- 
 sure, and make the angles b d c, a d c, &c. respec- 
 tively equal to the angles B D C, AD C, &c. and 
 join a b; it is plain that this figure must be similar 
 to that required; therefore draw AB, fig. 4, plate 
 0, equal the given distance, and make AB C equal 
 to a b c, BAG to bac, and so on respectively; 
 join the points, and you have the distances re- 
 quired. 
 
 By calculatlo7i. in the triangle a d c, we have 
 do, a d c, and a c d, to find a d, a c ; in b c d, we 
 have in like malnner the three angles, and d c, to 
 find d b, d c. 
 
 In the triangle a d b, we have ad, b d, and the 
 angle a d b, to find a b. Hence by the nature of 
 similar figures, as ab to AB :: dc : DC :: ad 
 ; AD :: bd : BD ::bc : BC. 
 
 Problem 2. The distances of three objects A, B, 
 C, from each other ^ and the angles ADC, CDE^
 
 GEOMETRICAL PROBL'EMS. 1S5 
 
 C E D, C E B^ fi(. 6, fuify- 9, hem^ g'roen, to find 
 the sides AD, DC, D E, EC, WE B. 
 
 Assume any line d c, at pleasure, make the angle 
 c d e equal .the angle C D E, and angle c e d equal 
 to the angle C E D; also the angle c d a, equal to 
 the angle CD A, and the angle ccb etjual to the 
 angle C E B ; produee ad, be to interseet each 
 other at f, and join c f. 
 
 It is evident that the figures c d {^,Q, D F E, are 
 similar; therefore, on AC, fig. 7, phte 9, describe 
 a segment of a circle, containing an angle AFC 
 equal to afc; and on C B a segment capable of 
 containing an angle CFB, equal the angle cfb; 
 from the point of intersection F draw FA, FB, 
 F C ; . make the angle F C D equal the angle fed, 
 and F C E equal the angle f c e, which completes 
 the construction ; then by assuming d e equal to 
 any number, the rest may be found as before. 
 
 This method fails when AD is parallel to BE, 
 fig. 8, flate 9 ; therefore, having described the seg- 
 ments ADC, B C, draw C F, to cut off a segment 
 equal to the angle CDF, and the right line C G, 
 to cut off a segment equal to the angle C E G ; 
 G F vv'ill be in the right line D E; therefore, join 
 GF, and produce the line each way, till it inter- 
 sects the segments, and the points D, E, will be 
 the stations required. 
 
 Problem 3. Four points B, C, D, F, fig. 9, 
 plate 9, or the four sides of a quadriluteral figure, 
 with its ayigles being given, and the angles BA C, 
 3AE, A ED, DE F, known by observation, to find 
 the station point A E, and, consequently, the length of 
 the lines AB, AC, E D, E F. 
 
 By construction. 1 . On B C describe the seg- 
 ment of a circle, to contain an angle equal to 
 BAC. 2. From C draw the chord CM, so that 
 the angle B C M may be equal to the supplement
 
 186 CURIOUS AND USEFUL 
 
 of the angle BAE. 3. On T)F describe the seg- 
 ment of a circle capable of containing an angle 
 equal to D E F; join M N^ cutting the two circles 
 at A and E, the required points. 
 
 By calculation. In the triangle B C M, the angle 
 B C M, (the supplement of BAE,) and the angle 
 BMC, (equal B AC,) and the side B C are given, 
 whence it is easy to find M C. In the same man- 
 ner, DN in the triangle DNF may be found; 
 but the angle M C D (equal angle B C D, less an- 
 gle BCM) is known with the legs M C, CD; 
 consequently, M D, and the angle M D C, will be 
 readily found. 
 
 The angle MDN (equal angle CDF, less 
 CDM, less FDN,) and MD, DN, are known; 
 whence we find M M, and the angles D M N, 
 DNM. 
 
 The angle CM A, (equal DMC added to 
 DMN,) the angle MAC, (equal MAB added to 
 BAC,) and the side M C are given; therefore, by 
 calculation, MA, and AC will also be known. 
 
 In the triangle E D N, the side D N, and the 
 angles E and N are given ; whence we find E N, 
 ED, and, conscquentlv, AE equal MN, less 
 MA, less EN. 
 
 And in the triangle ABC, the angle A, with 
 its sides B C, AC, are known; hence AB, and 
 angle B CA, are found. 
 
 In the triangle E F D, the angle E, with the 
 sides E D, D F, being known, E F and the angle 
 E D F will be found. 
 
 Lastly, in the triangle ACD, the angle A CD, 
 (equal BCD, less BCA,) and AC, CD, are 
 given ; hence AD is found, as in the same manner 
 E C in the triangle E C D. 
 
 Note. That in this problem, and in problems 1 
 jjnd 2, if the two stations fall in a right line with
 
 GEOMETRICAL PROBLEMS. 187 
 
 either of the given objects, the problem Is inde- 
 terminate. As to the other cases of this problem, 
 they fall in with what has already been said. 
 
 The solution of this problem is general, and 
 may be used for the two preceding ones: for sup- 
 pose C D the same point in the last figure, it gi\''es 
 the solution of problem 2 ; but if B, C, be sup- 
 posed the same points D F, you obtain the solution 
 of problem 1. 
 
 Problem 4. Havhio; iJic distance and mafrjieUc 
 hearings of two 'points A and ^,Jjg. 10, plaie p, pro- 
 tracted at any station S, not very ohliqiie to AB, lo 
 Jindits distance from these points hy the needle. 
 
 At S, with a good magnetic needle, take the 
 bearings of A. and B in degrees, and parts of a de- 
 gree; then from these points, draw out their res- 
 pective bearings in the opposite direction towards 
 S; that is, if A bears exactly north, draw a line 
 from the point A exactly south; if it bears east, 
 10 or 20 degrees southward, draw the line west, 
 10 or 20 degrees northward; and so for any other 
 bearing, draw the opposite bearing of B in the 
 same manner, and S, the intersection of these two 
 points, will be the point of station, and SA, SB, 
 the distances required. 
 
 This is an easy method of finding the distance 
 of any station from two places, whose distances 
 have been accurately determined bcfbrc, and will 
 be found very convenient in the course of a sur- 
 vey, and on many occasions sufficiently exact, 
 provided the places are not too remote from the 
 station, nor the intersection of the bearings too 
 oblique. If the needle be good, a distance of 20 
 miles is not too far, when the angle subtended by 
 the two places, is not less tkxin 50 degrees, or 
 ?nore than 140.
 
 188 CURIOUS AND USEFUL 
 
 Problem 5. To reduce angles to the center of 
 the station. 
 
 In surveys of kingdoms, provinces, counties, 
 &c. where signals, churches, &c. at a distance are 
 used for points of observation, it very often hap- 
 pens that the instrument cannot be placed exactly 
 at the center of the signal or mark of observation; 
 consequently, the angle observed will be either 
 greater, less, or equal to that which would have 
 have been found at the center. This problem 
 shews how to reduce them to the center; the cor- 
 rection seldom amounts to more than a few se- 
 conds, and is, therefore, seldom considered, unless 
 where great accuracy is required. 
 
 The observer may be considered in three dif- 
 ferent positions with respect to the center, and the 
 objects; for he is either in a line with the center, 
 and one of these objects, or in an intermediate one, 
 that is, a line from this center to the observer pro- 
 duced, would pass between the objects; or he is, 
 lastly, in an oblique direction, so that a line from 
 the center to him would pass without the objects. 
 
 In the first position, 7?^. l\, plate 9, where the 
 observer is at 0, between the center and one of the 
 objects, the exteiior angle m o n is greater than the 
 angle men, at the center, by the angle emo; 
 therefore, taking emo from the observed angle, 
 you have that at the center. 
 
 If the observer is at ^^fg- H, plate g, the exte- 
 rior angle m a n is greater than that of m e n at the 
 center, by the value of n; therefore, taking this 
 from the observed angle, you obtain the angle at 
 the center. But if the observer is further from the 
 objects than the center, as at i, the observed angle 
 m i n is less than that at the center m e n, by the 
 pngle m; therefore, by adding m to the observed
 
 GEOMETRICAL PROBLEMS. IS^ 
 
 angle, you obtain the angle m e n at the center. 
 In the same manner, if the observer is at u, we 
 should add the angle n to the observed m u n, in 
 order to have the angle men at the center. 
 
 Case 2. When the observer is at o,Jig. \1, plate 
 9, draw a o, and the exterior angle d exceeds the 
 angle ii at the center by the angle m, and the ex- 
 terior angle c exceeds the angle at the center a, 
 by the angle n ; therefore, m o n exceeds the angle 
 at the center, by the value of the two angles m 
 and n : these, therefore, must be subtracted from 
 it to obtain the central angle. On the contrary, 
 if the observer is at a, the two angles m and n 
 must be added to the observed angle. 
 
 Case 3. Fig. 13, plate g. When the observer is 
 at o, having measured the angles m o n, m o e, the 
 angle i is exterior to the two triangles m o i, n e i; 
 therefore, to render men, equal to m i n, we 
 must add the angle n ; and to render the exterior 
 angle m i n, equal to the observed angle m o n, 
 we must take away the angle m ; therefore, add- 
 ing m to the observed angle, and subtracting w 
 from the total, we obtain the central angle men. 
 
 From what has been said, it is clear, that in the 
 first case, you are to add or subtract from the ob- 
 served angles, that of the angles m or n, which 
 is not in the direction of the observer. 
 
 In the second position, you have either to sub- 
 tract or add the two angles men. 
 
 In the third position, you add to the observed 
 angle, that of the two m or n, which is of the 
 same side with the observer, and subtract the 
 other. 
 
 To know the position of the observer, care must 
 be taken to measure the distance of the instru- 
 strument from the center, and the angle this cen- 
 ter makes with the objects.
 
 IQO CURIOUS AND USEFUL 
 
 An inspection of the figures is sufficient to shew 
 how the value of the angles m n may be obtained. 
 Thus, in the triangle ni o e, we have the angle at 
 o, the distance o e, and the distances c in, o m, 
 (which are considered as equal,) given. 
 
 OP THE llEDUCTIOX OP TRIANGLES, FROM ONE 
 
 PLAIN TO ANOTHER 
 
 * 
 
 After the reduction of the observed angles to 
 the center of each respective station, it is gene- 
 rally necessary to reduce the parts of one, or of 
 several triangles to the same level. 
 
 Case 1. Let us suppose the three points A, P, E, 
 ^g. 15, plate g, to be equally distant from the cen- 
 ter of the earth, and that the point R is higher 
 than these points by the distance or quantity RE; 
 now it is required to reduce the triangle AP R to 
 that APE. 
 
 By the following rule, you may reduce the 
 angles RAP, R PA, which have their summits 
 in the plain of reduction, to the angle EPA, 
 EAP. 
 
 Rule The cosine of the reduced angle is equal 
 to the cosine of the observed angle, divided by 
 the cosine of the angle of elevation. 
 
 These two angles being known, the third E is 
 consequently known; we shall, however, give a 
 rule for finding AEP, independent of the other 
 two. 
 
 Rule. The cosine of the reduced angle is equal 
 to the cosine of the observed angle, lessened by 
 the rectangle of the sines of the angles of eleva- 
 tion, divided by the rectangle of the cosine of the 
 same ana:les. 
 
 ^b' 
 
 * Traite de Trigonometrie, par Cagnoli.
 
 GEOMETRICAL PROBLEMS. IQl 
 
 The reduction of the sides can be no difEculty. 
 
 Case 1, Fig. 17, plate 9. Let ARr be the tri- 
 angle to be reduced to the plain AE e, the points 
 E, e, of the vertical lines R E, r e, being supposed 
 equally distant from the center of the earth. 
 
 Prolong the plain AE e to P, that is, till it meets 
 the line Rr produced to P; and the value of 
 Ex\e will be found by the following formulae. 
 
 1. Tangent \ (PAR+PAr)=tangenti Rx\rX 
 
 tan£jent i (RAE + r Ae) x^ • .1 i ir _ 
 
 — £ :— ^ ! L. Knowms: the hair sum 
 
 tangent ^ ( R A E — r A c) ^ 
 
 and half difterence of PAR and PAr, we obtain 
 the value of each of the angles; the value of 
 P AE and PAe, may be then obtained by the first 
 of the two preceding rules, and the difference be- 
 tween them is the angle sought. 
 
 Let C, fg. 16, 'plate 9, be the center of the 
 earth, let AB be the side of a triangle reduced to 
 a common horizon by the preceding methods ; if 
 it be required to reduce this to the plain DE, as 
 these planes are parallel, the angles will remain the 
 same; therefore, the sides only are to be reduced, 
 the mode of performing which is evident from the 
 
 figure. 
 
 ... 7 
 Method of referring a series of triangles to a me- 
 
 riclian line, and another line perpendicular to it. 
 
 This method will be found somewhat sim lar to 
 one used l)y Mr. Gale, and described at length in 
 the article of surveying; it is a mode that should 
 be adopted wherever extreme accuracy is re- 
 quired, for whatever care is taken to protract a 
 series of triangles, the protractor, the pouits of the 
 compasses, the thickness of the line, the inequality 
 of the paper, &c. w^ill produce in the fixing of the 
 points of a triangle an error, which, though small 
 ^ first, will have its influence on those thai sue-
 
 ig2 CURlOirs AND USEFUL 
 
 ceed, and become very sensible, in proportion as 
 the number of triangles is augmented. This nuil- 
 tiplieation of errors is avoided by tlie following 
 problem. 
 
 Let AB,77^. 14, plate Qy be the meridian, CD 
 the perpendicular, and the triangles oad, d a e, 
 d e g, e g i, g i 1, those that have been observed ; 
 from the point o, (which is always supposed to be 
 on a meridian, or whose relation to a meridian is 
 known) observe the angle Boa, to know how 
 much the point a declines from the meridian. 
 
 In the right-angled triangle o B a, we have the 
 angle Boa, and the right angle, and consequent- 
 ly, the angle o a B, together with the side o a, to 
 iind O B, and B a. 
 
 For the point d, add the angle B o a to the ob- 
 served angle a o d, for the ^ d o b, or its equal 
 odm, and the complement is the angle mod^ 
 whence as before, to fmd om and m d. 
 
 For the point G, add the angles m d o, o d a, 
 a d e, and e d g, which subtract from sOo, to ob- 
 tain the angle g d r, of the right-angled triangle 
 gd r; hence we also readily, as in the preceding 
 triangles, obtain rg=m t, which added to mo, 
 gives to the distance from the meridian. Then 
 we obtain r d, from which taking d m, you obtain 
 rm, equal gt, the distance from the perpen- 
 dicular. 
 
 For the point e, take the right angle r d f, from 
 the two angles rdg, gde, and the remainder is 
 the angle fd e of the right-angled triangle dfe; 
 hence we obtain f"e d and d f, which added to d b, 
 gives b f, equal to x e, the distance from the me- 
 ridian.; from the same right-angled triangle we 
 obtain f e, which added to f n, equal to d m, gives 
 c a, the distance from the perpendicular.
 
 GEOMETRICAL PROBLEMS. lg3 
 
 For the point i, add together the angles rgd^ 
 d g e, e g i, and from the sum subtract the right 
 angle rg h, and you obtain the angle g h i of the 
 right-angled triangle h g i_, and consequently the 
 angle i; hence also we get hi, equal tp, which 
 added to t o^ gives o p distance from the meridian^ 
 and g h^ from which subtracting gt, we obtain t h, 
 equal p i distance from the perpendicular. 
 
 For the point 1, the angle g h i, added to the 
 angle 1 g i, gives the angle 1 g k of the right-angled 
 triangle g k 1, and of course the angle g 1 k, whence 
 we obtain k 1 or t y, which added to t o, gives o y 
 distance from the meridian ; hence we also obtain 
 gk, which taken from gt, gives kt, equal ly dis- 
 tance from the perpendicular. 
 
 If, before the operation, no fixed meridian was 
 given, one may be assumed as near as possible 
 from the point o; for the error in its position will 
 not at all influence the respective position of the 
 triangles. 
 
 To the mathematical student, who may have a desire to proceed on a com- 
 plete course of the mathematical sciences, the following eminent authors are 
 recommended. 
 
 Algebra, by Simpson, 8vo. 1755; Maclaurin, 8vo. 1771; Bonnycastle's 
 Introduction, lamo. 1796.- 
 
 (Jtometiy. Euclid's Elements, by Dr. Simson, 8vo. 1791 ; Elements, by 
 ■Simpson, 8vo. 1768, and Emerson, 8vo. 1763. 
 
 Trigon-jmetry, by Simpson, 8vo. 1 765; Emerson, 8vc), 1788; Traite, by 
 Cngnoli, 410. Paris, 1786; and a Treatise now in the press, 8vo. bj' T. Keith. 
 
 Conic Sections, by Hamilton, 4to. 1758; Hutton, 8vo, 1787; Elements, by 
 Vince, 8vo. 1781, and Newton, 8vo. 1794. 
 
 Fluxions, by Simpson, i\o. I'j'jd, and Maclaurin, 4to. 2 vols. 1742; Intro- 
 iluction, by Rowe, ^wo. 1767, and Emerson, 8vo, 1768. 
 
 Logarithmic Tables, by Taylor, to every second of the quadrant, 4to, 1792; 
 and Hutton, 8vo. 1794.^ 
 
 Men Mr at ion, by Hutton, 8vo, I788. 
 
 Mathematical Dictionary, by Hutton, in i voU. 4to. 1 796. 
 
 To these may be .added, Emerson's Mathematical Works, in lO vols. i\9, 
 HQd a aunt now piwiuhinji at C^mbxid^e by Vi.nv;« and Wood, in — vol*. 8vo.
 
 i m ] 
 
 OF 
 
 SURVEYING. 
 
 The practice of surveying may be considered 
 as consisting of four parts. 1. Measuring strait 
 lines. 2. Finding the position of strait lines with 
 respect to each other. 3. Laying down, or plan- 
 ning upon paper these positions and measures. 
 4. Obtaining the superficial measure of the land 
 to be surveyed. 
 
 We may, therefore, define land surveying to be 
 the art which teaches us to find how many times 
 any customary measure is contained in a given 
 piece of ground, and to exhibit the tiiie boun- 
 daries thereof in a plan or map. 
 
 A station line is a strait line, whose length is ac- 
 curately ascertained by a chain, and the bearing- 
 determined by some graduated instrument. 
 
 An offset is the distance of any angular point in 
 the boundary from the station line, measured by 
 a line perpendicular thereto. 
 
 The curvature of the earth within the limits of 
 an ordinary survey, is so inconsiderable, that its 
 surface may be safely considered by the land sur- 
 veyor as a plain. In a large extent, as a province, 
 or a kingdom, the curvature of the earth's surface 
 becomes very considerable, and due allowance 
 must be made for it. 
 
 All plains, how many sides soever they consist 
 of, may be reduced into triangles, and may there- 
 fore be considered as composed thereof; and, con- 
 sequently, what is required in surveying, arc such 
 instruments as will measure the length of a side, 
 and the quantity of an angle of a triangle.
 
 E 195] 
 
 GENERAL RULES. 
 
 A few general observations, or hints, can only 
 be expected in this place; for, after all that we can 
 s'ay, the surveyor must depend on his own judg- 
 ment for contriving his work, and his own skill in 
 discriminating, among various methods^ that which 
 is best. 
 
 The first business of the surveyor is to take 
 $uch a general view of the ground to be surveyed, 
 as will fix a map thereof in his mind, and thence 
 determine the situations for his station lines, and 
 the places where his instruments may be used to 
 the greatest advantage. 
 
 Having settled the plan of operations, his next 
 business is to examine his instruments, and see 
 that they are all in proper order, and accurately 
 adjusted. He should measure carefully his chain, 
 -and if there be any errors therein, correct them ; 
 prepare staves, marks, &c. for distinguishing the 
 several stations. 
 
 The fewer stations that can be made use of, the 
 less will be the labour of the survey; it will also be 
 more accurate, less liable to mistakes while in the 
 jield, or errors when plotting the work at home. 
 
 The station lines should always be as long as 
 possible, where it can be done without rendering 
 the offsets too large; where great accuracy is re- 
 quired, these lines should be repeatedly measured, 
 the first great point being the careful mensuration 
 of your station lines; the second, to determine the 
 situation of places adjoining to them. For every 
 station line is the basis of the succeeding opera- 
 tions, and fixes the situation of the different parts. 
 
 The surveyor should so contrive his plan, as to 
 avoid the multiplication of small errors, and par* 
 
 o %
 
 igQ GENERAL RULE^ 
 
 ticularly those that by communication will extend 
 themselves through th^ whole operation. If the 
 e'statc be large, or if it be subdivided into a great 
 number of fields, it would be improper to survey 
 the fields singly, and then put them together, nor 
 could a survey be accurately made by taking all 
 the angles and boundaries that inclose it. If pos- 
 sible, fix upon, for station points, two or more 
 eminent situations in the estate, from whence the 
 principal parts maybe seen; let thefe be as far 
 distant from each other as possible; the work will 
 be more accurate when but few of these stations 
 are made use of, and the station lines will be more 
 convenient, if they arc situated near the bounda- 
 ries of the estate. 
 
 Marks should be erected at the intersection of 
 all hedges with the station line, in order to know 
 where to measure from, when the fields are sur- 
 veyed. All necessary angles between the main 
 stations should be taken, carefully measuring in a 
 right line the distance from each station, noting 
 down, while measuring, those distances where the 
 lines meet a hedge, a ditch, &c If any remark- 
 able object be situated near the station line, its 
 perpendicular distance therefrom should be ascer- 
 tained; in the same manner, all offsets from the 
 ends of hedges, ponds, houses, &c. from the main 
 station line should be obtained, care being taken 
 tliat all observations from the station line, as the 
 measure of angles, &c. be always made from 
 points in the station line. 
 
 When the main stations, and every thing ad- 
 joining to them, have been found, then the estate 
 may be subdivided into two or three parts by new 
 station lines, fixing the stations where the best 
 views can be obtained ; these station lines must be 
 accurately measured, and the places where they
 
 Ton SURVEYING. ip7 
 
 intersect hedges be exactly ascertained, and all the 
 necessary offsets determined. 
 
 This effected, proceed to snrvey the adjoining 
 fields, by observing the angles that the sides make 
 vvich the station lines at their intersections there- 
 with ; the distances of each corner of the held 
 from these intersections, and that of all necessary 
 offsets. 
 
 Every thing that could be determined from 
 these stations being found, assume more internal 
 stations, and thus continue to divide and sub- 
 divide, till at last you obtain single fields, repeat- 
 ing the same operations, as well for the inner as 
 for the exterior work, till all be finished. 
 
 Every operation performed, and every observa- 
 tion made, is to be carefully noted down, as the 
 data for fixing the situations upon the plan. The 
 work should be closed as often as convenient, and 
 in as few lines as possible; what is performed in 
 one day shoidd be carefully laid down every 
 night, in order not only to discover the regular 
 process of the work, but to iind whether any cir- 
 cumstance has been neglected, or any error com- 
 rnitted, noticing in the field-book, how one field 
 lies by another, that they may not be displaced in 
 the draft. 
 
 If an estate be so situated, that the whole can- 
 not be surveyed together, because one part can- 
 not be seen from the other, divide it into three or 
 four parts, and survey them separately, as if they 
 were lands belonging to different persons, and at 
 last join them together. 
 
 As it is thus necessary to lay down the work as 
 you proceed, it will be proper to find the most 
 convenient scale for this purpose: to obtain this, 
 measure the whole length of the estate in chains, 
 th.en consider how many inches long your plao
 
 "igS ADVANTAGEOUS CIRCUMSTANCES 
 
 is to be, and from these conditions, you will as- 
 certain how many chains you have in an inch, and 
 thence choose your scale. 
 
 In order that the surveyor may prove his work, 
 and see daily that it goes on right, let him choose 
 some conspicuous object that may be seen from 
 all, or 2;reater part of the estate he is surveying, 
 and then measure with accuracy the angle this 
 object makes, with two of the most convenient 
 stations in the first round, entering them in the 
 iield-book or sketch where they were taken ; when 
 you plot your first round, you will find the true 
 situation of this object by the intersection of the 
 angles. .Measure the angle this object makes, 
 with one of your station lines in the second, third, 
 &c. rounds: these angles, when plotted one day af- 
 ter anotlier, will intersect each other in the place of 
 the object, if the work be right; otherwise some 
 mistake has been committed, which must be cor- 
 rected before the work is carried any further. 
 
 Fields plotted from measured lines only, are aU 
 ways plotted nearest the truth, when those lines 
 form at their meeting, angles nearly approaching 
 to right angles. 
 
 OF THE MOST ADVANTAGEOUS CIRCUMSTANCES 
 
 FOR A SERIES OF TRIANGLES. 
 
 > 
 
 , The three angles of every triangle should al- 
 ways, if possible, be measured. 
 • As it isimpossibleto avoid some degree of error 
 in taking of angles, we should be careful so to or- 
 iler Qur operations, 'that this error may have the 
 least possible infiuence on the sides, the exact 
 measure of which is the end of the operations. 
 
 Now, in a right-lined triangle, it is necessary to 
 h^ve at least one side measured mediately or ini"
 
 FOR A SERIES OF TRIANGLES. IQQ 
 
 TTicdiately; the choice of the base is therefore the 
 fiindainental operation; the detenninations will 
 be most accurate to find one side, when the base 
 is equal to the side required; to find two sides, an 
 equikiteral triangle is most advantageous. 
 
 In general, when the base cannot be equal to 
 the side or sides sought, it sliould be as long as 
 possible, and the angles at the base should be 
 equal. 
 
 In any particular case, where only tAvo angles 
 of a triangle can be actually observed, they should 
 be each of them as near as possible to 45°; at any 
 rate their sum should not differ much from 90°, for 
 the less the computed angle differs from 90°; the 
 less chance there will be of any considerable error 
 in the intersection. 
 
 .DESCJRIPTION OF THE VARIOUS INSTRUMENTS 
 USED IN SURVEYING, AND THE METHOD OP 
 APPLYING THEM TO PRACTICE, AND EX- 
 AMINING THEIR ADJUSTMENTS- 
 
 The variety of instruments that are now made 
 use of in surveying is so great, and the improve- 
 ments that have been made within these few years 
 are so numerous, that a particular description of 
 each is become necessary, that by seeing their re- 
 spective merits or defects, the purchaser may be 
 enabled to avail himself of the one, and avoid the 
 other, and be also enabled to select those that arc 
 best adapted to his purposes. 
 
 The accuracy of geometrical and trigonometri- 
 cal mensuration, depends in a great degree on the 
 exactness and perfection of the instruments made 
 use of; if these are defective in construction, or 
 difficult in use, the surveyor will either be subject 
 to error, or embarrassed with continual obstacles. 
 If the adjustments, by which they are to be ren-
 
 ^00 LIST OF INSTRUMENTS 
 
 dered fit for observation, be troublesome and in- 
 convenient, they will be taken upon trust, and the 
 instrument will be used without examination, and 
 thus subject the surveyor to errors, that he can 
 neither account for, nor correct. 
 
 In the present state of science, it may be laid 
 down as a maxim, that every instrument should 
 be so contrived, that the observer may easily ex- 
 amine and rectify the principal parts; for how^ever 
 careful the instrument-maker may be, however 
 perfect the execution thereof, it is not possible 
 that any instrument should long remain accurate- 
 ly fixed in the position in which it came out of the 
 maker's hands, and therefore the principal parts 
 should be moveable, to be rectified occasionally 
 by the observer. 
 
 AN ENUMERATION OP INSTRUMENTS NECESr 
 SARY FOR A SURVEYOR; 
 
 Fewer or more of which will be wanted, ac- 
 cording to the extent of his work, and the accu- 
 racy required. 
 
 A case of good pocket Instruments. 
 
 A pair of beam compasses. 
 
 A set of feather-edged plotting scales. 
 
 Three or four parallel rules, either those o(Jjg. 
 
 A, B and C, plate 2, or fg. F G H, plate 2, 
 A pair of proportionable compasses. 
 A pair of triangular ditto. 
 A pantagraph. 
 A cross staff. 
 A circumfcrcntor. 
 An Hadlcy's sextant, 
 An artificial horizon. 
 A theodolite. 
 A surveying compass,
 
 FOR A SURYEYOn. 201 
 
 Measuring chains, and measuring tapes. 
 
 King's surveying quadrant. 
 
 A perambulator, or measuring wheel. 
 
 A spirit level with telescope. 
 
 Station staves, used with the level. 
 
 A protractor, with or without a nonius. • 
 
 7(9 he added for county and marine surveying: 
 
 An astronomical quadrant, or circular instru- 
 ment. 
 
 A good refracting and reflecting telescope. 
 A copying glass. 
 
 For marine surveying. 
 
 A station pointer. 
 
 An azimuth compass. 
 
 One or two boat compasses. 
 
 Besides these, a number of measuring rods, iron 
 pins, or arrows, &c. will be found very convenient, 
 and two or three offset staves, which are strait 
 pieces of wood, six feet seven inches long, and 
 about an inch and a quarter square; they should 
 be accurately divided into ten equal parts, each of 
 which will be equal to one link. These are used 
 for measuring offsets, and to examine and adjust 
 the chain. 
 
 Five or six staves of about five feet in length, 
 and one inch and an half in diameter, the upper 
 part painted white, the lower end shod with iron, 
 to be struck into the ground as marks. 
 
 Twenty or more iron arrow^s, ten of which are 
 always wanted to use with the chain, to count the 
 number of links, and preserve the direction of the 
 chain, so that the distance measured may be really 
 m a strait line. 
 
 The pocket measuring tapes, in leather boxes, 
 jire often very convenient and useful. They arc
 
 202 XNSTKUMENTS FOR 
 
 made to the different lengths of one, two, three, 
 four poles, or sixty-six feet and 100 feet; di- 
 vided, on one side into feet and inches, and on the 
 other into links of the chain. Instead of the lat- 
 ter, are sometimes placed the centesimals of a 
 yard, or three feet into 100 equal parts. 
 
 OF THE INSTRUMENTS USED IN MEASURING 
 STRAIT LINES. 
 
 OF THE CHAIN. 
 
 The length of a strait line must be found me- 
 chanically by the chain, previous to ascertaining 
 any distance by tri2;onomctry: on the exactness 
 of this mensuration the truth of the operations 
 will depend. The surveyor, therefore, cannot be 
 too careful in guarding against, rectifying, or 
 making allowances for every possible error; and 
 the chain should be examined previous and subse- 
 quent to evei-y operation. 
 
 For the chain, however useful and necessary, is 
 not infallible, it is liable to many errors. 1. In 
 itself. 2. In the method of using it. 3. In the 
 uncertainty of pitching the arrows; so that the 
 surveyor, who wishes to obtain an accurate survey, 
 will depend as little as possible upon it, using it 
 only where absolutely necessary as a basis, and 
 then with every possible precaution. 
 
 If the chain be stretched too tight, the rings 
 will give, the arrow incline, and the measured base 
 will appear shorter than it really is; on the other 
 hand, if it be not drawn sufficiently tight, the 
 measure obtained will be too long. I have been 
 informed by an accurate ajid very intelligent sur- 
 vcyor, that when the chain has been much used,
 
 MEASCEIXG &TRAIT LINES. *203 
 
 lie lias gcnc]-ally tbund it necessary to shorten it 
 every second or third day. Chains made of 
 strong wire are preferred. 
 
 Gmiters chain is the measure universally adopted 
 in this kingdom for the purpose of land surveying,^ 
 being exceedingly well adapted for the mensura- 
 tion of land, and affording very expeditious me- 
 thods of casting up what is measured. It is sixty- 
 six feet, or four poles in length, and is divided into 
 100 links, each link with the rings between them 
 is 7.92 inches long, every tenth link is pointed out 
 by pieces of brass of different shapes, for the more 
 readily counting of the odd links. 
 
 The English acre is 4840 square yards, and 
 Guntej*'s chain is 22 yards in length, and divided 
 into 100 links; and the square chain, or 22 mul- 
 tiplied by 22, gives 484, exactly the tenth part of 
 an acre; and ten chains squared are equal to one 
 acre; consequently, as the chain is divided into 
 100 links, every superficial chain contains 100 
 multiplied by 100, that is 10.000 square links; 
 and 10 superficial chains, or one acre, contains 
 100.000 square links. 
 
 If, therefore, the content of a field, cast up in 
 square links, be divided by 100.000, or (which is 
 the same thing) if from the content we cutoff the 
 five last figures, the remaining figure towards the 
 left hand gives the content in acres, and conse- 
 quently the number of acres at first sight; the 
 remaining decimal fraction, multiplied by 4, gives 
 the roods, and the decimal part of this last pror- 
 duct multiplied by 40, gives the poles or perches. 
 
 Thus, if a field contains l6,54321 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 
 lt5,013{)O^ or 6 poles, 91 3(io parts.
 
 204 INSTRUMENTS FOR STRAIT LINKS* 
 
 Direc firms for using the chain. Marks are first t{> 
 be set up at the places whose distances are to be 
 obtained ; the place where you begin may be called 
 your first station; and the station to which you 
 measure, the second station. Two persons are 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 been all put down, the 
 leader must wait till the follower brings him the 
 arrows, then proceeding onwards as before, but 
 without leaving an arrow at the tenth extention of 
 the chain. In order to keep an account of the 
 number of times which the arrows are thus ex- 
 changed, they should each tie a knot on a string, 
 carried for that purpose, and which may be fas- 
 tened to the button, or button-hole of the coat; 
 they should also call out the number of those ex- 
 changes, that the surveyor may have a check on 
 them. 
 
 It is very necessary that the chain bearers should 
 proceed in a strait line; to this end, the second, 
 and all the succeeding arrows, should always be 
 ?o placed, that the next foregoing one may be in 
 a line with 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 rectili- 
 near direction, as to prevent mistakes. 
 
 All distances of offsets from the chain line to 
 any boundary which are less than a chain, are 
 most conveniently measured by the offset staff; 
 the measure must always be obtained in a direct, 
 lion perpendicular to the chain.
 
 king's surveying quadrant. 205 
 
 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.* 
 
 BESCRIPTION AND USE OP THE SURVEYING 
 • aUADRANT, FOR ADJUSTING AND REGULAT- 
 ING THE MEASURES OBTAINED BY THS 
 CHAIN WHEN USED ON HILLY GROUND, IN- 
 VENTED BY R. King, surveyor. 
 
 There are two circumstances to be considered 
 in the measuring of lines in an inclined situation : 
 ihcjirst regards the plotting, or laying down the 
 measures on paper; tlie second, the area, or super- 
 ficial 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, there- 
 fore, 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 necessary to reduce the hypothenusal 
 
 * The best method of surveying by the chain, and now gene- 
 rally 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 eveiy fence and line to be laid down, with 
 offsets from these lines when necessary. The peculiar advan- 
 tage of this method is, that, after three lines are measured and 
 laid down, every other line proves itself a 
 
 upon application. Thus^ if the trian- 
 gle ah c he. laid down, and the points 
 d and c given in the sides, when the 
 line lie has been measured for the pur- 
 pose of taking a fence contiguous 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 \vi 
 surveying roads or very detached parts of estates ; in such cases 
 recourse must be had to the theodolite, or other angular instru,' 
 ment. Edit,
 
 20a KING S SURVEYING QUADRANT. 
 
 lines to liorizontal, which is easily effected by 
 Mr. Kmg'a quadrant. 
 
 With respect to the area, tlicre is a difference 
 aiTionx surveyors; sonic contending that it should 
 be made according to the hypothenusal ; others, 
 accordino- to the horizontal lines: but, as i\l\ have 
 agreed to the necessity of the reduction for the lirst 
 purpose, we need not enter minutely into their 
 reasons liere; for, .even if we admit that in some 
 cases more may be grown on the hypothenusal 
 plain than the horizontal, even then the area 
 should be given according to both suppositions, as- 
 the hill}' and uneven ground requires more labour 
 in the working. 
 
 The quadrant AB, Jig. \, plate 14, is fitted to a 
 wooden square, which slides upon an offset staff, 
 and may be fixed at any height by means of the 
 §crcw 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 square answers the purpose of a cross 
 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 
 liinb D E of the square, by the screw G. 
 
 When the several lines on the limb of the quad- 
 rant have their first division coincident with their 
 respective index divisions, the axis of the level is 
 parallel to the staff. 
 
 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.
 
 king's SUVEYINQ aUADRANT. 207 
 
 The second, or middlemost line, shews the 
 number of links the chain is to be drawn forward, 
 to render the hypothenusal measure the same a^ 
 the horizontal. 
 
 The third, or uppermost line, gives the perpen- 
 dicular 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 
 (juadrant may be upright, then move the quad^ 
 rant till the bubble stands in the middle, and ou 
 the several lines you will have, 1. The horizontal 
 length gone forward in that chain. 2. The linka 
 to be drawn forward to complete the horizontal 
 chain. 3. The perpendicular height or descent 
 made in going forward one horizontal chain. 
 
 The two first lines are of the utmost impoitance 
 in surveying land, which cannot possibly be plan- 
 ned with any degree of accuracy without having 
 the horizontal 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 performed, and he saves by it more 
 time in plotting his survey, than he can lose in the 
 iicld ; for as he completes the horizontal chain as 
 he goes forwards, ihe offsets are aliz^ays m their right 
 places, and the field-book being kept by horizon- 
 tal measure, his lines are always sure to close. 
 
 If the superficial content by the hypothenusal 
 measure be required for any particular purpose, he 
 has that likewise by entering in the margin of his 
 iield-book the links drawn forward in each chain, 
 having thus the hypothenusal and hori;2;oataL 
 length of every line.
 
 208 OF THE PERAMEULATOR, OR 
 
 The third lino, which is the perpencHcuIar 
 height, may 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 fine, how much 
 the tape is to be drawn forward to complete the 
 distance of 100 horizontal feet; and the line of 
 perpendiculars 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 stafi"', place the staff in such a posi- 
 tion, 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 J to which add or subtract the perpen- 
 dicular height of the place from the foot of the 
 tree, and ) ou obtain the height required. 
 
 OP THE TERAMBULATOR, OR IMPROVED MEA- 
 SURING WHEEL; THE WAY-WISER, AND THE 
 PEDOMETER. 
 
 Fijr, (), pJaie 1 7 , 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 pi- 
 nion rod 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- 
 tlire;}dccl worm once round; the worm takes into
 
 IMPROVED MEASURING WHEEL. 200 
 
 a wliccl of 80 teeth, and turns it once round in 80 
 tevolutions ; 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 
 another 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 l6o teeth, whose socket carries an in- 
 dex that makes one revolution in 80 furlongs, or 
 10 miles. On the dial plate, see. Jig. 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 with- 
 out shake, and, as they have only very light indices 
 to carry, move with little or no friction, and arc, 
 therefore, not liable to wear or be soon out of or - 
 dfer; which is not the case with the general num- 
 ber of those that ace 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 arc 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. 
 The instrument is sometimes made with a double 
 wheel for steadiness when using, and also with a 
 bell connected to the wheel-work, to strike the 
 number of miles gone over. 
 
 This instrument is v<iry useful for measuring 
 roads, commons, and every thing where expedi- 
 tion is required; one objection is however made 
 
 p
 
 210 THE WAY-WISER AND PEDOMETER. 
 
 to it, namely, that it gives a measure somewhat: 
 too long by entering into hollows, and going over 
 small hills. This is certainly the ease; the mea'- 
 siiring wheel is not an infallible mode of ascer- 
 taining the horizontal distance between any two 
 places: but then it may with propriety 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 as liable to error, as that from the 
 wheel. 
 
 The ivay-wiser 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 in jf^. 8, 
 plate 17. A piece of plate iron A is screwed to 
 the inside nave of the wheel; this being of a cur- 
 vilineal 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 
 communicates 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 im- 
 pulse -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 accurately measured, and that multi- 
 plied by the number shewn on the dial of the 
 way-wiser gives the distance run. 
 
 By means of rods, universal joint, &c. it is often 
 made to act within the carriage, so that the person
 
 THE SURVEYING CROSS. 21 1 
 
 may at any moment, without the trouble of getting 
 out, see the number of the revolutions of the 
 wkeel. If the instrument is to be always applied 
 to one wheel, a table may easily be constructed to 
 shew the distance in miles and its parts by inspec- 
 tion only. 
 
 The pedometer is exactly the same kind of in- 
 strument 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 waist- 
 band or at the knee, the number of steps a man 
 takes in his regular paces are registered, from the 
 action of the string upon the internal wheel -work, 
 at every step, to the amount of 30,000. It is ne- 
 cessary to ascertain the distance walked, that the 
 average length of one pace be previously known, 
 and that multiplied by the number of steps regis- 
 tered on the dial plate. 
 
 OP THE SURVEYING CROSS, ^g. 2, p/ate 14. 
 
 TJ>e cross consists. of two pair of sights, placed 
 at right angles to each other: these sights are 
 sometimes pierced out in the circumference of a 
 thick tube of brass about 22 inches diameter, see 
 Jig. 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 
 jpoint at the bottom to stick in the ground; one of 
 this kind is represented at ^g. 2, plate 14. The 
 four sights screw off to make the instrument con- 
 venient for the pocket, and the staff which is about 
 Ah or five feet in length (for both the crosses) un- 
 ■screwsinto three parts to go into a portmantcau,&c. 
 
 The surveying cross is a very useful instrument 
 for placing of offsets, or even for measuring §mall
 
 21'2 THE SURVEYING CROSS, 
 
 pieces of ground; its accuracy depends on the 
 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, iirst to measure abase or station line in 
 the longest direction of the piece of ground, and 
 while measuring, to find by the cross the places 
 where perpendiculars 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 cor- 
 ner; then measuring the length of the said perpen- 
 dicular. To be more particular, let A, h, i, k, 1, m, 
 
 jig. 35, plate g, be a crooked hedge or river; mea- 
 sure a strait line, as AB, along the side of the 
 foregoing line, and w hile measuring, observe when 
 you are opposite to any bend or corner of the 
 hedge, as at c, d, e; from thence measure the per- 
 pendicular offsets, as at c h, d i, &c. with the 
 offset staff", if they arc not too long; if so, with the 
 chain. The situation of the offsets are readily 
 
 -found, as above directed, by the cross, or King's 
 
 * I have mack some additions to the box cross staft", which 
 have been found useful and convenient for the pocket, where 
 great accuracy is not, required. Seejf^. 6. A compass and nee- 
 dle at the top A to give the bearings, aud a moveable graduated 
 base at B, by rack -work and pinion C, to give aii angle to 5' of 
 ;<i degree by the nonius divided on the box above. Thus the sur- 
 veyor may have a small theodolite, circumferentor, and cross 
 staff all in one insti'ument. Edit.
 
 IMPROVED BY JOXES. 213 
 
 surveying quadrant; they arc to be registered in 
 the field-book. 
 
 Of surveying with the chain and cross. What 
 has been denominated by many writers, surveying 
 by the chaiyi oily, is in fact surveying by the cross 
 and chain; for it is necessary to use the cross, or 
 optical square", for determining their perpendicu- 
 lar lines, so that all that has been said, even by 
 these men, in favour of the chaitt alone, is founded 
 in fallacy. To survey the triangular field AB C, 
 Jig. 22, plate 9, by the chain and cross : 1 . Set up 
 marks at the corners of the field. 2. Bcirinnincr, 
 suppose at A, measure on in a right line till you 
 are arrived near the point D, where a perpendicu- 
 lar 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 
 \t does not coincide at C with the mark, the cross 
 must be moved backwards or forwards, till by 
 trials one pair of the sights exactly coincide with 
 the mark at C, and the other with A or B. 4. Ob- 
 serve how many chains and links the point D is 
 from A, suppose 3. O. 3. which must be entered in 
 the field-book. 5. Measure the perpendicular D C, 
 643. 7. finish the measure of the base line, and the 
 work is done. This mode is used at present by many 
 surveyors, prohahly 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 AB equal 1 1 . 41. AD equal 
 to 3. 0. 3. on the point D erect the perpendicular 
 PC, and make it equal 6.43. then draw AC, 
 3 C, and the triangle is formed.
 
 [ 214 ] 
 
 OF THE OPTICAL SQUARE, J^^. Ay plate 14. 
 
 This instrument has the two principal glasses of 
 Hadleys quadrant, 7,was contrived by my father; it 
 is in most, if not in all respeofe, superior to the 
 common surv^cying 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 mirrors, so disposed, that an object seen by 
 reflexion from both, will appear to coincide with 
 another object seen by direct vision whenever thd 
 two objects subtend a right angle from the center 
 of the instrument, and serves therefore to raise or 
 let fall perpendiculars on the ground, as a square 
 does on paper, of which we shall give some exam- 
 ples. Its application to the purposes of surveying 
 will be evident from these, and what has been al- 
 ready said concerning the cross. 
 
 Fig. 4, plate 14, is a representation of the instru- 
 ment without 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 directly, always coincides with the object seen 
 by reflexion, when they are at right angles to each 
 other*. 
 
 From a given point in a given line, to raise A 
 Perpendimlar. 1. The observer is to stand with 
 this instrument over the given point, causing a 
 person to stand with a m^rk, or fixing one at some 
 convenient distance on the given line. 2. An as- 
 sistant must be placed at a convenient distance, 
 with a mark somewhere near the line in wbich it 
 is supposed the perpendicular will fall; then if on 
 looking at one of the objects, the other be seen in 
 
 ■* See the description of a considerable improvement upon it, 
 after the description of the Hadley's sextant^ /^/u^t; 19^ edit.
 
 ^HE OPTICAL SQUARE. 215 
 
 a line with it, the place where the mark of the as- 
 sistant is fixed is the required point. 
 
 From a given point over a given line to let fall 
 a perpendicular. Every strait line is limited and 
 determined by two points, through which it is sup- 
 posed to pass; so in a field the line is determined 
 by two fixed objects, as steeples, trees, marks 
 erected for the purpose, &c. For the present ope- 
 ration, the two objects that determine must be on 
 one side the point where the perpcndiculai* falls; 
 or, in other words, the observer must not be be- 
 tween the objects, he must place himself over the 
 Hue, in which he will always be when the two ob- 
 jc:cts coincide; he must move himself backwards 
 on this line, till the mark, from whence the per- 
 pendicular is to be let fall, seen by direct vision, 
 coincides with one of the objects which determine 
 the givcj line seen by reflexion, and the instru- 
 ment will be over the required point. 
 
 To measure huicesslble distances by the optical 
 square. 
 
 Required the distance from the steeple A, fig. 
 20, plate 9, to B. Let the observer stand with his 
 instrument at B, and direct an assistant to move 
 about C with a staff as a mark, until he sees it 
 coincide by direct vision with the object at A ; let 
 Jiim fix the staff there; then let the observer walk 
 along the line until A and B coincide in the instru- 
 ment, and BD will be perpendicular to AC; 
 measure the three lines B C, D C, B D, and then 
 the following proportion will give the required dis- 
 tance, for as D C is to D B, so is B C to AB. 
 
 Second method. 1. Make B C, /^. 21, plate Q, 
 perpendicular to BA. 2. Divide BC into four 
 equal parts. 3. Make C D perpendicular to CB. 
 4. BringFEA into one line, and the distance 
 from C to F will be t of the distance from B to A.
 
 C 216 ] 
 
 OF THE CIRCUMFERENTOR,^^. \, plate 15. 
 
 This instrument consists of a compass box, a 
 magnetic needle, and two plain sights, perpen- 
 dicular to the meridian line in the box, by which 
 the bearings 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 la|-ge 
 tracts of ground, overstocked 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 un- 
 derwood, the circumferentor is chiefly used. 
 
 The circum.ferentor, stt fg. \, plate \5, con- 
 sists of a brass arm, about 14 or 15 inches long, 
 with sights at each end, and in the middle thereof 
 a circular box, with a glass cover, of about 51- 
 inches diameter; within the box is a brass gradu- 
 ated circle, the upper surface divided into 36o 
 degrees, and numbered 10. 20. 30. to 360; every 
 tenth degree is cut down on the inner edge of 
 the circle. The bottom of the box is divided in- 
 to four parts or quadrants, each of which is sub- 
 divided into QO degrees numbered from the me«- 
 ridian, 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 center 
 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 rule, and perpendicu- 
 lar thereto, sights are fixed ; in each sight there is
 
 THE CIP>.CUMFERENTOR. 'Jl/ 
 
 & large and small aperture, or slit, one over the 
 other, these arc alternate; that is, if the aperture 
 be uppermost 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 in- 
 strument ma}' 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 tliis 
 purpose the box, till the ends of the needle am 
 equidistant from the bottom, and traverse or play 
 with freedom. 
 
 Occasional variations 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 more convenient for carriage; sometimes 
 they are made to slide on and off with a dovetail; 
 sometimes 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 cen- 
 ter pin, and press the cap close against the glass, 
 to preserve the point of the center pin from being 
 blunted by the continual friction of the cap of the 
 needle. 3. In the needle itself, which is made of 
 diflcrent forms. 4. A further variation, and for 
 the best, will be noticed under the account of the 
 improved circumferentor. 
 
 The surveying compass represented at J^g. 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 theie is 
 a brass circle divided into 36o degrees; in the 
 center of the box is a pin to carry a magnetic
 
 31S THE CIRCUMFEREI^TOR. 
 
 needle; a telescope is fixed to one side of the 
 box, in such a manner as to be parallel to the 
 north and south line; the telescope has a vertical 
 motion for viewing objects in an. inclined plain ; 
 at the bottom of the box is a socket to receive a 
 stick or staft for supporting the instrument. 
 
 To use the c'lrcumferentor. Let ABC, 7?^'. 19, 
 flate 9, be the angle to be measured. 1 . The in- 
 strument being fixed on the staff, place its center 
 over the point B. 1. 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 compass 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, sup- 
 pose 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 
 1*25. 6. Take the former number of observed de- 
 grees 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 in the second on the other, add what 
 one wants of 36o", to what the other is past 36o°, 
 and the sum is the required angle. 
 
 This general idea of the use of the circumferen- 
 tor, it is presumed, will be suf^cient for the pre- 
 sent; it will be more particularly treated of here- 
 after. 
 
 When in the use of the circumferentor, you 
 l6ok through the upper sights, from the ending of 
 the station to the beginning, it is called a back 
 sights but when you look through the lower slit 
 from the beginning of the station towards the end, 
 it is termed the fore sight. A theodolite, or any 
 instrument which is not set by the needle^ must
 
 THte IMPROVED CIRCUMFERENTOH. 2,19 
 
 be fixed in its place, by taking back and fore sights 
 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, who- 
 soever works by the circumferentor, 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 tho, 'variation of the needle. This variation is 
 different at different places, and is also different at 
 different times; this difference in the variation is 
 called the variation of the variation; but the in- 
 crease and decrease thereof, both with respect to 
 time and place, proceeds by such very small in- 
 crements or decrements, as to be altogether insig- 
 nificant and insensible, within the small limits of 
 an ordinary survey, and the short time required 
 for the performance thereof.* 
 
 OF THE IMPROVED CIRCUMFERENTOR. 
 
 The excellency and defects of the preceding in- 
 strument both originate in the needle; from the 
 regular direction thereof, arise all its advantages; 
 the unsteadiness of the needle, the difficulty of as- 
 certaining with exactness the point at which it set- 
 tles, arc some of its principal defects. In this 
 improved construction these are obviated, as will 
 
 * The present variation, or, more properly, the dccVmaUon of 
 the needle, is near 22" W. of the north at London; or two points 
 in general may be allowed on an instrument to the E. to fix the 
 just meridian. Its hidhiahoii, or dip, was about 72° of the 
 north pole below the horizt)n in the year 17/5. The inclination, 
 as well as the declination of the needle, is found to be continually 
 Varying:; and, from the observations and h^'potheses hitherto 
 made, not to develope any law by which its petition can be dc- 
 tenjiined for any future time, ilpix..
 
 220 THE IMTROVED CIRCUMFERENTOR. 
 
 be evident from the following description. One of 
 these instruments is represented ?it fg.1, plate 15. 
 
 A- pin of about three quarters of an inch diame- 
 ter goes through the middle of thebox^ and forms 
 as it were a vertical axis, on which the instrument 
 may be turned round horizontally ; on this axis 
 an index AB is fastened, moving in the inside of 
 the box, having a nonius on the outer end to cut, 
 and subdivide 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 re- 
 moves the difficulties, which would otherwise arise 
 if the needle should at any time happen to be acted 
 upon, or drawn out of its ordinary position by ex- 
 traneous matter; there is a pin beneath, whereby 
 the index may be fastened temporarily to the bot- 
 tom 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 instrument, and the index, 
 may be either turned round together, or the one 
 turned round, and the other remain fixed, as oc- 
 casion shall require. A further improvement is 
 that of preventing all hori%ontal motion of the ball in 
 the socket; the ball has a motion in the socket every 
 possible way, and every one of these possible mo- 
 tions is necessary, except the horizontal one, which 
 is here totally destroyed, and every other possible 
 motion left perfectly free.* 
 
 * The instrument is made to turn into a vertical position, and 
 by the addition of a spirit level to take altitudes and depression*. 
 The index AB has been found to interfere too much with the 
 free play of the needle. In the year 179^ I contrived an external 
 nonius piece a, jig. 6, to move against and round the graduated 
 circ'e h, either with or without rack -work or pinion. The cir- 
 cle and compass plate are fixed, and the nonius piece and outside 
 rim and sight carried round together when in use. This has 
 been generally approved of. Edit,
 
 l 221 ] 
 
 GENERAL IDEA OF THE USE OP THESE 
 INSTRUMENTS. 
 
 For this purpose, Ictj^^'-. 1, plate 18, represent 
 a lield to be surveyed. 1 . Set up the eircumferen- 
 tor at any corner, as at A, and therewith take the 
 course or bearing, or the angle that such a Hne 
 makes with the magnetic meridian shewn 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 tlie needle settles, it will point 
 out on the graduated limb the course or number 
 of degrees which the line bears from the magnetic 
 meridian. 
 
 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 tix 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 index. This done, fasten the 
 body of the instrument to the under part again, 
 and helving set the instrument up at B, turn the 
 sights in a direct line back from B to A, anU 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
 
 ^2 USES OF THE CiRCUMFEREXTOR, kc, 
 
 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, un- 
 less the needle should happen to be drawn out of 
 its course by extraneous matter; but, in this case, 
 •theindex will not only shew the course or bearing, 
 'but will likewise shew how much the needle is so 
 ilrawn 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, 
 fDE, EF, FG, GA, 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 Icl, us suppose to be as the fol- 
 lowing : 
 
 1. 
 
 AB 
 
 North 
 
 7 
 
 West 
 
 21. 
 
 00. 
 
 2. 
 
 BC 
 
 North 
 
 55 
 
 15 East 
 
 18. 
 
 20. 
 
 3. 
 
 CD 
 
 South 
 
 62 
 
 30 East 
 
 14. 
 
 40. 
 
 4. 
 
 X)E 
 
 South 
 
 40 
 
 West 
 
 11. 
 
 
 5. 
 
 EF 
 
 South 
 
 4 
 
 15 East 
 
 14. 
 
 
 C. 
 
 FG 
 
 North 
 
 73 
 
 45 West 
 
 12. 
 
 40, 
 
 7. 
 
 GA 
 
 South 
 
 52 
 
 West 
 
 9- 
 
 17. 
 
 JV. B. By north 7° west, is meant seven degree* 
 to the westward, or left hand, of the north, as 
 shewn by the needle; by north 55* 15' east, lifty- 
 iive degrees 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 east- 
 ward, or left hand of the south; and by south 
 ♦40° west, forty degrees to the westward, or right 
 Jiand of the south.
 
 THE COMMON THEODOLITE. Q23 
 
 The 21 chains, 18 chains 20 links, &c. are the 
 lengths or distances of the respective sides, as mea- 
 sured by the chain. 
 
 Fig. A, plate 15, represents a small circumferen- 
 tor, or theodolite; it is a kind that was much used 
 by General Roy, for delineating the smaller parts of 
 a survey. The diameter is 4 inches. It is better 
 to have the sight pieces double, as shewn in^^, 2. 
 
 OP THE COMMON THEODOLITE. 
 
 The error to which an instrument is liable^ 
 where the whole dependance is placed on the nee- 
 dle, soon rendered some other hivention necessary 
 to measure angles with accuracy; among these, the 
 common theodolite, with four plain sights, took the 
 lead, being simple in construction, and easy in use. 
 
 The common theodolite is represented Jjg. 5, 
 pl-ate 14; it consists of a brass graduated circle, a 
 moveable index AB; 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 
 plain of the instrument. There arc two more sights 
 £ F, which are fitted to the graduated circle at the 
 points of 360^ and 1 80° ; they all take on and off 
 for the convenicncy of packing. In each sight 
 there is, as in the circumferentor, a large and a 
 small aperture 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 center thereof, is a sprang to fit 
 on the pin of the ball and socket, which fixes on a. 
 three-legged staff. 
 
 The circle is divided into degrees, which are 
 all numbered one way to 36o°, usually from the 
 left to the right, supposing yourself at the center
 
 224 THE COMMON TIIEOD OHLlT]?. 
 
 of the instrument; on the end of the index is ^ 
 nonius division, by which the degrees on the liml> 
 are subdivided to five minutes; the divisions on 
 the ring of the compass box are numbered in a 
 contrary direction to those of tlie Hmb. 
 
 As much of geometrical mensuration depends 
 on the accuracy of the instrument, it behoves every 
 surveyor to examine them carefully ; different me- 
 thods v^'ill 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 
 Lo each other, in order to level the instrument, but 
 it appears, to me much better to depend on the 
 needle : 1 . Because the bubbles, from their size, 
 are seldom accurate. 2. Because the operator 
 cannot readily adjust them, or ascertain when they 
 indicate a true level. 
 
 To examine the instrument; oi>,an extensive 
 plain set three marks to form a triangle; with, your 
 theodolite 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 
 wPiere the needle settles, and then remove it from 
 that situation, by placing a pieee of steel or mag- 
 net 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 in any situation of the box a deviatioit
 
 THE COMMON THEODOLITE, 225 
 
 is observed, the error is most probably occasioned 
 by some particles 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 de- 
 gree of the theodolite, and if the last division of the 
 nonius always terminates precisely, at each appli- 
 cation, with its respective degree, then the divi- 
 sions are accurate. 
 
