portrait of William Leybourn Vera Effigies Gulielmi Leybourn, Philom. anno Aetatis. 27. THE COMPLETE SURVEYOR: Containing The whole ART of Surveying of Land, BY THE Plain Table, Theodolite, Circumferentor, AND PERACTOR: After a more easy, exact and compendious manner, then hath been hitherto published by any: the PLAIN TABLE being so contrived, that it alone will conveniently perform whatsoever may be done by any of the Instruments, or any other yet invented, with the same ease and exactness; and in many cases much better. Together with the taking of all manner of Heights and Distances, either accessible or in-accessible, the Plotting and Protracting of all manner of Grounds, either small Enclosures, Champion plains, Wood-lands, or any other Mountainous and un-even grounds. Also, How to take the Plot of a whole Manor, to cast up the content, and to make a perfect Chart or Map thereof. All which particulars are performed three several ways, and by three several Instruments. Hereunto is added, the manner how to know whether Water may be conveyed from a Spring head to any appointed place or not, and how to effect the same: With whatsoever else is necessary to the Art of SURVEYING. By WILLIAM LEYBOURN. LONDON: Printed by R. & W. LEYBOURN, for E. BREWSTER and G. SAWBRIDGE, and are to be sold at the sign of the Bible upon Ludgate hill, near Fleet-bridge. MDCLIII. TO HIS MUCH HONOURED FRIEND EDMUND WINGATE, of Gray's Inn, Esq SIR, THis Treatise being finished, and ready to see the light, I could not bethink myself of a fit Patron than yourself to protect it; Your knowledge in, and affection to the Sciences Mathematical, as also the civil respect which You usually vouchsafe such as affect those Studies, arm me with this confidence. I foresee that this my presumption in exposing this Work to public view, may meet with some Detractors, but Your approbation thereof, will both convince them of their Error, and plentifully satisfy me for the pains I have taken therein. Howbeit, what reception soever it may obtain with the Vulgar, my intention (I doubt not) will give me support and encouragement, my aim therein being nothing else but the public good, and this my Dedication an evidence to let You know how much I am, SIR, Your most humble and obliged Servant, WILLIAM LEYBOURN. TO THE READER: Courteous Reader, ABout three years since there came into the World a little Pamphlet entitled Pla●…etria, or the whole art of Surveying of Land, under the name of Oliver W●…by, of which I confess myself to be the Author, that name being only the true letters of my own name transposed. I was indeed very unwilling the world should know me to be the Author thereof, it being so immaterial a Treatise, and too particular for a Subject of so large an extent, but that was occasioned by over much 〈◊〉, for (being urged thereunto) it was not above six weeks conceived before it was brought forth; and therefore must needs be little less, then monstrous: yet the good acceptance which that pamphlet received, occasioned me to prosecute that Subject more at large. Now as the opinions of men in the world are various, so I know this work will be variously censured, and therefore it might (per chance) be expected by some, that I should make an apology for myself, as to crave pardon 〈◊〉 excusse for whatsoever any man shall be pleased to object against, him I mean to make no excuse, see I know of nothing that needs it, neither did I ever know any Book the more favoured for the Authors bespeaking it besides, the subject of the ensuing Treatise being Geometry, needeth no such thing, for [Demonstration] the grand supporter thereof, is able to with stand all opposers, and silently with Lines and Figures to 〈◊〉 the most malevolent tongue or pen that shall 〈◊〉 speak or ●…rice against its. But to the judicious Reader I shall say thus much, As I dare not think my do free from all exception, so I do not know of any thing herein contained worthily deserving blame. Some small oversights which may possibly have crept in by chance, I must entreat the friendly Reader to over see or wink at, as for the understanding Reader, I am sure he will 〈◊〉 to cavil at every ●light mistake or literal fault in the printing, as for material faults I know of none in the whole Work, although I have diligently examined the printed sheets. In the following Treatise I have endeavoured to proceed methodically, and to insert every particular Chapter as it ought to be read and practised, and have omitted nothing that might any way tend to make a man in short time become an exquisite proficient in the Geometrical part of Surveying. The first part of this Book consisteth of Geometry only, and containeth such Problems as are meet and necessary to be known and practised by any man that intendeth to exercise himself in this employment: by help of these Problems the plot of any piece of Land may be enlarged or diminished according to any assigned proportion, and separation and division thereof made, if need be, by Rule and Compass only, and also by Arithmetic. In the second Book, you have a general description of all the most necessary Instruments used in Surveying, as of the Theodolite, Circumferentor, Plain Table, and the like, and more particularly of those which I make use of in the prosecuting of this discourse. Also I have given such directions for the making of the Plain Table, and furnished the Index and other parts thereof with divers necessary lines for several occasions, so that it being made according to the directions there given, it is the most absolute and universal Instrument yet ever invented; for by it may be performed whatsoever may be done by the Theodolite, Circumferentor, or Peractor, with the same facility and exactness, and in many cases better, as in the particular uses thereof will plainly appear. The third Book is of trigonometry, or the Doctrine of the dimension of Plain Triangles, by Sines, Tangents, and Logarithms, by which the nature and reason of the taking of all manner of Heights and Distances may the better appear, and for that reason I have in this third Book added short Tables of Sines and Logarithms, namely a Table of Sines to every 10 minutes of the Quadrant, and a Table of Logarithms from 1 to 1000, by which more Questions may be resolved in the space of one hour, then by the usual ways taught by others can be performed in six, if the like exactness be required. And for a further abreviation of these Calculations, I have also shown how to resolve all such Cases in Plain Triangles as may at any time come in use in the practice of Surveying by the lines of Artificial Numbers, Sins, and Tangents, whereby all such Cases may be resolved without pen, ink or paper. In the fourth Book is shown the use of all the Instruments in the practice of Surveying, and first, in the taking of all manner of Heights and Distances either accessible or inaccessible, in the practice whereof the young practitioner will take much delight, and receive no small satisfaction. There is also taught how to take the plot of any field or other enclosure several ways, both by the Plain Table, Theodolite and Circumferentor, by which will appear what congruity and harmony there is between these several Instruments, for if you take the plot of any field by any one of them, and then by another of them, and plot your work by the same Scale as both your observations, you shall (if you be careful) find that these two Plots will agree together as exactly as if they had been both taken by one and the same Instrument. And for this reason I have made one Scheme or figure serve for three several Chapters, which hath much abreviated the number of Diagrams, and will (I persuade myself) give better satisfaction to the Learner, than variety of figures could have done. In the manner of protracting, when you have reserved your degrees out by the Needle in the Circumferentor or the Index of the Peractor, I have (because the practice thereof is very usual and no less difficult) in pag 233 inserted a figure so plain and perspicuous, that the very sight thereof will be enough (if there were no words used) to explain the use thereof. After the plot of any field is taken and protracted according to any of the former directions, I come to show how the content thereof may be attained several ways, that is, to find how many Acres, Roods and Perches are contained in any field thus plotted. Also there is taught how to measure mountainous and uneven grounds, and to find the area or content thereof. You are also taught in this fourth Book how to take the time plot of a whole Manor, or of divers severals, both by the Plain Table, Theodolite, Circumferentor or Peractor, with the manner how to keep account in your Field-book after the most sure and exactest way. Also how to reduce your Plot, to draw a perfect draught thereof, and to deck and beautify the same. And in the last place there is an example of Water-levelling, by which you may know whether water may be conveyed from a Springhead to any determinate place or not. Thus have I given you some general intimation of the principal heads contained in the following Treatise, which you may see more aparent in the following Analysis, but best of all in the Book itself, unto which I chiefly refer you, wishing that you may take the same delight and pleasure in the practice of those things therein contained, as I did in the composing of them, so shall I think my labour well bestowed, and be the more animated to present thee with some other Mathematical Treatise, who am A Friend to all that are Mathematically affected. WILLIAM LEYBOURN. A GENERAL SURVEY Of the whole WORK. The following Treatise is divided into four Books. I. Of Geometry, which consisteth of 1. Definition, page 3. 2. Theorems, 10. 3. Problems concerning 1. Raising and letting fall of Perpendiculars 11. 2. The making of equal angles, and drawing of parallel lines, 13. 3. The dividing of right lines equally. 14. 4. The constituting of right lined figures, 16. 5. The working of proportions by lines, 17. 6. The dividing of right lines proportionally, 18. 7. The dividing of Triangles according to proportion, both Arithmetically and Geometrically, by a line drawn 1. From any angle, 19 2. From a point in any side, 21. 3. Parallel in any side, 22. 8. The power of Lines and Superficies, 25. 9 The reducing of figures from one form to another, as Four Five Six solid figures into Triangles, 28. 10. The dividing of any plain Superficies into two or more parts, according to any proportion, by lines drawn either from any angle, or from a point in any side, 30. II. Of Instruments, as, 1. In general, 37. 2. Of the Theodolite, 39 3. Of the Circumferentor, 40. 4. Of the Plain Table, 42. 5. Of Chains, and chiefly of Master Rathborns, 46. Master Gunters, 47. 6. Of the Protractor, 50. 7. Of Scales, Plain, and diagonal, 52. 8. Of a Field-book, 53. 9 Of the Parallelogram, 54. III. Of trigonometry and 1. Of the description and use of the Tables of Sines, 57 and Logarithms, 63. 2. The application of these Tables, as also of the lines of Numbers, Sins and Tangents, in resolving of Plain Triangles, Right angled, 74. and Obliqne angled. 79. iv The use of Instruments, and, 1. Of the Scale in taking therefrom laying down lines and angles of any quantity, 179. 2. Of the Protractor in laying down finding the quantity of any Angle, 182. 3. Of the Plain Table, Theodolite, Circumferentor, to find an angle in the field therewith, 163. 4. Of the Label, thereby to observe an horizontal line, or line of level, an angle of Altitude, 166. 5. Of taking Distances accessible, or inaccessible, by the Plain Table, 187. Theodolite, 189. Circumferent. 190. and to protract the same, 191. 6. Of the taking of accessible inaccessible altitudes by the Label and Tangent line, 192. and to protract the same, 195. 7. Of taking divers distances at once, by the Plain Table, 196. and Theodolite; 198. and to protract the same, 199. 8. To take the plot of a Field at one station taken in the middle thereof by the Plain Table, 201. Theodolite, 203. Circumferentor, 205. and to protract the same, 206. 9 To take the plot of a Field at one station taken in any angle thereof, by the Plain Table, 208. Theodolite, ibid. Circumferentor, 210. and to protract the same, 210. 10. To take the plot of a field at two stations taken in any parts thereof, by the Plain Table, 212. Theodolite, 214. Circumferentor, 216. and to protract the same, 216. 11. To take the plot of a field at two stations taken in any parts thereof, only measuring the stationary distance, by the Plain Table, 218. Theodolite, 220. Circumferentor, 220 and to protract the same, 222 12. Of Large Champion plains or Woods, to take Plots thereof by the Plain Table, 223. Theodolite, 226. and to protract the same, 228. With a way to prove the truth thereof, 230. 13. To take the plot of any Field, Woodpark, Chase, Forrest, or other large Champion plain, by the Circumferentor, 230. And to protract the same, 233. With divers cautions for the exact performance thereof. 14. Of the Peractor, contrived by Master Rathborn, how to make the Plain Table to do the work thereof better than the Peractor itself, 236. 15. To take the plot of any piece of Land by the Peractor, 236. and to protract the same, 240. 16. Of finding the Area or superficial content of any piece of Land, the plot thereof being first taken, and chief of The Geometrical Square, 241, The Long Square, 242. The Triangle, 242. The Trapezia, 243. Any irregular plot of a Field, 244. The Circle, 245. 17. The manner of casting up the content of any piece of Land in Acres, etc. by Mr. Rathborns Chain, 246 Mr. Gunter's Chain, 249 18. To reduce Acres into Perches, and the contrary, 248. 19 The use of a Scale of Reduction necessary for finding the Fraction parts of an Acre 250 20. Divers compendious rules for the ready casting up of any plain Superficies, with divers other Compendiums in Surveying, by the line of Numbers, 251. 21. Of Statute and Customary measure to reduce one to the other at pleasure, 254. 22. Of the laying out of common fields into furlongs, 255. 23. Of Hills and Mountains, how to find the lengths of the horizontal lines on which they stand several ways, 257 24. Of mountainous and uneven grounds, how to protract or lay the same down in plano after the best manner, giving the area or content thereof, 258. 25. How to take the Plot of a whole Manner by the Plain Table three several ways, 260. Circumferentor 266. or Peractor, 266. With the keeping an account in your Field-book after the best and most certain manner, 270. and to protract any observations so taken, 271. 26. Of enlarging or diminishing of Plots according to any possible proportion by Two Semicircles. Mr. Rathborns Ruler. A Line into 100 parts. The Parallelogram. 273. 27 Of conveying of water, 276. FOrasmuch as the whole Art of Surveying of Land is performed by Instruments of several kinds, and that the exact and careful making and dividing of all such Instruments is chiefly to be aimed at, I thought good to intimate to such as are desirous to practise this Art, and do not readily know where to be furnished with necessary Instruments for the performance thereof, that all, or any of the Instruments used or mentioned in this Book, or any Mathematical Instrument whatsoever is exactly made by Mr. Anthony Thompson in Hosier lane near Smithfield, London. THE COMPLETE SURVEYOR. The First Book. THE ARGUMENT. THis first Book consisteth of divers Definitions & Problems Geometrical, extracted out of the Writings of divers ancient and modern Geometricians, as Euclid, Ramus, Clavius, etc. and are here so methodically disposed, that any man may gradually proceed from Problem to Problem without interruption, or being referred to any other Author for the Practical performance of any of them. Only the Demonstration is wholly omitted; partly, because those Books, out of which they were extracted, are very large in that particular, and also for the avoiding of many other Propositions and Theorems, which (had the ensuing Problems been demonstrated) must of necessity have been inserted. Also, the figures would have been so encumbered with multiplicity of lines, that the intended Problems would have been thereby much darkened. And besides it was not my intent in this place to make an absolute or entire Treatise of Geometry, and therefore I have only made choice of such Problems as I conceived most useful for my present purpose, and come most in use in the practice of Surveying, and aught of necessity to be known by every man that intendeth to exercise himself in the Practice thereof, and those are chief such as concern the reducing of Plots from one form to another, and to enlarge or diminish them according to any assigned Proportion, also divers of the Problems in this Book will abundantly help the Surveyor in the division and separation of Land, and in the laying out of any assigned quantity, whereby large parcels may be readily divided into divers severals; and those again subdivided if need be. Also for the better satisfaction of the Reader, I have performed divers of the following Problems both Arithmetically and Geometrically. GEOMETRICAL DEFINITIONS. 1. A Point is that which cannot be divided. A Point or Sign is that which is void of all Magnitude, and is the least thing that by mind and understanding can be imagined and conceived, than which there can be nothing less, as the Point or Prick noted with the letter A, which is neither quantity nor part of quantity, but only the terms or ends of quantity, and herein a Point in Geometry differeth from Unity in Number. 2. A Line is a length without breadth or thickness. A Line is created or made by the moving or drawing out of a Point from one place to another, so the Line AB, is made by moving of a Point from A to B, and according as this motion is, so is the Line thereby created, whether straight or crooked. And of the three kinds of Magnitudes in Geometry, viz. Length, Breadth, and Thickness, a Line is the first, consisting of Length only, and therefore the Line AB, is capable of division in length only, and may be divided equally in the point C, or unequally in D, and the like, but will admit of no other dimension. 3. The ends or bounds of a Line are Points. This is to be understood of a finite Line only, as is the line AB, the ends or bounds whereof are the points A and B: But in a Circular Line it is otherwise, for there, the Point in its motion returneth again to the place where it first began, and so maketh the Line infinite, and the ends or bounds thereof undeterminate. 4. A Right line is that which lieth equally between his points. As the Right line AB lieth straight and equal between the points A and B (which are the bounds thereof) without bowing, and is the shortest of all other lines that can be drawn between those two points. 5. A Superficies is that which hath only length and breadth. As the motion of a point produceth a Line, the first kind of Magnitude, so the motion of a Line produceth a Superficies, which is the second kind of Magnitude, and is capable of two dimensions, namely, length and breadth, and so the Superficies ABCD may be divided in length from A to B, and also in breadth from A to C. 6. The extremes of a Superficies are Lines. As the extremes or ends of a Line are points, so the extremes or bounds of a Superficies are Lines, and so the extremes or ends of the Superficies ABCD, are the lines AB, BD, DC, and CA, which are the terms or limits thereof. 7. A plain Superficies is that which lieth equally between his lines. So the Superficies ABCD lieth direct and equally between his lines: and whatsoever is said of a right line, the same is also to be understood of a plain Superficies. 8. A plain Angle is the inclination or bowing of two lines the one to the other, the one touching the other, & not being directly joined together. As the two lines AB and BC incline the one to the other, and touch one another in the point B, in which point, by reason of the inclination of the said lines, is made the Angle ABC. But if the two lines which touch each other be without inclination, and be drawn directly one to the other, than they make no angle at all, as the lines CD and DE, touch each other in the point D, and yet they make no angle, but one continued right line. ¶ And here note, that an Angle commonly is signed by three Letters, the middlemost whereof showeth the angular point: As in this figure, when we say the angle ABC, you are to understand the very point at B: And note also, that the length of the sides containing any angle, as the sides AB and BC, do not make the angle ABC either greater or lesser, but the angle still retaineth the same quantity be the containing sides thereof either longer or shorter. 9 And if the lines which contain the angle be right lines, then is it called a right lined angle. So the angle ABC is a right lined angle, because the lines AB and BC, which contain the said angle, are right lines. And of right lined Angles there are three sorts, whose Definitions follow. 10. When a right line standing upon a right line maketh the angles on either side equal, then either of those angles is a right angle: and the right line which standeth erected, is called a perpendicular line to that whereon it standeth. As upon the right line CD, suppose there do stand another right line AB, in such sort that it maketh the angles on either side thereof equal, namely, the angle ABDELLA on the one side, equal to the angle ABC on the other side: then are either of the two angles ABC, and ABDELLA right angles, and the right line AB, which standeth erected upon the right line CD, without inclining to either part thereof, is a perpendicular to the line CD. 11. An Obtuse angle is that which is greater than a right angle. So the angle CBE is an obtuse angle, because it is greater than the angle ABC, which is a right angle; for it doth not only contain that right angle, but the angle ABE also, and therefore is obtuse. 12. An Acute angle is less than a right angle. So the angle EBBED is an acute angle, for it is less than the right angle ABDELLA (in which it is contained) by the other acute angle ABE. 13. A limit or term is the end of every thing. As a point is the limit or term of a Line, because it is the end thereof, so a Line likewise is the limit and term of a Superficies; and a Superficies is the limit and term of a Body. 14. A Figure is that which is contained under one limit or term or many. As the Figure A is contained under one limit or term, which is the round line. Also the Figure B is contained under three right lines, which are the limits or terms thereof. Likewise, the Figure C is contained under four right lines, the Figure E under five right lines, and so of all other figures. ¶ And here note, that in the following work we call any plain Superficies whose sides are unequal, (as the Figure E) a Plot, as of a Field, Wood, Park, Forrest, and the like. 15. A Circle is a plain Figure contained under one line, which is called a Circumference, unto which all lines drawn from one point within the Figure, and falling upon the Circumference thereof are equal one to the other. As the Figure ABCDE is a Circle, contained under the crooked line BCDE, which line is called the Circumference: In the middle of this Figure is a point A, from which point all lines drawn to the Circumference thereof are equal, as the lines AB, AC, OF, AD: and this point A is called the centre of the Circle. 16. A Diameter of a Circle is a right line drawn by the Centre thereof, and ending at the Circumference, on either side dividing the Circle into two equal parts. So the line BADE (in the former Figure) is the Diameter thereof, because it passeth from the point B on the one side of the Circumference, to the point D on the other side of the Circumference, and passeth also by the point A, which is the centre of the Circle. And moreover it divideth the Circle into two equal parts, namely, BCD being on one side of the Diameter, equal to BED on the other side of the Diameter. And this observation was first made by Thales Miletius, for, saith he. If a line drawn by the centre of any Circle do not divide it equally, all the lines drawn from the centre of that Circle to the Circumference cannot be equal. 17. A Semicircle is a figure contained under the Diameter, and that part of the Circumference cut off by the Diameter. As in the former Circle, the figure BED is a Semicircle, because it is contained of the right line BAD, which is the Diameter, and of the crooked line BED, being that part of the circumference which is cut off by the Diameter: also the part BCD is a Semicircle. 18. A Section or portion of a Circle, is a Figure contained under a right line, and a part of the circumference, greater or less than a semicircle. So the Figure ABC, which consisteth of the part of the Circumference ABC, and the right line AC is a Section or portion of a Circle greater than a Semicircle. Also the other figure ACD, which is contained under the right line AC, and the part of the circumference ADC, is a Section of a Circle less than a Semicircle. ¶ And here note, that by a Section, Segment, Portion, or Part of a Circle, is meant the same thing, and signifieth such a part as is either greater or lesser than a Semicircle, so that a Semicircle cannot properly be called a Section, Segment, or part of a Circle. 19 Right lined figures are such as are contained under right lines. 20. Three sided figures are such as are contained under three right lines. 21. Four sided figures are such as are contained under four right lines. 22. Many sided figures are such as have more sides than four. 23. All three sided figures are called Triangles. And such are the Triangles BCD. 24. Of four sided Figures, a Quadrat or Square is that whose sides are equal and his angles right. [As the Figure A.] 25. A Long square is that which hath right angles but unequal sides. [As the Figure B] 26. A Rhombus is a Figure having four equal sides but not right angles. [As the Figure C.] 27. A Rhomboides is a Figure whose opposite sides are equal, and whose opposite angles are also equal, but it hath neither equal sides nor equal angles. [As the Figure D.] 28. All other Figures of four sides (besides these) are called Trapezias. Such are all Figures of four sides in which is observed no equality of sides or angles, as the figures A and B, which have neither equal sides nor equal angles, but are described by all adventures without the observation of any order. 29. Parallel, or equidistant right lines are such which being in one and the same Superficies and produced infinitely on both sides, do never in any part concur. As the right lines AB and CD are parallel one to the other, and if they were infinitely extended on either side would never meet or concur together, but still retain the same distance. Geometrical Theorems. 1. ANy two right lines crossing one another, make the contrary or vertical angles equal. 2. If any right line fall upon two parallel right lines, it maketh the outward angles on the one, equal to the inward angles on the other, and the two inward opposite angles on contrary sides of the falling line also equal. 3. If any side of a Triangle be produced, the outward angle is equal to the two inward opposite angles, and all the three angles of any Triangle are equal to two right angles. 4. In equiangled Triangles, all their sides are proportional, as well such as contain the equal angles, as also the subtendent sides. 5. If any four Quantities be proportional, the first multiplied in the fourth, produceth a Quantity equal to that which is made by multiplication of the second in the third. 6. In all right angled Triangles, the square of the side subtending the right angle, is equal to both the squares of the containing sides. 7. All parallellograms are double to the triangles that are described upon their bases, their altitudes being equal. 8. All triangles that have one and the same Base, and lie between two parallel lines, are equal one to the other. GEOMETRICAL PROBLEMS. PROBLEM I. Upon a right line given, how to erect another right line, which shall be perpendicular to the right line given. THe right line given is AB, upon which from the point E it is required to erect the perpendicular EH. Opening your Compasses at pleasure to any convenient distance, place one foot in the assigned point E, and with the other make the marks C and D, equidistant on each side the given point E. Then opening your Compasses again to any other convenient distance wider than the former, place one foot in C, and with the other describe the arch GG; also (the Compasses remaining at the same distance) place one foot in the point D, and with the other describe the arch FF, then from the point where these two arches intersect or cut each other (which is at H) draw the right line HE which shall be perpendicular to the given right line AB, which was the thing required to be done. PROB. II. How to erect a Perpendicular on the end of a right line given. LEt OR be a line given, and let it be required to erect the perpendicular RS. First, upon the line OR, with your Compasses opened to any small distance, make five small divisions beginning at R, noted with 1, 2, 3, 4, 5. Then take with your Compasses the distance from R to 4, and placing one foot in R, with the other describe the arch PP. Then take the distance R 5, and placing one foot of the Compasses in 3, with the other foot describe the arch BB, cutting the former arch in the point S. Lastly, from the point S, draw the line RS, which shall be perpendicular to the given line OR. PROB. III. How to let fall a perpendicular, from any point assigned, upon a right line given. THE point given is C, from which point it is required to draw a right line which shall be perpendicular to the given right line AB. First, from the given point C, to the line AB, draw a line by chance, as CE, which divide into two equal parts in the point D, then placing one foot of the Compasses in the point D, with the distance DC, describe the Semicircle CFE, cutting the given line AB in the point F. Lastly, if from the point C you draw the right line CF, it shall be a perpendicular to the given line AB, which was required. PROB. IU. How to make an angle equal to an angle given. LEt the angle given be ACB, and let it be required to make another angle equal thereunto. First, draw the line OF at pleasure, then upon the given angle at C, (the Compasses opened to any distance) describe the ark AB, also, upon the point F (the Compasses un-altered) describe the ark DE: than take with your Compasses the distance AB, and set the same distance from E to D. Lastly, draw the line DF, so shall the angle DFE be equal to the given angle ACB. PROB. V A right line being given, how to draw another right line which shall be parallel to the former, at any distance required. THe line given is AB, unto which it is required to draw another right line parallel thereunto, at the distance AC, or BD. First, Open your Compasses to the distance AC or AD, then placing one foot in A, with the other describe the ark C; also, place one foot in B, and with the other describe the arch D. Lastly, Draw the line CD so that it may only touch the arks C and D, so shall the line CD be parallel to the line AB, and at the distance required. PROB. VI To divide a right line given into any number of equal parts. LEt AB be a line given, and let it be required to divide the same into four equal parts. First, From the end of the given line A, draw the line AC, making any angle, then from the other end of the given line, which is at the point B, draw the line BD parallel to AC, or make the angle ABDELLA equal to the angle CAB; then upon the lines AC and BD set off any three equal parts (which is one less than the number of parts into which the line AB is to be divided) on ●ace line, as 1 2 3, then draw lines from 1 to 3, from 2 to 2, and from 3 to 1, which lines, crossing the given line AB, shall divide it into four equal parts as was required. PROB. VII. A right line being given, how to draw another right line parallel thereunto, which shall also pass through a point assigned. LEt AB be a line given, and let it be required to draw another line parallel thereunto which shall pass through the given point C. First, Take with your compasses the distance from A to C, and placing one foot thereof in B, with the other describe the ark DE; then take in your compasses the whole line AB, and placing one foot in the point C, with the other describe the ark FG, crossing the former ark DE in the point H. Lastly, if you draw the line CHANGED it shall be parallel to AB. PROB. VIII. Having any three points given, which are not situate in a right line, how to find the centre of an arch of a Circle which shall pass directly through the three given points. THe three points given are A B and C, now it is required to find the centre of a Circle, whose circumference shall pass through the three points given. First, Opening your Compasses to any distance greater than half BC, place one foot in the point B, and with the other describe the arch FG, then, the Compasses remaining at the same distance, place one foot in C, and with the other turned about make the marks F and G in the former arch, and draw the line FG at length if need be. Again, opening the Compasses to any distance greater than half AB, place one foot in the point A, and with the other describe the arch HK, then, the Compasses remaining at the same distance, place one foot in the point B, and turning the other about make the marks HK in the former arch. Lastly, draw the right line HK cutting the line FG in O, so shall O be the centre upon which if you describe a Circle at the distance of OA, it shall pass directly through the three given points A B C, which was required. PROB. IX. Any three right lines being given, so that the two shortest together be longer than the third, to make thereof a Triangle. LEt it be required to make a Triangle of the three lines A B and C, the two shortest whereof, viz. A and B together, are longer than the third line C. First, Draw the line DE equal to the given line B, then take with your Compasses the line