 Cautions in the use of the Instrument . 1 . Spread 
 the legs that support the theodolite rather wide, 
 and thrust them firmly into the ground^ that they 
 may neither yield, nor give unequally during the 
 observation. 2. Set the instrument horizontal. 
 3. Screw the ball firmly in its socket, that, in turn- 
 ing the index, the theodolite may not vary from 
 the objects to which it is directed. 4. Where 
 accuracy is required, the angles should always be 
 taken twice over, ot'tner where great accuracy is 
 material, and the mean of the observation must 
 be taken for the true angle. 
 
 To measure an angle with the theodolite. Let 
 AB, B C, Jig. IQ, plate g, represent two station 
 lines; place the theodolite over the angular point, 
 and direct the fixed sights along one of the lines, 
 till you see tl>rough the sights the mark A ; at this 
 screw the instrument fast; then turn the moveable 
 index, till through its sights you see the other 
 mark C; then the degrees cut by the index upon 
 the graduated limb, or ring of the instrument, 
 shew the quantity of the angle. 
 
 The fixed sights are always to be directed to the 
 last station, and those on the index to the next. 
 If the beginning of the degrees are towards the 
 surveyor, when the fixed sights are directed to an 
 object, and the figured or N. point towards him in 
 directing the index, then that end of the index to-
 
 I'lG THE COMMON PLAIN TABLET. 
 
 wards the surveyor will point out the anglc^ and 
 the south end of the needle the bearing; the ap- 
 plication of the instrument to various cases that 
 may occur in surveying, will be evident from what 
 we shall say on that subject in the course of this 
 v.ork.. 
 
 OF THE com:\ion PLAIN T Ai&LE, Jrg.l , plate 17. 
 
 The tabular part of this instrument is usually 
 made of two well-seasoned boards^, forming a pa- 
 rallelogram of about 15 inches long^ and 12 inches 
 broad; the size is occasionally varied to suit the 
 intentions of the operator. 
 
 The aforesaid parallelogram is framed with a 
 ledge on each side to support a box frame, which 
 frame confines the paper on the table, and keeps 
 it close thereto; the frame is therefore so contrived, 
 that it may be taken off and put on at pleasure, 
 either side upwards. Each side of the frame is 
 graduated; one side is usually divided into scales 
 of equal parts, for drawing lines parallel or per- 
 pendicular to the edges of the table, and also for 
 inore conveiviently shifting the paper; the other 
 face, or side of the frame, is divided into 3()0°, from 
 a brass center in the middle of the table, in order 
 that angles may be measured as with a theodolite; 
 on the same face of the frame, and on two of the 
 edges, fu-e graduated 180°; the center of thescr 
 degrees is exactly in the middle between the two 
 ends, and about ith part of the breadth from one 
 of the sides. 
 
 A magnetic needle and compass box, covered 
 with a glass and spring ring, slides in a dovetail on 
 the under side of the tabkyand is fixed there by a 
 finger screw; it serves to point out the direction) 
 ijndbe a check ti])on the sights. ' -
 
 drltE COMMOX PLAIN TABLE. 227 
 
 There is also a brass index somewhat larger than 
 the diagonal ot the table, at each end of which a 
 sight is fixed ; the vertical hair, and the middle of 
 the edge of the index, are in the same plain; this 
 edge is chamfered, and is usually called the fiducial 
 edge of the index. Scales of different parts in an 
 inch are usually laid down on one side of the index. 
 
 Under the table is a sprang to fit on the pin of 
 the ball and socket, by which it is placed upon a 
 thrce-le2:2:ed staff. 
 
 7o place the pap-er on the table. Take a sheet of 
 paper that will cover it, and wet it to make it ex- 
 pand, then spread it fiat upon the table, pressing 
 down the frame upon the edges to stretch it, and 
 keep it in a fixed situation ; when the paper is dry 
 it will by contracting become smooth and flat. 
 
 To shift the paper on the plain table. When the 
 paper on the table is full, and there is occasion for 
 more, draw a line in any manner through the far- 
 thest point of the last station line, to which the 
 work can be conveniently laid down; then take 
 off the sheet of paper, and fix another on the table ; 
 draw a line upon it in a part most convenient for 
 the rest of the work; then fold, or cut the old 
 sheet of paper by the line drawn on it, apply the 
 edge to the line on the new sheet, and, as they lie 
 in that position, continue the last station line upon 
 the new paper, placing upon it the rest of the 
 measures, beginning where the old sheet left off, 
 and so on from sheet to sheet. 
 
 To fasten all the sheets of paper together, and 
 thus form one rough plan, join the aforesaid lines 
 accurately together, in the same manner as when 
 the lines were traiuterred from the old sheets to 
 the new one. But if the joining lines upon the 
 old and new sheets have not the same inclination to 
 the i^ide of the table, the needle will not point to 
 
 Q 2
 
 '228 THE COMMON PLAIX TABLE. 
 
 the original degree when the table is rectified. If 
 the needle therefore should respect the same de- 
 gree of the compass the easiest way of drawing 
 the line in the same position is to draw them both 
 parallel to the same sides of the table, by means of 
 the scales of equal parts on the two sides. 
 
 7\ use the plain table. Fix it at a convenient 
 part of the ground, and make a point on the paper 
 to represent that part of the ground. 
 
 Run a tine steel pin or needle through this 
 })oint into the table, against \\'hieh you must ap- 
 ply the fiducial edge of the index, moving it round 
 till you perceive some remarkable object, or mark 
 set up for that purpose. Then draw a line from 
 the station point, along the fiducial edge of the, 
 index. 
 
 Now set the sights to another mark, or object, 
 and draw that station line, and so proceed till you 
 have obtained as many angular lines as are neces- 
 sarv from this station. 
 
 Tlie next requisite, is the measure or distance 
 from til e station to as many objects as may be ne- 
 cessary by the chain, taking at the same time the 
 offsets, to the required corners or crooked parts of 
 the hedges; setting off all the measures upon their 
 respective lines upon the table. 
 
 Now remove the table to some other station, 
 whose distance from the foregoing was previously 
 measured; then lay down the objects which ap- 
 pear from thence, and continue these operations 
 till your work is finished, measuring such lines as 
 are necessary, and determining as many as you 
 can by intersecting lines of direction, drawn from 
 different stations. 
 
 It seems to be the universal opinion of the best 
 surveyors, that the plain table is not an instrument 
 to be trusted to in large survxys, or on hilly situa-
 
 THE IMPROVED PLAIX TABLE. 22^ 
 
 tions; that it can only be used to advantage in 
 planning the ground plot of buildings, gardens, or 
 a tew small pareels of land nearly on a level. 
 
 Mr. Gardiner, whose authority as a surveyor is 
 inferior to no one, asserts, that the plain table sur- 
 veyors, when they find their work not to close 
 right, do often close it wrong, not only to save time 
 and labour, but the acknowledgement of an error; 
 which they are not sure they can amend- 
 In uneven ground, where the table cannot in all 
 stations be set horizontal, or uniformly in any one 
 place, it is impossible the work should be true in 
 all parts. 
 
 The contraction and expansion of the paper ac- 
 cording to the state of moisture in the air, is a 
 source of many errors in plotting; for between 
 a dewy morning and the heat of the sun at noon, 
 there is a great difference, which may in some de- 
 gree be allowed for in small w^ork, but cannot be 
 remedied in surveys of considerable extent. 
 
 OF THE IMPROVED PLAIN T A.'Rl.'E,fg\2, plate IJ . 
 
 To remedy some of the inconveniences, and 
 correct some of the errors to which the common 
 plain table is liable, that Mhich we are now going 
 to describe has been constructed. It is usually 
 called Be'ightons plain table, though differing in 
 many respects from that described by him in the 
 rh'tlosophica I Transactions . 
 
 It is a plain board, l6 inches square, with a 
 frame of box or brass round the edge, for the pur- 
 pose of being graduated. On the sides, AB, C D, 
 are two grooves and holdfasts for confining.#i-mly, 
 or easily removing the paper; they are disengaged 
 by turning the screws under the table from the 
 right towards the l«ft, and drawn down and madt;
 
 230 THE IMPROVED PLAIN TABLE. 
 
 to press on the paper by turning the screws the 
 contrary way. When the holdfasts are screwed 
 down, their surface is even with that of the table. 
 There are two pincers under the table, to hold 
 that part of the paper, which in some cases lies 
 beyond the table, and prevent its flapping about 
 with the wind. 
 
 The compass box is made to fit either side of 
 the table, and is fixed by two screws, and does 
 not, when fixed, project above one inch and an 
 half from the side of the table. 
 
 There is an index with a semicircle, and teles- 
 copic sight, EFG; it is sometimes so con- 
 structed, as to answer the purpose of a parallel 
 rule. The figure renders the whole so evident_, 
 that a greater detail would be superfluous. 
 
 The papers, or charts for this table, are to be 
 either of fine thin pasteboard, fine paper pasted on 
 cartridge paper, or two papers pasted together, 
 cut as square as possible, and of such a length that 
 they may slide in easily, between the upright 
 parts, and under the flat part of the holders. 
 
 Any one of these charts may be put into the table 
 at any of the four sides, be fixed, taken out, and 
 changed at pleasure; any two of them maybe joined 
 together on the table, by making each of them 
 meet exactly at the middle, whilst near one half of 
 each will hang over the sides of the table; or, by 
 doubling them both ways through the middle, four 
 of them may be put on at one time meeting in the 
 6enter of the table. For this purpose, each chart 
 is always to be crossed quite through the middle; 
 by these means the great trouble and inaccuracy 
 in shifting the papers is removed. **-• 
 
 The charts thus used, are readily laid together 
 by corresponding numbers on their edges, and 
 thus make up the whole map in one view; and,
 
 nvIPROVED THEODOLITES. 231 
 
 bciiif^ in squares, arc portable, easily copied, en- 
 larged, or contracted. 
 
 The line of sight in viewing objects may, if that 
 method be preferred, be always over the center of 
 the tabic, and the station lines drawn parallel to 
 those measured on the land. Underneath the ta- 
 ble is a sprang to fit on the socket of a staff, with 
 parallel plates and adjusting screws. 
 
 Mr. Searle contrived a plain table, whose size 
 (which renders it convenient, while it multiplies 
 every error) is only five inches square, and con- 
 sists of two parts, the table and the frame; the 
 frame, as usual, to tighten the paper observed 
 upon. In the center of the table is a screw, on 
 MJiich the index sight turns; this screw is tightened 
 after taking an obscrvatiou. 
 
 THEODOLITES WITH TELESCOPIC SIGHTS. 
 
 In proportion as science advances, we find our- 
 selves standing upon higher ground, and are ena- 
 bled to see further, and distinguish objects better 
 than those that went before us : thus the great 
 advances in dividing of instruments have rendered 
 observers more accurate, and more attentive to the 
 necessary adjustments of their instruments. In- 
 struments arc not now considered as perfect, unless 
 they are so constructed, that the person who uses 
 them may either correct, or allow for the errors to 
 wliich they arc liable. 
 
 Theodolites with telescopic sights arp, without 
 jdoubt, the most accurate, commodious, and uni- 
 versal instruments -for the purposes of surveying, 
 and have been recommended as such by the most 
 expert practitioners and best wa-iters on the sub- 
 ject, as Gardiner, Hammond^ Ciom, IStoney IVjUl^ 
 fVudd'mgton, &c.
 
 '232 IMPROVED THEODOLITES. 
 
 The leading requisites in a good theodolite are, 
 1. That the parts be limily eonneeted^, so that 
 they may always preserve the same figure. 1. The 
 circles must be truly centered and accurately gra- 
 duated. 3. The extremity of the line of sight 
 should describe a true circle. 
 
 Fig 1, plate l6, represents a theodolite of the 
 second best kind; the principal parts are, 1 . A te- 
 lescope and its level, C, C, D. 1. The vertical 
 arc, BB. 3. The horizontal limb and compass, 
 A A. 4. The staff with its parallel plates, E. 
 The limb A A is generally made about seven 
 inches in diameter. 
 
 An attentive view of the instrument, or draw- 
 ing, compared with what has been said before, 
 will shew that its perfect adjustment consists in 
 the following particulars. 
 
 1 . The horizontal circle A A must be truly level. 
 1. The plain of the vertical circle B B must be truly 
 perpendicular to the horizon. 3. The line of 
 sight, or line of collimation,* must be exactly in 
 the center of the circles on which the telescope 
 turns. 4. The level must be parallel to the line 
 of collimation. 
 
 Of the telescope CC. Telescopic sights not only 
 enable the operator to distinguish objects better, 
 but direct the sight with much greater accuracy 
 than is attainable with plain sights; hence also we 
 can make use of much finer subdivisions. The 
 telescope, generally applied to the best instru- 
 ments, is of the achromatic kind, in order to ob- 
 tain a larger field, and greater degree of magnify- 
 ing power. In the focus of the eye-glass are two 
 
 * The line of collimation is the line of vision, cut by the in- 
 tersecting point of the cross hairs in the telescope, answering 
 to the visual line, by which we directly^ point at objects with 
 plain sights,
 
 IMPROVED THEODOLITES. ^233 
 
 very fine hairs, or wires, at right angles to each 
 other, whose ititersection is in the plain of the ver- 
 tical arc. The object glass may be moved to 
 different distanecs from the eye glasses, by turn- 
 ing the milled nut a, and may by this means be 
 accommodated to the eye of the observer, and tlic 
 <listance of the object. The screws for moving 
 -and adjusting the cross hairs are sunk a little 
 within the eye tube, and at about one inch from 
 the eye end: there are four of these screws, two 
 of which arc exactly opposite to each other, and 
 at right angles to the other two. By easing one 
 of the screws, and tightening the opposite one, 
 the wire connected with it may be moved in op- 
 posite directions. On the outside of the telescope 
 are two metal rings, which are ground perfectly 
 true; these rings are to lay on the su})porters c, c, 
 called Y's, which are fixed to the vertical arc. 
 
 Of the verl'ical arc B B. This arc is finnly fixed 
 to a long axis which is at right angles to the plain 
 of the arc. This axis is sustained by, and movea- 
 ble on the two supporters, which are fixed firmly 
 to the horizontal plate: on the upper part of the 
 vertical arc are the two Y's for holding the teles- 
 cope; the inner sides of these Y's are so framed, 
 as to be tangents to the cylindrie rings of the te- 
 lescope, and therefore bear only on one part. 
 The telescope is confined to the Y's bv two loops, 
 which turn on a joint, and may therefore be rea- 
 dily opened and turned back, when the two pins 
 are taken out. 
 
 One side of the vertical arc is graduated to 
 every half degree, which are subdi\ ided to every 
 minute of a degree by the nonius. It is numbered 
 each way from O to C)0°, towards the eve end, for 
 angles of altitude; from O to 50°, towards the ob- 
 ject end, for angles of depression. On the other
 
 234 IMPROVED THEODOLITES. 
 
 Side of the vertical arc are two ranges of divisions, 
 the lowermost for taking the upright heiglit of 
 timber in 100th parts of the distance the instru- 
 ment is placed at from the tree at the time of ob- 
 servation. The uppermost circle is for reducing 
 hypothenusal lines to horizontal, or to shew the 
 difference between the hypothenuse and base of a 
 right-angled triangle, always supposing the hypo- 
 thenuse to consist of 100 equal parts; conse- 
 quently, it gives by inspection the number of links 
 to be deducted from each chain's length, in mea- 
 suring up or down any ascent or descent, in order 
 to reduce it to a true horizontal distance^ simitar 
 to those on K'mg's quadrant, p. 205. 
 
 The vertical arc is cut with teeth, or a rack, 
 and may be moved regularly and wdth ease, by 
 turning the milled nut b; there is sometimes 
 placed about the nonius a steady pin, by which it 
 may be fixed when at the o, or zero point of the 
 divisions. 
 
 Of the compass. The compass is fixed to the 
 upper horizontal plate; the ring of the compass is 
 divided into 36o°, which are numbered in a direc- 
 tion contrary to those on the horizontal limb. 
 The bottom of the box is divided into four parts, 
 or quadrants, each of which is subdivided to every 
 10 degrees, numbered from the meridian, or north 
 and south points each way to the east and west 
 points. In the middle of the box is a steel pin 
 jSnely pointed, on which is placed the magnetic 
 needle; there is a Avire trigger for throw^ing the 
 needle off its point when not in use. 
 
 Of the hofhontal limh A A. This limb consist^ 
 of two plates, one moveable on the other; the 
 outside 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, ver-
 
 IMPROVED THEODOLITES. *235 
 
 tica! arc, and telescope, arc easily turned round by 
 a pinion fixed to the screw c; d is a nut lor fixing 
 the index to any part of' the hmb, and thereby 
 making it so secure, that there is no danger of its 
 being moved out of its place, while the instrument 
 is removed from one station to another. The ho^ 
 rizontal limb is divided to half degrees, and num- 
 bered from the right hand towards the left, 10, 
 '20, 30, &c. to 360 ; the divisions are subdivided by 
 the nonius scale to every minute of a degree. 
 
 On the upper plate, opposite to the nonius, arc 
 a few divisions similar to those oil the vertical arc, 
 giving the 100th parts for me^isuring the diameter 
 of trees, buildings, &c. 
 
 The whole instrument fits on the conical ferril 
 of a strong brass headed staff', with three substan- 
 tial wooden legs; the top, or head of the staff^ 
 consists of two brass plates E, parallel to each 
 other; four screws pass through the upper plate, 
 and rest on the lower plate; by the action of these 
 screws the situation of the plate may be varied, so 
 as to set the horizontal 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 pin and 
 ball arc so framed, as to be always perpendicular to 
 the plate, and, consequently, to the horizontal 
 limb. 
 
 To adjust the theodolile. As so much of survey- 
 ing depends on the accuracy of the instruments, 
 it is absolutely necessary that the sur\'eyor should 
 be very expert in their adjustments, without which 
 he cannot expect the instruments will properly 
 answer the purposes they were designed for, or 
 that his surveys will have the requisite exactness.
 
 236 IMPROVED THEODOLITES. 
 
 The necessary adjustments to the theodolite, 
 which we have just described, are, 1 . That the line 
 of sight, or colliination, be exactly in the center 
 of the cylindric rings round the telescope, and 
 which lie in the Y's. 2. That the level be paral- 
 lel to this line, or the axis of the above-mentioned 
 rings. 3. The horizontal limb must be so set, 
 that when the vertical arc is at zero, and the up- 
 per part moved round, the bubble of the level will 
 remain in the middle of the open space. 
 
 Previous to the adjustments, place the' instru- 
 ment upon the staff, and set the legs thereof firmly 
 upon the ground, and at about three feet from 
 each other, so that the telescope may be at a pro- 
 per height for the eye, and that two of the screws 
 on the staff that are opposite to each other may be 
 nearly in the direction of some conspicuous and 
 distant object. 
 
 To adjust the Tine of coUimatloii. Having set up 
 the theodolite agreeable to the foregoing direction, 
 direct the telescope to some distant object, pla- 
 cing 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 whether the horizontal hair still coincides 
 with the object; 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 hair coincide 
 with the object, by moving the vertical arc; re- 
 verse the telescope again, and if the hair docs not 
 coincide with the same part of the object, you must
 
 IMPROVED THEODOLITES. 237 
 
 repeat the foregoing operation, till in both posi- 
 tions 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 ho- 
 rizontal 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 ex- 
 actly with the same point of the object, while the 
 telescope is turned quite round; and the hairs are 
 not properly adjusted, till this is effected. 
 
 ylJ jus Intent of the level. To render the level 
 parallel to the line of collimation, place the verti- 
 cal 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 out of the Y's, 
 and turn it end for end, that is, let the eve end 
 lay where the ol^ject end was placed ; and it", when 
 in this situation, the bubble remains in the middle 
 as before, the level is well adjusted; if it does not, 
 that end to which the bubble runs is too high; the 
 position thereof nuist be corrected by turning 
 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. Return the te- 
 lescope to its former position, and if the adjust- 
 ments have been well made, the bubble will re- 
 main in the middle; if otherwise, the process of 
 altering the level and the staff" screws, with the re- 
 versing, nmst be repeated, till it bears this proof of 
 its accuracy. In some instruments there is s pro- 
 vision for raising- or lowering the Y's a small de-
 
 238 Ajsr IMPROVED theodolitf: 
 
 grcc, in order more conveniently to make the bub- 
 ble continue in its place when the vertical arc is 
 at o, and the horizontal limb turn round. 
 
 To adjust ihe level of the horizontal Ihiih. Place 
 the level so that it may be in a line with two of 
 the staff screws, then adjust it, or cause the bub- 
 ble 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 re- 
 mains in the middle as before, the level is well ad- 
 justed; if not, correct half the error by the screws 
 at the end of the level, and the other half bv 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 al- 
 tering must be pursued till the error is annihilated. 
 See this adjustment in the description oi Ramsclen& 
 theodolite. 
 
 When the bubble is adjusted, the horizontal 
 limb may always be levelled by means of the staff 
 screws. 
 
 OF THE THEODOLITE, AS IMPROVED BY 
 RAMSDEX, fg. 2, flale 16. 
 
 Among the improvements the instruments of 
 science have received from Mr. Ramsdeii, and the 
 perfection with which he has constructed them, we 
 arc to rank those of the theodolite; 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 instrument, and the various methods by which 
 the parts concur to give the most accurate result. 
 
 The principal parts of this instrument are how- 
 ever so similar to the foregoing, that a description 
 thereof must, in some degree, be a repetition of
 
 BY RAMSDEM. 239 
 
 what has been ah'cady described, and requires less 
 detail here. 
 
 F F represents the horizontal limb, of six, 
 seven, or eight inehes in diameter, but gene- 
 rally of seven inches, so called, because when in 
 use, it ought always to be placed parallel to the 
 horizon. It consists, like the former, of tw^o plates, 
 the edges of these two are chamfered, so that the 
 divisions and the nonius are in the same plaiii, 
 which is oblique to the plain of the instrument. 
 The limb is divided into half degrees, and subdi- 
 vided by the nonius to every minute; it is num- 
 bered to 3()0° from the north towards the east: be- 
 sides 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 hori- 
 zontal plate, by means of three capstan headed 
 screws, passing through three similar screws. By 
 the action of these, the vertical arc may be set per- 
 pendicular to the horizontal limb, or be made to 
 move in a vertical plain. On this 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 tele- 
 scope; the Y's arctangents to the cylindric rings 
 of the telescqpe, which rings are turned, and then 
 ground as true as possible, and are prevented from 
 tnoving backwards or forwards, by nieans of two 
 shoulders. The telescope is achromatic, and 
 about twelve inehes in length, and may be ad- 
 justed to the eye of the observer, or the distance, 
 of the object, by turning the milled nut B. The
 
 240 ' AX mntovED theodolite 
 
 Imirs are adjusted by the screws in the eye tube 
 at A. Under the teleseopc is fixed a spirit level C^ 
 the distance of whose ends from the telescope may 
 be rcQ-ulated bv tlie screws c, c. 
 
 Beneath the horizontal limb there is a second 
 or auxiliary telescope, w hieh has both an horizon- 
 tal and vertical motion: it is moved horizontally 
 by the milled screw H, and wdien directed to any 
 object, is fixed in its situation by another milled 
 screw; it moves vertically on the axis; there is an 
 adjustment to this axis, to make the line of colli- 
 mation move in a vertical plain. By the horizon- 
 tal motion, this telescope is easily set to what is 
 called the backset stations; the under telescopa 
 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 
 sci'cw L, and when tightened, a small motion of 
 the adjusting screw K will move the telescope a 
 few degrees, in order to set it with great accuracy. 
 Beneath these is the stafl^", the nature of which 
 will be sufficiently evident from w hat w^as said 
 thereon in the description of the last theodolite, 
 or by inspection of the figure. 
 
 To a{] just the levels of the horizontal plate. 1. 
 Place the instrument on its staff, with the legs 
 thereof at such a distance from each other, as will 
 give the instrument a firm footing on the ground. 
 2. Set the nonius to 36o, and move the instru- 
 ment roun.d, 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 bub- 
 ble in the level to become stationary in the middle 
 of the glass. 4. Turn the horizontal limb by the 
 milled nut half round, or till the nonius is at 180,
 
 BY RAMS DEN. 241 
 
 and if the bubble remains in the middle as before, 
 the level is adjusted; if it does not, correct the 
 position of the level, by turning one or both the 
 screws which pass through its ends, 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 screws of the parallel plates. 
 5. Return the horizontal limb to its former posi- 
 tion, 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 
 screws 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 pro- 
 per 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 con- 
 tinues to occupy the middle of the glass it is in its 
 right position ; if not, correct one half of the er- 
 ror by the capsttfn 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 middle in 
 both positions. 
 
 To adjust the line of coU'miation. 1 . Direct the 
 telescope, so that the horizontal wire may coincide 
 with some well defmed part of a remote object. 
 2. Turn the telescope so that the bubble may be 
 uppermost; if the wire does not coincide with the 
 same part of the object as before, correct half the 
 
 R
 
 242 VARIATION IN THEODOLITES. 
 
 difference by moving the vertical circle, and the 
 other half by moving the wire, which is effected 
 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 exactly 
 with the same point of an object, while the teles- 
 cope 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 w^ork, in order to be af- 
 forded at a lower price; one of these is represented 
 2iXfig. 5, pJate 15. It is clear from the figure, that 
 the difference consists principally in the solidity 
 of the parts, and in there being no rack-work to 
 "•ive motion to the vertical are and horizontal limb. 
 The mode of using it is the same with the other, 
 and the adjustment for the line of collimation, and 
 the level under the telescope, is perfectly similar 
 to the same adjustments in the instruments already 
 described, a further description is therefore un- 
 necessary. 
 
 A larger kind is also made without rack-work, 
 similar to j/f^. l, plate l6. 
 
 A small one, about four inches in diameter, 
 was invented by the late Mr. Be?ij. Martin, in 
 which the telescope, about six inches in length, 
 with a level, has no vertical motion, but the ho- 
 rizontal motion is given by a pinion; and it may 
 be turned into a vertical position to take angles 
 of altitudes or depressions. The divisions by the 
 nonius arc to five minutes.
 
 [ 243 ] 
 
 THE Following account of theobolites 
 
 IS ADDED BY THE EDITOR. 
 
 It generally happens, that the observer has oc- 
 casion to take the vertical and horizontal angles at 
 the same time, by portable as well as by larger theo- 
 dolites; the following is,therefore, recommended as 
 the most complete and portable instrument hitherto 
 made, and is in truth almost the best theodolite in 
 miniature. 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.y,p/A 4, IS a representationof theinstrumcnt. 
 The graduated limb and index plate A, A, arc 
 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 minutes of a degree. The achromatic te- 
 lescope C is about six inches in length, and con- 
 tains 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 at D. This arc is necessarily of a short length, 
 but adm.its about 30 degrees motion on each side 
 of 0, 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 
 adjustment 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 theodolite is produced, when the plates are 
 properly set by the screws. The telescope rests 
 upon a cradle, and by opening the two semicircles 
 
 R 2
 
 144 DESCRIPTION OF THEODOLITES, 
 
 a, a, it may be reversed, in order to adjust the 
 spirit level, or prove its truth to the axis of 
 the telescope. The adjustments of this little 
 instrument, being in all respects similar to^ 
 and made, as in those just described, they will be 
 evident to the reader, and quite unnecessary to 
 repeat here. 
 
 The instrument, exclusive of its staff, packs 
 into a pocket mahogany case, of 6 inches in 
 length, 4t inches- in breadth, and sf inches in 
 depth. 
 
 In surveys of very great extent and importance, 
 or in great trigonometrical operations, a larger 
 instrument is required, in order that the subdivi- 
 sions may be greater in number, or the angle taken 
 more accurately, to five, two, or even one second 
 of a degree. Several plans have been suggested, 
 but I do' not at present see any better principle to 
 adopt than that of the great one by Mr. Ramsden 
 hereafter to be described. A proportionate re- 
 duction of its size, as well as simplifying its ma- 
 chinery and movements, necessary only for the 
 grand purpose that it was applied to, would ac- 
 commodate the practitioner with as complete an 
 instrument as he could desire. The diameter of 
 the horizontal circle I would recommend to be 
 from about 15 to 20 inches, and the other parts 
 in proportion. The price, according to the work- 
 manship, would be from about 6o to 120 guineas, 
 stand, cases, &c. included. 
 
 In the preceding impression of this work, the 
 ingenious author^ now deceased, made the fron- 
 tispiece plate a representation of a new theodolite 
 of his own contrivance, the adjustments of which 
 he thought to be more perfect than those of any 
 other, and annexed the description and mode of 
 its adjustments to his preface. Future trials, how-
 
 BY THE EDITOR. 245 
 
 ever, on his own part, as well as by the hands of 
 others, gave me reason to conclude that it was 
 not answerable to the intended improvements; 
 although more complicated and costly than that 
 represented onfg. 1, plate l6, yet it was less sus- 
 ceptible of accuracy, and not so simple and easy 
 in the adjustments; I have, therefore, thought 
 proper to dispense with it here, and substitute a 
 short description of the largest, most accurate and 
 elegant theodolite ever made. 
 
 It is hardly necessary to acquaint the intelligent 
 reader, that the theodolite is a kind of general , 
 angular instrument, not useful merely to ascertain 
 angles for the surveyor, but also for many pur- 
 poses in practical astronomy, and other sciences 
 that have trigonometry as their fundamental basis. 
 Some years ago it was found necessary to in- 
 stitute a course of trigonometrical operations in 
 this country and in France, in order to determine 
 with precision the distance between the Royal 
 Observatories of Greenwich and Paris. The late 
 General Roy was deputed as the chief manager in 
 this country. A very accurate theodolite to take 
 angles, and other instruments, were essentially ne- 
 cessary; and the General was fortunate enough to 
 obtain the best articles and assistance that was 
 ever afforded in any mathematical undertaking 
 whatever. 
 
 The frontispiece to this book gives a general 
 view of the theodolite, the reader must not ex- 
 pect from this a complete representation of all the 
 minuter parts. In the General's account in the 
 Philosophical Transactions, containing 26 quarto 
 pages, and six large plates, replete with explanatory 
 figures, he confined himself only to the describ- 
 ing of the principal parts ; and the limits of this
 
 '24(5 THE GREAT THEODOLITE, 
 
 work will only admit of a summary description to 
 convey some idea of its plan, and to shew the in- 
 genious operator the superior utility and accuracy 
 of the instrument. If a further and more particu- 
 lar knowledge be desired, it will be best obtained 
 by a reference to page 135, vol. 80 of the Trans- 
 actions before cited. A A. a brass circle three feet 
 diameter; B the principal, or transit achromatic 
 telescope of 36 inches focal length, and 2^ inches 
 aperture, admitting its adjustment by inversion 
 on its supports, as performed by the transits in 
 fixed observatories; C a small lanthorn fixed to 
 an horizontal bar for giving light to the axis of 
 the telescope upon an illuminator that reflects 
 light on the wire in nocturnal observations; D a 
 semicircle of six inches radius, attached to the axis 
 of the transit. Each degree being divided into 
 two parts, or 30', and one revolution of the mi- 
 crometer head moving the wire in the field of the 
 microscope at a three minutes : therefore lOrevolu- 
 tions produce 30', which are shewn by a scale of 
 10 notches in the upper part of the field of the 
 microscope, each notch corresponding to three 
 minutes, or 180 seconds, and the head being di- 
 vided into three minutes, and each minute into 
 12 parts; therefore 12 parts is exjual to five 
 seconds. When the angles of altitude and de- 
 pression to be determined, are very small, they are 
 measured by the motion of an horizontal wire in 
 the focus of the eye glass of the telescope at^. Two 
 spirit levels are used to this telescope; one to level 
 the axis, making the long conical axis of the in- 
 strument truly vertical, not shewn in the plate, 
 and the other level E is suspended on a a rod at- 
 tached to the telescope, and serves to make it ho- 
 rizontal when vertical angles arc to be taken,
 
 BY nAMSDEX. 24/ 
 
 The vertical bar F extending across the top of the 
 axis, is supported by two braces GG that come from 
 the cone, Id above the plain of the instrument. 
 
 The great divided circle is attached by 10 brass 
 conical tubes, or radii, to a large vertical conical 
 axis H, of 24 inches in height, called the exterior 
 axis. Within the base of this hollow axis, a cast 
 steel collar is strongly driven ; and on its top is 
 inserted a thick bell-metal plate, with sloping 
 cheeks, which by means of five screws can be 
 raised or depressed a little. 
 
 The instrument rests on three feet, one of 
 which is shewn at I, united at the center by a 
 strong round plate of bell-metal, upon which rises 
 another vertical hollow cone, going into the 
 other, H, and is called the interior axis; a cast 
 steel pivot in its top, with sloping cheeks, passes 
 through the bell-metal plate at the top of the ex- 
 terior axis, being ground to fit one another. The 
 bell-metal base of this interior axis is also ground 
 to fit the steel collar of that without it. When 
 put together, the circle is to be lifted up by laying- 
 hold of its radii, and the exterior placed upon the 
 interior axis, the cheeks at top, adjusted to their 
 proper bearings, will then turn round smoothly 
 and steadily, and free from any central shake ; the 
 great circle, exterior axis, and upper telescope, 
 therefore, are moveable, independent of the lower 
 parts. 
 
 The feet of the mahogany stand K, form a square 
 of about three feet four inches at bottom, and by 
 the separation of the legs, make an octagon at the 
 top or the first plain; in the center of which is 
 an opening, nine inches in diameter. On the 
 top of this lies another mahogany octagonal plain, 
 of rather greater dimensions than the former, with
 
 0,48 THE GREAT THEODOLITE, 
 
 a circular curb about -'- an inch within the plain of 
 its sides. This hath in its center an open conical 
 brass socket, three inches in diameter; and on 
 four of its opposite sides there are fixed four 
 screws, acting against pieces of brass on the top 
 of the stand. The plain, with every thing upon it, 
 may, therefore, be moved in four opposite direc- 
 tions, until the plummet L, is brought to coincide 
 with the station points underneath, in order to 
 level the stand. The third or uppermost plain of 
 mahogany is part of the instrument, being con- 
 nected by screws, and carrying the handles 
 whereby it is lifted up for use. In the middle of 
 this bottom to the instrument there is another co- 
 nical brass socket, i inches in diameter, that turns 
 easily on that in the center of the octagon un- 
 derneath. 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 occasionally to raise or lower the plum- 
 met. To the three feet there are screws, such as 
 at N, for levelling the instrument; and also three 
 blocks of box wood, and three brass conical rol- 
 lers under the feet screws, fixed to the lower sur- 
 face of the mahogany, to give the whole a per- 
 fectly eavSy motion; O, O, O, are three of the four 
 screws attached to the octagonal plain, for accu- 
 rately centering the instrument by the plummet. 
 P and Q represent two positions of screws to 
 give a circular motion to the entire machine, 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, consisting 
 of a brass cock fixed to, and projecting from the 
 curb of the instrument; the cock being acte4
 
 BY RAMSDEX. 249 
 
 upon by two screws working in opposite direc- 
 tions, and which arc clamped to the curb of the 
 octagon. 
 
 The curb upon which the feet of the instru- 
 ment rest, carries the mahogany balustrade R R, 
 fitted to receive a mahogany cover, that guards 
 the whole instrument. In this cover are four 
 small openings, one for each of the vertical mi- 
 croscopes S, S, one for the clamp of the circle, 
 and one for the socket of the Hook'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 particu- 
 larly lanthorns used at night, for reading off the 
 divisions on the limb of the instrument. 
 
 There is a lower telescope T, lying exactly un- 
 der the center 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 17 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, 
 and likewise on its object, the lower telescope 
 by help of this adjustment is brought accurately 
 to the same object, from the point of commence- 
 ments from which the angles are to be measured. 
 
 There are three flat arms, one of which is re- 
 presented 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 towards the circumference of the circle, 
 going beyond it about li inches. One arm, lying 
 directly over one of the fect_, is that to which is at- 
 tached wheels and screws moved by Hook's joint.
 
 250 THE GREAT THEODOLITE, 
 
 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 oppo- 
 site to it, become the diameter of the circle, hav- 
 ing their extremities terminated 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 without di- 
 minishing strength. 
 
 The angles are not read off in this instrument 
 by a nonius as common to others, but with micros- 
 copes, and which form the most essential part of 
 the instrument. But a short account of them can 
 be given here, an adequate idea can only be ob- 
 tained by a reference to the Philosophical Trans- 
 actions, page 145 and 149. That horizontal mi- 
 croscope for the vertical angles has been already 
 mentioned. The two vertical ones S, S, are used 
 for reading off the divisions on the opposite sides 
 of the circle immediately under them. Each mi- 
 croscope 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 necessary for adjust- 
 ment, 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 for- 
 mer, of sufficient thickness to permit a micro- 
 meter screw of about 72 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 micrometer head. This head is di- 
 vided into 60 equal parts, each of which represents 
 one second or angular motion of the telescope. 
 This steel slide is attached by a chain to the
 
 BY BAMSDEX. ^51 
 
 spring of a watch coiled up within a small barrel 
 adjacent to it in the frame. By this no time what- 
 ever is lost, the smallest motion of the head being: 
 instantly shewn by a proportionable motion of 
 the wire in the field of the microscope. 
 
 Each microscope is supported between its pil- 
 lars, and can be a little raised or depressed in re- 
 spect to the plain of the circle by two levers. By 
 this motion distinctness is obtained of the wires, 
 and by the motion of the proper screw of the ab- 
 ject lensc, which follows that given to the whole 
 microscope, the scale is so adjusted, that ] 5 revo- 
 lutions of the head shall move the wire over 15', 
 or one grand division of the limb, equal to 900", 
 each degree on the circle being only divided into 
 four parts. To effect this, at the same time the 
 fixed wire must bisect the dot on a gold tongue, 
 the moveable wire must also bisect the dot at 180° 
 on the limb, as well as a first notch in the magni- 
 fied scale at the bottom of the plate. In this ad- 
 justment there is another circumstance to be at- 
 tened to, viz. that 60 on the micrometer head 
 should stand nearly vertical, so as to be conveni- 
 ently seen ; a few seconds of inclination are of no 
 consequence, because the dart, or index, being 
 brought to that position, whatever it may be, must 
 always remain there, any derangement of the in- 
 strument excepted. But if, when the wires coin- 
 cide with their respective dots, and the first notch 
 60 on the micrometer head should happen to be 
 underneath, or so far from the vertex side as to be 
 seen with difficulty, then the gold tongue is to be 
 moved a little by capstan head screws, which act 
 against each other on the opposite extremities of 
 the axis. Thus, by repeated trials the purpose 
 will be effected, viz. the 60, to which the dart is 
 to be set, will stand in a place easily seen. It is 
 not to be expected that each microscope will give
 
 2.52 THE GREAT THEODOLITE. 
 
 just 900 seconds for the run of 15 minutes; with- 
 out loss of time this cannot be done; besides, two 
 observers will adjust the microscopes differently. 
 After several trials of the runs in measuring 15 mi- 
 nutes in the different parts of the limb, one mi- 
 croscope gave S96", while the other gave at a me- 
 dium 901''; in a year afterwards, the former gave 
 900", while the latter gave 894". These ^.iffe 
 rences were allowed for in the estimation of angles 
 for computation. 
 
 The gold tongue mentioned is extremely thin, 
 and goes close to the surface of the circle. This 
 contrivance of a tongue with a dot was to guard 
 against any error from any accidental motion given 
 to the instrument between the observations, and 
 if any, it immediately detected them. This was also 
 a severe check upon the divisions of the instru- 
 ment. General Roy observes, that it rarely hap- 
 pens that two observers, reading off with the op- 
 posite m.icroscopes, differ more than half a second 
 from each other at the first reading; and judges, 
 that in favourable weather for repeating the observa- 
 tion with the telescope, a wonderful degree of accu- 
 racy in the measure of the angles may be obtained.* 
 
 For the auxiliary apparatus, such as the \00 feet 
 steel chain, portable scaffold, tripod ladder, common 
 ffag-staff, tripod for white lights, portable crane, &c. 
 the reader will see the account of in the Trans- 
 actions before cited. The horizontal angles taken 
 by the instrument as regulated by the General, 
 since deceased, are to the tenth of a second. 
 
 * The weighr of the whole instrument was about 200 pounds, and the 
 price, as 1 have been informed, about 350 guineas. By the completion of the 
 measurements and the necessary calculations, the difference of the two meri- 
 dians made <)' z6", as before fixed by Dr. Maske/ync. 
 
 A second instrument has since been made, and is now using by Col. !Vi:liiims, 
 Cap. jMucige, and Mr. Dnlby, from whose skill and ingenuity it is expected a 
 very arrwute survey of this country will be made. In this instrument, the 
 great circle is divided to 10 minutes, improvements made in the microscopes, 
 Ac. by Mr. Ranfsden. Sec Philos^jphicai Traruactions for 1 795.
 
 [ ^53 J 
 DESCRIPTION, USE, 
 
 AND METHOD OF ADJUSTING 
 
 IIADLEY's QUADRANT.* 
 
 " At the appointed time, when it pleased the 
 Supreme Dispenser ot' every {^ood gift to restore 
 light to a bewildered world, and more particularly 
 to manifest his wisdom in the simplicity, as well 
 as in the grandeur of his works, he opened the 
 glorious scene with the revival of sound astro- 
 nomy." This observ^ation of an excellent philo- 
 sopher and physician -|- is verified in every instance 
 of the progress of science; in each of which we 
 may trace some of the steps of that vast plan of 
 Divine Providence to which all things arc con- 
 verging, namely, the bringing all his creatures to 
 a state of truth, goodness, and consequent happi- 
 ness; an end worthy of the best and wisest of 
 beings, and which we may perceive to be gradu- 
 ally effecting, by the advancement of knovvledge, 
 the diffusion of liberty, and the removal of error, 
 that truth and virtue may at last shine forth in all 
 the beauty of their native colours. 
 
 It is thus that the discovery of the compass gave 
 rise to the present art of navigation; and when 
 this art grew of more importance to mankind, 
 Divine Providence blessed them w^ith the invention 
 of Hadleys quadrant, and in our own day and our 
 own time has further improved both it and the art 
 
 * This account of Hadley's quadrant, &c. is extracted from, 
 a small tract I published thereon sometime since, 8vo. 
 
 f Sir John Pringle's Six Discourses to the Royal Society,
 
 254 iiadley's quadrant. 
 
 of navigation, by the present method of finding 
 the longitude, which enables the mariner to ascer- 
 tain with certainty his situation on the unvaried 
 face of the ocean. 
 
 Hadley's quadrant or sextant is the only known 
 instrument, on which the mariner can depend for 
 determining with accuracy and precision his lati- 
 tude and longitude. It is to the use of this instru- 
 ment that navigation is indebted for the very great 
 and rapid advances it has made within these few 
 years. It is easy to manage, and of extensive use, 
 requiring no peculiar steadiness of hand, nor any 
 such fixed basis as is necessary to other astrono- 
 mical instruments. It is not the science of navi- 
 gation only which is indebted to this instrument, 
 but its uses are so extensive in astronomy, that it 
 may, accompanied with an artificial horizon, with 
 propriety be called 2i portable ohservafory, and in this 
 work we shall exemplify its application to surveying. 
 
 Mankind are ever desirous of knowing to whom 
 they are indebted for any peculiar or useful disco- 
 very; it. is the tribute of gratitude, and a reward 
 to merit. In the present instance there is no dif- 
 ficulty in giving the information; the respective 
 claims of the inventors arc easily decided. The 
 first thought originated with the celebrated Dr. 
 Hooke, it was completed by Sir Isaac Newton, and 
 published by Mr. Had ley. 
 
 Notwithstanding, however, the manifest supe- 
 riority of this instrument over those that were in. 
 use at the time of its publication, it was many 
 years before the sailors could be persuaded to 
 adopt it, and lay aside their imperfect and inaccu- 
 rate instruments: so great is the difficulty to re- 
 move prejudice, and emancipate the mind from 
 the slavery of opinion.
 
 PROPERTIES OF HADLEy's QUADRANT. 255 
 
 No instrument has undergone, since the origi- 
 nal invention, more changes than the quadrant of 
 Hadley; of the various aherations, many have had 
 no better foundation than the conceit and caprice 
 of the makers, who by these attempts have often 
 rendered the instrument more complicated in con- 
 struction and more difficult in use, than it was in 
 its original state. 
 
 ESSEXTIAL PROPERTIES OF HADLEy's 
 Q.UADRANT. 
 
 It is not my intention under this head to enu- 
 merate all the advantages of this instrument; but 
 barely to point out one or two of those essential 
 properties which distinguish it from every other 
 instrument of the kind, and rank it among one of 
 the greatest improvements in the practice of navi- 
 gation. 
 
 It is an essential property of this instrument, 
 derived from the laws of reflection, that half de- 
 grees on the arc answer to whole ones in the an- 
 gles measured: hence an octant, or the eighth 
 part of a circle, or 45 degrees on the arc, serves to 
 measure QO degrees ; and sextants will measure an 
 angular distance of 120 degrees, though the arc of 
 the instrument is no more than 6o degrees.* It 
 is from this property that foreigners term that in- 
 strument an octant, which we usually call a qua- 
 drant, and which in etlect it is. This property 
 reduces indeed considerably the bulk of the in- 
 strument ; but at the same time it calls for the 
 utmost accuracy in the divisions, as every en'or 
 on the arc is doubled in the observation. 
 
 * For a concise explanation of the theory, Sec, see my pam- 
 phlet on this instrumtnt. 8\-o. Kdit.
 
 136 DESCRIPTION OP 
 
 Another essential, and indeed an invaluable pra* 
 perty of this instrument, whereby it is rendered 
 peculiarly advantageous in marine observations, is, 
 that it does not require any peculiar steadiness ot 
 the hand, nor is liable to be disturbed by the 
 ship's motion ; for, provided the mariner can see 
 distinctly the two objects in the field of his in- 
 strument, no motion nor vacillation of the ship 
 will hinder his observation. 
 
 Thirdly, the errors to which it is liable are easily 
 discovered, and readily rectified, while the appli- 
 cation and use of it is facile and plain. 
 
 The principal requisites in a good sextant or 
 quadrant, are, 1. That it be strong, and* so con- 
 structed as not to bend across the plain. 2. That 
 it be accurately divided. 3. That the surfaces of 
 the glasses be perfectly plain and parallel to each 
 other. 4. That the index turn upon a long axis. 
 5. That the motion be free and easy in every part, 
 and yet without the least shake or jerk. 
 
 DESCRIPTION OF HADLEy's QUADRANT. 
 
 Fjg. I, ^late 19, represents a quadrant, or oc- 
 tant, of the common construction. The following 
 parts are those which require the particular atten- 
 tion of the observer. 
 
 I. BC the arc. 
 
 II. A D the index, a b the nonius scale. 
 
 III. E the index-glass. 
 
 IV. F the fore horizon-Q-lass. 
 
 V. G the back horizon-fflass. 
 
 o 
 
 VI. K the dark glasses or screens. 
 
 VII. H I the vanes or sights. 
 
 VIII. The arcBC is called the //"/«/' or qtiadrantal 
 
 arc, the arc cd lying from o, towards 
 the right is called the arc of excess.
 
 [ 257 ] 
 
 OF THE QUADRANT. 
 
 The quadrant consists of an arc B C^ firmly at- 
 tached to two radii, or bars, AB, AC, which are 
 strengthened and bound together by the two 
 braces LM. 
 
 Of the mdex. The index D is a flat tar of 
 brass, that turns on the center of the octant; at 
 the lower end of the index there is an oblong 
 opening, to one side of this opening a nonius scale 
 is fixed to subdivide the divisions of the arc ; at 
 the bottom or end of the index there is a piece of 
 brass, which bends under the arc, carrying a 
 spring to make the nonius scale lie close to the 
 divisions; it is also furnished with a screw to fix 
 the index in any desired position. 
 
 The best instruments have an adjusting screw 
 fitted to the index, that it may be moved more 
 slowly, and with greater regularity and accuracy 
 than by the hand. It is proper however to ob- 
 serve, that the index must be previously fixed 
 near its right position by the abov^e-mentioned 
 screw, before the adjusting screw is put in motion. 
 SeeBC,7%-. 4. 
 
 The circular arcs on the arc of the quadrant 
 are drawn from the center on which the index 
 turns: the smallest excentricity in the axis of the 
 index would be productiv^e of considerable errors. 
 
 The position of the index on the arc, after an ob- 
 servation, points out the number of degrees and 
 minutes contained in the 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 
 
 s
 
 258 DESCEIPTION' OF 
 
 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 
 either of the iwo horizon glasses F and G, accord- 
 ing to the nature of the observation. 
 
 The brass frame with the glass is fixed to the 
 index 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 advantages; but if the glasses be well se- 
 lected, there is little danger to be apprehended of 
 error, from a want of parallelism ; more is to be 
 feared from the surfaces not being flat. 
 
 Of the horizon glasses F, G. On the radius 
 AB of the octant, are two small speculums. The 
 surface of the upper one is parallel to the index 
 glass, when the counting division of the index is at 
 o on the arc; but the surface of the lower one is 
 perpendicular to the index glass, when the index is 
 at o degrees on the arc: these mirrors receive the 
 reflected rays from the object, and transmit them 
 to the obsei-ver. 
 
 The horizon glasses are not entirely quicksil- 
 vered; the upper one F, 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 unsiivered part of the glass. 
 The (^i\gG, of the foil of this glass is nearly parallel 
 to the plane of the instrument, and ought to b^ 
 very sharp, and without a flaw.
 
 Ihadley's quadrant. Ibg 
 
 The other horizon glass G is silvered at both endfi; 
 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 brass 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 turned a few degrees on its axis, in order to set 
 it parallel to the index glass. The lever has a 
 contrivance to turn it slowly, and a button to fix 
 it. To set the glasses perpendicular to the plane 
 of the quadrant, there are two sunk screws, one 
 before and one behind 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 mirror, may be altered, and 
 thus be set perpendicular to the plane of the in- 
 strument. For the lever, &c. see Jig. 13. 
 
 Of the shades^ or dark glasses^ K. There are t\A'o 
 red or dark glasses, and one green one; they are 
 used to prevent the bright rays of the sun, or the 
 glare of the moon, from hurting the eye at the 
 time of observation. They are each of them set 
 in a brass frame, which turns on a center, so that 
 they may be used separately, or together, as the 
 brightness of the sun may require. The green 
 glass 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 
 from a fixed star. 
 
 When these glasses are used for the fore obser- 
 vation, they are fixed as atK in/^. 1 ; when used 
 for the back obsei-vation, they are removed to N. 
 
 Of the tivo 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. 
 
 s 2
 
 Q,60 DESCRIPTION OF 
 
 That which 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 middle of the transparent part of the 
 mirror, for those objects which are bright enough 
 to be reflected from the unsilvered part of the 
 mirror. 
 
 Of the divisions o?i the limh of the quadrant, and of 
 the nonius on the index. For a description of these 
 divisions, see page 127- 
 
 Directio7is to hold the instrument. It is recom- 
 mended to support the weight of the instrument 
 by the right hand, and reserve the left to govern 
 the index. Place the thumb of the right hand 
 against the edge of the quadrant, under the swel- 
 ling part on which the fore sight I stands, extend- 
 ing the fingers across the back of the quadrant, 
 so as to lay hold on the opposite edge, placing the 
 fore finger above, and the other fingers below the 
 swelling part, or near the fore horizon glass; thus 
 you may support the instrument conveniently, in 
 a vertical position, by the right hand only; by 
 resting the thumb of the left hand against the side, 
 or the fingers against the middle bar, you may 
 move the index gradually either way. 
 
 In the back observation, the instrument should 
 be supported by the left band, and the index be 
 governed by the right. 
 
 Of the axis of vision, or line of sight. Of the two 
 objects which are made 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 horizon glass to the eye. This 
 ray is called the visual ray ; but when it is consi- 
 dered merely as a line drawn from the middlQ of
 
 HADLEy's QUADRA.NT. 26 1 
 
 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 obser- 
 vation is made, a circumstance absolutely neces- 
 sary, 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 eye glass tube may be brought paral- 
 lel to the plane of the quadrant, so that by bring- 
 ing the object to the middle between them, you 
 are certain of having the axis of vision parallel to 
 the plane of the quadrant. 
 
 OF THE NECESSARY ADJUSTMENTS. 
 
 It is a peculiar excellence of Hadlcy's quadrant, 
 that the errors to which it is liable are easily de- 
 tected, and soon rectified; the obseryer may, 
 therefore, if he will be attentive, always put his 
 instrument in a state fit for accurate observation. 
 The importance of this instrument to navigation 
 is self-evident; yet much of this importance de- 
 pends on the accuracy with which it is made, and 
 the necessary attention of the observer; and one 
 would hardly think it possible that any observer 
 would, to save the trifling sum of one or two gui- 
 neas, prefer an imperfect instrument to one that 
 was rightly constructed and accurately made; or 
 that any consideration should induce him to neg- 
 iect the adjustments of an instrument, on whose
 
 '^6'2 ADJUSTMENTS OP 
 
 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; 
 active about trifles, indolent where his v/elfare and 
 happiness are concerned. The adjustments for 
 the fore observation are, 1. To set the fore horizon 
 glass parallel to the index glass; this adjustment 
 is of the utmost importance, and should alv^^ays be 
 made previous to actual observation. The second 
 is, to see that the plane of this glass is perpendi- 
 cular to the plane of the quadrant. 3. To see 
 that the index glass is perpendicular to the plane 
 of the instrument. 
 
 To adjust the 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 INSTRUMENT. It is SO 
 
 to place the horizon glass, that the index may shew 
 upon the arc the true angle between the objects : 
 for this purpose, set the index line of th'6" nonius 
 exactly against o on the limb, and fix it there by 
 the screw at the under 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 reflection 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 be adjusted, and the horizon glass to be 
 parallel to the index glass. But if the horizons 
 do not coincide, then loosen the milled nut on 
 the under side of the quadrant, and turn the ho- 
 rizon glass on its axis, by means of the adjusting 
 lever, till you have made them perfectly coincide; 
 then fix the lever firmly in the situation thu^
 
 HADLEy's QUADr.AXT. 263 
 
 obtained, by tightening the milled nut. This 
 adjustment ought to be repeated before and after 
 every astronomical observation. 
 
 So important is this rectification, that experi- 
 enced observers, and those who are desirous of 
 being very accurate, will not be content with the 
 preceding mode of adjustment, but adopt another 
 method, which is usually called finding the index 
 error; a method preferable to the foregoing both 
 for ease and accuracy. 
 
 7(9 find the index error. Instead of fixing the 
 index at o, and moving the horizon glass, till the 
 image of a 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 observe whether the index division 
 on the nonius agrees with the o line on the arc; 
 if it does not, the number by which they differ 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 90 
 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 way, or, which 
 is the same thing, you may read them off in the 
 usual way, and then their complement to 2,0 min. 
 will be the real number of degrees and minutQS to 
 be added to the observation. ' •'jt 
 
 * This adjustment may be made more accurately, or the 
 error better found, by using the sun instead of the horizon 3 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 instru- 
 ments, and this method of adjustment is explained in the follow- 
 icg description of the sextant.
 
 2^4 ADJUSTMEK'TS OP 
 
 To make the index glass and fore horizon glass 
 perpendicular to the plane of the instrument . Though 
 these adjustments are neither so necessary nor 
 important- as the preceding one; yet, as after being 
 once performed, they do not require to be repeated 
 for a considerable time, and as they add to the 
 accuracy of observation, they ought not to be 
 neglected; and further, a knowledge of them ena- 
 bles the mariner to examine and form a proper 
 judgment of his instrument. 
 
 To adjust the index glass. This adjustment con- 
 sists in setting the plane of the index glass perpen- 
 dicular to that of the instrument. 
 
 Method 1. By means of the two adjusting 
 tools, 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- 
 mentioned tools near one end of the arc, the other 
 at the opposite end, the side with the lines towards 
 the index glass ; then look down the index glass, 
 directing the sight parallel to the plane of the in- 
 strument, you will 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 rightly fixed ; if 1iot, it must be restored to its 
 proper situation by loosening the screw c, and 
 tightening the screw d; or, ^vice versa, by tighten- 
 ing 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 sec the arc of the 
 quadrant by direct vicWp and by reflection at tbc
 
 hadley's quadrant. 265 
 
 same time. If they join in one direct line, and 
 the arc seen by reflection forms an exact plane 
 with the arc seen by direct view, the glass is per- 
 pendicular to the plane of the quadrant; if not, 
 the error must be rectified by altering the posi- 
 tion of the screws behind the frame, as directed 
 above. 
 
 To ascertain whether the fore horizon glass he 
 'perpendicular to the plane of the instrument. Having 
 adjusted 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 image of 
 the horizon at the edge of the silvered part coin- 
 cide with the object seen directly, the glass is 
 perpendicular to the plane of the instrument. If 
 it fall above or below, it must be adjusted. If the 
 image seen by reflection be higher than the object 
 itself seen directly, release the fore screw and 
 tighten the back screw; and, 'v'lce 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 
 moving the index you can make the image and 
 the object 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 con- 
 tinues to be seen in both parts of the glass. If, 
 when the instrument is thus inclined, the edge oi 
 the sea continues to form one unbroken line, the 
 quadrant is perfectly adjusted; but if the reflected 
 horizon be separated from that seen by direct 
 vision,' the speculum is not perpendicular to the 
 plane of the quadrant. And if the observer is in-
 
 ^66 ADjrSTMENTS OP 
 
 clined to the right, with tlie face of the quadrant 
 upwards, and the reflected sea appears higher than 
 ,the real sea, you must slacken the screw before 
 the horizon glass, and tighten that which is be- 
 hind it; but if the reflected sea appears 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 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 quadrant may be 
 held in a vertical position; if the image seen by 
 reflection appears to the right or left of the object 
 seen directly, then the glass must be adjusted as 
 before by the two screws. 
 
 OF THE ADJUSTMENTS FOR THE BACK 
 OBSERVATION. 
 
 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 di- 
 rectly against the o divison of the arc. 
 
 The method, therefore, of adjusting it consists 
 in measuring the distance of two objects nearly 
 1 80 degrees 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.
 
 hadley's quadrant. 267 
 
 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 
 opposite edge of the sea will appear by reflection 
 inverted, or upside down. By moving the lever 
 of the axis, if necessary, the edges may be made 
 to coincide, and the quadrant is adjusted. 
 
 There is but one position in which the quadrant 
 can be held with the limb downwards, without 
 causing 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 up- 
 right, observe which part of the reflected horizon 
 is lowest. 
 
 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 con-r 
 trary must be performed: this adjustment is, 
 however, of much less importance than the pre^ 
 ceding, as it does not so much aflcct the angle 
 measured. 
 
 INCONVENIENCIES AND IXACCURACIES OF THE 
 BACK OBSERVATION. 
 
 The occasions on w^hich 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 adjacent shores; or when the angular distance 
 between the moon and sun, or a star, exceeds 90 
 
 * This is according to the height of the obseiver's eye above 
 fjiesee. Sec liobcrimii Navi^i'iio'i.
 
 268 DEFECTS IN BACK OBSERVATIONS. 
 
 degrees, and is required to be measured for obtain- 
 ing the longitude 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 dis- 
 tance, without having recourse to observation: 
 and again, if the distance of the land in miles ex- 
 ceed 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 obser- 
 vation may be vised; and lastly, if the land be too 
 near to use the fore observation, its extreme points 
 will in general be so far asunder, as to prevent the 
 adjustment, by taking away the back horizon. In 
 the case of measuring the angular distances of the 
 heavenly bodies, the very great accuracy required 
 in these observations, makes it a matter of impor- 
 tance that the adjustments should be well made, 
 and frequently examined into. But the quantity 
 of the dip is varied by the pitching and rolling of 
 the ship; and this variation, which is perceptible 
 in the measuring altitude by the fore observation, 
 is doubled in the adjustment for the back observa- 
 tion, and amounts to several minutes. It is like- 
 wise exceeding difficult, in a ship under way, to 
 hold the quadrant for any length of time, so that 
 the two horizons 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 adjustment uncertain ; the fore obser- 
 vation is subject to none of these inconveni- 
 encies,* 
 
 * Nicholson's Navigator's Assistant.
 
 [ 269 ] 
 
 DESCRIPTION, USE, 
 AND METHOD OF ADJUSTING 
 
 HADLEY's SEXTANT. 
 
 As the taking tlie angular distances of the 
 moon and the sun, or a star, is one of the best and 
 most accurate methods of discovering the longi- 
 tude, it was necessary to enlarge the arc of the 
 octant to the sixth part of a circle; but as the ob- 
 servations for determining the longitude must be 
 made with the utmost accuracy, the framing of the 
 instrument was also altered, that it might be 
 rendered more adequate to the solution of this 
 important problem. Hence arose the present 
 construction of the sextant, in the description of 
 which, it is presumed that the foregoing pages have 
 been read, as otherwise we should be obliged to 
 repeat the same observations. 
 
 Sextants are mostly executed (some trifling 
 variations excepted) on two plans ; in the one, all 
 the adjustments are left to the observer: he has it 
 in his power to examine and rectify every part of 
 his instrument. This mode is founded on this 
 general principle, that the parts of no instrument 
 can be so fixed 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 ad- 
 justment, or that of the horizon glass for the 
 index error, is rejected; and this rejection i$ 
 grounded upon, two reasons ;
 
 270 T5ESCRIPTI0N OP 
 
 l.That from the nature of the adjustment it 
 frequently happens that a sextant will alter even 
 during the time of an observation, without any 
 apparent eause whatever, or without being sensi- 
 ble at what period of the observation such altera- 
 tion took plaee, and consequently the observer is 
 unable to allow for the error of adjustment. 
 
 2. That this adjustment is not in itself sufh- 
 ciently exact, it being impossible to adjust a sex- 
 tant 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 accu- 
 rate observers have always directed the index error 
 to be found and allowed for, which renders the 
 adjustment of the horizon 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 expe- 
 rience: many sextants, whose indices had been 
 determined previous to their being carried out to 
 India, have been found to remain the same at their 
 return. 
 
 Notwithstanding the probable certainty of the 
 horizon glass remaining permanently in its situa- 
 tion, the observer ought from time to time to 
 examine the index error of his instrument, to sec 
 if it remains the same; or make the proper allow- 
 ances for it, if any alteration should have taken 
 place. 
 
 One material point in the formation af a sextant 
 is so to construct it, that it may support its own 
 weight, and not be liable to bend by any inclina- 
 tion of the plane of the instrument, as every flexure 
 would alter the relative position of th^ mirrors, on 
 which the determi-nation of an angle depends.
 
 hadley's sextant. tJ7f 
 
 Besides the errors now mentioned, to which the 
 sextant or quadrant, &c. is Uable, 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 
 center. 
 
 To obviate this error, let the observer be care- 
 ful alwajs 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 indiffe- 
 rent; but as the common practice in observing is 
 to finish the observations by a motion of the index 
 in that direction which increases the angle, that 
 is, in the fore observ^ation 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 degree at least in the required direc- 
 tion. 
 
 DESCRIPTION OF THE SEXTANT. 
 
 Fig. 4, j^late IQ, represents the sextant so 
 framed, as not to be liable to bend. The arc A A, 
 is divided into 120°, each degree is divided into 
 three parts, of course equal to 20 minutes, which 
 are again subdivided by the nonius into every half 
 minute, or 30 seconds: see the nature of the 
 nonius, and the general rule for estimating the 
 value thereof, in the preceding part of these Essays. 
 Every second division or minute on the nonius, is
 
 272 DESCRIPTION OP 
 
 cut longer than the intermediate 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 towards the 
 right hand being to be eonsidered as the index 
 division. 
 
 In order to observe Avith aceuracy, 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 screw C. Care should be taken not to 
 force the adjusting screw when it arrives at either 
 extremity of its adjustment. When the index is 
 to be moved any considerable quantity, the screw 
 C at the back of the sextant must be loosened; 
 but when the index is brought nearly to the di- 
 vision required,this back screwshouldbe tightened, 
 and then the index may be moved gradually by 
 the adjusting screw. iV small shade is sometimes 
 fixed to that part of the index where the nonius is 
 divided, this being covered with white paper, re- 
 fleets the light strongly upon the divisions. 
 
 There are four tinged glasses at D, each of which 
 is set in a separate frame turning on a center: they 
 are used to screen and save the eye from the 
 brightness 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 E, to weaken the rays of the 
 sun or moon, when they are viewed directly 
 through the horizon glass. The paler glass is 
 sometimes used in observing altitudes at sea, to 
 take off the strong glare of the horizon. 
 
 The frame of the index glass I, is firmly fixed 
 by a strong cock to the center plate of the index.
 
 HADLEY S SEXTANT. 2^3 
 
 rhc horizon glass F, is fixed in a frame that turns 
 on the axes or pivots, which move in an exterior 
 frame: the holes in which the pivots move may be 
 tightejied by four screws in the exterior frame ; G 
 is a screw by which the horizon glass may be set 
 perpendicular to the plane of the instrument ; 
 should this screw become loose, or move too easv, 
 it may be easily tightened 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, ^f^. 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 
 natural position ; the longer one, jig. (J, shews 
 them inverted. It has a large field of view, and 
 other advantages; and a little use will soon 
 accustom the observer 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 is more perfectly distin^ 
 guished; and by the place of the images in the 
 field of the telescope, it is easy to perceive whether 
 the sextant is held in the proper plane for observa- 
 tion. 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. 
 
 The telescopes arc 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 sextant. The exterior ring is fixed 
 
 T
 
 ^74 DESCRIPTION OP 
 
 on a triangular brass stem that slides in a socket, 
 and by means of a screw at the back of the sex- 
 tant, may be raised or lowered so as to move the 
 center of the telescope to point to that part of the 
 horizon glass which shall be judged the most fit 
 for observation. Fig. 8, is a circular head, with 
 tinged glasses to screw on the eye end of either of 
 the telescopes, or the plain tube. The glasses 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, or any of the tinged 
 glasses, may be brought between the eye glass of 
 the telescope and the eye. 
 
 Fig. g, a small screw driver. Fig. 10, a mag- 
 nifying glass, to read oflfthe divisions by. 
 
 To find the index error of the sextant. To find 
 the index error, is, in other words, to shew what 
 immber of degrees and minutes' is indicated by 
 the nonius, \vhen the direct and rcfiected images 
 of an object coincide with each other. 
 
 The most general and most certain method of 
 asccrtainino- this error, is to measure the diameter 
 of the sun, by bringing the limb of its image to 
 coincide with the limb of the sun itself seen di- 
 rectly, hoth on the quadrantal arc, and on the arc 
 of excess. 
 
 If the diameter taken by moving the index for- 
 ward on the quadrantal arc be greater than that 
 taken on the arc of excess, then half the difilTcnce 
 is to be snbtracted\ but if the diameter taken on 
 the arc of excess be greater than that by the qua- 
 drantal arc, half the difference is to be added. If 
 the numbers shewn on the arc be the same in both 
 cases, the glasses arc truly parallel, and there is no
 
 HADLEY S SEXTAXT. ^75 
 
 index error; but if the numbers be different^ then 
 half the differenec is the index error.* 
 
 It is however to be observed, that when theindex 
 is on the arc of excess, or to the right of o, the 
 complement 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, whiclf 
 will give the index error to very great exactness. 
 
 Example. 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 complement of this to 20 is 15, which, added 
 to 20, gives 35 minutes; and, secondly, that the 
 number shewn by the index, when on the left of 
 zero, and the opposite limbs are in contact, be 20 
 minutes, and by the nonius 9' 30", which makes 
 together 29' 30"; this subtracted from 35' gives 
 5' 30", which -divided by 2, gives I' Ao" 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 
 timeof 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-defined 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 
 
 * In other words, the difference between the degree and mi- 
 nute shewn by the index : first, when the lower retiected limb of 
 the sun is exactly in contact with the upper limb of the sun; 
 and secondly, when the upper edge of the image is in contact 
 with the lower edge of the object, divided by 2, will be the index 
 frror. 
 
 T 2
 
 2/6 DESCRIPTION OF 
 
 sextant, by turning the screw, and the images will 
 pass exactly over each other, and the mirror will 
 then be adjusted in this direction. 
 
 To set the axis of the telescope parallel to the pla7ic 
 of the sextimt. We have already observed, that in 
 measuring angular distances, the line of sight, or 
 plane of oLssrvation, should be parallel to the plane 
 of the instrum-ent, 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 telescope is made use of, in whose field 
 there are placed two wires parallel to each other, 
 and equidistant from the center. These 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 certain 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 ob- 
 jects, as the sun and moon, or the moon and a 
 j^tar, whose distance should, for this purpose, ex- 
 ceed QO degrees; the distance of the sun and 
 moon is to be taken great, because the same de- 
 viation 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, exactly in 
 contact with that wire which is nearest to the plane 
 of the sextant; fix the index there; then, by al- 
 tering a little the position of your instrument, 
 make the images appear on the wire furthest from 
 the sextant. If the nearest limbs be now precisely 
 in contact as they were before, then the axis of 
 the telescope is in its right situation. But if the
 
 hadley's sextaxt. 277 
 
 limbs of the two objects appear to separate at the 
 wire that is furthest from the plane of the instru- 
 ment, it shews that the objeet end of the telescope 
 incHnes towards the plane of the instrument, and 
 must be rectified by tightening the screw nearest 
 the sextant, having previously unturned the screw 
 furthest from it. If the images overlap each 
 other at the wire furthest from the sextant, the 
 object end of 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 left: 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 in- 
 strument. 
 
 To examine the glasses of a sexta?it, 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 appears singly, and well defined about the 
 edges, it is a proof that the surflices 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 specukim, particularly 
 if that image be the sun, be viewed through a small 
 telescope, the examination will be more perfect. 
 
 2. To find whether the surface of a reflecting 
 glass be plane. Chusetwo distant objects, nearly 
 on a level with each other; hold the instrument 
 in an horizontal position, view the left hand object 
 directly through the transparent part of the hori- 
 zon glass, and move tlie index till the reflected 
 ijnage of the other is seen below it in the silvered
 
 278 DESCRIPTION OP 
 
 part; make the two images unite just at the line 
 of separation, then turn the instrument round 
 slowly on its own plane, so as to make the united 
 images move along the line of separation of the 
 horizon glass. If the images continue united 
 without receding from each other, or varying their 
 respective position, the reflecting surface 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 dark- 
 ening glass are parallel and perfectly plane. It is 
 diffiicult, nay, almost impossible, to procure the 
 shades perfectly parallel and good; they will 
 therefore, according to their different combina- 
 tions, give difl'erent 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 removed; then take away the smoaked 
 glass, and view the sun through each shade and 
 the several combinations 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 ad- 
 justment, there will be no error in the observed 
 angle,* 
 
 * The dark glasses are generally left to turn in their cells ; 50 
 that, alter one observation has been made, turning either of them 
 half round, after another observation^ half the ditFerence thus 
 given is the error. Edit.
 
 [ 279 J 
 
 OF HADLEY S SEXTANT, AS USED IN' 
 SURVEYING. 
 
 No instrument can be so conveniently used for 
 taking anc^lcs in maritime surveying as Hadley's 
 sextant. It is used with equal facility at the mast 
 head, as upon deck, by which its sphere of obser- 
 vation 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 instrument. But by thi-, angles may 
 be taken at the mast head from the one visible 
 object with great exactness; and further, taking 
 angles from heights, as hills, or a ship's mast 
 head, is almost the only way of exactly describing 
 the figure and extent of the shoals. 
 
 It has been objected to the use of Hadley's sex- 
 tant for surveying, that it docs not measure the 
 horizontal angles, by which alone a plan can be 
 laid down. This observation, however true in 
 theory, may be obviated in practice by a little 
 caution. 
 
 If an angle be measured between an object on 
 an elevation, and another near to it in a holloA\ . 
 the difference between the base, which is the hori- 
 zontal angle, and the hypothenuse, which is the 
 angle obsen'cd, mav be very great; but if these 
 objects are measured, not from eacli other, but 
 from some very distant object, the difi'erence be- 
 tween the angles of each fVom the very distant 
 object, will be very near the same as the horizon- 
 tal angle. This may be still further corrected, by 
 measuring the angle not between an object on a 
 plane and an object on an elevation, but between 
 the object on a plane and some object in the same 
 direction as the elevated objectj of which the eye 
 is suliicicntly able to judge.
 
 280 OF hadley's sextaxt, 
 
 How to ohsewc the hor'i%oni al angle^ or angular 
 distance^ hetiveen Hvo 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 deg. 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 object, 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, and by the inclination of the plane of the 
 sextant when necessary, till they unite as one, or 
 appear to join in one vertical line in the middle of 
 the line ^^hich divides the transparent and reflect- 
 ing 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 objects subtend at the naked 
 eye. This angle is always double the inclination 
 of the planes of the two reflecting 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 subtended by 
 them at the eye will be twice that number of de- 
 grees 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 120 
 degrees, as before observed. 
 
 The angle found in this manner between two 
 objects that are near the observer, is not precise; 
 and may be reckoned exact only when the objects 
 ure above half a mile off. For, to get the angle 
 truly exact, the objects should bevicwedfroni th$
 
 A3 USED IX SURVEYING. 281 
 
 center of the index glass, and not where the sight 
 vane is plaeed; therefore, except the objects are so 
 remote, that the distance between the index glass 
 and sight vane vanishes, or is as nothing compared 
 to it, the angle will not be quite exact. This in- 
 accuracy 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 occasion 
 to regard it; but if it shall happen that great ac- 
 curacy 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, ob- 
 serve the angle between near objects, when the 
 sextant has been first properly adjusted by a dis- 
 tant 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 be 
 the true angle between them. If one of the ob- 
 jects is near, and the other distant, and no re- 
 mote 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 found. 
 
 Example. To measure the horizontal angle 
 K^Q,Jig. \Qf, plate 9, with the sextant. 
 
 1. Set up such marks at A and C, as mav be 
 seen when you are standing at B. 2. Set o on 
 the index to coincide \\ithoon the quadrantal 
 arc. 3. Hold the sextant in an horizontal posi- 
 tion, look through the sight and horizon glass at 
 the mark A, and observe whether the image is di- 
 rectly under the object; if not, move the index 
 till they coincide. 4. if the index division of the
 
 1S1 OF hadley's sextant. 
 
 nonius is on the arc of excess, the indicated quan- 
 tity is to be added to the observed angle ; but if it 
 be on the quadrantal arc, the Quantity 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 
 index till they unite as one, or appear in one ver- 
 tical line, and the index will shew the angle sub- 
 tended 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 subdivided by marks, and then 
 measured. 
 
 No instrument can be more convenient or ex- 
 peditious than the sextant, for setting of offsets. 
 Adjust the instrument, and set the index to 90 de- 
 grees; walk along the station line with the octant 
 in your 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 
 offset 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 point in the boundary 
 through which that offset passes; on the other 
 hand, if you wish an ofl'set 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 octant, 
 and that will be the point where an offset from the 
 proposed point or bend will fall. 
 
 The manner of using this instrument for the 
 solution of those astronomical problems that are 
 necessary in surveying, will be shewn in its proper 
 place.
 
 [283 ] 
 
 DESCRIPTION OP A NEW I'OCKET BOX SEX- 
 TANT, AND AN ARTIFICIAL HORIZOX, BY 
 THE EDITOR. 
 
 Fig. 1 1 , plate 1 9, is a representation of a very con- 
 venient pocket sextant, and contains a material im- 
 provement on the retiecting cross staff" before de- 
 scribed, ?,Q.Qjig.A,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. AC 
 is the index turning an index glass within the box. 
 a, a, arc 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 
 through an hole in the side of the box about D, 
 upon and through the horizon glass and the se- 
 cond 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 before the sight hole, not 
 shewn in the figure; by pushing the pin at h^ a 
 dark screen for the sun is interposed between the 
 index and horizon glasses. I is an endless screw, 
 sometimes applied to give a very accurate motion, 
 like the tangent screw to the index of a sextant. 
 Or a racked arc and pinion may be applied at 
 about c, d, which I think in some respects better. 
 
 The following table is sometimes engraved upon 
 the cover that goes over the box when shut up. 
 By the sextant being set to any of the angles con- 
 tained in this table, an height or distance of acces- 
 sible or inaccessible objects is obtained in a very 
 simple and expeditious manner.
 
 128* 
 
 TABLE USED WITH THE 
 
 Mul. 
 
 Angle. 
 
 Angle. 
 
 Div. 
 
 1 
 
 4.5° 00' 
 
 45° 00' 
 
 1 
 
 2 
 
 63 26 
 
 26 34 
 
 2 
 
 3 
 
 71 34 
 
 18 26 
 
 o 
 
 4 
 
 75 53 
 
 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 glasses to coincide with the mark; then, if 
 the angle be greater than 45°, multiply the dis- 
 tance by the corresponding tigure 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 backwards and forwards, 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 index to any of the lesser 
 angles, and go backwards in a line from the ob- 
 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 th^ 
 distance^ multiply the height of the object by the 
 numbers against either of the angles made use of,
 
 NEW POCKET BOX SEXTANT, &C. 285 
 
 and the product will be the distance of the object 
 from the place where such angle was used. 
 
 If the index is set at 45° the distance is th-e 
 height of the object, and vice versa. The index 
 set to 90° becomes a reflecting cross staff, and is 
 used according to the directions in page 282. 
 
 The sextants, as before described by the author, 
 of the best kind, are made of brass, or other me • 
 tal. 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 horizon, can only be obtained by an ar- 
 tificial one. There have been a variety of these, 
 sort of instruments made, but the kind now to be 
 described is allowed to be the only one that can 
 be depended upon. Fig. 12, plate 19, represent 
 the horizon fixed up for use. A is a wood or 
 metal framed roof containing two true parallel 
 glasses of about 5 by 3 2 inches, 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 sextant, the reflected image of the 
 sun, moon, or other object, is brought to coii>- 
 cide with the same object reflected from the 
 glasses of the sextant; half the angle shewn upon 
 the limb is the altitude above the horizon or 
 level required. It is necessary in a set of obser- 
 vations that the roof be always placed th^ same
 
 285 TO SURVEY WITH 
 
 way. When done with, the roof folds up fiat- 
 ways, and, with the quicksilver in a bottle, &c. 
 is packed into a portable flat case. 
 
 TO SURVEY WITH THE CHAIN ONLY. 
 
 The difficulties that occur in measuring with 
 accuracy a strait line, render this method of sur- 
 veying altogether insufficient for measuring a 
 piece of ground of any extent; it would be not 
 only extremely 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 quadrant, 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 instrument, as 
 the surveying cross and optical square, to their 
 aid. Little more need be said, as it is evident, 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 pro- 
 per for level ground and small inclosures ; and that 
 even then, it is better to go round the field and 
 measure the angles thereof, taking offsets from 
 the station lines to the fences. That this work 
 may not be deemed imperfect, we shall introduce 
 an example or two selected from some of the best 
 writers on the subject; observing, however, that 
 fields that are plotted from measured lines ^ are always 
 ■plotted nearest to the truth, ivhen those lines form at 
 their junction angles that approach nearly to a right 
 
 antrle. 
 
 Example 1. To survey the triangular field 
 AB Cjfig. 2,2, plate Q, by the chain and cross. Set 
 up marks at the corners, then begin at one of them, 
 and measure from A to B, till you imagine that
 
 THE CHAIN ONLY. 287 
 
 you are near the point D, where a perpendicular 
 would fall from the angle C, then letting the chain 
 lie in the line AB, 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 Uiove it backwards 
 and forwards on the line AB, 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 AD, 
 and complete the measure of AB, by measuring 
 from D to B, 11 .4 1 . Set down this measure, then 
 return to D, and measure the perpendicular D C, 
 6.43. Having obtained the base and perpendi- 
 cular, the area is readily found: it is on this prin- 
 ciple that irregular fields may be surveyed by the 
 chain and cross; the theodolite, or Hadleys sextant , 
 may even here be used to advantage for ascertain- 
 ing 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 2. To measure the four-sided figures, 
 AB C D, Jig. 34, plate 9. 
 
 AE 
 
 214 
 
 DE 210 
 
 AF 
 
 362 
 
 BF 306 
 
 AC 
 
 592 
 
 
 Measure either of the diagonals, as AC, and the 
 two perpendiculars DE, B F, as in the last pro- 
 blem, which gives you the above data for com- 
 pleting the figure. 
 
 Example 3. To survey the irregular field, fi^. 
 22, plate 13. Having set up marks or station 
 staves wherever it may be necessary, walk over
 
 283 OP SURVEYING BT 
 
 the ground, and consider how it can be most con-i 
 vcniently divided into triangles and trapeziums^ 
 and then measure them by tlic last two problems. 
 
 It is best to subdivide the field into as few sepa- 
 rate triangles as possible, but rather into trape - 
 ziums, by drawing diagonals from corner to corner, 
 so that the perpendicular may fall within the fi- 
 gure; thus the fi2:ure is divided into two trape- 
 ziums AB C G, G^D E F, and the triangle GCD. 
 Measure the dbgonal AC, and the two perpen- 
 diculars GM, BN, then the base GC,. and the 
 perpendicular Dq; lastly, the diagonal D F, and 
 the two perpendiculars, p E, O G, and you have 
 obtained sufficient for your purpose. 
 
 OF SURVEYING BY THE PLAIN TABLE. 
 
 We have already given our opinion of this in- 
 strument, and shewn how far only it can be de- 
 pended upon where accuracy is required; that 
 there are many cases where it may be used to ad- 
 vantage, there is no doubt; that it is an expedi- 
 tious mode of surveying, is allowed by all. I shall, 
 therefore, here lay down tlic general modes of sur- 
 veying with it, leaving it to the practitioner to 
 select those best adapted to his peculiar circum- 
 stances, recommending him to use the modes laid 
 down in example 3, in preference to others, where 
 they may be readily applied. Fie will also be a 
 better judge than I can be, of the advantages of 
 Mr. Break's method of using the plain table. 
 . Example 1 . To take by the plain table the plot 
 of a piece of land AB C D E, fg. 36, plate 9, at 
 one station near the middle, from whence all the 
 corners may be seen. 
 
 Let RTSV,j^^. 37,plafeg, represent the plain 
 table covered \\'ith a sheet of paper, on which the
 
 THE PLAIN TABLE. !28() 
 
 plan of the field, j^^. 36, is to be drawn; go round 
 the field and set up objects at all the corners 
 thereof, then put up and level your plain table, 
 turning it about till the south point of the needle 
 points to the N. point, or 3dO° 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 
 O, and direct the sight to the object at A, when 
 this is cut by the hair, draw a blank line along the 
 chamfered edge of the index from © towards A, 
 after this move the index round the point as a 
 center, 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 obscure lines by the edge of the 
 index to . 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 admit of, these measures on 
 their respective lines; join the points AB, B C, 
 CD, D E, EA, by lines for the boundaries of the 
 field, which, if the work be properly executed, 
 will be truly represented on the paper. 
 
 N. B. It is necessary, before the lines 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 Jig. 33, plate C), represent 
 the piece of ground to be surveyed from one sta- 
 tion point, whence all the angles may be seen, but 
 
 u
 
 igO OF SURVEYING BY 
 
 not so near the middle as in the foregoing instance; 
 go round the lield, and set up your objects at all 
 the corners, then plant the table where they may 
 all be conveniently 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 O, ^" coincides with //, and (■ 
 with I?; the table is planted thereon, making the 
 lengthway of the table correspond to that of the 
 lield. Make your point-hole and circle to repre- 
 >-cnt the place of the table on the land, and apply 
 tlie edge of the index thereto, so as to see through 
 the slit the mark at a cut by the hair; then with 
 your pointrcl draw a blank line from © towards a, 
 do the same by viewing through the sights the se- 
 veral marks c, d, e,f, g, keeping the edge of the 
 index always close to O, and drawing blank lines 
 from O towards each of these marks. 
 
 Find by a plumb-line the place on the ground 
 under O on the paper, and from this point mea- 
 sure 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 O , then set oft' 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 oft" their distances; then measure from © to d, 
 and from ^ to © ; and, lastly from © to f, and 
 .sett off" their distances. Then draw lines in ink 
 from each point thus found to the next for boun- 
 daries, and a line to cross the whole for a meridian 
 line. 
 
 Fig, v33, may be supposed to be two fields, and 
 the table to be planted in the N. E. angle of the 
 lower field, where the other angles of both fields 
 may be observed and measured to. 
 
 Example 3. To survey a field represented at 
 Jrg. 32, phjte g, by going round the same either
 
 THE PLAIN TABLE. 201 
 
 \vltliln or without, taking at the same thne offsets 
 to the boundaries; suppose inside. 
 
 Set up marks at a, b, c, d, at a small distance 
 from the hedges, but at those places which you 
 intend to make your station points. 
 
 Then, beginning at ©, plant your instrument 
 there, and having adjusted it, make' a fine point 
 O 1 on that part of the paper, where it will be 
 most probable to get the whole plan, if not too 
 large, in one sheet; place the index to O 1, and 
 direct the sight to the mark at 6, draw a blank 
 line from © 1 to O 6, then direct the index to the 
 tree near the middle of the field, and afterwards to 
 the mark at 2, then dig a hole in the ground 
 under 1 in the plan, and taking up the table, 
 set up an object in it exactly upright, and measure 
 from it towards 2, and find that perpendicular 
 against 218, the offset to the angle at the boun- 
 dary is 157 links, which set off in the plan; then 
 measuring on at 375 the offset is but 6, and con- 
 tinues the same to 698, at both which set off 6 in 
 the plan ; then measure on to 2, and find the 
 whole 1041 links, which set off in the blank line 
 drawn for it, and mark it 2; then taking out 
 the object, plant the table to have 2 over the 
 hole, when placed parallel to what it was on 1 ; 
 that is, the edge of the index-ruler touching both 
 stations, the hair must cut the object at © 1, and 
 then screw it fast. 
 
 Now setting up objects in the by-angles a and h, 
 first turn the index to view that at a, and draw a 
 blank line from 2 towards it; then do the same 
 towards h, Q 3 and the tree, which last crossing 
 that drawn towards it from 1, the intersection 
 determines the place of the tree, which being re- 
 markable, as seen from all the stations, mark it in 
 the plan; then measure to a 412 links, and from h 
 
 V 2
 
 ■igl OP SURVEYING BY 
 
 353, and set them off in the blank lines drawn 
 towards them; then set off the distance to the 
 boundaries in the two station lines produced, viz. 
 151 in the next produced backwards, and 15 in 
 the first produced forwards; after this, draw the 
 boundaries from the angle where the first offset 
 was made to the next, and so on round by a, h, 
 through the 151 to the next angle; then, taking 
 up the table, fix again the object as before, and 
 measure on the O 3, which set oft' 564 links, and 
 the offset to the angle in the boundary 27, and 
 then draw the boundary from it through the 15 to 
 the angle at meeting that last drawn. 
 
 Now, taking out the object at © 3, plant the 
 table so as to have O 3 over the hole, when 
 placed parallel to what it was at the former sta- 
 tions, and screwed fast; then turn the index to 
 make the edge touch the place of the tree and O 
 3 in the plan, and finding the hair cuts the tree, 
 turn the index to view © 4, and draw a blank line 
 towards it; then taking up the table, fix the object 
 as before^ and measure on to © 4, which set oft" 
 471 links, and the offset 23, and draw the boun- 
 dary from the last angle through it to the next; 
 then measure on in the station line produced to 
 the next boundary 207 links, -and the distance of 
 © 4, from the nearest place in the same boundary 
 173, both which set off and draw the boundary 
 from this last through the 207 to the angle. 
 
 Now taking out the mark, plant the table to 
 have © 4 over the hole, when screwed fast in the 
 same parallelism as at the other stations; then, 
 after viewing again the tree, turn the index to 
 view © 5, and draw a blank line towards it; then 
 taking up the table, fix the object as before, and 
 mcp^SLiring on towards © 5, at 225 the nearest 
 place of the boundary is distant 121, which set
 
 THE PLAIN TABLE. 293 
 
 off bearing forwards, as the figure shews; at 388 
 the perpendicular offset is t), and at 712 it is I'i, 
 both which set off in your plan ; then measure on 
 to O 5, and set it off at 912 links. 
 
 Take out the mark, plant the table to have 5 
 over the hole, when screwed fast in the same pa- 
 rallelism as before; then set up objects in the 
 by-angles c and J, and after viewing the tree, 
 turn the index to view the objects at c, d^ and 6, 
 and draw a blank line towards each ; then measure 
 to c 159, and from J 245, both which set off in 
 your plan, and also the distance to the boundary 
 ill the next station line produced backward 95 ; 
 and now make up the boundary round by the se- 
 veral offsets to the angles c and d\ then taking 
 up your table, fix the object as before, and mea- 
 suring towards 6, find at l()2 the offset is 32, 
 w^hich set off; measure on to 6, and set it off 
 at 7 08, and the offset from it to the boundary is 
 36 links. 
 
 Finding the blank line drawn from 1 to in- 
 tersect the point-hole here made for 6, do not 
 plant the table at 6, but begin measuring from 
 it towards 1, and finding at right angles to the 
 line at 6, the offset to the angle is 42, set that 
 off in your plan; then measuring to 1, 582, 
 which, measuring the same by the scale in the 
 plan, proves the truth of the work; the offset is 
 here also 42, which set off, and draw the boun- 
 dary from J, round by the several oflsets, through 
 this last to the angle; then measure on in the sta- 
 tion line produced to the next boundary 88 links, 
 and set that off also, and draw the boundary from 
 the angle at the first offset, taken through it at 
 the angle at meeting the last boundary;. .and then 
 if a meridian line be drawn, as in the former, the 
 rough plan is completed.
 
 294 OF SURVEYING BY 
 
 But if O 6 had not met in the intersection, or 
 its distance from 1 been too much, or too little, 
 you would very likely have all your work, except 
 the offsets, to measure and plot over again. 
 
 '' The plain table surveyors, says Mr. Gard'mer, 
 when they find their work not to close right, do 
 often close it wrong, not only to save time and 
 labour, but the acknowledging an error to their 
 assistants, which they are not sure they can amend, 
 because in many cases it is not in their power, 
 and may be more often the fault of the instrument 
 than the surveyor; for in uneven land, where the 
 table cannot at all stations be set horizontal, or in 
 any other one plane, it is impossible the work 
 should be true in all parts: but to prevent great 
 errors, at every © after the second, view wherever 
 it is possible, the object at some former ©, besides 
 that which the table was last planted at; because 
 if the edge of the index ruler do not quite touch, 
 or but very little covers that in the plan, whilst 
 it touches the you are at, the error may be 
 amended before it is more increased, and if it va- 
 ries much, it may be examined by planting again 
 the table at the former station, or stations. 
 
 If a field is so hilly, that you cannot, without 
 increasing the number of stations, see more than 
 one object backward, and another forward, and 
 there is nothing fit within the field, as the sup- 
 posed tree in f(r, 32, then set up an object on 
 purpose to be viewed from all the stations, if pos- 
 t:^ible, for such a rectifier. 
 
 The lengthening and shortening of the paper, 
 as the weather is moister or drier, often causes no 
 &mall error in plotting on the plain table; for be- 
 tween a dewy morning, and the sun shining hot 
 at noon day, there is great difference, and care 
 should be taken to allov/ for it; but that cannot
 
 THE PLAIN TABLE. 2Q5 
 
 be done in large surveys, and so ought not to be 
 expected; indeed, those working by the degrees, 
 without having their plan on it, are not hable to 
 this error, though they are to the former; but 
 both ways are liable to another error, which is, 
 that the station lines drawn, or the degrees taken, 
 are not in the line between the objects, nor pa- 
 rallel thereto; neither will this error be small in 
 short distances, and may be great, if each O on 
 the plan, or the center used with the degrees, is 
 not exactly over the station holes; but to be most 
 exact, it is the line of their sights that should be 
 directly over the hole. 
 
 Mr. Beighton made such improvements to his 
 plain tables, by a conical ferril fixed on the same 
 staves as his theodolite, that the above errors, ex- 
 cept that of the paper, are thereby remedied ; for 
 the line of the sights, in viewing, is always over 
 the center of the table, which is as readily set 
 perpendicular over the hole, as the center of the 
 theodolite, and the station lines drawn parallel to 
 those measured on the land; and the table is set 
 horizontal with a spirit level by the same four 
 screws that adjust the theodolite; therefore some 
 choose to have both instruments, that they may 
 use either, as they shall think most convenient. 
 
 Let j'^o-. 32 now be a wood, to be measured and 
 plotted on the outside \ if on coming round to the 
 iirst O, the lines meet as they ought, the plan will 
 be as truly made, as if done on the inside; but 
 here having no rectifier of the work as you go on, 
 you must trust to the closing of the last measured 
 line; and if that does not truly close with the first, 
 you must go over the work again ; and, without a 
 better instrument than the common plain table, 
 you cannot be sure of not making an error in this 
 case.
 
 '^9^ OF SURVEYING BY 
 
 Suppose the table planted at 1 on the out- 
 side, with paper fixed on it, and objects set up at 
 all the other stations on the outside, and dry blank 
 lines drawn from l on the paper towards d 
 and 2; these done, take up the table, and set 
 up an object at 1 ; then, measuring from it to- 
 wards 2, you find at 20 the offset to the first 
 angle is 38, then at 280 the offset to the next an- 
 gle is 26 links, both of which set off in the plan ; 
 then at 394 the perpendicular offset to the next 
 angle is 206 ; then at 698 the distance of the same 
 angle is 366 bearing backward, as may be seen 
 in the figure, that by the intersection of these 
 two ofiset lines the angle may be more truly 
 plotted; then measure the distance from this an- 
 gle to the next 323, and from that to 698 place in 
 the station line 280, which is the perpendicular 
 offset; then by the intersection of these, that an- 
 gle will be well plotted; then at 77^ the offset 
 to angle a is 48, and at 1012 the offset to b is 22, 
 both which set off' in the plan, and at 1306 you 
 make 2; now draw the boundaries from the 
 first offset to the next, &V, to the angle b. As 
 there is no difiiculty in taking the offsets from 
 the other station lines, we shall not proceed far- 
 ther in plotting it on the outside; for a sight of 
 the figure is sufficient. 
 
 Some surveyors would plant their table at a 
 place between 394 and 698 in the first station 
 line, and take the two angles, which are here 
 plotted by the intersection of lines, as the by- 
 angles (2 and /-were taken at 2 within the 
 field; but if the boundary should not be a strait 
 line from one angle to the other, then their dis- 
 tance should be measured, and offsets taken to 
 the several bends in it,
 
 THE PLAIN TAELfi. IQ? 
 
 You plot an Inaccessible distance in the same 
 manner as the tree in Jjg. 32; for if you could 
 come no nearer to it than the station line, yet you 
 might with a scale measure its distance from O 
 1, 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 distance from one another, that 
 the lines drawn towards the tree may intersect 
 each other a& near as possible to right angles, 
 drawing a line from each to O 1 ;* writing down 
 the degree the north end of the needle points to, 
 as it should point to the same degree at each sta- 
 tion ; remove the table, and set up a mark at l. 
 
 The imperfections in all the common methods 
 of using the plain table, are so various, so tedious, 
 and liable to such inaccuracies, that this instru- 
 ment, so much esteemed at one time^ is now dis- 
 regarded by all those who aim at correctness in 
 their work. Mr. Break has endeavoured to re- 
 medy the evils to which this instrument is liable, 
 by adopting another method of using it; a method 
 which I think docs him considerable honour, and 
 which I shall therefore extract from his complete 
 " System of Land Surveying^'' for the inforijiation 
 of the practitioner. 
 
 Example 1. To lake the plot of afeld KQQT>^Y, 
 Jjg. 1 1, plate \3, from one station therein. 
 
 Choose a station from whence you can see every 
 corner of the field, and place a mark at each, 
 numbering these with the figures 1, 2, 3, 4, &c. 
 at this station erect your plain table covered with 
 
 * In general the mark © always denotes a station or place 
 where the instrument is planted. The dotted lines leading from 
 one station to another, are the station lines; the l)lack lines, the 
 boundaries; the dotted lines frunl the boundary to the station 
 line ai'« cffsets.
 
 298 OP SURVEYING BY 
 
 paper, and bring the south point of the needle to 
 the flower-de-luce in the box; then draw a circle 
 O P Q R upon the paper, and as large as the pa- 
 per will hold. 
 
 Through the center of this circle draw the 
 line NS parallel to that side of the table which is 
 parallel to the meridian line in the box, and this 
 will be the meridian of the plan. 
 
 Move the chamfered edge of the index on , 
 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 protracted the several bearings, the dis- 
 tances must be measured as shewn before. 
 
 To draw the plan. Through the center ©and the 
 points 1, 2, 3, &c. draw lines © A, © B, © C, 
 © D, © E, © F, and make each of them equal 
 to its respective measure in the field; join the 
 points A, B, C, &c. and the plan is finished. 
 
 Example 2. To take the plot of a field AB C D, 
 &c.from several stations, fig. 14, plate 13. 
 
 Having chosen the necessary stations in the 
 field, and drawn the circle P QR, which you 
 must ever observe to do in every case, set up 
 your instrument at the first station, and bring the 
 needle to the meridian, which is called adjusting 
 the instrument; move the index on the center 
 © , and take an observation at yl, B, C, © 2, H, 
 and the fiducial edge thereof will intersect the 
 circle in 1,2, 3, &c. Then remove your instru- 
 ment to the second station in the field, and ap- 
 plying the edge of the index to the center © and 
 the mark © 2 in the circle, take a back sight to 
 the first station, and fasten the table in this posi- 
 tion; then move the index on the center ©, and 
 direct the sights to the remaining angular marks, 
 so will the fiducial edge thereof cut the circle in
 
 THE PLAIN TABLE. 
 
 m 
 
 the points 4, 5, 6, &c. The several distances 
 being measured with a chain, the work in the 
 lield is finished; and entered in the book thus: 
 
 THE FIELD BOOK EXPLAINED. 
 
 No. 
 
 Dist. 
 
 No. 
 
 Dist. 
 
 1 
 2 
 3 
 
 02 
 8 
 
 ©1 
 520 
 344 
 360 
 730 
 386 
 
 3 
 4 
 5 
 6 
 
 7 
 8 
 
 ©2 
 
 370 
 470 
 550 
 550 
 
 
 To draw the Plan. 
 
 Having chosen 1 upon paper to represent 
 the first station in the field, lay the edge of a pa- 
 rallel ruler to 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 1, and, ex- 
 tending the other edge to 1, draw thereby the 
 line 1,2=344, which gives the corner B, as the 
 line 1,1 does the corner A. After the same man- 
 ner 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 give 
 the point or corner D. Thus project the remain- 
 ing corners E, F, G, and the plan is ready for 
 closing. 
 
 Examples. To take the plot of several fields 
 ABCD, BECF, DIHK, and \Y QYi, from
 
 300 OF SURVEYIXG BY 
 
 Stations chosen at or 7iear the jniddle of each, fig. 
 15, plate 13. 
 
 Adjust your plain table at the first station in 
 Jl BCD, and draw the protracting circles and me- 
 ridians N S; then by proposition 1, project the an- 
 gles, or corners o^ A B CD, 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 center and the mark 
 2, and direct the sights by turning the table 
 to the second station, then move the index till you 
 observe the third station, and the edge thereof 
 will cut the circle in 3. Then remove the in- 
 strument to the third station, lay the index on the 
 center and the mark 3, and take a back 
 observation to the first station ; after which by the 
 last proposition find the points 7j 8? 4, 9, in the 
 circle P QR. As to the measuring of distances, 
 both in this and the two succeeding propositions that 
 shall be passed over in silence, having sufficiently 
 displayed the same heretofore; what I intend to 
 treat of hereafter, is the method of taking and 
 protracting the bearings in the field, Avith the man- 
 ner of deducing a plan therefrom. 
 
 To draw the Plan. 
 
 Choose any point 1 for your first station; 
 apply the edge of a parallel ruler to the center 
 and the point I, 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 jB, C,D, together with 2; which being 
 joined, finishes the field AB CD. After the same . 
 method construct the other fields B E F C^ 
 DIRK, IFGII, and you have done.
 
 THE PLAN TABLE. 
 
 301 
 
 THE FIELD BOOK. 
 
 REMARKS. 
 
 N^. 
 
 Dl3T. 
 
 REMARKS. 
 
 The open 
 Sir Will, Jones's 
 
 1 
 
 2 
 
 ©2 
 3 
 4 
 
 2 
 5 
 6 
 
 7 
 3 
 
 ©3 
 
 7 
 
 ©4 
 8 
 
 9 
 
 4 
 
 6 
 10 
 
 8 
 
 ©1 
 370 
 
 380 
 580 
 403 
 440 
 
 In J BCD. 
 
 Field. 
 
 BEFI. 
 
 DIHK. 
 Ground. 
 
 
 ©2 
 
 In BEFI. 
 
 John Simpson's 
 
 BEFI doses at 
 Return to 
 
 400 
 400 
 
 Ground. 
 IFGH. 
 
 ©1 
 600 
 
 In J BCD. 
 
 
 ©3 
 
 In DIHK. 
 IFGH. 
 
 Field. 
 Ground. 
 
 South 
 
 John Sjiencer's 
 
 DIHK closes at 
 
 630 
 432 
 400 
 
 
 
 Ed. Johnstone's 
 
 South 
 
 IFGH doses at 
 
 ©4 
 
 In IFGH. 
 
 Ground. 
 Field. 
 
 380 
 
 
 
 Example 4. 7(9 fake the plot of a field ABCD 
 E F, hy going round the same, fig. 12, plate 13. 
 
 Set up your plain table at the first station in the 
 field; move the fiducial edge of the index on the 
 center 0, and take an observation at the mark 
 placed at the second station, then will the same 
 fiducial edge cut the circle OP QR'in the point 1, 
 Then remove your instrument to the second sta- 
 tion, and placing the edge of the index on O and 
 the point 1 , take a back sight to the first, or last
 
 302 
 
 OP SURVEYING BT 
 
 station ; then directing the index on the center (£) 
 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 in- 
 dex directed forward to the next succeeding sta- 
 tion, will giv^e the protracted points 3, 4, 5> 6. 
 
 THE FIELD BOOK. 
 
 REMARKS. 
 
 0. 
 
 qL. 
 
 0. 
 
 REMARKS. 
 
 
 
 ©1 
 
 
 In JBCDEF. 
 
 
 70 
 
 
 
 
 
 
 50 
 
 250 
 
 
 
 
 85 
 
 550 
 
 
 
 « 
 
 ©2 
 
 In Ditto. 
 
 
 84 
 
 
 
 
 
 Corner 
 
 65 
 
 440 
 
 
 
 
 ©3 
 
 In Ditto. 
 
 
 
 
 
 
 
 
 60 
 
 465 
 
 
 
 
 ©4 
 
 In Ditto. 
 
 
 72 
 
 
 
 
 
 
 58 
 
 365 
 
 
 
 
 80 
 
 750 
 
 05 
 
 
 In Ditto. 
 
 
 -40 
 
 
 
 
 
 
 68 
 
 302 
 
 
 
 
 60 
 
 680 
 
 
 
 
 ©6 
 
 In Ditto. 
 
 
 58 
 
 
 
 
 
 
 50 
 
 355 
 
 
 
 
 67 
 
 663 
 
 to© I 
 
 Close Ditto.
 
 THE PLAIN TABLE. 303 
 
 To draw the Plan, 
 
 Choose any point 1, to denote the first sta- 
 tion. Lay the edge of a parallel ruler on the cen- 
 ter O and the point 1, and extend the other edge 
 till it toueh 1, and draw by the side thereof the 
 line 1,2=550; then apply the rider to and the 
 mark 2, and extend the other edge to 1, and 
 draw thereby the line 2,3=440; again, lay the 
 edge of the ruler to and the point 3, and the 
 other edge being extended to 3, draw the line 
 3,4=463 ; after the same method lay down the 
 remaining stations, and the trav^erse is delineated. 
 As for drawing the edges, that shall be left for the 
 learner's exercise. 
 
 Example 5 . To take the plot of several fields 
 A, B, C, D, hy circidation,fig. \Q, plate 13. 
 
 From the projecting point by last example, 
 project the stations in A, into the points 1, 2, 3, 
 4 ; then the instrument being planted at the se- 
 cond station, from the same projecting point 
 project that station the second in A into the point 
 2^, 2^ denoting the instrument being planted a 
 second time at that station, which is done thus : 
 lay the index to and the point 2, and take a 
 back sight to the first station, that being the sta- 
 tion immediately preceding that you are at in the 
 field book ; then on the center take a fore ob- 
 servation at the next succeeding station, and the 
 index will cut the circle in the point 2^. Thus 
 project every other remaining station.
 
 304 
 
 OF SURVEYING BY 
 
 THE FIELD BOOK. 
 
 REMAPxKS. 
 
 0. 
 
 ©L, 
 
 0. 
 
 REMARKS. 
 
 
 
 ©1 
 
 
 In Field A. 
 
 
 53 
 
 
 
 
 
 
 80 
 
 290 
 
 
 
 Coi-ner 
 
 85 
 60 
 
 610 
 695 
 
 
 
 
 ©2 
 
 In Ditto. 
 
 
 70 
 
 
 
 
 
 
 65 
 
 560 
 
 
 
 
 ©3 
 
 In Ditto, 
 
 
 60 
 
 
 
 
 
 
 75 
 
 368 
 
 
 
 
 55 
 
 680 
 ©4 
 
 
 In Ditto. 
 
 
 50 
 
 
 
 
 
 
 48 
 
 440 
 
 tool 
 
 Close here A. 
 
 Return to 
 
 ©2 
 
 In^. 
 
 
 
 
 
 
 
 Corner 
 
 60 
 50 
 
 50 
 60 
 50 
 
 80 
 402 
 
 05 
 
 
 
 340 
 
 650 
 
 into 
 
 B. 
 InB. 
 
 
 ©6 
 
 In Ditto. 
 
 
 50 
 
 
 
 
 
 Ag. Hedge 
 
 51 
 
 500 
 
 
 
 
 
 560 
 
 by© 
 
 for closing Cj 
 
 Closes at Cor. of ^ 
 
 50 
 
 620 
 
 z= 
 
 the Hedge, 
 
 
 
 682 
 
 to© 3 
 
 in A. 
 
 Return to 
 
 ©4 
 
 In A. 
 
 
 
 
 
 
 
 
 
 60 
 
 70 
 
 
 
 
 into C. 
 
 
 
 
 
 380 
 
 60 
 
 
 
 682 
 
 68 

 
 THE PLAIN TABLE. 
 
 ;o5 
 
 THE FIELD BOOK. 
 
 REMARKS. 
 
 0. 
 
 ©L. 
 
 0. 
 
 REMARKS. 
 
 
 
 ©7 
 
 
 InC. 
 
 
 
 
 
 40 
 
 
 
 
 300 
 
 60 
 
 
 
 
 720 
 
 50 
 
 
 
 ©8 
 
 In Ditto. 
 
 
 
 
 
 70 
 
 
 
 
 400 
 
 24 
 
 
 
 
 780 
 
 63 
 
 Close C. 
 
 
 
 Cross 
 
 hedge 
 
 
 
 
 820 
 
 bye 
 
 © at 560I. 6, 4. 
 
 Return to 
 
 ©8 
 
 In Ditto. 
 
 
 
 
 
 
 
 
 
 60 
 
 50 
 
 Corner. 
 
 
 
 into 
 
 2). 
 
 
 
 
 220 
 
 50 
 
 
 
 
 554 
 
 40 
 
 
 
 ©9 
 
 InD. 
 
 
 
 
 
 40 
 
 
 
 
 250 
 
 38 
 
 
 
 10 
 
 In Ditto, 
 
 
 
 
 
 
 
 
 
 64 
 
 50 
 
 
 
 
 530 
 
 64 
 
 Close D at Cor. B. 
 
 
 
 into 
 
 B. 
 
 
 
 
 570 
 
 to©6 
 
 mB. 
 
 The bearing being protracted, the plan may be 
 readily drawn from what has been already de- 
 scribed. 
 
 GENERAL METHODS OP SURVEYING DETACH£P 
 PIECES OF GROUND. 
 
 To survey the triangular field, fig. 12, plate 9, 
 v:ith any instrument used for vieasiiring afigles. 
 I. Set up marks at the three corners A, B, C, 
 
 X
 
 >06 OP SURVEYING BY 
 
 i. Measure the angle ACB. 3. Measure the two 
 sides AC, C B. 
 
 This method of measurins; two sides and the in- 
 cluaed angle, is far more accurate than the old 
 method of going round the field, and measuring 
 all the angles. It was first introduced into prac- 
 tice by Mr. Tallwt. 
 
 If the field contain four sides. Jig. 23, plale 9, 
 begin at one of the corners A. 1. Measure the 
 angle BAD. 2. Measure the side AB. 3. Take 
 the angle ABC. 4. Measure the side BC, and 
 angle BCD. 5. Measure the side C D, and the 
 angle CDA, and side DA, the dimensions arc 
 finished; add the four angles together, and if the 
 sum makes 360", you may conclude that your 
 operations are correct; the above figure may be 
 measured by any other method as taught before, 
 by measuring the diagonal, &c. 
 
 If the field contain more than four sides, Jig. 
 24, 25, 26, pJafe 9, having set up your marks, en- 
 deavour to o;ct an idea of the laro;est four-sided 
 figure, that can be formed in the field you are 
 going to measure; this figure is represented in 
 the figures by the dotted lines. 
 
 Then beginning at A, take the angle BAD, 
 measure in a right line towards B, till you come 
 against the angle/, there with your sextant, or 
 cross, let fall the perpendicular/^, as taught in the 
 method of the triangle, observing at how many 
 chains and links this offset or perpendicular falls 
 from the beginning or point A, which note in your 
 field book, and measure ef, noting it also in your 
 field book; then continue the measure of the line 
 AB to B; take the angle ABC, and measure the 
 line B C; take the angle BCD, and measure the 
 line CD; take the angle CDA, and measure the 
 line DA; observing as you go round, to let fall
 
 THE PLAIN TABLE. 
 
 307 
 
 perpendiculars where neeessary, and measure them 
 as specified in the line AB. 
 
 E-xamplc tofg:. 26, containing seven sides. 
 
 FIELD BOOK. 
 
 IA = 
 
 86°„35' 
 
 
 Ae = 
 
 2,08 
 5,32 
 
 80=^/. 
 
 = 1st side AB. 
 
 zB = 
 
 88°„25' 
 
 
 
 4,80 
 
 = 2(1 side BC. 
 
 ZC = 
 
 93°,00' 
 
 
 c^= 
 
 1,20 
 4,92 
 
 88=gb. 
 
 = 3(1 side CD. 
 
 ZD = 
 
 cr2°„oo' 
 
 
 Di = 
 
 2,00 
 4,84 
 
 l,OS = ik. 
 
 =:4th side DA. 
 
 N. B. Set the offsets to the right or left of your 
 column of angles and chains, according as they 
 fall to the right and left of your chain line in the 
 field. 
 
 TO USE THE COMMON CIRCUMFERENTOE. 
 
 We have already observed that this instrument 
 should never be used where much accuracy is re- 
 quired, for it is scarcely possible to obtain with any 
 certainty the measure of an angle nearer than ta 
 two degrees, and often not so near; it has there- 
 fore long been rejected by accurate surveyors.* 
 
 * See Gardiner's Practical Surveying, p. 54. 
 X 2
 
 30§ TO USE THE 
 
 This instrument takes the bearing of objects 
 from station to station, by moving the index till 
 thcHneof the sights coincides with the next sta- 
 tion mark; then counting the degrees between 
 the point of the compass box marked N, and the 
 point of the needle in the circle of quadrants. 
 
 Thus let it be required to survey a large wood, 
 fg. 31, pidte g, by going round it, and observing 
 the bcarmg of the several station lines which en- 
 compass that wood.* 
 
 The station marks being set up, plant the cir- 
 cumferentor at some convenient station as at a, the 
 flower-de-luce in the compass box being from you ; 
 direct the sights to the next station, rod /', and set 
 down the division indicated by the north end of 
 the needle, namely 26o° 30', for the bearing of the 
 needle. 
 
 Remove the station rod b to r, and place the 
 circumfcrentor exactly over the hole where the 
 rod h was placed, measuring the station lines, and 
 the oifscts from them to the boundaries ; now 
 move the instrument, and place the center thereof 
 exactly over the hole from whence the rod i? was 
 taken. The flower-de-luce being from you, turn 
 the instrument till the hair in the sights coincides 
 with the object at the station c, then will the north 
 end of the needle point to 292° 12', the bearing 
 o{ b c\ the instrument being planted at c, and the 
 sights directed to d, the bearing of ccl, will be 
 33 1*^45'. In the same manner proceed to take 
 the bearing of other lines round the wood, observ- 
 ing 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. 
 
 * Wyld's Practical Surveyor, p. 77'
 
 COMMON CIRCUMFEREXTOR. 300 
 
 lines. 
 
 beari] 
 
 QgS. 
 
 links, 
 
 ab 
 
 260. 
 
 30 
 
 1242 
 
 be 
 
 292. 
 
 12 
 
 1015 
 
 cd 
 
 331. 
 
 45 
 
 1050 
 
 dc 
 
 5g. 
 
 00 
 
 1428 
 
 c f 
 
 112. 
 
 15 
 
 645 
 
 fa 151. 30 I8O6 
 
 Instead of planting the circumfercntor 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 fiowerdelace 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 lines a h^ and b c, would be the same as before 
 observed; also the bearings of the lines r^, and 
 cl e, might be observed at d, and ef, andy"<r/, 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^ 
 to 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 magnetic 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 b, the flower-de-luce from 
 you, the north end of the needle pointed to 260" 
 3Q'; now being come to h, direct the sights back 
 to a mark at ^, keeping the flowcr-de-lucc towards*
 
 310 OF MR. GALE S METHOD BY 
 
 you: so shall the north end of the needle point to 
 260^ 30', as before at a^ and then you may be 
 sure the bearing of the line ah\% truly observed. 
 
 But if the needle doth not point to the same 
 number of degrees, &c, there hath been some error 
 in that observation, which must be corrected. 
 
 OF THE IMPROVED CIRCUMFERENTOR, WITH 
 MR. gale's METHOD OF USING IT; A ME- 
 THOD THAT IS APPLICABLE WITH EQUAL 
 ADVANTAGE TO THE THEODOLITE, &C. 
 
 For the ^ake of perspicuity, it will be necessary 
 to give again the example before used in page 222, 
 and that not only because it will exhibit more 
 clearly the advantages of Mr. Gales method,* but 
 because we shall have occasion to refer to it when 
 we come to his improved method of plotting; and 
 further, because I have thought this mode so ad- 
 vantageous, and the tables so conducive to accu- 
 racy and expedition, that I have caused occasio- 
 nally the traversing quadrant to be engraved in 
 smaller figures under the usual one of the limb of 
 the theodolite. 
 
 Set the circumferentor up at B,j^^. 1, plate 18, 
 take the course and bearing of B C, and measure 
 the length thereof, and so proceed with the sides 
 CD, D E, E F, F G, GA, all the way round to 
 the place of be^ginning, noting the several courses 
 or bearings, and the lengths of the several sides in a 
 field book, which let us suppose to be as follow : 
 
 
 ^ ^ chains, links. 
 
 1. AB 
 
 North 7 West 21. 00. 
 
 2. BC 
 
 North 55 15 East 18. 20. 
 
 •" See page 219 of this work for a fuller account of this 
 method.
 
 THE IMPROVED CIKCUMFERENTOR. 311 
 
 chains, links. 
 
 3. 
 
 CD 
 
 o / 
 
 South 6'2 30 East 
 
 14. 
 
 40. 
 
 4. 
 
 DE 
 
 South 40 West 
 
 11. 
 
 
 5. 
 
 EF 
 
 South 4 15 East 
 
 14. 
 
 
 6. 
 
 FC 
 
 North 73 45 West 
 
 12. 
 
 40. 
 
 7. 
 
 GA 
 
 South 52 West 
 
 9- 
 
 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- 
 iive degrees 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, or left-hand of the south ; and by south 
 40° west, forty degrees to the westward, or right 
 hand of the south. 
 