C, and setting one foot in E, with the other describe the arch HG, also, take the given line A in your Compasses, and placing one foot in D, with the other describe the arch HF, cutting the former arch HG in the point H. Lastly, if from the point H you draw the lines HE and HD, you shall constitute the Triangle HDE, whose sides shall be equal to the three given lines A B C. PROB. X. Having a right line given, how to make a Geometrical Square, whose side shall be equal thereunto. THe line given is QR, and it is required to make a Geometrical Square whose side shall be equal to the line QR. First, Draw the line AB, making it equal to the given line QR, than (by the first or second Problem) upon the point B raise the perpendicular BC, making the line BC equal to the given line QR also. Then taking the line QR in your Compasses, place one foot in C, and with the other describe the arch D, also the Compasses so resting, place one foot in A, and with the other describe another arch crossing the former in the point D. Lastly, draw the lines DC and DA, which shall include the Geometrical Square ABCD. PROB. XI. Two right lines being given, how to find a third right line which shall be in proportion unto them. LEt the two given lines be A and B, and let it be required to find a third line which shall be in proportion unto them. First, Draw two right lines making any angle at pleasure, as the lines OPEN and ON, making the angle PON; then taking the line A in your Compasses, set the length thereof from O to S, also, take the line B in your Compasses, and set the length thereof from O to R, and also from O to D, then draw the right line SD, and from the point R draw the line RC parallel to SD, so shall OC be the third proportional required, for, As OS to ODD ∷ so OR to OC. As 8 to 12 ∷ so 12 to 18. PROB. XII. Three right lines being given, to find a fourth in proportion to them. THe three lines given are A B C, unto which it is required to find a fourth proportional line. This is to perform the rule of three in lines. As in the last Problem, you must draw two lines making any angle, as the angle DEF. Then take the line A in your Compasses, and set it from E to G, then take the line B in your Compasses and set that length from E to H. Then take the third given line in your Compasses, and set that from E to K, and through that point K draw the line KL parallel to GH, so shall the line EL be the third proportional required; for, As EGLANTINE to EH ∷ so EKE to EL. As 24 to 28 ∷ so 36 to 42. ¶ Here note that in the performance of this Problem, that the first and the third terms (namely the lines A and C) must be set upon one and the same line, as here upon the line ED, and the second term (namely the line B) must be set upon the other line OF, upon which line also the fourth proportional EL will be found. PROB. XIII. To divide a right line given into two parts, which shall have such proportion one to the other as two given right lines. THe line given is AB, and it is required to divide the same into two parts, which shall have such proportion one to the other, as the line C hath too the line D. First, from the point A, draw the line A, at pleasure, making the angle EAB; then take in your Compasses the line C, and set it from A to F, also take the line D, and set it from F to E, and draw the line EBB, then from the point F, draw the line FG parallel to EBB, cutting the given line AB in the point G; 〈◊〉 is the line AB divided into two parts in the point G, being in proportion one to the other, as the line C is to the line D; for, As A to AB ∷ so OF to AG. Arithmetically. LEt the line AB contain 40 Perches, and let the line C be 20, and the line D 30; and let it be required to divide the line AB into two parts, being in proportion one to the other, as the line C is to the line D. First, Add the lines C and D together, their sum is 50, then say by the Rule of Proportion: If 50 (which is the sum of the two given terms) give 40 the whole line AB, what shall 30, the greater given term give? Multiply and divide, and you shall have in the quotient 24 for the greater part of the line AB, which being taken from 40 the whole line, there remains 16 for the other part AGNOSTUS; for, As A to AB ∷ so FE to GB. As 50 to 40 ∷ so 30 to 24. PROB. XIV. How to divide a Triangle into two parts, according to any proportion assigned, by a line drawn from any angle thereof, and to lay the lesser part towards any side assigned. LEt ABC be a Triangle given, and let it be required to divide the same, by a line drawn from the angle A, into two parts, the one bearing proportion to the other, as the line F doth to the line G, and that the lesser part may be towards the side AB. By the last Problem divide the base of the Triangle BC in the point D, in proportion as the line F is to the line G, (the lesser part being set from B to D.) Lastly, draw the line AD, which shall divide the Triangle ABC in proportion as F to G; for, As the line F, is to the line G; So is the Triangle ADC; to the Triangle ABDELLA. PROB. XV. The Base of the Triangle being known, to perform the foregoing Problem by Arithmetic. SUppose the Base of the Triangle BC to be 40, and let the proportion into which the Triangle ABC is to be divided, be as 2 to 3. First, Add the two proportional terms together, 2 and 3, which makes 5, then say by the rule of proportion: If 5 (the sum of the proportional terms,) give 40 (the whole base BC,) what shall 3 (the greater term) give? Multiply and divide, and the quotient will give you 24 for the greater segment of the Base DC, which being deducted from the whole base 40, there will remain 16 for the lesser segment BD. PROB. XVI. How to divide a Triangle, whose area or content is known, into two parts, by a line drawn from an angle assigned, according to any proportion required. LEt the Triangle ABC contain 8 Acres, and let it be required to divide the same into two parts, by a line drawn from the angle A, the one to contain 5 Acres, and the other 3 Acres. First, Measure the whole length of the Base, which supppse 40, then say, If 8 Acres (the quantity of the whole Triangle) give 40, (the whole Base,) what parts of the Base shall 5 Acres give? Multiply and divide, the Quotient will be 25 for the greater segment of the base CD, which being deducted from 40 (the whole Base,) there will remain 15 for the lesser segment of the Base BD, then draw the line AD, which shall divide the Triangle ABC according to the proportion required. PROB. XVII. How to divide a Triangle given into two parts, according to any proportion assigned, by a line drawn from a point limited in any of the sides thereof: and to lay the greater or lesser part towards an angle assigned. THe Triangle given is ABC and it is required from the point E, to draw a line that shall divide the Triangle into two parts, being in proportion one to the other as the line I is to the line K, and to lay the lesser part towards B. First, From the limited point E, draw a line to the opposite angle at A; then divide the base BC in proportion as I to K, which point of division will be at D, then draw DF parallel to A Lastly, from F, draw the line FE, which will divide the Triangle into two parts being in proportion one to the other as the line I is to the line K. PROB. XVIII. To perform the foregoing Problem Arithmetically. IT is required to divide the Triangle ABC, from the point E, into two parts in proportion as 5 to 2. First, Divide the base BC according to the given proportion, than (because the lesser part is to be laid towards B) measure the distance from E to B, which admit 30, then say by the rule of Proportion; If EBB 30, give DB 15, what shall AGNOSTUS 29 (the perpendicular of the Triangle) give? Multiply and divide, the Quotient will be 14½, at which distance draw a parallel line to BC, namely F, then from F draw the line FE, which shall divide the Triangle according to the required proportion. PROB. XIX. How to divide a Triangle, whose area or content is known, into two parts, by a line drawn from a point limited in any side thereof, according to any number of Acres, Roods and Perches. IN the foregoing Triangle ABC, whose area or content is 5 Acres, 1 Rood, let the limited point be E in the base thereof, and let it be required from the point E to draw a right line which shall divide the Triangle into two parts between M and N, so that M may have 3 Acres, 3 Roods thereof, and N may have 1 Acre and 2 Roods thereof. First, reduce the quantity of N (being the lesser) into perches, which makes 240, than (considering on which side of the limited point E this part is to be laid, as towards B) measure that part of the base from E to B 30 Perches, whereof take the half, which is 15, and thereby divide 240, the part belonging to N; the quotient will be 16, the length of the perpendicular FH, at which parallel distance from the base BC cut the side AB in F, from whence draw the line FE which shall cut off the Triangle FBE, containing 1 Acre, 2 Roods, the part belonging to N, then will the Trapezia AFEC (which is the part belonging to M) contain the residue, namely, 3 Acres, 3 Roods. PROB. XX. How to divide a Triangle according to any proportion given, by a line drawn parallel to one of the sides. THe following Triangle ABC is given, and it is required to divide the same into two parts by a line drawn parallel to the side AC, which shall be in proportion one to the other, as the line I is to the line K. First (by the 13 Problem) divide the line BC in E, in proportion as I to K, than (by the 24 Problem following) find a mean proportional between BE and BC, which let be BF, from which point F, draw the line FH parallel to AC, which line shall divide the Triangle into two parts, viz. the Trapezia AHFC, and the Triangle HFB, which are in proportion one to the other as the line I is to the line K. PROB. XXI. To perform the foregoing Problem Arithmetically. LEt the Triangle be ABC, and let it be required to divide the same into two parts, which shall be in proportion one to the other, as 4 to 5, by a line drawn parallel to one of the sides. First, Let the base BC containing 54 be divided according to the proportion given, so shall the lesser segment BE contain 24, and the greater EC 30; then find out a mean proportional line between BE 24, and the whole base BC 54, by multiplying 54 by 24, whose product will be 1296, the square root whereof is 36, the mean proportional sought, which is BF, then, by the rule of proportion say: If BF 36 give BE 24, what AD 36? the answer is HG 24, at which distance draw a parallel line to the base, to cut the side AB in H, from whence draw the line HF parallel to AC, which shall divide the Triangle as was required. PROB. XXII. To divide a Triangle of any known quantity, into two parts, by a line drawn parallel to one of the sides, according to any number of Acres, Roods, and Perches. THe Triangle given is ABC, whose quantity is 8 Acres, 0 Roods, 16 Perches, and it is required to divide the same (by a line drawn parallel to the side AC) into two parts, viz. 4 Acres, 2 Roods, 0 Perches, and 3 Acres, 2 Roods, 16 Perches. First, Reduce both quantities into perches (as is hereafter taught) and they will be 720, and 576, then reduce both those numbers, by abbreviation, into the least proportional terms, viz. 5 and 4, and according to that proportion, divide the base BC 54 of the given Triangle in E, then seek the mean proportional between BE and BC, which proportional is BF 36, of which 36 take the half, and thereby divide 576, the lesser quantity of Perches, the Quotient will be HG 32, at which parallel distance from the base, cut off the line AB in H, from whence draw the line HF parallel to the side AC, which shall divide the Triangle given according as was required. PROB. XXIII. From a line given, to cut off any parts required. THe line given is AB, from which it is required to cut off 3/7 parts. First, draw the line AC, making any angle, as CAB, then from A, set off any seven equal parts, as 1 2 3 4 5 6 7, and from 7 draw the line 7B. Now because 3/7 is to be cut off from the line AB, therefore from the point 3, draw the line 3D parallel to 7B, cutting the line AB in D, so shall AD be 3/7 of the line AB, and DB shall be 4/7 of the same line; for, As A7, is to AB ∷ so is A3, to AD. PROB. XXIV. To find a mean proportional between two lines given. IN the following figure, let the two lines given be A and B, between which it is required to find a mean proportional. Let the two given lines A and B, be joined together in the point E, making one right line, as CD, which divide into two equal parts in the point G, upon which point G, with the distance GC or GD, describe the Semicircle CFD; then, from the point E, (where the two lines are joined together) raise the perpendicular OF, cutting the Periphery of the Semicircle in F, so shall the line OF be a mean proportional between the two given lines A and B; for, As ED to OF ∷ so OF to CF. As 9 to 12 ∷ so 12 to 16. PROB. XXV. How to divide a line in power according to any proportion given. IN this figure let CD be a line given to be divided in power as the line A is to the line B. First, divide the line CD in the point E, in proportion as A to B, (by the 13 Problem:) then divide the line CD into two equal parts in the point G, and on G, at the distance GC or GD, describe the Semicircle CFD, and upon the point E, raise the perpendicular OF, cutting the Semicircle in F: Lastly, draw the lines CF and DF, which together in power shall be equal to the power of the given line CD, and yet in power one to the other as A to B. PROB. XXVI. How to enlarge a line in power, according to any proportion assigned. IN the former figure, Let CE be a line given, to be enlarged in power as the line B to the line G. First, (by the 13 Problem) find a line in proportion to the given line CE, as B is to G, which will be CD, upon which line describe the Semicircle CFD, and on the point E, erect the perpendicular OF; then draw the line CF, which shall be in power to CE, as G to B. PROB. XXVII. To enlarge or diminish a Plot given, according to any proportion required. LEt ABCDE be a Plot given, to be diminished in power as L to K. Divide one of the sides (as AB) in power as L to K, in such sort, that the power of OF, may be to the power of AB, as L to K. Then from the angle A, draw lines to the points C and D, that done, by F draw a parallel to BC, to cut AC in G, as FG. Again, from G, draw a parallel to CD to cut AD in H. Lastly, from H, draw a parallel to DE, to cut A in I, so shall the plot AFGHI be like ABCDE, and in proportion to it, as the line L, to the line K, which was required. Also, if the lesser Plot were given, and it were required to make a greater in proportion to it as K to L. Then from the point A, draw the lines AC and AD, at length, also increase OF and AI: that done, enlarge OF in power as K to L, which set from A to B, then by B draw a parallel to FG to cut AC in C, as BC. Likewise from C draw a parallel to GH, to cut AD in D, as CD. Lastly, a parallel from D to HI, as DE, to cut AI being increased in E, so shall you include the Plot ABCDE, like AFGHI, and in proportion thereunto, as the line K is to the line L, which was required. PROB. XXVIII. How to make a Triangle which shall contain any number of Acres, Roods and Perches, and whose base shall be equal to any (possible) number given. IF it be required to make a Triangle which shall contain 5 Acres, 2 Roods, 30 Perches, whose base shall contain 50 Perches, you must first reduce your 5 Acres, 2 Roods, 30 Perches, all into Perches in this manner. First, (because 4 Roods make one Acre) multiply your 5 Acres by 4 which makes 20, to which add the two odd Roods, so have you 22 Roods in your 5 Acres 2 Roods. Then (because 40 Perches make one Rood) multiply your 22 Roods by 40, which makes 880 Perches, to which add the 30 odd Perches, and you shall have 910, and so many Perches are contained in 5 Acres, 2 Roods, 30 Perches. Now to make a Triangle which shall contain 910 perches, & whose base shall be 50 Perches, do thus, Double the number of perches given, namely 910, and they make 1820, then because the base of the triangle must contain 50 Perches, divide 1820 by 50, the quotient will be 36⅖, which will be the length of the perpendicular of your Triangle. This done, From any equal Scale lay down the line AB equal to 50 Perches, then upon B, raise the perpendicular BD equal to 36⅖ perches, and draw the line CD parallel to AB then, from any point in the line CD (as from E) draw the lines EA and EBB, including the Triangle AEB, which shall contain 5 Acres, 2 Roods, 30 Perches, which was required. PROB. XXIX. How to reduce a Trapezia into a Triangle, by a line drawn from any angle thereof. THe Trapezia given is ABCD, and it is required to reduce the same into a Triangle. First, Extend the line DC, and draw the diagonal BD, then from the point A, draw the line A parallel to BD, extending it till it cut the side CD in the point E. Lastly, from B, draw the line BE, constituting the Triangle EBC, which shall be equal to the Trapezia ABCD. PROB. XXX. How to reduce a Trapezia into a Triangle, by lines drawn from any point in any of the sides thereof. LEt ABCD be a Trapezia given, and let H be a point in one of the sides thereof, from which point H let it be required to draw lines which shall reduce the Trapezia into a Triangle. First, Extend the side which is opposite to the given point, namely, the side CD, both ways to E and F, and then from the point H, draw lines to the angles C and D, as the lines HC and HD; also, draw the lines A and BF parallel to HC and HD, cutting the extended line CD in the points E and F. Lastly, If from the point H you draw the lines HE and HF, you shall constitute the Triangle HEF, which shall be equal to the Trapezia ABCD. PROB. XXXI. How to reduce an irregular Plot of five sides into a Triangle. THe irregular Plot given is ABCDE, and it is required to reduce the same into a Triangle. First, extend the side A both ways to F and G, and from the angle C, draw the lines CA and CE, to the angles A and E. Then from the point B, draw the line BF parallel to CA cutting the extended side A, in F; also, from the point D, draw the line DG parallel to CE, cutting also the extended side in G. Lastly, from the angle C, draw the lines CF and CG, constituting the Triangle CFG which is equal to the Plot ABCDE. PROB. XXXII. A Trapezia being given, how from any angle thereof of to divide the same into two parts being in proportion one to the other as two given right lines, and to set the part cut off towards an assigned side. LEt the Trapezia given be ABCD, and let it be required to draw a line from the angle B, which shall divide the Trapezia into two parts, being in proportion one to the other, as the line G is to the line H, and that the lesser part of the Figure cut off, may be towards the side AB. First (by the 29 Problem) reduce the Trapezia ABCD into a Triangle, by drawing the line BF from the assigned angle, thereby constituting the Triangle ABF, equal to the Trapezia ABCD: this done, divide the base of the Triangle OF in proportion as G to H, which will be in the point E. Lastly, draw the line BE, which shall divide the Trapezia in proportion as G to H. Now because the lesser part of the Trapezia was to be set towards the side AB, therefore the lesser part of the line must be set from A to E. Here note that the same manner of working is to be observed, if it had been required to divide the Trapezia by a line drawn from any of the other angles. PROB. XXXIII. A Trapezia being given, how, from a point limited in any side thereof, to draw a line which shall divide the same into two parts in proportion as two given lines. THe Trapezia given is ABCD, and it is required from the point H, to draw a line which shall divide the Trapezia in proportion as O to Q. First, Prolong the side CD, and reduce the whole Trapezia into the Triangle HEF by the 30 Problem, then divide the line OF in proportion as O to Q, which will fall in the point G, therefore draw the line HG which shall divide the Trapezia into two parts in proportion as O to Q, which was required, PROB. XXXIV. A Trapezia being given, how to divide the same into two parts in proportion as two lines given, and so that the line of partition may be parallel to any side thereof. THe Trapezia given is ABCD, and it is required to divide the same into two parts, which shall be in proportion one to the other as the line K is to the line L, and that the line of partition may be parallel to the side BD. Consider first, through which sides of the Trapezia the line of partition will pass, as in this Figure it will pass through the sides AB and CD (because parallel to BD,) therefore, extend the sides AB and CD, till they concur in E, then (by the 32 Problem) reduce the Trapezia ABCD into the Triangle BGD, whose base is GD, which line GD, divide in the point H in proportion as K to L; so that, As K to L ∷ So DH to HG. This done, find a mean proportional between ED and EH (by the 24 Problem) as ER. Lastly, through this point R, draw the line RF parallel to BD, which shall divide the Trapezia into two parts being in proportion one to the other, as the line K is to the line L, and with a line parallel to the side BD, which was required. But if it had been required to divide the Trapezia by a line drawn parallel to the side CD, than the lines CA and DB must have been extended, but the rest of the work must be performed as is before taught. PROB. XXXV. The figure of a Plot being given, how to divide the same into two parts, being in proportion one to the other as two given lines are, with a line drawn from an angle assigned. LEt the following Figure ABCDE represent the Plot of a Field or such like, and let it be required to divide the same into two parts, being in proportion one to the other as the line R is to the line S, by a line drawn from the angle B. First, Reduce the Plot ABCDE into the Triangle BFG, (by the 31 Problem) so shall the line FG be the base of a Triangle equal to the given Plot, than (by the 13 Problem) divide this line FG into two parts in the point H, in proportion one to the other, as the line R is to the line S; so that, As R to S ∷ so GH to HF. Lastly, draw the line BH, which shall divide your given Plot into two parts which shall have such proportion one to the other, as the line R hath to the line S. PROB. XXXVI. How to divide a Triangle into any number of equal parts, by lines drawn from a point given in any side thereof. LEt it be required to divide the Triangle ABC into five equal parts, by lines drawn from the point D. First, From the given point D, to the opposite angle B, draw the line DB, then divide the side AC of the Triangle into five equal parts, at E F G and H, and through each of those points draw lines parallel to DB, as EM, FL, GK, and HI: then from the point D, draw the lines DIEGO, DK, DL, and DM, which shall divide the Triangle ABC into five equal parts from the point D, as was required. PROB. XXXVII. How to divide an irregular Plot of six sides, into two parts, according to any assigned proportion, by a right line drawn from a point limited in any of the sides thereof. THe irregular Plot given is ABCDEF, and it is required to divide the same into two parts, being in proportion one to the other, as the line R is to the line S. First, Draw the right line HK, and (by the 30 Problem) reduce the Trapezia ABFG into the Triangle HGK, then divide ●he base thereof, namely HK, into two parts in proportion as R to ●…, which will be in the point O, then draw the line GO, which will divide the Trapezia ABFC into two parts in proportion one ●o the other, as the line R is to the line S. Secondly, From the point O (by the 31 Problem) reduce the Trapezia FCED into the Triangle OLM, and divide the base thereof, namely LM, in the point N, in proportion as R to S, and draw the line ON, which will divide the Trapezia FCED into two parts in proportion as R to S: and by this means is the whole Plot ABCDEF divided into two parts in proportion as R to S, by the lines GO and ON. But it is required to resolve the Problem by one right line only drawn from the point G, therefore, from the point G, draw the line GN, and through the point O, draw the line OPEN parallel to GN: and lastly, from G, draw the right line GP, which shall divide the whole Plot ABCDEF into two parts, being in proportion one to the other as the line T is to the line S. PROB. XXXVIII. How to divide an irregular Plot according to any proportion, by a line drawn from any angle thereof. LEt ABCDEFG be an irregular Plot, and let it be required to divide the same into two equal parts, by a line drawn from the angle A. First, draw the line HK, dividing the Plot into two parts, namely, into the five sided figure ABCFG, and into the Trapezia FCED, than (by the 31 Problem) reduce the five sided figure ABCFG into the Triangle HAK, the base whereof HK divide into two equal parts in O, and draw the line OA, which shall divide the five sided figure ABCFG into two equal parts. Then (by the 30 Problem) reduce the Trapezia FCDE into the Triangle OLM, and divide the base thereof LM into two equal parts in the point P, and draw the line OPEN, which will divide the Trapezia FCDE into two equal parts, and so is the whole Plot divided into two equal parts by the lines AO and OPEN, but to perform the Problem by one right line only, do thus, from the point A, draw the line AP, and parallel thereunto, through the point O, draw the line ON. Lastly, if you draw a right line from A to N, it shall divide the whole Plot into two equal parts. The end of the First Book. THE COMPLETE SURVEYOR. The Second Book. THE ARGUMENT. IN this Book is contained both a general and particular description of all the most necessary Instruments belonging to Surveying, as the Theodolite, Circumferentor and Plain Table, with all the appurtenances thereunto belonging, as the Staff, Sockets, Screws, Index, Label, and other necessaries. Now whereas these three Instruments are the most convenient for all manner of practices in Surveying, I have so ordered the matter, that in this Book, after the Theodolite and Circumferentor are particularly described, as they have usually been made; I come to the description of the Plain Table, and therein have showed how that Instrument may be ordered to perform the work of any of the other; so that whatsoever may be done by the Theodolite, Circumferentor, or any other Instrument, the same may be effected by the Plain Table only, as it is there contrived, with the same ease, dispatch, and exactness, and in many respects better, as in Chap. 1. doth plainly appear: so that this Instrument only is sufficient for all manner of practices whatsoever. And besides the Instruments for mensuration there is described divers other Instruments belonging thereunto, as Chains, Scales, Protractors, and the like; all which are described according to the best contrivance yet known. A DESCRIPTION OF INSTRUMENTS. CHAP. I. Of Instruments in general. THe particular description of the several Instruments that have from time to time been invented for the practice of Surveying, would make a Treatise of itself, and in this place is not so necessary to be insisted on, every of the inventors, in their several Books of the uses of them having been already large enough in their construction. To omit therefore the description of the topographical Instrument of Master Leonard Diggs, the Familiar Staff of Master John Blagrave, the Geodeticall Staff, and topographical Glass of Master Arthur Hopton, with divers other Instruments invented and published by Gemma Frisius, Orentius, Clavius, Stofterus, and others; I shall immediately begin with the description of those which are the ground and foundation of all the rest, and are now the only Instruments in most esteem amongst Surveyors, and those are chiefly these three, the Theodolite, the Circumferentor, and the Plain Table. Now, as I would not confine any man to the use of one particular Instrument for all employments, so I would advise any man not to cumber himself with multiplicity, since these three last named are sufficient for all occasions. And if I should confine any man to the use of any one of these Instruments (as, for a shift, any one of them will perform any kind of work in Surveying) yet in that I should do him injury, for in many cases one Instrument may make a quicker dispatch, and be altogether as exact as another: As in laying down of a spacious business, I would advise him to use the Circumferentor or Theodolite, and for Townships and small Enclosure the Plain Table, so altering his Instrument according at the nature or quality of the ground he is to measure doth require. These three special Instruments have been largely described already by divers, as namely, by Master Diggs, Master Hopten, Master Rathborne, and last of all in Planometria; yet in this place it will be very necessary to give a particular description of them again, because if any man have a desire to any particular Instrument, he may give the better directions for the making thereof. For the description which I shall make of these three Instruments in particular, it shall be agreeable to those Instruments as they are usually made; with some small addition or alteration: But when I come to the description of the Plain Table, after that I have described it according to the vulgar way, I will then show you a new metamorphosis of that Instrument, making it the most absolute and universal Instrument yet ever invented, so that having that one Instrument (made according to the following directions) you shall have need of no other for the due, exact, and speedy performance of any thing belonging to the Art of Surveying. The Plain Table used as the Theodolite. For, the Frame of the Table being graduated according to that description, will be an absolute Theodolite, and perform the work thereof with the same facility and exactness, and whatsoever may be done by the limb of the Theodolite, the same the degrees on the frame of the Table will as well perform. The Plain Table used as a Circumferentor. Likewise, the Index and Sights, together with the Box and Needle, being taken from the Table, and screwed to the Staff (as in the description thereof it is so conveniently ordered) will be an absolute Circumferentor, and in some respects better than the ordinary one hereafter described, because the Sights thereof stand at a greater distance, so that thereby the visual line may be the better directed. The plain Table, not one, but all Instruments. And this Instrument (as now contrived) though it be called the Plain Table only, yet you see that it contains both the other, and therefore in advising any man to the use thereof chiefly, I do not confine him to one, but to all Instruments, and therefore do not contradict my former expression. Besides, there is another great convenience which doth ensue by the degrees on the Tables frame; for, in taking the plot of a field according to the following directions by the Plain Table, you may at the same time perform the same work by the degrees on the frame of the Table, if at the drawing of every line you observe the degrees cut by the Index, and note them upon the paper. This I say is a great convenience, for at one observation you perform two works with the same labour, as in the uses of these Instruments severally will evidently appear. Many other conveniencies will redound to a Surveyor by this contrivance, which with small practice will appear of themselves. CHAP. II. Of the Theodolite, the description thereof, and the detection of an error frequently committed in the making thereof, with the manner how to correct the same. THe Theodolite is an Instrument consisting of four parts principally. The first whereof is a Circle divided into 360 equal parts, called degrees, and each degree subdivided into as many other equal parts as the largeness of the Instrument will best permit: For the diameter of this Circle, it may be of any length, but those usually made in brass are about twelve or fourteen inches, and the limb thereof divided as aforesaid into 360 degrees, and subdivided into other parts by diagonal lines drawn from the outmost and inmost concentrique Circles of the limb; in the drawing of which concentrique Circles, they use to draw them equidistant, which is erroneous, as shall appear hereafter. The second part of this Instrument is the Geometrical Square, which is described within the Circle, and the sides thereof divided into certain equal parts, but there are few of them made now with this Square, for the degrees themselves will better supply that want, it being only for taking of heights and distances. Yet if any man be desirous to have this Square upon his Instrument, there is a more convenient way to set it on then that which Master Diggs showeth, namely, upon the limb of the Instrument, the manner how is well known to the Instrument maker. The third part of this Instrument is the Box and Needle, so conveniently contrived to stand upon the centre of the Circle, upon which centre also the Index of the Instrument must turn about; and sometimes over the Box and Needle there is a Quadrant erected for the taking of heights and distances. The fourth part of this Instrument is a Socket, to be screwed on the back side of the Instrument, to set it upon a staff when you make use thereof. In the making of this Instrument, it were necessary to have two back Sights fixed at each end of one of the Diameters, for the readier laying out of any angle without moving of the Instrument. Now, forasmuch as in the dividing of the Degrees of any Circumference (as of a Quadrant, Theodolite, etc.) into Minutes, they usually draw the concentrique Circles equidistant, which is false, as Master Norwood plainly demonstrateth, pag. 81. Architecture Military: but because the way which he there showeth is trigonometrical, and sufficiently shown by him, I will pass that by, and show you another way how to perform the same Geometrically, as followeth. Let the angle BAC be a part of the circumference of any Instrument, to be divided into four equal parts by Diagonals, and let it be required to