 The 21 chains, 18 chains 20 links, &c. are the 
 lengths or distances of the respective sides, as 
 measured by the chain. 
 
 To survey a Jield, or tract of land, having irre- 
 gular howidaries. 
 
 When the boundaries of a survey have crooks 
 and bends in them, it is by no means necessary to 
 take a new course for every small bend; the best 
 and most usual way, is to proceed in a strait line 
 from one principal corner to another, and when 
 you are opposite to any bend in the boundary, to 
 measure the rectangular distance; termed the off- 
 set, from the strait line to the bend, noting the 
 same in the field book, together with the distance 
 on the strait line from whence such offsets were 
 made. The offsets, as already obsei-ved, arj ge- 
 nerally measured with an offset staff. 
 
 For the purpose of noting these offsets, it is ne- 
 cessary that the field book should be ruled intq
 
 312 OF MR. gale's method BY 
 
 FIVE COLUMNS. The middle columii to contaiii 
 the courses and distances; the adjoining columns 
 on the right and left hand to contain the measure 
 of the offsets, made to the right or left hand res- 
 pectively; and the outside columns on the right 
 and left hand, to contain remarks made on the 
 right and left hand, respectively, such as the 
 names of the adjoining fields, or the bearing of 
 any remarkable object, &c. 
 
 Example. \j€i.fig. 3, plate 18, represent afield 
 to be surveyed, whose boundaries are crooked. 
 
 Set up the instrument at, or near any conve- 
 nient corner, as at 1, and take the course, and 
 bearing, as before directed, north 7° west, note 
 this down in the middle column of the field book, 
 and measure with the chain as before directed, till 
 you come opposite to the first bend, so that the 
 bend be at right angles to the station line; note 
 the distance thus measured, 3 chains 6o links, in 
 the middle column, and measure the offset from 
 thence to the bend 40 links, noting the same in 
 the adjoining left hand column, because the 
 boundary is on the left hand of the station line, 
 and note in the outside column, the name or owner 
 of the adjoining field, proceeding in the same 
 manner all round the field, noting the courses, 
 distances, offsets, and remarks, as in the fol- 
 lowing :
 
 THE IMPROVED CIRCUMFEREXTOR. 313 
 
 FIELD BOOK. 
 
 REMARKS. 
 
 H. Derman's land 
 
 A corner 
 
 W. Higgln's land 
 
 A corner 
 
 D. Home's land 
 
 A corner 
 
 C, Ward's land 
 
 To a corner 
 
 To the above a:)rner 
 C. Ward's land 
 
 A corner 
 
 W.Humphrey's land 
 
 Oft- 
 
 sets. 
 
 Station 
 lines. 
 
 Otl- 
 sets. 
 
 1st. At the corner, 
 against Wni. Hum- 
 phreys, and H. Bar- 
 man's land. 
 
 REMjVRKS. 
 
 O 
 40 
 O 10 
 O 65 
 
 O O 
 
 3 60 
 
 8 45 
 
 15 60 
 
 O 21 00 
 
 O 
 
 
 2d. 
 
 N.55°15.E. 
 
 O O 
 
 6 10 
 
 18 20 
 
 O 
 6o 
 o o 
 
 o 
 
 o 
 
 
 O 40 
 95 
 
 68 
 
 1 2.5 
 O 
 
 3d. 
 
 S. 62°30.E 
 
 7 40 
 14 40 
 
 4th. 
 
 S. 40° W. 
 
 o o 
 
 8 00 
 11 00 
 
 
 
 90 
 
 
 
 5th. 
 
 S. 4° 15. E 
 
 O 66 
 
 7 50 
 
 14 00 
 
 
 
 6th. 
 
 X.73''4oV^^ 
 
 o 
 
 3 20 
 
 7 50 
 
 12 40 
 
 To the bound- 
 ary of the field. 
 
 To the bound- 
 ary of the field- 
 
 O O 
 
 30 To the bound- 
 
 1 25 arj'of the field, 
 O O A corner.
 
 314 
 
 OF MR. GALE S METHOD BY 
 
 FIELD BOOK. 
 
 REMARKS. 
 
 Off- 
 sets. 
 
 Station 
 lines. 
 
 Off- 
 sets. 
 
 REMARKS. 
 
 W.Humphrey's land 
 
 O O 
 
 7th. 
 
 S. 52° 
 O 
 4 
 
 9 
 
 W. 
 
 O 
 
 10 
 
 17 
 
 o 
 o 60 
 
 
 To the bound- 
 ary of the field. 
 
 To the place of beginning. 
 
 To survey a tract of land, consisting of any nuni- 
 her of 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 bounda- 
 ries of the several fields from the place where they 
 so butt on the outside bounds. 
 
 \j^\fig. A^ 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 remarks, as in the following field book. 
 
 FIELD BOOK. 
 
 REMARKS. 
 
 Off- 
 sets. 
 
 Station 
 lines. 
 
 Off- 
 sets. 
 
 REMARKS. 
 
 
 1st. At a comer 
 against Winterton 
 farm, and H. Smith's 
 land. 
 
 
 Lands of 
 Heniy Smith. 
 
 00 
 
 S. 87° W. 
 
 00 
 
 5 50 
 
 11 30 
 
 15 00 
 
 00 
 
 1 10 
 00 
 
 (i(j 
 
 East field. 
 
 To the corner of east 
 and west fields. 
 Corner of west field.
 
 THE IMPROVED CIRCUMPERENTOE. 315 
 FIELD BOOK. 
 
 REMARKS. 
 
 Off- 
 sets. 
 
 Station 
 lines. 
 
 Off- 
 sets. 
 
 REMARKS. 
 
 Hunterdon farm. 
 
 
 2d. 
 
 N.18°30.W 
 58 
 9 60 
 
 16 
 00 
 
 The above corner. 
 
 
 
 3d. 
 
 N.12°15.E. 
 O 00 
 4 10 
 8 90 
 
 00 
 60 
 00 
 
 West field. 
 
 
 00 
 
 4th. 
 
 N. 37° E. 
 5 40 
 
 17 00 
 
 40 
 00 
 
 To the comer of west 
 field and north field. 
 
 Lands of 
 Jacob Williams. 
 
 70 
 00 
 
 5th. 
 
 S.76°45.E. 
 
 6 50 
 
 16 00 
 
 00 
 
 North field. 
 
 Winlerton farm. 
 
 
 6th. 
 
 S.5°15.W. 
 
 9 60 
 
 14 30 
 
 19 00 
 
 25 00 
 
 56 
 10 
 
 70 
 
 00 
 
 To the corner of north 
 and east fields. N. B- 
 A gate into each field 
 20 links from the 
 corner. 
 
 
 
 7th. 
 
 S.74°30.W. 
 2 30 
 
 7 65 
 
 86 
 00 
 
 East field. 
 
 
 To the place ot" beginning. 
 
 
 
 8th, At the corner 
 of east and west field, 
 viz. The first offset on 
 the first station line. 
 
 
 West field. 
 
 
 North. 
 
 8 30 
 
 16 61 
 
 75 
 00 
 
 East field.
 
 310 TO SURVEY AN ES 
 FIELD BOOK. 
 
 TATE;, 
 
 REMARKS. 
 
 Ofi'- 
 
 sets. 
 
 Station 
 lines. 
 
 Otf- 
 
 sets. 
 
 REMARKS. 
 
 
 00 
 
 9th. 
 
 East. 
 
 O 
 
 5 40 
 
 13 77 
 
 00 
 60 
 00 
 
 East field. 
 
 To the corner of north 
 
 North field. 
 
 
 and eatt fields against 
 Winterton farm. 
 
 West field. 
 Corner of west 
 and north fields 
 against Hunten- 
 don farm. 
 
 Back 
 
 00 
 70 
 00 
 
 10th. 
 
 to the last stJ 
 N.62^03.W 
 
 
 3 40 
 7 92 
 
 ition. 
 00 
 00 
 
 North field. 
 
 To take a survey of an estate, manor ^ and lord- 
 ship. An estate, manor, or lordship, is in reality 
 a tract of land, consisting of a number of fields ; 
 it differs in no respect from the last article, ex- 
 cepting in the number of fields it may contain, and 
 the roads, lanes, or waters that may run through 
 it, and is of course surveyed in the same manner. 
 
 It is 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 
 boundaries are intersected by roads, lanes, or wa- 
 ters, the places where the boundaries of the res- 
 pective 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 no- 
 tice of. If, however, there should be a large stream 
 of water running through the estate, tl;#ercby di- 
 xnding it into two parts, and no bridge riear the 
 boundary, then it will be best to survey that part 
 which lies on one side of the stream first, and af-^ 
 tcrwards that part which lies on the other side
 
 MANOR, AND LORDSHIP. 317 
 
 thereof: if the stream be of an irregular breadth, 
 both its banks forming boundaries to several fields 
 should be surveyed, and its breadth, where it en- 
 ters, and where it leaves the estate, be determined 
 by the rules of trigonometry. 
 
 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 betw^een 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 tQ survey in the lane, because in 
 so doing, you can take the offsets and remarks 
 both on the rio;ht hand and the left, and therebv 
 carrv 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 sta- 
 tion 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 out- 
 side of the grounds, or on the lanes, &c. noting 
 ahvays every remarkable object in the field 
 book. 
 
 Example. Let fg. 5, plate 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 di- 
 rected from A to B, from B to C, and so on to 
 D, E, F, G, H, I, K, L, M and A, quite round 
 the estate.
 
 313 TO SUkVEY AN ESTATE, 
 
 Then proceed along the lane from A to N, O 
 and I, setting the courses, distances, offsets, and 
 remarks, as before. 
 
 This done, proceed to the internal divisions, 
 beginning at any convenient place, as at O, and 
 proceed, always taking your notes as before direct- 
 ed, from O to P, Q and R, so will you have with 
 the notes previously taken, the dimensions of the 
 north lield; go back to Q, and proceed from Gl 
 to H, and you ol)tain the dimensions of the copse. 
 Take E S and S P, and you will have the dimen- 
 sions of the home field; go back to S, and take 
 S T, and you W'ill have the dimensions of the land,^ 
 . applied to domestic purposes of buildings, yards^ 
 gardens, and orchards, the particulars and sepa- 
 rate 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 off- 
 sets 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 mea- 
 dow; then goto L, and take LX, which gives 
 you the dimensions of the east meadow, and of 
 the great field ; and lastly, take the internal divi- 
 sions of the land, appropriated to the domestic 
 purposes of buildings, yards, orchards, gardens, &c. 
 The method of taking the field notes is so in- 
 tirely similar to the examples already given,that they 
 xyould be altogether unnecessary to repeat hei-c.
 
 MANOR, AND LORDSHIP. 31^ 
 
 To the surveyor there can need no apology for 
 introducing, in this place, the method used by 
 Mr. Milne, one of the most able and expert sur- 
 veyors of the present day; and I think he will con- 
 sider hmiself obliged to Mr. Milne, for communi- 
 cating, with so much liberality, his deviations froin 
 the common 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 practi- 
 tioner. As every man can describe his own me- 
 thods, in the clearest and most intelligible manner, ' 
 I have left Mr. Milne's in his own language. 
 
 Mr. Milne's method of surveying. 
 
 " The method I 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 ou 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 instru- 
 ments, the latter by diligence and long practice. 
 
 From twenty years experience, in the course of 
 which I have tried various methods, tlie following 
 is what I at last adopted, as the most eligible for 
 carrying on an extensive survey, either in England, 
 Scotland, 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 mea- 
 sure a circuit of three or four miles, I begin at a 
 point convenient for placing the theodolite upon, 
 and making a small hole iu the ground, as at A^
 
 3'ia MR. MILXe's METHOD 
 
 pliife 20, one assistant leading the chain, the other 
 having a spade for making marks in the ground, I 
 measure in a direct line from the hole 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, 400 links, touching the 
 south side of the road, with an ofi'set of 40 links to 
 the other side, 500 links to the corner of a wood, 
 and 817 to the corner of another wood; which 
 last not allowing me to carry the line farther, I 
 make a mark at B, where I mean to plant the in- 
 strument, and beginning a new line, measure along 
 the side of the road 885 links to C, where I make 
 a mark in the gfound, writing the same in my field 
 sketch, taking care always to stop at a place from 
 which the last station can be seen, when a pole is 
 placed at it. Beginning a new^ line, and measur- 
 ing in a direct line towards D, I have 345 links to 
 a line of trees, SQO links opposite a corner of paled 
 inclosurcs, wliere I make a mark to have recourse 
 to, and 8C)3 links to the end of the line, where \ 
 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 theodo- 
 lite, I may take the depression thereof; continuing 
 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 th^ 
 left, and here also the line comes to the north 
 side of the road, and 1388 links to the end of the 
 line at E, 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. 'J'he holes, or marks made 
 in the ground, are represented by round dots on
 
 OF SURVEYING. 3'li 
 
 the field sketch ; steepness of ground is expressed 
 on the sketch by faint strokes of the pen, as at 
 D, F, &c. Figures ending a station line are writ- 
 ten larger than the intermediate ones; offsets arc 
 written down opposite the places they were taken 
 at, and are marked either 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 inserting 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 
 myself. 
 
 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 decli- 
 vities that have been measured up or down. 
 
 Having previously prepared a sheet of Dutch 
 paper with meridian lines drawn upon it, as in 
 f>lafe 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 nee- 
 dle may settle steady at the magnetic north, which 
 is indispensably necessary at first setting off, at 
 the same time taking: care that no iron is so near 
 the place as to attract the needle.
 
 321 MR. MILNE S METHOD 
 
 The best theodolite for this purpose is the large 
 onej set Jig. 2, 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 
 coincides with 180° and 36o°, and with the mag- 
 netic 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 in 
 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. 
 
 Then unscrewing the moveable index by the 
 aforesaid nut H, the first assistant having been 
 previously 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 teles- 
 cope 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 instrument F, find it to be 40*^ 55' 
 JS. AV. I then take my sheet of paper, and placing 
 the horn protractor upon the point A, plate 21, 
 along the meridian 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 off
 
 OF SURVEYING. 323 
 
 the length of the linc^ which I find to be, by my 
 first sketch, f>lale '20, 7690 Hnks ; and writing both 
 bearing and distance down, as in plate "Zl, and 
 again reading otf 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 see that it points on the same 
 object as at first, I read the angle upon the limb 
 of the theodolite, 48° 5b' S. E. and plotting it off 
 upon the sketch with the horn protractor, .draw a 
 strait line, and prick off by the scale the length to 
 B, 8I7 links; writing the same down on the field 
 sketch, and again reading the angle, to see that I 
 have wrote it down right, I screw fast the move- 
 able index to the limb, by means of the screw H, 
 and making a signal to the second assistant to pro- 
 ceed with his pole, and plant it at the third sta- 
 tion C, wliile I with the theodolite proceed to B, 
 leaving the first assistant with his pole where the 
 instrument stood at h. 
 
 Planting the instrument by means of a plumb- 
 line over the hole, which the pole made in tlie 
 ground at the second station, and holding the 
 moveable index at 48° 55' as before, and the limb 
 levelled, I turn the limb till the vertical wire in 
 the telescope cuts the pole at A nearly, and there 
 screw it flist. 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 re- 
 mark what distant object it points on, 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 
 
 Y 2
 
 .1'24 Mil. MILNE S METHOD 
 
 the index. Reading the degrees and minuter; 
 which the index now points to on the Hmb, I plot 
 it in my field sketch as before, the bearing or angle 
 being 48° 30', and the length 885 links. Sending 
 the theodolite to the next station C, the same ope- 
 ration 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 declivity 600 links below it, 
 which I perform thus: Having the instrument 
 and the double quadrant level, I 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 
 iit the bottom of the declivity, and making the 
 cross wires of the telescope cut the same part of his 
 body equal the height of the instrument, I find 
 the depression pointed out upon the quadrant to 
 ])e f of a link upon each chain, which upon six 
 chains is four links to be subtracted from 1388 
 links, the measured length of the line, leaving 
 1384 links for the horizontal length, which I 
 mark down in my field sketch, plafti 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 otFthe same in my field sketch, 'draw 
 a strait line; the same temple being seen from 
 station L, and station M, I take bearings to it; 
 from each; and from the extension of these three 
 bearings intersecting each other in the point O, is 
 a proof that the lines have been truly measured, 
 and the angles 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 
 tlepression ; and do the same with every consi- 
 derable rise and fall round the circuit*
 
 OF SURVEYING. 325 
 
 Coming to station N, and havino: the moveable 
 index at dl° 15' N. W. taking up the pole at M, 
 and making fast the limb, I loosen the index serew, 
 and turn it till I take up the pole at the first star 
 tion A. This bearing ought to be 40° 55' N. E. 
 being the same number of degrees and minutes it 
 bore from A south-west, and if it turn out so, or 
 within a minute or two, we eall 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 the circuit again, 
 till I find out the error. But an error will seldom 
 or never happen with such an instrument as here 
 described, if attention is paid to the lower teles- 
 cope; and besides, the needle will settle at the 
 same degree (minutes by it cannot be counted) 
 in the box, as the moveable index will point out 
 on the limb, provided it is a calm day, and no 
 extraneous matter to attract it. If it does not, I 
 then suspect some error has been conmiitted, 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 2\, 
 nearly to their just proportions; and then I am 
 ready to proceed upon surveying the interior parts 
 of the circuit. For doing this, the little light 
 theodolite represented at Jig. 4, plate 15, or the 
 jnore complete one, Jig, 7, plate M, will be suffi-
 
 326 MR. Milne's method 
 
 ciently accurate for common surveying. It is un- 
 necessary for me to describe farther the taking the 
 angles and mensurations of the interior subdivi- 
 sions of the circuit, but in doing of \\hich, 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 small ness 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 pciper, there would then be room for en- 
 tering 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 find 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; the thick or great wire drawing or 
 lengthening more than the small. 
 
 The careful surveyor, if he has more than two 
 or three days 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 off any link on 
 one side, 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 ; 
 beyond this^ an error from small beginnings in the
 
 OF SURVEYING. 
 
 327 
 
 ■mensuration becomes very sensible, notwitlistand- 
 ing the utmost care; therefore, surveying hu-gcr 
 tracts of countiy, as counties or kingdoms, requires 
 ii different process. Surveys of this kind are made 
 from a judicious scries of triangles, proceeding 
 from a base line, in Icn^rth not less than three 
 miles, measured upon an horizontal plane with the 
 greatest possible accuracy. Thomas Milne^ 
 
 The folLowiu^ Tables are inserted for the occasiorral use of the 
 yomig Surveyor. 
 
 1. A Tabic of Long Jllcasura. 
 
 7.g2 inch 
 
 1 link 
 
 
 
 
 
 
 
 12 
 
 lii 
 
 1 foot 
 
 
 36 
 
 4A 
 
 3 
 
 1 )'anl 
 
 
 igs 
 
 25 
 
 liil 
 
 '1 ' 
 
 1 pole 
 
 
 792 
 
 100 
 
 66 
 
 22 
 
 4 
 
 1 chain 
 
 
 7920 
 
 1000 
 
 660 
 
 220 
 
 40 
 
 10 
 
 l.furl. 
 
 
 63360 
 
 8000 
 
 52S0 
 
 1760 
 
 320 
 
 80 
 
 8 
 
 1 mile 
 
 
 2. A Tabic of Square Measures. 
 
 
 9 feet 
 
 \ yard ' 
 
 
 
 
 •^72i 
 
 30^ 
 
 1 perch 
 
 
 
 
 4356 
 
 484 
 
 , 16 
 
 1 chain 
 
 
 
 
 IO89O 
 
 1210 
 
 40 
 
 2} 
 
 1 ro(xl 
 
 
 
 43560 
 
 4840 
 
 160 
 
 10 
 
 4 
 
 1 acre 
 
 
 27878400 
 
 3097600 
 
 102400 
 
 6-^00 
 
 2560 , 
 
 640 
 
 1 mile 
 
 In these Tables, each perpendicular column is of one denomination through- 
 oirt; and all the lateral ones are equal, but of different denominations. 
 
 JvB. Perch the same as pole, and a s(juare chain i$ the tenthp^tof aiiac;e.
 
 [ 328 ] 
 
 OF 
 
 PLOTTING, 
 
 OR MAKING A DRAUGHT OF THE LAND FROM 
 THE FIELD NOTES, 
 
 OP PLOTTING, AND OF THE INSTRUMENTS 
 USED IN PLOTTING. 
 
 By plotting, we mean the making a draught of 
 the land from the field notes. As the instruments 
 necessary to be used by the surveyor in taking the 
 dimensions of land, are sueh wherewith he may 
 measure the length of a side, and the quantity of 
 an angle in the field; so the instruments eom- 
 inonly 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 parts for 
 laying down the lengths or distances, ^n^ protrac-^ 
 tors 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 I'l 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, I, 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 
 backwards and forwards; the commencement of
 
 OF PLOTTING. 32^ 
 
 tlie scale is about two or three of tlie larger divi- 
 sions from the fore end of tlie scale; these are 
 numbered backwards from o towards the left hand 
 with the iRimbers or figures 1, 2, 3, &c. these 
 scales are often sold in sets. Sccj^g. 2, plt^tt; 22. 
 
 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 pro- 
 tracting pin, point off from the scale the given 
 distance in chains, or chains and links. 
 
 Fig. F G H, f/ate 2, represents a new scale or 
 rather scales of equal parts, as several may be laid 
 down on the same instrument; each scale is di- 
 vided to every ten links, or tenths of a chain, 
 which are again subdivided by their respective no^ 
 nius divisions into single links; the protracting pin 
 is moved with the nonius by means of the screw ^, 
 so that the distances may be set off with such 
 great accuracy, as not to err a single link in set- 
 ting off any extent, which in the scale of four 
 chains in an inch, does not amount to more than 
 the 400th part of an inch. 
 
 F/g. I K L, plate 2, answers the same purpose as 
 the foregoing instrument, and may be used as a 
 j)rotractor also. 
 
 For Mr. Gales method of plotting, a method 
 which will recommend itself to every attentive 
 surveyor, two scales should be used, one of about 
 15 inches long, the other about 10 inches, each 
 divided on the edge from one end to the other; a 
 clasp should be fixed on the shorter scale, w hereby 
 it may at pleasure be so lixed to the other scale, 
 as to move along the edge thereof at right angles, 
 for the purpose of laying off perpendicular lines.
 
 S30 OF PLOTTING. 
 
 Fi^.F GH, plaU 2, represents an instrument 
 for this purpose, with different scales on the part 
 F G; boih thiS; and that represented at Jig. IK L, 
 form excellent parallel rules. They arc also made 
 only with the clasp, if desired. 
 
 The protractor is a circle or semicircle of thin 
 brass, divided into degrees, and parts of a degree, 
 on the outer edge. 
 
 The common semicircular protractor, Jig. 2, 
 ^hite 3, is of six or eight inches diameter, the limb 
 divided into 180 dcgTees, and numbered both 
 ways, 10, 20, 30, &c, to 180; each degree is sub- 
 divided mto two parts. In the middle of the dia- 
 meter is a small mark, to indicate the center of the 
 protractor; this mark must be always placed on 
 the given angular point. 
 
 The common circular protractor is more useful 
 then the semicircular one; the outside edge is di- 
 vided into 360 degrees, and numbered 10, 20, 30, 
 to 360 ; each degree is subdivided into halves. 
 In the middle of the diameter is a small mark, 
 which is to be placed on the angular point, when 
 an angle is to be protracted; the diameter, repre- 
 senting a meridian, must be placed on the meri- 
 dian of any plan, where the bearing of any object 
 is to be laid down. The application of this instru- 
 ment is so easy and simple, that examples of its 
 use are unnecessary here. 
 
 Fig. 7,p^iiic 1 1 J and 4 and 5, pJafe 17, represent 
 the three best circular protractors; the principle 
 is the same in all ; the difference consists in supe- 
 riority of execution, and the conveniences that 
 arise from the construction. Fig. 4, plate 17, is 
 a round protractor, the limb accurately divided to 
 360°, each degree is divided into two parts, which 
 are again subdivided to every minute by the no- 
 nius^ which moves round the limb of the protrac-
 
 OP PLOTTING. 331 
 
 tor, on <i conical center; that part of the index 
 "beyond the limb has a steel point tixed at the end, 
 Avhosc use is to prick oif the angles; it is in a di- 
 rect line with the center of the protractor, and the 
 index division of the nonius, all which is evident 
 from an inspection of the figure. 
 
 OF PLOTTING, OR MAKING A DRAUGHT OP THE 
 LAND FROM THE FIELD NOTES. 
 
 The common method of plotting is this : t:ikc a 
 sheet of paper of convenient size, draw a line 
 thereon, to represent the magnetic meridian, and 
 as^ign any convenient point therein to represent 
 the place where the survey commenced; lay the 
 edge of the protractor on this meridian line, and 
 bringing the center thereof to the point so assigned 
 to represent the place of beginning, mark off the 
 defrrces and minutes of the first course or bearins: 
 by the limb of the protractor, and draw a line 
 from the place of beginning through the point so 
 marked, laying off its proper length or distance 
 by the scale of equal parts; this line will reprcseiit 
 the first line of the survey. Through the point or 
 termination of the said first line of the survey, 
 draw another line, representing the magnetic me- 
 ridian, parallel to the former; and lay off the 
 course or bearing of the second line of the survey 
 by the protractor, and its length by the scale of 
 equal parts, as before; and so proceed, till the 
 whole be laid down; and 3-ou will have a plot or 
 draught of the survey. 
 
 For example. Let it be required to make a plot 
 or draught of the field notes. 
 
 1st. Draw any line, as NAS, fg. 1, plate 18, 
 to represent the magnetic meridian, and assiga
 
 33^ bP^ rLOTTING. 
 
 any convenient point therein, as A, to represent 
 the place of beginning the survey; lay the edge of 
 the protractor on the line NAS, with the center 
 thereof at the point A, and mark oW seven degrees 
 on the limb to the westward, or left hand of the 
 north, and draw the line AB through the point 
 so marked off, making the length thereof 21 
 chains by the scale of equal parts. 2nly. Draw 
 another meridian line N B S, through the point B, 
 parallel to the former.* Lay the edge of the pro- 
 tractor on this second meridian line N B S, with 
 the center thereof at the point B, and mark off 
 55" 15' on the limb to the right hand, or east- 
 ward of the north; and draw the line B C through 
 the point so marked off, making the length thereof 
 18 chains 20 links by the scale of equal parts. 
 3dly. Draw another meridian line N C S, through 
 the point C, parallel to the former; lay the edge 
 of the protractor to this third meridian line N C S, 
 with the center thereof at the point C, and lay 
 down the third course and distance C D, in the 
 same manner as before; and so proceed with all 
 the other lines, D E, E F, F G, GA; and if the 
 last line shall terminate in the place of beginning, 
 the work closes, as it is called, and all is right. But 
 if the last line do not terminate in the place of 
 beginning, there must have been a mistake either 
 in taking the notes, or in the protraction of them : 
 in such case, therefore, it will be necessary to go 
 over the protraction again, and if it be not founrl 
 then, it must of course be in the field notes, to 
 correct which, if the error is material, they must 
 be taken again. 
 
 "^ These parallel lines m^y be drawn with the least tronble 
 by a parallel rule.
 
 [ 333 ] 
 
 Mr. Gale s method op plotting. 
 
 The foregoing method of plotting is liable to 
 sonic inaccuracies of practice, on account of hav- 
 ing a new meridian for every particular line of the 
 survey, and on account of hiying off every new 
 Imcfrof/i the point of term'mat'vm of the preceeding 
 oiie, whereby any httle inaccuracy that may hap- 
 pen in laying down of one line is communicated 
 to the rest. But there is a second method of plot- 
 ting, by which these inconveniences are avoided, 
 and by which also the accuracy or inaccuracy of 
 the field work is decided with precision and cer- 
 tainty; I would, therefore, recommend this second 
 method to the practitioner, as far preferable to any 
 other I have seen. 
 
 'id Method. Take out from the first table in the 
 j4ppcndix to this Work the northings^ southings^ enst- 
 wgs, and zvestings, made on each of the several lines 
 of the survey, placing them in a kind of table in 
 their respective columns; and, if the sum of the 
 northings be equal to the sum of the southings, and 
 the sum of the eastings equal to the sum of the 
 westings, the work is right, otherwise not.* 
 
 Then in an additional column put the whole 
 quantity of northing or of southing made at the 
 termination of each of the several lines of the sur- 
 
 * The truth of this observation cannot but appear self-evi-^ 
 dent to the reader. For the meridians within the limits of an 
 ordinary survey having no sensible difference from parallelism, 
 it must necessarily follow, that if a person travel any way soe- 
 ver with such small limits, and at length come round to the 
 'place where he sat 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 surveyor will always find it 
 difficult to make his work close with this perfect degree of t^ 
 actnesa.
 
 334 MR. gale's method 
 
 vey; which will be determined by adding or sub- 
 tracting the northing or southing made on each 
 particular line, io ov from the northing or southing 
 made on the preceding line or lines. And in 
 another 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 determined in like manner by adding or sub- 
 tracting the easting or westing made on each par- 
 ticular line, to or from the easting or westing made 
 on the preceding line or lines. 
 
 The whole quantity of the northings or south- 
 ings, and of the eastings or ^vcstings, 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, however, to use a pair of scales with a 
 clasp,,* whereby the one may at pleasure be so fas- 
 tened to the other, as to move along the edge 
 thereof at right angles, so that the one scale may 
 represent the meridian, or north and south line, 
 the other, an east and west line. It may also be 
 observed, 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 represents the meridian ; or, in other words, 
 the stationary scale should represent an assiuned 
 merid'um, laying wholly on one side of the survey. 
 On this account it will be necessary to note in a 
 third additional column of the preparatory table, 
 the distances of each of the corners or termina- 
 tions of the lines of the survey, east or west from 
 such assumed meridian. In practice, it is rather 
 
 ■" Or one of those at f^. F G H, or IK L, plats 2.
 
 OF PLOTTING. 335 
 
 more convenient that this assumed meridian should 
 lay on the west side, than on the east side of the 
 survey. 
 
 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 mag- 
 netic meridian which passes through the place of 
 beginning the survey. In order, therefore, that 
 the assumed meridian should lav entirely without 
 the survey, and on the west side thereof, it will only 
 be necessary that its distance from that meridian, 
 which passes through the place of beginnina:, 
 should be somcwjiat greater than the greatest 
 quantity of westing 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 co- 
 lumn, be placed at the top of the tliird additional 
 column, to represent the distance of the place of 
 beginning of the survey from the assumed meri- 
 dian. Let the several eastings, contained in the 
 second additional colun^in, be added to this as- 
 sumed number, and the several westings subtracted 
 from it ; and these sums and remainders being 
 respectively placed in the third column, will shew 
 the distance of the several corners or terminations 
 of the lines of the survey from the assumed meri- 
 dian. 
 
 The preparatory table being thus made, take a 
 sheet of paper of a convenient size, or tv/o 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 Y/hich lay the stationary scale. Place the 
 moveable scale to any convenient point on th^
 
 33 G 
 
 Mk. GALE S METH01> 
 
 edge of the former, and point off by the edge of 
 the latter, at^cording to any desired number of 
 chains to an ineh, the assumed distance or num- 
 ber of chains contained at the top of the above- 
 mentioned third column of the preparatory table. 
 Move tlic moveable seale along the edge of the 
 stationary one, to the several north or south dis- 
 tances contained in the first of the above-menti- 
 oned additional colunms; so will the points thus 
 marked off represent the several corners or termi- 
 nations of the lines of the survey; and lines being 
 drawn fi-om one point to the other, will of course 
 represent the several lines of the survey. 
 
 Exanij>le. Let it be required to make a plot or 
 drdught of the field notes, p. 222, according to 
 Mr. Gules method. 
 
 The preparatory table will be as follows : 
 
 
 
 N. and S. 
 
 E. and W. 
 
 Distance^ 
 
 
 
 distances 
 
 distances 
 
 of the ends 
 
 Courses and distan- 
 
 Xorthings, southings, 
 eastings, and westings, 
 made on those respective 
 
 from thc 
 
 fr.ini tlic 
 
 of cacli 
 
 ces as taken in the 
 held. 
 
 place of 
 beginning 
 
 place of 
 beginning 
 
 line from 
 the as- 
 
 courses and distances. 
 
 to the end 
 
 ti> the end 
 
 sumed 
 
 
 
 of each 
 
 of each 
 
 meridian. 
 
 
 
 line. 
 
 line. ( 
 
 
 N. 
 
 S. 
 
 E. 
 
 W. 
 
 
 
 9 ch. 
 
 
 
 
 
 
 E. 3 00 
 
 I.N. 7 \V. 2 100 
 
 20 84 
 
 
 
 2 56 
 
 X. 20 86 
 
 \V. 2 56 E. 44 
 
 2. N. ^^ 15 E. 18 20' 10 37 
 
 
 '4 9' 
 
 
 N. 31 21 
 
 E. 12 39 15 39 
 
 3. S. 62 30 E. 1440 
 
 
 665 
 
 12 77 
 
 
 N. 24 56 
 
 E. 25 16 
 
 28 16 
 
 4. S. 40 W. 1 1 
 
 
 S43 
 
 
 7 07 N. 16 13 
 
 H. 18 09 
 
 21 09 
 
 ;;. S. 415 E. 14 
 
 
 13 96 
 
 I 04 
 
 |N. 2 17 
 
 E. 19 13 
 
 22 M 
 
 6. N. 73 4^ W.I 2 40 
 
 3 47 
 
 
 
 11 90! N, 5 64 
 
 E. 7 23 
 
 10 25 
 
 7. S. 52 "W. 917 
 
 
 5 (^4 
 
 
 723 
 
 00 
 
 00 
 
 3 00 
 
 
 34 68 34 6S 28 7O 2i5 76 
 
 
 
 
 The northings and southings, eastings and west- 
 ings in the above table, are taken from the first 
 table in the Appendix ; thus, first fmd the course 
 7 degrees in the table, and over against 21 chains, 
 in the column marked dist. you nave 20.843 in the 
 column marked N. S. which, rejecting the right 
 hand figure 3 for its insignificancy, is 20 chains
 
 OF PLOTTING. 33/ 
 
 84 links for the quantity of northing, and in the 
 coknnn marked E. W. you have '2.559, very near 
 1 ehains and 5(3 links for the westing made on that 
 course and distance. 2. Find the next course 
 55.15 in the table, and over against 18 ehains ia 
 the column dist. you have 10 chains 26 links in 
 the column marked N. S. and 14 chains 79 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 l6 links in the column 
 marked E, W. which, put together, make 10 
 chains 37 links for the northing, and 14 chains 
 95 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 each line, con- 
 tained in the next column, are deteVmined thus. 
 On the first course, 20 chains and 84 links of 
 northing was made; on the second course, lOchains 
 37 links of northing, which, added to the pre- 
 ceding, makes 31 chains 21 links of northing; on 
 the third course was made 6 chahis Qo 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, con- 
 tained in the next right hand column, are deter- 
 mined in the same manner; thus, on the first 
 course Avas 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 remains 12 chains 39 
 links of easting; on the third course was made 
 12 chains 77 links of easting, which added to the 
 preceding 12 chains 39 links, makes 25 ctiains 
 16 links of easting; and so of the rest.
 
 338 MR. GALE S METHOD 
 
 The distances of the end of each line from the 
 assumed meridian, contained in the next right 
 hand coUimn, are thus determined. The first as- 
 sumed nuniber may be taken at pleasure, provided 
 only that it exceeds the greatest quantity of west- 
 ing contained in the preceding column, whereby 
 the assumed meridian shall be entirely out of the 
 survey. In the foregoing example, the greatest 
 quantity of westing contained in the preceding 
 column is 2 chains 5(5 links; the nearest whole 
 number greater than this is 3 chains, which is ac- 
 cordingly taken and placed at the top, to repre- 
 sent the distance between the assumed meridian 
 and the place of beginning of the survey; from 
 this 3 chains subtract 2 chains 56 links of west- 
 ing, there remains 44 links for the distance be- 
 tween the terminations of the first line and the 
 assumed meridian. The 12 chains SQ links of 
 casting in the next step, is added to the assumed 
 3 chains, which make 15 chains 3() links for the 
 distance of the termination, by the second line 
 from the assumed meridian. The 25 chains l6 
 links of casting in the next step, being added in 
 like manner to the assumed 3 chains make 28 
 chains 1(3 links for the distance of the termination 
 of the third line from the assumed meridian ; and 
 so on, alu'ays adding the eastings and subtracting 
 the westinu-s from the first assumed number. 
 
 The preparatory table being completed, take a 
 sheet of paper or more, joined together if neces- 
 sary, and near the left hand edge thereof rule a 
 line as N. S. fig. 2, flate 18, to represent the as- 
 sumed meridian ; on this line lay the stationary 
 scale, and assuming a 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
 
 OP PLOTTING. 339 
 
 the last-mentioned column of the preparatory ta- 
 ble, viz. 3 chains, from a to A, and A will represent 
 the place of beginning of the survey. Move the 
 moveable scale along the stationary one to the first 
 north and south distance from the place of begin- 
 ning, viz. 20.84 from a to b, and lay off the cor- 
 responding distance from the assumed meridian, 
 0.44, from b to B, and draw AB, so will AB re- 
 present the first line of the survey; 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 off 
 the corresponding distance from the assumed me- 
 ridian, viz. 15.39, f^om c to C, and draw BC, so 
 will B C represent the second line of the survey. 
 Again move the moveable scale along the station- 
 ary one to the third north and south distance 
 from the place of beginning, viz. 24.56, from a 
 to d, and lay off the corresponding distance from 
 the assumed meridian, viz. 28.16, and draw CD, 
 which will be the third line of the survey. Pro- 
 ceed in the same manner till the whole be laid 
 down, and ABCDEFG will be the required 
 plot. 
 
 This inethod of ph/thig is hy far the most perfect^ 
 a)id the least liahle to error of any that has been con- 
 trived. It may appear to some to require more 
 labour than the common method, on account of 
 the computations required to be made for the 
 preparatory table. These computations are how- 
 ever made with so much ease and expedition, by 
 the help of the table in the appendix^ that this ob- 
 jection would vanish, even if the computation were 
 of no other use hut merely for plotting^ hut it must he 
 observed that these computatiojis are of much further 
 use in determining the area or quantity of land 
 contained in the survey, which cannot be J^scer- 
 
 z 2
 
 310 OBSERVATIONS ON PLOTTING, 
 
 taincd with equal accuracy in any other way. 
 When this is considered, it will be found that this 
 method is not only preferable on account of its 
 superior accuracy, but is attended with less labour 
 on the whole than the common method. 
 
 If a pair of scales, such as are above recom- 
 mended, 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 after- 
 wards laying off the corresponding east distances 
 at right angles thereto. 
 
 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, 
 represented \nfig, 3, pl.lQ, 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 
 fig. 3, a bare inspection of which will make the 
 work perfectly plain. 
 
 To plot the field notes, p. 314. 1. Lay down the 
 whole of the outside boundaries, the station lines 
 first, the offsets afterwards, as directed in the pre- 
 ceding articles; and then lay down the internal 
 divisions or boundaries of the respective fields, in 
 the same manner as is done in j*?^. 4. 
 
 OBSERVATIONS ON PLOTTING, BY Mr. MiLNE. 
 
 The protractor, whether a whole or a semicir- 
 cle, 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°.
 
 BY MR. MILNE. 341 
 
 It would be right to repeat the numbers the con- 
 trary way, when the breadth of the limb will ad- 
 mit of it. Others are numbered 10° 20°, &c. to 
 180°, and the contrary. Surveyors will suit them- 
 selves with that kind best adapted 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 
 off an angle mentally very readily, in which ever 
 way the instrument happens to be graduated. 
 
 For instance, I have in the field taken the an- 
 gles made with the magnetic meridian, in which 
 case I count no angle above 89° 59'; if the angle 
 comes to 90° O', I call it due E. or due W. if it is 
 89° 59', I write to it N. E. or S. E. or else N. W. 
 or S. W. as it happens to turn out. 
 
 Now 89° 59' N. E. upon a protractor numbered 
 to 360° O', and placina: 36o to the north, reads the 
 same; but 89° 59' S. E. reads 90° V, and 8Q° 59' 
 S. W. reads '269° 59'; 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° T S. E. it reads 
 179° 59'; and if 0° l' S. W. it is 180° l'; and so 
 of any other intermediate angle or bearing. 
 
 The great inducement to surveyors for taking 
 the angles by the bearings the lines make with 
 the magnetic meridian, is the having the r^eedle as 
 a check in the course of the survey ; and when the 
 circuit comes to be plotted, having it in his power 
 to prick off all the angles, or bearings of the cir- 
 cuit, at once planting of the protractor. 
 
 This mode of sui-veying, or taking angles, is 
 also more expeditious, and more accurate than 
 any other, provided the index of the instrument
 
 342 OBSERVATIONS ON PLOTTING, 
 
 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 survey- 
 ing 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 
 protractors for general use are graduated 10, 20, 
 &c. to 360, I shall here describe a very useful and 
 convenient one of this latter kind, and then pro- 
 ceed to give an example of plotting all the angles 
 of a circuit by it from one station. 
 
 This protractor is represented at j^^-. 5, plate 17. 
 Its diameter, from pointer to pointer, is 9^ inches; 
 the center point is formed by two lines crossing 
 each other at right angles, which are cut on a 
 piece of 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 fur- 
 nished 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 poin- 
 ter, 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 traverses but 
 one half of the protractor.* 
 
 * The protractor, represented^. T, plate 11, is circular, and 
 of the same diameter as the foregoing j the center 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 j 
 the index is moved round by teeth and pinion. The limb is 
 divided into degrees and half degrees, and subdivided to eveiy 
 minute by the nonius 5 the pointer maybe set at any convenient 
 distance from the center^ as the socket which carries it moves
 
 BY MR. MILXE. 343 
 
 Let^?^-. 5, plate 13, be the circuit to be plotted. 
 I draw the magnetic meridian N. S.Jig.6, plate\3, 
 and assigning a point therein for station 1, place 
 the center 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 conveniency of more easily reading off" de- 
 grees and minutes, I bring the nonius to the south 
 side of the protractor, or ^ide next to me; and 
 seeing by the field sketch, or eye draught, j^^)-. 5, 
 that the bearing from station 1 to station 2 is 87" 
 30' S. E. I readily conceive it wanting 2° 30' of 
 being due east, or 90° o'; therefore, adding men- 
 tally 2° 30' to 90° O', I make 92° 30' on this pro- 
 tractor, to which I bring the nonius, and then 
 make a prick by pressing down the pointer of the 
 protractor, and with my lead pencil mark it 1. 
 The bearing from station 2 to station 3 being 10° 
 15' N. E. sec Jig. 5, I bring the nonius, for the 
 sake of expedition and ease of reading off, to 10° 
 15' S. W. which is upon this protractor 190^ 15', 
 and pressing down the north pointer, it will prick 
 off 10° 1 5' N.E. 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 east; therefore I bring the nonius to 
 within that of due west, which upon this protrac- 
 tor is 265° 13', and pressing down the N.E. 
 pointer, it will prick off 85° 13' N.E. which I 
 mark 3. 
 
 upon the bar B C, and is fixed thereto by the nut D, At right 
 angles to the bar B C, and moveable with it, there is another bar 
 E F; on this bar different scales of equal parts are placed, ^othat 
 by moving a square against the inner edge thereof, angles may 
 be transferred to any distance from the center, containing the 
 same number of degrees marked out by the index; the use of 
 which will be evident from Mr. Milne's observations.
 
 344 OBSERVATIONS ON PLOTTING^ 
 
 The bearing from station 4 to station 5 being 
 1 1° 5' N. W. I bring the nonius to l68° 55', and 
 with the opposite, or north-west pointer, prick off 
 1 1° 5' N. W. which I mark 4. 
 
 In like manner I prick off the bearings from 
 station 5 to 6, from 6 to 7, and from 7 to 1 , which 
 closes the circuit; marking them severally with 
 my pencil 5, 6, 7. 
 
 Then laying aside the protractor, and casting 
 jny eye about the tract traced by the pointer of 
 the protractor for the bearing from station 1, 
 marked 1, I apply the ruler to it and station 1, and 
 drawing a line, mark off' its length by compasses 
 and scale, eight chains, which fixes station 2. 
 
 Again, casting my eye about the tract traced by 
 the pointer of the protractor, for the 1 bearing, 
 marked 2, I apply the parallel ruler to it and 
 station 1, and moving the ruler parallel 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 3. Thus the 
 bearing 10° 15' N. E. at station 1, is transferred to 
 station 2, as will easily be conceived by supposing 
 the points a, b, moved eastward, preserving their 
 parallelism till they fall into the points of station 
 2 and 3, forming in fig. 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 previ- 
 ous to the ingenious Mr. WiUiam Gardiner com- 
 municating this more expeditious and more accu- 
 rate mode to the ])ublic. However, its accuracy 
 depends entirely upon using a parallel ruler that 
 moves truly parallel, which the artist will do well
 
 BY MR. MILXE. 345 
 
 to look to, before he proceeds to this way of 
 plotting. 
 
 Station 3 being thus fixed, to find station 4, 
 I cast my eye about the tract traced by the pointer 
 of the protractor for the bearing marked 3, and ap- 
 plying 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-east- 
 ward, and by scale and compasses marking its 
 length, three chains, station 4 is found. 
 
 Now conceive the points c, d, moved north- 
 ward, 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 same manner I proceed to the next bear- 
 ing 4 ; and placing the ruler to the point made by 
 the pointer of the protractor at 4 and station 1, 
 and moving the ruler parallel till its edge touch 
 station 4 ; from thence dravv' 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 tlie 4th station, as e, or f, docs with the me- 
 ridian from the first station. 
 
 In like manner, applying the parallel ruler to 
 station 1, and the several other bearings of the cir- 
 cuit 5, 6, and 7 ; the points g, h, will be transferred 
 to 5th and 6th, and i, k, to 6th and 7th; and lastly, 
 the bearing from the 7th 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, &c. fig. 6, are drawn 
 there only for illustrating the operation: all that 
 is necessary to mark in ])ractice are the figures 
 1, 2, 3, &e. round the tract traced by the pointer 
 pf the protractor. T. Milne.
 
 [ 346 ] 
 
 OF 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 Enghsh acre. 
 
 2. The most simple surface for admensuratiorv 
 is the rectangidar parallelogram, which is a plane 
 four-sided figure, having its opposite sides equal 
 and parallel, and the adjacent sides rectangular to 
 one another; if the length and breadth be equal, 
 it is called a square \ but, if it be longer one way 
 than the other^ it is commonly called an ohlong. 
 
 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 me- 
 thod 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 de- 
 cimals more in the product than there arc both in 
 the multiplicand and multiplier', which answers the 
 Bame purpose as dividing the area into chains bylO; 
 
 * Euclid, Book i. Prop. 35 Scholium.
 
 TO DETERMINE THE AREA, ScC. 347 
 
 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 75 links; what is the area? 
 
 Multiply 6.75 
 
 by Q.ys 
 
 3375 
 4725 
 
 4050 
 
 Acres 4.55(325 
 
 4 Roods In an acre. 
 
 Roods 2.22500 
 
 40 Perches in a rood. 
 
 Perches 9. 00000 
 
 Acres, Roods. Perches. 
 Ans. 4 2 Q. 
 
 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? 
 
 Multiply 9.45 
 by 5.15 
 
 4725 
 
 4725 
 
 Acres 4.86675 
 4 
 
 Roods 3,46700 
 40 
 
 Perches 18.68000 
 
 A, R, P. 
 
 Ans. 4 3 18. (The decimal parts .QQ arc not 
 worth regarding.)
 
 548 TO DETERMINE THE AREA 
 
 If the parallelogram be not rectangular, but ob- 
 lique angled, the length must be multiplied by the 
 perpendicular breadth, in order to give the area.* 
 
 3. After the parallelogram, the next most simple 
 surface for admensuration is the triangle. 
 
 TO DETERMINE THE AREA OF A TRIANGLE. 
 
 Rule. 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.-|~ 
 
 Examples. In the triangle 
 ABC, right angled at B, let 
 the base A B be 8 chains and 
 25 links, and the perpendicu- 
 lar BC 6 chains and 40 links; 
 
 C 
 
 how many acres doth it con- 
 tain ? A 
 
 by (the 
 
 half 
 
 A. 
 2 
 
 Multiply 8.25 
 of 6.40, viz.) 3.20 
 
 16500 
 2475 
 
 
 Acres 2.6400O 
 4 
 
 
 Roods 2.56000 
 40 
 
 
 Perches 22.40000 
 
 Ans. 
 
 R. P. 
 
 2 22. 
 
 * See Euclid, Book i. Prop. 35 and 36, 
 
 f Euclid, Book i. Prop. 41, Scholium. Ever}' triangle is 
 equal to the one halt of a parallelogram, having the same base 
 and altitude.
 
 OF A TRIANGLE. 
 
 349 
 
 In the oblique an- 
 gled triangle D E F, 
 let the base D E be 
 12chainsancl25links, 
 and the perpendicu- 
 lar FG, let fall on 
 the base DEfrom the 33 
 ooposite angle 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 
 60'75 
 
 Acres 4.47125 
 4 
 
 Roods 1.88500 
 40 
 
 Perches 35.40000 
 
 A. R. P. 
 
 Ans. 4 1 35. 
 
 It would be the same thing, if the perpendicu- 
 lar 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 admensu- 
 ration 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 
 ihe base by the whole sum of the breadths at each
 
 350 TO DETERMINE THE AREA 
 
 end, and that 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. 
 6 
 
 4.751 
 7.35/ 
 
 2)12.10 sum 
 
 6.05 half sum, 
 
 lo.go 
 
 
 54450 
 6050 
 
 
 Acres 6.59450 
 4 
 
 
 Roods 2.378OO 
 40 
 
 
 Perches 15.12000 
 
 Ans. 
 
 R. P. 
 
 2 15. 
 
 5. Irregular four-sided figures are called irape- 
 zia. The area of a trapezium may be determined 
 by the following 
 
 Rule. Take the diagonal length from one 
 extreme corner to the other as a base, and multiply 
 it by half the sum of the perpendiculars, falling 
 thereon from the other two corners, and that pro- 
 duct will be the area. 
 
 Exajnple. In the trape- 
 zium ABCD, let the 
 diagonal AC be 11 
 chains and 90 links; the 
 perpendicular B F 4 A^ 
 chains and 40 links, and the perpendicular D B 
 3 chains and 90 links; what is the area?
 
 OP A TRIANGLE. 351 
 
 
 B F 4.40 
 DE 3.90 
 
 
 2)8.30 sura 
 
 
 4.15 half sum 
 AC 11.90 
 
 
 37350 
 415 
 415 
 
 
 Acres -1. 93850 
 4 
 
 
 Roods 3.75400 
 40 
 
 
 Perches .^O.ltXXW 
 
 A. 
 4 
 
 R. P. 
 3 30. 
 
 Ans. 
 
 If the diagonal AC were multiplied by the 
 whole sum of the perpendiculars, that product 
 would be double 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 consist, may be determined either by 
 dividing the given figure into triangles, or trapezia, 
 and measuring those triangles, or trapezia, sepa- 
 rately, the sum total of which will be the area re- 
 quired; or otherwise, the area of any figure may 
 be determined by a computation made from the 
 
 * The regular Jigurcs, polygons and circles, do not occur in 
 practical surveying. On this head, therefore, I shall only ob- 
 serve, that the method of determining the area of a regular poly- 
 gon, which is a figure containing any number of equal sides and 
 eqvial angles, is, to multiply the length of one of the sides by the 
 mimber of sides which the polygon contains, and then to multi- 
 ply the product by half the perpendicular, let fall from the center 
 to any one of the sides, and this last product will be the area. 
 
 The length of the perpendicular, if not given;, may be de-ter- 
 mined by trigOQometry, thus:
 
 352 
 
 TO DETERMINE THE AREA 
 
 courses and distances of the boundary lines, ac- 
 cording to an universal theorem, which will be 
 mentioned presently. 
 
 7. The method of determining the area by di- 
 viding the given figure into trapezia and triangles, 
 and measuring those trapezia and triangles sepa- 
 rately, as in article 5 and article 3 of this part, 
 is generally practised by those land 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 bases and per- 
 pendiculars of the several trapezia and triangles in 
 the field. 
 
 Thus, for exam- 
 ple, suppose the an- 
 nexed figure be a 
 field to be measured 
 according to this 
 method, the land 
 measurerwouldraea- ■ 
 sure with the chain 
 the base BG, and the 
 perpendiculars H C, 
 and lA, of the tra- ^ 
 pezium A B C G; 
 the base GD, and the 
 perpendiculars K F, 
 and LC, of the tra- 
 
 Divide 360 degrees by double the number of sides contained In 
 the polygon, and that quotient will be half the angle at the cen- 
 ter: then say, as the tangent, of half the angle at the center, is 
 to half the length of one of the sides j so is radius, to the per- 
 pendicular sought. 
 
 The area of a circle is thus determined j if the diameter be 
 given, say, as 1 : is to 3.141592 : : so is the diameter : to the cir- 
 cumference j or, if the circumference be given, say, as 3.14I5C)2 
 is to 1 :: so is the circumference : to the diameter. Then mul- 
 tiply the circumference by ^ of the diameter, and the product 
 will be the area.
 
 of AN IRREGULAR FIGURE. 3.53 
 
 pczium G C D F; and the base D F, and perpen- 
 dicular M E, of the triangle D E F. 
 
 In order to determine where on the base lines 
 the perpendiculars shall fall^ the land measurer 
 commonly makes use of a square, consisting of a 
 piece of wood about three inches square, with 
 apertures therein, made with a fine saw, from cor- 
 ner to corner, crossing each other at right angles 
 in the center; which, when in use, is fixed on a 
 staff, or the crosss; sccfgs. 2, 3, and 6, plale 14. 
 
 The bases and perpendiculars being thus mea- 
 sured, the area of each trapezium is determined as 
 in article 5, and the area of the triangle as in arti- 
 cle 3 of this part; which, being added together, 
 gives the area of the whole figure required. 
 
 This method of determining the area, \vhcrethe 
 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 mea- 
 sured; this method, however, is practicable only 
 in open or cleared countries; in countries covered 
 with wood, it would be altogether im])racticablc, 
 because it would be impossible to see from one 
 part of the tract to another, so as to divide it into 
 the necessary trapezia and triangles. But in 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 measur- 
 ing of all the several bases and perpendiculars 
 would be a very tedious business. The same ends 
 are as effectually answered, with infinitely less la- 
 bour, by taking the courses and distances of the 
 boundary lines^ in the manner mentioned p. 310. 
 
 A a
 
 354 AX UNIVETxSAL THEOREM. 
 
 8. Many surveyors who take their field notes in- 
 courses and distances, make a plot or map 
 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 
 something nearer the truth, but it falls short of that 
 accuracy which might be wished, because there 
 is no determining the lengths of the bases and per- 
 pendiculars by the scale, within many links; and 
 the smaller the scale shall be by which the plot is 
 laid down, the greater, of course, must be the in- 
 accuracy of this method of determining the area. 
 
 This method, however, notwithstanding its in- 
 accuracy, 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 Appendix to this 
 Tract, ought to be in the hands of every surveyor. 
 By these tables the area of any survey is deter- 
 mined with great accuracy and expedition, by an 
 easy computation from the following 
 
 UNIVERSAL THEOREM. 
 
 If the sum of the distances, on an east and 
 •west line, of the two e7ids of each line of the survey, 
 front any assumed meridian laying entireh out of the 
 survey, be midtiplied by the respective northing or 
 SOUTHING made on each respective line', the dif- 
 ference between the sum of the north pro-
 
 An universal theorem. 
 
 !55 
 
 DUCTS, and the swn of the south products, 'lcv'// 
 be double the area of the survey * 
 
 For example. Let it be required to determine 
 the area of the survey mentioned in p. 310, from 
 the field notes there given. 
 
 The northings, southings, eastings and westings, 
 as also the north and south distances, and the east 
 mid west distances made from the place of begin- 
 ning to the end of each line; and likewise the dis- 
 tances of the ends of each line from the assumed 
 
 * DcmpnstraUou. Let 
 AB C D E F be any survey, 
 and let the east and west 
 lines Aa, B b, C c, Dd, 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. laying J 
 entirely out of the survey. 
 
 Then, I say, the difference b etween th e sum of the products 
 Aa+Bbxab, Bb + Ccxb c, Cc + Dd xcd; an d the sum of 
 the products Dd + Eexde, Ee + Et"xef, Ff+Aaxfaj will be 
 double the area of AB C D E FA. 
 
 For, by article 4, the former products will be, respectively, 
 double the areas ot the spaces ABba, BCcb, and CDdc, and 
 their sum will of course be double the area of the space Aad 
 DCBA. In like manner, the latter products will be, respec- 
 tively, double the areas of the spaces DEed, EFfe, and FAaf> 
 and their sum will of course be double the area of the space 
 AadDEFA. But it is evident, that the diiference between 
 the areas of the spaces AadDEFA, and AadDCBA, must be 
 the area of the space ABCDEFA5 consequently, the dit- 
 ference between their doubles must be double the area of ABC 
 BE FA. 
 
 A a 2
 
 3D 
 
 6 
 
 AN UNITERSAL THEOREM. 
 
 meridian, have been already placed in their respec- 
 tive 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 determining the area; in order 
 that the whole, together with the respective uses 
 of the several parts, may appear under one view, 
 as follows. 
 
 Again, the diflerence be- 
 tween the sum of the products 
 Min + Nn X mn, Nn + Oox 
 no, Qq + Rrxqr; and the sum 
 of the products Oo + Ppxop, /^ 
 
 Pp+ Qqxpq, Rr + Mmxrm; 
 will be double the area of M N ^ 
 O P Q R M. 
 
 '721 
 
 For, reasoning as before, the sum of the fir st mentioned pro - 
 ducts will be double the area of QqoONMRQ + MRrmM; 
 and the sum of the last mentioned products will be double 
 the area of QqoOPQ-|- MRrmM. But it is evident, that 
 QqoOPQ— QqoQNM R Q=MNQPQRM: conseq uently, 
 QqoOPQ-l- MRrmM — QqoONMRa-l- MRrmM = M 
 NOPQ.RM; and, consequently, the difference of their baulks 
 must be equal to double MNOPQRM. And the like in every 
 ©ther possible figure.
 