find where the concentrique Circles E F and G must be drawn; so that lines drawn from the centre A through the points E F and G, shall divide the arch BC into four equal parts. First, BD is the outward Circle of the limb of the Instrument, and HD the inward Circle, between which, the other three must be drawn concentrical (that is, upon the same centre A) but not equidistant, therefore, (by the ● Problem of the 1. Book) draw the arch of a Circle which shall pass through the points B D A, then divide the part of that arch which lies between B and D into four equal parts in E F and G, through which points draw the three Circles E F and G, which shall be the true Circles that must cross your Diagonals, to divide the limb into four equal parts, whereas, if the Circles had been equidistant, the arch would have been unequally divided and this error is frequently practised, for in the making of any Instrument, they commonly divide the distance BH or CD into four equal parts, and through them draw the concentrique Circles, whereas by the figure you see that the farther the Circles are from the centre the closer they come together: but let this suffice for the correction of this Error. CHAP. III. The description of the Circumferentor. THis Instrument hath been much esteemed by many, for portability thereof, it being usually made to contain in length about eight inches, in breadth four inches, and in thickness about three quarters of an inch; one side whereof is divided into divers equal parts, most fitly of ten or twelve in an inch; so that it may be used as the Scale of a Protractor, the Instrument itself being fitting to protract the plot on paper by help of the Needle, and the degrees of angles, and length of lines taken in the field. On the upper side of this Instrument is turned a round hole, three inches, and a half Diameter, and about half an inch deep, in which is placed a Card divided commonly into 120 equal parts or degrees, and each of those into three, which makes 360 answerable to the degrees of the Theodolite, in which Card is also a Dial drawn to find the hour of the day, and Azimuth of the Sun; within the Box, is hanged a Needle touched with a Loadstone, and covered over with a clear glass to preserve it from the weather. On the upper part of this Instrument is also described a Table of natural Sins, collected answerable to the Card in the box, that is to say, if the Card be divided but into 120 parts, the Sins must be so also; but if into 360, the Sins must be the absolute degrees of the Quadrant. To this Instrument also belongeth two Sights, one double in length to the other, the longest containing about seven inches, being placed and divided in all respects, as those hereafter mentioned in the description of the Plain Table. On the edge of the shorter Sight toward the upper part thereof, is placed a small Wyer representing the Centre of a supposed Circle, the Semidiameter whereof is the distance from the Wire to the edge of the Instrument underneath the same, which parts is imaginarily divided into sixty equal parts, and according to those divisions is the right line of divisions on the edge of the Instrument divided, and numbered by 5, 10, 15, from the perpendicular point to the end thereof: And also from the same point on the upper edge of the Instrument is perfected the degrees of the Quadrant, supplying the residue of those which could not be expressed on the long Sight, from 28 to 90 by ten. There is also belonging to these divisions a little Ruler, at one end whereof is a little hole to put it upon the wire, on the edge of the shorter Sight; and at the other end of this Ruler is placed a small Sight, directly over the siduciall edge thereof; which edge is likewise divided according to those divisions on the edge of the Instrument. To this short Sight is added a plummet to set the Instrument horizontal. And this short Ruler, with the divisions thereof, and those on the edge of the Instrument serve for taking of altitudes chief, and for the reducing of hypothenusal to horizontal lines. CHAP. IU. A Description of the Plain Table, how it hath been formerly made, and how it is now altered, it being the most absolute Instrument of any other for a Surveyor to use, in that it performeth whatsoever may be done either by the Theodolite, Circumferentor, or any other Instrument, with the same ease and exactness. THe Table itself is a Parallelogram, containing in length about fourteen inches and a half, and in breadth eleven inches: it is composed of three several boards, which may be taken asunder for ease and convenience in carriage. For the binding of these three boards fast when the Table is set together, there belongeth a jointed frame, so contrived, that it may be taken off, and put on the Table at pleasure: this frame also is to fasten a sheet of paper upon the Table, when you are to describe the plot of any field, or other enclosure by the Table. This frame must have upon it, near the inward edge, Scales of equal parts on both sides, for the speedy drawing of parallel lines upon the paper; and also for the shifting of your paper, when one sheet will not hold your whole work. Unto this Table belongeth a Ruler or Index, containing in length about sixteen inches or more, it being full as long as from angle to angle of your Table, it ought to be about two inches in breadth, and one third part of an inch in thickness. Upon this Ruler or Index two Sights must be placed; one whereof is double in length to the other, the longer containing in length about twelve inches, the other six: on the top of this shorter Sight is placed a brass pin, and also a thread and plummet to place your Instrument horizontal. Through the longer Sight must be made a slit, almost the whole length thereof, These two sights thus prepared, are to be perpendicularly erected upon the Index; in such sort, that the Wire on the top of the shorter Sight, and the slit on the longer Sight stand precisely over the fiducial edge of the Index. The space or distance of these two Sights one from the other, is to be equal to the divided part of the longer Sight. Upon the longer Sight is to be placed a Vane of brass, to be moved up and down at pleasure, through which a small hole is to be made, answerable to the slit in the same Sight, and the edge of the Vane. By these Sights thus placed on the Index there is projected the Geometrical Square, whose side is the divided part of the long Sight (or the distance between the two Sights.) In the middle of the long Sight (through the whole breadth thereof) there is drawn a line called the line of Level, dividing the side of the projected Square into two equal parts: also the same side is on this Sight divided into a hundred equal parts, which are numbered upwards and downwards, from the line of Level, by five and ten to fifty, on either side, which divisions are called the Scale. There is also on the same Sight another sort of division, representing the hypothenusal Lines of the same Square, as they increase by Unites, and are likewise numbered upwards and downwards from the line of Level, from one to twelve, by 1, 2, 3, etc. sometimes signifying 101, 102, 103, etc. these divisions show how much any hypothenusal or slope line drawn over the same Square, exceedeth the direct horizontal line, being the side of the same Square. On this Sight there is a third sort of divisions, representing the degrees of a Quadrant (or as many as the same sight is capaple to receive, which are about 25) numbered from the line of Level upward and downward by five and ten to 25, which divisions are called the Quadrant. Unto this Instrument, as unto all others belong these necessary parts, as the Socket, the Staff, the Box, and Needle, etc. ¶ According to this description, have Plain Tables formerly been made, but if unto it be added these additional parts and alterations (which make it less cumbersome than before) it will be the most exact, absolute and universal Instrument for a Surveyour that was ever yet invented. First, Let the frame be so fitted to the Table, that it may go on easily either side being upwards; so that as one side is divided into equal parts, (as in the description) the other side may have projected upon it the 180 degrees of a Semicircle, from a Centre noted in the superficies of the Table, which degrees must be numbered from the left hand towards the right (when the Centre is next to you) by five and ten to 180, and then beginning again, set 190, and so successively to 360. These degrees thus inserted are of excellent use in wet or stormy weather, when you cannot keep a sheet of paper upon your Table, either in respect of rain or wind. Also these degrees will make the Plain Table to be an absolute Theodolite, so that you may work with these degrees as if they were the degrees of a Theodolite. Secondly, Upon the Index or Ruler before spoken of, (instead of the Sights before described) let there be placed two Sights, both of one length, and back-sighted; one having a slit below, and a thread above; and the other, a slit above, and a thread below, serving to look backward and forward at pleasure without turning about the Instrument, when the Needle is at quiet. The expedition that these back-sights will make, will best appear by practice; for using these you shall need (in going about a field) to plant your Instrument but at every second angle. Thirdly, for the ready taking of heights, and the reducing of hypothenusal to horizontal lines (instead of the divisions on the Sights before mentioned) let there be projected a Tangent line along the side of the Ruler, whose divisions must touch the very edge thereof, so that a Label or Ruler of Box or Brass, which is hanged on a pin sticking in the side of one of the Back-sights, and having another small Sight at the end thereof, may move justly along the side of the Index; then (the Instrument standing horizontal) if you look through this small Sight, and by the Pin on which the Label hangeth, moving the Label too and fro, till you espy the mark you look at, then will the Label show you what Degree of the Tangent line is cut thereby. This one line thus projected upon the side of the Ruler performeth all the uses of those divided Sights, and is far better, and less cumbersome than them or a Quadrant, (such as I formerly described in Planometria) because the degrees are larger. This line of Tangents is projected on the Index from the foot of the farthermost Sight, all along the Ruler to the foot of the nethermost Sight, and up the side thereof and is numbered from 1 to 90, by 10, 20, 30, 40, 50, etc. ending at the foot of the furthermost Sight; from whence the line proceeded. The use of this line of Tangents in taking of Heights is showed in the fourth Book, & is used with the Tables of Sines and Logarithms treated of in the third Book, without which Tables, (or something equivalent thereunto) this line of Tangents will be of little use, therefore it will be convenient to have upon the Index of your Table the lines of Artificial Numbers, Sins, and Tangents, by which you may work any proportion required very speedily and exactly, so that if you be destitute of your Tables, these Lines will sufficiently help you. There is yet another way by which you may take any altitude, or reduce hypothenusal to horizontal lines, only by Vulgar Arithmetic, without the help of Tables, by having a line of equal parts divided on the edge of the Index, and another line of the same equal parts on the Label, by which lines, and Vulgar Arithmetic, an Altitude may very well be taken. Now because I intent only to show in general the use of these equal parts, I will therefore do it in this place, because I shall have occasion to speak no more thereof hereafter: The use thereof (briefly) is thus. Now for the reducing of hypothenusal to horizontal lines, having measured the hypothenusal line with your Chain, the proportion will be: As the equal parts cut on the Label, Are to the equal parts cut on the Index; So is the length of the hypothenusal line measured, To the length of the horizontal line required. I thought good to give the Reader a view of the several ways there are to perform these conclusions, leaving every man at liberty to use that which he best liketh, or all if he please, for all the lines may very well be put upon one Instrument without any confusion of lines: but the way which I shall chief insist upon in the prosecuting of this Work, shall be by the line of Tangents, as being (in my opinion) the best of all. Now when I come to show you the use of this line of Tangents, with the Tables of Sines and Logarithms in the resolving of Triangles, I will also show you how to perform the same Propositions by the lines of Artificial Numbers, Sins, and Tangents, and therefore I would advise every man to have these so necessary lines upon his Index. Fourthly, Unto this Instrument also belongeth a Box and Needle, which is to be fastened to the side of the Table by help of two screws, so that it may be taken off and put on at pleasure. In the bottom of this Box must be placed a Card divided into 360 degrees numbered (if you please) after the usual manner, from the North Eastward, but the Card by which all the Examples in this Book were framed was numbered from the North Westward by 10, 20, 30, etc. to 360, contrary to the common custom. There belongeth also to this Instrument a Socket of Brass to be screwed on the back side of the Table, into which must be put the head of the three legged Staff; this Staff ought to be jointed in the middle, so that it may be the more portable▪ For the Socket, it may be a plain one, but a Ball and Socket with an endless screw is the best of all, for by help thereof you may place the Table (or any other Instrument) either horizontal, Vertical, or in any other position. ¶ Note, that this Instrument (if made according to these directions) is the most absolute Instrument for a Surveyor to use. CHAP. V Of Chains, the several sorts thereof. OF Chains there are divers sorts, as namely, Foot Chains, each link containing a Foot or 12 Inches, and so the whole Pole or Perch will contain 16½ Links or Feet, answering to the Statute denomination. Some Chains have each Pole divided into 10 equal parts, and these are called decimal Chains, and this gross division may be convenient in some practices. The Chains now used, and most esteemed amongst Surveyors, are especially two, namely, that generally used by Master Rathborne, which hath every Perch divided into 100 Links: and that of Master Gunter, which hath four Poles divided into 100 Links, so that each Link of Master Gunter's Chain, is as long as four of Master Rathborns. Now because these Chains are most esteemed of and used by Surveyors, I will therefore make a general description of them both, leaving every man at liberty to take his choice. Of Mr. RATHBORNS Chain. THe Chain which Master Rathborne ordinarily used (as himself saith) contained in length two Statute Poles or Perches, each Pole containing in length 16½ feet, which is 198 Inches, than each Pole was divided into 10 equal parts called Primes, every of which contained in length 19● Inches; again, every of those Primes was subdivided into 10 other equal parts called Seconds, so that every of these Seconds contained in length 1 49/50 Inch, so that the whole Pole, Perch, Unite, or Commencement (as he calleth it) was divided into 130 equal parts or Links, called Seconds. The Chain (or one Pole thereof) being thus divided, at the end of every 50 Links or half Pole, let a large Curtain ring be fastened, so shall you have in a whole Chain of two Perches long, three of these Rings, the middlemost being the division of the two Poles. Then at the end of every Prime, that is, at the end of every ten Links, let a smaller Curtain Ring be fastened. By this distinction of Rings, the Chain is divided into these three denominations, Unites, Primes, and Seconds, whose Characters are these, ◯ · ·, so that if you would express 40 Unites, 8 Primes, and 7 Seconds, they are thus to be written, 408̇7̇, by which you may perceive that those Figures which have no pricks over them are Unites or Intigers, and the figure under the first point Primes, and under the next Seconds: so also, three Unites, seven Primes, and two Seconds, will stand thus, 37̇2̇. Besides these divisions, Master Rathborn for his own use, sewed at the end of every two Primes and a half (which is a quarter of a Pole) a small red cloth, and at every seven Primes and a half (being three quarters of a Pole) the like of yellow, or other discernible colour, which much helped him in the ready reckoning of the several Rings upon the Chain, remembering this Rule: That if it be the next Ring short of the Red, it is two Primes, if the next over three, if the next short of the yellow, seven Primes; if the next over eight; if the next short of the great half Ring it is four, the next over six: and if the next short of the middle great Ring, it is nine, and if the next over one. ¶ But here is to be noted, that if you use this distinction by colours, you must always work with one end of the Chain from you. This Chain being thus divided and marked, you have every whole Pole equal to ten Primes, or 100 Seconds: every three quarters of a Pole, equal to seven Primes and a half, or 75 Seconds: every half Pole equal to five Primes, or 50 Seconds: and lastly, every quarter of a Pole equal to two Primes and a half, or 25 Seconds. And here is to be noted, that in the ordinary use of this Chain, for measuring and platting, you need take notice only of Unites and Primes, which is exact enough for ordinary use, but in case that separation or division of Lands into several parts, you may make use of Seconds. Of Mr. GUNTER'S Chain. AS every Pole of Master Rathborns Chain was divided into 100 Links, so Master Gunter's whole Chain (which is always made to contain four Poles) is divided into 100 Links, one of these Links being four times the length of the other. Now if this Chain be made according to the Statute, each Perch to contain 16½ Feet, than each Link of this Chain will contain 7 Inches, and 92/100 of an Inch, and the whole Chain 729 Inches, or 66 Foot. In measuring with this Chain, you are to take notice only of Chains and Links, as saying such a line measured by the Chain contains 72 Chains 48 Links, which you may express more briefly thus, 72,48, and these are all the Denominations which are necessary to be taken notice of in Surveying of Land. For the ready counting of the Links of this Chain, there ought to be these distinctions, namely, In the middle thereof, which is at two Poles end, let there be hanged a large Ring, or rather a plate of brass like a Rhombus, so is the whole Chain (by this plate) divided into two equal parts. Secondly, Let each of these two parts be divided into two other equal parts, by smaller Rings or Circular plates of brass, so shall the whole Chain be divided into four equal parts or Perches, each Perch containing 25 Links. Thirdly, At every ten Links let be fastened a lesser Ring than the former, or else a Plate of some other fashion, as a Semicircle, or the like. And lastly, at every fift link (if you please) may be fastened other marks, so by this means you shall most easily and exactly count the Links of your Chain without any trouble. The Chain being thus distinguished, it mattereth not which end thereof be carried forward, because the notes of distinction proceed alike on both sides from the middle of the Chain. ¶ Here note, that in all the examples in this Book, the lines are supposed to be measured by this four Pole Chain of Master Gunter, it being the best of any other: the manner how to cast up the content of any plot measured therewith shall be hereafter taught in its due place. Cautions to be observed in the use of any Chain. IN measuring a large distance with your Chain, you may casually mistake or miss a Chain or two in keeping your account, from whence will ensue a considerable error: Also in measuring of distances (when you go not along by a hedge side) you can hardly keep your Instrument, Chain, and Mark, in a right line, which if you do not, you must necessarily make your measured distance greater than in reality it is. For the avoiding of either of these mistakes, you ought to provide ten small sticks or Arrows, which let him that leadeth the Chain carry in his hand before, and at the end of every Chain, stick one of these Arrows into the ground, which let him that followeth the Chain take up, so going on till the whole number of Arrows be spent, and then you may conclude that you have measured ten Chains, without any further trouble, and these ten Chains (if the distance you are to measure be large) you may call a Change, and so you may denominate every large distance by Changes, Chains, and Links. Or you may at the end of every ten Chains set up another kind of stick, by which (standing at the Instrument) you may see whether your eye, the stick, and the Mark to which you are to measure be in a right line or not, and accordingly guide those that carry the Chain, with the more exactness to direct it to the Mark intended. How to reduce any number of Chains and Links, into Feet. IN the practice of many Geometrical Conclusions, as in the taking of Heights and Distances, hereafter taught, it is requisite to give your measure (in such cases) in Feet or Yards, and not in Poles or Perches; yet because your Chain is the most necessary Instrument to measure withal, I thought it convenient in this place to show you how to reduce any number of Chains and Links into Feet, which is thus. Multiply your number of Chains and Links together as one whole number, by 66, cutting off from the product the two last figures towards the right hand, so shall the rest of the product be Feet, and the two figures cut off shall be hundred parts of a Foot. EXAMPLE. Let it be required to know how many Feet are contained in 5 Chains, 32 Links. First, Set down your 5 Chains, 32 Links as is before taught, and as you see in the first Example, with a Comma between the Chains and Links, then multiplying this 5 Chains, 32 Links by 66, the product will be 35112, from which, cut off the two last figures toward the right hand with a Comma, then will the number be 351,12, which is 351 Feet and 12/100 parts of a foot, and so many Feet are contained in 5 Chains, 32 Links. Example I. 5,32 66 3192 3192 351,12 Example II. 9,05 66 5430 5430 597,30 But let the number of Chains be what they will, if the number of Links be less than 10, as in the second Example it is 9 Chains 5 Links, you must place a cipher before the five Links as there you see, and then multiplying that number (viz. 9,05) by 66, the product will be 59730, from which taking the two last figures, there will remain 597 Feet, and ●…/100 parts of a Foot. The like may be done for any other number of Chains and Links whatsoever. According to these Examples is made the Table following, which showeth how many Feet are contained in any number of Chains and Links, from 5 Links to 10 Chains, for every fift Link, which is sufficient for ordinary use, by which Table you may see that in 6 Chains 40 Links, is contained 422 Feet, and 40/100 of a Foot, Also in 5 Chains 55 Links is contained 366 Feet, and 30/100 parts of a Foot: and so of any other. A TABLE showing how many Feet, and parts of a Foot are contained in any number of Chains and Links between five Links and eight Chains. 0 1 2 3 4 5 6 7 0 66,00 132,00 198,00 264,00 330,00 396,00 462,00 5 3,30 69,30 135,30 201,30 267,30 333,30 399,30 465,30 10 6,60 72,60 138,60 204,60 270,60 336,60 402,60 468,60 15 9,90 75,90 141,90 207,90 273,90 339,90 405,90 471,90 20 13,20 79,20 145,20 211,20 277,20 343,20 409,20 475,20 25 16,50 82,50 148,50 214,50 280,50 346,50 412,50 478,50 30 19,80 85,80 151,80 217,80 283,80 349,80 415,80 481,80 35 23,10 89,10 155,10 221,10 287,10 353,10 419,10 485,10 40 26,40 92,40 158,40 224,40 290,40 356,40 422,40 488,40 45 29,70 95,70 161,70 227,70 293,70 359,70 425,70 491,70 50 33,00 99,00 165,00 231,00 297,00 363,00 429,00 495,00 55 36,30 102,30 168,30 234,30 300,30 366,30 432,30 498,30 60 39,60 105,60 171,60 237,60 303,60 369,60 435,60 501,60 65 42,90 108,90 174,90 240,90 306,90 372,90 438,90 504,90 70 46,20 112,20 178,20 244,20 310,20 376,20 442,20 508,20 75 49,50 115,50 181,50 247,50 313,50 379,50 445,50 511,50 80 52,80 118,80 184,80 250,80 316,80 382,80 448,80 514,80 85 56,10 122,10 188,10 254,10 320,10 386,10 452,10 518,10 90 59,40 125,40 191,40 257,40 323,40 389,40 455,40 521,40 95 62,70 128,70 194,70 260,70 326,70 392,70 458,70 524,70 CHAP. VI Of the Protractor. The Scale being thus divided, on the middle of the line AB, as at H, describe the Semicircle EGF, which divide into two Quadrants in the point G, by help of the perpendicular HG: then divide each of those Quadrants into 90 equal parts called degrees, so shall the whole Semicircle contain 180 degrees, which must be numbered by 10, 20, 30, 40, etc. to 180, from E by G to F, and the same way also from 180 to 360, as you see done in the Figure, the numbers of the first Semicircle from 00 to 180 being for the East side of the Protractor, and the other numbers from 180 to 360 for the West side. Now you are to note, that the line AB always representeth the Meridian line, and is sometimes noted with the letters S and N, for South and North, but than it is necessary that the Protractor be divided on either side the plate, which this double numbering avoideth: for the line AB being taken for the Meridian in general, the Semicircle of the Protractor may be turned any way (either upward or downward) and so one Semicircle being divided will be sufficient; yet if any man be desirous, he may have it made according to his own fancy, but this manner of numbering (in my opinion) is the best, it being most agreeable to your Instruments. To use with this Protractor in protracting, you must provide a fine needle, put into a piece of Box or Ivory neatly turned, this will serve to fix in your centre, note your degrees, and for other uses in drawing your Plot, and is called a Protracting pin. CHAP. VII. Of Scales. FOr the ready laying down of lines and angles according to any assigned quantity, you must provide divers Scales. The Scales now ordinarily used by Surveyors, are principally two: First, of equal parts, for the protracting of lines: and Secondly, of Chords, for the protracting of angles. Unto these may be added, Thirdly, a diagonal Scale, which is (indeed) no other than a Scale of equal parts more scrupulously divided. If you desire a convenient Scale, let it be made in this manner, to contain in length about 8 or 9 Inches, & in breadth one Inch and a quarter: on one side thereof let be placed divers Scales, as of 10, 11, 12, 16, 20, 24, and 30 in an Inch. ¶ Here is to be noted, that when I say a Scale of 12 in an Inch, you are to understand a part of a line divided into 10 equal parts, 12 of which parts would make an Inch, and the like is to be understood of any other number of equal parts whatsoever. On the same side of the Ruler let be placed a line of Chords extended up to 90, and numbered as you see in the figure, by 10, 20, 30, etc. to 90. This Scale will be of good use for many purposes, as to divide the circumference of a Circle, and to protract angles, in some cases better than the Protractor. On the other side of the Ruler let be drawn a diagonal Scale, of 10 in an Inch, which will be an excellent Scale for large Plots, out of which you may very well take the hundred part of an Inch, and this Scale will agree with your four Pole Chain exceeding well, for as your whole Chain contains 100 Links, so each Inch of this Scale contains 100 parts, so that out of it you may take any number measured by your Chain, to a Link, and lay it down upon paper. You may also have half an Inch divided into 100 parts, which Scale will be of good use also to lay down a small Plot. These Scales are many times put upon the Index of the Plain Table, because they should be ready at hand when you survey by the Table, and plot your work as you go; but if you use the degrees on the Frame of the Table, or the Circumferentor, and keep your account in a Book, than I would advise you to have your Scale of Brass or Box neatly and exactly divided. To use with this Scale, you must provide a pair of neat Compasses of Brass, with steel points, filled very small, and also a neat pair of Compasses with three points, and Screws to alter the points, so that you may draw lines or Circles with black lead, or any coloured Ink, which will be very necessary and convenient in beautifying of your Plots after Protraction. CHAP. VIII. Of a Field-Book. IT will be sufficient in this place only to describe the manner how a Field-Book ought to be ruled: Let the Book contain any quantity of paper, more or less, and in what volume you please. Let it be ruled, towards the left Margin of every page, with five lines in red ink, so shall you have four Columns, in the first whereof you must note down the degrees cut either by the Index on the frame of the Table, or else by the Needle on the Card, at every angle you observe, and the second Column is to note the minutes or parts of a Degree, for you are to note, that every degree on the frame of the Table, or in the Card of the Circumferentor, is supposed to be divided into 60 other parts called Minutes, which cannot be expressed by reason of the smallness of the Instruments, and therefore must only be estimated, yet if your Instrument be large enough, you may have each degree divided into 3 equal parts, so shall every part contain 20 minutes. The other two Columns serve to note down the lengths measured by your Chain, as the Chains & Links. The manner how a Field-Book ought to be ruled. Degrees. Minute's Chains Links. 326 45 16 87 Now suppose that making any observation in the Field either with the Degrees on the frame of the Table, or with the Circumferentor, and that observing any angle, (as is hereafter taught) you find the Index of the Plain Table, or the Needle in the Circumferentor, to cut 326 degrees, 45 minutes, these 326 degrees must be set down in the first Column of your Field-Book, and the 45 minutes in the second Column, as you see bear done. Also if you measure any length in the Field with your Chain, as suppose some distance measured to contain 16 Chains, 87 Links, the 16 Chains must be set in the third Column, and the 87 Links in the fourth Column, under their respective Titles, as you see here done. CHAP. IX. Of Instruments for Reducing of Plots. FOr the reducing of Plots from one form to another, there hath been divers Instruments invented by divers men. One that performeth that work very well, is a Ruler, having fixed at each end thereof a Semicircle divided into degrees, and another Ruler having two Semicircles to move thereon, upon the centres of all these Semicircles there are thin rulers of Brass to move from angle to angle of your Plot: but the manner of working by these Semicircles being very tedious, I pass it over. Another way is by having certain proportional Scales upon one and the same Ruler as Master Rathborn describeth, but this I shall also wave, and likewise that which I described in Planometria, as being too particular The best and most absolute is a Parallelogram, the making whereof is well known to the Instrument-maker. The end of the Second Book. THE COMPLETE SURVEYOR. The Third Book. THE ARGUMENT. THis Third Book is as it were a Key to those that follow, the subject whereof is Trigonometry. Now forasmuch as the whole Art of measuring heights and distances, and plotting and protracting of Land, and all other lineal and superficial dimensions are grounded upon the resolution of Plain Triangles, I hold it convenient (before I come to the practice of Surveying, or to show the use of any Instrument in taking of heights and distances) to say something concerning Plain Triangles (at least so much as is necessary for a Surveyor to know) though that subject be already handled by divers able Mathematicians already, whose Works are extant: viz. Pitiscus, Snelius, the Lord Nepair, Master Gunter, Master Norwood, Master Gellibrand, etc. Now because the readiest way of resolving Triangles is by Signs, Tangents, and Logarithmes, I have therefore added brief Tables for that purpose, viz. a Table of Sines to every tenth minute of the Quadrant, and a Table of Logarithmes from 1, to 1000, which will be large enough for ordinary use in Surveying, but those who desire to make a further scrutiny into Trigonometry, may peruse the forementioned Authors. In this Book I have only insisted upon such Cases as may come in use in Surveying, and therefore have omitted divers, yet those which I have insisted on, are performed both by the Tables following in this Book, and also by the Lines of Artificial Numbers, Sins and Tangents before spoken of in the description of the Index of the Plain Table in the last Book. trigonometry. CHAP. I. The Elplanation and Use of the Table of SINES. BEfore I come to the mensuration of Triangles, it will be necessary to explain and show the use of the Tables of Sines and Logarithms following, by which Tables the sides or angles of right lined Triangles may be readily and exactly measured, so that in any plain Triangle, if there be any three parts thereof given, a fourth may be easily discovered. The Table of Sines consisteth of two Rows or Columns, the first whereof showeth the Degrees and Minutes of the Quadrant, having over the head thereof these two letters, D. M, standing for Degrees and Minutes: In the second Column is the Artificial Sins answering to every Degree, and 10th Minute of the Quadrant, having the word Sine over the head thereof. The use of this Table will appear by the following Propositions. PROP. I. Any Degree and Minute being given, to find the Sine thereof. FIrst, Seek the Degree and minute in the first Column of the Table, under D. M. and right against it, in the next Column towards the right hand, under the word Sine, you shall have your desire. EXAMPLE. I. Suppose it were required to find the Sine of 20 degrees, First, you must seek 20 in the first Column of the Table under D. M. and right against 20 in the second Column under the word Sine, you shall find 9,534052, which is the Sine of 20 Degrees. In the same manner you shall find the Sine of 50 degrees to be 9,884254, and the Sine of 76 degrees to be 9,986904. EXAMPLE. II. Let it be required to find the Sine of 40 degrees, 30 minutes. First, you must find 40 30 (which is 40 degrees 30 minutes) in the first Column, under the letters D. M. and against it you shall find 9,812544, which is the Sine of 40 degrees, 30 minutes. Also the Sine of 62 degrees 10 minutes, will be found to be 9,946604, and the Sine of 86 degrees 30 minutes will be 9,999189, and in this manner may you find the artificial Sine of any number of Degrees and minutes expressed in the Table. PROP. II. Any Sine being given, to find the number of degrees and minutes thereunto belonging. EXAMPLE. LEt 9,866470 be a Sine given, and let it be required to find the degree and minute of the Quadrant answering thereunto. First, seek in the second Column amongst the Sins for 9,866470, and against it (on the left hand) you shall find 47 degrees 20 minutes, which is the arch of the Quadrant answering thereunto. Again, Let it be required to find the arch answering to this Sine 9,821264, having found 9,821264 in the second Column under the word Sine, against it you shall find 41 degrees 30 min. and that is the arch of degree answering thereunto. ¶ But in case you have a number given which you cannot exactly find in the Table, you must then instead thereof, take the nearest in the Table. As if your number given were 9,675859, if you look in the Table for this number it cannot be found there, but the nearest thereunto is 9,676328, which is the Sine of 28 degrees 20 minutes, which you must take instead thereof. The Table of Sines. D. M. Sines 0 0 0,000000 10 7,463726 20 7,764754 30 7,940842 40 8,065776 50 8,162681 1 0 8,241855 10 8,308794 20 8,366777 30 8,417919 40 8,463665 50 8,505045 2 0 8,542819 10 8,577566 20 8,609734 30 8,639679 40 8,667689 50 8,693998 3 0 8,718800 10 8,742259 20 8,764511 30 8,785675 40 8,805852 50 8,825130 4 0 8,843584 10 8,861283 20 8,878285 30 8,894643 40 8,910404 50 8,925609 5 0 8,940296 10 8,954499 20 8,968249 30 8,981573 40 8,994497 50 9,007044 6 0 9,019235 10 9,031089 20 9,042625 30 9,053859 40 9,064806 50 9,075480 7 0 9,085894 10 9,096062 20 9,105992 30 9,115698 40 9,125187 50 9,134470 8 0 9,143555 10 9,152451 20 9,161164 30 9,169702 40 9,178072 50 9,186280 9 0 9,194332 10 9,202234 20 9,209992 30 9,217609 40 9,225092 50 9,232444 10 0 9,239670 10 9,246795 20 9,253761 30 9,260633 40 9,267395 50 9,274049 11 0 9,280599 10 9,287048 20 9,293399 30 9,299655 40 9,305819 50 9,311899 12 0 9,317879 10 9,323780 20 9,319599 30 9,335337 40 9,340996 50 9,346579 13 0 9,352088 10 9,357524 20 9,362889 30 9,368185 40 9,373414 50 9,378577 14 0 9,383675 10 9,388711 20 9,393685 30 9,398600 40 9,403455 50 9,408254 15 0 9,412996 10 9,417684 20 9,422317 30 9,426899 40 9,431429 50 9,435918 16 0 9,440338 10 9,444720 20 9,449054 30 9,453342 40 9,457584 50 9,461782 17 0 9,465935 10 9,460446 20 9,474115 30 9,478142 40 9,482128 50 9,486075 18 0 9,489982 10 9,493851 20 9,497682 30 9,501476 40 9,505234 50 9,508955 19 0 9,512642 10 9,516294 20 9,519911 30 9,523495 40 9,527046 50 9,530565 20 0 9,534052 10 9,537507 20 9,540931 30 9,544325 40 9,547689 50 9,551024 21 0 9,554329 10 9,557606 20 9,560855 30 9,564075 40 9,567269 50 9,570435 22 0 9,573575 10 9,576689 20 9,579777 30 9,582840 40 9,585877 50 9,588890 23 0 9,591878 10 9,594842 20 9,597783 30 9,600700 40 9,603594 50 9,606465 24 0 9,609313 10 9,612148 20 9,614944 30 9,617727 40 9,620488 50 9,623229 25 0 9,625948 10 9,628647 20 9,631326 30 9,633984 40 9,636623 50 9,639242 26 0 9,641842 10 9,644423 20 9,646984 30 9,649527 40 9,652052 50 9,654558 27 0 9,657047 10 9,659517 20 9,661970 30 9,664406 40 9,666824 50 9,669225 28 0 9,671609 10 9,673977 20 9,676328 30 9,678663 40 9,680982 50 9,683284 29 0 9,685571 10 9,687842 20 9,690098 30 9,692339 40 9,694564 50 9,696774 30 0 9,698970 10 9,701151 20 9,703317 30 9,705469 40 9,707606 50 9,709730 31 0 9,711839 10 9,713935 20 9,716017 30 9,718085 40 9,720140 50 9,722181 32 0 9,724210 10 9,726225 20 9,728227 30 9,730216 40 9,732193 50 9,734157 33 0 9,736109 10 9,738048 20 9,739975 30 9,741889 40 9,743792 50 9,745683 34 0 9,747562 10 9,749429 20 9,751284 30 9,753128 40 9,754960 50 9,756781 35 0 9,758591 10 9,760390 20 9,762177 30 9,763954 40 9,765720 50 9,767474 36 0 9,769219 10 9,770952 20 9,772675 30 9,774388 40 9,776090 50 9,777781 37 0 9,779463 10 9,781134 20 9,782796 30 9,784447 40 9,786088 50 9,787720 38 0 9,789342 10 9,790954 20 9,792557 30 9,794149 40 9,795733 50 9,797307 39 0 9,798872 10 9,800427 20 9,801973 30 9,803510 40 9,805038 50 9,806557 40 0 9,808067 10 9,809569 20 9,810061 30 9,812544 40 9,814019 50 9,815485 41 0 9,816943 10 9,818392 20 9,819832 30 9,821264 40 9,822688 50 9,824104 42 0 9,825511 10 9,826910 20 9,828301 30 9,829683 40 9,831058 50 9,832425 43 0 9,833783 10 9,835134 20 9,836477 30 9,837812 40 9,839140 50 9,840459 44 0 9,841771 10 9,843079 20 9,844372 30 9,845662 40 9,846944 50 9,848218 45 0 9,849485 10 9,850745 20 9,851997 30 9,853242 40 9,854480 50 9,855710 46 0 9,856934 10 9,858150 20 9,859360 30 9,860562 40 9,861757 50 9,862946 47 0 9,864127 10 9,865302 20 9,866470 30 9,867631 40 9,868785 50 9,869933 48 0 9,871073 10 9,872208 20 9,873335 30 9,874456 40 9,875571 50 9,876678 49 0 9,877780 10 9,878875 20 9,879963 30 9,881045 40 9,882121 50 9,883191 50 0 9,884254 10 9,885311 20 9,886361 30 9,887406 40 9,888444 50 9,889476 51 0 9,890503 10 9,891522 20 9,892536 30 9,893544 40 9,894546 50 9,865542 52 0 9,896532 10 9,897516 20 9,898494 30 9,899467 40 9,900433 50 9,901391 53 0 9,902349 10 9,903298 20 9,90424 30 9,905179 40 9,906111 50 9,907037 54 0 9,907958 10 9,908873 20 9,909782 30 9,910686 40 9,911584 50 9,912477 55 0 9,913364 10 9,914246 20 9,915123 30 9,915994 40 9,916859 50 9,917719 56 0 9,918574 10 9,919424 20 9,920268 30 9,921107 40 9,921940 50 9,922768 57 0 9,923591 10 9,924409 20 9,925222 30 9,926029 40 9,926831 50 9,927628 58 0 9,928420 10 9,929207 20 9,929989 30 9,930766 40 9,931537 50 9,932304 59 0 9,933066 10 9,933822 20 9,934574 30 9,935320 40 9,936062 50 9,936799 60 0 9,937531 10 9,938257 20 9,938980 30 9,939697 40 9,940409 50 9,941116 61 0 9,941819 10 9,942517 20 9,943210 30 9,943898 40 9,944582 50 9,945261 62 0 9,945935 10 9,946604 20 9,947269 30 9,947929 40 9,948584 50 9,949235 63 0 9,949881 10 9,950522 20 9,951159 30 9,951791 40 9,952419 50 9,953042 64 0 9,953660 10 9,954274 20 9,954883 30 9,955488 40 9,956088 50 9,956684 65 0 9,957276 10 9,957862 20 9,958445 30 9,959023 40 9,959596 50 9,960165 66 0 9,960730 10 9,961290 20 9,961846 30 9,962398 40 9,962945 50 9,963488 67 0 9,964026 10 9,964560 20 9,965090 30 9,965615 40 9,966136 50 9,966653 68 0 9,967166 10 9,967674 20 9,968178 30 9,968678 40 9,969173 50 9,969665 69 0 9,970152 10 9,970634 20 9,971112 30 9,971588 40 9,972058 50 9,972524 70 0 9,972986 10 9,973443 20 9,973897 30 9,974346 40 9,974792 50 9,975233 71 0 9,975670 10 9,976103 20 9,976532 30 9,977956 40 9,977377 50 9,977794 72 0 9,978206 10 9,978615 20 9,979019 30 9,979419 40 9,979816 50 9,980208 73 0 9,980596 10 9,980980 20 9,981361 30 9,981737 40 9,982109 50 9,982477 74 0 9,982842 10 9,983202 20 9,983558 30 9,983910 40 9,984259 50 9,984603 75 0 9,984943 10 9,985280 20 9,985613 30 9,985942 40 9,986266 50 9,986587 76 0 9,986904 10 9,987217 20 9,987526 30 9,987832 40 9,988133 50 9,988430 77 0 9,988724 10 9,989014 20 9,989299 30 9,989581 40 9,989860 50 9,990134 78 0 9,990404 10 9,990671 20 9,990934 30 9,991193 40 9,991448 50 9,991699 79 0 9,991947 10 9,992190 20 9,992430 30 9,992666 40 9,992898 50 9,993127 80 0 9,993351 10 9,993572 20 9,993789 30 9,994003 40 9,994212 50 9,994418 81 0 9,994620 10 9,994818 20 9,995012 30 9,995203 40 9,995390 50 9,995573 82 0 9,995753 10 9,995928 20 9,996100 30 9,996269 40 9,996433 50 9,996594 83 0 9,996751 10 9,996904 20 9,997053 30 9,997199 40 9,997341 50 9,999998 84 0 9,997614 10 9,997732 20 9,997873 30 9,997996 40 9,998106 50 9,998232 85 0 9,998344 10 9,998453 20 9,998558 30 9,998659 40 9,998757 50 9,998851 86 0 9,998941 10 9,999927 20 9,999110 30 9,999189 40 9,999265 50 9,999336 87 0 9,999404 10 9,999469 20 9,999529 30 9,999586 40 9,999640 50 9,999689 88 0 9,999735 10 9,999778 20 9,999816 30 9,999851 40 9,999882 50 9,999910 89 0 9,999934 10 9,999954 20 9,999971 30 9,999983 40 9,999993 50 9,999998 CHAP. II. The Explanation and Use of the Table of LOGARITHMS. THe Table of Logarithms following consisteth of two Rows or Columns, the first of which (namely that towards the left hand, having the word Num. at the head thereof) containeth all absolute numbers increasing by a Unite in continual proportion from 1, to 1000 In the other Column is placed the Logarithms of those absolute numbers; which Logarithms are numbers so fitted to proportional numbers, that themselves retain equal differences. By this Table, the Logarithme of any absolute number under 1000, maybe readily found: Or if any Logarithme, whose absolute number exceedeth not 1000, be given, this Table will plainly discover what absolute number answereth thereunto. The use of this Table will appear by the Propositions following. PROP. I. A number being given, to find the Logarithme thereof. LEt it be required to find the Logarithm of 223, First, seek 223 in the first Column of the Table under the word Num. and against it in the second Column you shall find 2,348305. which is the Logarithm thereof. Also, Let it be required to find the Logarithm of 629, if you seek 629 in the first Column, against it in the second you shall find 2,798651, which is the Logarithm thereof. PROP. II. A Logarithme being given, how to find the absolute number thereunto belonging. LEt 2,731589 be a Logarithm given, whose absolute number you require: you must first seek this number in the second Column of the Table, under the word Logar, against which you shall find 539, which is the absolute number answering to that Logarithme. ¶ But in this Table, as in the Table of Sines, if you cannot find the direct Logarithm which you look for, in the Table, you must take the nearest thereunto. The Table of Logarithms Num. Logarith. 1 0,000000 2 0,301030 3 0,477121 4 0,602060 5 0,698970 6 0,778151 7 0,845098 8 0,903090 9 0,954242 10 1,000000 11 1,041393 12 1,079181 13 1,113943 14 1,146128 15 1,176091 16 1,204120 17 1,230449 18 1,255272 19 1,278754 20 1,301030 21 1,322219 22 1,342423 23 1,361728 24 1,380211 25 1,397940 26 1,414973 27 1,431364 28 1,447158 29 1,462398 30 1,477121 31 1,491362 32 1,505150 33 1,518514 34 1,531479 35 1,544068 36 1,556302 37 1,568202 38 1,579783 39 1,591065 40 1,602060 41 1,612784 42 1,623249 43 1,633468 44 1,643453 45 1,653212 46 1,662758 47 1,672098 48 1,681241 49 1,690196 50 1,698970 51 1,707570 52 1,716003 53 1,724276 54 1,732394 55 1,740363 56 1,748188 57 1,755875 58 1,763428 59 1,770852 60 1,778151 61 1,785330 62 1,792392 63 1,799341 64 1,806180 65 1,812913 66 1,819544 67 1,826075 68 1,832509 69 1,838849 70 1,845098 71 1,851258 72 1,857332 73 1,863323 74 1,869232 75 1,875061 76 1,880814 77 1,886491 78 1,892095 79 1,897627 80 1,903089 81 1,908485 82 1,913814 83 1,919078 84 1,924279 85 1,929419 86 1,934498 87 1,939519 88 1,944483 89 1,949390 90 1,954242 91 1,959041 92 1,963788 93 1,968483 94 1,973128 95 1,977724 96 1,982271 97 1,986772 98 1,991226 99 1,995635 100 2,000000 101 2,004321 102 2,008600 103 2,012837 104 2,017033 105 2,021189 106 2,025306 107 2,029384 108 2,033424 109 2,037426 110 2,041393 111 2,045323 112 2,049218 113 2,053078 114 2,056905 115 2,060698 116 2,064458 117 2,068186 118 2,071882 119 2,075547 120 2,079181 121 2,082785 122 2,086359 123 2,089905 124 2,093422 125 2,096910 126 2,100371 127 2,103804 128 2,107209 129 2,110589 130 2,113943 131 2,117271 132 2,120574 133 2,123852 134 2,127105 135 2,130334 136 2,133539 137 2,136721 138 2,139879 139 2,143015 140 2,146128 141 2,149219 142 2,152288 143 2,155336 144 2,158362 145 2,161368 146 2,164353 147 2,167317 148 2,170262 149 2,173186 150 2,176091 151 2,178977 152 2,181844 153 2,184691 154 2,187521 155 2,190332 156 2,193125 157 2,195899 158 2,198657 159 2,201397 160 2,204119 161 2,206826 162 2,209515 163 2,212187 164 2,214844 165 2,217484 166 2,220108 167 2,222716 168 2,225309 169 2,227887 170 2,230449 171 2,232996 172 2,235528 173 2,238046 174 2,240549 175 2,243038 176 2,245513 177 2,247973 178 2,250420 179 2,252853 180 2,255273 181 2,257679 182 2,260071 183 2,262451 184 2,264818 185 2,267172 186 2,269513 187 2,271842 188 2,274158 189 2,276462 190 2,278754 191 2,281033 192 2,283301 193 2,285557 194 2,287802 195 2,290035 196 2,292256 197 2,294466 198 2,296665 199 2,298853 200 2,301029 201 2,303196 202 2,305351 203 2,307496 204 2,309630 205 2,311754 206 2,313867 207 2,315970 208 2,318063 209 2,320146 210 2,322219 211 2,324282 212 2,326336 213 2,328379 214 2,330414 215 2,332438 216 2,334454 217 2,336459 218 2,338456 219 2,340444 220 2,342227 221 2,344392 222 2,346353 223 2,348305 224 2,350248 225 2,352183 226 2,354108 227 2,356026 228 2,357935 229 2,359835 230 2,361728 231 2,363612 232 2,365488 233 2,367356 234 2,369216 235 2,371068 236 2,372912 237 2,374748 238 2,376577 239 2,378398 240 2,380211 241 2,382017 242 2,383815 243 2,385606 244 2,387389 245 2,389166 246 2,390935 247 2,392697 248 2,394452 249 2,396199 250 2,397940 251 2,399674 252 2,401401 253 2,403121 254 2,404834 255 2,406540 256 2,408239 257 2,409933 258 2,411619 259 2,413299 260 2,414973 261 2,416641 262 2,418301 263 2,419956 264 2,421604 265 2,423246 266 2,424882 267 2,426511 268 2,428135 269 2,429752 270 2,431364 271 2,432969 272 2,434569 273 2,436163 274 2,437751 275 2,439333 276 2,440909 277 2,442479 278 2,444045 279 2,445604 280 2,447158 281 2,448706 282 2,450249 283 2,451786 284 2,453318 285 2,454845 286 2,456366 287 2,459889 288 2,459392 289 2,460898 290 2,462398 291 2,463893 292 2,465383 293 2,466868 294 2,468347 295 2,469822 296 2,471292 297 2,472756 298 2,474216 299 2,475671 300 2,477121 301 2,478566 302 2,480007 303 2,481443 304 2,482874 305 2,484299 306 2,485721 307 2,487138 308 2,488551 309 2,489958 310 2,491362 311 2,492760 312 2,494155 313 2,495544 314 2,496929 315 2,498311 316 2,499687 317 2,501059 318 2,502427 319 2,503791 320 2,505149 321 2,506505 322 2,507856 323 2,509203 324 2,510545 325 2,511883 326 2,513218 327 2,514548 328 2,515874 329 2,517196 330 2,518514 331 2,519828 332 2,521138 333 2,522444 334 2,523746 335 2,525045 336 2,526339 337 2,527629 338 2,528916 339 2,530199 340 2,531479 341 2,532754 342 2,534026 343 2,535294 344 2,536558 345 2,537819 346 2,539076 347 2,540329 348 2,541579 349 2,542825 350 2,544068 351 2,545307 352 2,546543 353 2,547775 354 2,549003 355 2,550228 356 2,551449 357 2,552668 358 2,553883 359 2,555094 360 2,556303 361 2,557507 362 2,558709 363 2,559907 364 2,561101 365 2,562293 366 2,563481 367 2,564666 368 2,565848 369 2,567026 370 2,568202 371 2,569374 372 2,570543 373 2,571709 374 2,572872 375 2,574031 376 2,575188 377 2,576341 378 2,577492 379 2,578639 380 2,579784 381 2,580925 382 2,582063 383 2,583199 384 2,584331 385 2,585461 386 2,586587 387 2,587711 388 2,588832 389 2,589949 390 2,591065 391 2,592177 392 2,593286 393 2,594393 394 2,595496 395 2,596597 396 2,597695 397 2,598790 398 2,599883 399 2,600973 400 2,602059 401 2,603144 402 2,604226 403 2,605305 404 2,606381 405 2,607455 406 2,608526 407 2,609594 408 2,610660 409 2,611723 410 2,612784 411 2,613842 412 2,614897 413 2,615950 414 2,617000 415 2,618048 416 2,619093 417 2,620136 418 2,621176 419 2,622214 420 2,623249 421 2,624282 422 2,625312 423 2,626340 424 2,627366 425 2,628389 426 2,629409 427 2,630428 428 2,631444 429 2,632457 430 2,633468 431 2,634477 432 2,635484 433 2,636488 434 2,637489 435 2,638489 436 2,639486 437 2,640481 438 2,641475 439 2,642465 440 2,643453 441 2,644439 442 2,645422 443 2,646404 444 2,647383 445 2,648360 446 2,649335 447 2,650308 448 2,651278 449 2,652246 450 2,653213 451 2,654177 452 2,655138 453 2,656098 454 2,657056 455 2,658011 456 2,658965 457 2,659916 458 2,660865 459 2,661813 460 2,662758 461 2,663701 462 2,664642 463 2,665581 464 2,666518 465 2,667453 466 2,668386 467 2,669317 468 2,670246 469 2,671173 470 2,672098 471 2,673021 472 2,673942 473 2,674861 474 2,675778 475 2,676694 476 2,677607 477 2,678518 478 2,679428 479 2,680336 480 2,681241 481 2,682145 482 2,683047 483 2,683947 484 2,684845 485 2,685742 486 2,686636 487 2,687529 488 2,688419 489 2,689309 490 2,690196 491 2,691081 492 2,691965 493 2,692847 494 2,693727 495 2,694605 496 2,695482 497 2,696356 498 2,697229 499 2,698101 500 2,698970 501 2,699830 502 2,700704 503 2,701568 504 2,702480 505 2,703291 506 2,704151 507 2,705008 508 2,705864 509 2,706718 510 2,707570 511 2,708421 512 2,709269 513 2,710117 514 2,710963 515 2,711807 516 2,712649 517 2,713491 518 2,714329 519 2,715167 520 2,716003 521 2,716838 522 2,717671 523 2,718502 524 2,719331 525 2,720159 526 2,720986 527 2,721811 528 2,722634 529 2,723456 530 2,724276 531 2,725095 532 2,725912 533 2,726727 534 2,727541 535 2,728354 536 2,729165 537 2,729974 538 2,730782 539 2,731589 540 2,732394 541 2,733197 542 2,733999 543 2,734799 544 2,735599 545 2,736397 546 2,737192 547 2,737987 548 2,738781 549 2,739572 550 2,740363 551 2,741152 552 2,741939 553 2,742735 554 2,743509 555 2,744293 556 2,745075 557 2,745855 558 2,746634 559 2,747412 560 2,748188 561 2,748963 562 2,749736 563 2,750508 564 2,751279 565 2,752048 566 2,752816 567 2,753583 568 2,754348 569 2,755112 570 2,755875 571 2,756636 572 2,757396 573 2,758155 574 2,758912 575 2,759668 576 2,760422 577 2,761176 578 2,761928 579 2,762679 580 2,763428 581 2,764176 582 2,764923 583 2,765669 584 2,766413 585 2,767156 586 2,767898 587 2,768638 588 2,769377 589 2,770115 590 2,770852 591 2,771587 592 2,772322 593 2,773055 594 2,773786 595 2,774517 596 2,775246 597 2,775974 598 2,776701 599 2,777427 600 2,778151 601 2,778874 602 2,779596 603 2,780317 604 2,781037 605 2,781755 606 2,782473 607 2,783189 608 2,783904 609 2,784617 610 2,785329 611 2,786041 612 2,786751 613 2,787460 614 2,788164 615 2,788875 616 2,789581 617 2,790285 618 2,790988 619 2,791691 620 2,792392 621 2,793092 622 2,793791 623 2,794488 624 2,795185 625 2,795880 626 2,796574 627 2,797268 628 2,797959 629 2,798651 630 2,799341 631 2,800029 632 2,800717 633 2,801404 634 2,802089 635 2,802774 636 2,803457 637 2,804139 638 2,804821 639 2,805501 640 2,806179 641 2,806558 642 2,807535 643 2,808211 644 2,808886 645 2,809559 646 2,810233 647 2,810904 648 2,811575 649 2,812245 650 2,812913 651 2,813581 652 2,814248 653 2,814913 654 2,815578 655 2,816241 656 2,816904 657 2,817565 658 2,818226 659 2,818885 660 2,819543 661 2,820201 662 2,820858 663 2,821514 664 2,822168 665 2,822822 666 2,823474 667 2,824126 668 2,824776 669 2,825426 670 2,826075 671 2,826723 672 2,827369 673 2,820015 674 2,828659 675 2,829304 676 2,829947 677 2,830589 678 2,831229 679 2,831869 680 2,832509 681 2,833147 682 2,833784 683 2,834421 684 2,835056 685 2,835691 686 2,836324 687 2,836957 688 2,837588 689 2,838219 690 2,838849 691 2,839478 692 2,840106 693 2,840733 694 2,841359 695 2,841985 696 2,842609 697 2,843233 698 2,843855 699 2,844477 700 2,845098 701 2,845718 702 2,846337 703 2,846955 704 2,847573 705 2,848189 706 2,848805 707 2,849419 708 2,850033 709 2,850646 710 2,851258 711 2,851869 712 2,852479 713 2,853089 714 2,853698 715 2,854306 716 2,854913 717 2,855519 718 2,856124 719 2,856729 720 2,857332 721 2,857935 722 2,858537 723 2,859138 724 2,859739 725 2,860338 726 2,860937 727 2,861534 728 2,862131 729 2,862728 730 2,863323 731 2,863917 732 2,864511 733 2,865104 734 2,865696 735 2,866287 736 2,866878 737 2,867467 738 2,868056 739 2,868643 740 2,869232 741 2,869818 742 2,870404 743 2,870989 744 2,871573 745 2,872156 746 2,872739 747 2,873321 748 2,873902 749 2,874482 750 2,875061 751 2,875639 752 2,876218 753 2,876795 754 2,877371 755 2,877947 756 2,878522 757 2,879096 758 2,879669 759 2,880242 760 2,880814 761 2,881385 762 2,881955 763 2,882525 764 2,883093 765 2,883661 766 2,884229 767 2,884795 768 2,885361 769 2,885926 770 2,886491 771 2,887054 772 2,887617 773 2,888179 774 2,888741 775 2,889302 776 2,889862 777 2,890421 778 2,890979 779 2,891537 780 2,892095 781 2,892651 782 2,893207 783 2,893762 784 2,894316 785 2,894869 786 2,895423 787 2,895975 788 2,896526 789 2,897077 790 2,897627 791 2,898176 792 2,898725 793 2,899273 794 2,899821 795 2,900367 796 2,900913 797 2,901458 798 2,902003 799 2,902547 800 2,903089 801 2,903633 802 2,904174 803 2,904716 804 2,905256 805 2,905796 806 2,906335 807 2,906874 808 2,907411 809 2,907949 810 2,908485 811 2,909021 812 2,909556 813 2,910051 814 2,910624 815 2,911158 816 2,911690 817 2,912222 818 2,912773 819 2,913284 820 2,913814 821 2,914343 822 2,914872 823 2,915399 824 2,915927 825 2,916454 826 2,916980 827 2,917506 828 2,918030 829 2,918555 830 2,919078 831 2,919601 832 2,920123 833 2,920645 834 2,921166 835 2,921686 836 2,922206 837 2,922725 838 2,923244 839 2,923762 840 2,924279 841 2,924796 842 2,925312 843 2,925825 844 2,926342 845 2,926857 846 2,927370 847 2,927883 848 2,918396 849 2,928908 850 2,929419 851 2,929929 852 2,935439 853 2,930949 854 2,931458 855 2,931966 856 2,932474 857 2,932981 858 2,933487 859 2,933993 860 2,934498 861 2,935003 862 2,935507 863 2,936011 864 2,936514 865 2,937016 866 2,937518 867 2,938019 868 2,938519 869 2,939019 870 2,939519 871 2,940018 872 2,940516 873 2,941014 874 2,941511 875 2,942008 876 2,942504 877 2,942999 878 2,943495 879 2,943989 880 2,944483 881 2,944976 882 2,945468 883 2,945961 884 2,946452 885 2,946943 886 2,947434 887 2,947924 888 2,948415 889 2,948902 890 2,949390 891 2,949878 892 2,950365 893 2,950851 894 2,951338 895 2,951823 896 2,952308 897 2,952792 898 2,953276 899 2,953759 900 2,954243 901 2,954725 902 2,955207 903 2,955688 904 2,956168 905 2,956640 906 2,957128 907 2,957607 908 2,958086 909 2,958564 910 2,959041 911 2,959518 912 2,959995 913 2,960471 914 2,960946 915 2,961421 916 2,961895 917 2,962369 918 2,962842 919 2,963315 920 2,963788 921 2,964259 922 2,964731 923 2,965202 924 2,965672 925 2,966142 926 2,966611 927 2,967079 928 2,967548 929 2,968016 930 2,968483 931 2,968949 932 2,969416 933 2,969882 934 2,970347 935 2,970812 936 2,971276 937 2,971739 938 2,972203 939 2,972666 940 2,973128 941 2,973589 942 2,974050 943 2,974512 944 2,974972 945 2,975432 946 2,975891 947 2,976349 948 2,976808 949 2,977266 950 2,977724 951 2,978181 952 2,978637 953 2,979093 954 2,979548 955 2,980003 956 2,980458 957 2,980912 958 2,981366 959 2,981819 960 2,982271 961 2,982723 962 2,983175 963 2,983626 964 2,984077 965 2,984527 966 2,984977 967 2,985426 968 2,985875 969 2,986324 970 2,986772 971 2,987219 972 2,987666 973 2,988113 974 2,988559 975 2,989005 976 2,989449 977 2,989895 978 2,990339 979 2,990783 980 2,991226 981 2,991669 982 2,992111 983 2,992554 984 2,992995 985 2,993436 986 2,993877 987 2,994317 988 2,994756 989 2,995196 990 2,995635 991 2,996074 992 2,996512 993 2,996949 994 2,997386 995 2,997823 996 2,998259 997 2,998695 998 2,999133 999 2,999565 1000 3,000000 CHAP. III. The use of the Tables of Sines and Logarithms in the resolving of Plain Triangles. BEfore I come to show how the quantity of the sides and angles of any Triangle may be found by help of the former Tables, it will be convenient first to deliver these following considerations and Theorems, as necessaries thereunto. 1. A Triangle is a figure consisting of three sides and three angles, as is the figure ABC. 2. Any two sides of a Triangle are called the sides of the angle comprehended by them, as the sides AB and AC are the sides containing the angle CAB. 3. The measure of an Angle, is the quantity of an arch of a Circle described on the angular point, and cutting both the containing sides of the same angle, as in the Triangle following, the arch CB, is the measure of the angle at A; the arch KD is the measure of the angle at E; and the arch FG is the measure of the angle at H; each of these arches are described on the angular points A, H, E, and cut the containing sides. 4. A Degree is the 360 part of any Circle 5. A Semicircle containeth 180 degrees. 6. A Quadrant containeth 90 degrees. 7. The compliment of an angle less than a Quadrant, is so much as that angle wanteth of 90 degrees, as if the angle HAE should contain 50 degrees, the compliment thereof would be 40 degrees, for if you take 50 from 90 there will remain 40. 8. The compliment of an angle to a Semicircle, is the remainder thereof to 180 degrees. 9 An angle is either Right, Acute, or Obtuse. 10. A Right angle is that whose measure is a Quadrant. 11. An Acute angle is less than a right angle. 12. An Obtuse angle is greater than a Quadrant. 13. A Triangle is either Right angled, or Obliqne angled. 14. A Right angled Triangle is that which hath one right angle, as the Triangle AHE is right angled at E. 15. In every right angled Triangle, that side which subtendeth or lieth opposite to the right angle, is called the hypothenusal, and of the other two sides, the one is called the Perpendicular, and the other the Base, at pleasure, but most commonly the shortest is called the Perpendicular, and the longer is called the Base. So in the former Triangle, the side AH is the hypothenusal, HE the Base, and A the Perpendicular. 16. In every right angled Triangle, if you have one of the acute angles given, the other is also given, it being the compliment thereof to 90 degrees. As in the Triangle AHE, suppose there were given the angle AHE 40 degrees, then by consequence the angle HAE must be 50 degrees, which is the compliment of the other to 90 degrees. 17. The three angles of any right lined Triangle whatsoever, are equal to two right angles, or to 180 degrees: so that if in any right lined Triangle, you have any two of the angles given, you have the third angle also given, it being the compliment of the other two to 180 degrees. So in this Triangle ABC, if there were given the angle BAC 30 degrees, and the angle ACB 130 degrees, I say by consequence there is also given the third angle ABC 20 degrees, it being the compliment of the other two to 100 degrees: for, the two given angles 30 and 130 being added together, they make 160, which being taken from 180, there remains 20, the quantity of the third angle ABC. 18. In all plain Triangles whatsoever, the sides are in proportion one to the other, as the Sins of the angles opposite to those sides. So in the Triangle ABC, the Sine of the angle ACB, is in such proportion to the side AB, as the Sine of the angle CAB is to the side BC, and so of any other. CHAP. IU. Containing the doctrine of the dimension of right lined Triangles, whether right angled or obliqne angled, and the several Cases threin resolved, both by Tables, and Lines of Artificial Numbers, Sins, and Tangents. HAving in the foregoing Chapters of this Book explained and shown the use of the Tables of Sines and Logarithms, and also delivered divers necessary Theorems relating to the mensuration of plain Triangles, I come now to show how a plain Triangle may be resolved, that is, by having any three of the six parts of a plain Triangle given, to find a fourth, both by the Tables of Sines and Logarithms, and also by the lines of Artificial Numbers, Sins and Tangents on the Index of your Table, so that when your Tables are not ready at hand, you may make use of these Lines, which will sufficiently supply the want of them. In all the cases following, I have made use but of two Triangles for Examples, one right angled, and the other obliqne angled, but in either of them I have expressed all the varieties that are necessary, so that any three parts being given in any of them, a fourth maybe found at pleasure. The several cases of the right angled Triangle will best be applied in the taking of heights, as is showed in the next Book, and the obliqne angled Triangle for the taking of distances there also taught; so that if the line CA in the right angled Triangle were a Tree, Tower, or Steeple, and that you would know the height thereof, you must observe with your Instrument the angle CBA, and measure the distance BASILIUS; so have you in the right angled Triangle ABC the Base AB, and the angle at the Base CBA, then may you (by the 1. Case) find the side CA, which is the height of the thing required. In the resolving of plain Triangles, there are several Cases, of which; I will only insist on those that have most relation to the work in hand. And first, Of Right angled plain Triangles. CASE I. In a right angled plain Triangle, the Base and the angle at the Base being given, to find the Perpendicular. IN the right angled Triangle following ABC, there is given, the Base thereof BASILIUS, 400 foot, and the angle at the Base CBA 30 degrees, and it is required to find the perpendicular CA Now because the angle CBA is given, the angle BCA is also given; it being the compliment of the other to 90 degrees; and therefore the angle BCA is 60 degrees. Then to find the perpendicular CA, the proportion is, As the Sine of the angle BCA, 60 degrees (which is) 9,937531 Is to the Logarithm of the side BASILIUS, 400 foot (which is) 2,602059 So is the sine of the angle CBA 30 degrees (which is) 9,698970 the sum of the second and third numbers added 12,301029 the first number substracted from the sum 9,937531 To the Logarithm of the side CA (which is) 2,363498 The nearest absolute number answering to this Logarithm is 231 ferè, and that is the length of the side CA in feet which was the thing required. A general Rule. In all proportions wrought by Sines and Logarithms, you must observe this for a general rule, viz. to add the second and third numbers together, and from the sum of them to subtract the first number, so shall the remainder answer your question demanded, as by the former work you may perceive, where the Logarithm of the side BASILIUS 2,602059 (which is the second term) is added to the sine of the angle CBA 9,698970, (which is the third term) and from the sum of them (namely from 12,301029) is substracted 9,937531, the sine of the angle BCA, which is the first number, and there remaineth, 2,363498, which is the Logarithm of 231 almost, and that is the length of the side required in feet. The same manner of work is to be observed in all the Cases following as will plainly appear. How to perform the same work, by the lines of Sines and Numbers. These kind of proportions are wrought more easily by help of the lines of artificial Numbers, Sins and Tangents on the Index of your Table, and exact enough for any ordinary occasion, for the proportion being, As the sine of the angle BCA, 60 degrees, Is to the Logarithm of the side BASILIUS 400 feet, So is the Sine of the angle CBA, 30 degrees, To the Logarithm of the side AC 231 feet, ferè. Therefore, if you set one foot of your Compasses at 60 degrees in the line of Sines, and extend the other foot to 400 in the line of Numbers; the same extent of the Compasses will reach from the sine of 30 degrees to 231 in the line of Numbers, which is the length of the side AC, which was required. Or otherwise, Extend the Compasses from the sine of 30 degrees to the sine of 60 degrees, in the line of Sines, the same extent will also reach from 400, in the line of Numbers, to 231 as before. And thus by these Artificial Lines, the work is much abreviated, there being need neither of pen, ink, paper or Tables, but only of your Compasses. CASE II. The Base, and the angle at the Base being given, to find the hypothenusal. IN the same Triangle ABC let there be given (as before) the Base AB 400 foot, and the angle ABC 30 degrees, and let it be required to find the hypothenusal BC. Now because the angle CBA is given, the other angle BCA is also given, and the proportion is, As the Sine of the angle BCA, 60 degrees, 9,937531 Is to the Logarithm of the side BASILIUS, 400 foot 2,602059 So is the Sine of the angle CAB, 90 degrees, 10,000000 the sum of the second and third numbers added 12,602959 the first number substracted from the sum 9,937531 To the Logarithm of the side BC: which is, 2,665428 The absolute number answering to this Logarithm is 462, and so many feet is the hypothenusal BC. By the lines of Sines and Numbers. The manner of work is altogether the same with the former, for the proportion being, As the Sine of the angle BCA 60 degrees, Is to the length of the side BASILIUS 400 foot; So is the sine of the angle CAB 90 degrees, To the length of the side CB 462. Extend the Compasses from the sine of 60 degrees to 400 in the line of Numbers, the same extent will reach from the Sine of 90 degrees to 462 in the line of Numbers, and that is the length of the side BC. Or you may extend the Compasses from the Sine of 60 degrees to the Sine of 90 degrees; the same extent will also reach from 400 to 462, as before. CASE III. The hypothenusal, and angle at the Base being given, to find the Perpendicular. IN the same Triangle, let there be given the hypothenusal BC 462 feet, and the angle at the Base CBA 30 degrees, to find the perpendicular CA The number answering to this Logarithme is 231 ferè, and that is the length of the side CA in feet. Here the Work is somewhat abreviated, for the angle CAB being a right angle, and being the first term, when the second and third terms are added together, the first is easily substracted from it by cancelling the figure next your left hand, as you see in the example; and so the rest of that number is the Logarithme of the number sought. By the lines of Sines and Numbers. Extend the Compasses from the Sine of 90 degrees to 462, the same extent will reach from the Sine of 30 degrees to 231. Or extend the Compasses from the Sine of 90 degrees to the Sine of 30 degrees, the same extent will reach from 462 to 231; and that is the side CA CASE IV, The hypothenusal, and angle at the Base being given, to find the Base LEt there be given in the former Triangle the Hypothenusal BC, and the angle at the Base CBA, and by consequence the angle BCA the compliment of the other to 90; then to find BASILIUS, the proportion is, As the Sine of the angle CAB, 90 degrees 10,000000 Is to the hypothenusal BC, 462 2,664642 So is the Sine of the angle BCA, 60 degrees, 9,937531 To the Logarithm of the Base BASILIUS, 12,602173 The nearest number answering to 2,602173, is the Logarithm of 400, and so long is the Base BASILIUS. By the lines of Sines and Numbers. As before, Extend the Compasses from the Sine of 90, to 462, the same extent will reach from the Sine of 60 degrees, to 400 in the line of Numbers. Or, extend the Compasses from the Sine of 90, to the Sine of 60, the same extent will reach from 462 to 400, which is the length of the Base BASILIUS. CASE V The Perpendicular, and angle at the Base being given, to find the hypothenusal. IF the Perpendicular CA be given 231, and the angle at the Base CBA 30 degrees, the hypothenusal BC may be found thus; for, As the Sine of the angle CBA, 30 degrees, 9,698970 Is to the Logarithm of the Perpendicular CA 231 12,363612 So is the Sine of the angle CAB, 90 degrees, 10,000000 To the Logarithm of the hypothenusal BC 2,664642 ¶ Here, because the angle CAB is a right angle, or 90 degrees, and comes in the third place, I therefore only put an unite before the second term, and from that second term subtract the first term, and the remainder is 2,664642, the absolute number answering thereunto is 462, the side BC. By the lines of Sines and Numbers. Extend the Compasses from the Sine of 30 degrees, to 231, the same extent will reach from the sine of 90 degrees to 462. Or, the distance between the Sine of 30 degrees and 90 degrees, will be equal to the distance between 231, and 462, which giveth the side required. CASE VI The hypothenusal and Perpendicular being given, to find the angle at the Base. IN the foregoing Triangle there is given the hypothenusal BC 462 feet, and the perpendicular CA, 231 feet, and it is required to find the angle CBA, the proportion is, As the Logarithm of the hypothenusal BC 462 2,664642 Is to the right angle BAC, 90 degrees, 10,000000 So is the Logarithm of the perpendicular CA, 231, 12,363612 To the sine of the angle CBA, 30 degrees. 