 FIELD BOOK EXI-ARGED. 
 
 357 
 
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 358 EXPLANATION OP 
 
 The three last columns relating to the areay are 
 the only ones that rpmain to be 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 to the foregoing universal theorem, are 
 7nultipliers for determining the area. These are 
 formed by adding each two of the successive num- 
 bers in the preceding left hand column together. 
 Thus 3.00 is the distance of the place 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 
 28.16 makes 43.55 for the third multiplier; and 
 so of the rest. 
 
 The north products contained in the next co- 
 lumn, are the products of the multiplication of 
 the several norihhigs contained in the columns 
 marked N, by their corresponding multipliers in 
 the last mentioned column. Thus 7.1 6896 is the 
 product of the multiplication of 20.84 by 3.44; 
 an additional place of decimals being cut off, to 
 give the product in acres. 
 
 The south products contained in the next right 
 hand column are, in like manner, the products of 
 the multiplication of the several southings in the 
 column marked S, by their corresponding multi- 
 pliers. Thus 28.96075 is the product of the muU 
 tiplication of 6.65 by 43.55; an additional place 
 of decimals being cut oiF, as before, to give the 
 product in acres; and so of the rest. 
 
 The north products being then added together 
 into one sum, and the south products into another 
 sum, and the lesser of these sums being subtracted 
 frpm the greater, the remainder, by the above
 
 THE FIELD BOOK. 35() 
 
 universal theorem, is drjiihle the area of the survey : 
 which being divided by 2, gives the area required: 
 viz. 51.73087, equal, when reduced, to 51 acres, 
 2 roods, 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, east- 
 ings 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 the plotting are 
 formed by a simple addition or subtraction. And 
 the multipliers, for determining tlie 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 mul- 
 tipliers by the scale of equal parts. 
 
 The superior accuracy and ease with which 
 every part of the process is performed, cannot, it 
 is imagined, fail to recommend it to every practi- 
 tioner, into whose hands this tract shall fall. 
 
 10. If the boundary lines of the survey shall 
 have crooks and bends in them, which is gcnerall}'' 
 the case in old settled countries, those crooks and 
 bends are taken by making offsets from the station 
 lines; as explained in page 311. In these cases, 
 the area comprehended within the station lines is 
 determined as in the last example; and the areas 
 
 * It must be observed, that, in order to tktermine the area 
 by this method of computation, it is indispensibly 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 field, because, without such arrange- 
 ment, the northings, or southings made on each line, and the 
 distances from the assumed meridian, would not correspond with 
 jL»ue another.
 
 3f50 EXAMPLE TO THE 
 
 comprehended between the station lines and the 
 boundaries are determined separately, as in the 
 third 'd\\(\ fourth articles of this part, from the off- 
 sets and base lines, noted in the field book; and 
 are added or subtracted respectively, according as 
 the station lines shall happen to be •within or with- 
 out the field surveyed; which, of course, gives the 
 area required. 
 
 For example. Let it be required to determine 
 the area of the survey mentioned in the example, 
 page 312, from the field book there given.* 
 
 The bearings and lengths oi '^Q, station lines arc 
 the same as in the last example; and, conse- 
 quently, the area comprehended within those sta- 
 tion lines must be the same; viz. 51.73087- 
 
 Then, for the area comprehended between the 
 first station line and the boundary of the field, we 
 have (see the field book, page 313) first, a base 
 line 3.6o, and a perpendicular oflset 0.40; which, 
 according to article 3 of this part, contains an area 
 of 0.07200. Secondly, a base line 4.85, (being the 
 diiFerence between 3.6o and 8.45 on the station 
 line, at which points the offsets were taken,) and 
 an offset or breadth at the one end 0.40, and at 
 the other end 0.10; which, according to article 4 
 of this part, contains an area of 0. 1 2 1 25 . Thirdly^ 
 a base line 7-15, (being the difference between 
 8.45 and 15.6o on the station line,) and an offset 
 or breadth at the one end 0.10, and at the other 
 end 0.65; which according to the 4th article of 
 this part, contains an area of 0.26812. Fourthly^ 
 a base line 5.40, (being the difibrence between 
 ]5.60 and 21.00 on the station line,) and a per- 
 pendicular offset 0.65 ; which, according to the 
 3d article of this part, contains an area of 0.17550, 
 These added together make 0.63687 for the area 
 
 •(*■ A plot of this survey is given in fig, 3, pJaie 18,
 
 UNIVERSAL THEOREM. 
 
 361 
 
 of the offsets on the first station line; and, as this 
 station Hne is within the survey, this area must, of 
 coui-se, be added to the above-mentioned area com- 
 prehended within the station hues. 
 
 Next, for the area comprehended between the 
 second station Hne and the boundary of the fit Id, 
 we have a base Hne 18.20, and a perpendicular or 
 offset 0.60 ; which, according to article 3 of this 
 part, contains an area of 0.54600. And, as this 
 station line is without the survey, this area must of 
 course hcsuhtracted (voin the above-mentioned area. 
 
 The areas of all the other ofisets are determined 
 in the same manner, and are respectively as fol- 
 lows; viz. 
 
 On \.\\cjirst station line 
 On the sttond - 
 On the third 
 On the four lb - 
 On ihcjiftb 
 On the sixth 
 On the scvL7itb - 
 
 Totals - - 
 
 The area comprehended within the station lines, 
 
 as above A 51.7308/ 
 
 Otisets to be added 2.06S12 
 
 Sum 53.81bC).9 
 
 Offsets to be subtracted 2.15660 
 
 Remains the area of the survey required A 51.6623p 
 
 4 
 
 Roods 2.64956 
 
 40 
 
 Perches .. 25.^8240 
 
 Areas ot" the ( )tiscts. j 
 
 To be added. 
 A. O.636S7 
 
 To be sub- 
 tracted. 
 
 0.546(X) 
 O.64S0O 
 
 
 0.36250 
 1 .O8875 
 
 
 0.68750 
 0.27510 
 
 
 
 2.08812 
 
 2.15660
 
 36*2 EXAMPLE TO THE 
 
 II. When the survey consists of a number ot 
 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-whieh, 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. 
 
 Bat here it must be observed, that the boun- 
 daries of each field must be arranged for the com- 
 putation, so as to proceed regularly one after ano- 
 ther all the way round 'the field; because, as was 
 observed in the note to the Qth 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 as- 
 certain the area of the survey mentioned in page 
 314, of the several fields therein contained, from 
 the field book there given.* 
 
 The bearings and lengths of the station lines on 
 the outside boundaries are as follow; viz. 
 
 ch. I. 
 
 o / 
 
 1st. S. 87 W 15.00 
 
 2d. N. 18 30 W 9.60 
 
 3d. N. 12 15 E 8.90 
 
 4th. N. 37 E 17.00 
 
 5th. S. 76 45 E l6.(X) 
 
 6th. S. 5 15 W 25.00 
 
 7th. S. 74 30 W 7.(55 
 
 The area comprehended within these station llne.1, 
 computed according to the above-mentioned univer- 
 sal theorem, is A. 66.90493 
 
 The area of the offsets to be added 0.56"/ \5 
 
 (computed as in the last example) ■ 
 
 Sum 67.4720s 
 
 * See the plot of this survey mfg. 4, plate 18,
 
 UNIVERSAL THEOREM. 363 
 
 (brought forward) Sum 67.472O8 
 
 The area of the offsets to be subtracted 2.50/15 
 
 The area of the whole surv'cy 
 
 Acres 64.90493 
 
 4 
 
 Roods 3.61973 
 
 40 
 
 Perches 24.78880 
 
 Then, for determining the area of the east field, 
 \vc have the following lines, viz. 
 
 1st. Part of the first station line In ch. 1. 
 
 the field book, S. 87° W. 5.50 
 
 2d. The offset to the corner of the 
 
 field, N. 30° \V. 1.10 
 
 3d. The eighth station line in the 
 
 field book, North — 16.61 
 
 4th. The ninth station line in the 
 
 Held book, East — 13-77 
 
 5th, The offset from the corner of the 
 
 field to the sixth station line in the 
 
 field book, S. 84° 45' E. 0.56 
 
 Gth. 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° 30' W. 7.65 
 
 The area comprehended within these lines, com- 
 jmted 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 
 
 Hemains the area of the east field. Acres '20.6g3S4 
 
 4 
 
 Roods 2.79036 
 
 40 
 
 Perches ...... 31.81440
 
 304 EXAMPLE TO THE 
 
 For ascertaining the area of the west field_, \vc 
 have the following lines, viz. 
 
 1st. The offset from the south-east- 
 erly corner to the first station line ch.l. 
 
 in the field book, S. 3' E. 1.10 
 
 2d. Part of the first station line in 
 
 the field book, S. 87° W. Q.SO 
 
 3d. 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. 8.9O 
 
 5th. Part of the fourth station line 
 
 in the field book, N. 37° E. 5.40 
 
 eth. The offset to the comer, S. 53° E. 0.40 
 
 7th. The tenth station line in the 
 
 field book, reversed, S. 62° 30' E. 7.92 
 
 6th. The eighth station line in the 
 
 field book, reversed, South — 16.61 
 
 The area comprehended within these lines, com- 
 puted according to the foregoing universal theo- 
 rem, is A. 22.18048 
 
 The areas of the offsets to be added, computed as 
 in che last example, 0.6228/ 
 
 Sum 22.80335 
 
 - The areas of the offsets to be subtracted 1 .159OO 
 
 Remains the area of the west field. Acres 21.64435 
 
 4 
 
 Roods 2.57740 
 
 40 
 
 Perches 23.O96OO 
 
 For ascertaining the area of the north jfield, we 
 have the following lines, viz. 
 
 Jst. The offset from the south-west- 
 erly corner to the fourth station ch. 1, 
 line in the field book, N. 53° W, 0.40 
 
 2d. Part of the fourth station line in 
 
 the field book, N.37° E. II.60 
 
 3d. The fifth station line in the field 
 
 book, S. 76=45'E. 16.00
 
 UNIVERSAL THEOREM. 365 
 
 4th. Part of the sixth station line in ch.l. 
 
 field book, S. 5° 15' W. g.tjO 
 
 .'Jth. The offset to the corner, N. 84° 45' W. 0,5(5 
 
 0"th. The ninth station line in the 
 
 field book, reversed, West — 13.77 
 
 7th. 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 foregoing universal theo- 
 rem, is . A. 21 .81422 
 
 The areas of the offsets to be added, computed as 
 in the last example, 1 .25030 
 
 Sum 23.064o2 
 
 The areas of the offsets to be subtracted 0.5()08O 
 
 Eemains the area of the north field. Acres 22.5637'i 
 
 4 
 
 Roods 2.25488 
 
 40 
 
 Perches 10.1C)520 
 
 A. R. P- 
 
 The area of the east field, 20 2 3^ 
 
 West field, 21 2 23 
 
 North field, 22 2 lO 
 
 Total A. 64 3 24 
 
 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 computations. 
 
 For other methods of surveying, and a new 
 plan of a field book, see the jiddejida to this 
 work.
 
 [ 366 J 
 
 OF 
 
 MARITIME SURVEYING. 
 
 It was not my intention at first to say any thing 
 concerning maritime surveying, as that subject 
 had been already very well digested by Mr. Mur- 
 doch Alacketizie, whose treatise on marit'pne survey- 
 ing ought to be in every person's hands who is 
 engaged in this branch of surveying, a branch 
 which has been hitherto too much neglected. A 
 few general principles only can be laid down in 
 this work; these, however, it is presumed, will be 
 found sufficient for most purposes; when the prac- 
 tice is seen to be easy, and the knowledge thereof 
 readily attained, it is to be hoped, that it will con- 
 stitute a part of every seaman'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 station staff.* 
 
 Q,. Choose some level spot of ground upon which 
 a right line, called a fundamental base, may be 
 measured 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 
 
 * See Nicholson's Navigator's Assistant,
 
 MARITIME SURVEYING. 067 
 
 situation must be such, that the whole, or most 
 part of the station -staves may be seen from both 
 ends thereof; and its length and direetion must, 
 if possible, be sueh, that the bearing of any station- 
 staif, taken from one of its ends, may ditfcr at 
 least ten degrees from the bearing of the same staff 
 taken from the other end; station-staves must be 
 set at eaeh end 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 con- 
 struction, may be taken as the fundamental base. 
 
 If the sand measured has a sensible and gradual 
 declivity, as from high-water mark to low-w^ater, 
 then the length measured may be reduced to the 
 horizontal distance, which is the proper distance, 
 ])y making the perpendicular rise of the tide one 
 side of a right-angled triangle; the distance mea- 
 sured along the sand, the hypothcnuse; and from 
 thence finding the other side trigonometrically, or 
 by protraction, on paper; which will be the true 
 length of the base line. If the plane measured be 
 on the. dry land, and there is a sensible decliviry 
 there, the height of the descent must be taken by 
 a spirit-level, or by a quadrant, and that made ths 
 perpendicular side of the triangle. 
 
 If in a bay one strait line of a sufficient length 
 cannot be measured, let two or three lines, form- 
 ing angles with each other, like the sides of a po- 
 lygon, 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 strait distance betweeen the two far- 
 thest extremities of the first and last line. 
 
 3. Find the bearing of the fundamental base by 
 the compasses, as accurately as possible, with Had- 
 Icy's quadrant, or any other instiument equally
 
 368 MAniTIME SURVEYING. 
 
 exact; take the angles formed at one end of the 
 base, between the base line and lines drawn ta 
 eaeh of the station-staves; take likewise the an- 
 gles fonned between the base line and lines drawn 
 to every remarkable object near the shore^ as 
 houses, trees, windmills, eharches, &e. 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 objects; if any angle be greater than 
 the arc of the quadrant, measure it at twice, by 
 taking the angular distance of some intermediate 
 object from each extreme object; enter all these 
 angles in a book as they are taken. 
 
 4. Draw the fundamental base upon paper from 
 a scale of equal parts, and from its ends respectively 
 draw unlimitted lines, forming with it the angles 
 taken in tlie survey, and mark the extreme of each 
 line with the letter of the station to which its an- 
 gle 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 rough draught, that letter 
 belongs; through, or near all the points of inter- 
 section which represent statioji staves draw a wav- 
 ing line with a pencil to represent the coast. 
 
 5. At low water sail about the harbour, and 
 take the soundinij-s, obscrvino: whether the ground 
 be rocky, sandy, shelly, &:c. These soundings 
 jnay be entered by small numeral figures in the 
 charty by taking at the same time the bearings of 
 two remarkable objects; in this excursion, be par- 
 ticular in examining the ground off 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 islands, &:e. observe the set 
 jiud velocity of the tide of flood, by heaving the
 
 MARITIME SURVE'i'ING. 3$^ 
 
 log while at anchor, 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 shad- 
 ing, the TigurGS upon which usually shew the 
 depth at low water in feet; good anchoring 
 places are marked by a small anchor. Upon com- 
 ing near the shore, care must be taken to ex- 
 amine and correct the outline of the chart, by 
 observing the inflections, creeks, &c. more mi- 
 nutely. 
 
 6. In a small sailing vessel go out to see,, and 
 take drawings of the appearance of the land, with 
 its bearings; sail into the harbour, observe the 
 appearance of its entrance, and particularly whe- 
 ther there be any false resemblance of an entrance 
 by which ships may be deceived into danger; re- 
 mark the signs or objects, by attending to which, 
 the harbour 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, &c. on shore, 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 leagues, which must be 
 taken from the same original scale as the funda- 
 mental base. 
 
 Charts or plans may be neatly drawn with In- 
 dian ink, or the pen and common ink, and are as 
 useful as any others, but they are frequently done 
 
 Bb
 
 370 MARITIME SURVEYING. 
 
 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. 
 
 An example to illustrate the foregohig precepts. 
 Let AB C 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. 
 
 Tb make the actual ohserval'ions on shore. By sailing 
 or walking along the shore, a rough sketch is made, 
 and station staves set up at the points of land, A, B, 
 C, D, E, F, G, H, and also at I, K, L, on the island, 
 precept 1. During this operation, it is observed, 
 that no proper place offers itself at which a fun- 
 damental base can be drawn, so as to command 
 at once a view of all the stations; it will;, there- 
 fore, be necessary to survey the coast in two se- 
 parate parts, by making use of two base lines; if 
 the coast had been more extended, or irregular, a 
 greater number of base lines might have been 
 necesscuy. Between the points B and C, the 
 ground is level, and a considerable number of the 
 southernmost points may be seen from both points, 
 conformable to precept 2; make B C, therefore, 
 the first fundamental base, its length by admea- 
 surement is found to be 812 fathoms, and its 
 bearing by compass from B to C, is N. 1 1° 14' E. 
 from each end of this ])ase, measure with Hadley^ 
 quadrant the angles formed between it, and lines 
 drawn to each station in -sight, precept 3, cntf'r,'6r 
 tabulate them as follows: 
 
 First fundamental base B 0=812 fathoms, 
 bearing of C from B, N. 11° 14' E.
 
 MARITIME SURVEYING. 
 
 371 
 
 Angle A B C = i35 07 
 LB C=: 85 00 
 KB C=: 56 28 
 EB C= 43 32 
 DBC= 13 30 
 
 Angle A C B = 22 5>i 
 LCB=: 71 02 
 KC B= 91 40 
 EC B = 113 50 
 DCB = 145 20 
 
 After having made these observations, it will 
 be requisite to proceed to the northern part of the 
 coast; in all cases where a coast is surveyed in se- 
 parate parts, it is best to measure a new funda- 
 mental base for each part, when it can be conve- 
 niently done; a line from the station E drawn to- 
 wards F, is well adapted to our purpose; let E P, 
 therefore, be the second base line, its length by 
 admeasurement is found to be 778 fathoms, and 
 its bearing by compass from E to P, N. 38° 20' E. 
 Measure the angles formed by lines drawn from 
 each end of this base, as before directed, and tabu- 
 late them as follows: 
 
 Second fundamental base E P= 77 8 fathoms, 
 bearing of P, from E. N. 38°20'E. 
 
 Angle H E P=8S 30 
 I EP=7l 32 
 GEP=43 59 
 FEP=27 5l 
 
 Angle PI P E = 73 00 
 I PE= 28 54 
 G P E = 1 1 1 00 
 F PE = 127 18 
 
 It is sufficiently apparent, that the connection 
 between the two parts of this survey is preserved 
 by the second fundamental base being drawn from 
 the point E, whose situation was before determined 
 by observations from the first base line; if this 
 particular position of the second base had not been 
 (Convenient, and it had been taken at a distance 
 from every point determined in situation from the 
 
 B b 2
 
 372 GEOMETRICAL CONSTRUCTION. 
 
 first base, the connection would have required an 
 observation of the bearing of one of the said points 
 from each end of the second base: thiis^ 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 
 E, from I and P. 
 
 All the observations which are required to be 
 made on shore, being now completed, it will be 
 advihcable to construct the chart before we pro- 
 ceed to the other observations on the water, for, 
 by this method, an opportunity is offered of draw- 
 ing the waving line of the coast with more correct- 
 ness than could otherwise be obtained. 
 
 GEOMETRICAL CONSTRUCTION,^^. 12, pi. 23. 
 
 On the point B as a center, describe the circle 
 T) e s w, the diameters n s and w e being at right 
 angles to each other, and representing the meri- 
 dian and parallel ; from n set off the arc n N, 
 equal to the quantity of the variation of the com- 
 pass ; draw the diameter N S, and the diameter 
 W E at right angles to the same; these lines will 
 represent the magnetic meridian, and its parallel 
 of latitude. 
 
 From the center B, draw the first base line 
 BC=81-2 fathoms, N. 11° 14' E. by compass; 
 from one extremity B, draw the unlimited line»- 
 Ba, Bl, Bk, Be, Bd, forming, respectively with 
 the base, the angles observed from thence ; from 
 the other extremity C, draw the unlimited lines 
 C a, CI, C k, C e, C d, forming, respectively with 
 the base, the angles observed from thence ; the 
 intersections of lines terminated by the same let- 
 ter, will give the stations A, L, K, E, D, pre- 
 cept 4.
 
 GEOMETRICAL CONSTRUCTION. 373 
 
 Draw the line B Q, N. 38° 10' E. by compass, 
 and from E, draw EP=778 fathoms in the same 
 direction or parallel to it, E P will be the second 
 base line; from one extremity E, draw the un- 
 limited lines Eh, Ei, Eg, E f, forming, respec- 
 tively with the base, the angles observed from 
 thence; from the other extremity P, draw the un- 
 limited lines Ph, Pi, Pg, Pf; the intersections 
 of lines terminated by the same letter will give 
 the stations H, I, G, F, precept 4. 
 
 Near the points A, B, C, D, E, F, G, H, draw 
 a waving or irregular line to represent the coast ; 
 draw also by the help of the points I, K, L, a line 
 to represent the coast of the island. 
 
 The chart being thus far delineated, it will be 
 proper to make the observations which are re- 
 quired on the water. 
 
 TO MAKE THE ACTUAL OBSERVATIONS ON THE 
 WATER. 
 
 Precept 5, sufficiently explains the business to 
 which it is necessary to attend on the water ; no- 
 thing more, therefore, need be added here, except 
 an elucidation of the method, by the place at 
 which any observation is made, may be found on 
 the chart; the place of an observer may be deter- 
 mined by the bearings of two known objects; the 
 intelligent practitioner will find reasons for oc- 
 casionally preferring each of these methods; but, 
 as the distance in the second method is known 
 only by estimation, we have given the preference 
 to the first, in the present example; all the sound- 
 ings, &c. are supposed to be laid down by these 
 methods, but, to prevent confusion and prolixity, 
 the lines of bearing are drawn only from the points 
 M, N, and O.
 
 374 TO MAKE THE OBSERVATIONS 
 
 From the southern extremity M, of the reef of 
 rocks, which runs off the point L, the bearing of 
 the stations C and B are taken by the compass^ as 
 follows, viz. C, N. 62° E. and B, N. 85° E. 
 
 On the chart, therefore, from C, parallel to the 
 rhumb B R, draw the opposite rhumb C m, S. 6'2° 
 W. and from B, draw the opposite rhumb B m, 
 S. 85° W. their intersection M, will be the ex- 
 tremity of the reef of rocks. 
 
 Again, from the sand head N, the bearings of 
 the stations L and K are taken by the compass, 
 as follows, viz. L, N. 17° W. and K, N. 5° E. 
 
 On the chart draw the rhumb B n, N. 17° W. 
 and from L, parallel to the same, draw the op- 
 posite rhumb Ln; draw likewise the rhumb BT, 
 N. 5° E. and frpm K, parallel to the same, draw 
 the opposite rhumb K n; the intersection of these 
 two last lines will be the sand head, and the situa- 
 tion of the point of land O, which lies so that it 
 cannot be seen from either of these base lines, is 
 determined in like manner from the bearings of L 
 and K, though in practice it may have been more 
 convenient to settle its place from a single bear- 
 ing and distance of the station A. 
 
 The position of the compass is given from that 
 of the circle NESW, made use of in the con- 
 struction, and the scale of geographical miles is 
 found from the consideration, that 1021 fathoms 
 make one geographical mile; take therefore 1021 
 from the same scale of equal parts, as was used in 
 laying down the fundamental bases, and it will 
 give one primary division for the intended scale 
 of miles. 
 
 7^0 measure a strait Vine on the surface of the 
 sea. First, prepare a measuring line of strong 
 chord, two or three hundred yards in length, with 
 small pieces of cork of equal thickness made fast 
 to it at small distances, all along, like a fishing-
 
 ON THE WATER. 375 
 
 net, SO that it may float strait on the surface of 
 the water: if the line has been well stretched, or 
 much used before, it is the better; also prepare 
 two ropes somewhat longer than the greatest 
 depth of the water to be measured, with a pig of 
 lead or iron ballast, which we call an anchor, 50 
 or 60 pounds weight, tied by the middle to one 
 end of each rope, that when it is at the bottom it 
 may be able to anchor a boat, and bear to be 
 stretched strait without shifting the place of the 
 anchor. Let the measuring line be thoroughly 
 wet immediately before you begin to use it, and 
 then stretched on the water close by the shore, 
 and its length measured there with a pole. Then, 
 in the direction intended to be measured, take two 
 remarkable sharp objects on the land in a line, 
 one near the shore, the other as far up in the coun- 
 try as you can: if such are not to be had, place 
 buoys on the water at proper distances in that 
 direction. 
 
 2. Take the objects, or buoys, in a line, and 
 holding one end of the measuring line fast on the 
 shore, carry out the other in a boat, in that direc- 
 tion, till it is stretched strait at its full length by 
 one man in the boat, and exactly at the end of the 
 line let another man drop the anchor, which will 
 mark one length of it. There keep the boat, and 
 the end of the measuring line, close to the anchor 
 rope, drawn tight up and down, till another boat 
 takes in the other end which was on the shore, and 
 rows farther on, and lays it strait in the direction 
 of the land marks, or buoys, and there drops ano- 
 ther anchor, which will mark the second length 
 of the measuring line. Go on thus till the whole 
 proposed distance is measured; and immediately 
 after let the measuring line be again measured with 
 a pole on the water near the shore, as at first, and 
 \[ the lengths differ, take the mean between them
 
 87^ TO MAKE THE OBSERVATIONS 
 
 for the true length. It is obvious, that to measure 
 with any exactness this way, the sea must not only 
 be smooth, but void of a swell, and of all stream of 
 tide; either of which will hinder the line from ly- 
 ing strait. This method of measuring a strait line 
 miay be convenient on some occasions; and if care 
 is taken to keep the anchor rope right up and 
 down when the measuring line is applied to it, 
 will be found sufficiently exact for many purposes, 
 but not for a fundamental base line from which 
 pther distances are to be deduced. 
 
 There is another way of measuring a strait line, 
 mechanically, on the sea, which is so well known 
 to seamen, that it is needless to describe it partis 
 cularly here: and that is, by heaving the log over 
 a ship's stern while she is under sail, and observ- 
 ing how many knots of the log line run out in half 
 a minute; for the line is so divided that the ship 
 will run, or is supposed to run, so many miles in 
 an hour, in a strait course; and twice as much in 
 two hours, and so on. But this conclusion is 
 founded on three suppositions, neither of which is 
 certain, viz. that the log remains in the same 
 place during the whole half minute that the line is 
 running out from the ship's stern; that the ship 
 continues to sail with the same velocity, and also 
 in the same direction, during an hour, or two, 
 that she did during the half minute; the contrary 
 of which is more likely in most cases. For the 
 log line may shrink, or stretch, while it is running 
 out; or may drag after the vessel by the weight 
 of the line, or by not running easily and readily 
 off the reel ; the swell of the sea may alter the place 
 of the log; and currents, or streams of tide, 
 stronger orweaker below the surface than on it, an 
 lunsteady helm, lee-way, and varying winds, may 
 change the direction, or celerity of the ship's mo- 
 t}on; for neither of which can any certain allov,-
 
 ON THE WATER. 377 
 
 ance be made. This way, therefore^ of measuring 
 a .strait line, or distanee, is not to be depended on 
 as exact; but is mentioned here, because rocks, 
 shoals, or islands, sometimes lie so far from the 
 coast, that there is no other way of forming any 
 notion of their distance. If any such distance is 
 to be measured after this manner, let the log-line 
 be thoroughly wet when it is measured; let the 
 length between each knot be 51 feet, which is the 
 1 '20th part of a geometrical mile, as half a minute 
 is ri^th part of an hour. Choose neap-tide, as 
 much slack water as can be got, and a moderate 
 breeze of following wind; let the line be run off 
 the reel so as never to be stretched quite strait; 
 and if the half minute is measured by a watch that 
 shews seconds, rather than by a glass, it will ge- 
 nerally be more exact. Perhaps one second should 
 be allowed for the loss of time in callina: out at 
 the beginning, and stopping it at the end of the 
 time; except the person who holds the watch can 
 contrive to observe the going out of the red rag at 
 the beginning, and also to stop the line himself at 
 the end of the time; which does not seem a diffi- 
 cult matter. 
 
 To find the distance of two 'places hy the flash and 
 report of a gun. Sound moves 1142 English feet 
 in one second of time, or 6l20 feet, which makes 
 a geographical mile, in 5t"' nearly ; therefore, let 
 a gun be fired at one place at an appointed time, 
 and observe the time that elapses between the 
 flash and report, and so many seconds as you ob- 
 serve, so many times 1142 feet are you distant 
 from the place; the operation should be repeated 
 two or three times for greater certainty. The dis- 
 tance to be measured in this way should never be 
 less than two miles, on account of errors that may 
 ftjrise in taking the time.
 
 L 378 ] 
 
 TO TRANSFER A PLAN FROM ONE PAPER TO 
 ANOTHER. 
 
 Method 1 . By points. Lay the rough plan upon 
 the clean paper, on which you intend to draw the 
 fair copy, press them close together by weights, 
 and keep them as flat as possible; then with a 
 ' pointrel, or needle point, prick through all the 
 corners of the plan to be copied; separate the 
 papers, and join by lines the points on the clean 
 paper. This metho^l can only be used in plans, 
 whose figures are small, regular, and bounded by 
 Strait lines. 
 
 Method 2. By tracing faper. Rub the back of 
 the rough plan with black-lead powder, and having 
 wiped off the superfluous lead, lay the blacked 
 part upon the clean paper, or place a sheet of 
 black tracing paper between the rough plan and 
 the clean paper; weights are to be placed as in 
 the former method, to maintain the papers in the 
 same position. 
 
 Then, with a blunt point of brass, steel, or 
 ivory, trace exactly the lines of the plan, pressing 
 the paper so much, that the black lead under the 
 lines maybe transferred to the clean paper; when 
 the whole of the plan has been thus delineated, go 
 over the black-lead marks with common, or Indian 
 ink. 
 
 Method"^. By squares. See prob. 65, page 90. - 
 
 Method A. By a copying glass. This is a large 
 square, or rectangular piece of looking glass, fixed 
 in a frame of wood, which can be raised to any 
 angle, like a desk, the lower end resting upon the 
 table; a screen of blue paper fits to the upper edge, 
 and stands at right angles to it.
 
 TO TRANSFER A PLAN. 37^ 
 
 Place this frame at a convenient angle against a 
 strong light: fix the old plan and clean paper 
 firmly together by pins, the clean paper upper- 
 most, and on the face of the plan to be copied; 
 liiy them with the back of the old plan next the 
 glass, namely, that part which you intend to copy 
 first. The light through the glass will enable you 
 to perceive distinctly every line of the plan upon 
 the clean paper, and you can easily trace over them 
 with a pencil ; and having finished that part which 
 covers^ the glass, slide another part over it, and 
 copy t3ii^, and thus continue till the whole be 
 copied. 
 
 Method 0. By tlie assistance of proporl'ionahle and 
 triangular compasses, Jig. A and N, plate 1, and 
 fig. 12, plates. These will, in many instanstan- 
 ces, assist the draughtsman v(-'ry much, and lessen 
 the labour of copying. 
 
 Method 6. By the pantographcr. There is no 
 method so easy, so expeditious, nor even so accu- 
 rate, as the pantographcr. It is an instrument as 
 useful to the experienced draughtsman, as to those 
 who have made but little progress in the art. It 
 saves a great deal of time cither in reducing, en- 
 larging, or copying of the same size, giving the 
 outlines of any drawing, however crooked or com- 
 plex, with the utmost exactness; nor is it confined 
 to any particular kind, but may with equal facility 
 be used tor copying figures, plans, sea charts, 
 maps, proliles, landscapes, &c. 
 
 DESCRIPTION AND USE OF THE PANTOGRA- 
 PHER, OR PANTAGRAPH. 
 
 I have not been able to ascertain who was the 
 inventor of this useful instrument. The earliest 
 account I find, is that of the Jesuit Scheiuer^ about
 
 380 DESCRIPTION AND USE 
 
 the year l631, in a small tract entitled, Fantagra- 
 fh'ice, s'lve Ars nova DeUneaiidi. The principles 
 are self-evident to every geometrician; the mecha- 
 nical construction was first improved by my fa- 
 ther, about the year i730. It is one, among 
 other scientific improvements completed by him, 
 that others have many years after, assumed to 
 themselves. 
 
 The pantographer is usually made of wood, or 
 brass, from 12 inches to two feet in length, and 
 consists of four flat rules, Jig. IQ, plate ^^, two 
 of them long and two short. The two longer 
 are joined at the end A by a double pivot, 
 which is fixed to one of the rules^ and works in 
 two small holes placed at the end of the other. 
 Under the joint is an ivory castor to support this 
 end of the instrument. The two smaller rules are 
 fixed by pivots at E and H, near the middle of the 
 larger rules, and are also joined together at their 
 other end, G. 
 
 By the construction of this instrument, the four 
 \ rules always form a parallelogram. There is a 
 ^ sliding box on the longer arm, and another on the 
 shorter arm. These boxes may be fixed at any 
 part of the rules by means of their milled screws; 
 each of these boxes are furnished with a cylindric 
 tube, to carry either the tracing point, crayon, 
 or fulcrum. 
 
 The fulcrum, or support K, is a leaden weight; 
 on this the whole instrument moves when in use. 
 To the longer instruments are sometimes placed 
 two moveable rollers, to support and facilitate the 
 motions of the pantographer; their situation may 
 be varied as occasion requires. 
 
 The graduations are placed on two of the rules, 
 B and D, with the proportions of t^ t; i, &c. to tV, 
 marked on them,
 
 OP THE PANTOGRAPHER. 381 
 
 The pencil holder, tracer, and fulcrum, must in 
 all cases 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, there is 
 an error either in the setting or graduations. 
 
 The 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 afterwards 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 up occasionally, and thus pre- 
 vented from making false or improper marks upon 
 the copy. 
 
 To reduce in any of the froport'ions t, i, i, I, &c, 
 as marked on the two bars B, and D. Suppose, for 
 example, i is required. Place the two sockets at -s 
 on the bars B and D. Place the fulcrum, or lead 
 weight at B, the pencil socket, with pencil at D, 
 and the tracing point at C. Fasten down upon a 
 smooth board, or table, a sheet of white paper un- 
 der the pencil D, and the original map, &;c. under 
 the tracing pointC, allowing yourself room enough 
 for the various 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 D 
 will describe exactly the same figure as the origi- 
 nal, but 2 the size. In the same manner for any 
 other proportion, by only setting the two sockets 
 to the number 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 pa- 
 per, or to increase the strength of its mark. 
 
 There is a silken string fastened to the pencil- 
 holder, in order that the pencil may be drawn up
 
 382 TO USE THE PAXTOGHAPHER. 
 
 off the paper, to prevent false marks when cross* 
 ing the original in the operation. 
 
 If the original should be so large, that the in- 
 strument will not extend over it at any one opera:-' 
 tion, 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 on the original, the pencil falls on 
 the three corresponding points upon the copy. In 
 this manner, by repeated shiftings, a pentagraph 
 may be made to copy an original of ever so large 
 dimensions. 
 
 To enlarge in ayiy of the proportions i^ t, t, &V. 
 Suppose 2. You set the two sockets at ■h, 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, }>iU reversed. Place 
 the two sockets at •§, the fulcrum at D, and the 
 pencil at B. 
 
 There are sometimes divisions of 100 unequal 
 parts laid down on the bars B and D, to give any 
 intermediate proportion, not shewn by the frac- 
 tional numbers commonly placed. 
 
 Pentagraphs of a greater length than \\\o feet 
 are best made of hard wood, mounted in brass, 
 with steel centers, upon the truth of which de- 
 pends entirely the equable action of this useful 
 instrument.
 
 [ 383 ] 
 
 OF 
 
 LEVELLING. 
 
 The necessity of finding a proper channel for 
 conveying water occurs so often to the surveyor, 
 that any work on that subject, which neglected to 
 treat on the art of levelling, would be manifestly 
 imperfect; I shall therefore endeavour to give the 
 reader a satisfactory account of the instruments 
 used, and the mode of using them. 
 
 A DESCRIPTION OF THE BEST SPIRIT LEVEL. 
 
 Fig. 3, plate 17, represents one of the best con- 
 structed spirit levels, mounted on the most com- 
 plete staves, similar to those affixed to a best the- 
 odolite. 
 
 The achromatic telescope, A, B, C, is movea- 
 ble, either in the plane of the horizon, or with a 
 small inclination thereto, so as to cut any object 
 whose elevation, or depression, from that plane, 
 does not exceed \1 degrees; the telescope is about 
 two feet long, is furnished with fine cross wires, 
 the screws to adjust which are shewn at a, for de- 
 termining the axis of the tube, and forming a just 
 line of sight. By turning the milled screw B, 
 on the side of the telescope, the object glass is 
 moved outwards, and thus the telescope suited to 
 different eyes. 
 
 The tube c c with the spirit 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 adjustment. The two supporters D, E, on
 
 384 DESCRIPTION OP THE 
 
 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 cyHndric ring of the telescope. 
 The lower ends of these supporters are let perpen- 
 dicularly into a strong bar; to the lower end of 
 the support E, is a milled nut F, to bring the in- 
 strument accurately to a level; and at the other 
 end of tlie bar at H, is a screw for tightening the 
 support D at any height. Betw^een the two sup- 
 ports is a compass box G, divided into four quar- 
 ters of 90° each, and also into 36o°, with a mag- 
 neticul needle, and a contrivance to throw the 
 needle off the center 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 coni- 
 cal brass fcrril K, which is fitted to the bell-metal 
 frustum 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 plates, with four adjusting screws, 
 h, h, h, b, which arc used for adjusting the hori- 
 zontal 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 ex- 
 actly the same as those applied to the best theo- 
 dolites, 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 
 tlie intersection of the wires in the telescope, so 
 that it shall coincide with the axis of the cylindri-
 
 BEST SPIlllT LEVEL^ 385 
 
 cal rings on which the telescope turns. 2. To ren- 
 der the level parallel to this axis. 3. To adjust for 
 the horizontal motion quite round upon the staves. 
 
 To adjust the cross ivin's. It, while looking 
 through the telescope at any object, the intersec- 
 tion of the v/ires does not cut precisely the" same 
 part during a revolution of the telescope on its 
 axis, their adjustment is necessary, and is easily 
 obtained by turning the little screws a, a, a. The 
 two horizontal screws are to move the vertical 
 wire, and act in opposite directions to each other; 
 one of which is to be tightened as the opposite is 
 slackened, (tliis not being attended to, will en- 
 danger the breaking of the wire) till the wire has 
 been moved sufficiently. The upper screw to the 
 horizontal wire is generally made with a capstan 
 head, so tliat by simply turniug it to the right or 
 letl hand, the requisite motion of the wire is pro- 
 duced, and thus the intersection brought exactly 
 in the axis of the telescope. 
 
 To adjust the spirit level at only one station. 
 When the spirit level is adjusted to the telescope, 
 the bubble of a^r will settle in the middle, or 
 nearly so, whether the telescope be reversed or 
 not on its supports D, E, which in this case are 
 uot to be moved. The whole level being placed 
 firmly on its staves, the bubble of air brought to 
 the middle by turning the screw F, the rims/,/, 
 of the Y's open, and when the telescope is taken 
 off and laid the contrary way on its supports, 
 should the bubble of 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 
 adjustment. The end to which the bubble 
 of air goes must be noticed, and the distance 
 of the bubble case, and height of one support, 
 so altered by turning the screws at c and F, 
 
 c c
 
 380 DESCRIPTION OF THE 
 
 till by trial the bubble comes to the middle hi 
 both positions of the telescope. This very facile 
 mode of adjustment of the level is one great im- 
 provement 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 
 instrument into the field, and at two distant sta- 
 tions observing both forwards and backwards 
 the deviation upon the station staves, and cor- 
 recting accordingly; which occasioned no small 
 degree of time and trouble not necessary by the 
 improved level. 
 
 To adjust for the lior'izontal motion. The level 
 is said to be completely adjusted, when, after the 
 two previous ones, it may be moved entirely round 
 upon its staft-head, without the bubble changing 
 materially its place. To perform which, bring the 
 telescope over two of the parallel plate screws ^, l\ 
 and make it level by unscrewing one of these 
 screws while you are screwing up the opposite one, 
 till the air bubble is in the middle, and the screws 
 up firm. Then tm-n the instrument a quarter 
 round on its staff, till the telescope is directly over 
 the other two screws h, h, unscrewing one screw, 
 and screwing up at the same time the opposite 
 one, as before, till the bubble settles again in the 
 middle. The adjustment of the staff plates are 
 thus made for the horizontal motion of the instru- 
 ment, and the telescope may be moved round on 
 its staff without any material change of the 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.
 
 BEST SPIRIT LEVEL. 38^^ 
 
 The telescope is generally glassed to shew ob- 
 jects erect. In the common sort of levels with 
 a shorter telescope from 12 to 20 inches in length, 
 and without a circumferentor, they are glassed to 
 shew the objects inverted. To an expert observ^er 
 this will make no difference; and, there being 
 but two eye-glasses instead of four, as in the 
 other, about three or four 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 
 object end c of the telescope, and a screw-driver, 
 and steel pin for the capstan-headed screws arc 
 packed in the same case with the instrument.* 
 
 Two mahogany station staves often accompany 
 this instrument; they consist of two parts, each 
 part is about five feet ten inches long, so that 
 when tliey are pulled out to their greatest extent, 
 they form a ten feet rod; and every foot is divided 
 into 100 equal parts. To each of these staves 
 
 * F/g. 10, is a representation of a brass mounted pocket spirit 
 level, about six inches in length, with sights, having a ground 
 bottom, intended for ordinary purposes either in levelling a 
 plane or determining 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. 1 1, is one that 1 have constructed with some improvement. 
 The brass perpendicular piece A, is made to slide on occasionally 
 by a dovetail. By this piece, and the bottom of the level toge 
 ther, any standing square pillar, or other object, may be set to the 
 level, and perpendicular at the same time. The case of the spirit 
 level B, swings on two pivots; the horizontal position, therefore, 
 of a ceiling or under surface of a plane may be as readily ob- 
 tained as any inferior plane in the common way ; the spirit level 
 thereby always remaining the same, and the position of the base 
 only changing to admit cf 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, oi* 
 made, by bringing the bubble in the middle i;pon any table or 
 
 C C 2
 
 38S OF LEVELLING. '' 
 
 there is a sliding vane A, fig. Q, plate 17, with a 
 brass wire across a square hole made in the vane; 
 when this wire coincides with the horizontal wire 
 ot" the telescope, it shews the height of the appa- 
 I'ent 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 eaith is, than another given point on 
 the same surface; or in other words, the difference 
 in their distance from the center of the earth. 
 
 Those points are said to be level, which are 
 cqui-distant from the center of the earth. The 
 art of levelling consists, therefore, 1st. In finding 
 and marking two, or more level points that shall 
 be in the circumterence of a circle, wdiose center 
 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 
 center. 
 
 hctfig. 1, plate 23, represent the earth; A, its 
 center; the points B, C, D, E, F, upon the cir- 
 cumference thereof are level, because they are 
 equally distant from the center; such are the wa- 
 ters of the sea, lakes^ &c. 
 
 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 
 be so raised or depressed as to cause the bubble to stand the re- 
 versing, at the same time altering the inclination of the plane on 
 which the level is tried. If a bubble stand the reversion of the 
 level upon the plane, both level and plane are right, and it is 
 most expeditious to adjust small levels on a true horizontal plane; 
 if the bubble does not stand the reversion, both bubble and plane 
 are inclined, and both require to be corrected by half the whole 
 angle of the deviation shewn by reversion. Edit.
 
 OF LEVELLING. 38Q 
 
 To know Jiozv much higher the fo'inl B, fig. 2, 
 flate 23, is than C, and C lower ^han D. We must 
 lind, and mark the level points, E, F, G, upon the 
 radii AB, AC, AD, thereby comparing B with E, 
 C with F, and D with G; we shall discover how 
 much B is nearer the circumference of the circle 
 than C, and, consequently, how much further 
 from the center of the earth, and so of the other 
 points. 
 
 OF THE DIFFEREXT METHODS OF MARKING 
 OUT THE LEVEL POINTS. 
 
 The first, which is the most simple and inde- 
 pendent, is by the tangent of a circle, for the two 
 extremities of the tangent give the true level 
 points, wdien the point of contact is precisely in 
 the middle of the line. But if the point of con- 
 tact with the circumference be at one of the extre- 
 mities of the line, or i/Tany other part except the 
 middle, it will then only shew the apparent level, 
 as one of its extremities is further from the cir- 
 cumference than the other. Thus the tangent 
 BC, j^^. 3, plate 23, marks out two true level 
 points at B and C, because the point of contact 
 D is exactly in the middle of the line B C, and its 
 two extremities arc equally distant from the cir- 
 cumference and the center A. 
 
 But the tangent B T>,fig. 7, plate 23, marks two 
 points of apparent level, because the point B, 
 where it touches the circumference, is not the 
 middle of the line; and, therefore, one of its ex- 
 tremities B, is nearer to the center than the other 
 D; D is further from the center, in proportion as 
 it is more distant from the point of contact B ; 
 which constitutes the difference between the true 
 and apparent level, of w hich we shall speak pre-
 
 SQO DIFFERENT METHODS OP 
 
 gently. Every point of the apparent level, except 
 the point of contact, is higher than the true level. 
 
 As the tangent to a circle is perpendicular to 
 the radius, we may make use of the radius to de- 
 termine the tangent, and thus mark the level 
 points. Thus let A,fy. 5, ^late 23, represent the 
 center of the earth, AB the radius, and C, B, D, 
 the tangent; the two extremities C, D, are equally 
 distant from the point of contact B, consequently 
 the angles B C A, B DA, will be equal ; the angles 
 nt the tangent point are right angles, and the ra- 
 dius common to both triangles, and the sides C A, 
 DA, are equal, and the points C D, are two level 
 points, because equally distant from the center. 
 
 It is evident from this, that if from any point of 
 the radius two lines be drawn, one on each side, 
 making equal angles with it, and being of an equal 
 length, the extreme points of these lines will be 
 level points. Thus, if from B, of the radius BA, 
 two equal lilies BC, KDt, fig, 6, '^late 23, be 
 drawn, making equal angles, CBA, DBA, then 
 w^ill C and D be equally distant from the center; 
 though the level may be obtained by these oblique 
 lines, yet it is far easier to obtain it by a line per- 
 pendicular to the radius. 
 
 When the level line is perpendicular to the ra- 
 dius, and touches it at one of its extremes, the 
 other extremity will mark the apparent level, and 
 the true level is found by knowing how much the 
 apparent one exceeds it in height. 
 
 To find the height of the apparent above the 
 true level for a certain distance, square that dis- 
 tance, and then divide the product by the diame- 
 ter of the earth, and the quotient will be the re- 
 quired difference; it follows clearly, that the 
 heights of the apparent level at different distances 
 ^re as the squares of those distances; and, conse^. 
 icjucntly, that the difference is greater, or smaJIerj
 
 MARKING LEVEL POINTS. 
 
 391 
 
 in proportion to the extent of the line; for the ex- 
 tremity of this line separates more from the circum- 
 ference of the circle, in proportion as it recedes 
 from the point of contact. Thus, let A, fg, 7, 
 plate 23, be the center of the earth, BC the arc 
 which marks the true level, and B, E, D, the tan- 
 gent that marks the apparent level; it is evident, 
 that the secant AD exceeds the radius AB, by CD, 
 which is the difference between the apparent and 
 true level; and it is equally evident, that if the line 
 extended no farther 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 neg- 
 lected; but if it be 50, 100, &c. yards, then the 
 error resulting from the difference will become 
 sensible and require to be noticed. 
 
 A TABLE, 
 
 Which shews the quantity of curvature below the apparent level, 
 in inches, for every chain up to 100. 
 
 
 Inches 
 
 
 
 Inches 
 
 
 
 Inches 
 
 
 
 3- 
 
 Inches 
 
 B' 
 
 
 5' 
 
 
 
 
 5' 
 40 
 
 
 1 
 
 0,00125! 
 
 14 
 
 0,24 
 
 \v 
 
 0,91 
 
 2,00 
 
 2 
 
 0,005 
 
 15 
 
 0,28 
 
 \2S 
 
 0,98 
 
 45 
 
 2,28 
 
 3 
 
 0,01125 
 
 16 
 
 0.32 
 
 j2p 
 
 1,05 
 
 50 
 
 3,12 
 
 4 
 
 0,02 
 
 17 
 
 0,36 
 
 '30 
 
 1,12 
 
 55 
 
 3,78 
 
 5 
 
 0,03 
 
 18 
 
 0,40 
 
 ;31 
 
 |32 
 
 1,19 
 
 , 60 
 
 4,50 
 
 6 
 
 0,04 
 
 IP 
 
 0,45 
 
 1,27 
 
 ; 65 
 
 5,31 
 
 7 
 
 0,06 
 
 20 
 
 0,50 
 
 l33 
 
 1,35 
 
 70 
 
 6,12 
 
 8 
 
 0,08 
 
 21 
 
 0,55 
 
 l34 
 
 1,44 
 
 75 
 
 7.03 
 
 9 
 
 0,10 
 
 22'0,60 
 
 ;35 
 
 1,53 
 
 80 
 
 8,00 
 
 10 
 
 0,12 
 
 23io,67 
 
 36 
 
 1,62 
 
 85 
 
 9,03 
 
 11 
 
 0,15 
 
 24,0,72 
 
 37 
 
 1,71 
 
 90 
 
 10,12 
 
 12 
 
 0,18 
 
 25,0,73 
 
 38 
 
 1,80 
 
 95 
 
 11,28 
 
 13 
 
 0,21 
 
 260,84 
 
 1 
 
 39 
 
 1,91 
 
 100 
 
 12,50
 
 392 DIFFERENT METHODS OP 
 
 The first method of finding and marking two 
 level points, is by a tangent whose point of con- 
 tact is exactly in the middle of the level line; this 
 metliod may be practised without regarding the 
 difference between the apparent and true level; 
 but, to be used wdth success, it would be well to 
 place your instrument as often as possible at an 
 equal distance from the stations; for it is clear, 
 that if from one and the same station, the instru- 
 ment remaining at the same height, and used in 
 the same manner, two or more points of sight be 
 observed, equally distant from the eye of the ob- 
 server, they will also be equi-distant from the cen- 
 ter of the earth. 
 
 Thus, let the instrument, jig. 8, flate 23, be 
 placed at equal distances from C, D; E, F, the 
 two points of sight marked upon the station-staves 
 C G, DH, will be the level points, and the diffe- 
 rence in the height af the points E, F," will shew 
 how much one place is higher than the other. 
 
 Second method. This consists in levelling from 
 one point immediately to another, placing the in- 
 strument at the stations where the staves were 
 lixed. This may also be performed without no- 
 ticing the difference between the true and apparent 
 level; but then it requires a double levelling, 
 made from the first to the second station, and re- 
 ciprocally from the second to the first. 
 
 Thus, let the two stations be B E, jig. g, fhte 
 0,3, the station staves C B, D E, which in the prac- 
 tice of levelling may be considered as parts of the 
 radii AB, AE, though they be really two perpen- 
 diculars parallel to each other, without the risk of 
 any error. In order to level by this method, plant 
 the instrument at B, let the height of the eye, at 
 the first observation, beat F, and the point of sight 
 found be G; then rcmovp the instrument to Ej
 
 MARKING LEVEL POINTS. 393 
 
 and fix it so that the eye may be at G; then, if 
 the Hne of sight cuts F, the points F and G are 
 level, being equally distant from the center of the 
 earth, as is evident from the figure. 
 
 But if the situation of the two vStations 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. 10, flate '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 I, will be parallel, and their ex- 
 tremities 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 obtained 
 by further observations; bitt, 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 tVom four to six inches ; a four 
 inch fall in a strait 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; but 50 or 6o yards are to be preferred. 
 
 OF THE PRACTICE OF LEVELLING. 
 
 1st. Of s'nuple 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 AB, fig. l, plate 2A,^h^. the station 
 points of the level, C, D, the two points ascer- 
 tained, and let the height
 
 394 OF THE PRACTICE 
 
 Feet Inches. 
 
 from A to C be 6 O O 
 from B to D be 9 O O 
 
 their difference is 3 O 
 shewing that B is three feet lower 
 than A. 
 
 In this example, the station points of the level 
 are below the line of sight and level points, as is 
 generally the case ; but if they were above, as in 
 jig. 2, plate 24, and the distance of A to C be six 
 feet, and from B to D nine feet, the difference will 
 be still three feet, that B is higher than A. 
 
 2d. Of compound Levelling. 
 
 Compound levelling is nothing more than a 
 collection of many single or simple operations 
 compared together. To render this subject clearer, 
 we shall suppose, that for a particular purpose it 
 were necessary 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 Belann. 
 
 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. Le Fehvre. 
 
 Stakes should be driven down at A and N, ex- 
 actly 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 operator in the conducting of his business; by 
 this he will discover that the shortest way from A 
 to N, is by the dotted line AQ GH^ H N, and
 
 OP LEVELLING. 3^5 
 
 he will from hence also determine the necessary 
 number of stations, and distribute them properly, 
 some longer than the others, accordmg to the na- 
 ture of the case, and the situation of the ground. 
 In this instance 12 stations are used; stakes should 
 then be driven at the limits of each station, as at 
 
 A, B, C, D, E, F, &c. they should be driven 
 about 18 inches into the 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, he. 
 
 Things being thus prepared, he may begin his 
 work; the first station will be at 1, equi-distant 
 from the two limits A, B; the distance from A to 
 
 B, l66 yards; and, consequently, the distance on 
 each side of the instrument, or from the station 
 stake, will be 83 yards. 
 
 Write down 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 in- 
 dicated at the station staff B, namely, 5. O. O. 
 Lastly, in the fifth column, the distance from one 
 station staff to the other, in this instance l66 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 
 down 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 dis- 
 tance between B and C. 
 
 As, from the inequality of the ground at the 
 third station, it is not possible to place the instru- 
 j]icnt in the middle between the two station staves.
 
 3g6 OF THE PRACTICE, hc. 
 
 find the most convenient point for your station, as 
 at 3 ; then measure exactly how far this is from 
 each station staif, and you will find from 3 to C, 
 160 yards; from 3 fo 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 ; 
 therefore, mark out 80 yards from the station staff" 
 D, to the point 4; and 160 yards from 4 to E; 
 and this must be carefully attended to, as by such 
 compensations the work may be much facilitated. 
 
 Proceed in the same manner with the eight re- 
 maining stations, as in the four former, observing 
 to enter every thing in its respective column ; 
 when the whole is finished, add the sums of each 
 column together, and then subtract the less from 
 the greater, thus from 82 : 2 : 5, take 76 : 9 : 7, 
 and the remainder 5:4: 8, is what the ground at 
 N is lower than the ground at A. 
 
 
 f. 
 
 in. 
 
 
 
 
 f. 
 
 in. 
 
 
 yards 
 
 A 
 
 7 
 
 6 
 
 
 B 
 
 
 6 
 
 
 
 
 
 166 
 
 B 
 
 4 
 
 6 
 
 
 C 
 
 
 5 
 
 6 
 
 2 
 
 250 
 
 C 
 
 12 
 
 8 
 
 6 
 
 D 
 
 
 8 
 
 4 
 
 
 
 240 
 
 D 
 
 
 
 
 
 
 
 E 
 
 
 4 
 
 1 
 
 
 
 250 
 
 E 
 
 6 
 
 10 
 
 
 
 F 
 
 
 2 
 
 11 
 
 
 
 250 
 
 F 
 
 7 
 
 
 
 4 
 
 G 
 
 
 4 
 
 8 
 
 6 
 
 300 
 
 G 
 
 7 
 
 7 
 
 6 
 
 H 
 
 
 10 
 
 
 
 
 
 250 
 
 H 
 
 4 
 
 6 
 
 4 
 
 I 
 
 
 8 
 
 10 
 
 
 
 110 
 
 I 
 
 6 
 
 3 
 
 
 
 K 
 
 
 10 
 
 
 
 
 
 130 
 
 K 
 
 6 
 
 4 
 
 3 
 
 L 
 
 
 5 
 
 8 
 
 (5 
 
 250 
 
 L 
 
 7 
 
 
 
 
 
 M 
 
 
 8 
 
 4 
 
 3 
 
 250 
 
 M 
 
 6 
 
 5 
 
 
 
 N 
 
 
 7 
 
 10 
 
 
 
 250 
 
 
 76~ 
 
 9 
 
 7 
 
 82 
 
 2 
 
 5 
 
 268O 
 
 
 
 
 
 82 
 
 2 
 
 5 
 
 
 
 
 
 
 
 
 76 
 
 9 
 
 7 
 
 
 
 
 
 5 
 
 4 
 
 6 

 
 [ 397 ] 
 
 OF THE PROFILE OP THE LEVEL. 
 
 The next thing to be obtained is a section of 
 this level; for this purpose draw a dotted Hnc, as 
 o o, fig. 4, flate 24, either above or below the 
 plan, which line may be taken for the level or ho- 
 rizontal line; then let fall perpendiculars upon 
 this line from all the station points and places, 
 where the station staves were fixed. 
 
 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 ^/, parallel to the dotted line 
 o o, which will cut the third perpendicular at b, 
 the second station staff; set off from this point 
 downwards six feet to B, which shews the second 
 limit of the first operation, and that the ground at 
 B, is one foot six inches higher than at A; place 
 your instrument between these two lines at the 
 height of the level line, and trace the ground ac- 
 cording to its different heights. 
 
 Now set off on the second station staff B, four 
 feet six inches to C, the height determined by the 
 level at the second station ; and from c, draw a line 
 parallel to o o, which will cut the fifth perpen- 
 dicular at d, the third station staff from this point 
 set off 5 feet 6 inches fV downwards to C, which 
 will be our second limit with respect to the pre- 
 ceding one, and the third with respect to the 
 first; then draw your instrument in the middle 
 between B and C, and delineate the ground with 
 its different inequalities. Proceed in the same 
 manner, from station to station, to the last N, and 
 you will have the profile of the ground over which 
 the level was taken.
 