9,698970 By the lines of Sines and Numbers. Extend the Compasses from 462, to the sine of 90, the same extent will reach from 231 to the sine of 30 degrees. Or, Extend the Compasses from 462 to 231, the same extent will reach from the sine of 90 degrees, to the Sine of 30 degrees, which is the quantity of the enquired angle CBA. Of Obliqne angled plain Triangles. CASE VII. Having two angles, and a side opposite to one of them given, to find the side opposite to the other. IN the Triangle QRS, there is given the angle QSR 24 degrees 20 minutes; and the angle QRS 45 degrees 10 minutes, and the side QS 303 feet, and it is required to find the side QR. ¶ Here note, that in obliqne angled plain Triangles, as well as in Right angled, the sides are in proportion one to the other, as the sins of the angles opposite to those sides. Therefore, As the sine of the angle QRS 45 deg. 10 min. 9,850745 Is to the Logarithm of the side QS 303 feet, 2,481443 So is the sine of the angle QSR 24 degrees 20 min. 9,614944 the sum of the second and third terms 12,096387 the first term substracted 9,850745 To the Logarithme of the side QR, 2,245642 The nearest absolute number answering to this Logarithm is 176, and so many feet is the side QR. By the lines of Sines and Numbers. The lines of Sines and Numbers will resolve these Triangles by the same manner of work as in the other before. For, If you extend the Compasses from the sine of 45 degrees 10 min. to 303, the same extent will reach from the sine of 24 degrees 20 minutes, to 176, and so much is the side QR. Or, Extend the Compasses from the Sine of 45 degrees 10 min. to 24 degrees 20 minutes, the same extent will reach from 303, to 176, the length of the inquired side. In like manner, if the angle RQS 110 degrees 30 minutes, and the angle QRS 45 degrees 10 min. and the side QS 303 feet, had been given, and the side RS required, the manner of work had been the same; for, As the sine of the angle QRS 45 degrees 10 min. 9,850745 Is to the Logarithm of the side QS 303 feet, 2,481443 So is the sine of RQS 110 deg. 30 min. (or 69 de. 30 m.) 9,971588 the sum of the second and third terms 12,453031 the first term substracted 9,850745 To the Logarithm of the side RS, 2,602286 The absolute number answering to this Logarithm is 400, and so much is the side RS. ¶ In this case, because the angle RQS is more than 90 degrees, you must therefore take the compliment thereof to 180 degrees, so 110 degrees 30 minutes, being taken from 180 degrees, there remains 69 degrees 30 min. whose Sine is the same with 110 deg. 30 min. and being used in stead thereof, will effect the same thing. By the lines of Sines and Numbers. Extend the Compasses from the Sine of 45 degrees 10 min. to 303, the same extent will reach from the sine of 69 deg. 30 min. to 400. which is the side RS required. Or the Compasses being opened to the distance between the sine of 45 deg. 10 min. and 69 deg. 30 min. the same distance will reach from 303 to 400 as before. CASE VIII. Two sides and an angle opposite to one of them being given, to find the angle opposite to the other. IN the same Triangle, let there be given, the side QS 303, and QR 176, together with the angle QSR 24 degrees 20 minutes, and let it be required to find the angle QRS, the proportion is, As the Logarithm of the side QR 176, 2,245513 Is to the sine of the angle QSR, 24 deg. 20 min. 9,614944 So is the Logarithm of the side QS 303, 2,481443 the sum of the second and third numbers 12,096387 the first number substracted from the sum 2,245513 To the sine of the angle QRS, 9,850374 The nearest degree answering to this sine is 45 degrees 10 min. which is the quantity of the angle QRS, required. By the lines of Sines and Numbers. Extend the Compasses from 176, to the sine of 24 degrees 20 minutes, the same extent will reach from 303 to 45 deg. 10 min. the angle QRS. Or, the distance between 176 and 303, will be equal to the distance between 24 degrees 20 minutes, and 45 deg. 10 min. CASE IX. Having two sides, and the angle contained by them given, to find either of the other angles. THis Case will seldom come in use in Surveying, because the thing required is an angle, which are most commonly given, they being observed by Instrument, and therefore in this place may be omitted, partly because the proposition is not wrought by Sines and Logarithms, but by Tangents and Logarithms, and there is no Tables of Tangents in this Book, to work the proportion by: Yet those that are desirous to resolve all kind of Triangles by the proportional lines, may have added to the lines of artificial sins and Numbers, a line of artificial Tangents, and these three lines together, will resolve all Cases in Spherical, as well as in plain Triangles. For the performance of this Problem, suppose there were given the side QS 303, and the side RQ 176, and the angle comprehended by them; namely, the angle RQS 110 degrees 30 minutes, and it were required to find either of the other angles. First, Take the sum and difference of the two given sides, their sum is 479, and their difference is 127. Then knowing that the three angles of all right lined Triangles are equal to two right angles or 180 degrees, (by the 17. Theor. of Chap. 3.) therefore the angle RQS being 110 degrees 30 minutes, if you subtract this angle from 180 degrees, the remainder will be 69 deg. 30 min. which is the sum of the two unknown angles at R and S, the half whereof is 34 deg. 45 min. The side QS, 303 The side QR, 176 The sum of the sides, 479 The difference of the sides 127 The half sum of the two unknown angles 34 deg. 45 min. The sum and difference of the sides being thus found, and also the half sum of the two unknown angles, the proportion by which you must find the angles severally is, As the Logarithm of the sum of the sides, 479, 2,680335 Is to the Logarithm of the difference of the sides, 127, 2,103804 So is the Tangent of the half sum of the two unknown angles 34 degrees, 45 minutes, 9,841187 the sum of the second and third numbers 11,944991 the first number substracted 2,680335 To the Tangent of 10 degrees 25 minutes, 9,264656 These 10 degrees 25 minutes, being added to the half sum of the two unknown angles, namely, to 34 degrees 45 minutes; the sum will be 45 degrees 10 minutes, the quantity of the angle QRS, which is the greater angle of the two: Also, these 10 degrees 25 minutes, being substracted from the same half sum, there remaineth 24 degrees 20 minutes for the angle QSR, which is the lesser of the unknown angles: and thus are either of the enquired angles easily found. By the lines of Tangents and Numbers. Extend the Compasses from the sum of the sides 479, to the difference of the sides 127, the same extent upon the line of Tangents will reach from the Tangent of 34 degrees 45 minutes (which is the half sum of the two unknown angles) to the Tangent of 10 degrees 25 minutes, and these 10 degrees 25 minutes, added to, and substracted from the half sum, as before is showed, will give the quantity of either of the two unknown angles. CASE X. The three sides of a right lined plain Triangle being given, how to find the Area, or the superficial content thereof. EXAMPLE. Let the Triangle given be ABC, the sides thereof being 20, 13, 11, how much is the superficial content thereof? The sum of the sides is 44, the half sum is 22, the differences betwixt each side and that half are 2, 9, 11, which numbers rank in this order following. The half sum 22 1,342423 The differences, 2 0,301030 9 0,954243 11 1,041393 The sum of the Logarithms 3,639089 The Area or Content required, 66. 1,819544 And this Area, or superficial Content thus found, is always of the same nature with the sides of the Triangle, that is to say, if the sides of the Triangle be given in feet, then is the content found in feet; also, if the sides be Perches, you shall have the content in perches, and so of any other measure whatsoever. I might add hereunto divers other Cases, but in this place at present let these suffice. The end of the Third Book. THE COMPLETE SURVEYOR. The Fourth Book. THE ARGUMENT. Our business hitherto hath been to provide necessary Instruments and to learn such things which of necessity ought to be known before we enter the Fields to Survey. Being thus provided we come now to apply them several ways: First, in taking of Heights and Distances whether accessible or in-accessible; and then in Surveying of Land. In this Book every kind of work is performed three several ways, by three several Instruments, viz. the Plain Table, the Theodolite, and Circumferentor, by which the congruity and harmony of the several Instruments may be easily discerned, and the truth of every Example may the better appear. Here is also divers ways of Surveying by one and the same Instrument, that is, to take the Plot of a Field several ways, and to measure all kind of Grounds whatsoever, whether Woodland or other. Here is also shown how to take the Plot of a whole Manor, and to keep your account in your Field-Book after the best and most easiest manner: with divers Rules, Cautions, and Directions, throughout the whole Book inserted. THE APPLICATION AND USE of the several Instruments (before described) in the practice of SURVEYING. CHAP. I. Of the use of the Scale. HAving before described the several Instruments belonging to Surveying, I will now show the use of them: and first, of the Scale. The Scale is principally intended for the laying out of lines, for which purpose the several Scales of equal parts are there divided, some of greater and some of lesser quantities: the uses of all the lines being the same, for each line is divided into 11 equal parts, representing 11 Chains, and these grand divisions are numbered with Arithmetical Figures by 1, 2, 3, etc. to 10, than the uppermost large division is again divided into ten other smaller parts, each part containing 10 links of your Chain, each of which smaller parts you may suppose to be again divided into ten other lesser parts, representing single Links of your Chain. 1. Any length being measured by your Chain, how to lay down the same distance upon paper. Suppose, that measuring along a hedge with your Chain, you find the length thereof to contain 5 Chains 60 Links: Now to take this distance from your Scale, and lay it down upon paper, do thus. First, Draw a line as AB, then place one foot of your Compasses upon your Scale at the figure 5, for your five Chains, and extend the other foot to six of the small divisions (which represents the 60 Links) then set this distance upon the line drawn from A to B, so shall the line AB contain 5 Chains 60 Links, if you take the distance from the Scale of 10 in an Inch. But if you would have your line shorter, and yet to contain 5 Chains 60 Links, then take your distance from a smaller Scale, as of 12, 16, 20, or 24 in an Inch, so shall the 5 Chains 60 Links end at C, if taken from the Scale of 12 in an inch, or at D, by the Scale of 16, or at E by the Scale of 24: either of which lines will contain 5 Chains 60 Links, and be in proportion one to the other as the Scales from whence they were taken. And in this manner may any number of Chains and Links be taken from any of the Scales. 2. A right line being given, to find how many Chains and Links are therein contained, according to any Scale assigned. Suppose AB were a line given, and it were required to find how many Chains and Links are contained therein, according to the Scale of 10 in an inch. Take in your Compasses the length of the line AB, and applying it to your Scale of 10 in an Inch, you shall find the extent of the Compasses to reach from 5 of the great divisions to fix of the lesser divisions, wherefore the line AB contains 5 Chains and 60 Links: The like must be done for any other line, and also by any of the other Scales. Upon the Ruler there is (besides the several Scales of equal parts) a Line or Scale of Chords, which is numbered by 10, 20, 30, etc. to 90, and this line serveth to protract or lay down angles; but in all the prectise of Surveying a Protractor is much more convenient, yet for other uses this line may be very serviceable, and when a Protractor is wanting, it may supply that defect: the manner how to use it is thus. 3. How to lay down upon paper, an angle containing any number of degrees and minutes, by the Line of Chords. Draw a line at pleasure, as AB, and from the point A, let it be required to protract an angle of 40 degrees 20 minutes. First, extend your Compasses upon the line of Chords, from the beginning thereof to 60 degrees always, and with this distance, setting one foot upon the point A, with the other describe the pricked arch BC, then with your Compasses take 40 degrees 20 minutes (which is the quantity of the inquired angle) out of the line of Chords, from the beginning thereof to 40 degrees 20 minutes, than (the Compasses so resting) if you set one foot thereof upon B, the other will reach upon the arch to C. Lastly, draw the line AC, so the angle CAB shall contain 40 degrees 20 minutes. 4. Any angle being given, to find what number of degrees and minutes are contained therein. Suppose CAB were an angle given, and that it were required to find the quantity thereof. Open your Compasses (as before) to 60 degrees of your Chord, and placing one foot in 〈◊〉, with the other describe the arch CB, then take in your Compasses the distance CB, and measuring that extent upon the little of Chords from the beginning thereof, you shall find it to reach to 40 degrees 20 minutes, which is the quantity of the required angle. If any angle given or required shall contain above 20 degrees, you must then protract it at twice, by taking first the whole line, and then the remainder. CHAP. II. Of the use of the Protractor. ALthough the chief uses of the Protractor may be performed by the line of Chords last spoken of, yet for avoiding of superfluous lines and arches (which must otherwise be drawn all over your Plot) the Protractor is far more convenient, the 〈◊〉 ●ereof is, 1. To lay down upon paper an angle of any quantity. First, draw a right line at length as AB, then on any part thereof, as on C, place the centre of the Protractor, in which point also fix your protracting pin, and turn the Protractor about upon the centre, till the Meridian line of the Protractor (noted in the description thereof with OF) lie directly on this line AB, the Semicircle of the Protractor lying upwards (or from you) then close to the edge of the Semicircle, at the division of 50 degrees, mark the point D with your protracting pin; and draw the line CD, so shall the angle DCA, contain 50 degrees. 2. Any angle being given, to find the quantity thereof by the Protractor. Suppose DCB were an angle given, and that it were required to find the quantity thereof by the Protractor. First, you must apply the centre of the Protractor to the point C, and the Meridian line thereof directly upon the line DC, then shall you find the line CB to lie directly under 130 degrees of the Protractor, and such is the quantity of the angle DCB required. CHAP. III. Of the Plain Table, how to set the parts thereof together, and make it fit for the field. WHen you would make your Table fit for the field, lay the three boards thereof together and also the ledges at each end thereof in their due pla●…●ccording as they are marked. Then lay a sheet of white paper 〈◊〉 over the Table, which must be stretched over all the boards by putting on the Frame, which binds both the paper to the boards, and the board's one to another. Then screw the Socket on the back side of the Table, and also the Box and Needle in its due place, the Metidian line of the Card (which is in the Box) lying parallel to the Meridian or Diameter of the Table; which diameter is a right line drawn upon the Table from the beginning of the degrees through the centre, and so to the end of the degrees. Then put the Socket upon the head of the Staff, and there screw it. Also, put the sights into the Index, and lay the Index on the Table, so is your Instrument prepared for use as a Plain Table or Theodolite, the difference only being in placing of the Index, for when you use your Instrument as a Plain Table, you may pitch your centre in any part of the Table, which you shall think most convenient for the bringing on of the work which you intent: But if you use your Instrument as a Theodolite, than the Index must be turned about upon the Centre of the Table, for which purpose there is a piece of wire which goes through a small hole of brass fastened to the Index, and so into the centre, by which means the Index keeps his constant place, only moving upon the centre. Your Instrument being thus ordered, you may use it either as a Plain Table or a Theodolite, but if you would use it as a Circumferentor, you need only screw the Box and Needle to the Index, and both of them to the head of the Staff, with a brass screw-pin fitted for that purpose, so that the Staff being fixed in any place, the Index and fights may turn about at pleasure without moving of the Staff, and now is your Instrument a good Circumferentor, nay better than that before described in the second Book. Also, when you have occasion to measure any Altitude, hang the Label upon the farther Sight, and thus are you exactly fitted for all occasions. CHAP. IU. How to measure the quantity of any angle in the field, by the Plain Table, Theodolite, and Circumferentor: and also to observe an angle of Altitude. YOu must understand that when I mention the Plain Table, or perform any work thereby, that I mean the Table when it is covered with a sheet of paper, upon which, all observations of angles that are taken upon the Table in the field do agree exactly in proportion with those of the field itself, but are not denominated by their quantities, but by their symmetry or proportion. Secondly, When I mention the Theodolite, or work by that Instrument, I do not mean the Theodolite before described in the 2 Chapter of the 2 Book, but I mean the degrees described on the frame of the Table, which supplies the use thereof. Thirdly, When I mention or make use of the Circumferentor, I mean the Index with the Box and Needle screwed to the Staff. ¶ Having thus given you a sufficient description of the several Instruments and their parts, I come now to the use of them, showing how any angle in the field may be measured by any of them. And, 1. How to observe an angle in the Field by the Plain Table. Suppose EKE and KG to be two hedges, or two sides of a field, including the angle EKG, and that it were required to draw upon your Table, an angle equal thereunto. First, place your Instrument as near the angular point K, as conveniency will permit, turning it about till the North end of the Needle hang directly over the Flower-de-luce in the Box, and then screw the Table fast. Then upon your Table, with your protracting pin or Compass point, assign any point at pleasure upon the Table, and to that point apply the edge of the Index, turning the Index about upon that point, till through the sights thereof you espy a mark set up at E, or parallel to the line EKE, and then, with your protracting pin, or Compass point, or Black-lead, draw a line by the side of the Index to the assigned point upon the Table. Then (the Table remaining ) turn the Index about upon the same point, and direct the sights to a mark set up at G, or parallel thereto, that is, so far distant from G, as your Instrument is placed from K, and then, by the side of the Index, draw another line to the assigned point, so shall you have drawn upon your Table two lines, which shall represent the two hedges EKE and KG, and those lines shall include an angle equal to the angle EKG, and although you know not the quantity of this angle yet you may (by the 1 or 2 Chapters of this Book, find the quantity thereof if there were any need, for in working by this Instrument, it is sufficient only to give the symmetry or proportion of angles and not their quantities, as in working by the Theodolite or Circumferentor it is. Also, in working by the Plain Table, there needeth no protraction at all, for you shall have upon your Table the true figure of any angle or angles which you observe in the field, in their true positions, without any farther trouble. 2. How to find the quantity of an angle in the field by the Theodolite. Let it be required to find the quantity of the angle EKG by the Theodolite: place your Instrument at K, laying the Index on the diameter thereof, then turn the whole Instrument about (the Index still resting on the Diameter) till through the sights you espy the mark at E, then screwing the Instrument fast there, turn the Index about upon the centre, till through the sights you espy the mark at G, then note what degrees (on the frame of the Table) are cut by the Index, which you will find to be 114 degrees, and that is the quantity of the angle EKG. 3. How to find the quantity of any angle in the field, by the Circumferentor. If it were required to find the quantity of the former angle EKG, by the Circumferentor; First, place your Instrument (as before) at K, with the Flower-de-luce, in the Card, towards you; then direct your sights to E, and observe what degrees in the Card are cut by the South end of the Needle, which let be 296, then turning the Instrument about the staff (the Flower-de-luce always towards you) direct the sights to G, noting then also what degrees are cut by the South end of the Needle, which suppose 182, this done (always) subtract the lesser number of degrees out of the greater, as in this Example 182 from 296, and the remainder is 114 degrees, which is the true quantity of the angle EKG. Again; the Instrument standing at K, and the sights being directed to E, as before, suppose that the South end of the Needle had cut 79 degrees; and then directing the sights to G, the same end of the needle had cut 325 degrees, now, if from 325, you subtract 79, the remainder is 246, but because this remainder 246 is greater than 180, you must therefore subtract 246 the remainder, from 360, and there will remain 114, the true quantity of the inquired angle, and thus you must always do, when the remainder exceedeth 180 degrees. ¶ This adding and substracting for the finding of angles, may seem tedious to some, but here the Reader is desired to take notice, that for quick dispatch, the Circumferentor is as good an Instrument as the best, for in going round a field, or in surveying of a whole Manor, you are not to take notice of the quantity of any angle, but only to observe what degrees the needle cutteth, which in those cases is sufficient, as will appear hereafter, but in taking of distances by the Circumferentor it is altogether necessary, as may appear by the 7 Chap. following, and for that reason I have here showed how to find an angle by the Circumferentor, and also that you might thereby perceive what congruity and harmony there is in all the three Instruments. 4. How to set the Index and Label horizontal upon the Staff. When you have screwed the Index and sights to the Staff as a Circumferentor, before you put the Label upon the brass pin or wire, you must hang a line and plummet upon that pin, and then put on the Label, then move the Index up and down till the thread and plummet hang directly upon a line which is gauged from under the pin all along the Sight, and then doth the Instrument stand horizontal or level, which it must always do when you take an altitude therewith. 5. How to observe an angle of Altitude. The Label which is to be hanged on one of the sights of the Circumferentor (as was intimated in the description thereof) and the Tangent line on the edge of the Index, is only for the finding of angles of Altitude, and is therefore only useful in taking of heights, and in surveying of mountainous and uneven grounds. The manner how to observe an angle of Altitude by this Label, and the Tangent line on the Index, is thus. Suppose CA to be a Tree, Tower or Hill, whose height were required. Your Instrument being placed at B, exactly level, direct the sights thereof towards CA, and there fix it, hanging the Label on the farthermost fight, upon a pin for that purpose; then move the Label too and fro, along the side of the Index, till through the sight at the end of the Label, and by the Pin on which the Label hangeth, you espy the very top of the object to be measured at C, then note what degree of the Tangent line is cut by the Label, which suppose 30, and that is the quantity of the angle of Altitude, it being equal to the angle CBA, Thus by the Rules in this Chapter delivered, may the true quantity of any angle be easily taken, and this is the most convenient use to be first placed, I will now show how by your several Instruments you may take all manner of heights and distances, whether accessible or inaccessible, several ways, with divers other necessary conclusions incident thereunto. CHAP. V How to take an inaccessible Distance at two stations by the three forementioned Instruments, and first, by the Plain Table. YOu are taught in the last Chapter how to make observation of any angle in the field by the several Instruments before mentioned, as the Plain Table, Theodolite, and Circumferentor, and also an angle of Altitude by the Index, and the Label thereunto annexed. I conceive it now convenient to show how all manner of heights and distances may be readily and exactly measured, several ways, whether they be accessible or inaccessible: and first of distances. ¶ You may remember that I formerly intimated, that the measuring of a Height or Distance is only to resolve a Triangle, so that when you make any observation either of Height or Distance, the observation of angles which you make are the angles of some Triangle, and the lines which you measure on the ground, are the sides of the same Triangle, and these are the given parts of the Triangle. The manner how to take a distance by the Plain Table is thus. Suppose you were standing in a field at R, and that at S were some eminent mark (as a Tree, Church, House, or such like) and that it were required to find the distance between R and S. First, place your Table at R, and thereon assign any point at pleasure, unto which point apply the edge of your Index, turning it about upon that point, till through the sights you espy the mark at S, and draw a line by the side of ●he Index, as RS. Then in some other convenient place of the field (as at Q) let a staff or other mark be erected, and the Table remaining as before, turn the Index about, till through the sights you espy the mark at Q, drawing a line by the side thereof, as RQ, so have you described upon your Table an angle equal to the angle QRS. Then (with your Chain) measure the distance QR, which let be 176 foot, then take with your Compasses 176 out of any Scale, and set it upon your Table from R to Q, so shall this point Q upon your Table, represent the mark at Q in the field. This done, set up a staff a R, and remove your Table to Q, laying the Index upon the line QR, and holding it fast there, turn the whole Table about till through the sights you espy the mark set up at your former place of standing at R: then screw the Table fast, and lay the Index on the point Q, turning it about, till through the sights you espy your mark at S, then draw a line by the side of the Index, which will cut the line RS (first drawn) in the point S. By this means shall you have upon your Table a Triangle equal to the Triangle QRS, the correspondent sides and angles thereof being proportionally equal with those in the field: therefore, if with your Compasses you take the length of the side RS, and apply that distance to the same Scale from whence you took the side QR, you shall find it to contain 400 foot, and that is the distance between R and S. Likewise, if you take with your Compasses the length of the line QS, and apply it to the same Scale, you shall find it to contain almost 303, and so many foot is the distance QS. ¶ In this manner may the distance between any two places be measured, although they be so situated, that by reason of water or other impediments you cannot approach near unto them. And here note, that when you take your second station, that you take it as large as the ground will permit, so shall your work be so much the truer, by now much the distance taken is the larger. CHAP. VI How to take an inaccessible distance at two stations by the Theodolite. IN the former Diagram, let R and Q be two stations, from either of which it is required to find the distance to S. First, place your Instrument at R, laying the Index and sights upon the Diameter thereof, turning the whole Instrument about, till through the sights you espy your second station at Q, and there screw it fast, then turn the Index about upon the centre, till through the sights you espy the mark at S, noting the degrees cut by the Index, which suppose 45 degrees 10 minutes. Then remove your Instrument to Q, laying the Index on the Diameter thereof, and holding it there, turn the whole Instrument about, till through the sights you espy your mark at S, and fixing the Instrument there, turn the Index about till through the sights you see the mark set up at your former station at R, noting the degrees there cut, which let be 110 degrees 30 minutes. This done, measure the distance of your two stations Q R, which let be 176 feet, 10 in the Obliqne angled Triangle QSR, you have given, (1) the angle SRQ, 45 degrees 10 minutes, the angle observed at your first station. (2) the angle RQS, 110 degrees 30 minutes, which was the angle observed at your second station. And (3) you have given the side RQ 176 foot, which is the distance of your two stations: and you are to find the two other sides RS, and QS which you may find by the 7 Case of the 4 Chapter of the 3 Book, in this manner: for, Having the two angles QRS, and RQS given, you have also the third angle RSQ given, 24 degrees 20 minutes, it being the compliment of the other two to 180 degrees. (by the 17 of Chap. 3, Lib. 3.) Then to find the other two sides, the proportion is; I. For the side QS. As the sine of the angle RSQ, 24 degrees 20 minutes, Is to the Logarithm of the side RQ 176 foot, So is the sine of the angle QRS 45 degrees 10 minutes, To the Logarithm of the side QS, 303 foot ferè. II. For the side RS. As the sine of the angle QRS ●… degrees 10 minutes, Is to the Logarithm of the side QS, 303 foot, So is the sine of the angle RQS 110 deg. 30 min. (or 69 d. 30 m.) To the Logarithm of the side RS, 400 foot. Which is the distance required. ¶ I have been larger upon this particular than I intended (having sufficiently insisted thereon before in the dimension of plain Triangles) but that the Reader may fully understand these necessary conclusions, I have in this example used all the perspicuity I could imagine, so that in the subsequent Chapters I may be the briefer, for this being well understood, he may easily apprehend any of the other at the first view. CHAP. VII. How to take an in-accessible distance at two stations by the Circumferentor. LEt it be required to find the distance from R and Q to S. First, place your Instrument at R, and direct the sights to S, observing what degrees the South end of the Needle cutteth, which let be 315 degrees 30 min. then turning the Instrument about, direct the sights to Q observing what degrees the needle there cutteth, which let be 270 degrees 20 minutes, therefore from 315 degrees 30 minutes, subtract 270 degrees 20 minutes, and there will remain 45 degrees 10 minutes, which is the quantity of the angle SRQ. Then remove the Instrument to Q, and direct the sights to R, the Needle cutting 91 degrees 00 minutes, also, direct the sights to S, the needle cutting 340 degrees 30 minutes, now if you subtract 91 degrees 00 minutes, from 340 degrees 30 minutes, the remainder is 249 degrees 30 minutes, which (because it exceedeth 180 degrees) subtract from 360 degrees, and there remains 110 degrees 30 min. the true quantity of the angle RQS. Having thus obtained the two angles RQS and SRQ, you must measure the stationary distance QR 176 foot, so have you given in the Triangle QRS, (1) the angle RQS 110 degrees 30 minutes, (2) the angle QRS, 45 degrees 10 minutes, (3) the angle QSR, 20 degrees 10 minutes, (the compliment of the other two to 180 degrees, and (4) the stationary distance QR 176 foot, whereby you may find the other sides QS and RS, according to the doctrine delivered in the foregoing Chapter. dg. min. First station at R, degrees cut 315 30 270 20 The quantity of the angle QRS 45 10 Second station at Q, degrees cut 340 30 91 00 249 30 360 00 The quantity of the angle RQS 110 30 The stationary distance 176 foot. Having these things given, if you resolve the Triangle QRS, you shall find the side RS to contain 400 foot, and the side QS 303 foot ferè, as in the last Chapter. CHAP. VIII. How to protract or lay down a Distance taken, according to the directions of the two last Chapters, upon paper, by help of your Protractor or line of Chords. WHen you make any observations in the field, by the Theodolite or Circumferentor, you are to note down the quantities of the several lines and angles observed in the field, in a Book or paper, so that they may be ready at hand when you come to protraction, and this is the usual way. Suppose it were required to draw upon paper or pasteboard the true symmetry or proportion of the distance taken in the last Chapter. CHAP. IX. How to take the altitude of any Tower, Tree, Steeple, or the like (being accessible) by the Label and Tangent line. HAving in the 5 Section of the 4 Chapter of this Book, shown how to observe an angle of Altitude by the Label and Tangent line, we now come to the further use thereof. Suppose therefore that the line CA were a Tree Tower, Steeple, or other thing, whose height were required. This proportion being wrought according to the former directions, the side CA will be found to contain almost 231 foot, and that is the height of CA required. CHAP. X. How to protract or lay down upon paper, the observation made in the last Chapter. HAving drawn a line upon your paper as BA, place the centre of the Protractor upon B, now (because when you made your observation at B, the degrees cut were 30) turn the Protractor about till the line BASILIUS lie just under 30 degrees, than (with your protrocting pin) make a mark by the edge of your Protractor against 00 degrees, and draw the line BC, so shall the angle CBA contain 30 degr. Then (because the measured distance BASILIUS was 400 foot,) take 400 from any of your Scales of equal parts, and set that distance from B to A, and from the point A, erect the perpendicular AC, which perpendicular being taken in your Compasses, and measured upon the same Scale from whence the 400 foot was taken, you shall find it to contain almost 231 foot, and so much is the altitude CA as before. CHAP. XI. How to take an in-accessible Altitude, by the Label and Tangent line. Then must you make a second work in the Triangle BCD, in which you have given, 1. The angle BDC, 64 degrees, 2. The angle DBC 26 degrees, 3. The side DB, 633 foot, And you are to find the side BC, the altitude required, wherefore say again, As the sine of the angle BCD, 90 degrees, Is to the Logarithm of the side DB 633 foot; So is the sine of the angle BDC 64 degrees, To the Logarithm of the Altitude BC: Which according to the former Doctrine will be found to be 569 foot. CHAP. XII. How to Protract the observation taken in the last Chapter. WHen you have made your observation as in the last Chapter, and noted down in a Book or otherwise, that the degrees cut at your first station at A were 50, and the degrees cut at the second station at D were 64, and that your stationary distance AD was 200 foot, you may immediately find the Altitude BC by protraction, thus. First, draw a line as AC, in which line let A represent your first station, whereon lay the centre of your Protractor, and make the angle BAC to contain 50 degrees (as hath been several times before shown:) and draw the line AB. Then upon the line AC set off the distance of your two stations 200 foot from A to D, then bring your Protractor to D (which represents your second station) and placing the centre of your Protractor thereupon, set off an angle of 64 degrees, as BDC, and draw the line DB, then where these two lines AB and DB intersect or meet, which is in the point B, from that point let fall the perpendicular BC, the length whereof being measured upon the same Scale from whence you took the distance AD, will give you 569 foot, and that is the altitude of AB, which was required. CHAP. XIII. How to take the distance of divers places one from another, according to their true situation, in plano, and to make (as it were) a Map thereof, by the Plain Table. THis Proposition is of good use to describe in plano the most eminent places in a Town or City, and to make (as it were) a Map thereof. Let A B C D E F G, be certain eminent places situate in some Town or City, and let it be required to describe all those places upon paper, by which the distance of any of them one from another, may be readily found. At some convenient distance from the City, Town, or Field, make choice of two other convenient places as K and L, from either of which you may plainly discern all the marks which you intent to describe in your Map. Then, at one of these places, (as at K) place your Table, and near one of the sides thereof draw a line parallel to the edge of the Table; In this line assign any point, as K, for your first station, and laying the Index upon this line, turn the Table about, till through the sights you espy the other place which you intent for your second station, which found, screw the Table fast there. Then laying the Index to the point K, turn it about, till through the sights you espy your first mark at A, and by the side of the Index draw the line AK. Secondly, turn the Index to the second mark at B, and draw the line BK. Thirdly, direct your sights to C, and draw the line CK. Fourthly, direct your sights to D, and draw the line DK. Fifthly, direct the sights to E, and draw the line EKE. Sixtly, direct the sights to F, and draw the line KF. Lastly, direct the sights to G, and draw the line KG, so have you finished your work at your first station. This done, with your Chain, measure the distance of your two stations K and L, which suppose to contain 800 foot, and removing your Table to L, lay the Index upon the line KL, turning the Table about, till through the sights you see your first station at K, and there screw it fast so that it altar not so long as your work continueth. Then laying the Index to the point L, direct your sights to the several marks as before, namely, to A C B F D E G, and from each of those marks draw lines by the side of the Index, as ALL, CL, BL, FL, DL, EL, and GL, so is your work finished at your second station also. Having thus done, first observe where the line KA crosseth the line LA, which is at A, at which point you may draw the figure, or write the name of the thing which it representeth. Secondly, observe where the line KB crosseth the line LB, which is at B, at which point writ the name of the place as before. Thirdly, observe where the lines KC and LC intersect, which is at C, at which point also note the place. Fourthly, at the intersection of KD and LD which is at D, writ the name of the place as before. Do thus with all the rest of the places be they never so many, so shall the several points of intersection ABCDEFG upon your Table, represent the respective places in the Town or City. Now to know the distance of any of these places one from another, you must take the distance required in your Compasses, and apply it to the same Scale by which the stationary distance KL was laid down, and it will there show you the distance required. CHAP. XIV. How to perform the work of the last Chapter by the Theodolite. AS in the last Chapter, make choice of two places, from either of which you may conveniently see all those Marks which you intent to describe, which two places let be K and L. Then placing the Instrument at K, lay ●he Index on the Diameter thereof, and turn the whole Instrument about till through the sights you espy your second station at L: then fixing the Instrument there, direct your sights to the several marks A B C D E F G, observing what degrees the Index cutteth when directed to any of the marks intended. As, suppose, your Instrument being fixed at K, and the sights directed to A, the Index cuts 83 degrees 50 minutes; at B, 97 degrees 55 minutes; at C, 114 degrees 10 minutes; at D, 123 degrees 40 minutes; at E, 134 degrees 35 minutes; at F, 138 degrees 30 minutes; and at G, 155 degrees 20 minutes. Then removing your Instrument to L, lay the Index on the Diameter thereof, and turn it about till through the sights you espy your former station at K, as is before taught: Then directing the sights to your first mark A, the Index cuts 33 degrees 50 minutes; at C, 43 degrees 40 minutes; at B, 54 degrees 10 minutes; at F, 64 degrees; at D, 73 degrees 20 minutes; at E 87 degrees 15 minutes; and at G, 113 degrees 40 minutes. These several observations of the degrees cut by the Index at both stations, aught to be noted in a Book or paper, together with the stationary distance, as in this example. deg. min. First Station A 83 50 B 97 55 C 114 10 D 123 40 E 134 35 F 138 30 G 155 20 The stationary distance 800 Foot. Second Station A 33 50 C 43 40 B 54 10 F 64 00 D 73 20 E 87 15 G 113 40 By help of this Table of your observations, you may at any time protract the same upon paper, and making a Scale of equal parts answerable to the parts of your stationary distance, you may with your Compasses measure the distance of any of these marks or places one from another, or from either of your stations. CHAP. XV. How to protract the former Observations upon paper, and to make a Scale to measure any of the Distances. YOur paper or parchment being provided, draw thereupon a line at length, and therein assign two points as K and L, representing your two stations, then upon your first station at K, lay the Centre of your Protractor, with the Meridian line thereof (which is noted with OF) directly upon the line KL. Then lay the Table of your observations before you, and seeing that at your first observation the Index cut 83 degrees 50 minutes, you must therefore with your protracting pin make a mark against 83 degrees 50 minutes of your Protractor. Again, seeing that at your second observation the Index cut 97 degrees 55 minutes, therefore, with your protracting pin, make a mark upon your paper, against 97 degrees 55 minutes of your Protractor. And thirdly, seeing that at your third observation your Index cut 114 degrees 10 minutes, you must likewise make a mark against 114 degrees 10 minutes, and thus must you do with all the rest of your observations, be they never so many. Which being done, from the point or station K, you must draw the straight lines KA, KB, KC, KD, etc. Then remove your Protractor to L, which signifies your second station, laying the Meridian line thereof upon the line KL, and then by your Table, note the angles of your observations made at your second station in all respects as you did those of your first station: so shall you find that at the first observation at your second station, the Index cut 33 degrees 50 minutes, therefore, with your protracting pin make a mark upon the paper against 33 degrees 50 minutes of the Protractor. Again, the degrees cut at your second observation were 43 degrees 40 minutes, therefore make a mark against 43 degrees 40 minutes of your Protractor. Also, the degrees cut at your third observation were 54 degrees 10 minutes, against which likewise make a mark, dealing with all the rest of your observations in the same manner: then through these several points, from your station L, draw straight lines till they intersect those lines before drawn from K, which will be the points A B C D E F and G, which points bear a just proportion to the Marks which you observed. Now to find the distance of any of these marks one from another, you must divide a line into such equal parts, so that your stationary distance KL may contain 800 of them. Your Scale being thus made, take in your Compasses the distance between any two marks or places here described, and apply it to your Scale so shall it exactly show you the true distance between the two places so taken, in the same parts as the the line KL was divided. In this manner may you with speed and exactness attein the true distance and situation of any Mark or Marks far remote, without approaching near any of them: and thus in overgrown land, where you can neither go about it, nor measure within it, this Chapter will be of excellent use. CHAP. XVI. How to take the true plot of a field at one station taken within the same field, so that from thence you may see all the angles of the same field, by the Plain Table. WHen you enter any field to survey, your first work must be to set up some visible mark at each angle thereof, or let one go continually before you to every angle, holding up a white cloth, or the like, to direct you: which being done, make choice of some convenient place about the middle of the field, from whence you may behold all your Marks, and there place your Table covered with a sheet of paper, the needle hanging directly over the Meridian line of the Card (which you must always have regard unto, especially when you are to survey many fields together.) Then make a mark about the middle of your paper, which shall represent that part of the field where your Table standeth, and laying the Index unto this point, direct your sights to the several angles where you before placed your marks, and draw lines by the side of the Index upon the paper; then measure the distance of every of these marks from your Table, and by your Scale set the same distances upon the lines drawn upon the Table, making small marks with your Protracting pin or Compass point at the end of every of them; then lines being drawn from one to another of these points, you shall have upon your Table the exact plot of your Field, all the lines and angles upon the Table being proportional to those of the Field. Suppose you were to take the plot of the Field ABCDEF. Having placed marks in the several angles thereof, make choice of some convenient place about the middle of the Field, as at L, from whence you may behold all the marks before placed in the several angles, and there place your Table, then turn your Instrument about, till the needle hang over the Meridian line of the Card, the North end of which line is noted with a Flower-de-luce, and is represented in this figure by the line NS. Your Table being thus placed, with a sheet of paper thereupon, make a mark about the middle of your Table which shall represent that place in the field where your Table standeth: then, applying your Index to this point, direct the sights to the first mark at A, and the Index resting there, draw a line by the side thereof to the point L, then with your Chain measure the distance from L, the place where your Table standeth, to A your first mark, which suppose to be 8 Chains 10 Links, then take 8 Chains 10 Links from any Scale, and set that distance upon your Table from L to A, and at A make a mark. Then directing the sights to B your second mark, draw a line by the side of your Index as before, and measure the distance from your Table at L, to your mark at B, which suppose 8 Chains 75 links, this distance must be taken from your Scale, and set upon your Table from L to B, and at B make another mark. Then direct the sights to the third mark C, and draw a line by the side of the Index, measuring the distance from L to C, which suppose 10 Chains 65 links; this distance being taken from your Scale and applied to your Table from L to C, shall give you the point C, representing your third mark. In this manner you must deal with the rest of the marks at D E and F, and more, if the field had consisted of more angles. Lastly, when you have made observation of all the marks round the Field, and found the points A B C D E and F upon your Table, you must draw lines frnm one point to another till you conclude where you first began: as draw a line from A to B, from B to C, from C to D, from D to E, from E to F, and from F to A, where you began: then will ABCDEF be the exact figure of your Field, the sides and angles of the said figure bearing an exact proportion to those in the Field, and the line NS, in this and the following figures, always representeth the Meridian line. CHAP. XVII. How to take the plot of a field at one station taken in the middle thereof by the Theodolite. PLace marks at the several angles of the Field as before, and make choice of some convenient place about the middle thereof, as L, from whence you may see all the marks, and there place your Instrument, the Needle hanging directly over the Meridian line in the Card. This done, direct your sights to the first mark at A, noting what degrees the Index cutteth, which let be 36 degrees 45 minutes, these 36 degrees 45 minutes must be noted down in your Field-book in the first and second Columns thereof. Then measure the distance from L the place of your Instrument, to A your first mark, which let contain 8 Chains 10 Links, these 8 Chains 10 Links must be placed in the third and fourth Column of your Field-book, as hath been directed in the description thereof. Then direct the sights to B your second mark, and note the degrees cut by the Index, which let be 99 degrees 15 minutes, and the distance LB 8 Chains 75 Links, the 99 degrees 15 minutes must be noted in the first and second Columns of your Field-book, and the 8 Chains 75 Links in the third and fourth Columns. Then direct your sights to C, your third mark, and note the degrees cut by the Index, which let be 163 degrees 15 minutes, and let the distance LC be 10 Chains 65 Links; the 163 degrees 15 minutes must be noted in the first and second columns of your field-book, and the 10 Chains 65 Links in the third and fourth columns thereof. Then direct your sights to D, your fourth mark, and note the degrees cut by the Index; which let be 212 degrees: ¶ And here you must note that in using the degrees on the frame of the Table, that after the Index hath passed 180 degrees, which is at the line NS (representing always the Meridian line) you must then count the degrees backward, according as they are numbered on the frame of the Table, from 190 to 360. Then measure the distance LD, which let be 8 Chains 53 Links; the 212 degrees must be noted in the first Column of your field-book, and the 8 Chains 53 Links in the third and fourth Columns thereof. Then direct your sights to E, the Index cutting 287 degrees 15 minutes, and the distance LE being 8 Chains 15 Links, these must be noted in your field-book as before, the 287 degrees 15 minutes in the first and second columns, and the 8 Chains 15 Links in the third and fourth. Lastly, direct the sights to F, your last mark, the Index cutting 342 degrees, and the distance LF being 9 Chains 55 Links, these must be noted down in your field-book in all respects as the former, viz, the 342 degrees in the first column, and the 9 Chains 55 Links in the third and fourth: then will your observations noted in your Field-book stand as in this Table following. Degrees Minute's Chains Links A 36 45 8 10 B 99 15 8 75 C 163 15 10 65 D 212 00 8 53 E 287 15 8 15 F 342 00 9 55 CHAP. XVIII. How to take the plot of a Field at one station taken in the middle thereof by the Circumferentor. THere is little difference between the work of this and the last Chapter: for, the marks being placed in the several angles of the field, and the station appointed at L, place there the Instrument, and turning it about, direct the sights to A (the Flower-de-luce of the Card being always towards you) the South end of the Needle cutting 36 degrees 45 minutes, the same which the Index of the Theodolite did in the last Chapter, then measuring the distance from L to A, you will find it to contain, as before, 8 Chains 10 Links, which you must note down in your Field-book as in the last Chapter. Then turning the whole Instrument about (as before) direct the sights to B, the South end of the Needle cutting 99 degrees 15 minutes, and the distance LB will contain 8 Chains 75 Links, which note down in your Book also. In this manner must you direct the sights to all the other angles C D E and F, and you shall find the South end of the Needle always to cut the same degrees in the Card as the Index of the Theodolite did, and the measured lines LC, LD, LE, and LF, will be likewise the same, so that the Table of observations in the last Chapter will serve to protract either this or the other work, as is taught in the next Chapter. CHAP. XIX. How to protract any observations taken according to the directions in the last Chapter. FIrst, draw upon your paper or parchment a line at length, which shall represent the Meridian line NS in the figure, then make choice of some point or other in that line, which shall represent your station or place of standing in the Field, as K: upon this point place the centre of your Protractor, so that the Meridian line OF of the Protractor, may lie directly upon the Meridian line NS of this figure. Then laying your Field-book before you; seeing that at your first observation at A, the Index of the Theodolite, or the Needle of the Circumferentor, cut 36 degrees 45 minutes, you must therefore against 36 degrees 45 minutes of your Protractor make a mark upon your paper. 2. Seeing the degrees cut at your second observation were 99 degrees 15 minutes, you must make a mark upon your paper against 99 degrees 15 minutes of your Protractor. 3. The degrees cut at your third observation were 163 degrees 15 minutes, therefore agaigst 163 degrees 15 minutes make a mark upon your paper. 4. The degrees cut by the Index or Needle at your fourth observation being 212 degrees,— ¶ Now because 212 degrees is greater than 180 degrees, you must therefore turn the Semicircle of the Protractor downwards, yet the line OF thereof must lie directly upon the Meridian line NS, as before. — you must against 212 degrees of the Protractor make a mark upon your paper. 5. Seeing the degrees cut at your fifth observation were 287 deg. 15 minutes, therefore make a mark against 287 degrees 15 minutes of the Protractor. Lastly, the degrees cut at your last observation were 342, therefore against 342 degrees of your Protractor make a mark with your Protracting pin, as before. This done, you must observe by your Field-book the length of every line. As the line LA at your first observation was 8 Chains 10 Links, therefore, 8 Chains 10 Links being taken from your Scale, and set upon your paper from L to A, it shall give you the point A upon your paper. 2. The length of your second line being 8 Chains 75 Links, you must take 8 Chains 75 Links from your Scale, and set it upon your paper from L to B. 3. The line LC being 10 Chains 65 Links, you must therefore take 10 Chains 65 Links from your Scale, and set it upon your paper from L to C. And thus must you deal with all the rest of the lines, as LD, LE, and LF. Lastly, draw the lines AB, BC, CD, DE, OF, and FAVORINA, so shall you have the exact figure of the Field upon your paper. ¶ In these four last Chapters you are taught how to take the plot of any field at one station taken in the midst thereof, both by the Plain Table, Theodolite, and Circumferentor, and also how to protract the same. This way of plotting of a field is seldom, or never, used in surveying of divers parcels, but for one particular field it is as good as any, but divers other varieties will appear in the following Chapters. CHAP. XX. How to take the plot of a Field at one station taken in any angle thereof, from whence all the other angles may be seen, by the Plain Table. PLace your Table in some convenient angle in the Field to be measured, and turn it about till the Needle hang directly over the Meridian line in the Card, and there fix it: then draw a line parallel to the side of your Table, as NS, in which line assign any point at pleasure, as H, which shall represent your station or place of standing, unto this point apply the Index, and direct the sights to A and draw a line upon your paper as HA'; and measure the distance HA' (as was directed before in Chap. 16.) Then direct the sights to B, your second mark, and there likewise draw a line HB, measuring the distance HB, as was taught in the forementioned Chapter. In like manner direct the sights to C D E F and G, drawing lines by the side of your Index at every observation, and measure with your Chain the distance from H (the place where your Instrument standeth) to the several angles of the Field A, B, C, D, E, F, and G; which distances being taken in your Compasses, from any Scale, and set upon your Table from H upon the several lines HA', HB, HC, HD, HE, HF, and HG, so shall you have upon your Table the points A, B, C, D, E, F, and G, by which marks draw the lines HA', AB, BC, CD, DE, OF, FG, and GH, which lines will include the exact figure of the Field upon your Table. CHAP. XXI. How to take the plot of a Field at one station taken in any angle thereof by the Theodolite. IN the same figure following, having placed your Instrument at H, as is taught in the foregoing Chapter, direct the sights to A, your first mark, noting the degrees cut by the Index, which suppose 22 degrees 15 minutes, these degrees and minutes must be noted in the first and second columns of your Field-book (as hath been before sufficiently taught.) Then with your Chain measure the distance from your station at H to the angle A, which let be 8 Chains 46 Links, which you must place in the third and fourth columns of your Field-book, according to the former directions. 2. Direct your sights to B, noting the degrees there cut, which suppose 42 degrees, 45 minutes, these degrees and minutes place in the first and second Columns of your Field-book, and measure the distance HB, 15 Chains 21 Links, and note them down in the third and fourth Columns thereof. 3. Direct your sights to C, the degrees cut being 66 degrees 30 minutes, and the distance HC 16 Chains 64 Links, note these also in your Field-book as before. And in this manner must you deal with the other marks D, E, F, and G, so having noted them all in your Field-book they will stand as followeth. Degrees Minute's Chains Links A 22 15 8 46 B 42 45 15 21 C 66 30 16 64 D 86 45 16 23 E 122 30 16 68 F 130 15 15 22 G 162 00 7 73 CHAP. XXII. How to take the Plot of a field at one station taken in any angle thereof, from which all the rest may be seen, by the Circumferentor. PLace your Instrument at H, and direct the sights to A (observing the cautions formerly delivered in the use of this Instrument) the Needle cutting 22 degrees 15 min. and the distance HA' containing 8 Chains 46 Links, which agrees exactly with the first observation in the last Chapter: these degrees and minutes, together with the measured distance HA', must be noted down in the several Columns of your Field-book, and if you make observation round about the field, from angle to angle, and measure the length of every line from H, to B C D E F and G, you shall find the degrees cut by the Needle, to be the same with those (in the last Chapter) cut by the Index, and the measured distances to be likewise equal: and if you make a Table of your observations, you shall find it the same with that in the last Chapter. CHAP. XXIII. How to Protract any observation taken according to the Doctrine of the two last Chapters. FIrst, draw the meridian line NS, and make choice of a point therein representing your stationary angle, as at H, to which point apply the centre of your Protractor, the Semicircle upwards. Then laying your Field-book before you, you may perceive that at your first observation (which was at A) the Index of the Theodolite, or the Needle of the Circumferentor cut 22 degrees 15 minutes, therefore make a mark against 22 degrees 15 minutes, and draw the line HA. 2. The degrees cut at your second observation at B, being 42 degrees 45 minutes, make a mark likewise against 42 degrees 45 min. of your Protractor, and draw the line HB. 3. The degrees cut at your third observation being 66 deg. 30 mi. make a mark against 66 degrees 30 minutes, and draw the line HC. And in this manner must you proceed with the rest of your observations, D, E, F, and G. Having thus protracted your angular observations, proceed now to your lineal, namely, to the length of your lines, noted in the third and fourth Columns of your Field-book. 1. Seeing that the length of your first line HA' was 8 Chains 46 Links, you must take 8 Chains 46 Links from your Scale, and apply it to your paper from H unto A. 2. The length of your second line HB, being 15 Chains 21 Links, take 15 Chains 21 Links from your Scale, and apply that distance to your paper from H unto B. 3. The distance of your third mark HC being 16 Chains 64 Links, take that distance from your Scale, and apply it to your paper from the point H unto C. In all respects as before, you must proceed with the measuring of all the other lines about the field, were they never so many. Lastly, if from these points A B C D E F G and H, you draw the lines AB, BC, CD, DE, OF, FG, and GH, you shall have upon your paper the exact figure of your field. ¶ And herein you may receive abundant satisfaction, to see your several Instrumental operations, and your Geometrical protraction so exactly to agree: and if at any time you make several observations of any one piece of ground, according to the directions of the foregoing Chapters, or the like, if you find them not exactly to agree, you may be sure you have failed in one or other of your observations, and therefore, before you proceed further, it is best to reform your first error. CHAP. XXIV. How to take the Plot of a Field at two stations taken in any parts thereof, by measuring from either of the stations to the visible angles, by the Plain Table. THis manner of work is chiefly to be used in such Fields which are so irregular that from any one part thereof you cannot discern all the angles, or else in such whose largeness will not permit a sufficient view of all the angles at once. The manner of work will be the very same with that in the 16 Chap. only the Instrument, in this, must be placed in two several places, whereas, in that, the same thing was effected at once placing of the Instrument. Suppose then that ABCDEFGHIKL and M, were such an irregular Field as is before spoken of. Having made choice of two places within the same for your two stations, as O and Q, from which you may conveniently see all the angles. First, place your Table at O, turning it about till the needle hang directly over the Meridian line in the Card, represented in this figure by the line NOS. Then fixing the Table there, you must (1.) direct the sights to A, and by the side of the Index draw the line AO, containing 7 Chains 46 Links. (2.) direct the sights to B, and draw the line BOY, containing 7 Chains 18 Links. (3.) direct the sights to C, and draw the line OC, containing 7 Chains 21 Links. (4.) direct the sights to D, and draw the line ODD, containing 6 Chains 33 Links. (5.) direct the sights to E, and draw the line OE, containing 5 Chains 57 Links. (6.) direct the sights to K, and draw the line OK, containing 7 Chains 83 Links. (7.) direct the sights to L, and draw the line OL, containing 9 Chains 95 Links. (8.) direct the sights to M, and draw the line OM, containing 5 Chains 8 Links. Having thus made observation of these angles which are all that can conveniently be seen from your first station at O, and drawn the several lines OA, OB, OC, ODD, OE, OK, OL, and OM, and upon them set the several lengths as you found them by measuring, as from O to A, 7 Chains 46 Links, from O to B, 7 Chains 18 Links, etc. you must then lay the Index again to the point O, and direct the sights to your second station at Q, drawing the line OQ, then measure the distance from O to Q, which let contain 8 Chains 89 Links. Then remove your Instrument to Q, and lay the Index upon the line OQ, turning the table about till through the sights you espy your first station at O, then will the Needle hang directly over the Meridian line in the Card as before, and your Instrument is truly situated in the same position as before, so that you may now deal with the angles F, G, H, and I. (which before you could not conveniently see) as you did with those on the other side of the field, by laying the Index to the point Q, and directing the sights, (1.) to E, and drawing the line QE, containing 5 Chains 10 Links. (2.) to F, and drawing the line QF, containing 7 Chains 64 Links. (3.) to G, and drawing the line QG, containing 6 Chains 40 Links. (4.) to H, and drawing the line QH, containing 5 Chains 33 Links. (5.) to I, and drawing the line QI, containing 6 Chains 95 Links. (6.) to K, and drawing the line QK, containing 7 Chains 61 Links. These angles being observed and the lines measured as the former were, you shall find the several points E, F, G, H, I, and K, on this side of the Field also, so that you may draw the lines AB, BC, CD, DE, OF, FG, GH, HI, IK, KL, LM, and MA, which shall represent upon your Table the exact figure of the field to be measured. ¶ And here note, that in this Example I make observation of the angles E and K at both stations, but there was no deed thereof, only this satisfaction will accrue thereby, for when you have measured your stationary distance OQ, and removed your Instrument to Q, and there fixed it, when you direct the sights to E or K, and measure the distance QE or QK, and set it off from Q, you shall find the points E and K to fall directly upon the same points E and K formerly drawn, if there be no error in your work. And in this manner may you make three four or five stations for one field if need so require, remembering always, that at every station the Needle hang directly over the Meridian line, or the same degree of the Card at every station. CHAP. XXV. How to take the true Plot of a Field at two stations taken in any parts thereof, from whence the angles may be seen by the Theodolite. YOur stations O and Q being chosen, place your Instrument in the Field at O, and turn it about till the Needle hang over the Meridian line, and there fixing it, direct the sights to A, the Index cutting 19 degrees 10 minutes, and the line OA containing 7 Chains 46 Links, the 19 