 2Q8 OP THE PROFILE 
 
 To trace more 'particularly the profile of each 
 station. It is necessary to observe in this place, 
 that if the object of the operation be only to know 
 the reciprocal height of the two extreme terms, as 
 in the preceding example, then the method there 
 laid down for the profile or section will be suffi- 
 cient; but if it be necessary to have an exact de- 
 tail of the ground between the said limits, the 
 foregoing method is too general; we shall, there- 
 fore, institute another example, in which we shall 
 suppose the level to have been taken from A to N 
 by another route, but on more uniform ground, 
 and less elevated above the level of the two rivers, 
 in order to form a canal, marked O, P, Q, R, S, T, 
 U, X, Y, for a communication from one river to 
 the other. 
 
 Draw at pleasure a line Z Y,fig. 5, plate 14, to 
 represent the level line and regulate the rest; then 
 let fall on this line, perpendiculars to represent the 
 staves at the limits of each station, taking care 
 that they be fixed accurately at their respective 
 tlistances from each other. 
 
 As the difference between the extreme limits 
 ought to be the same as in the former example, is 
 5 feet 4 inches to, set off this measure upon the 
 perpendicular at o, the first limit, and from o, pro- 
 longing the perpendicular, mark off at a, the height 
 determined at the first station staff, as in the pre- 
 ceding example; then do the same with the se- 
 cond and third, and so on with the following, till 
 this part of the work is finished; there then re- 
 mains only to delineate in detail the ground be- 
 tween the station staves; the distances are as- 
 sumed larger in this instance on account of the 
 detail. 
 
 To get the section of the ground between O and 
 P, place your instrument at one of the limits as P,
 
 OP THE LEVEL. ^QQ 
 
 fixing It so that the cross hairs may answer to the 
 point b; then look towards the first limit o, rais- 
 ing or depressing the vane, till it coincides with 
 the intersection of the cross hairs, and the line of 
 ^ight, from one point to the other, will mark the 
 level or horizontal line. 
 
 Now to set off the height of the brink of the 
 river above the first limit, drive a stake down 
 close to the ground at a, upon which place your 
 station staff, and observe at what height the hairs 
 intersect the vane, it will be at 4. 10; then laying 
 off upon the lirte o z, the distance from the first 
 term to the first stake, from whence let fall a per- 
 pendicular, and set oft' thereon 4. 10 to a, which 
 gives the height at the first stake, or, which is the 
 same, the height from the edge of therivcrabove the 
 ftvu-face of the water, as is plain from the section. 
 
 Drive a second stake at b, in a line between the 
 limits; place the station staff' upon this stake, and 
 observe the height 4. 6. intersected by the cross 
 hairs, the instrument still remaining in the same 
 situation; set off on the level line the distance 
 from the first stake a to the second b, and then 
 let fall a perpendicular, and mark upon it 4. 6. to 
 /', which gives us the height of the ground at this 
 place. 
 
 To mark out the small hollow c, drive down a 
 third stake even with the ground, in the middle 
 of it at t'; but as before in the station line, mark 
 upon the level line the exact distance from the 
 second stake /», to the third c; then let fall a per- 
 pendicular from c, and set off thereon 6:8:0 
 pointed out by the cross hairs on the staff, which 
 determines the depth of the hollow, as may be 
 seen by the section. 
 
 With respect to the ground between each stake, 
 as the distances are now very shorty it will be
 
 400 OP THi: PROFILK 
 
 easily expressed by the operator, whose judgment 
 will settle the small inequalities by a comparison 
 with those already ascertained. 
 
 Proceed thus with the other stations, until yon 
 arrive at the last, and you will obtain an accurate 
 section in detail of your work; by such a section, 
 it is easy to form a just estimation of the land to 
 be dug away, in order to form the canal, by adding 
 thereto the depth to be given to the canal. 
 
 Another exiunph of coinpowui Jevellhig. In the 
 levelling exhibited m plate 25, we have an exam- 
 ple taken wliere the situation was so steep and 
 mountainous, that it was impossible to place the 
 staves at equal distances from the instrument, or 
 even to make a reciprocal levelling from one sta- 
 tion to another. 
 
 Such is the ease between the first point A, taken 
 from the surface of a piece of water, which falls 
 from the mountains, and the last point K, at the 
 bottom of a bason, where it is proposed to make a 
 fountain, and the height is required, at which a 
 jet d'eau will play by conducting the water from 
 the reservoir A, to the point K of the bason, by 
 tubes or pipes properly made, and disposed with 
 all the usual precaution. 
 
 From the manner in which the operator must 
 proceed in this instance, it is evident, that the in- 
 strument should be adjusted with more than or- 
 dinary care, as the true distance from one moun- 
 tain to the other cannot be attained without much 
 trouble. 
 
 The height is here so great, that it will be ne- 
 cessary to go by small ascents from A to D, and 
 it will probably be commodious in some part of 
 the work to use a smaller instrument; underneath 
 is a table of the different level points as ascertain- 
 ed, which, widi the profile and plan, render this
 
 OF THE LEVEL. 
 
 401 
 
 part of the worL sufficiently clear, without further 
 explication. 
 
 
 f. 
 
 in. 
 
 
 
 f. 
 
 in. 
 
 
 yards 
 
 A 
 
 21 
 
 6 
 
 
 
 c 
 
 
 
 .9 
 
 
 
 90 
 
 C 
 
 4 
 
 3 
 
 
 
 D 
 
 
 
 3 
 
 
 
 40 
 
 D 
 
 3 
 
 9 
 
 
 
 E 
 
 16 
 
 3 
 
 
 
 3aO 
 
 E 
 
 5 
 
 
 
 
 
 F 
 
 17 
 
 9 
 
 
 
 250 
 
 F 
 
 10 
 
 6 
 
 
 
 G 
 
 5 
 
 
 
 
 
 3/5 
 
 G 
 
 5 
 
 
 
 
 
 H 
 
 19 
 
 
 
 
 
 300 
 
 H 
 
 5 
 
 
 
 
 
 K 
 06 9 
 
 47 
 
 3 
 
 
 
 1000 
 
 
 55 
 
 
 
 
 
 106 
 
 9 
 
 
 
 2405 
 
 
 
 
 1 
 
 
 
 
 
 
 
 Difference 
 
 55 
 
 
 
 
 
 
 
 51 9 
 
 
 
 
 Only two Icvellings are made here between A 
 and D, though it is evident from the plan, that 
 more would have been necessary ; but as our de- 
 sign is only to shew the manner of proceeding in 
 this particular case, and as more would have con- 
 fused the plan and section, they are here neglected. 
 
 In the fourth, the height found was l6. 8 ; but as 
 the distance E, 350 yards, was considerable, it was 
 necessary to reduce the apparent level to the true 
 one; the same was also done in all other cases 
 where it was necessary ; at the last limit we first 
 get the height from N to o, then from o to I, then, 
 from I to K, which added together, and then cor- 
 rected for the curvature, gives 47.3.0. Now, 
 by adding each column together, and then sub- 
 tracting one from the other, we obtain 51 feet 
 nine inches for the height, which the point A is 
 above the bottom of the bason, and which will 
 cause the jet d'cau to rise about 45 feet, 
 
 D d
 
 402 OP THE PROFILE 
 
 The general seclion of this operation Is delineated 
 under the plan at j^^^. '2, and is suffieicntly plain 
 from what has been already said in the preeeding 
 instances. But an exact profile of the mountain 
 is not so easy, as it would require many opera- 
 tions, as may be seen in the section; some of these 
 niigjit however be obtained, by measuring from 
 the level line already mentioned, without moving 
 the instrument, as 'dtj^g. 3. 
 
 Example 3, fg. \, plate 26, Mr. Le Febirre 
 gives us in that of a river, being one part of the 
 river Haynox, from Ltgnehruk to Vtllelmrg, and 
 the mode he observed in taking this level. 
 
 The first operation was that of having stakes 
 driven at several parts of the river, even with the 
 water's edge; the first stake A, a little above the 
 mills at L'lgnehruk, shews the upper w^ater mark 
 when the water is highest, and is our first limit ; 
 the stake h shews the low water mark at the same 
 mills, the stake B is the second limit. 
 
 The stakes C and D, above and below the mills 
 of Mazurance, shew the height of the waters 
 when at the highest and lowest, and their dif- 
 ference; these stakes form our third and fourth 
 limits. Lastly, the stakes at E and F, above and 
 below the mills at ViUehoiirg, mark as before the 
 difference between the highest and lowest stations 
 of the water, and are also the last limits of the 
 operation. 
 
 Particular care was taken, that the marks should 
 iill be made exactly even with the edge of the 
 water, and they were all made at the different 
 parts of the river, as nearly as possible at the same 
 instant of time. 
 
 The principal limits of the levelling being now 
 detemaincd and fixed, it only remains to bind the 
 level between the limits, according to the methods
 
 OF THE LEVEL. 403 
 
 already pointed out, using every advantage that 
 may contribute to the success of the work, and at 
 the same time avoiding all obstacles and difficul- 
 ties that might retard, or injure the operations. 
 
 The first rule is always to take the shortest pos- 
 sible way from one limit to the other. 
 
 However, this rule must not be followed if 
 there are considerable obstacles in the way, as 
 hills, woods, marshy ground, &c. or, if by going 
 aside, any advantage can be obtained ; thus, in the 
 present instance, it was found most convenient to 
 go from A 2 to B, by the dotted line A c d e f g h 
 i k B, which, although it appears the longest, was 
 in effect the shortest, as you have only to level 
 from one pond to the other, at Ac, d e, {g, h i, 
 k B (at the top of the plate,) the distances c d, c f, 
 gh, i k, the surfaces of the several ponds being 
 assumed as level lines, thereby abridging the work 
 without rendering it less exact ; more so, as it was 
 not the length of the river that was required, but 
 only the declivity. 
 
 Having levelled from A to B, proceed from B 
 to C, following the dotted line B I m n o C, whence 
 we obtain the difFerence in height between the 
 surface of the water at A, and that at C. 
 
 The next step was to level between C and D, 
 above and below the mills, to fmd the difi'crence 
 between the water when at its highest and lowest 
 situations. From D, levels were taken across the 
 countr)^ to p; leaving y) on account of the pond or 
 lake which was assumed as level, we began at q, 
 from thence to r, where we left off; beginning 
 again at s, then levelling from thence to /, and so 
 on to L, above the mills at ViUcbourg, and finished 
 at F, below them. 
 
 By these operations we obtained the knowledge 
 of how much the v/aters above and below the mills 
 D d 2
 
 404 THj; PiioFiJ.r. of the level. 
 
 of Lifiehrnk arc higher than those of Mazurance^ 
 mid these, than those of Vlllebourg, with all the 
 necessary conse(]uences. From this example, the 
 importance of a thorough knowledge of the ground 
 in order to carry on such a work is very evident; 
 this piece of levelling was near five German miles 
 in lengiii, in a strait line, and nine or ten with the 
 bendings of the river. For the profile or section 
 of the foregoing operation, fg. 2, flate 26, first 
 dra^v the dotted line AG, on which let fall per- 
 pendiculars from the principal limits ABCD 
 produced; then beginning at the highest water- 
 mark 'dt Lig}iehruk^ set off three feet to six, for the 
 difference between high and low water; fi'om b 
 draw the dotted line be, parallel to AG. From 
 the point set off on the perpendicular four feet to 
 B, the difference found between b and B; from B 
 drawBd, parallel also to AG; then set off' three 
 feet from d to E downwards, for the difference 
 found between B and C, and A\ feet from C to D, 
 for the difference in height of the mills at Mazu- 
 rance. From D draw the line D e parallel to AG, 
 and from the point c to L, set off three feet for the 
 difference of the level between D and F. And 
 lastly, from E to F, set off one foot six inches for 
 the difference between the higher and lower wa- 
 ters at the mills of ViUehoiirg^ shewing that there 
 are 1() feet difference which the upper waters at 
 Lingebriik are higher than the lower waters of 
 f-lllehourg. 
 
 The pocket measuring tape of 100 feet in 
 length, with the centesimals of a yard, is found to 
 be a useful article in the practice of levelling.
 
 [ 'i03 ] 
 
 ASTRONOMICAL OBSERVATIONS AT LAND, WITH 
 HADLEy's OCTANT AND SEXTANT. 
 
 The portability of the scxtaiU, its cheapness com- 
 pared with other instruments, the ease with which 
 it is used, the accuracy of the observations made 
 with it, strongly recommend it to the attention of 
 surveyors, &c. the only addition necessary to em- 
 ploy it advantageously in astronomical observations 
 at lipid, is an artificial or reflecting horizon. 
 
 The best artificial horizon is quicksilver or water; 
 but as these arc always more or less afieeted bv the 
 air, the trough which contains them should be 
 covered with a roof, consisting of two parallel 
 planes of glass. Sec page '285. 
 
 If these are parallel, it matters little at what an 
 g\c they are set, and if any error be suspected, the 
 frame may be placed sometimes with one side, 
 sometimes with the other, foremost, taking a mean 
 -of the observations. 
 
 Reflecting surfaces, whether of glass or metal, 
 circidar levels, floating planes of glass, &c. arc not 
 to be depended upoji, as they always give a differ- 
 ent altitude or diameter from that observed from 
 the surface of the mercury, orwater, proved by tak- 
 ino- the sun's meridian altitude, or its diameter 
 successively from these different surfaces; this 
 arises from the imperfection of the surface of me- 
 tal or glass, which has never been ground j)er- 
 fectly/^/. 
 
 Tlie parallelism of a glass may be readily exa- 
 mined, and its defects easily discovered; whereas 
 the want of a fiat surfice has scarce been suspected. 
 The parallelism of a glass may be discovered by 
 looking at the moon with it, and receiving her 
 rays in a very oblicjue manner, so as to make the 
 angle between the direct and reflected rays as ob- 
 tuse as possible; if it appears single and well de-
 
 406 ASTTvOXOMICAL OBSERVATIONS 
 
 fined, the glass is parallel. Or a glass may be ae- 
 CLirately examined by laying it on a pieee of paper, 
 and viewing the top of a wall or chimney, &e. 
 about 15 or 20 yards distant ; for, if the two sur- 
 faces be not parallel, the object will appear double 
 and surrounded with a light fringe, and thus may 
 every part of the glass be examined, and its de- 
 fects discovered; the examination will be more 
 perfect if a small telescope be used. 
 
 To examine whether a surface be a perfect 
 plane, take the sun's diameter very accurately 
 with your sextant, when its altitude is considera- 
 ble; then examine in the surface you wish to try 
 the two images, without altering the index ; if it 
 be concave, the two images will lap over, if con- 
 vex, they will separate, and the quantity of this 
 error may be found by the sextant. If, therefore, 
 you use glass, &c. as a reflecting surface for an 
 artificial horizon, you must either allow for the 
 error, which makes the given altitude too great, if 
 the glass be convex, and too small, if concave, or 
 you must make both your adjustments and obser- 
 vations from the same reflecting surface. But this 
 will not entirely obviate the difficulty, as the sur- 
 face is apt to vary from the sun's heat during a 
 long course of observations. 
 
 The angle, observed by means of the artificial 
 horizon, is always double the altitude of the star, 
 &c. above the horizon ; consequently, you cannot 
 take an altitude of more than 45° with an octant_, 
 or of 6o° by the sextant. 
 
 Every thing being ready, and the instrument 
 properly adjusted, move backward, till you sec the 
 reflected image of the sun in the water. If this 
 image be bright, turn one or more of the dark 
 glasses behind the horizon glass.
 
 WITH HADLEY S 5EXTAXT. 407 
 
 Hold now the sextant in a vertical plane, and 
 direct the sight to the sun's image in the artificial 
 horizon. Then move the index tell you see the 
 other image reflected from the mirrors come down 
 to the sun's image seen in the horizon, so as to 
 touch, but not pass it; then bring the edges in 
 contact in the middle, between the wires of the 
 telescope, as before directed, and the divisions on 
 the arc will shew the double altitude. 
 
 Correct the double altitude for the index error, 
 before you halve it. Then to this half altitude add 
 the sun's semi-diameter, and subtract the correc- 
 tion for refraction, and you will have the true alti- 
 tude of the sun's center above the real horizon. 
 
 The altitude of a star must betuken in the same 
 manner as that of the sun: the double altitude 
 must be corrected for the index error, if any, then 
 halved, and this half corrected for refraction gives 
 the true altitude above the real horizon. 
 
 In taking the sun's altitude, whether for the 
 purpose of calculating time, or for double alti- 
 tudes, it is best to fix the index to some particular 
 division of the instrument with great nicety, and 
 then wait till the sun is risen or fallen to that al- 
 titude. 
 
 This is much better than observing its altitude 
 and moving the screw to it, as the screw when thus 
 suddenly moved is very apt to alter a small trifle 
 by the inequality and pressure of the threads, after 
 the hand is removed from it; whereas, when it is 
 iixed to some division previous to the observation, 
 it may be repeatedly tried and examined before 
 the observation is taken. 
 
 An accurate observer will And that the error of 
 his sextant will vary according as he takes it, by 
 moving the index backwards or forwards in taking 
 the O's (sun's) diameter on the quadrantal arc of
 
 408 ASTRONOMICAL OBSERVATIONS, &C. 
 
 excess; this is owing to some spring in the index, 
 or inequality in ♦the adjusting screw, which it is 
 very difficult totally to obviate. 
 
 The best way to correct this, is always to move 
 the index the same way in making your observa- 
 tions, as you did in taking the error of adjust- 
 ment; though, where a great number of observa- 
 tions are taken, it were best both to settle the ad- 
 justment, and take the observations alternately by 
 moving the index backwards and forwards, or by 
 setting the objects open, and making them lap 
 over alternately. A mean of all these will cer- 
 tainly be the most free from error, as the errors 
 w^ill counteract each other. 
 
 This may also correct a faulty habit which an 
 observer may have contracted, in terming the con- 
 tact between the two objects; and though there 
 may seem to be some impropriety in the mode, 
 yet, a mean of them will be much nearer the truth 
 than any single observation, where a person pre- 
 fers seeming to real accuracy. 
 
 The lower limb of the O or C always comes 
 first into contact, when you move the index for- 
 ward, and the index shews the double altitude of 
 the O's upper limb, if the moveable sun is upper- 
 most, but of the lower limb when the moveable 
 sun is lowermost. 
 
 At sea, they generally take the altitude of the 
 0's lower limb, because it is most natural to bring 
 that to sweep the horizon. But, by land, it is 
 most correct to take the altitude of the 0's upper 
 limb; ]. Because it is highest, and less liable to 
 be affected by refraction. '1. Because the semi- 
 diameter and refraction are both subtractive, and 
 the operation is more direct than when one is plus, 
 the other minus.
 
 [ 409 ] 
 
 TO TAKE THE ALTITUDE OF THE SUN BY 
 THE FORE OBSEllVATION. 
 
 Observations taken by means of the fore horizon 
 glass arc called fore obsenations, because in them 
 both objects are before the observer. 
 
 Previous to every observation, the insti'ument 
 should be examined, in order to see whether the 
 index or horizon glasses be firm, or whether any 
 of the screws be loose; the horizon glass must also 
 be adjusted. 
 
 One or two of the dark glasses should be placed 
 before the horizon" glass, always proportioning the 
 strength of the shades to the lirightness of the sun's 
 rays, that the image may be looked at without in- 
 juring the eye. 
 
 Hold the quadrant in a vertical position, the arc 
 downwards, either by the braces or the radii, as 
 may be most convenient, or still better according 
 to the foregoing directions. Let the eye be at the 
 upper hole in the sight vane, and the lower part of 
 the limb against the breast. 
 
 Turn yourself towards the sun, and direct the 
 sight to that part of the horizon that lies directly 
 under it, keeping the quadrant, as near as you can 
 judge, in a plane passing through the sun's center 
 and the nearest part of tlie horizon, moving at the 
 same time the index a\ ith the left hand, so as to 
 bring the image of the sun down towards tlic ho- 
 rizon; then swing the quadrant round in a line 
 parallel to the line of sight; by this means the 
 image of the sun may be made to describe tlie arc 
 of a circle, with the convex side downwards. 
 Now if that edge of the sun which is observed, 
 just grazes upon the horizon, or if the horizon just 
 touches it like a tangent, without cutting it, the
 
 410 TO TAKE A 
 
 observation is rightly made, and the degrees and 
 minutes pointed out by the nonius on the arc 
 shew the apparent altitude of the sun. But if the 
 sun's edge dip below, or cut the horizon, the in- 
 dex must be moved backward; if, on the contrary, 
 it falls short of it, the index must be moved for- 
 ward, until it just grazes the horizon. 
 
 Dr. Maskelyne gives the following advice: that 
 in taking the sun's altitude, the observer should 
 turn his quadrant round upon the axis of vision, 
 and at the same time turn himself upon his heel, 
 so as to keep the sun always in that part of the 
 horizon glass which is at the same distance as the 
 eye from the plane of the quadrant; and that un- 
 less care be taken to observe the objects in the 
 proper part of the horizon glass, the measured an- 
 gles cannot be true. In this method the reflected 
 sun will describe an arc of a parallel circle round 
 the true sun, whose convex side will be down- 
 wards, and consequently when by moving the in- 
 dex, the lowest point of the arc is made to touch 
 the horizon, the quadrant will stand in a vertical 
 plane, and the altitude above the visible horizon 
 Avill be properly observed. 
 
 Great care should be taken that the situation of 
 the index be not altered, before the quantity it 
 makes is read off. 
 
 The observed or apparent altitude of the sun 
 requires three corrections, in order to obtain the 
 true altitude of the sun's center above the ho- 
 rizon. 
 
 1 . The first correction is to obtain the observed 
 altitude of the sun's center. 
 
 All astronomical calculations respecting the 
 heavenly bodies, are adapted to their centers; but 
 in taking altitudes of the sun, it is usual to bring 
 his lower limb in apparent contact with the hori-
 
 fOTLE OBSERVATION- 411 
 
 zon. In this case it is evident, that a quantity 
 eqnal to the scmidiamcter of the sun must l)e ad- 
 ded to the observed altitude, to give the altitude 
 of his center. But if on any occasion, as from 
 clouds, the altitude of the upper limb betaken, the 
 semidiametcr of the sun must be subtracted. 
 
 The mean semidiametcr of the sun is l6 mi- 
 nutes, which may be taken as a constant quantity 
 in common o1)servations, as the greatest variation 
 from this quantity scarcely exceeds one quarter of 
 a minute. 
 
 2. The second correction is, to rectity the er- 
 rors arising from refraction. 
 
 One of the principal objects of astronomy is to 
 fix the situation of the several heavenly bodies. 
 It is necessary, as a first step, to understand the 
 causes which occasion a variation in the appear- 
 ance of the place of those objects, and make us 
 suppose them to be in a different situation from 
 what they really are: among these causes is to be 
 reckoned the fol lowing. *= 
 
 The rays of light, in their passage from the ce- 
 lestial luminaries to our eyes, are bent from their 
 true direction by the atmosphere; this bending is 
 called refraction, and they are more or less re- 
 fracted, according to the degree of obliquity with 
 which they enter the atmosphere, that is accord- 
 ing to the altitude of the object; from this cause, 
 their apparent altitude is always too great ; the 
 quantity to be subtracted from the observed alti- 
 tude may be found in any treatise on navigation. 
 
 The following pleasing and easy experiment 
 will give the reader an idea of what is mcfmt by 
 the refraction of the rays of light; a wonderful 
 
 * Adams's Astronomical and Geographical Essaj-s.
 
 412 TO TAKE A FORE OBSERVATIOX. 
 
 property, to which we are indebted for nil the 
 advantages of vision, and the assistance we receive 
 from telescopes, &c. 
 
 Experiment. Into any shallow vessel, a bason, 
 put a shilling, and retire to such a distance, as 
 that you can just see the farther edge of the shil- 
 ling, but no more; let the vessel, the shilling, and 
 your eye, remain in the same situation, while an 
 assistant fills up the vessel with water, and the 
 whole shilling will become visible, the rays com- 
 ing from the shilling being lifted or bent upwards 
 in their passage through the water. For the same 
 reason, a strait stick put partly into water appears 
 bent. 
 
 3. The third correction is for the dip, or de- 
 pression of the horizon. 
 
 The dip of the horizon is the quantity that the 
 apparent horizon appeal's below the true horizon, 
 and is principally occasioned by the height of the 
 observer's eye above the water; for, as he is ele- 
 vated above the level of the sea, the horizon he 
 views is belov/ the true one, and the observed alti- 
 tude is too great, by a quantity proportioned to the 
 height of the eye above the sea: the quantity to be 
 subtracted from the altitude will be found in any 
 Treatise upon Navigation.* 
 
 Mr. Nicholson says, that observers at sea gene- 
 rally choose to stand in the ship's waist when they 
 take altitudes, because the height of the eye above 
 the water is not so much altered by the motion of 
 the ship; but this is of no consequence, for in 
 rough vveather the edge of the sea beheld from a 
 small elevation is made uneven by waves, whose 
 altitudes amount to two or three minutes, or more; 
 
 * See W. Jones's Description of the Hadley's Quadrant,
 
 MERIDIONAL ALTITUDES, &C. 413 
 
 which circumstance prochiccs as great an uncer- 
 tainty as the -rise and fall of the object seen from 
 the poop, when the ship pitches. These are mi- 
 nute causes of erro.r, but not to be disregarded by 
 those who wish to obtain habits of accuracy and 
 exactness.* 
 
 MERIDIONAL ALTITUDES. 
 
 Tlie meridian altitude of the suii'i' is found by 
 attending a few minutes before noon, and taking 
 his altitude from time to time; when the sun's 
 altitude remains for some time without any con- 
 siderable increase, the observer must be attentive 
 to mark the coincidence 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 meridional 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 desirous to obtain. His quadrant being properly 
 adjusted, 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, 
 agreeable to the foregoing directions. 
 
 3. Look, through the sight vane and the trans- 
 parent part of the horizon glass, strait up to the 
 star, which will coincide with the image seen in 
 
 - Nicholson's Navigators Assistant. 
 t At places where it rises and sets.
 
 414 RULES FOP. FIXDIXG 
 
 the silvered part, and form one star; but, as soon 
 as you 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 of the horizon glass, as the motion of the in- 
 dex depresses it, until it comes down exactly to the 
 edge of 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 pemimbra, or indistinct shad- 
 ing 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 horizou, the other f(3r refraction. 
 
 RULES FOR PINDI>XG THE LATITUDE; THE SUn's 
 ZENITH, DISTAXCE, AND DECLIXATION AI' 
 NOON BEING GIVEN. 
 
 The first subject for consideration is, whether 
 the sun's declination be north or south ; and, se- 
 condly, 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 denomination; but if one be north, and the 
 other south, they are said to be of different deno- 
 minations. Thirdly, take the given altitude 
 from 90° 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 demomination is the sam& 
 with the declination, if it be greater than the ze- 
 nith distance; but the latitude is of a contrary
 
 THE LATITUDE. 415 
 
 denomination, if the zenith distance be less than 
 the declination. 
 
 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 de- 
 clination. 
 
 Or, by the two following rules you may find the 
 latitude of the place from the altitude and decli- 
 nation ffiven. 
 
 Rule 1. If 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 re- 
 mainder is the latitude, and is of the same name as 
 the declination. 
 
 Rule 2. If the meridian altitude and declina- 
 tion are of the same denomination, that is, both 
 north or both south, then add the declination and 
 altitude together, and subtract that sum from go 
 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 al- 
 titude. 
 
 EXAMPLES FOR FINDING THE LATITUDE BY 
 MERIDIAN OBSERVATION.* 
 
 Example 1. Being at sea, July 29, 1779? the 
 meridian altitude of the sun's lower limb was ob- 
 served to be 34° 10' N. the eye of the observer 
 being 25 feet above the sea. The latitude of the 
 place is required. 
 
 * Nicholson's Navigator's Assistant,
 
 4l6 TO FIXD THE lATITUDE EY 
 
 The sun's declination for the third year * after 
 leap year on Julv 2g, is found in Table IIlT'to'^e 
 18" 46' N. the dip for 25 feet elevation is live mi- 
 nuteS;, the refraction for 34" is one minute; there- 
 fore, 
 
 Altitude of sun's lower limb 34°10'N. 
 Add semidiaiueter l(> 
 
 34 26 
 
 Subtract dip and refraction 0' 
 
 Correct altitude 34 20 
 
 Zenith distance or co-alt. 55 40 
 
 Subtract declination IS 46 N. 
 
 Remains latitude 36 54 S. or of the contrary 
 name to the declin. 
 
 Example 1. October 26, 1780, sun's meridi- 
 onal altitude, lower limb 62° 09' S. required the 
 latitude. Height of the eye 30 feet. 
 
 17 80 is leap year, and the sun's declination in 
 October 26, is 12° Ao S. The dip for 30 feet 
 elevation is six minutes, the refraction for 62° is 
 5 minute. Therefore, 
 
 ■* The annual course of the seasons, or the natural year, con- 
 sisting of nearly 305 days six hours, and the current year being 
 reckoned 3t)o days, it is evident that one whole day would be 
 lost in four years if the six hours were constantly rejected. To 
 avoid this inconvenience, which, if not attended to, would cause 
 the seasons to ^;hift in process of time through all the months of 
 the year, an additional day is added to the month of P'ebruary 
 every fourth yearj 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 hrst, second, or third years 
 after leaj) year, according as the remainder is 1,2, or 3. 
 
 The Nautical Ahnanack, and Robirtsoii's Treatise on Naingahon^ 
 contain tlie best tables uf the sun's decliuation, Sec. &:c,.
 
 MERIDIONAL OBSERVATION. 417 
 
 Sun's apparent alt. 1. I, 62° Og' S. 
 -f semidiam.---dip and refrac, f) i 
 
 Correct alt, 62 18 i 
 
 Zenith dis. 27 41 ^ S. 
 Sun's declination 12 45 S. 
 
 Difference is lat. 14 56 ^ N. or of the con- 
 trary name. 
 
 Examples. Jan. 7, 1776, altitude of sun's 
 lower limb at noon 87° lO' S. height of the eye 
 30 feet, required the latitude. 
 
 Sun's apparent alt. 1.1. 87° 10' 
 
 + semidiam. — dip. 10 refraction being inconsl- 
 
 derablo. 
 
 Correct alt. 87 20 
 
 Zenith distance 2 40 S. 
 Sun's declinat. I9 13 N. 
 
 Sum is latitude 21 53 N. 
 
 Example 4. In the year 1778, July 30, the 
 sun's meridian altitude lower limb was 84° 10' N. 
 required the latitude, the height of the eye being 
 30 feet. 
 
 Sun's apparent alt. 1. 1, 84° 10' 
 + semidiam. — 10 
 
 Correct alt. 84 20 
 
 Zenith dist. 5 40 N. 
 Sun's declination 18 28 N. 
 
 Difference is lat. 12 48 N. 
 
 Example 5. Being at sea in the year 1777> 
 close weather prevented tjie meridian observation 
 of the sun, but the night proving clear, the nor- 
 thermost star in the square of the constellation of 
 
 E e
 
 41S TO TAKE THE MOON S ALTITUDE. 
 
 the Great Bear was observed to come to its least 
 altitude 30"" lO'. Required the latitude; the 
 height of the eye being 20 feet. 
 
 Star's altitude apparent 30° 10' 
 — dip and refraction 6 i 
 
 Correct altitude 30 03 i 
 Star's co-declination 2/ 03 N. 
 
 Sum is latitude 57 06 i N. by Prob. II, 
 
 Example Q. June 11, 1770, the sun's meri- 
 dian altitude of the upper limb below the pole 
 was observed to be 2° 08'. The height of the ob- 
 server's eye being l6 feet; required the latitude. 
 
 Sun's apparent altitude upper limb 2° 08' 
 — dip, retraction^ and semidiam. 38 
 
 Correct alt. 1 30 
 Sim's declin. 23°08'N. its comp. QQ 52 N. 
 
 Sum is latitude 68 22 N, 
 
 TO TAKE THE ALTITUDE OF THE MOON AT SEA, 
 
 The enlightened edge of the moon, or that 
 edge which is round and well defined, must be 
 brought in contact with the horizon, whether it 
 be the upper or under edge; in other respects, the 
 same method is to be used in taking the altitude of 
 the moon as was directed for the sun. 
 
 Between new and full moons the enlightened 
 limb is turned towards the west; and durinsr the 
 time from the full to the new moon, the enligh- 
 tened limb is turned towards the east. 
 
 If that telescope, which shews "objects inverted, 
 be used, then the upper edge or limb of the moon 
 will appear the lower, the left side will appear the 
 night, and the contrary.
 
 to TAKE THE MOON S ALTITUDE. 4l() 
 
 V 
 
 The wires of the telescope should be turned 
 parallel to the plane of the instrument, as by keep- 
 ing them in a perpendicular direction, they will 
 serve at night as a guide for holding the sextant 
 in a vertical position, which cannot otherwise at 
 that time be readily ascertained. 
 
 The moon is generally bright enough to be 
 seen by reflection from the unsilvered part of the 
 glasses; if not, the telescope must be removed 
 nearer to the plane of the instrument. 
 
 The observed altitude of the moon requires fouf 
 corrections, in order to obtain the true altitude of 
 her center above the horizon. 
 
 Correction \. For the semi-diameter. This is 
 to be found in the Nautical Almanac, page 7, 
 of every month for ev^ry noon and midnight at 
 Greenwich. If the lower limb was observed, add 
 the semi-diameter thus found. If the upper limb 
 was observed, it must be subtracted. 
 
 Correction 2. For the dip of the horizon to be 
 subtracted. 
 
 Correction 3. For refraction. This is to be 
 subtracted. 
 
 Correction 4. The moon's parallax in altitude 
 This is to be added in the observed altitude. It 
 is to be found in the Requisite Tables to the Nau- 
 tical Almanac. 
 
 TO DETERMINE THE LONGITUDE AT SEA, BY 
 TAKING THE ANGULAR DISTANCE BETWEEN 
 THE MOON AND ANY CELESTIAL OBJECT. 
 
 The latitude is obtained at sea without diffi-^ 
 culty, and with as much accuracy as is requisite for 
 nautical purposes; but the motion of th6 earth on 
 its axis prevents our ascertaining the longitude 
 with the same facility ; and hence it is that most 
 
 E e 2
 
 4'iO TO FIND THE DISTANCE 
 
 methods of determining the longitude by celestial 
 observation, consists in discovering the difference 
 of apparent time between the two places under 
 consideration. 
 
 The angular motion of the moon being much 
 greater than that of any other celestial body, the 
 observation of its place is much better adapted to 
 discover small differences of time, than similar 
 observations made with any other instrument. 
 The only practical method of observing its place 
 at sea, is that of measuring the angular distance 
 between it and the sun and a fixed star. 
 
 GENERAL DIRECTIONS. 
 
 The most obscure, or rather, the least luminous 
 of the two objects must be viewed directly, and 
 the other must be brought by reflection in appa- 
 rent contact with it. 
 
 The well-defined image of the moon must be 
 always made use of for the contact, even though it 
 should be necessary for that purpose to make the 
 reflected image pass beyond the other. 
 
 In the night-time it is necessary to turn down 
 one or more of the green screens, to take oft' the 
 glare of the moon, which would otherwise prevent 
 the star from being seen. 
 
 TO FIND THE DISTANCE BETWEEN THE MOON 
 AND THE SUN. 
 
 The central distances of the sun and moon every 
 three liours of time, at Greenwich, on such days 
 as this method is practicable, are set down in the 
 Nautical Almanac. From these distances you 
 are to compute roughly the distance between^theiF 
 nearest limbs at the time of observation.
 
 BETWEEN THE MOON AND SUN. A2} 
 
 Mr. Ludlani says, the moon should be viewed 
 directly through the unsilverecl part of the horizon 
 glass, but the sun by reflection; and, if it be very 
 bright, from the unsilvered part of the glass. 
 
 If the sun be to the left hand of the moon, the 
 sextant must be held with its face downwards; but 
 with the face upwards, if the sun be to the right 
 hand of the moon. ^^' 
 
 Set the index to the distance of the nearest limbs 
 of the sun and moon, computed roughly as before; 
 and, placing the face of the sextant agreeable to the 
 foregoing rules, direct the telescope to the moon, 
 putting the sextant into such a position, that if you 
 look edge-ways against it, it may seem to form a 
 line passing through the sun and moon, a circum- 
 stance that can be only obtained by practice, the 
 parent of aptness ; then give the sextant a sweep 
 or swing round a line parallel to the axis of the 
 telescope, and the reflected image of the sun \\\\\ 
 pass by the moon to and fro, so near that you can- 
 not fail oi seeing it. 
 
 The nearest edges, or limbs, may now be 
 brought into exact contact, by moving the index, 
 and then using the adjusting screw; observing, 
 first, that on giving the sextant a motion round 
 the axis of the telescope, the images of the sun and 
 moon only touch at their external edges, and that 
 the body of the sun must not pass over, or be upon 
 the body of the moon. And, secondly, that the 
 edge of the sun touch the round or enlightened 
 edge of the moon. Then will the index point out 
 the observed or apparent distance of the nearest 
 edges of the sun and moon. 
 
 But the obsei-ved distance requires several cor- 
 rections, before the true distance of the centers of 
 the objects, as seen from the center of the earthy 
 can be found.
 
 42'2 TO TAKE THE DISTANCE 
 
 Correction 1. Is the sam, or their semi-diame- 
 ters, which is to be added, to give the apparent 
 distance of the centers of the sun and moon. 
 
 The semi-diameter of the sun for every sixth day, 
 and of the moon for every noon and midnight, at 
 Greenwich, are to be found in the Nautical Ahna- 
 nac; from these their semi-diameters are to be 
 computed at the time of observation, by the rules 
 to be found in the same work. 
 
 Correction 1. Is to free the apparent distance 
 of the effect of refraction and parallax, which will 
 then be the true distance of the centers of the sun 
 and moon, as seen from the earth. 
 
 For this purpose, two sets of tables, with direc- 
 tions how to use them, are to be found among the 
 Requisite Tables to the Nautical Almanac; being 
 a set of tables published for that purpose by the 
 Board of Longitude. 8 vo. 1781. 
 
 TO TAKE THE DISTANCE BETWEEN THE MOON 
 AND SUCH STARS AS ARE SELECTED IN THE 
 NAUTICAL ALMANAC, FOR THE PURPOSE OF 
 FINDING THE LONGITUDE AT SEA. 
 
 The distance of these stars from the moon's 
 center for every three hours at Greenwich, is given 
 in the Nautical Almanac, from whence their dis- 
 tance from the enlightened edge may be roughly 
 computed as before. 
 
 The star must be viewed directly; the moon is 
 generally bright enough to be seen by reflection 
 from the unsilvered part of the glass; the proper 
 shade to take oflT the glare of the moon is soon 
 found. When the star is to the left hand of the 
 moon, the sextant must be held with its face up-- 
 wards; but if the star be to the right hand of the 
 piopn, with its face downwards.
 
 BETWEEN THE MOON AND STARS. 423 
 
 Set the index to the distance roughly computed, 
 and placing the face of the octant by the foregoing 
 rules, direct the telescope to the star. Then place 
 the sextant so that, if seen edge-ways, it may seem 
 to form a line passing through the moon and star, 
 and give it a sweep round a line parallel to the axis 
 of the telescope, and the reflected image of the 
 moon will pass so near by the star, that you will 
 see it in the field of the telescope; a proof that the 
 sight is directed to the right star. - 
 
 The enlightened edge of the moon, whether east 
 or west, must then be brought into contact with 
 the star, by moving the index. To know whether 
 the contact is perfect, let the quadrant gently vi- 
 brate in a line parallel to the axis of vision, for the 
 star should just graze the edge, without entering 
 at all within the body of the moon ; when this is 
 the case, the index will shew the apparent distance 
 of the moon from the star, which, when corrected, 
 gives the true one. 
 
 Correction 1. For the semi-diameter of the 
 moon. This may be found in the Nautical Alma- 
 nac for every noon and midnight, at Greenwich ; 
 and from thence computed, by the rules there 
 given, for the time of observation. If the observed 
 or enlightened limb be nearest the star, the semi- 
 diameter thus found is to be added; if the enlight- 
 ened edge be the furthest from the star, then the 
 semi-diameter is to be subtracted. 
 
 Correction 2. Is for refraction and parallax, to 
 be found from the table as directed before for the 
 sun and moon. 
 
 These corrections being properly made, you 
 have the true distance of the moon's center from 
 the star, as seen from the center of the earth. 
 From this distance, and the time of observation, 
 tlic longitude may be found.
 
 424 TO OBSERVE THE SUN S 
 
 The star to be observed is always one of the 
 brightest, and lies in a line nearly perpendicular 
 to the horns of the moon, or her longer axis; but 
 if you have any doubt whether the sight be di- 
 rected to the proper star, set the index to the sup- 
 posed distance as before, hold the sextant as near 
 as you can judge, so that its plane, seen edge- 
 ways, may coincide with the line of the moon's 
 shorter axis, and moving it in that plane, seek the 
 reflected image of the moon through the telescope. 
 Having found the reflected image of the moon, 
 turn the sextant round the incident ray, that is, a 
 line passing from the moon to the instrument, and 
 you will perceive through the telescope all those 
 stars which have the distance shewn by the index; 
 " but the star to be observed lies in a line nearly 
 perpendicular to the horns of the moon, there 
 will, therefore, be no fear of mistaking it. 
 
 TO OBSERVE CORRESPONDING ALTITUDES OP 
 
 THE SUN, 
 
 The basis of all astronomical observation is the 
 determination of the exact time of any appearance 
 in the heavens. By corresponding altitudes this 
 time may be determined, without the apparatus of 
 a fixed observatory; they are also useful in finding 
 a nieridian line, and may be easily and accurately 
 made by a sextant. 
 
 For these observations it is necessary to be pro- 
 vided with a clock. These altitudes should be 
 observed, in our latitude, at least two hours dis- 
 tant from noon. The best time is when the sun is 
 on or near the prime vertical^ that is, the east or 
 west points of the compass. 
 
 About these times in the forenoon, take several 
 Rouble altitudes of the sun^ write down the dcr
 
 CORRESPONDING ALTITUDES. 425 
 
 grces, minutes, and seconds shewn on the arc, and 
 also the exact time shewn by the clock at each ob- 
 servation; and let the different observations made 
 in the forenoon be written one below the other in 
 the order they are made. 
 
 In the afternoon, set the index to the same de- 
 gree and minute as the last morning observation ; 
 note very exactly the time shewn by the clock 
 when the sun is come down to the same altitude, 
 and write down the time on the right hand of the 
 last morning observation; proceed in the same 
 manner to find the time by the clock of all the al- 
 titudes corresponding to those taken in the morn- 
 ing, and write down each opposite to that morn- 
 ing one to which it corresponds. 
 
 Take now the first pair of corresponding alti- 
 tudes, add them together, and to half their sum 
 add six hours; this being corrected for the change 
 of the sun's declination between the morning and 
 evening observations, you will have the time of 
 solar noon derived from this pair of observations. 
 Do the same for each pair, and take the mean of 
 the times thus found from each pair, and you 
 \vill have the exact time shewn by the clock at 
 solar noon. 
 
 The time by the clock of solar or apparent noon 
 being thus obtained, the time of mean noon may 
 be had by applying the proper equation of time. 
 
 Or thus. Add 12 hours to the time of the 
 afternoon observation, from which subtract the 
 time of the forenoon one,* and add half the diffe- 
 rence to the time of the forenoon or morning ob- 
 
 * This gives the whole interval betwreen the obsen^ations, to 
 which, if your clock is known to vary much in that time, you 
 must add the clock's loss, or subtract its gain, during that interval, 
 and halt this corrected (if necessary) interval n-.ust be added to the 
 time of forenoon. Sec.
 
 426 TO OBSERVE THE SUn's^ &C. 
 
 servatioii; this will give the time of apparent noon 
 nearly. 
 
 Having this time nearly, it must be corrected by 
 the table of equations for equal altitudes, on ac- 
 count of the sun's change in declination, in the 
 interval between the observations; and you must 
 also apply the equation of time found in the Nau- 
 tical Almanac with contrary signs, subtracting 
 when it is +, and adding when it is — . 
 
 Example. Equal altitudes taken June 1782. 
 
 East Azlmu 
 
 th. 
 
 
 
 West Azimuth. 
 
 h. 
 
 m. 
 
 s. 
 
 
 
 
 h. m. 
 
 s. 
 
 a 8 
 
 55 
 
 46 
 
 
 
 c 
 
 3 8 
 
 44 
 
 b 8 
 
 57 
 
 41 
 
 
 
 b 
 
 3 6 
 
 48 
 
 c 8 
 
 59 
 
 27 
 
 
 
 a 
 
 3 4 
 
 58 
 
 h. m. 
 
 s. 
 
 
 h. 
 
 m. 
 
 s. 
 
 h. 
 
 m. s. 
 
 15 8 
 
 44 
 
 
 15 
 
 6 
 
 48 
 
 15 
 
 4 58 
 
 8 55 
 
 46 
 
 
 8 
 
 57 
 
 41 
 
 8 
 
 59 V 
 
 6 12 
 
 58 
 
 6 
 
 9 
 
 7 
 
 6 
 
 5 31 
 
 3 6 
 
 29 
 
 
 3 
 
 4 
 
 33i 
 
 3 
 
 2 45i 
 
 12 2 
 
 15 
 
 12 
 
 2 
 
 141 
 
 12 
 
 2 \1\ 
 
 As the seconds differ add them together, i5 
 and divide the sum by 3, by which you |^f 
 obtain a mean. "^^ 
 
 3)42(14 
 8 equation for six hours. 
 
 h. m. s. 
 Therefore;, 12 2 14 the mean 
 
 12 2 14 8 
 
 1 55 1 equation of time sub. 
 
 12 O 19 7 clock near 20" too fast for 
 equal time. 
 
 For accurate Tables of Ecpiations to Equal Altitudes, and the 
 Method of finding the Longitude at Sea by Time-keepers, I refer the 
 reader to Mr. Waless, pamphlet. 8vo. 1794,
 
 ( 427 ) 
 
 MERIDIONAL PROBLEMS BY THE STARS, AND 
 VARIATION OP THE MAGNETIC NEEDLE BY 
 THE SUN.* 
 
 To fix a meridian Vine by a star, when it can 
 be 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 
 occasionally along the rod: let the moveable 
 plummet hang four or five feet northward of the 
 iixed 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 always see both in one, and 
 just touching the star. When the star becomes 
 stationary, or moves not beyond the moveable 
 plummet, set up a light on a staff, by signals, about 
 half a mile off, ]jreeisely in the direction of both 
 the plummets. Near 12 hours afterwards, when 
 the star comes towards its greatest elongation on 
 the other side of the pole, with your eye a foot or 
 two behind the fixed plummet, follow it with the 
 moveable plummet, till you perceive it stationary 
 as before; mark the direction of the plummets 
 then by a light put up precisely in that line: take 
 the angle between the places of the two lights with 
 Hadleys quadrant, or a good theodolite; the cen- 
 ter 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 fixe4 
 plummet. 
 
 '■ ]VIackcnzie"s ^Maritime Siirveying, p. -i/.
 
 428 MERIDIONAL PROBLEMS. 
 
 This observation will be more accurate if the 
 eye is steadied by viewing the plumb-lines and staf 
 through a small slit in a plate of brass stuck up- 
 right on a stool, or on the top of a chair-back. 
 
 This problem depends on no former observa- 
 tions whatever; and, as there is nothing in the 
 operation, or instruments, to affect its accuracy, 
 but what any one may easily guard against, it may 
 be reckoned the surest foundation for all subse- 
 quent celestial observations 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 Jix a meridian line by two circimipolar 
 stars that have the same right ascension, or differ 
 precisely 180 degrees. 
 
 Fix on two stars that do not set, and whose 
 right ascensions are the same, or exactly 1 SO de- 
 grees different: 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 ex- 
 actly in the meridian. 
 
 In order to place a distant light exactly in the 
 direction of the plumb-line and stars, proceed in 
 the following manner. Any night before the ob- 
 servation is to be made, when the two stars ap- 
 pear to the eye to be in a vertical position, set up 
 a staff in the place near which you would have the 
 plummet to hang; and placing your eye at that 
 staff, direct an assistant to set another staff up- 
 right in the ground, 30 or 40 yards off', as near in 
 a line with the stars as you can. Next day set 
 the two staffs 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.
 
 MERIDIONAL PROBLEMS. 42Q 
 
 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 the better, and half an hour before the stars 
 appear near the same vertical, have a lighted Ian- 
 thorn ready tied to the top of a pole, and set up- 
 right 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 
 plumnlet. At the same time, let another pole, or 
 rod, six or eight feet long, be supported hori- 
 zontally 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 ground, 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 
 not shake by a small motion of the air. Then 
 shift the plumb-line to one hand or the other, till 
 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 same side of the plumb 
 line also; and then the plummet and light will 
 be exactly in the meridian. 
 
 There are not t\yo remarkable stars near the 
 north pole, with the same right ascension pre- 
 cisely, or just a semi-circle's difference: but there 
 are three stars that are very nearly so, viz. the pole 
 star, s in Ursa Major, and y in Casiopeia. If
 
 430 MERIDIONAL PROBLEMS. 
 
 either of the two last, particularly f, Jvre taken in 
 the same vertical with the pole star, they will 
 then he so very near the meridian, that no greater 
 exactness need he desired for any purpose in sur- 
 veying. At London y is about l' west of the 
 meridian then; and s. much nearer it, on the west 
 side likewise. Stars towards the S. pole proper 
 for this observation are, 7, in the head of the 
 cross; fj, in the foot of the cross; and c, in the 
 head of the phenix. 
 
 To find a meridian line hy a circimipoJar star, 
 •when it is at its greatest azimuth from the pole. By 
 a circumpolar star is here meant, a star wdiose 
 distance from the pole is less than either the lati- 
 tude, or co-latitude of the place of observation. 
 
 1. Find the latitude of the place in which you 
 arc to observe the star. 
 
 2. Fix on a star whose declination is known, 
 and calculate its greatest azimuth from the north, 
 or elevated 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 427, nntil 
 the star is stationary; then set up a light half a 
 mile, or a mile off, 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 a pole where the light stood, 
 with a flag flying at it : tlKni with a theodolite, 
 or HadJeys quadrant, set the index to the degree 
 and minute of the star's azimuth from the north, 
 fountl 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 fiag, as the pole of 
 the world was from it, so as at the plumb-line these
 
 MERIDIONAL PROBLEMS. 431 
 
 two lines may make an angle equal to the azimuth 
 of the star, and the plumb-line and staff 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 east-side of it. 
 
 The pole of the world is always between the 
 pole star and Ursa-Major: so that when Ursa- 
 IMajor 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 pro- 
 portion. 
 
 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 observing when f, in Ursa-Major, and y, 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 upright piece, Hke the let- 
 ter T; when the plummet is at rest, and both 
 stars are seen touching the upper edge of the cross 
 piece, then they are both horizontal. It is ano- 
 ther conveniency in making use of this star, that 
 it changes its azimuth much slower than other 
 stars, and therefore affords more time to take its 
 direction exact. 
 
 On the south side of the equator, the head of 
 the cross is the most convenient for this observa-
 
 422 MEKIDIOXAL PROBLEMS. 
 
 fion, being nearest to the S. pole; and the time of 
 its coming to the horizontal diameter of the diur- 
 nal circle, when it appears in a horizontal line with 
 the foot of the cross, or the head of the phcnix; 
 and at the equator its greatest azimuth is one hour 
 16 minutes before or after that, as it is E. or W. 
 of the meridian. The pole of the world is be- 
 tween the head of the cross and the last of these 
 stars. 
 
 7o find when 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 proposed; their difference will shew the dif- 
 ference between their times of coming on the me- 
 ridian in the south, or between tlie 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 below the pole. 
 
 Given the latitude of the place, and the decimations 
 and right ascensions of two 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, ami the 
 precise time of the observation. 
 
 Case 1. JVlien the two stars are northward of 
 the zenith, plate 16, fig. 4. 
 
 Let Z P O be a meridian, Z the place of obser- 
 vation, H O its horizon, and 2iF a vertical circle 
 passing through two 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 ZP the co-latitude of the 
 place, P s and P S the co-declination of the two 
 stars respectively, and the angle s P S the nearest
 
 MERIDIONAL PROBLEMS. 433 
 
 distance of their cireles of right ascension ; VO the 
 arc of the horizon between the vertical circle and 
 the meridian; and d Q the arc of the equator be- 
 tween the star S and the meridian. When, in the 
 triangle PZs, the angles PZs and sPZ, mea- 
 sured by the arcs VO and D E, are found, the 
 problem is solved. 
 
 Solution 1. Begin with the oblique-angled tri- 
 angle P s S, in which are given two sides P s and 
 P S, the co-declinations of the two stars respec- 
 tively, and the included angle SPs; which angle 
 is the difterence of their right ascensions when it 
 is less than 180 degrees; but if their difference is 
 more than 1 80 degrees, then the angle S P s is 
 equal to the lesser rigjht ascension added to what 
 the p-reater wants of 36o deg-rees. From hence 
 find the angle S s P by the following proportions : 
 
 As radius is to the co-sine of the given angle 
 SPs, so is the tan. of the side opposite the re- 
 quired angle P s, to the tangent of an an 2:1 e, 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 P S, adja- 
 cent the required angle, and M; call it N. Then, 
 
 Sine N : sine M :: tan. of the given angle SPs 
 : 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 than P S. 
 
 * Two arcs or angles are said to be like, or of the same kind, 
 when both are less than 90°, or both more than t)0^; but are said 
 to be unlike, when one is greater, and the other less than pO°i 
 and are made like, or unlike to another, by taking the supple- 
 ment to 180° degrees of the arc, or angle, produced in like pro- 
 Jjortion, in pla«e of what the proportion brings out. 
 
 F f
 
 431 MEniDIONAL PROBLEMS. 
 
 2. Next ill the oblique-angled triangle PsZ 
 there arc given two sides, P Z, the co-latitude of 
 the place of observation ; P S, the co-deelination 
 of the stars ; and the angle P s Z opposite to one 
 of them, which is the supplement of P s S, last 
 found to 180°: from thence iind the angle PZs 
 opposite the other side, by the following propor- 
 tion : 
 
 Sine of PZ : sine PsZ :: sine PZs, either 
 acute or obtuse. 
 
 This angle PZs, measured by the arc of the 
 horizon VO, is therefore equal to the horizontal 
 distance of the vertical of the two stars from the 
 meridian. 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 day, the degrees and minutes 
 in the arc VO, may be added to it by Hudlefs 
 quadrant, and a pole set up there, which will be 
 in the direction of the meridian from the plumb- 
 line. 
 
 3. Last, in the same triangle PsZ find the angle 
 s P Z between the given sides, by the following 
 proportions: 
 
 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 Ps, is to the co-t. of the other side PZ; so 
 is the co-s. M, to the c6-s. of an angle, which 
 call N. 
 
 N is like tl^e side opposite the given angle, if 
 that angle is acute; but unlike the side opposite 
 the givei> angle, if that angle is obtuse.
 
 MERIDIONAL PROBLEMSi 435 
 
 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. 
 
 The angle sPZ, thus found, added to s P S, 
 and their sum subtracted from 180'^ will leave the 
 angle d P Q, or the arc d Q that measures it, 
 which is the arc of the equator the star S must 
 pass over in coming from the vertical, ZV, to the 
 meridian. Which converted into time, and mea- 
 sured 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 and sun, 
 at what time that star should come to the meridian 
 in the north, the night of the observation ; subti-act 
 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 bet- 
 ter; for then an ordinary watch will serve to mea- 
 sure the time sufficiently exact. It is still more 
 advantageous 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 are 
 preferable to those that are southward of it» 
 
 F f 2
 
 436 MERIDIONAL PROBLEMS. 
 
 If the two stars are past the meridian when they 
 are observed in the same vertical, then the arc d Q 
 gi,ves the time S took to come from the meridian 
 to the vertical, and must be added to the time 
 when that star was on the meridian, to give the 
 time of the obseiTation ; and the arc of the horizon 
 VO must be marked on the ground on the side of 
 the vertical, contrary to what it would have been 
 in the foregoing supposition ; that is, eastward l;e- 
 loiv the pole, and westward above it. 
 
 When two stars come to the vertical line near 
 the meridian, it maybe difficult to judge on which 
 side of it they are at that time; for determining 
 this, the following rules may serve. 
 
 The right ascension of two stars may be either 
 each less, or each mo?-e than 180 degrees, or one 
 more and the othe?- less. 
 
 When the right ascension of each of the stars 
 is either less, or more, than 180 degrees, they will 
 come to the same vertical on the east-side of the 
 meridian when the star, with the greatest right 
 ascension is the lowest-, but on the ivest-side o^ the 
 meridian when it is highest. 
 
 IVlien the right ascension of one of the stars is more 
 thaii 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 right ascension of the higher star is less than 
 that excess, they come to the vertical on the west- 
 side of the meridian. 
 
 , If the right ascension of the higher star is inore 
 than 180 degrees, and that excess is less than the 
 right ascension of the lower star, then they come to 
 the vertical on the west-side of the meridian : but if
 
 MERIDIONAL PROBLEMS. 437 
 
 the excess of the higher star's right ascension 
 above 180° is more that the right ascension of the 
 lower star, then they come to the same vertical on 
 the east-side of the meridian. 
 
 Case 2. When the two stars are in the same 
 vertical, southward of the zenith, plate \Q, Jig. 3, 
 
 When two stars are observed in the same ver- 
 tical line southzvard of the zenith, or toward the 
 depressed pole, the operations and solutions are 
 nearly the same as in Case 1 . For let P Z O be a 
 meridian, Z the place of observation, O H its ho- 
 rizon, ZV a vertical circle passing through the 
 two stars S and s, P the pole, Q E the equator, 
 PD and Pd two circles of declination, or ri^rht 
 ascension, passing through the two stars respec- 
 tively; then is ZP the co-latitude of the place, 
 P S and P s the co-declination of the tvi^o stars re- 
 spectively, the angle 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. S P s 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 foimd, 
 will give the solution of the problem, as in Case 1 . 
 For the supplement sZB to 180° degrees is the 
 angle VZO, or its measure OV, 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
 
 438 TO FIND THE VAEIATION- 
 
 which the solution in Case 1, properly applied, 
 will serve. 
 