degrees 10 minutes must be placed in the first and second columns of your Field-book, and the 7 Chains 46 Links in the third and fourth columns thereof. Then direct the sights to B, the Index cutting 53 degrees 30 minutes, and the line OB containing 7 Chains 18 Links, which note down in your Field-book as before. In this manner proceed with the rest of the lines and angles, namely, so many as you intent to observe at your first nation, viz. A, B, C, D, E, K, L, and M: which done, direct the signs to your second station at Q, the Index cutting 18 degrees 15 minutes, which note down in your Field-book by itself: Also measure the stationary distance OQ, 8 Chains 89 Links, as before, this also must be noted in your Field-book. Having thus finished one part of the Field, remove your Instrument to Q, and laying the Index upon 18 degrees 15 minutes, (which is the inclination or difference of Meridian's between your two stations) turn it about till through the sights you espy your first station at O, then will the Needle hang over the Meridian line, and the Instrument will be truly situate. Deg. Min. Chai. Links A 19 10 7 46 The first station at O. B 53 30 7 18 C 95 15 7 21 D 132 00 6 33 E 166 30 5 57 K 251 30 7 83 L 282 00 9 95 M 304 30 8 05 The stationary distance OQ is 8 Chains 89 Links, and the angle OQN 18 degrees 15 minutes, the inclination or difference of Meridian's. E 52 15 5 10 The second station at Q. F 99 30 7 64 G 148 30 6 40 H 232 30 5 33 I 275 00 6 95 K 321 30 7 61 CHAP. XXVI. How to take the Plot of a Field at two stations taken in any parts thereof, by the Circumferentor. THe use of this Instrument in taking the plot of a field by observing the lines and angles in the midst thereof, is sufficiently shown already in Chap. 18. and the work of this Chapter differeth nothing therefrom, only in this you make observation in two places. Therefore placing the Instrument at O, and directing the sights to A B C D E K L and M, you shall find the degrees cut by the Needle to be the same with those collected in your Field-book at your first station at O. Also, your Instrument being removed to Q, and observation made of the several angles there, namely of the angles E F G H I and K, they will likewise be found the same with those observed by the Theodolite at your second station in the last Chapter, and therefore to make repetition thereof again in this place, were superfluous. ¶ Here note, that the Plain Table and Theodolite are the most convenient Instruments for these kind of practices hitherto treated of, and not the Circumferentor, I only have hinted the use thereof, that the agreement of the several Instruments might be taken notice of, the Circumferentor serving chiefly for large Champion plains and Wood-lands, as will appear hereafter. CHAP. XXVII. How to protract any observations taken according to the directions of the two last Chapters. DRaw upon your paper the Meridian line NOS, the point O representing your first station: upon this point O place the centre of your Protractor, laying the line OF thereof, directly upon the Meridian line N S. Then laying your Field-book before you, observe the degrees there noted, namely, (1.) at A, 19 degrees 10 minutes, the line OA containing 7 Chains 46 Links. (2.) at B, 53 degrees 30 minutes, the line OB containing 7 Chains 18 Links. (3.) at C, 95 degrees 15 minutes, the line OC containing 7 Chains 21 Links. And so of the rest, against which degrees and minutes make marks by the edge of your Protractor, and daw lines from O through those marks, as OA, OB, OC, ODD, OE, OK, OL, AM, and upon those lines set off the lengths from O, as you find them collected in your Field-book. Having thus protracted the observations of your first station (before you move your Protractor) make a mark against 18 degrees 15 minutes which is the inclination or difference of Meridian's, and draw the line OQ, setting off 8 Chains 89 Links the length thereof from O to Q. Then upon the point Q, place the centre of the Protractor as before, moving it up and down till the line OQ lies just under 18 degrees 15 minutes, and holding it there, lay your Field-book before you, and prick down by the side thereof the several degrees and minutes as by your Instrument you observed them, together with the lengths of the lines as they were measured, drawing lines through those points also, as the lines QE, QF, QG, QH, QI, and QK. Lastly, draw the lines AB, BC, CD, DE, OF, etc. so shall you have upon your paper the exact plot of your field, in which (if there be no error in your work) the line MA being drawn will close exactly with the line BASILIUS in the point A. CHAP. XXVIII. How to take the Plot of a field at two stations taken in the middle thereof, from either of which all the angles in the field may be seen, with the measuring of one line only, by the Plain Table. NEcessity may sometimes require the plotting of a field according to the directions which I shall deliver in this Chapter, yet I would have as little use made thereof as possible can be, in regard of the acuteness of the angles, which is more liable to error then any of the ways formerly taught, although it be grounded upon as firm a Geometrical principle, as any of them. Let ABCDEFGH be the figure of a Field, and let the two stations taken within the same be O and Q. Having placed your Instrument at O, your first station, the Needle hanging directly over the Meridian line of the Card, you must, (1.) direct the sights to A, and draw the line OA. (2.) direct the sights to B, and draw the line OB. (3.) direct the sights to C, and draw the line OC. (4.) direct the sights to D, and draw the line ODD. (5.) direct the sights to E, and draw the line OE. (6.) direct the sights to F, and draw the line OF. (7.) direct the sights to G, and draw the line OG. (8.) direct the sights to H, and draw the line OH. This done, direct the sights to your second station at Q, and draw the line OQ upon your Table: then (with your Chain) measure out your stationary distance OQ, which is 7 Chains, and removing your Instrument to Q (the needle hanging over the Meridian line of the Card as before) make observation as you did at O; As, (1.) direct the sights to A, and draw the line QA. (2.) direct the sights to B, and draw the line QB. (3.) direct the sights to C, and draw the line QC. (4.) direct the sights to D, and draw the line QD. (5.) direct the sights to E, and draw the line QE. (6.) direct the sights to F, and draw the line QF. (7.) direct the sights to G, and draw the line QG. (8.) direct the sights to H, and draw the line QH. Now you may plainly perceive by the figure where the correspondent lines at each station intersect or cross each other; as, (1.) the lines OA and QA intersect each other at A. (2.) the lines OB and QB, intersect each other at B. (3.) the lines OC and QC, intersect each other at C. (4.) the lines ODD and QD, intersect each other at D. (5.) the lines OE and QE, intersect each other at E. (6.) the lines OF and QF, intersect each other at F. (7.) the lines OG and QG, intersect each other at G. (8.) the lines OH and QH, intersect each other at H. Therefore, if from one to another of these points successively you draw lines, you shall have upon your paper the exact symmetry or proportion of your field, as namely, the lines AB, BC, CD, DE, etc. In this kind of plotting you cannot but perceive a wonderful quick dispatch, you being to measure nothing but the distance between your stations, but by reason of the acuteness of the angles (without exact and curious drawing of your lines, and observing the precise points of intersection) you may run into gross absurdities and mistakes. CHAP. XXIX. How to take the Plot of a field at two stations taken in any part thereof, from either of which all the angles in the field may be seen, and measuring only the stationary distance, by the Theodolite or Circumferentor. YOu may perceive by what hath been said in the foregoing Chapters, that the manner of work is the same both with the Theodolite and Circumferentor, and therefore in this place I make but one example for both Instruments. Now to take the plot of the field ABCDEFG and H, by either of these Instruments, place your Instrument at O your first station, and turn it about till the needle hang over the Meridian line NS, and fixing it there, (1.) direct the sights to A, the Index or Needle cutting 21 degrees 30 minutes. (2.) direct the sights to B, the Index or Needle cutting 69 degrees 15 minutes. (3.) direct the sights to C, the Index or Needle cutting 124 degrees 45 minutes. (4.) direct the sights to D, the Index or needle cutting 168 degrees 10 minutes. (5.) direct the sights to E, the Index or Needle cutting 202 degrees 30 minutes. (6.) direct the sights to F, the Index or Needle cutting 237 degrees 30 minutes. (7.) direct the sights to G, the Index or needle cutting 307 degrees 00 minutes. (8.) direct the sights to H, the Index or needle cutting 328 degrees 30 minutes. This done, measure your stationary distance OQ, which suppose to contain 7 Chains, and remove your Instrument to Q, turning it about till the Needle hang directly over the Meridian line as before, and there fix it; then, (1.) direct the sights to A, the Index or Needle cutting 11 degrees 00 minutes. (2.) direct the sights to B, the Index or Needle cutting 35 degrees 30 minutes. (3.) direct the sights to C, the Index or Needle cutting 79 degrees 45 minutes. (4.) direct the sights to D, the Index or Needle cutting 153 degrees 15 minutes. (5.) direct the sights to E, the Index or Needle cutting 224 degrees 30 minutes. (6.) direct the sights to F, the Index or Needle cutting 279 degrees 30 minutes. (7.) direct the sights to G, the Index or Needle cutting 329 degrees 00 minutes. (8.) direct the sights to H, the Index or Needle cutting 347 degrees 30 minutes. Having thus made observation of all the angles round about the field at both stations and noted the degrees cut by the Index of the Theodolite or the Needle of the Circumferentor, and noted them down in your Field-book, together with the distance between your two stations, you may proceed to protract your work as is taught in the next Chapter. CHAP. XXX. How to protract any observations taken according to the directions of the last Chapter. FIrst draw the Meridian line NS, upon which line assign any point at pleasure, as O, for your first station, unto which point apply the centre of your Protractor, with the line OF thereof upon the Meridian line NS. Then look into the Field-book for the degrees observed at your first station at O, and make marks against those degrees by the edge of your Protractor, and when you have marked them all, draw lines from O through every of them, as the lines OA, OB, OC, etc. Then from your Scale take 7 Chains (which is your stationary distance) and place it from O to Q, which represents your second station, upon this point Q, place the centre of your Protractor, and laying your Field-book before you, prick down the degrees by the edge of the Protractor, as you find them noted in your Field-book at your second station at Q, and through those points draw the lines QA, QB, QC, etc. The line QA crossing the line OA in the point A. The line QB crossing the line OB in the point B. The line QC crossing the line OC in the point C. The line QD crossing the line ODD in the point D. The line QE crossing the line OE in the point E. The line QF crossing the line OF in the point F. The line QG crossing the line OG in the point G. The line QH crossing the line OH in the point H. Therefore if you draw the lines AB, BC, CD, DE, OF, FG, GH, and HA', it shall be the exact plot or figure of the field required. ¶ I might now proceed to show you the manner of taking the plot of any field without approaching nigh the same; but in regard the performance thereof differeth nothing at all from that which is already taught in the 13, 14, and 15 Chapter of the fourth Book, I shall therefore in this place pass it over as superfluous. CHAP. XXXI. How to take the Plot of a Wood, Park, or other large Champion plain by the Plain Table, by measuring round about the same, and making observation at every angle, HItherto we have showed how the plot of any plain and even ground, or any small enclosure may be taken several ways, as being the easiest for a practitioner to try experience upon, I now come to show how the plot of any large Champion plain, or overgrown wood may be measured, for in such kind of grounds the former directions will be of little validity, for the largeness of the plain, or the thickness of the wood may many times hinder both your sight and measuring; therefore the best way to measure these kind of Lands is to go about them, and make observation at every angle. Suppose the following figure ABCDEFG to be a large Wood or other Champion plain, whose Plot you desire to take upon your Plain Table. 1. Place your Instrument at the angle A, directing your sights to the next angle at B, and by the side thereof draw a line upon your Table, as the line AB, then measure by the hedge side from the angle A to the angle B, which suppose 12 Chains 5 Links, then from your Scale take 12 Chains 5 Links, and set that distance upon your Table from A to B. 2. Remove your Instrument from A, and set up a mark where it last stood, and place your Instrument at the second angle at B; then laying the Index upon the line AB, turn the whole Instrument about till through the back-sights you see the mark which you set up at A, and there screw the Instrument: then laying the Index upon the point B, direct your sights to the third angle at C, and draw the line BC upon your Table, then measuring the distance BC 4 Chains 45 Links, take that distance from your Scale and set it upon your Table from B to C. 3. Remove your Instrument from B, and set up a mark in the room thereof, and place your Instrument at C, laying the Index upon the line CB, and turn the whole Instrument about till through the back-sights you espy your mark set up at B, and there fasten the Instrument: then laying the Index on the point C, direct the sights to D, and draw upon your Table the line CD, then measure from C to D 8 Chains 85 Links, and set that distance upon your Table from C to D. 4. Remove your Instrument to D (placing a mark at C where it last stood) and lay the Index upon the line DC, turning the whole Instrument about till through the back-sights you espy the mark at C, and there fasten the Instrument: then lay the Index on the point D, and direct the sights to E, and draw the line DE, then with your Chain measure the distance DE 13 Chains 4 Links, and set that distance upon your Table from the point D unto E. 5. Remove your Instrument to E (placing a mark at D where it last stood) and laying the Index upon the line DE, turn the whole Instrument about till through the back-sights you see your mark at D, and there fasten the Instrument: then lay the Index on the point E, and direct the sights to F, and draw the line OF, then measure the distance OF 7 Chains 70 Links, which take from your Scale, and set it on your Table from E to F. 6. Remove your Instrument to F (placing a mark at E where it last stood) and lay the Index upon the line OF, turning he Instrument about, till through the back-sights you see your mark set up at E, and there fasten the Instrument: then laying the Index on the point F, direct the sights to G, and draw the line FG upon your Table, then measure the distance FG 5 Chains 67 Links, and set that off upon your Table from F to G. In this manner may you take the plot of any Chamption plain be it never so large, and here note, that many times, hedges are of such a thickness that you cannot come near the sides or angles of the field, either to place your Instrument ot measure your lines; therefore, in such cases, you must place your Instrument, and measure your lines parallel to the side thereof, and then your work will be the same as if you measured the hedge itself. Note also, that in thus going about a field, you may much help yourself by the Needle, for look what degree of the Card the needle cuts at one station, if you remove your Instrument to the next station, and with your back-sights look to the mark where your Instrument last stood, you shall find the Needle to cut the same degree again, which will give you no small satisfaction in the prosecution of your work. CHAP. XXXII. How to take the Plot of a Wood, Park, or other large Champion plain, by going about the same, and making observation at every angle thereof, by the Theodolite. PLace your Instrument at the angle A, and lay the Index on the diameter thereof, turning the whole Instrument about till through the sights you espy the second angle at B, then fastening it there, turn the Index about till through the sights you see the angle at G, the Index cutting 130 degrees 00 minutes, which is the quantity of the angle GAB, and the line AB containing 12 Chains 5 Links, which you must note down in your Field-book as formerly. 2. Remove your Instrument to B, laying the Index on the diameter, and turn it about till through the sights you see the third angle at C, and there fasten it, then turn the Index backward till through the sights you see the angle at A, the degrees cut by the Index being 120 degrees 30 minutes, the quantity of the angle ABC, and the line BC containing 4 Chains 45 Links, which you must note in your Book as before. 3. Remove your Instrument to C, and lay the Index on the diameter thereof, turning the Instrument about till through the sights you see the fourth angle at D, and there fixing it, direct the sights back again to B, the Index cutting 137 degrees 30 minutes, and the line CD being 8 Chains 85 Links. 4. Place your Instrument at D, and lay the Index on the Diameter, turning the Instrument about, till through the sights you espy the fift angle at E, and there fixing it, turn the Index backward towards C, the degrees cut thereby being 120 degrees 30 minutes, and the line DE 13 Chains 4 Links, which must be noted in your Field-book. 5. Remove your Instrument to E, and lay the Index on the Diameter thereof, turning the Instrument about till through the sights you see the angle at F, and there fixing it, turn the Index backward to D, the degrees cut being 121 degrees 30 minutes, and the line OF 7 Chains 70 Links, which note down also. 6. Place your Instrument at F, and lay the Index on the Diameter thereof, turning the Instrument about till through the sights you see the angle at G, and there fixing it, turn the Index till through the sights you espy the former angle at E, the degrees cut being 126 degrees 30 minutes, and the length of the line FG being 5 Chains 67 Links. 7. Lastly, Place the Instrument at G, and lay the Index on the Diameter, turning the whole Instrument about till through the sights you espy the angle at A, and there fixing it, direct the sights back again to F, the degrees cut by the Index being 143 degrees 30 minutes, and the length of the line GA' 7 Chains 87 Links. Having thus made observation at every angle of the field in this manner, and collected the quantity of every angle, and the length of every line in your Field-book, you shall find them to stand as followeth. Degrees Minute's Chains Links A 130 00 12 5 B 120 30 4 45 C 137 30 8 85 D 120 30 13 4 E 121 30 7 70 F 126 30 5 67 G 143 30 7 87 CHAP. XXXIII. How to protract or lay down any observations taken according to the doctrine of the last Chapter. COnsider which way your Plot will extend, and accordingly upon the paper that you would have the Plot of your Field described, draw a line at pleasure, as the line GA. Then place the centre of your Protractor upon the point A, and (because the angle at your first observation at A, was 130 degrees 00 minutes) turn it about till the line AGNOSTUS lie directly under 130 degrees, and then at the beginning of the Protractor (which is at 00 degrees, noted (in the figure thereof pag. 51.) with the letter E,) make a mark, and through it draw the line AB, setting 12 Chairs 5 Links (the length of the same line) from A to B. 2. Lay the centre of your Protractor upon the point B, and seeing the degrees cut at B were 120 degrees 30 minutes, therefore turn the Protractor about till the line AB lies directly under 120 degrees 30 minutes, and then at the beginning of the degrees make a mark, and through it draw the line BC, the length thereof being 4 Chains 45 Links. 3. Lay the centre of the Protractor on the point C, turning it about till the line BC lies directly under 137 degrees 30 minutes, (which were the degrees cut at your observation at C,) and then making a mark at the beginning or 00 degrees of your Protractor, through it draw the line CD, setting 8 Chains 85 Links thereon from C to D. 4. Bring the centre of your Protractor to the point D, turning it about till the line CD lies directly under 120 degrees 30 minutes, and then making a mark at the beginning of the Protractor, through it draw the line DE, and upon it set 13 Chains 4 links, from D to E. In this manner must yond deal with all the rest of the angles, and when you come to protract the angle at F, which is the last angle, and have drawn the line FG, you shall find it to cut the line AGNOSTUS first drawn in the point G, leaving the line AGNOSTUS to contain 7 Chains 87 Links, and the line FG 5 Chains 67 Links; and in this, practice is better than many words, and the sight of the figure better than a whole Chapter of information, in which figure you may see the Protractor lie at every angle in its true position. This work may be performed otherwise, by protracting your last observation first, so having drawn the line AGNOSTUS, lay the centre of the Protractor on G, and the Meridian line thereof (namely OF) on the line GA', then (because the degrees cut at your observation at G were 143 degrees 30 minutes) make a mark with your protracting pin against 143 degrees 30 minutes, and through it draw the line GF, upon which line from G to F, set 5 Chains 67 Links. Then placing the centre of your Protractor on the point F, and the Meridian line thereof upon the line FG, making a mark by the edge of the Protractor against 126 degrees 30 minutes (which were the degrees cut by the Index at your observation at F) and through that point draw the line FE, setting 7 Chains 70 Links thereupon from F to E. And in this manner must you proceed with the rest of the lines and angles, and at last you shall find the plot of your field to close at A, as before it did at G, and if the sides and angles were never so many, the manner of the work would be the same. ¶ Here note that if in going about a field, and measuring the angles thereof with the Theodolite or degrees on the frame of the Table (as in Chap. 32.) that if you meet with any angle that bendeth inwards in the Field, you must reckon that angle to be so much above 180 degrees as the bending is, and when you note the degrees of such an angle in your Field-book, you may make this > or the like mark against them for a remembrance when you come to protract, and in protracting you must turn the Semicircle of the Protractor the contrary way to what you do in protracting of other angles. CHAP. XXXIV. How to know whether you have taken the angles of a Field truly in going round about the same with the Theodolite, as in Chap. 33, whereby you may know whether your Plot will close or not the sides being truly measured. HAving made observation of all the angles in the Field with your Instrument, and noted them down in your Field-book as is done in the latter end of Chap. 32. collect the quantity of all the angles found at your several observations into one sum, and multiply 180 degrees by a number less by two then the number of angles in the field, and if the product of this multiplication be equal to the total sum of your angles, then is your work true, otherwise not. EXAMPLE. In the work of the 32 Chap. the angel's found were as in the margin, the sum of them being 900 degrees 00 minutes. Now, because the Field consisted of 7 angles, you must therefore multiply 180 degrees by 5, (which is a number less by two then the number of angles in the Field) and the product will be 900, deg. min. 130 00 120 30 137 30 120 30 121 30 126 30 143 30 900 00 which exactly agreeing with the sum of all the angles in the Field as you found them by observation, you may conclude that your work is exactly performed. CHAP. XXXV. How to take the Plot of any Wood, Park, or other large Champion plain, by going about the same, and making observation at every angle thereof, by the Circumferentor. Suppose then that ABCDEFGHK were a large field or other enclosure to be plotted by the Circumferentor. 1. Placing your Instrument at A (the Flower-de-luce towards you) direct the sights to B, the South end of the Needle cutting 191 degrees, and the ditch, wall or hedge AB containing 10 Chains 75 Links, the degrees cut, and the line measured, must be noted down in your Field-book as in the foregoing examples. 2. Place your Instrument at B, and direct the sights to C, the South end of the Needle cutting 279 degrees, and the line BC containing 6 Chains 83 Links, which note down in your Field-book as before. 3. Place the Instrument at C, and direct the sights to D, the Needle cutting 216 degrees 30 minutes, and the line CD containing 7 Chains 82 Links. 4. Place the Instrument at D, and direct the sights to E, the needle cutting 325 degrees, and the line DE containing 6 Chains 96 Links. 5. Place the Instrument at E, and direct the sights to F, the Needle cutting 12 degrees 30 minutes, and the line OF containing 9 Chains 71 Links. 6. Place the Instrument at F, and direct the sights to G, the Needle cutting 342 degrees 30 minutes, and the line FG containing 7 Chains 54 Links. 7. Place the Instrument at G, and direct the sights to H; the Needle cutting 98 degrees 30 minutes, and the line GH containing 7 Chains 52 Links. 8. Place the Instrument at H, and direct the sights to K, the Needle cutting 71 degrees, and the line HK containing 7 Chains 78 Links. 9 Place the Instrument at K, and direct the sights to A (where you began) the Needle cutting 161 degrees 30 minutes, and the line KA containing 8 Chains 22 Links. Having gone round the field in this manner, and collected the degrees cut, and the lines measured in the several columns of your Field book according to former directions, you shall find them to stand as followeth, by which you may protract and draw the plot of your Field as in the next Chapter. Degrees. Minutes. Chains. Links. A 191 00 10 75 B 279 00 6 83 C 216 30 7 82 D 325 00 6 96 E 12 30 9 71 F 342 30 7 54 G 98 30 7 54 H 71 00 7 78 K 161 30 8 22 In going about a field in this manner, you may perceive a wonderful quick dispatch, for you are only to take notice of the degrees cut once at every angle, and not to use any back-sights as in the fore going work of the Theodolite: but to use back-sights with the Circumferentor is best for to confirm your work; for when you stand at any angle of a field, and direct your sights to the next, and observe what degrees the South end of the needle cutteth, if you remove your Instrument from this angle to the next, and look to the mark or angle where it last stood, with your back-sights, the Needle will there also cut the same degree as before, which ought to be done, and may be, without much loss of time. So the Instrument being placed at A if you direct the sights to B, you shall find the Needle to cut 191 degrees, then removing your Instrument to B, if you direct the back-sights to A, the Needle will then also cut 191 degrees. Now for dispatch and exactness (if the Needle be good, the Card well divided, and the degrees (by a good eye) truly estimated) the Circumferentor, for large and spacious grounds is as good as any, and therefore observe well the manner of protracting. CHAP. XXXVI. How to protract any observations taken by the Circumferentor, according to the doctrine of the last Chapter. ACcording to the largeness of your Plot provide a sheet of paper or skin of parchment, as LMNO, upon which draw the line LM, and parallel thereto, draw divers other lines, quite through the whole paper or parchment, as the pricked lines in the figure drawn between LM and NO, and let the distance of each of these parallels one from another be somewhat less than the breadth of the Scale of your Protractor. These parallel lines thus drawn do represent Meridian's, and are hereafter so called, upon one or other of these lines (or parallel to one of them) the Meridian line of your Protractor, noted in the figure thereof pa. 51, with OF) must always be laid when you protract any observations taken by the Circumferentor as in the Chapter before going. Your paper or parchment being thus prepared, assign any point upon any of the Meridian's, as A, upon which point place the centre of your Protractor, laying the Meridian line thereof just upon the Meridian line drawn upon your paper, as you see it lie in the figure annexed. Then look in your Field-book what degrees the needle cut at A, which were 191 degrees, now, because the degrees were more than 180, you must therefore lay the semicircle of the Protractor downwards, and holding it there, with your protracting pin make a mark against 191 degrees, through which point, from A, draw the line AB, which contains 10 Chains 75 Links. 2. Lay the centre of the Protractor on the point B, with the meridian line thereof parallel to one of the pricked Meridian's drawn upon the paper, and seeing the degrees cut at B were more than 180, viz. 279, therefore the Semicircle must lie downwards, and so holding it, make a mark against 279 degrees, and through it draw the line BC, containing 6 Chains 83 Links. 3. Place the centre of the Protractor on the point C, the Meridian line thereof lying parallel to one of the pricked Meridian's drawn on the paper, than the degrees cut by the Needle at your third observation at C being above 180, namely 216 degrees 30 minutes, therefore must the Semicircle lie downwards, then making a mark against 216 degrees 30 minutes, through it draw the line CD, containing 7 Chains 82 Links. 4. Lay the centre of the Protractor upon the point D, the degrees cut by the Needle at that angle being 325, which, being above 180, lay the Semicircle of the Protractor downwards, and against 325 degrees make a mark with your protracting pin, through which point, and the angle D, draw the line DE, making it to contain 6 Chains 96 links. 5. Remove your Protractor to E, laying the Meridian line thereof upon (or parallel to) one of the Meridian's drawn upon your paper, and because the degrees cut by the Needle at this angle were less than 180, namely, 12 degrees 30 minutes, therefore, lay the Semicircle of the Protractor upwards, and make a mark against 12 degrees 30 minutes, through which draw the line OF, containing 9 Chains 71 Links. 6. Lay the centre of the Protractor upon the point F, and because the degrees to be protracted are above 180, viz. 342 degrees 30 minutes, lay the Semicircle of the Protractor downwards, and make a mark against 342 degrees 30 minutes, drawing the line FG which contains 7 Chains 54 Links. And in this manner must you protract all the other angles G, H, and K, and more, if the field had consisted of more angles, always observing this for a general rule, to lay the meridian line of the Protractor upon (or parallel to) one of the Meridian's drawn upon your paper (which the small divisions at each end of the Scale of the Protractor will help you to do,) and if the degrees you are to protract be less than 180 (as those at G H and K are) to lay the Semicircle of the Protractor upwards, or from you; and if they be above 180 degrees (as those at A B C and D are) to lay the Semicircle downwards, as you see done in the figure. CHAP. XXXVII. How to take the Plot of any Park, Forrest, Chase, Wood, ot other large Champion plain, by the Index and Needle, together with the degrees on the frame of the Table, most commodiously supplying the use of the Peractor. THe use of the Plain Table, Theodolite and Circumferentor, hath been sufficiently taught in the preceding Chapters, and their agreement in all kind of practices fully intimated, so that you may perceive by what hath been hitherto delivered, that for some kind of works one Instrument is better than another, and for large and spacious businesses, the Circumferentor is the best (the Needle being good, and no impediment near to hinder the playing or virtue thereof) there being only this objection to be made against it, viz. that the degrees in the Card are (for the most part) so small that they cannot be truly estimated, and so may occasion the greater error in protraction. For the salving of this grand inconvenience, Master Rathborn hath a contrivance in his Book of Surveying (by an Instrument which he calleth a Peractor, which is no other than a Theodolite, only the Box and Needle is so fitted to the centre of the Instrument, that when the Instrument is fixed in any position whatsoever, the Index may be turned about, and yet the Box and Needle remain . The benefit of this contrivance is, that whereas in the Circumferentor the degrees are cut by the Needle, here the same degrees are cut by the Index, and therefore are larger, the use whereof is thus. Place the Peractor at any angle of a field, and turn it about till the Needle hang directly over the Meridian line in the Card, then fix the Instrument there, and turn the Index about till through the sights you espy the mark or angle you would look at, then shall the Index cut the same degrees and minutes upon the Limb of the Peractor, as the Needle would have cut upon the Card of the Circumferentor, if used as is before taught: yet notwithstanding this contrivance, you see you must be beholding to the Needle, the convenience only being, that the degrees which you are to note in your Field-book, are larger upon the limb of the Instrument then in the Card, which (I confess) is something considerable. Let ABCDE be a Field to be measured by the Index and Needle on the Plain Table, supplying the use of the Peractor. 