 On the south-side of the zenith, when the highest 
 of the two stars has the least right ascension, they 
 come to the same vertical on the east-side of the 
 meridian; but when the highest star has the 
 greatest right ascension, they come to the same 
 vertical on the ivest-sicle of the meridian. 
 
 Problem. To find the suns ampUtude at ris- 
 ing, or setting, and from thence the variation of the 
 magnetic needle, with an azimuth compass. 
 
 Make the needle level with the graduated cir- 
 cle in the box. Then, when the sun's loAver edge 
 is a semi-diameter above the horizon, take the 
 bearing of its center, from the N. or S. whichever 
 is nearer, through the sights making the thread 
 bisect the sun's disk, and that subtracted from 90°, 
 will be the sun'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 co-sine of the latitude is to radius, so is 
 the sine of the sun's declination at setting or ris- 
 ing to the sine of his amplitude from the W. or E. 
 
 Which will be N. or S. as the sun's declination 
 is N. orS: and the distance in degrees and mi- 
 nutes between the true E. or W. and the mag- 
 netic, is the variation of the needle. 
 
 An easy and sure way to prevent mistakes, 
 which the unexperienced are 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 sub- 
 traction, and on which side of the north it lies. 
 f'/or example:
 
 OF THE MAGNETIC XEEDLE. 43Q 
 
 Plate 16, Jig. 5. Suppose the variation was 
 Bought at sun-setting. Draw bv 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 
 Avest; 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 0, representing 
 the direction of the sun's center at setting, and 
 another line C w, for the magnetic west, cither on 
 the north or south side of © ? as it was observed to 
 be, and at its judged distance. Then by observ- 
 ing the situation of these lines, it will easily occur 
 ■whether the magnetic amplitude and true ampli- 
 tude are to be added, or subtracted, to give the 
 variation ; and on which side of the true north the 
 variation lies. In the present supposition, w 
 is the magnetic amplitude, and 0W the true 
 amplitude; ©W therefore must be subtracted 
 from © w, to give w W the distance of the one 
 from the other; and n, the magnetic N. QO^from 
 w, must be westward of N, the true north. 
 
 This method is sufficiently exact for finding the 
 variation, but it is not exact enough for fixing a 
 precise meridian line; because of the uncertainty 
 of the refraction, and of the sun's center: but if 
 the sun ascends, or descends, with little obliquity, 
 the error then will be very little. 
 
 To f7id the Sim s a%hnuth^ and from thence to find 
 the variation of the needle. 
 
 First, let the latitude of the place be exactly 
 found- Next, let the quadrant be carefully ad- 
 justed for observation. Then, two or three hours 
 before or after mid-day, take the altitude of the 
 sun's center as exactly as possible, making the ver- 
 tical wire of the telescope bisect the sun's disk; 
 and, without altering the plane of the quadrant m
 
 440 TO FIND THE VARIATION 
 
 the least, move the telescope vertically till you sec 
 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 center was taken. 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 follow- 
 ing rule. 
 
 Add the complement of the latitude, the com- 
 plement of the altitude, and the complement of 
 the sun's declination to QO'' together, and take the 
 half of that sum, and note it down ; subtract the 
 complement of the declination from the half sum, 
 and take the remainder; then take the comple- 
 ment arithmetical * of the sines of the comple- 
 ment of the altitude, and of the complement of the 
 latitude, and add them together, and to them add 
 the sines of the fore-mentioned half sum and re- 
 mainder: half the sum of these four logarithms is 
 the co-sine of half the azimuth required. There- 
 fore, find by the tables what angle that co-sine an- 
 swers to, double that angle, and that will be the 
 sun's azimuth from the north. 
 
 If the sun's declination is S. in north latitude, or 
 IS. south latitude^ in place of taking the comple- 
 ment of the declination to 90°, 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. 
 
 •' The complement arithmetical of a logarithm is found thus : 
 begin at the left hand of the logarithm, and subtract each figure 
 from p, and the last figure from 10, setting down the several re- 
 mainders in a line; and that number will be the arithmetical 
 complement renuired.
 
 OP THE MAGNETIC NEEDLE. 441 
 
 Then, to find the variation, pkicc your needle 
 below the center 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 the 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 center will not be sensible in the azi- 
 muth. But, when the sun does not rise high on 
 the meridian, this method is not to be relied on, 
 when great exactness is necessary; for then 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 center of a theodolite, or Had ley & quadrant, 
 where the center 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 center of the 
 quadrant. 
 
 N. B. The sun's declination in the tables must 
 be corrected by the variation arising from the dif- 
 ference 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 Roherisons, Moore Sy and other trea- 
 tises on navigation; it is, thercfoi'c needless to be 
 inore particular here.
 
 [ 442 ] 
 
 A COURSE 
 
 O F 
 
 PRACTICAL GEOMETRY* 
 
 ON THE GROUND, 
 BY ISAAC LANDMAN,, 
 
 PROFESSOR OF FORTIFICATION- AND ARTILLERY TO TKt 
 ROYAL MILITARY ACADEMY AT WOOLWICH. 
 
 It is as easy to trace geometrical figures on the 
 ground, as to describe them on paper; there is, 
 however, some small difference in the mode of 
 operation, because the instruments are dift'erent. 
 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, instead of 
 a rule; the same cord, by fixing one of 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 fi- 
 gure necessary in practice, may be easily traced on 
 the ground. 
 
 Problem 1 . To draw upon the ground a strait 
 Tine through two given points, A, ^,Jig. 1 , plate 27. 
 
 Plant a picket, or staff, at each of the given 
 points A, B, then fix another, C, between them, 
 in such manner, that when the eye is placed so as 
 
 * I have been obliged to omit in this Course, so liberally 
 communicated by Mr. I-andman, the calculations that illustrate 
 the examples} this, I hope, will not in the least lessen its use, a.s 
 this work will tall into the hands of very few who are ignorant 
 of the nature and application of logarithms.
 
 PRACTICAL GEOMETRY. 443 
 
 to sec the edge of the staff A, it may coincide 
 with the edges of the staves B and C. 
 
 The line may be prolonged by taking out the 
 staff A, and planting it at D, in the direction of 
 B and C, and ^o on to any required length. The 
 accuracy of this operation depends greatly upon 
 fixing the staves upright, and not letting the eye 
 be too near the staff, from whence the observation 
 is made. 
 
 Problem 2. To measure a strait Vine. 
 
 We have already observed, that there is no 
 operation more difficult than that of measuring a 
 strait 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- 
 incide with that of the first rod ; now let the first 
 man lift up his, and count one^ and then lay it 
 down at the end of the second rod; the second 
 man is now to lift up his, and count two ; and thus 
 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 
 already describc<I, page 204. 
 
 Problem 3. To measure distances hy pacing, 
 and to make a scale erf paces, ivhich shall agree with 
 ajiother, containing fathoms, yards, ox 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 re- 
 connoitre fortified places, where great accurac}' is 
 not required, or where circumstances will not al-
 
 444 PRACTICAL GEOMETRf 
 
 low the use of instruments ; in this case it is ne- 
 cessary to be well accustomed to measuring by the 
 common pace, which is easily eifccted by a little 
 practice; to this end measure on the ground 
 300 feet, and as the common pace is ih. 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 finish it more minutely, by laying down the 
 roads, single houses, hills rocks, marshes, &c. by 
 measuring with the common pace; take the scale 
 belonging to the map, and make another relative 
 to it, in the following manner, whose parts are 
 paces. Let the scale of the map be 200 fathoms, 
 draw a line AB,^^.3, plate 27, equal to this scale, 
 and divide it into four equal parts, AC, CD, &c. 
 each of which v^dll represent 50 fathoms; bisect 
 AC at E, divide AE into five equal parts, Ae, ef, 
 fg, &c. each of which will be five fathoms; draw 
 GH parallel to AB, and at any distance there- 
 from; then through the points of division A, ef, 
 gh, E, C, &c. draw lines perpendicular to AB, 
 and cutting G H, which will be thereby divided 
 into as many equal parts as AB. Two hundred 
 fathoms, at l\ feet per pace, is 480 paces; there- 
 fore v^^rite at H, the last division of G H, 480 paces, 
 at I 360, at K 240, and so on. To lay down on 
 the plan any distance measured in paces, take the 
 number of fathoms from the line AB, correspond- 
 ing to the number of paces in G H, which will be 
 the distance to be laid down on the plan. 
 
 When plans, or maps, are taken by the plain 
 table, or surveying compass, this is an expeditious 
 method of throwing in the detail^ and a little 
 practice soon renders it easy.
 
 ON THE GROUND. 
 
 445 
 
 A TABLE, 
 
 For reducing the common pace of 2^ feet Into 
 feet and inches. 
 
 
 Rices 
 
 I 
 
 3 
 4 
 
 Feet Inc. 
 
 P. 
 
 Feet Inc. 
 
 P. 
 
 J I 
 
 33 
 
 H 
 
 Feet Inc. 
 
 !p. 
 
 .47 
 '48 
 
 '49 
 
 i^ 
 
 5- 
 5i 
 
 Feet Inc. 
 
 P. 
 
 Feet Inc. 
 
 
 2 6 
 
 5 
 
 7 6 
 
 10 
 
 16 
 
 17 
 
 19 
 
 40 
 42 6 
 
 45 : 
 
 47 6 , 
 
 77 6 
 So 
 82 6 
 
 i55 
 
 115 
 117 6 
 120 
 122 6 
 
 125 
 127 6 
 130 
 132 6 
 
 61 
 
 62 
 
 63 
 
 64 
 
 65 
 66 
 67 
 68 
 
 152 6 
 155 
 
 157 6 
 160 
 
 162 6 
 165 
 167 6 
 170 
 
 v 
 
 6 
 8 
 9 
 
 lO 
 2 I 
 t2 
 
 13 
 H 
 15 
 
 12 6 
 
 15 
 17 6 
 20 
 
 20 
 21 
 22 
 -3 
 
 50 
 
 ^-2 6 . 
 
 'S5 ! 
 57 6 
 
 ?5 
 
 57 
 38 
 
 87 6 
 90 
 92 6 
 95 
 
 
 22 6 
 
 25 
 
 27 6 
 30 
 
 1 
 
 '2^ 
 26 
 
 >7 
 
 60 
 62 6 
 65 
 67 6 
 
 39 
 +0 
 41 
 4- 
 
 97 6 
 100 
 102 6 
 105 
 
 '54 
 55 
 56 
 57 
 
 135 
 137 6 
 140 
 142 6 
 
 69 
 70 
 71 
 72 
 
 172 6 
 175 
 177 6 
 180 
 
 
 32 6 
 35 
 37 6 
 
 28 
 
 29 
 30 
 
 70 
 72 6 
 75 
 
 43 
 44 
 
 45 
 
 107 6 
 no 
 112 6 
 
 5'^ 
 59 
 60 
 
 145 
 147 6 
 150 
 
 73 
 74 
 
 11 
 
 182 6 
 185 
 
 187 6 
 
 Distances of a certain extent may 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 75 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 
 encampment, &c. It may be performed very well 
 on horseback, having first exercised the horse so a§
 
 446 PRACTICAL GEOMETRY 
 
 to make him pace a given space in a determinate 
 time, and this may be elFectcd in more or less 
 time, according as he is trained to walk, trot, or 
 gallop the original space. 
 
 Problem 4. To iralk hi a strait line from a 
 proposed point to a given object, Jig. 2, plate IT , 
 
 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 are 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 strait line. 
 
 This problem is of great use in measuring dis- 
 tances, and surveying by the pace; particularly in 
 reconnoitring either on foot or on horseback. It 
 may also be used when a battalion is ordered to 
 take a position at the distance of 3 or 400 paces, 
 and parallel to that in which it is standing; to ef- 
 fect this, let two non-commissioned officers, who 
 are well exercised in the practice of this problem, 
 step out from the extremities, or wings of the 
 battalion, place themselves square with the front 
 of the battalion, then fix upon two objects strait 
 before them, and on a signal given, set out and 
 step the required number of paces, then halt, thus 
 becoming two guides for placing the battalion. 
 
 Prob LEM 5. To place a troop E F, hetijoeen two 
 gill e7i points A, B, fig. 4, plate 27- 
 
 To perform this, let two persons, each with a 
 pole in his hand, separate about 50 or 6o paces 
 from each other; and then move on till the poles 
 are situated in the direction A, B, i. e. so that the 
 person at C may see the point D in a direction
 
 ON THE GROUND. 44/ 
 
 with B, at the same time that the person at D sees 
 C in a direction with the object A. 
 
 Method %^ jig. 5. To perform the same with 
 one person. Take a staff C F, fig. 5, plate 27, 
 three or four feet long, sht the top thereof, and 
 place a strait 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 rale forward or backward till it is 
 in the direction of D, E. 
 
 Problem 6. To raise a perpericficuJar from a 
 point C, to the give?i line AB, fig. 6, plate 2^ . 
 
 Set off two points E, D, in AB, equally distant 
 from C; fold a cord in two equal parts, place a 
 slafJ'in the middle at F, fasten the two ends to the 
 staves E and D; then stretch the cord tight, and 
 the point F will be the required point, and the 
 line CF will be perpendicular to AB. 
 
 Problem 7* Froin a given point F, out of the 
 l/ne AB, to let fall a perpendicular C Yyfig.J^pl.'lJ. 
 
 Fold your cord into two equal parts, and fix the 
 middle thereof to the point F, stretch the two 
 halves to A, B, and where they meet that line plant 
 two staves, as at E, D, divide the distance ED 
 into two equal parts at C, and the line C F will be 
 perpendicular to AB. 
 
 Problem 8. To raise a perpendicular B C r?/ 
 the end B of the line B F, fig. 8, plate 27. 
 
 From D taken at pleasure, and with the length 
 D B, plant a staff at the point A, in the direction 
 F B; with the same length set off from D towards 
 
 C, plant a stafi^'C in the direction DA, and BC 
 w ill be perpendicular to F B. 
 
 Method 2, fig. 9, plate 27- 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, an^
 
 448 PRACTICAL GEOMETRY 
 
 plant a staff at D, take a length of three feet and 
 five fcetj or yards, aceording as the former mea- 
 sure was feet or yards; fasten one end of the cord 
 at C, and the other at D; then stretch the cord so 
 that three of these parts may be next to the point 
 C, and five next to D; plant a staff at H, and C H 
 will be perpendicular to CA.* 
 
 Prob LEM (). To draw a line C F parallel to the 
 line AB, and at a given distance from it, jig. 10, 
 p>}ate 27. 
 
 At A taken at pleasuse in AB, raise a perpendi- 
 cular AC, then from B taken also at pleasure in 
 AB, draw BF perpendicular to AB; 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 
 required line. 
 
 Problem 10. To make an angle abc on the 
 ground equal to a given angle ABC, fig. 1 1 and 12, 
 plate 27. 
 
 Set off any number of equal parts from B to C, 
 and from B to A, and with the same parts mea- 
 sure AC; describe on the ground with these three 
 lengths a triangle a b c, and the angle abc thereof 
 will be equal to the angle ABC. 
 
 Problem 11. To prolong the line AB, 7Tot~ 
 ivithstanding the ohstacle G, and to obtain the length 
 thereof, fig. I'i, plate 'I7 . 
 
 At B raise the perpendicular B C, draw C D 
 perpendicular to B C, and E D to C D, make E D 
 equal to BC; then raise a perpendicular EFto 
 E D, and E F will be in a strait line with AB. 
 
 The measures of the several lines A B, CD, EF, 
 added together, give the measure of the line AF. 
 
 This problem is particularly useful in mining. 
 
 J''. Much traiible and lime viiay be saved by using across-staff, 
 before described J seep, 211. }Li)ir.
 
 ON THE GROUND. 44Q 
 
 Problem 12. To draw a line B C, 'parallel to 
 the inaccessible face b c of the bastion a, b, c, d, e, m 
 order to place a battery at C, 'where it will produce 
 the greatest effect^ jig. 14, plate 2/. 
 
 Place a staff at the point A, out of the reach of 
 musket shot^ and in a line with be; from A draw 
 AG, perpendicular to bA; set off from A to G 
 about 375 paces, and at G raise GB perpendicular 
 to AG, and produce this line as far as is requisite 
 for placing the battery, i. 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 
 equal to the given angle AB C,fg.l5 and l6, pl.27. 
 
 Set off 36 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 equal parts taken from any scale, describe the 
 arc D H, then with 52 parts from the same scale, 
 mark off the distance DH; draw GH, and the 
 angle D G H is equal to the angle ABC. 
 
 Problem 14. To measure from the outside an 
 angle AB Q,, formed by tvco walls AB^ CBj fig. 17, 
 plat el"]. 
 
 Lay off 30 feet from B to E in the direction 
 AB, and plant a staff at E; set off" the same mea- 
 sure from B to F in the direction B C, and measure 
 FE, 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 
 inaccessible flanked angle BCD of the bastion AB 
 C D E, fig. 18, plate 'if. 
 
 Place a staff at F, in the direction of the face 
 B C, but out of the reach of musket shot; make 
 
 G g
 
 450 PRACTICAL GEOMETIvY 
 
 F G perpendicular to F B, and equal to about 250 
 paces; at G make GH perpendicular to GF, 
 and produce it till it meets, at H. the direction of 
 the face DC; place a staff at I in a line with DH, 
 and the angle I H K v^'ill be equal to the required 
 angle B C D, and the value of I H K may be ob- 
 tained by Problem 13. 
 
 PfioBi.EM lO. To ascerla'in the length of the 
 line A Bj acressihle only at the P'jco extremities A, B, 
 Jig. m, plate 27. 
 
 Choose a point O accessible to A and B, draw 
 AO, OB, prolong AO to D, and make OD equal 
 to AO; prolong B O to C, and make O C equal 
 to B O, and C D will be equal to AB, the required 
 line. 
 
 Problem 17- To let fall a p erpe7idi cut ar from 
 the inaccessible 'point D, upon the right line AB, fg, 
 1\, plate 17. 
 
 From A and B draw AD, B D, let fall the per- 
 pendiculars AH, BF, upon the sides AD, BD, 
 plant a staff at their intersection C, and another E, 
 in the direction of the points C, D, and E C D 
 will be the required perpendicular. 
 
 Problem 18. To find the breadth AB, of a 
 river, fg. 20, plate 17 . 
 
 Plant a staff at C in a line; with AB, so that 
 BC may be about I of the length of AB, draw a 
 line C E in any direction, the longer the better. 
 Set up a staff at D, the middle of C E, find the 
 intersection F of EB, DA; draw DG parallel to 
 BC; measure BF, FE; and you will find AB 
 by the following proportion; as FE — BF : 5BC 
 o'rDG :i BF : AB. 
 
 Inaccessible distances may be obtained in other 
 modes, when circuin stances and the grouud will 
 permit, as in the following problem.
 
 ON THE GJROUNT). 461 
 
 Problem I9. Let it he required to measure the 
 distance A B of the produced capital of the ravelin^ 
 from the head of the trenches B, to the angle A, 
 fig. 11, flat e IT. 
 
 Draw BE perpendicular to AB, and set off 
 from B to E 100 feet, produce B E to C, and make 
 E C equal one eighth or tenth part of B E ; at C 
 raise the perpendicular C D, 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 AB by the following 
 proportion; as EC : CD :: BE : AB; thus if 
 B E = 100 feet, E C = 10 feet, and C D = 38 feet, 
 then as 10 : 100 :: 3$ : 380 feet, the distance of 
 AB. 
 
 Problem 20. To measure the inaccessible dis- 
 tance AB, fg. '23, flate 27. 
 
 Plant a staff at C, a paint from whence you can 
 readily see both A and B. By the preceding pro- 
 blems, find the length of C A, C B, make C D as 
 many parts of C A, as you do C E of C B, and join 
 DE; then as CD : DE :: CA ; AB.* 
 
 Problem 21. To determine the direction of the 
 capital of a hastion 'produced, fig. 24, plate 27. 
 
 Upon the produced lines B D, B E, of the two 
 faces C B, AB,- place the staves E and D; find the 
 lengths of EB and DB, by Problem IQ, and 
 measure D E. Then as the capital G B divides 
 
 * A7i easy method of finding the distance of forts or other ohjects.— 
 If vou go off at any angle, as gO, and continue in that direction 
 until you bring the object and your first station under an angle 
 of 63, the distance measured from the first station is'equal to half 
 the distance the object is from your first station. But should the 
 ground not admit of going od' at an angl^ of gOj go off at an an- 
 gle of 45, and whatever distance in that direction you find the 
 object and the first station under an ang^fcbf 10S:|, is half the 
 distance of the object and first station. See aUo pag^i 215 
 and 263 of these Essays. 
 
 G g2
 
 Aol' , PRACTICAL GEOMETRY 
 
 the angle ABC, and its opposite E B D, into two 
 equal parts, we have as D B : B E : : D F : E F, 
 and as D B + B E : B E : : D E : E F, and conse- 
 quently the point F, whieh will be in the direction 
 of the capital G B produced. 
 
 Problem 22. Through the 'po'mt C to draiv a 
 line IK parallel to the inaccessible line AB, Jig. 25, 
 plate 27. 
 
 Assume any point D at pleasure, find a point E 
 in the line AD, which shall be at the same time in 
 a line with CB; from E draw EG parallel to 
 DB, and through C draw GCF parallel to AD, 
 meeting B D in F. Plant a staff at H in the line 
 E G, so that H may be in a line with FA, and a 
 line I, H, C, K, drawn through the points B C, 
 will be parallel to AB. 
 
 Problem 23. To take a plan of a place A, B, 
 C, D, E, hy similar triangles, fg. 26 and 27, pi. IJ . 
 
 1. Make a sketch of the proposed place, us Jig. 
 27. 2. Measure the lines AB, BC, CD, DE, 
 EA, and write the measures ol^tained upon the 
 corresponding lines a b, b c, c d, d e, e a, of the 
 sketch. 3. Measure the diagonal lines AC, AD, 
 and write the length thereof on a c, ad, and the 
 figure will be reduced to triangles whose sides are 
 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 measures on the sketch. 5. Proceed in 
 the same manner with the other building-s, &c. 
 6. Draw the figure neatly from a scale of equal 
 parts. 
 
 Problem 24. To take a plan of a wood, or 
 marshy ground, by measuring roimd about it. Jig. 28 
 and-lQ, plate 27. 
 
 Make a rouglifsketch, j?^. 2p, of the wood, set 
 Jip staves at the angular points, so as to form the 
 circumscribing lines AB, BC, CDj and measure
 
 ON THE GROUND. 453 
 
 these lines^ and set down the measures on tlie cor- 
 responding lines in the sketch, then find the value 
 of the angles by Problems 13 and 14. 
 
 Pr o B L K M 2 5 . To take a plan of a river, fg. 3 O 
 a?id:M, plate 17. 
 
 1 . Draw a sketch of the river. 2. Mark out a line 
 AB,j^^.30. 3. At C, upon AB, raise the perpendi- 
 cular CH. 4. Measure AC, CH. 5. Measure 
 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 AB, till you have obtained the principal 
 bendings of the river, writing down every measure 
 when taken on its corresponding line in the sketch, 
 and you w^ill thus obtain sufficient data for draw- 
 ing the river according to any proportion. 
 
 Problem 26. To take a phm of the. neck ofland^ 
 A, B, C, D, E, F, G,/^. 32 and 33, plate 27. 
 
 Take a sketch of the proposed spot, divide the 
 figure into triangles AB C, AC D, AD E, AE G, 
 by staves or poles placed at the points A, B, C, D, 
 E, F, G, measure the sides AB, B C, AC, CD, 
 AD, AE, AF, F G, AG of the triangles, writing 
 down these measures upon the corresponding lines 
 ab, be, ac, &c. then measure AH on AB, 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, ik, kl, &c. Proceeding thus, you will 
 ascertain a sufficient number of points for laying- 
 down your plan by a scale of equal parts. 
 
 PRACTICAL TRIGONOMETRY. 
 
 Problem 27. To ascertain the height of a 
 building, fig. 34, plate 28. 
 
 Measure a line F E, from the foot of the build- 
 ing, so that the angle C DA may be neither too
 
 454 PRACTICAL TRIGONOMETRY 
 
 acute nor ohtuse; thus suppose EF=130 fee\j 
 place your theodolite at D, and measure the angle 
 ADC=34°56^ ' 
 
 Then as radius to tangent 34'' 56', so is E F = 
 130 feet, to AC. 
 
 Which, by workingthe proportion, you will find 
 to be 89 feet 156 parts, or 89 feet 7-92 inches, to 
 which adding four feet for D E, or its equal C F, 
 you obtain the whole height 93 feet 7-92 in-^hes. 
 
 Problem 28. To find the angle formed hy the 
 line of aim, and the axis of a 'piece of ordnance pro- 
 duced, the caliber and dimensions of the piece being 
 hiown, fig. 35, plate 28. 
 
 Suppose the line B I to he drawn through B, 
 the summit of the swelling of the muzzle, and pa- 
 rallel to C D, the axis of the piece ; the angle A B I 
 will be equal to the angle AE C, formed by the 
 line of aim A F, and the axis CD; then in the 
 right-angled triangle AI B, we have the sides AI 
 end BI to find the ang-le ABI, which we obtain 
 by this proportion; as I B : A I :: radius, tangent 
 of angle ABI equal to AE C required. 
 
 Problem 29. The elevation, three degrees, of a 
 light twdve-pounder being given, to find the height 
 to which the line of aim rises at the distance of 1200 
 yards, which is about the range of a twelve-pounder, 
 with an elevation of three degrees, fig. 35, plate 38. 
 
 The line of aim, which we suppose to be found 
 by the preceding problem, forming with the axis 
 of a light twelve-pounder \° 24', will make with 
 the horizon an angle of 1° 36'; thus the height at 
 the horizontal di|tance of 1200 yards will be the 
 second side of a^ght-angied triangle, where the 
 angle adjacent to the side of 1200 yards is 1° 36', 
 and may therefore be obtained by the followmg 
 proportion; as radius to the tangent of 1" 36', so is 
 1200 to 33 yards 1 foot 6 inches=FO,
 
 ON THE GROU>TD» 455 
 
 Problem 30. The first embrasure K of a r'l- 
 cochee battery being direct, to find the inclination of 
 the seventh embrasure B, i. e. the angle formed by 
 the line of direction B C, and the breast-zvork A B, 
 at the seventh embrasure A B ; it is supposed that all 
 the pieces are directed towards a point Q, at the dis- 
 tance of 1500 feet, fi.g. 36, plate 28. 
 
 The line of direction A C of the first embrasure 
 is supposed to be perpendicular to the bre.ist-work 
 AB; therefore we have to find the angle AB C of 
 the right-angled triangle BAG, 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 em-brasure to another be 20 feet, 
 this multiplied by 6, will be 120 feet, and equal to 
 A B; then as A B to AC, 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 ouj^ht 
 to be laid against the breast- work, it will make an 
 angle AFD, which was found by the preceding 
 problem to be 85° 26'; therefore, knowing the 
 length D E of the hurler, and consequently its 
 half D F, it will be easy to calculate the distance 
 BF from the breast-work where the middle F of 
 the hurler ought to be placed, upon the line of 
 direction of the gun. 
 
 PplOBLEM 32. To ascertain the height of a build- 
 ing from a given point, front whence it is impossible to 
 measure a base in any direction, the points A and F 
 being supposed to be in the same horizontal line, fig, 
 38, plate 'IS. 
 
 Measure the angles C E D, AE D, let C E D be 
 equal 43° 12', the angle AED equal 2° 26', and 
 the height EF from the center of the instrument
 
 456 PRACTICAL TRIGONOMETRY 
 
 to its foot five feet; then in the right-angled tri- 
 angle ADE we have AD=:zEF=5 feet, the an- 
 gle D E A=2'' 26' to find D E, which may be ob- 
 tained by the following proportion ; as radius : 
 co-tangent of angle DEA, so is AD to DE 118 
 feet; then in the right-angled triangle CDE we 
 have DE 118 feet, angle C ED 33° 12' to find 
 CD, which is found by the following proportion; 
 as radius to tangent of angle C E D, join D E to 
 D C 77 feet, which added to E F 5 feet, is 82 feet, 
 the height of the tower. 
 
 Problem 33. The distance AC, 135 toiseSy 
 from the point C, to the fianked angle of the bastion 
 being given, and also AB, 186 toises, the extefior side 
 of the polygon, to find B C, fg. SQ, plate 28. 
 
 1. Find the angle B by the following propor- 
 tion; as AB is to the sine of angle C, so is AC 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 angle BAC is also known. 
 
 2. B C is found by the following proportion ; a$ 
 sine angle C is to AB, so is sine of angle BAC : 
 B C, 213 toises, three feet. 
 
 Pr o B L E M 3 4 . To find the height of the building 
 AC, fg. 40. 
 
 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-f-BDC. 
 
 Then as sin. ZBCD : BD :: CDB : BC, 
 B C being found, we have in the right-angled tri- 
 angle FBC, the side B C and angle F B C, to find 
 F C, which is found by this proportion ; as radius 
 to sin. angle F B C, so is B C : F C, F C added to 
 AF, the height of the instrument, giycs the height 
 pf the tower.
 
 ON THE GROUND. 457 
 
 Problem 35. To ascertain the distance hetwecn 
 Hvo inaccessible objects, C D, jig. 41, plate 28. 
 
 \. Measure a base AB, from whose extremities 
 you can see the two objects C, D. 2. Measure 
 the angles CAB, DAB, DBA, CBA, CBD. 
 3. In the triangle BAC we have the side AB and 
 angles ABC, BAC, to find B C, which is found 
 by the following proportion; as sine angle ACB : 
 AB :: sine BAC : B C. 4. The angles CAB, 
 CBA, added together and subtracted from 180, 
 gives the angle AC B. 5. In the triangle ABD 
 we have the angles DAB, ABD, and conse- 
 quently ADB, and the side AB to find B D, 
 which is found by the following proportion ; as the 
 sine of angle AD B is to AB, so is the sine of angle 
 D AB : B D. 6. In the triangle CBD we have 
 the two sides B C, B D, and the angle CBD, to 
 find the angle C D B, and the side CD; to find 
 CDB we use this proportion; as the sum of the 
 two given sides is to their difference, so is the tan- 
 gent of half the sum of the two unknown angles 
 to the tangent of half their difference; the angle 
 CDB being found, the following proportion will 
 give CD; as sine angle CDB to B C, so is sine 
 angle CBD to CD. 
 
 Problem 36. To draw a line through the point 
 B, parallel to the inaccessible line C D, fg. 41, pi. 28. 
 
 Find the angle B C D by the preceding problem, 
 and then place your instrument at B, and with 
 B C make an angle C B E equal BCD, and E B 
 will be parallel to C D. 
 
 Problem 37- To ascertain several points in the 
 same direction, though there are obstacles ivhich pre- 
 'vent one extremity of the line being seen from the other, 
 fg. 42, plate 28. 
 
 1. Assume a point C at pleasure, from which 
 J:he two extremities of the line AB may be seen.
 
 458 PRACTICAL TRIGONOMETRY 
 
 2. Measure the distances CB, CA, ai'd the angle 
 ACBj and find the value of the angle CAB. 
 
 3. Measure the angle ACD, and then m the tri- 
 angle AC D we have the side AC, and the two 
 angles DAC, ACD, and therefore the third to 
 find C D, and as sine angle AD C to AC, so is sine 
 angle CAD to CD. 4. Set off CD, making 
 with AC an angle equal the angle ACD, equal 
 the measure thus found, and the point D will be 
 in a line with AB; and thus as many more points, 
 as G, may be found as you please; in this manner 
 a mortar battery 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 richo- 
 chee battery M, fig, 43, so as to be upon the cur- 
 tain AD produced. 
 
 Problem 38. To measure the height of a hill^ 
 •whose foot is inaccessihle, fig.AA^ plate 28. 
 
 1 . Measure a base F G, from whose extremities 
 the point A is visible. 2. Measure the anj^Ies 
 ABC, ACB, ACD. 3. In the triangle ABC 
 we have B C, and the angles ABC, ACB, to find 
 AC; but as sine angle BAC is to BC, so is sine 
 angle ABC to AC. 4. In the right-angled tri- 
 angle ADC we have the side AC, and the angle 
 AC D, to find AD; but as radius is to sine angle 
 AC D, so is AC to AD, the height required. 
 
 Problem 39. To take the map of a country y 
 fig. A5, plate 28. 
 
 First, choose tv/o places so remote from each 
 other, that their distance niay serve as a common 
 base for the triangle to be observed, in order to 
 form the map. 
 
 Let A, B, C, D, E, F, G, H, I, K, be several re- 
 markable objtr-ts, whose situations are to be laid 
 down in a map.
 
 ON THE GfiOUND,. 45^ 
 
 Make a rough sketch of these objects, according 
 to their positions in regard to each other; on this 
 sketch, the different measures taken in the course 
 of the observations are to be set down. 
 
 Measure the base A B, whose length should be 
 proportionate to the distance of the extreme ob- 
 jects from A and B; from A, the extremity of the 
 base, measure the angles EAB, Fx\B, GAB. 
 CAB, DAB, formed at A with the base AB, 
 
 From B, the other extremity of the base, ob- 
 serve the angles EBA, FBA, GEA, CBA, 
 DBA. 
 
 If any object cannot be seen from the points A 
 and B, another 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 stations, be- 
 cause its position 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 triangles 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, trigono- 
 metrical calculations are absolutely necessary. 
 
 But where the distance is moderate, after having 
 measured a base and observed the angles, instead 
 of calculating the 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, ansv/ers sufficiently for most mili- 
 ■ tary operations. 
 
 Problem 40. The use of the sur-v eying compass, 
 fS' ^i i>^'^i^ 15; in determining the particulars of va-
 
 460 PRACTICAL TRIGONOMETRY 
 
 rious ohjects to he inserted in a plan, fig. 46, 
 plate 28. 
 
 The first station being at A, plant staves at the 
 requisite places, and then place the compass at A, 
 directing the telescope to C; observe the number 
 of degrees north AC, made by the needle and the 
 telescope, or its parallel line drawn through the 
 center of the compass, and mark this angle in your 
 sketch, seej^^, 46; observe and mark in the same 
 manner the other angles, north AO, north AP, 
 north AQ; then measure AC, and at the second 
 station C, observe the angles north C M, north 
 C Q, north C O, north C P, north C D, and so 
 on at the other stations; in observing the angles, 
 attention must be paid, when the degrees pass 180, 
 to mark them properly in the sketch, in order to 
 avoid mistakes in protracting. 
 
 Problem 41. To raise •perpendiculars^ and form 
 angles equal to given angles by the surveying compass , 
 fg.AJ, plate 1Q. 
 
 Let it be required to trace out the field work 
 AD C B, at the head of a bridge. 
 
 The direction of the capital F E being given, 
 describe the square AB CD in the following man- 
 ner; place your compass at F, and as AB is to be 
 perpendicular to F E, direct the telescope to E, 
 and observe, when the needle is at rest, the num.- 
 ber of degrees it points to; then turn the compass 
 box upon its center, till the needle has described 
 an arc of 90°, and place a staff in the direction of 
 the telescope towards A, and AF will be perpen- 
 dicular to F E; continue AF towards B, and make 
 AF, BF, each equal 30 toises at E; in the same 
 manner raise the perpendicular D C, and make 
 ED, EC, each equal 30 toises; join DA, CB, 
 ^nd you have the square AB C D.
 
 ox THE GROUND. 4.6l 
 
 Make CG, DK, each equal to § of AB, 
 through G and K draw the line GK, set ofFGH, 
 K I, each equal to i: of AB or C D, draw the lines 
 of defence I C, H D, place the compass at I, and 
 the telescope in the direction I H ; observe where 
 the needle points when at rest, and turn the com- 
 pass till the needle has described an arc of 100, the 
 value of the angle L I H of the flank, place a staff 
 at L in the direction of the telescope, and at the 
 same time in the line I H, which gives the length 
 of the face D L and the flank I L; the face C M 
 and flank H M are ascertained in the same manner. 
 
 Make B Q equal to i of AB, and AR equal to 
 
 \ R 
 
 , and draw D R, C Q, through Z the middle 
 
 of AD, draw OZ, make RT equal to t AB; at 
 T and with T D, form an angle DTU, equal to 
 105°; plant a staff" at U, so that it may be in the 
 line Z O, and at the same time in the direction 
 T U, which gives the flank T U and face O U. 
 
 Make QS equal y A B, at S form an angle CSP 
 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 tete de pont required. 
 
 THE USE OF THE PLAIN TABLE IN MILITAPyY 
 OPERATIONS. 
 
 Problem 42. To fake the flan of a caiiip, fig. 2, 
 fhte 29. 
 
 Place the table at A, where you can conveniently 
 seethe 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 al- 
 ways to be applied against the needle.
 
 462 USE OF THE PLAIN TABLE 
 
 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 AF; then point it to K, the right 
 wing of the cavalry K L, and draw AK, then to 
 L, and draw AL, 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 south of your compass, to represent 
 the magnetic meridian. 
 
 Remove the plain table to C, planting a staff at 
 Q, measure x\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 line AC, and 
 turn the table till the line of sight coincides with 
 A, fasten the table, point the index to F, and draw 
 CF, intersecting AF in F, and determining the 
 position of the windmill F; from C diawthe mde- 
 finite lines CM, CK, CL, &c. which will de-- 
 tcrmine the points M K L, &c. draw K L and 
 m n parallel thereto, to represent the line of ca- 
 valry. 
 
 Remove the table from C to D, setting up a 
 staff at C, measure C D, and set off the distance on 
 CB from your scale, place the needle at D, the 
 index on CD, and turn the table till C coin- 
 cides 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, hills, &c.
 
 IN MILITARY OPERATIONS, 4^3 
 
 Problem 43. Ti> talie a pla?i of the irenchss of 
 an attack. Jig. I, plate ig. 
 
 The plan of trenches, taken with accuracy, gives 
 a just idea of the objects, and shews how you may 
 close more and more upon the enemy, and be co- 
 vered from their enfilade fire, and also how to pro- 
 ceed in the attack without multiplying useless 
 works, which increase expense, augment the la- 
 bour, and occasion a ereat loss of men. 
 
 Measure a long line AB, 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 AB, fasten the table, and fix a needle 
 at the point A, direct the line of sights to the 
 flanked angle of the ravelin C, and draw AC; 
 proceed in the same manner v/ith the flanked an- 
 gles D, Ej F, G, to draw the lines at the opening 
 of the trenches, plant staves at H and R, from A 
 draw a line on the table in the direction AH, mea- 
 sure AH, and set off that measure from your scale 
 upon the line AH on 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 AB, and turn the table about till the staff 
 at A coincides v/ith the line of sight, then fasten 
 the table, direct the sights to C, . and draw B C, 
 intersecting A at C; in the same manner ascertain 
 the flanked angles D, E, F, G, draw a line in the 
 direction 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 RB, then turn 
 the table about till the line of sight is in the direc- 
 tion BR, screw the table fast, direct the sights to- 
 wards P, and draw on the table the line R P, and 
 by the scale lay off its measure on that line.
 
 464. OF LEVELLIXC. 
 
 Remove tlic 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 
 PQ.; measure 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. 
 
 Having removed the table to the station S, and 
 duly placed it with regard to Q, from the point S, 
 draw the lines S V, ST, SU, setting off from your 
 scale their lengths, corresponding to their measures 
 on the ground. 
 
 Having thus taken the zigzags RPQ, STW, 
 and the parts U U WZ of the parallels, you remove 
 the plain table to H, 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. 
 
 OP LEVELLING, J/g. 48, plclfe 2S. 
 
 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 center of the earth than the other; when 
 a line has all its points equally distant from the 
 center, it is called the line of true level; whence, 
 because the earth is round, that line must be a 
 curve, and make a part of the earth's circumfe- 
 rence, as the line ABE D, all the points of which 
 are equally distant from the center C of the earth; 
 but the line of sight AG, 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 more extended; for 
 which reason, the operations which wc shall give^
 
 OF LEVELLINGi 465 
 
 arc only of an apparent level, which must be cor- 
 rected to have the true level, when the line of sight 
 exceeds 300 feet. 
 
 Suppose, for example, that AB was measured 
 upon the surface of the earth to be 600O feet, a9 
 the diameter of the earth is near 42018240 feet, 
 you will find B F by the following proportions. 
 
 * 42018240 : 60OO :: 60OO : BF equal to 
 0,85677 f. which is 10,28124 in. that is to say, be- 
 tween two objects A and F, 60OO feet distant from 
 each other, and in the same horizontal line, 
 the difference B F of the true level, or that of 
 their distance from the center of the earth, is 
 10,28124 in. 
 
 When the difference between the true and ap- 
 parent level, as of B F, has been calculated^ it will 
 be easy to calculate those which answer to a less 
 distance; for we may consider the distances E F, 
 b f, as almost equal to the lines A I, A i, which are 
 to each other as the squares of the chords, or of 
 the arcs AB, a b, because in this ease the chords 
 and the arcs may be taken one for the other. 
 
 Thus to find the diiferenee f b of level, which 
 answers to 5000 feet, make the following propor- 
 tion; 6000 f. : 5000 f. :: 0,85^77 : fb, which will 
 be equal to 0,71399 f. or 8,56788 in. 
 
 The point F, which is in the siime horizontal 
 line with A, is said to be in the apparent level of A, 
 and the point B is the true level of F; so that B F 
 is the difference of the true level from the appa- 
 rent. 
 
 Problem 44, ^^.49, j^lafe 28. The above no- 
 tion.s being supposed to know the difference of 
 
 * As the arc AB = CiOOO feet is hut very small, it may be con- 
 sidered equal to the tangent AF, and in this respect, AF is a 
 mean proportional between the whole diametei';, or twice tlia ra- 
 dius B C, and the exterior part B F. 
 
 H h
 
 466 dF LEVELLING. 
 
 level between points B and A, which are not in the 
 same horizontal line; then at A make use of arv 
 instrument proper to take the angle BCD, and 
 having measured tlie distance G D, or C I, by a 
 chain which must be kept horizontal in different 
 parts of it on the ground ALVB, you may in the 
 triangle C D B^ considered as rectangular in D^ 
 calculate B D, to which add the height CA of the 
 instrument, and calculate the difference 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, which is shewn by the 
 following method. 
 
 Problem 45. Use of the sp'irli or ivate)- level , 
 fig. 50, plate 2S. 
 
 Place the level at E, at equal distances from B 
 and G, fix one station staff" at B, the other at G; 
 vour iiistrument 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 H, 
 and the difference in height shewn by the vanes 
 on the two staves, will be the difference in the le- 
 vel between the two points B and G. Thus, sup- 
 pose the vane at G was at 4 f . 8 in. and at B 3 f. 
 gin. subtract one from the other, and the re- 
 mainder 1 1 inches, will be the difference in the 
 level between the two points B and G; you may 
 proceed in the same manner with the other points; 
 but more need not be said on this head, as I have 
 already treated this subject very fully in the fore- 
 going part of this work.
 
 AN 
 
 Essay on perspective; 
 
 AND A DESCRIPTION OF 
 
 SOME INSTRUMENTS, 
 
 JPOR FACILITATING THE PRACTICE OP TPIAT 
 USEFUL ART. 
 
 DEFINITIONS. 
 
 Definition 1. Perspective is the art of delineating 
 the representations of bodies upon a plane^ and 
 has two distinct branches, linear and aerial. 
 
 Definition 1. 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. 
 
 Definition Z. Aerial perspective gives rules to 
 fill up this drawing with colours, lights, and 
 shades, such as the objects themselves appear to 
 have, when viewed at that point where the eye of 
 the spectator is placed. 
 
 To illustrate these definitions, suppose the pic- 
 ture to be a plate of glass Inclosed in a frame 
 PLN, fig. I, f late 30, through which let the 
 eye of the spectator, placed at E, view the object 
 QRST; from the given point E, let the visual 
 lines E Q, E R, E S, &c. be drawn, cutting ^:he 
 glass, or picture, in q, r, s, &c. these points of 
 intersection will be the perspective representations 
 n h 2
 
 46s AN^ ESSAt 
 
 of the oris2:ii]al Q R S T, &c. and if they arc joinetL 
 qrst will be the true pcrspeetive delineation of 
 the original figure. And lastly, if this out-line be 
 so coloured in cverv part, as to deceive the eye of 
 a spectator, viewing the same at E, in such a man- 
 ner, that he cannot tell whether he views the real 
 object itself, or its representation, it may be truly 
 called the picture of the object it is designed for. 
 
 Definition A. When the eye, or projecting pointy 
 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, niili- 
 tary perspective. 
 
 Definilion 5. If this system of parallel rays be 
 perpendicular to the horizon and to the picture, 
 the projection is called a plane. 
 
 Defifiifion 6. If the parallel rays be horizontal, 
 and the picture upright, it is called an elevation. 
 
 Definition 7. A right line Ee, fig. 2, plate 30, 
 from the eye, E, cutting the plane of the picture 
 at right angles, and terminating therein, ate, is 
 called the distance of the picture. 
 
 Definition 8. The point e, where this central 
 ray cuts the picture, is called the center of the 
 picture. 
 
 Definition Q. The point X, fig. 2, plate 30, \Ahere 
 any original line cuts the picture, is called the in- 
 tersection of that line. 
 
 Definition 10. The scat F, fig. 2, plate 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 F G N at F, this point F is 
 called the seat of the point E upon the ground 
 plane.
 
 ON PERSrECTIVE. 46p 
 
 Defimt'ion 11. Apparent niagnitudc is iriCnsured 
 by the degree of opening of two radials passing 
 through the extremes of bodies whose apparent 
 magnitudes are eompared; thus, the apparent 
 magnitude of Q R, fg. 3, plate 30, to an eye at E, 
 is measured by the optic angle Q E R. Hence it 
 is evident, tliat all objects viewed under the same 
 iingle, have the same apparent magnitude. 
 
 Definition 12. The intersection GN, fg-1, 
 plate 30, of the picture with the ground plane, is 
 called the ground line. 
 
 Definition 13. A plane passing through the eve, 
 and every where parallel to the ground (or ground 
 plane, as it is commonly called, because it is sup^- 
 posed ev^ery where flat and level) is called the ho- 
 rizontal plane, as E h n, fig. 2, plate 30. 
 
 Definition 14. The intersection of the horizon- 
 tal plane with the picture is called the horizontal 
 Tine; thus h e n, fig. 1, plate 30, is called the hori- 
 zontal line, being the intersection of the horizontal 
 plane E h e n, parallel to the ground plane F G N. 
 
 Definition 15. The center of any line is where a 
 perpendicular from the eye cuts it. 
 
 Definition l6. A plane i E Fgh^ fig.1, no. 2, 
 plate 30, passing through the eye E, perpendicu- 
 lar to the ground, is called XXie 'vertical plane; 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 
 GFN, or its parailels., as depths are hv ^g, or its 
 parallels. 
 
 Proposition 1. Parallel aad equal strait lines ap- 
 pear less, as they are farther removed from the eye. 
 
 Let O P, Q R, J%.3, plate 30, be t^ o equal and 
 parallel strait lines, viewed at the point E; QI^ 
 t;eing the farthest off, will appear the least.
 
 470 AN ESSAY 
 
 For, draw E Q and E R, cutting O P in r; then 
 Q R and O r have the same apparent magnitude, 
 (definition 1 1 J but O r is only a part of O P. 
 
 Corollary. Similar figures parallelly situated, 
 appear less, the farther they are removed from the 
 eye. 
 
 Proposition 1. All original parallel strait lines, 
 P R, Jig. 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 paral- 
 lel 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) approaches to the fixed point O, and the case 
 is the same with any other line parallel to PR; 
 and, therefore, if all the parallels be indefinitely 
 produced, that is, at least till the strait lines mea- 
 suring their distance become invisible to the eye, 
 they will all appear to vanish together in this point, 
 w^hich, in consequence thereof, is called the va- 
 nishing point of all those parallels. 
 
 Corollary 1. Hence all original parallel planes, 
 PRSP, intersecting the picture, as in Pp, seem 
 to converge to a right line therein, viz. that line 
 Oo, which is the intersection of the picture, with 
 a plane QOEoq, parallel to them all passing 
 through the eye. 
 
 For, since p S, fig.^, 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 
 since both the points R, S, equally appear to tend 
 to the respective points Oo, therefore the line R S 
 also appears to approach to the line Oo; and this 
 is the case with all planes parallel to P R S p, and
 
 ON .rERSPECTlYE. 4^1 
 
 iherctbrc if they are produced till thev l)Cconic in- 
 visible, they will appear to meet in Oo, whciicc 
 this line is called the vanishing ofall those parallel 
 planes. 
 
 Corollary 2. The representations of original lines 
 pass through their intersections and vanishing- 
 points; and of original planes, through their inter- 
 sections 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 A. Hence the representations of plane 
 figures parallel to the picture, are similar to their 
 originals. 
 
 Let q r s t u, fig. 4, plate 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, st, &c. being res- 
 pectively parallel to their originals Q R, R S, S T, 
 &c. as well as the diagonals, t q, T Q, &c. there- 
 fore the inscribed triangle q t u, and QTU, are 
 similar to each other, and so of all the other tri- 
 angles; 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, 
 P2 q r, &e. ar,e similar; therefore, as E q is to q u, 
 so is E Q. to QU; that is, as q u is to Q U, so is 
 the distance of the picture to the distance from 
 the plane QU. 
 
 Problem 1. Having the center ami distance of 
 .the picture given., to find the representation of any 
 given point thereon., fig. 5. plate 30. 
 
 From the eye E, to the center of the picture e, 
 ,4ravy E e, and parallel to it Q d ii om the given
 
 47'2 AN ESSAY 
 
 point Q, cutting the picture in d; join ed, and 
 draw E Q., intersecting e 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 e; therefore q must be 
 somewhere in this line, ^r. 1, cor. 2 ; it must also 
 be somewhere in EQ, and therefore in the point q, 
 where they intersect. 
 
 Problem 2. Tke center and distance of the 
 ■picture 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 R Q X, cutting the picture 
 in V, the vanishing point of the line RX; draw 
 E R and E Q, cutting V X the indefinite repre- 
 sentation of R X, in r and q, and r q is the repre- 
 sentation of R Q. 
 
 This needs no demonstration. 
 
 Pkoblem 3. Having the center and distance of 
 the picture, to find the representation of an original 
 plane, whose positiofi 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 these points a line being 
 drawn will be the vanishing line required. 
 
 This is evident from prop. 2, cor. 1 . 
 
 Problem 4. Having the center and distance of 
 the picture given, to find the vanishing point of lines 
 perpendicidar to a plane whose representation is 
 given, fig. 7, plate 30. 
 
 Let P L be the vanishing line of the given re- 
 presentation, X its center, e that of the picture, 
 and cE its distance; join EX, and perpendicular 
 to it dra\v E f^ cutting X e in f.
 
 ox PERSPECTIVE. 473 
 
 For, EPL is the original plane, producing the 
 Vanishing line PXL, and Ef being a visual ray, 
 parallel to all the original lines that are perpendi- 
 cular to the plane EPL, or its parallels; f is 
 therefore the vanishing point of all of li'ieni. 
 
 Corollary 1. When the vanishing line passes 
 through the center of the picture; that is, wJ:en 
 the parallel planes are perpendicular to that of the 
 picture, the points X and e coinciding with £,i 
 become perpendicular to E e, or parallel to the 
 picture, and therefore the lines will have parallel 
 representations. 
 
 Corollary 2. If the origirjal lines were desired to 
 make any other angle with the original plane than 
 a right one, it is only making xEf equal to it. 
 
 PnoBLEM 5. The center e, aiul distance Ee, of 
 the picture being given, and the vanishing point f, of 
 a line parallel to E f, to find the vanishing line of 
 planes perpendicular to that line whose parallel E f 
 produces the vanishing point f, fig. 7, plate 30. 
 
 Join f e, and make E X perpendicular to, cut- 
 ting f e, produced in X, draw P L perpendicular 
 to ex, and PL will be the vanishing line re- 
 quired. 
 
 The planes may form any angle instead of a 
 right one, if that f E X be made equal to the same. 
 
 Pro B L E M (5. The center and distance of the pic- 
 ture being given, and the vanishing point of a line, 
 to find the vanishing line of planes, perpendicular to 
 the line whose vanishing point is given, fig. J , pi. 30. 
 
 From f, the given vanishing point, through c, 
 the center of the picture, draw feX, and through 
 the eye E draw E f, perpendicular to which draw 
 EX, cutting fe in X; make PXL i)erpendicular 
 to X e f, and P L will be the vanishing line. 
 
 Problem 7- Having given the center and dis- 
 tance of the picture, the inclination of two planes, the
 
 .47-1 AN ESSAY, &V. 
 
 'vanishing line of one of them^ and the -vanishing point 
 of their common i}i.ter section^ to find the vanishing line 
 of the other plane. 
 
 Case 1. Let the inclination of the ])lancs be a 
 right angle, and let PL be the vanishing line, P 
 the vanishing point of their common intersection. 
 By Problem 5 find f, the vanishing point of lines 
 perpendicular to the plane, whose vanishing line 
 is PL; join Pf, which is the vanishing line re- 
 quired. 
 
 For, since this last plane is perpendicular to the 
 former, f will be the vanishing point of one line in 
 it, (prop. 1) and P being the vanishing point of 
 another, therefore Pf is the vanishing line. 
 
 Case '1. When the inclination n c m, fg. 8, 
 plate 30, is greater or less than a right angle. 
 Let N X be the vanishing line of one plane, x the 
 vanishing point of its intersection with the other 
 plane; from the eye E draw Ex, and perpendi- 
 cular thereto, the plane N E M intersecting xN 
 in N, and the picture in NfM, make the angle 
 NEM equal to n e m the given inclination, join 
 X M, which will be the required vanishing line. 
 
 For, the planes being parallel to the original 
 planes by construction, M x, N x are their vanish- 
 ing lines, cor. '1, pr. 2. 
 
 Problem 8. To find the projection of any solid 
 figure. 
 
 Find the representation of any one of its faces 
 bv 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 
 <lrawn evenly through them will represent the 
 curved superficies required; or squares, or other 
 regular figures may be inscribed or circumscribed 
 about the original figures, and these plain figures 
 being projected by the foregoing methods, together
 
 GENERAL REMARK. 475 
 
 ^^ ith a few points therein, by which means the 
 curves may be similarly drawn about or within 
 these projected squares, &c. by this means also may 
 tangents be ch'awn to the representations of" all 
 kinds of curve lines, in all kinds of situations. 
 
 GENERAL REMARK. 
 
 So far I have endeavoured to render tlie prin- 
 ciples of perspective obvious by a mere inspection 
 of the figures. For the voung artist, whose mind 
 seldom conforms to mathematical reasoning, 
 may, in all the foregoing problems, suppose the 
 plane of the picture placed upright upon thai of 
 the paper, which paper he mav consider as the 
 ground plane, the operator's eye E, being al- 
 ways in its proper situation with respect to 
 the picture, as well as to the planes of the original 
 objects. The dnta, or things required to be known 
 before objects can be put into pci'spectivc, arc 
 their plans and elevations, see Deiin. 5 and 6, 
 which must be actually laid down by a convenient 
 scale adapted to the size you mean your picture 
 should be, which maybe v^ery easily accomplished; 
 for you must remember, that the two extreme vi- 
 sual rays, that is, those which pass from the eye to 
 the two opposite borders of the picture, must not 
 make an angle less than t\i:o-t}iirth, 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 qunntit}^; 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 piclurc be
 
 476 GENERAL REMARK. 
 
 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 qua- 
 drangle G N c d, jig. 9, 710. 2, 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; therefore this side of the figure should be 
 made exactly equal to the breadth of the picture, 
 and the other two adjacent sides should, if pro- 
 duced, meet at the distance of the picture, as at F 
 the foot of the observer, as is represented in^^. p, 
 no. 3. The problems already given are sufficient 
 for all cases that can happen in putting objects in 
 whatsoever position into perspective; and though 
 they are perhaps solved in such a manner, as to 
 give the clearest ideas of the genuine practice of 
 perspective, yet others may possibly prefer some 
 of the following methods, which are however ea- 
 sily deduced from the preceding system. 
 
 Exainph 1. tiaving given the center e of the 
 picture P L N f, jig. 2, no. 2, plate 30, its distance 
 eE 12 inches, height of EF 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 fN, take fn=five inches, 
 and having drawn 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 scat 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 fN in t, draw to per- 
 pendicular to fN, and join TE, intersecting to 
 in o, the representation of the point T required. 
 
 Otherwise, if the point T be situated on the 
 ground plane, draw FT, cutting fN in t, and ts
 
 GEXERAL REMARK, 47/ 
 
 perpendicular to i"N, cutting the visual ET.in o^ 
 the point required. 
 
 If the point T has any elevation perpendicular 
 over the same point, as at S, make T S equal to the 
 Jieight 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. 
 perpendicularly over one another, &c. 
 
 Method 2. In the horizontal line e E, Jig. 2, 
 fio. 3, phite 30, from the center of the picture e, 
 take e E equal to the distance of the picture; from 
 the given point Q. draw Q,d perpendicular to the 
 ground line G N, make d Q "^ equal to d Q, and 
 join Q. "" E and ed, intersecting each other in q, 
 the representation of the point Q. 
 
 Md/Jiod3. Perpendicularly over the center e, 
 Jig. 2, no. 4, p/ate 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 com- 
 paring the figures with each other, being marked 
 with the same letters for that purpose. It may be 
 fiirther remarked, that both may be alternately 
 used in the same piece, remembering to use that 
 which you judge will give the bluntest intersec- 
 tion, that is, whichever makes the angle e q E the 
 greatest. 
 
 And now supposing a clear knowledge both of 
 the theory and practice to be obtained, it may not 
 be amis's to shew how naturally a practice more 
 elegant and simple may be deduced, viz. by sup- 
 posing all the planes to coincide with the paper, 
 or plane uf the picture.
 
 478 GENERAL REMARK. 
 
 Thus, y/g. 5, plafe 30, if the triaiifrlcs q e E^ 
 q d Q. revolve roiiiul d q c till thc}' full into the 
 plane PLN, no change takes place in any of the 
 comparative distances E e, E Q, &:c. and the 
 point q preserves its situation on the picture, as in 
 Jig. 2, no. 4; and again, if \njjg. 2, 7io. 4, cE be 
 moved round on the point e, till it falls into the 
 horizontal line at n, and dQ 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 N f s r, fg. '2, no. 5, ^late 30, 
 be the ground line, f e E perpendicular to it, pass- 
 ing through the center of the picture e, and let 
 f E ^ equal to f N, be the distance of the picture^ 
 N P being perpendicular to N F, and the repre- 
 sentation of the point Q, \vhose distance from Nr 
 is rQ. 
 
 Draw E "" Q, cutting Nr in s, taking Nm equal 
 to Qr, and draw em, cutting N P in o, draw oq 
 parallel to N r, and s q to N o, intersecting each 
 other in q, the point required. 
 
 Remark. The lines E ^ Q e m need not be drawn, 
 but a dot made at s, where the ruler crosses n r 
 and e m ; 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 trans- 
 ferred to q, the representation of the given point 
 obtained without drawing any lines over the pic- 
 ture. (This method was first discov^cred by Mr. 
 Beck, an ingenious artist, well known for many 
 useful contrivances.) It is also more accurate 
 
 * ^ccfig. 20, pLiic31. This is a ver}' useful article in draw- 
 ing; a iiilcr a, is fixed as a square to f'; there i* also a moveable 
 piece c. This ruler applied close to the side of a true drawing 
 board, will admit of parallel lines being drawn, as well as oblique 
 ones, with more ease and expedition than by the common parallel 
 rulers. Ei>it.
 
 . GENERAL REMARK". 47p 
 
 tliaii the last method in general, as the intersec- 
 tions arc more obtuse. 
 
 But nevertheless, when the two points in repre- 
 sentation are projected very near together, it is 
 l!ie best way to work by the intersections and va- 
 nishing points of the original lines; for then hav- 
 ing 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 
 ihe ground line in I, in fe E ^ take eE equal to 
 E " f, tliat is, equal to the distance of the picture; 
 draw E V parallel to I Q, intersecting the horizon- 
 tal line h o in V the vanishing point of I Q and all 
 its ])arallcls. 
 
 Rxamplel. To put any plane figure, as QRST, 
 or M N O P, into perspective, flate 30, Jig. 9. 
 
 This is, in fiict, only a repetition of the last; for 
 by finding 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 
 m p o n, the representation of the quadrangle 
 M PON, &e. But if any line, as p o, happens 
 to be very short, so that a small error in point o 
 would considerably alter the direction of it, iind 
 the intersection and vanishing point of its origi- 
 nal; or, if the vanishing point falls at too great a 
 distance, as it very frequently does in practice, 
 tind the intersection 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 me- 
 thods 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 FY per- 
 pendicular to S X, and take Y k equal to Y P, join
 
 480 GENERAL P.EMArxK. 
 
 Ek and c Y, intersecting in p, which is a better 
 intersection than the third method would give, 
 viz. by drawing EP; but yet inferior to that in 
 the fourth method. 
 
 Example 3. To put any sohd body into per- 
 spective. Find the seats of all the ])oints upon 
 the ground plane, and project them as before; 
 let G be the seat of one of the points, g its pro-' 
 jection, produce eg to the ground line aE % and 
 make E " H perpendicular, and equal to the height 
 of the given point, from its seat join He, and it 
 will cut the perpendicular g g '^ , in g % the repre- 
 sentation of the required point. 
 
 Or, by the fourth method, G ^ S being equal to 
 to the distance of G from S X. 
 
 Make G "^ G ^ perpendicular and equal to the 
 given height, draw G ^ e, cutting S U in e ^, then 
 e^ g* parallel to SX, will cut the perpendicular 
 ^g^ in the point g^ required; or the point g 
 may be transferred to g " by the T square, with- 
 out drawing gg^ or e ^ g ^. 
 
 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 
 strait or curved lines according to your original, by 
 which means you may instantly see the connectioa 
 of your work at all times without confusion. 
 
 Remarks. 1 . Sometimes when there is a great 
 number of small parts in a body to be perfectly 
 made out, it may not be amiss to draw squares 
 over the ground plane of the object, and iiud the 
 seats of its several elevated points; then by turn- 
 ing these squares into persspective, the positions of 
 the several points will likewise be found by in- 
 spection, and the horizontal row of squares will 
 serve as a scale for the altitudes of bodies, wiiose 
 seats lie in that row, or both plans and elevations 
 may be used, as in Jig. 10, ^late 30.
 
 GENERAL REMARK!. 
 
 481 
 
 1. Perspective may also be practised without 
 having any recourse to ground plans; for, by tak- 
 ing 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. 
 
 Florizontal. 
 
 Vert. 
 
 Up. 
 
 Left. 
 
 Ridit. 
 
 Down. 
 
 In protracting which angles you must make the 
 distance of the picture radius and lay down the 
 angles by a line of tangents adapted to that radius, 
 which is therefore best done by means of the sec- 
 tor; or more expeditiously by the fourth method, 
 page 4/8; for, if a protractor be fixed at E% 
 Jig. 9, plaU 30, horizontal angles will cut the 
 ground line S X in the same places, as if the lines 
 forming them passed over the original points in 
 the plan ; and if the protractor be fixed at e, the 
 same may be said of S U with respect to eleva- 
 tions. 
 
 Solid bodies may also be put into perspective, 
 by drawing lines in particular directions, as from 
 the center of a circle, or of concentric ones, and 
 finding the representations of the same, raise per^ 
 pendiculars of the proper heights, always suppos- 
 ing the bodies to be transparent, fg, lo, ^laie 30. 
 
 I i
 
 r 
 
 482 
 
 OF SHADOWS. 
 
 Luminous bodies, as the sun, moon, lamps, &c. 
 arc generally considered as points; b^it the artist 
 takes the advantage of their not being perfectly 
 so, by softening the extremities of his shadows, 
 which are so softened in nature, and for this rea- 
 son, which may be thus explained. 
 
 Let R S, fig. 11, no. 2, flatb 3 1, be the radius of 
 a luminous body, whose scat is rs, and center is S; 
 from the opaque body O, draw x r, touching the 
 extremities of both the bodies on both sides, by 
 which means a penumbra, or semi-shade, qxn, 
 will be formed on each side of the main shadow, 
 which becomes 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 luminous be greater than 
 that of the opaque one, measured in the direction 
 of their centers, converge to a point, as n n, fig. i\, 
 plate 3\. 
 
 N. B. The opaque body which casts the sha- 
 dow is called the shading body; and those that 
 are immersed in the shadow, arc called shadowed 
 bodies. 
 
 Problem C). To find the shado-w of any object 
 ■npon a plane. 
 
 Through the luminous body diaw planes touch- 
 ing all the illumined planes of the object, and the 
 intersections of these planes with the given plane, 
 will give the boundary of the shadow required. 
 
 RxauLple \ . Let the luminous point be the sun, 
 the plane of jtrojection ARSB, fig. I'i, plate 2>\, 
 and the object the parallelogram ABCD, 
 
 The sun's rays, on account of his distance, may 
 be supposed parallel, therefore the planes O K A,
 
 OP SHADOWS. 483 
 
 O S B, arc 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 BA the base of the parallelogram. 
 
 Therefore find the seat of one of the rays, as 
 OAR, and make the angle ADR equal to the 
 complement of the sun's altitude, and make the 
 parallelogram B R, and the thing is done. 
 
 Example 1. Let the luminous body be a lamp 
 placed at O, fg. 13, plate 31; the object, a pa- 
 rallelogram ABCD, standing upon the plane 
 ARSD. 
 
 Having drawn O G perpendicular to the plane 
 ARSD, and the rays O B R, O C S ; from G, the 
 seat of the lamp on the plane AS, draw GAR, 
 G D S, intersecting the rays OB, OC, in R and S, 
 and ARSD 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'', B', B", 
 &c. f^, 14, plate o\, and where AG, BH, cut the 
 body GHIK, draw KG A", I H B, making the 
 angles A'GA", BH B', equal to the inclination of 
 the plane G I to G B, proceed in the sam.e man- 
 ner with every new plane M N; or if the object is 
 curvilinear, tangents will always pass through the 
 lines C A' and D B respectively, except when they 
 are perpendicular. 
 
 Example 4. Let a lamp O, fig. 15, flate 31, 
 throw a shadow on the body RSUT, and let 
 QTU be the central line of the shadow from the 
 parallelogram, AB, upon the ground; draw O B^ 
 GP, parallel to QT, and from T to O P draw 
 tP, touching the plane Us of the body; from P 
 
 I i 2
 
 484 OF MILITARY PERSFECTIVE. 
 
 draw PSu, PwU, cutting the extremities of the 
 shadow in U and ii, and w U u S will be the sha- 
 dow of AB upon the face US; proceed in like 
 manner with all the other illumined faces. 
 
 N. B. If t P never meets O P, it denotes the 
 face of the body to be parallel to AB, and the sha- 
 dow 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 reflections of ohjecis 
 upon polished su faces. 
 
 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 AB, fg. 18, plate 31, be any 
 object placed on the water; from B draw Bb, 
 perpendicular to the surface Ab, which continue 
 till the angles BEb and CEb are equal; that is, 
 (Eb being a horizontal line) till be is equal to 
 b B, and b C will be the reflection of AB. 
 
 Example 2. When objects are upright, the lines 
 may be produced below the horizontal line, as 
 much as the real ones are above it. 
 
 OF PARALLEL, OR MILITARY PERSPECTIVE. 
 
 In this kind of projections, the eye is supposed 
 to be placed at an indefinite distance from the ob- 
 ject in the diagonal, and looking down upon it in 
 an angle of 45°, so that the top, one side, and one 
 end, are seen under the same angle, and therefore 
 appear in their true proportions with respect to 
 each other; and therefore heights, lengths, and 
 breadths must be laid down by the same scale, 
 and all parallel lines made parallel, see flg. A^ 
 plate 30.
 
 [ 485 ] 
 
 OF AERIAL PERSPECTIVE. 
 
 Before we can give rules for regulating the force 
 of lights and shades in a picture, we must consi- 
 der what degree of it the bodies themselves are 
 endued with, according to their several positions 
 with respect to the illuminating body. 
 
 Proposition 3. The intensity of light upon any 
 plane is reciprocally as the square of the distance 
 of that plane from the illuminating body. 
 
 Let AB C D, /^. l6, plate 3 1, be the shadow of 
 the square abed upon a plane parallel to it, which 
 projection will therefore be a square, and in pro- 
 portion to abed as the square upon O A to that 
 upon O a; therefore since the real quantity of light 
 is the same as would be received upon ABCD, 
 the intensity of it is reciprocally as the square 
 upon AB to the square upon ab, or as square O A 
 to square O a. 
 
 For example, if parallel planes are at the dis- 
 tance of one, two, and three feet from a luminous 
 point, the intensity of light upon them would be 
 one, one-fourth, one-ninth, &c. 
 
 Corollary, All parallel planes are equally illu- 
 minated by the sun at the same moment. 
 
 For, his rays being parallel, the squares abed, 
 and AB C D arc equal. 
 
 Proposition 4. If the sun's beams fall perpendi- 
 cular upon one face AB, fg. 17, plate "il, of an 
 object, and inclined upon another AC, the inten- 
 sity of light on the faces, is 'as radius to the sine of 
 the angle of incidence. 
 
 Produce AB to b, the quantity of light Ab re- 
 ceives is the same as w^ould be received on AC if 
 b A were away; therefore the brightness is as AC 
 .to Ab, that is, the brightness of Ab, or AB is tq
 
 4S6 OF AERIAL PERSPECTIVE. 
 
 that of AC as AC to Ab, or as radius to sine of 
 the angle of incidence ACb. 
 
 Proposition 5. A plane uniformly enlightened, 
 does not appear so to an eye in different situations. 
 
 For, as all bodies are porous, the little exube- 
 rances will have their light and dark sides, and 
 the eye will view more of the former, as it is more 
 nearly situated in a line with the rays of light, and 
 more of the latter, the more it faces them. 
 
 The subject of this proposition is one great 
 cause of the graduation of light upon the faces of 
 buildings and other planes, and not altogether 
 owing to a greater teint of air, as the artists call it, 
 on that part which is the farthest off. 
 
 Remark, It is very necessary to observe, that 
 transparent and polished bodies are not included 
 among those mentioned in this proposition, for 
 they seem most illuminated in that part which 
 jnakes the angle of reflection equal to that of in- 
 cidence; but if bodies of this kind are not flat, as 
 water when just broken by small rippling waves, 
 then the light is reflected from some part of al- 
 most every wave, and so is extended to a great 
 space, but is strongest perpendicular under the lu- 
 minary, and gradually decreases on each side. 
 
 The case is the same in the sky, which is 
 brightest near the sun's apparent place, and gra- 
 duates into a deeper azure as it retires farther off', 
 land for a reason nearly the same; for the pellucid 
 particles floating above us, having large interstices 
 between them, act in the same manner as the rip- 
 pling waves in disturbed water; and, therefore, 
 the more obliquely the light strikes upon them, 
 the more united their force will be to an eye situ- 
 ated in the proper angle of reflection. 
 
 Proposition^. All shades and shadowing objects 
 would be equally dark and indistinguishable, if 
 they received no secondary or reflected li^ht.
 
 OF AERIAL PERSPECTIVE. 48/ 
 
 For, light is not visible of itself, but by striking 
 upon other bodies renders them so, and these en- 
 lightened bodies serve as lights to bodies other- 
 wise in shade, and such lights are called secondary 
 or reflected ones, thcchief of which is the sky. 
 
 Proposiiio7i 7. Every body participates of the 
 colour of the light by which it is illumined; for, 
 blue rays thrown upon a yellow body will produce 
 a green; red rays, purple; and purple rays, that 
 is, blue and red, black. 
 
 Corollary. Hence shadows arc often observed 
 green in the morning or evening, for the sky is 
 always very green at those times compared with 
 other times of the day, owing to the warm rays 
 being more copiously reflected downwards by the 
 sun's beams striking more obliquely on the at- 
 mosphere, which partly acts as a prism, and 
 the shadows become more blue, as the sky be- 
 comes so; but clouds are of all colours, and as 
 they are denser than the blue part of the sky, they 
 throw stronger reflections, and cause many acci- 
 dental teints in the shadows of bodies; therefore, 
 as the shadow of every body is partially enlight- 
 ened by all the bodies surrounding it, it must par- 
 take of the colours of all of them; and this is the 
 grand source of harmony in painting, of which 
 system the colour of the original light serves as a 
 key, and is to be attended as nicely to in painting, 
 as in nuisic. 
 
 Proposition 8. Bodies partake more of the colour 
 of the skv, as they are farther ofl"". 
 
 For, the sky being only a body of air every 
 where surrounding us, its natural colour supposed 
 to be blue, the farther off" any body is, the more 
 of this blue air is intercepted between us and the 
 body, and therefore tlie bluer it is, and that in 
 proportion to its distance.
 
 488 INSTRUMENTS FOR 
 
 OP INSTRUMENTS FOR DRAWING IN 
 PERSPECTIVE. 
 
 Various have been the methods used to facilitate 
 the practice of perspective, as well for those who 
 understand, as those who are ignorant of that art; 
 and, though some have supposed that the warmth 
 of imagination and luxuriance of fancy, which im- 
 pels the mind to the cultivation of the fine arts, is 
 not to be confined to mechanical modes, yet upon 
 enquiry they will find, that the most able and ac- 
 complished artists are often obliged to have re- 
 course to some rules, and to use some mechanical 
 contrivances to guide and correct their pencil. So 
 great is the difficulty, and so tedious the operation 
 of putting objects in true perspective, that they 
 trust mostly to their eye and habit for success; 
 how well they succeed, we may decide from the 
 portraits drawn by the best artists, and the dif- 
 ferent judgments formed concerning them. Mr. 
 Eckhirdt has well observed, that there is no artist 
 who will be hardy enough to say, that he can deli- 
 neate by the eye the same object twice with ex- 
 actness, arijd preserve a just and similar proportion 
 of parts in each. In one of the figures, we shall 
 find some of the parts larger than in the other — 
 both cannot be right: yet, supposing them per- 
 fectly the same, neither may be conformable to 
 nature. Add to this, many situations of an object 
 occur, which no eye, however habituated, can re- 
 present with accuracy. 
 
 On this account, I have a long time endea- 
 voured to complete an instrument that should give 
 the out-line of an object with accuracy. These 
 Essays have now swelled so far beyond my inten- 
 tions, that I must be as concise as possible. I nuist, 
 however, acknowledge the valuable hints commu- 
 » jested by M.vMej'woodj and other ingenious men.
 
 DRAWING IN PErvSPECTIYE. 48^ 
 
 The methods most generally in use arc, 1. The 
 camera obscura. 1. The glass medium or plane. 
 3. A frame of squares. The inconveniences and 
 inaccuracies which attend these expedients, in- 
 duced Sir Christopher IVren, Mr. Fergrxson, Mr. 
 Hirst, My Father, Mr. IVatt, Mr. Eckhardt, Fere 
 Toussaint, and others, to have recourse to different 
 contrivances to remedy their defects; of which 
 those by the Rev. Mr. Hirst, My Father, Mr. JFatt, 
 and Mr. Eckhardt, are undoubtedly the best; 
 Mr. Eckhardfs and Mr. Hirsfs vary but little 
 from each other. 
 
 Those represented ?itfg. 1 and 1, plate 32, ap- 
 pear to me far superior to any that have been 
 hitherto contrived; the object is delineated- on an 
 horizontal plane, the pencil B,may be moved in any 
 direction, whether curved or strait, v^ith the ut- 
 most freedom. By cither, the artist may be sure 
 of obtaining the measure of every part of the ob- 
 ject with exactness; and this is performed without 
 any loss of time. The instrument may be moved 
 from any place, and brought back to the same with 
 great exactness; and the outline may be formed 
 either of a number of points, or one continued 
 line, at the pleasure of the draftsman. 
 
 That represented u.tjig. 2, is the simplest of the 
 two instruments: j'?^. 1, though more complex, 
 merits, for its contrivances and motions, the at- 
 tention of the mechanic, as well as the draftsman. 
 They both move with facility in every direction, 
 find the whole operation consists in looking 
 through the sight C, which may be placed in any 
 convenient situation, and moving the pencil B, so 
 that the apex A, of the triangle may go over the 
 object, whose outline will be delineated at the same 
 tune by the pencil B. 
 
 To lessen the expense, and render the instru- 
 ment more portable, I have consti'uctcd an instru-
 
 4Q0 INSTRUMENTS, ScC. 
 
 mcnt somewhat similar to that represented at 
 fig. 1j plate 2>'l, but which moves only in a vertical 
 plane, the board on which the drawing is made 
 being in the same plane with the triangle. 
 
 To these may be added the parallel rule, and 
 the perspectiv^c compasses. The distance of the 
 rale from the eye, as it has no sights, must be re- 
 gulated by a piece of thread tied to it, and held 
 between the teeth. 
 
 Fig. 3, plate 32, is a pair of pocket brass perspec- 
 tive compasses, by Mr. Jones, that have been found 
 very useful and convenient for taking readily the re- 
 lative proportions of buildings landscapes, &c. and 
 protracting them on the drawing. A, A, are the 
 two legs, made of small tubes about six inches in 
 length; B, B, are two sliding legs moving to dif- 
 ferent distances out of the tubes A, A; D is a 
 sight piece with a small hole ; E, E, are two steel 
 points to take the sights by ; F, F, are two more 
 small steel points at the ends of the sliders to mark 
 down the distance on the paper, after an observa- 
 tion ; C is an arc with teeth fixed on one leg, by 
 which, and the pinion G, the other leg is moved to 
 the proper angle while observing. This arc is 
 sometimes divided into degrees, and otherways sub- 
 divided by a set of figures, so as to give distances by 
 inspection, &c. according to the pleasure of the 
 purchaser. The sight C, turns down; the sliders 
 B,B, go inwards; the arc takes oiF, and the whole 
 packs into a small narrow case. 
 
 FINIS.
 
 ADDENDA 
 
 THE EDITOR. 
 
 As this treatise is designed to comprehend a ge- 
 neral collection ot' the most approved methods of 
 surveying, I tJiink the following method of sur- 
 veying a large estate by Mr. Eincrson, and the 
 new method of surveying and keeping a field 
 book by Mr. Rodhum, as publislied in Dr. Hultoiis 
 Mathematical Dictionary , 1 vols. 4to. IZQO, will be 
 of real information to many surveyors ; and, in my 
 opinion, as deserving of practice as any other me- 
 thod I am acquainted with. 
 
 " TO SURVEY A LORDSHIP, OR LARGE ESTATE 
 OF LAND. 
 
 '• If the estate be very large, and contains a 
 great number of lields, it cannot be done by sur- 
 veying all the fields singly, and then putting them 
 together; nor can it be done by taking all tlie an^ 
 gles and boundaries that inclose it. For in these 
 cases, any small errors will be so multiplied as to 
 render it very much distorted. 
 . 1. Walk over the lordship two or three times, 
 in order to get a perfect idea of it, and till you can. 
 carry the map of it in your head. And to help
 
 -iQI Emerson's method 
 
 your memory, draw an eye draught of it on paper, 
 to guide you, or at least of the prineipal part^ 
 of it. 
 
 2. Choose two or more eminent places in the 
 estate for your stations, from whenee you ean see 
 all the principal parts of it; and the fewer stations 
 you have to command the whole, the more exact 
 your work will be; and let these stations be as far 
 distant from one another as possible; and they will 
 be fitter for your purpose, if these stationary lines 
 be in or near the boundaries of the ground, and 
 especially if two lines or more proceed from one 
 station. 
 
 3. Take what angles, between the stations, you 
 think necessary, and measure the distances from 
 station to station, always in a right line; these 
 things must be done, till you get as many angles 
 and lines as are sufficient for determining all your 
 points of station. And in measuring any of these 
 stationary distances, mark accurately where these 
 lines meet with any hedges, ditches, roads, lanes, 
 paths, rivulets, &c. and where any remarkable ob- 
 ject is placed, by measuring its distance from the 
 stationary line; and where a perpendicular from it 
 cuts that line. And always mind, in any of these 
 observations, that you be in a right line, which you 
 will know by taking backsight and foresight, along 
 your stationary line; which you must never omit. 
 And thus as you go along any main stationary line, 
 take offsets to the ends of all hedges, and to any 
 pond, house, mill, bridge, &c. omitting nothing 
 that is remarkable, and all these things must be 
 noted down, for these are your data, by which the 
 places of such objects are to be determined upon 
 your plan. And be sure to set marks up at the 
 intersections of all hedges with the stationary line, 
 that you may know where to measure from, when
 
 OP SURVEYING. 4^3 
 
 you come to survey these particular fields, which 
 must immediately be doue, as soon as you have 
 measured that stationary line, whilst they are fresh 
 in memory. By these means all your stationary 
 lines are to be measured, and the situation of all 
 places adjoining to them determined, which is the 
 tirst grand point to be obtained. I would have 
 you lay down 'your work upon paper every night, 
 when you go home, that you may see how you 
 go on. 
 
 4. As to the inner parts of the estate, they must 
 be determined in like manner by new stationary 
 lines. For, after the main stations are determined, 
 and every thing adjoining to them ; then the estate 
 must be subdivided into two or three parts by new 
 stationary lines; takmg inner stations at proper 
 places, where you can have the best view; and 
 measure these stationary lines as you did the first, 
 and all their intersections with hedges, and all off- 
 sets to such objects as appear; then you may pro- 
 ceed to survey the adjoining fields, by taking the 
 angles that the sides make with the stationary line, 
 at the intersections, and measuring the distances to 
 each corner, from the intersections. For every 
 stationary line will be a hash to all the future ope- 
 rations; the situation of all parts being entirely de- 
 pendent thereon; and therefore they should be 
 taken as long as possible; and are best to run 
 alono* some of the hedsres or boundaries of one or 
 more fields, or to pass through some of their angles. 
 All things being determined for these stations, you 
 must take more jnner stations, and continue to di- 
 vide and subdivide; till at last you come to single 
 iields; repeating the same work for the inner sta- 
 tions, as for the outer ones, till all be done. And 
 close the work as oft as you can, and in as few 
 lines as possible. And as it may require some
 
 4^4 EMERSOI^ S JfETHOD 
 
 JQcigmcnt to choose stations the most conveni- 
 ently, so as to cause the least labour; let the sta- 
 tiontiry lines run as far as you can along some 
 hedges, through as many corners of the iielcls, and 
 other remarkable points, as you can. And take 
 notice how one iield lies by another; that you may 
 not misplace them in the draught. 
 
 5. An estate may be so situated, that the whole 
 cannot be surveyed together; bscause one part of 
 the estate cannot be seen from another. In this 
 case, yoa may divide it into three or four parts, and 
 survey the parts separately, as if they were lands 
 belonging to different persons; and at last join 
 them together. 
 
 6. As it is necessary to protract or lay down 
 your work as you proceed in it, you must have a 
 acale of a due length to do it by. To get such a 
 scale, you must measure the whole length of the 
 estate in chains; then you must consider hov/ many 
 inches long tlie map is to be; and from these you 
 will know how many chains you must have in an 
 inch, and make your scale, or choose one already 
 made accordingly. 
 
 7. The trees in every hedge row must be placed 
 in their j)roper situation, which is soon done by the 
 plain table; but may be done by the eye without 
 an instrument; and being thus taken by guess, in 
 a rough draught, they will be exact enough, being 
 only to look at; except it be such as are at any 
 remarkable places, as at the ends of hedges, at 
 stiles, gates, &e. and these must be measured. 
 But all this need not be done till the draught be 
 finished. And observe in all the hedges, what side 
 the gutter is on, and to whom the fences belong. 
 
 8. When you have long stations, you ought to 
 have a srood instrument to take angles with, which 
 should be exact to a quarter of a degree at least j
 
 OF SURVEYING. 405 
 
 and hardly any common surveying Instrument will 
 come nearer. And though the plain table is not at 
 all a proper instrument to survey a whole lordship 
 with, yet it may very properly be made use of to 
 take the several small internal parts; and such as 
 cannot be taken from the main stations; and is a 
 very quick and ready instrument.* 
 
 Example. Walking over the lordship, I pitch 
 upon the four stations A, B, C, D, fg. 1, phite 33, 
 from which I can conuiiand the greatest part of it, 
 there I set up marks. Then I measure along AB, 
 which is a right line, and the boundary on one side 
 of the land. In measuring along, I set down the 
 distances measured, when 1 come at the corners of 
 tjic fields a, a, a, a, where the hedges come in, and 
 likewise where I cross the brook bh. Then bav- 
 in"; ffot to B, I set down the whole lenjjth of AB. 
 
 Next I measure from B to C, and in my way, 
 I set down how far I have measured when I cross 
 the hedges at c^ c^c^c; and likewise where I cross 
 the brook h h again. Thus I measure forward till 
 1 come at C, and then I set down the length of the 
 stationary line B C. 
 
 After the same manner I measure along the sta- 
 tionary line CA, observing to set down the inter- 
 sections with the hedges, as before; till I come at 
 A, where I set down the length of C A. Then the 
 three points, A, B, and C, are determined; and 
 may be laid down in the plan; and all the foresaid 
 points. 
 
 "'■•■ The angles by this method require to be taken very' correctly; 
 and, as instruments are now constructed with an extraordinary 
 degree of perfection, to a skillful observer the angles, however 
 numerous, can be of no reasonable objection; for, the chain itself, 
 and oftentimes the manner of using it, may create as many errors 
 as might be found by taking a multitude of angles by the tUeQ' 
 dolite^ bic.
 
 4g6 Emerson's method 
 
 Being come to A again, I go from A towards B, 
 and in my way I survey every single field adjoin- 
 ing to the stationary line AB. To do which the 
 shortest way, I take the angles at every intersec- 
 tion a, that the sides of each field make with the 
 stationary line AB; and then I measure their 
 lengths; by which every field is easily laid down. 
 In the same manner I proceed from B to C, and 
 measure every field adjoining to BC. And then I 
 go to A, and measure every field in my way thither. 
 
 Next I go from A towards D, and set down, as 
 before, all my crossing of the hedges; and the 
 length AD, when I come at D. And in like 
 manner I measure along D C, setting down all the 
 crossings of the hedges as before, with whatever 
 else is remarkable, as where a highway crosses at d. 
 
 Having finiiihed all the main stations, we must 
 begin to make inner stations. Therefore I take F 
 and G for two stations, being in the lines AB and 
 B C, 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 /", 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 F to G, we can take all the angles that 
 the sides of the fields make with the stationary 
 line FG, and then measure their lengths; by 
 which these fields may be laid down on paper. 
 
 Then I take another inner station at I, and mea- 
 suring from A to 0, I come to the opposite corner 
 of the field; then measuring on to p, 1 cross a 
 hedge; then I proceed to my station I. Then I 
 i^easure from 1 to F, and take an offset to «, where 
 the hedge crosses the brook. Then I come to the
 
 Op SUfvVEYING. 497 
 
 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 I 
 toF. 
 
 All this being done, take a new station H, and 
 measuring from B towards H, 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 tw^o 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 the left. 
 Then going on still further, we have a windmill to 
 the right; and afterwards a cross hedge to the left, 
 and then we measure on to the station H. 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 H 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 sur- 
 veying, WITH THE PLAN OF THE FIELD- 
 BOOK, fhte 34. 
 
 " The field book is ruled into three columns. 
 In the middle one are set down the distances on 
 the chain line at which any mark, off'set, or other 
 observation is made; and in the right and left 
 hand columns are entered the offsets and obser- 
 vations made on the right and left hand respec- 
 tively of the chain line. 
 
 K k
 
 498 Rodham's new method 
 
 It is of great advantage, both for brevity and 
 perspicuity, to begin at the bottom of the leaf and 
 write upwards; denoting the crossing of fences, 
 by hues drawn across the middle column, or only 
 a part of such a line on the right and left opposite 
 the figures, to avoid confusion; aud the corners 
 of iields, and other remarkable turns in the fences 
 where offsets are taken to, by lines joining in the 
 manner the fences do, as will be best seen by com- 
 paring the book with the plaiT annexed to the 
 licld-book, as shewn in plate 34. 
 
 The marks called, a, b, c, &c. 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 
 so on, 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 be- 
 ginning of every line, is the mark or place mea- 
 sured /ro;/z; 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 such a distance 
 from one ?;/^r^ towards anotlier:, and where a mark 
 is not measured to, the exact place is ascertained 
 by saying, turn to the right or left hand, such a dis- 
 tance to such a mark, it being always understood 
 that those distances are taken in the chain line. 
 
 The characters used, are P for tuni 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
 
 OF SURVEYING. ' 49^ 
 
 some strait fence; being chiefly used when cross- 
 ing 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 
 determined either by former work (as in the case 
 of producing a given line, or measuring from one 
 known place or mark to another) or by itself (as 
 in the third side of a triangle) it is called a fast 
 line, and a double line across the book is drawn at 
 the conclusion of it; but if its position is not de- 
 termined (as in the second side of a triangle) it is 
 called a loose line, ^nd a single line is drawn across 
 the book. When a line becomes determined in 
 position, and is afterwards continued, a double 
 line half through the book is drawn. 
 
 When a loose line is measured, it becomes 
 absolutely necessary to measure some line that 
 will determine its position. Thus, the first line 
 a h, being the base of a triangle, is always deter- 
 mined; but the position of the second side hj, 
 does not become determined, till the third &\^ejb 
 is measured; then the triangle may be constructed, 
 and the position of both is determined. 
 
 At the beginning of a line, to fix a loose line 
 to the mark or place measured from, the sign of 
 turning to the right or left hand must be added 
 (as at ;■ in the third line;) otherwise a stranger, 
 when laying down the work, may as easily con- 
 struct the triangle hjh on the wrong side of the 
 line a h, as on the right one: Hut 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 pB r, by the 
 angle at B bein^ very obtuse, a small deviation, 
 
 K k 2
 
 500 KEITH S IMPROVED 
 
 from truth, even the breadth of a point at f or r, 
 would make the error at B, when constructed, 
 very considerable; but by constructing the trian- 
 gle />B^, such a deviation is of no consequence. 
 
 Where the words, leave off, are written in the 
 field-book, it is to signify that the taking of off- 
 sets is from thence discontinued; and of course 
 something is wanting between that and the next 
 offset." 
 
 Mr. Keith's improved parallel scale. 
 
 Mr. Thomas Keith, Teacher of the mathematics, 
 has considerably improved the German parallel 
 ruler, or that of Mr. Marquois, see pages 27 and 
 28. By making the hypothenuse, and the perpen- 
 dicular hne to it from the opposite angle, in the 
 ratio of 4 to 1, and adding several scales, he. its 
 uses are considerably extended for drawing plans 
 of fortifications, and other branches of the mathe- 
 matics. 
 
 Fig. 2, 3, 4, and 5, plale 33, represent the two 
 faces of the ruler, and the triangle of half the 
 real dimensions. 
 
 The slider is a right-angled triangle, the per- 
 pendicular is divided into inches and tenths, and 
 the base has three indices, and other divisions re- 
 quisite to be used with the scale. 
 
 The ruler contains lO different scales, which, 
 bv the help of the slider may be increased to 20, 
 without guessing at halves and quarters. The fi- 
 gures 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 con- 
 tains the names of the pol vgons, the angles at their 
 centers, and a scale of chords by which all poly- 
 gons may be readily constructed.
 
 PARALLEL SCALE. 501 
 
 ITS USE AS A COMMOX PARALLEL RULEK. 
 
 To draw a i'lne 'parallel to the slider. Lay the 
 hypothenuse, or sloped edge of the triaiig-le in the 
 position you intend to have your line, place the 
 scale against the base of the triangle, and draw a 
 line along the slope edge, keep the scale fixed, 
 and mrve the slider to the left or right hand, ac- 
 cording as you want a parallel line above or below 
 the other. 
 
 To draw a line parallel to the scale. Lay the 
 scale in the position you intend to have your lines, 
 and draw a line along the edge of it. Place the 
 base of the triangle against this edge, the middle 
 index standing at O on the scale, and make a 
 mark against any one division on the perpendi- 
 cular; turn the triangle the other side uppermost, 
 and make a mark against the same division; join 
 these marks, and this line will be parallel to the 
 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 Tine parallel to the slider at a great 
 distance. Draw a line along the slope edge of the 
 triangle, and fix the scale as before, make a mark 
 against 2 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 triangle, 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 
 distance. Draw a line along the edge of the scale, 
 place the base of the triangle against that edge.
 
 501 Keith's improved 
 
 the middle index being at O, and make a mark 
 against 2 on the perpendicular; turn the triangle 
 the other side uppermost, and make a mark against 
 the same division; takeaway 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 re- 
 quired. . 
 
 ITS USE IN ERECTING PERPENDICULARS. 
 
 If a line is wanted perpendicular to the scale, 
 apply the base of the slider to it, and draw a line 
 from the scale along the perpendicular of the tri- 
 angle; should a longer perpendicular be wanted, 
 apply the perpendicular of the triangle to the scale, 
 and draw a line along the base. 
 
 If a perpendicular is wanted to a line, which has 
 been drawn along the hypothenuse of the triangle, 
 keep the scale fixed, and apply the hypothenuse 
 of the triangle to it, then draw a line along the 
 perpendicular of the triangle, 
 
 ITS USE 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 polygon, make angles equal to half the re- 
 mainder, 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 trigonometry.
 
 PARALLEL SCALE. 503 
 
 JTS USE IN DRAWING PARALLEL LINES AT ANY 
 GIVEN DISTANCE, WITHOUT THE ASSIST- 
 ANCE OP A PAIR OF COMPASSES. 
 
 If the side of your polygon zvas taken from th' 
 scale C. Move the slider from O to 1, 2, 3, &c. 
 on the scale D, and you will draw a parallel \u\c of 
 the width of 1, 2, 3, &c. divisions on the scale C. 
 This scale is 20 fathoms to an inch. 
 
 In a similar manner the scales G, L^ and P, arc 
 to be used. 
 
 If the side of your polygon was taken from the 
 scale C, calling the divisions two 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. 
 
 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 
 middle index stands at O, the divisions marked C 
 on the slider will make straight lines with 5 and 6 
 on the scale D. By moving the slider to the 
 right or left till the other divisions thereon 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. divisions on the scale C. This scale 
 is 6o fathoms to an inch. 
 
 Similar directions must be observed in using the 
 scales G, L, and P. 
 
 If the side of your polygon was takeyi from the 
 scale A. Move the slider from O, to 1, 2, 5, &c. 
 on the scale B, and you will draw a parallel line of 
 the width of 1, 2, 3, &c. divisons on the scale A. 
 This scale is 10 fathoms to an inch'.
 
 504 KEITH S IMPROVED 
 
 Similar directions must be observed in using the 
 scales E, I, and N. 
 
 If 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 fathoms to an inch. 
 
 In making use of H, M, and Q, similar direc- 
 tions must be observed. , 
 
 Note. There is no absolute necessity for always 
 moving the slider from O, and it may be used ei- 
 ther side uppermost. Any one of the three in- 
 dices may likewise be made use of, &c. 
 
 To render the scale still more perfect;, Mr. 
 Keith has made the following additions. 
 
 1 . The divisions on the scale A, and the fourth 
 scale from A, have been subdivided for the purpose 
 of constructing sections, &c. Ten of the divi- 
 sions on the scale A make one inch ; if, therefore, 
 you 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 divisions in a successive order from the 
 left hand division 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, 7? 9? oi' H 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 
 2i 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
 
 PARALLEL SCALE. 505 
 
 of the sdVcral polygons under which they stand, 
 tlic 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 octagon, the exterior sides being 120 toises? 
 as 180 : 235 • 18 :: 120 : 156 ' 78 Ans. . The 
 remaining 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.oOs, a toise six French feet, a fathom six 
 English feet. The English foot is to the Rhyn- 
 land foot as 9715 is to 10000; the Rhynland foot 
 is to the French foot as 1033 is to 1068; the Rhyn- 
 land rod is 12 Rhynland feet. 
 
 It may not be amiss to remark, that cither the 
 scale or the slider will erect a perpendicular in- 
 stantaneously, for the long divisions axrross the 
 scale make right angles with the edges. 
 
 OF THE G.UXNER's CALLIPERS, OR COMPASSES. 
 
 This article is generally included in the ma- 
 gazine case of instruments for the military of- 
 ficer, or engineer; is a very useful mathematical 
 instrument in the artillery service, and, as its de- 
 scription was omitted 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 diameters of common siiot, the bore or caliber 
 of a piece of ordnance, estimate the weight of shot, 
 quantity of powder, &c. for guns of given dimen- 
 sions, and other particulars in practical gunnery. 
 
 Fig. 6 and 7, are the representations of the two 
 faces of the callipers marked A, B, C, D. They 
 consist of two thin flat bra3s rulers moving on a
 
 506 DESCRIPTION OF 
 
 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 center, 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 Fig. 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 ne- 
 cessary 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 giving the diame- 
 ters of the calibers of guns in inches. 
 
 4. A line marked Guns, contiguous to the pre- 
 ceding, shewing the nominal pounders, or weight 
 of shot for the respective bores of the guns in 
 inches. 
 
 5. On the semicircular head of the rule is a 
 semicircle divided into degrees, figured in contrary 
 directions, to measure angles by, and give the ele- 
 vation of cannon, &c. 
 
 6. Next to the preceding, is a circular scale 
 marked Shot Diameters, which, with the cham- 
 fered edge marked Index, shews the convex dia- 
 meter 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 marked Brass Guns, 
 shewing the quantity of powder necessary for the 
 proof and service charges of brass guns.
 
 THE gunner's callipers. 507 
 
 8. The line of lines, marked Lin. 
 
 9. Two circular scales on the head of the rider, 
 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 pow^lcr 
 necessary for proof and service charges of iron 
 guns, from \ to 42 pounders. 
 
 11. To these are sometimes added various fi- 
 gures of a circle, cube, &c. with numbers. 
 
 12. A table of the weights and specific gravi- 
 ties of a cubic foot of various metals, ivory, wood, 
 waters, &c. 
 
 EXPLANATION OF THE LINES AND TABLES. 
 
 1. The line of inches, graduated contiguous 
 to the exterior edj^es of the sides A and B, 
 
 Jig. 6, when opened to a straight line, makes a 
 measure of 12 inches and tenths for the purpose 
 of a common rule. 
 
 2. The table shewing the quantity of powder 
 necessary for mortars and howitzers. This table 
 is adapted for both sea and land. Thus, by in- 
 spection merely is shewn that a 13 inch brass 
 mortar at sea requires 30 pounds of powder; by 
 land only 10 pounds. Mortars and howitzers 
 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 
 continues on to 1 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
 
 50S DESCRIPTION OF 
 
 possible distance; the callipers then being taken 
 out, and inspected \^'hcrc in the scale one exter- 
 nal edge of the rule is upon the other scale, that 
 division will give in inches and quarters the cali- 
 ber of the gun required. 
 
 4. The line shewing the weight of shot for the 
 bores. Adjoining to the preceding line is the one 
 marked Guns^ 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 
 4i inches shews a 9-inch shot, and i-'ice versa. 
 
 5. The semicircular graduation on the joint- 
 head. The degrees are figured to ISO 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, &c. 
 
 First. Jo lay down any angle, suppose 30 degrees. 
 Open the legs till the chamfered edges cut 30, 
 and two lines drawn against the outside edges 
 will be at the inclination, or angle of 30 degrees, 
 tending to a proper center. If a center point be 
 desired, cross the legs till the other chamfered 
 edge cuts 30; the outside edges will then be at 
 the angle of 30 degrees, and the place of their in- 
 teiseclion the angular point. 
 
 Secondly. To measure an entering or internal 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 semicircle on the outward set of figures 
 "will point out the degrees contained in the given 
 angle. 
 
 Tiiirdly. To measure a salient, or external angle. 
 Place the legs of the callipers over each other_,
 
 THE GUNXEll S CALLIPERS. 50^ 
 
 and make their outside edges coincide with the 
 legs or sides of the angle to be measured; then 
 will the chamfered edge cut the degree required 
 on the inner semicircle, equal to the angle mea- 
 sured. 
 
 Fourthly. To deternime 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 t;he legs of the callipers extended, so as to 
 touch both the string and rod, the chamfered 
 tCi^Q, will cut the degrees equal to the comple- 
 ment of 90° of the true angle of the elevation of 
 the cannon, figured on the outer semicircle. So 
 that if tlje edge had cut 30 degrees, the true 
 elevation would have been ()0° the complement 
 
 6. The circular line of inches on the head next 
 to the preceding, marked Shot D'lam. This scale 
 extends from O to 10 inches, and is subdivided 
 into quarters. The cjhamfered edge marked Inches^ 
 is the index to the divisions. 
 
 To measure by this scale the convex diameter of 
 sliot^ &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 shot rather less than by th|e scale No. 4, on 
 account of a customary allowance of what is called 
 Windage. 
 
 7 . The table marked Brass Guns, to shew the 
 quantity of powder necessary for proof and service 
 charges of brass guns. This table is arranged for 
 the size of the piece, called either Heavy, Middle^
 
 5 10 DESCRIPTION OF 
 
 or Light, into their respective columns. Ex. Sap- 
 pose it is required to know the quantity of pow- 
 der for a proof and service charge of a heavy 42 
 pounder. Under 42 are 31 lb. 8 oz. for the proof 
 charge, and 21 lb. for the service charge. 
 
 8. The line of lines, marked L'm. 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 re- 
 fer the reader to the description of that instrument 
 already given in this work. 
 
 g. Two circular scales on the joint head for 
 /giving the weight of iron shot, marked Shot. 
 These scales are engraved upon the chamfered 
 part of the head of the callipers; they are, in fact, 
 but one continued scaler only separated for the 
 adv_antage of being more conspicuous. They 
 shew the weights of iron shot from h to 42 pounds. 
 The diameter of the shot is to be taken by the 
 points of the callipers, and then by an index en- 
 graved 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 quan- 
 tity requisite for a service charge; and all these 
 adjusted from \ to 42 pounders. 
 
 11. The mathematical figures. The most va- 
 cant part of the callipers are sometimes filled up 
 by six mathematical figures, with numbers an- 
 nexed to each to assist the learner's memory, and 
 indicating as follows.
 
 THE GUNNEll S CALLIPERS. 511 
 
 1. A circle with two diametrical lines, about 
 which are the numbers 7- 22, 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 
 therefore readily worked by them. 
 
 Required the circnmfere7ice of a circle to any giroen 
 diameter. 
 
 As 7 : 22 :: given diameter : the circumference 
 required. 
 
 Or 113 : 355 :: diameter : the circumference 
 jnore 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 figure are annexed the numbers 28. 22. 
 14. 11. denoting that the larger square is 28, the 
 inscribed circle is 22. The area inscribed at the 
 square in that circle 14, and the area of the 
 smaller inscribed circle 11. 
 
 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 diame- 
 ter 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, the area. 
 
 Or as 14 : 11 :: 144 : 113.1. 
 
 3. Represents a cube inscribed in a sphere; the 
 affixed number 89! shews that a cube of iron in- 
 scribed in a sphere of 12 inches in diameter, weighs 
 BQt pounds.
 
 51'2 DESCRIPTION 0S» 
 
 4, Represents a sphere inscribed in a cube, witll 
 the numl3ers 243 affixed; 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, Represents a cylinder and cone, the diame- 
 ter and height of which are one foot. To the cy- 
 linder is affixed the number 3()4,5, the weight in 
 pounds of an iron cylinder of the above dimen- 
 sions. The cone shewn by the number 121,5 
 that it is the weight 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 to 3. 
 
 d, > Represents an iron cube, whose side is 12 
 inches, to weigh 4()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. 154f. 
 
 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 ciibe of iron being two feet, re^ 
 quired the iveight. 
 
 The sixth figure shews that a cube of iron, 
 whose sides are each 1 foot, weighs 4()4,5. There- 
 fore, 
 
 As 1 (the cube of 1 foot) : 464,5 :: 8 (the cube 
 of 2 feet) : 37 1 6, the weight of an iron cube whose 
 •sides are 2 feet.
 
 THE GUNNER S CALLIPERS. 513 
 
 2. The diameter of an iron shot being six inches, 
 required the weight. 
 
 By the fourth figure a one foot iron ball weighs 
 243 pounds^ and 6 inches=;,5 the tV of a foot. 
 
 Therefore, as 1 (the cube of 1 foot) : 243 (the 
 weight) :; ,125 (cube of ,5 orrV) : 30,375 pounds, 
 the weight required. 
 
 Another rule. Take \ the cube of the diameter 
 in inches, and \ of that eighth, and their sum will be 
 the weight required in pounds exactly. Or, the 
 weight of a four inch shot being nine pounds, the 
 proportions 64 : Q may be used with equal ex- 
 actness. 
 
 3. The tveight of an iron hall heing given, to find 
 its diameter. 
 
 This rule is the converse of the preceding, the 
 same numbers being used; as 243 (the weight) : 1 
 (cube of 1 foot;) or, as 9 pounds : 64 (cube of 4.) 
 
 Another rule. Multiply the weight by 7? ^nd 
 to the product add h 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 evident to require worked examples. 
 
 4. The following example will make up a suffi- 
 cient number, by which the learner may know 
 how to apply readily any number from the figures 
 just desbribed. 
 
 A parcel of shot and cannon weighing fii>e tons, or 
 1 1 200 founds is to he melted, and cast into shot of 
 three and five inches diameter, and weight of each 
 sort to he the same ; required the tmniber there will 
 he of each. 
 
 1st. Half of 11200 is 560 pounds. 
 
 2d. Cube of 3 inches, or cube i of a foot =:-sV. 
 
 Then, as 1 (cube of 1 foot) : 243 (the weight 
 of a 1 football) :: 1/4 : 3,793 weight of a 3 inch 
 iron shot. 
 
 l1
 
 514 DESCRIPTION OF 
 
 N. B. In this proportion 64 divides 143 on ac- 
 count of the fraction. 
 
 Now 5 inches =T2r of a foot, its cube -iW^. 
 
 Therefore, as I : 243 :: tW? : 17,(^7, weight of 
 an iron 5 inch shot. 
 
 Dividing 5000 by 3,79 gives 1504, the number 
 required of the 3 inch shot. 
 
 Dividing 5600 by 17357 gives 313, the required 
 number of the 5 inch shot. 
 
 A table of specific gravities and weights of bo- 
 dies is added, or not, at the pleasure of the 
 purchaser. It is not essential to the general uses 
 of the callipers, although many curious and useful 
 problems relative to the weights and dimensions 
 of bodies may be obtained from it in the most ac- 
 curate manner. =^ 
 
 The quantity of lines placed upon the callipers 
 may be increased or arranged at pleasure. The 
 following 19 WTre 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. The weights of iron shot from given diameters. 
 
 4. The weight of iron shot proper to given gun 
 bores. 5. The degrees of a semicircle. 6. The 
 proportion of Troy and Averdupoise weight. 7 • The 
 proportion of English and French feet and pounds. 
 
 8. Factors useful in circular and spherical figures. 
 
 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 
 
 * See Mr. Adams's Lectures, five vols. 8vo. a new and im- 
 proved edition of which is now in the press, and under my cor- 
 riEction and augmentation.
 
 GUNNER*S QUADRANT^ &C. 5l5 
 
 or mortars. 15. A line of inches. l6. 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 su- 
 perficies. 19. A sectoral line of solids. 
 
 gunner's guadrant. 
 
 Fig. 8, plate 33, is a representation of a qua- 
 drant used for elevating a cannon, or mortar, in 
 the most expeditious manner. The bar A, is 
 placed in at the mouth ; the index B, brought to 
 the arc till the bubble of the spirit level settles in 
 the middle. The angle is then read off to mi- 
 nutes upon the arc by the nonius at C. 
 
 gunner's perpendicular. 
 
 Fig. 9, plate 33, is a representation of a small 
 level and perpendicular. It is used to find the 
 center line of a piece in the operation of pointing 
 it to an object, or to mark the point for a breech 
 hole, &c. The spirit level A, determines the po- 
 sition on the gun, and the spring index point B, 
 serves to mark the necessary points upon the sur- 
 face to obtain the line by. 
 
 SHOT GAUGES. 
 
 Are a set of brass rings, all connected to one 
 center, with holes suitable to the diameters, or 
 pounders of iron shot, from four to 42 pounders; 
 being all respectively marked, and are too evident 
 to need a description here. 
 
 err a t a. 
 
 Page 6, line 2, Preface, for desighed read designed. 
 
 ■ • 89, — 2, Note, for desiderata read desideratum. 
 
 • 380, — II, for plate 28 read plate 31. 
 
 •— 500, — 14, for By making, &c. read By raaking the base of the triangJe 
 
 and a perpendicular liae lo it, drawn fro;.-: the opp?si-e angle, in ti.e 
 
 ratio of 4 to i ,
 
 A LIST 
 
 Of tbi; pr'mdpal Listrtiments described hi this Work, and ibeir Prias, 
 as made and sold hy W. atid S. Jones, Holborn, London. 
 
 PLATE I. 
 
 Pahtial, or complete magazine cases of instruments, 
 
 as represented in the plate, from ll. 18s. to — 10 10 O 
 
 PLATE IL 
 Parallel rulers, jig. A, B, C, D, and E, according to 
 
 the length and mounting, from 2s. fid. each, to 2 12 6 
 N. B. Oifgs. A, or B, or C, or D, or E, one of them 
 
 is included in the cases above, to order. 
 Improved ruler, FGH — — — — — — 
 
 Ditto IKL — — — — — — — — 
 
 Ditto represented at M, from 10s. 6d. to — — 
 
 PLATE III. 
 F:g. 1. Marquols's parallel scales — — — — 
 
 2 and 3. Protractors, wood or ivory, from 2s. to 
 
 4. Sectors, according tp the length and mate- 
 
 .rials they are made of, from 2s. 6d. to — 
 
 /. Brass proportional compasses — — — 
 
 Without the adjusting rod. Jig. A, flate 1 
 
 ^ and 9 a. Elliptical, sector, and beam compasses 3 13 
 
 10. Beam compasses from ll. 4s. to — — 
 
 ■ 11. Sisson's beam compasses — — — — 
 
 ——12. Triangular compasses — — — — — 
 
 Ditto represented at./^. N, flate 1 — — 
 
 13. Calliper and beam, small size — — — 
 
 PLATE X. 
 Bevel, &c. rulers, Jigs. 4, 12, 15, iG, 17, and 18, va- 
 rious prices from 10s. 6d. to — — — — 3 13 6 
 
 PLATE XI. 
 Kg. 1. Suardi's geometric pen — — — — — 4 40 
 Ditto, with a great variety of .vt'heels, &c. to 7 7 
 
 1 
 
 11 
 
 6 
 
 1 
 
 1(5 
 
 
 
 
 
 16 
 
 
 
 
 
 9 
 
 
 
 
 
 7 
 
 
 
 3 
 
 13 
 
 6 
 
 2 
 
 2 
 
 o 
 
 1 
 
 11 
 
 6 
 
 3 
 
 13 
 
 6 
 
 4 
 
 4 
 
 
 
 3 
 
 3 
 
 
 
 1 
 
 1 
 
 
 
 
 
 13 
 
 
 
 1 
 
 11 
 
 6
 
 [ 517 ] 
 
 Fig. 2. Elliptical drawing board — — — — 
 
 3. Elliptical brass compasses — — — — 
 
 4. Parabolic drawing machine — — — — 
 
 5. Cyclograph for drawing circles of large radii 
 
 Q. Spiral machine — — — — — — 
 
 7- Protractor, perpendicular, &c. — — — 
 
 PLATE XIV. 
 
 Fig. 1 . King's surA'eying quadrant with sights — 
 
 2. Portable surveying brass cross and staff" — 
 
 ' 3. Brass pocket. box cross and staff" — — 
 
 Improved ditto, from ll. lis. 6d. to — — 
 
 Optical squares — — — — — — 
 
 5. Ten inch common theodolite and staves — 
 
 7- ^sst universal pocket theodolite — — — 
 
 PLATE XV. 
 Fig. X. Common circumferentor and staves — — 
 
 2. Improved ditto 4l. 14. 6d. to — — — 
 
 3. Surveying compass with telescope — — 
 
 1. Six inch pocket common theodolite and staves 
 
 5. Six inch theodolite by rack work and staves 12 12 
 
 Ditto without rack work, common — — 
 
 PLATE XVI.. 
 
 Fig. 1. Second best 7 or 8 inch theodolite and staves 22 
 
 2. Very best improved ditto, ditto — — — 31 
 
 N. B. For large theodolite see the frontis- 
 piece, from sol. to — — — — 300 
 
 PLATE X^ai. 
 Fig. 1. Plane table and staves in a box — — — 4 4 
 
 2. Beighton's improved ditto — — — — 6 6 O 
 
 3. Spirit level of the best kind with telescope 
 
 and staves, according to the adjustments 
 and finishing, from 8l. Ss. to — — 12 12 O 
 Common ditto, without compass, from 
 
 31. 13. 6d. to — — — — — — 6 6 O 
 
 -- " •i. Six inch circular protractors ll. 18s. to — 3 10 Q 
 
 2 12 
 
 6 
 
 2 12 
 
 6 
 
 2 12 
 
 6 
 
 4 14 
 
 Q 
 
 4 14 
 
 6 
 
 5 5 
 
 
 
 1 18 
 
 
 
 1 18 
 
 
 
 18 
 
 
 
 3 3 
 
 
 
 1 1 
 
 
 
 4 14 
 
 6 
 
 10 10 
 
 
 
 4 4 
 
 
 
 5 15 
 
 6 
 
 2 2 
 
 
 
 3 3 
 
 
 
 12 12 
 
 o 
 
 8 8 
 
 
 
 22 1 
 
 
 
 31 10 
 

 
 [ 518 ] 
 
 Fig. 5. Ditto by rack work — — — — — 4 14 6 
 
 Common circular brass protractors 6s. to — 18' O 
 
 » ' ■ ( j. Perambulator, or measuring wheel — — 6 6 O 
 
 • 3. Way wiser for carriages 61. 6s. to — — 15 15 O 
 
 1 <^g. Station staves, the pair — — — — 212 6 
 
 ■ 10. Pocket spirit levels, from 10s. 6d. each, to 1 11 6 
 
 ■ 1 1 . Improved universal ditto — — — — 1116 
 
 PLATE XIX. 
 
 Fig. I. HadIey'squadpant,ebonyandbrass,2l.l2s.6d.to3 13 6 
 
 ■ 1 . Best metal sextant, 8 to 10 inches radius — 13 2 6 
 
 Brass stand and counterpoise for a ditto — 4 4 
 
 Second best ditto 8l, 8s. to — — — — 10 10 
 
 11. Pocket box sextant from 2l. 2s. to — — 3 3 
 
 12. Artificial horizon coftiplete — — — 2 2 
 
 PLATE XXII. 
 Fig. 2. Feather edged 12 inch box scales, each — 2 6 
 Ditto ivory, each — — — " — — — 080 
 
 PLATE XXXI. 
 
 F,g. 19. Pantagraph, 2 feet, best — — — — 414 6 
 Common ditto, 12 to 18 inches, from ll. 4s. to 3 13 6 
 
 PLATE XXXII. 
 
 Fig. 1 . Perspective machine — — — — — 660 
 
 2. Ditto, ditto — — -^ — — — — 550 
 
 w 3. Perspective compasses — — — — — 118 
 
 PLATE XXXIII. 
 Fig. 2, 3, 4, and 5. Keith's improved parallel scales 
 
 in a case — — — — — — — OIO6 
 
 Ditto in wood and ivory, or all ivory, ll. 5s. to 2 2 O 
 
 •K 6 and 7, Gunner's callipers, from 2l. 12s. 6d. to 6 6 
 
 8. Gunner's quadrant — — — — — 212 6 
 9- \t\t\ and perpendicular — — 1 16 O 
 
 For farther particulars see W. and S. Jones's general catalogue. 
 
 U
 
 UNIVERSITY OF CALIFORNIA LIBRARY 
 
 Los Angeles 
 This book is DUE on the last date stamped below. 
 
 Form L9-lC0m-9,'52(A3105)444
 
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