1. Place your Instrument at A, laying the Index and sights with the Box and Needle screwed thereto upon the Diameter of the Table, than the Index so lying, turn the whole Instrument about till the Needle hang directly over the Meridian line in the Card, then screw the Instrument fast, and turn the Index about upon the centre, till through the sights you espy your second angle at B, than you shall see that the South end of the Needle will cut upon the Card in the Box, about 218 degrees, and the Index (at the same time) upon the Table will cut 218 degrees 10 minutes, which must be noted down in your Field book as hath been several times before taught, and measure the distance AB, 9 Chains 65 Links, which you must note down in your Field-book also. ¶ By this you may see the convenience of counting the degrees cut by the Index rather than by the Needle, as here you see 10 minutes are lost in estimation, which the Index giveth more precisely, nay sometimes you may possibly miss half or a whole degree by the Needle. 2. Place your Instrument at B, laying the Index on the diameter thereof, and turn the Instrument about till the Needle hang over the Meridian line in the Card, then fixing the Instrument there, turn the Index and sights to C, so shall both the Needle in the Box, and the Index on the frame of the Table cut 298 degrees 30 minutes, and measuring the distance BC, you shall find it to contain 9 Chains 28 Links, the degrees and minutes, and the length of the line measured, must be noted down in your Field-book as before. 3. Place your Instrument at C, and lay the Index and sights upon the diameter thereof, then turn the Instrument about till the Needle hang over the Meridian line, then fixing it there, turn the Index about till through the sights you espy the fourth angle at D, then will both the Needle and Index cut 15 degrees 40 minutes, these degrees and minutes, with the measured distance CD 5 Chains 70 Links, must be set down in your Field-book. 4. Your Instrument being placed at D, with the Index on the diameter thereof, turn it about till the Needle hang over the Meridian line, and there fixing it, turn the Index about till through the sights you see the next angle at E, then will both the Needle and Index cut 68 degrees, and the distance CE will be 8 Chains 72 Links, which note in your Field-book as before. 5. Lastly, place your Instrument at E (observing all the former cautions) and direct the sights to A, where you shall find both the Needle and Index to cut 142 degrees 45 minutes, and the measured distance EA to be 7 Chains 11 Links, which note down in your Field-book. And thus may you go about any field, let it consist of never so many sides and angles, observing always this general rule, to lay the Index with the Box and Needle, on the diameter of the Table, and to turn the Table about till the Needle hangs directly over the meridian line in the Card, and then fixing the Table, turn the Index about till through the sights you espy the mark you look for, then will both the Index and the Needle cut the degrees which you must note in your Field-book, so will the collected notes of this example stand as followeth. Degrees Minute's Chains Links A 218 10 9 65 B 298 30 9 28 C 15 40 5 70 D 68 00 8 72 E 142 45 7 11 Having thus collected your several observations, you may proceed to protract your work as is taught in the next Chapter, which differeth nothing from that in the 36 Chap. ¶ It will be here objected by the affectors of the Peractor, that here it is required that the Needle should play twice at each observation, to which I answer, it is true, but if you neglect the latter of them; it is both as speedy and as exact as the Peractor, and if you have opportunity to observe both (which you may conveniently do) it will then be better. CHAP. XXXVIII. How to protract any observation taken as in the last Chapter. YOu must first rule your paper or parchment all over with parallel lines or Meridian's, as is taught in the 36 Chapter, and upon one of these Meridian's assign any point at pleasure, as A, then laying your Field-book before you, place the centre of the Protractor upon the point A, the Scale thereof lying upon, or parallel to, one of the meridians ruled on your paper, and because the degrees cut at A were above 180 degrees, viz. 218 degrees 10 minutes, therefore lay the Semicircle of the Protractor downwards, and against 218 degrees 10 minutes of your Protractor make a mark, through which mark and the point A draw the line AB, containing 9 Chains 65 Links. 2. Remove your Protractor to the point B, which represents your second station or angle, laying the Meridian line thereof upon (or parallel to) one of the Meridian's drawn upon the paper, and because the degrees cut at Bare above 180, lay the Semicircle downwards as before, and against 298 degrees 30 minutes make a mark, and through it draw the line BC containing 9 Chains 28 Links. 3. Bring your Protractor to C, and lay it parallel to some one of your Meridian's, and because the degrees observed at C were under 180, namely 15 degrees 40 minutes, lay the Semicircle upwards, and against 15 degrees 40 minutes make a mark, drawing the line CD containing 5 Chains 70 Links. 4. Place your Protractor as before upon the point D, with the Semicircle upwards, and against 68 degrees thereof make a mark, and draw the line DE containing 8 Chains 72 Links. Lastly, Remove your Protractor to E, placing it as before, and against 142 degrees 45 minutes (which were the degrees observed at your station at E) make a mark, and through it and the point E draw the line EA, which (if your work be true) will pass through the point A, and will contain 7 Chains 11 Links. CHAP. XXXIX. How to find how many Acres, Roods and Perches, are contained in any piece of Land, the plot thereof being first taken by any Instrument. HAving shown how to take the plot of any field or other enclosure several ways, and also to protract the same upon paper, it is now necessary to show how the content thereof may be attained, that is to say, how many Acres, Roods and Perches, any field so plotted doth contain: In the performance hereof you must consider that the original of the mensuration of all superficial figures, such as Land, Board, Glass or the like, doth depend upon the exact measuring of certain regular figures, as the Geometrical Square, the Long Square or Parallelogram, the Triangle, the Trapezia, and the Circle: therefore, if any plot of Land to be measured be not one of these figures, it must (before it can be measured) be reduced into some of these forms: I will therefore in the first place show how to measure any of these figures severally by themselves, and afterwards how to reduce any other irregular figure into some of these regular forms, and lastly to measure them by the same rules: and first, Of the Geometrical Square. A Geometrical Square is a figure consisting of four equal sides and angles, as is the Square ABCD, whose sides are all equal to the line QR, which containeth six equal parts, which may be attributed either to Inches, Feet, Yards, Perches, Chains, or any other measure whatsoever. Now, to find the superficial content of such a Square, you must multiply one of the sides in itself, and the product of that multiplication shall be the content of the Square. EXAMPLE. Suppose the Square ABCD to be a piece of Land, and the side thereof to contain 6 Perches, therefore multiply 6 in itself, and the product will be 36, & so many Perches doth the square piece of Land contain. Of the long Square. A Long Square is a figure consisting of four sides, as the figure ABCD, the two opposite sides whereof are equal, as the sides AB, and CD, and likewise AC and BD, each of the shorter sides containing 7 Perches, and the longer sides 13 Perches. To find the superficial content of this long Square or Parallelogram, you must multiply one of the longer sides by one of the shorter, and the product will show the superficial content thereof. Example, The longer side of the Square contains 13 perches, and the shorter 7 perches, now if you multiply 13 by 7, the product will be 91, and that is the content of the square in perches. Of the Triangle. ALthough there be several kinds of Triangles, yet in respect they are all measured by one and the same rule, I will therefore add one example for all, which is general. Half the length of the Base being multiplied by the length of the perpendicular, shall be equal to the area of the triangle. Or, Half the length of the Perpendicular being multiplied by the whole Base, will be the content of the triangle. EXAMPLE. Suppose you were to find the area or content of the triangle BCD, the Base thereof DB contaiking 58 perches, and the perpendicular CA 24 perches. Now if you multiply 12 (which is half the length of the perpendicular CA) by 58 (the length of the whole base DB) the product will be 696 and that is the area or content of the Triangle. Or, If you multiply 24 (the whole length of the perpendicular) by 29 (the length of half the base) the product will be 696 as before. Or again: If you multiply 58 (the whole length of the base) by 24 (the whole length of the perpendicular) the product will be 1392, the half whereof is 696, the area or content of the Triangle, as before. Of the Trapezia. A Trapezia is a figure consisting of four unequal sides, and as many unequal angles, as is the figure ABCD. To measure this Trapezia, you must first draw the diagonal line BD, for by this means the figure is reduced into two Triangles, as ADB, and CDB, then if you let fall the perpendiculars from the points A and C, you may measure them by the last examples, as two Triangles; the sums whereof being added together will be the area or content of the whole Trapezia. EXAMPLE. Having drawn the line BD, and so reduced the Trapezia into two Triangles, and let fall the perpendiculars A and CF, upon the line BD, which is the common base to both the Triangles, you may find the area of the whole Trapezia, thus. Suppose the perpendicular CF, were 102 perches, the perpendicular A 118 perches, and the base BD (which is common to both Triangles) 300 perches. Now, if according to former directions, you multiply 300 the base, by 59 half the perpendicular A, the product will be 17700, for the content of the Triangle ABDELLA. In like manner, if you multiply 300 the Base, by 51, half the perpendicular FC, the product will be 15300, for the content of the Triangle BCD. Now if you add the contents of these two Triangles together; namely 17700, and 15300; the sum of them will be 33000, and that is the content of the whole Trapezia ABCD. But this work may be performed with more brevity, thus. In respect the Base BD is common to both the Triangles, you may therefore add the two perpendiculars together, the half of which being multiplied by the whole Base, the product will show the content of the whole Trapezia. EXAMPLE. The two perpendiculars 118 and 102 being added together, the sum of them is 220, the half whereof is 110, this number being multiplied by 300 (the whole length of the common base) giveth 33000 the content of the whole Trapezia. OR, You may multiply the sum of the perpendiculars by the length of the Base, and half that product will be the content of the Trapezia also. Of irregular Figures, how to reduce them into Triangles or Trapezias, and to cast up the content thereof. LEt ABCDEFGH be the figure of a Field drawn upon your Plain Table, or otherwise protracted upon paper, according to any of the former directions. In regard that the Field is irregular, that is to say, it is neither Square, Triangle, or Trapezia, it must therefore (before it can be measured) be reduced into some of these forms, which to effect do thus: draw lines from one angle to other, as the lines AD, DB, A, OF, and FH, then will the whole figure be reduced into six Triangles, as 1. the Triangle BCD, 2. the Triangle ADB, 3. the Triangle ADE, 4. the Triangle AEF, 5. the Triangle AFH, 6. the Triangle FGH. These six Triangles being measured severally, according to the former directions, and the contents of them all added together into one sum, will show the area or content of the whole field. As, Suppose the Triangle BCD should contain 72 Perches. Suppose the Triangle ADB should contain 84 Perches. Suppose the Triangle ADE should contain 110 Perches. Suppose the Triangle AEF should contain 121 Perches. Suppose the Triangle AFH should contain 165 Perches. Suppose the Triangle FGH should contain 66 Perches. These six numbers being added together make 618 perches, and that is the area or content of the whole Field in perches. But for an abreviation of this work, you need not to find the area of every Triangle, but of every Trapezia, as is before taught, for the figure is as well divided into Trapezias as Triangles, namely, into the Trapezias ABCD, ADEF, AFGH. By this means you need but to find the area or content of these three Trapezias, which will abreviate nigh half of the Arithmetical work, for if you measure the three Trapezias severally, as hath been taught in this Chapter, you shall find The Trapezia ABCD to contain 156 Perches. The Trapezia ADEF to contain 231 Perches. The Trapezia AFGH to contain 231 Perches. These three numbers being added together produce 618 exactly agreeing with the former. ¶ Here note, that at any time when you reduce any irregular plot into Triangles, your number of Triangles will be less by two then the number of the sides of your plot; as in this figure, the plot consisted of 8 sides, and you see it is reduced into 6 Triangles. Of the Circle. THe proportion of the circumference of any Circle is to its diameter, as 7 to 22. Now to find the area or content of any Circle, you must multiply the diameter thereof in itself, and multiply that sum by 11, which product being divided by 14, shall give you the area of the Circle. EXAMPLE. In this Circle ABCD, let the diameter thereof DB be 28, which multiplied in itself giveth 784, this number multiplied by 11 giveth 8624, which being divided by 14, the quotient will be 616, and that is the area of the Circle. The Circumference of a Circle being given, to find the Diameter. MUltiply the Circumference by 7, and divide the product by 22, the Quotient shall be the length of the Diameter. EXAMPLE. Let the circemference of the Circle ABCD be 88, this multiplied by 7, giveth 616, which being divided by 22, giveth 28 for the Diameter DB. CHAP. XL. Of the manner of casting up the content of any piece of Land in Acres, Roods and Perches, by Master Rathborns Chain. IN the 5. Chapter of the 2 Book, you have a description of Chains in general, and more particularly of Master Rathborns and Master Gunter's. In the measuring of Land by Master Rathborns Chain, you call every Pole or Perch thereof (which is divided into 100 Links) a Unite, and every ten of those Links you call a Prime, and every single Link you call a Second. Now because there are divers that fancy this Chain rather than any other, because it giveth the content of any Superficies measured therewith in its smallest denomination, namely in Perches and parts of Perches, so that when any Superficies is cast up and brought to Perches, it may easily be reduced into Roods and Acres. Now (for their sakes that affect this Chain) I will show the use thereof, and afterwards of Master Gunter's Chain, leaving every man to take his choice, and use that which liketh him best. Suppose that the figure B were a piece of Land lying in a long square, which being measured by Master Rathborns Chain should contain in length 16 Unites, 2 Primes; and in breadth 1 Unite, 3 Primes,, 2 Seconds, and that it were required to find the area or content thereof in Perches, which to effect you must multiply the length by the breadth as is taught in the last Chapter, therefore, the length being 16 Unites 2 Primes, and the breadth 1 Unite, 3 Primes, 2 Seconds, these two numbers multiplied together shall produce the area. Set your numbers down as you are taught in the 5 Chapter, of the 2 Book, or as you see them stand in this Example, with a prick over the head of every fraction: 162̇ 13̇2̇ 324 486 162 21384 under these numbers draw a line, and multiply them together in all respects as if they were whole numbers, and then the work will stand thus, the product of your multiplication being 21384. Now because in your two numbers, viz. your multiplicand and your multiplyer, there are three fractions, namely, one in your multiplicand, and two in your multiplyer, you must therefore (with a dash of your pen) cut off the three last figures of the product towards your right hand, and then will your product stand thus, 21/384 the three last figures whereof are the numerator of a fraction, whose denominator is 1000, and the other two figures towards your left hand are Integers of your multiplication; so that the sum of this multiplication is 21 perches, and 384/1000 parts of a perch, which is something more than a third part of a perch. But to express the exact quantity of these fractions in a business of this nature were superfluous, only observe this one Rule for all, namely, that if the figures cut off come near to a Unite, that is, when the figures cut off are near as much as those underneath them, or the first figure cut off is either 7, 8, or 9, you may then increase your whole number by a Unite, and not at all regard the fraction. But for your further practice take another Example, which let be a piece of land containing in breadth 5 Unites, 6 Primes, 3 Seconds, and in length, 15 Unites, 4 Primes and 2 Seconds, which place as before. Now if you multiply these numbers one by another as if they were whole numbers, then will they stand as in the margin, the product being 868146, 154̇2̇ 56̇3̇ 4626 9252 7710 868146 from whence take the four last figures (because there are four fractions in your two numbers) there remains 86 perches, and 9146/10000 parts of a perch; now because 8146 is near to 10000, I add 1 to 86, making it 87 perches, dis-regarding the excess as immaterial. In like manner, suppose the perpendicular of a Triangle should contain 1 Unite, 3 Primes, 2 Seconds, and half the length of the base should contain 16 Unites, 2 Primes, these numbers being placed as those before, and multiplied one by another, will produce this product 21384, from whence cut off the three last figures (because there were three fractions in your numbers multiplied) and there will remain 21 perches, and 384/1000 parts of a perch, which being but of small value you may reject. CHAP. XLI. How to reduce any number of Perches into Roods and Acres, or any number of Acres and Roods into Perches. BY a Statute made the 33. of Edw. 1. an Acre of ground ought to contain 160 square Perches, and every Rood of Land 40 square Perches, and every Perch was to contain 16 foot and a half. Now, if any number of Perches be given to be turned into Acres, you must divide the number given by 160 (the number of perches contained in one Acre) and the quotient shall show you how many Acres are contained in that number of Perches, and if any thing remain (if it be under 40) it is Perches; but if the remainder exceed 40, than you must divide it by 40 (the number of perches contained in one Rood) and the quotient shall be Roods, and the remainder perches. EXAMPLE. Let 5267 perches be given to be reduced into Acres, first, divide 5267 by 160, and the quotient will be 32, and 147 remaining, which divide by 40, the quotient will be 3, and 27 remaining, so that the whole amounteth to 32 Acres 3 Roods and 27 perches. Again, let 5496 perches be given to be reduced into Acres, first, divide 5496 by 160, the quotient will be 34, and 56 remaining, which 56 being divided by 40, the quotient will be 1, and 16 remaining, so that the whole will be 34 Acres 1 Rood and 16 perches. To reduce Acres into Perches. THis is but the converse of the former, for (as before) to reduce perches into Acres, you divided by 160, you must now, to reduce Acres into Perches, multiply by 160. EXAMPLE. Let 32 Acres 3 Roods and 27 perches, be given to be reduced into Perches: first, multiply the 32 Acres by 160, and the product will be 5120, then multiply the 3 Roods by 40, the product is 120, these two products, and the 27 perches being added together, the sum will be 5267, 5120 120 27 5267 and so many perches are contained in the foresaid number of Acres, Roods and perches: and thus much concerning the use of Master Rathborns Chain. CHAP. XLII. How to cast up the content of any piece of Land in Acres, Roods and Perches, by Master Gunter's Chain. IN measuring by Master Gunters Chain you are in your account only to take notice of Chains and Links, as was before intimated in the description thereof Cap. 5. Lib. 2. Suppose then that the figure B were a piece of Land lying in a long square, and that being measured by Master Gunter's Chain should contain in length 9 Chains 50 Links, and in breadth 6 Chains 25 Links. Set your numbers down as before is taught and as here you see, drawing a line under them, then multiplying them together, you shall find the product to be 593750, 9,50 6,25 4750 1900 5700 593750 from which product you must always cut off the five last figures towards the right hands with a dash of your pen, then will the product stand thus, 593750, so is the 5 towards the left hand complete Acres, and the 93750 hundred thousand parts of an Acre, which to reduce into Roods and Perches is easy, by help of this Table. For, if you look for 90000, under the title Links (which is the first figure with Ciphers added) Links. R. P. 100000 4 0 90000 3 24 80000 3 8 70000 2 32 60000 2 16 50000 2 0 40000 1 24 30000 1 8 20000 0 32 10000 0 16 9375 0 15 8750 0 14 8125 0 13 7500 0 12 6875 0 11 6250 0 10 5625 0 9 5000 0 8 4375 0 7 3750 0 6 3125 0 5 2500 0 4 1875 0 3 1250 0 2 624 0 1 you shall find against it 3 Roods, 24 Perches, then look for 3750, and against it you shall find 6 perches, all which being added together as here you see, the area or content of the whole piece will be 5 Acres, 3 Roods and 30 Perches. A. R. P. 5 00 00 3 24 6 5 03 30 Another Example. Suppose the base of a Triangle should contain 16 Chains 56 Links, and half the perpendicular of the same Triangle 4 Chains 32 Links, these being multiplied one in the other will produce the area or content of the whole Triangle. Set your numbers down as in the margin is done, and multiply one by the other, so will the Product be 715392, 16,56 4,32 3312 4968 6624 715392 from which cutting off the five last figures towards the right hand, there will be left before the line of partition 7, which is 7 complete Acres, and behind the line there will be 15392, which are hundred thousand parts of an Acre, and how much that is, the Table will easily show; for, if you look in the first column for 10000, against it you shall find 00 Roods 16 Perches; then looking for 5392, you find it not, but the nearest thereto is 5625, against which there standeth 9 perches, all these numbers being added together will produce 7 Acres, 00 Roods, 27 Perches, A. R. P. 7 00 00 16 9 7 00 27 which is the area of the Triangle. Thus may you find the area of any Triangle or Parallelogram very easily by one multiplication and addition, which is much easier than the way of casting up by Master Rathborns Chain. By this manner of work if the length and breadth of a long Square or Parallelogram given should be 9 Chains 75 Links, and 6 Chains 25 Links, the area of such a long Square would be found to be 6 Acres, 00 Roods 15 perches. Or, the length and breadth being 12 Chains 42 Links, and 1 Chain 36 Links, the area or content will be found to be 1 Acre, 2 Roods, 30 perches. Also, the length and breadth being 12 Chains 86 Links, and 5 Chains 25 Links, the area will be found to be 6 Acres, 3 Roods, 00 perches. But lest you should be destitute of this Table when you have need thereof, you may have it put upon some spare place of your Instrument, or rather (instead of this Table) a Scale, which I will now show you the use of, which performeth that work far better and more easily than the Table, and may conveniently be graduated upon the Index of your Table, the dividing and numbering whereof is well known to the Instrument maker. The Scale consisteth of two parts, one whereof is square perches, the other square Links, the Scale of square perches proceedeth gradually from 1 to 40 with sub-divisions, and is numbered by 5, 10, 15, 20, etc. to 40. The Scale of square Links proceedeth gradually from 1 to 25000, and is also subdivided and numbered by 1000, 2000, etc. to 25000, equal to 1 Rood or 40 perches. The use of the Scale of Reduction. We will instance in the second example beforegoing, where the length and breadth of the long Square was 16 Chains 56 Links, and 4 Chains 32 Links, these being multiplied together produce 715392, and the five last figures being cut off, there is 7 Acres and 15392 remaining, now to find how many Roods and Perches this is, look in the Scale of Square Links for fifteen thousand three hundred ninety two, and against it, in the Scale of square Perches you shall find 24 Perches and above half a perch. Another Example. Let us take the first example beforegoing, where the numbers multiplied were 9.50, and 6.25, these being multiplied one by another produce 593750, and the five last figures being cut off, there will be 5 Acres, and 93750 remaining: now to know how many Roods and Perches are contained therein by the Scale, ¶ You must consider that 25000 square Links are equal to one Rood or 40 Perches, as appeareth by the Scale itself, and also by the Table, then is 50000 equal to 2 Roods, and 75000 equal to 3 Roods; therefore, if your number remaining exceed 25000, and be under 50000, you may conclude 1 Rood and odd perches to be contained therein. If it exceed 50000, and be under 75000, you may conclude 2 Roods and some odd perches to be therein. If above 75000, you may then conclude 3 Roods and odd perches to be therein. Now in this example, the number remaining is 93750, which because it exceedeth 75000, I conclude there is 3 Roods contained therein, which I set to the 5 Acres, and subtract 75000 from 93750, the ramainder being 18750, this number, eighteen thousand seven hundred and fifty, I seek in the Scale of Square Links, and right against it I find 30 perches, which added to the former, giveth 5 Acres, 3 Roods and 30 Perches, A. R. P. 5 3 30 which is the area or content required. Thus you see with what celerity and exactness the Scale effecteth your desire, and therefore let it be graduated upon the Index of your Table that it may always be ready at hand when you have need thereof. The construction of this Reducing Scale I received of my honoured friend Master S. F. deceased. CHAP. XLIII. Containing divers compendious rules, for the ready casting up of the content of any plain superficies, and other necessary conclusions incident to Surveying, by the line of Numbers. THe line of Numbers is of singular use in casting up of the content of any Superficies, and for land measuring especially Master Gunter hath several propositions, like unto which, I will insert seven other propositions which will be of singular use in the practice or Surveying. 1. The length and breadth of a right angled Parallelogram or long Square being given in Perches, to find the content thereof in Perches. As 1 perch, is to the breadth of the Parallelogram in perches; So is the length in perches, to the content in perches. In this long Square or Parallelogram ABCD, if the breadth thereof CB be 36⅖ perches, and the length thereof AB 50 perches, the content will be found to be 1820 perches: for, If you extend the Compasses from 1 to 36⅖ the length, the same extent will reach from 50 the breadth, to 1820, the area or content in perches, which you may reduce into Acres as is taught in the 41 Chap. 2. The length and breadth of a long Square being given in Perches, to find the content in Acres. As 160, to the breadth in perches; So the length in perches, to the content in Acres. So in the former figure, if the length thereof AB be 50 perches, and the breadth thereof 36●, the content will be found to be 11 Acres 40 parts, which is 1 Rood 20 perches; for, If you extend the Compasses from 160 to 36⅖, the same extent will reach from 50 to 11 Acres 40 parts. 3. The length and breadth of a Parallelogram being given in Chains, to find the content in Acres. As 10, to the breadth in Chains; So the length in Chains, to the content in Acres. So the length of the long Square AB being 12 Chains 50 Links, and the breadth BC 9 Chains 10 Links, the area will be found to be 11 Acres 37 parts, or 1 Rood 20 perches, for, If you extend the Compasses from 10, to 9 Chains 10 Links, the same extent will reach from 12 Chains 50 Links, to 11 Acres 37 parts. 4. Having the Base and perpendicular of a Triangle given in Perches, to find the content in Acres. As 320, to the Perpendicular; So the length of the Base, to the content in Acres. So in the Triangle LABERNELE, if the line BD be taken for the perpendicular of the Triangle, than the length of the base being 50 perches, and the perpendicular 36 2/5, the area will be found to be 5 Acres 22 parts, which is 2 Roods 30 perches, then, If you extend the Compasses from 320 to 36 2/5 the perpendicular, the same extent will reach from 50 the length of the base, to 5 Acres 22 parts. 5. The Base and perpendicular of a Triangle being given in Chains, to find the content in Acres. As 20, to the perpendicular; So the Base, to the content in Acres. So in the former figure, if AB 12 Chains 50 Links be taken for the Base, and BD 4 Chains 55 Links for the perpendicular of the Triangle ALB, the area (by this proportion) will be found to be 5 Acres 68 parts, that is, 5 Acres 2 Roods 30 perches, therefore, If you extend the Compasses from 20 to 4 Chains 55 Links, the same extent will reach from 12 Chains 50 Links, to 5 Acres 68 parts, which is 2 Roods 30 perches. 6. The Area or superficial content of any piece of Land being given according to one kind of Perch, to find the content thereof accoading to cnother kind of Perch. As the length of the second perch, To the length of the first perch; So the content in Acres, To a fourth number: And that fourth number to the content in Acres required. Suppose the figure B were a piece of Land, which being plotted and cast up by a Chain of 16 foot and a half to the Perch, should contain 8 Acres, and that it were required to find how much the same piece would contain if it were measured with a Chain of 18 foot to the perch: if you work according to the proportion here delivered, you shall find it to contain 6 Acres 72 parts: for, If you extend the Compasses from 18 to 16½, that extent will reach from 8 to 7.30, and from 7.30 to 6.72, and so many Acres would the figure B contain if it were measured by a perch of 18 foot. 7. Having the length of the Furlong, to find the breadth of the Acre. As the length of the furlong in Perches, to 160; So is 1 Acre, to the breadth in Perches. So if the length of the furlong be 50 perches, the breadth for one Acre will be 3.20: for, If you extend the Compasses from 50, the length of the furlong in perches, the same extent will reach from 1 Acre to 3.20 perches. But if the length of the Furlong be given in Chains, then, As the length of the Furlong in Chains, is to 10; So is 1 Acre, to the breadth of the furlong in Chains. So the length of the Furlong being 12 Chains 50 Links, the breadth thereof will be found to be 00 Chains 80 Links: for, If you extend the Compasses from 12 Chains 50 Links, to 10, that extent will reach from 1 Acre to 80 Links, which is the breadth of the furlong required. CHAP. XLIV. How to reduce one kind of measure into another, as Statute measure to Customary measure. BY the 6 Prop. of the last Chapter you may perform this work by the line of Numbers as is there taught, but however, it will not be amiss in this place to show how to perform the same Arithmetically, that the reason thereof may the better appear. Now whereas (by the forementioned Statute) an Acre of ground was to contain 160 square perches, measured by the Pole or Perch of 16 foot and a half, but in many places of this Nation (through long custom) there hath been received other quantities, called Customary, as namely, of 18, 20, 24, and 28 foot to the Pole or Perch. It is therefore necessary for a Surveyor to know how readily to reduce Customary measure to Statute measure, and the contrary. Suppose then, that it were required to reduce 5 Acres, 2 Roods, 20 Perches, measured by the 18 foot Pole into Statute measure, you must seek out the least proportional terms between 18 foot, and 16 foot and a half, which to perform do thus. Because 16 and a half beareth a fraction, reduce 16 and a half into halves, and that both your numbers may be of one denomination, you must reduce 18 (the customary Pole) into halves also, then will your numbers stand thus 33/36, which abreviated by 3, by saying how many times 3 in 33? the quotient will be 11, and again, how many times 3 in 36? the quotient will be 12, so will the two proportional terms between 16 and a half and 18, be 11 and 12. This done, reduce your given quantity (5 Acres, 2 Roods, and 20 perches) into perches, which makes 900 perches: Now considering that what proportion the square of 11, which is 121, bears to the square of 12, which is 144, the same proportion doth the Acre of 16 foot and a half to the Perch, bear to the Acre of 18 foot to the Perch. Now (because the greater measure is to be reduced into the lesser) multiply the given quantity 900 perches, by 144, the greater square, and the product will be 129600, which divided by 121, the quotient will be 1071 9/●… perches, which being reduced into Acres, giveth 6 Acres, 2 Roods, 31 perches, and 9/●… parts of a perch, according to statute measure. But on the contrary, suppose it had been required to reduce Statute measure into Customary measure, than you must have multiplied 900 perches (your given quantity) by 121 the lesser Square, (because the lesser measure is to be reduced into the greater) the product will be 108900, which divided by the greater Square 144, the quotient will be 756¼ perches, which reduced into Acres is 4 Acres, 2 Roods 36 perches and a quarter. The same manner of work is to be observed in the reducing of any Customary quantity whatsoever. CHAP. XLV. How to lay out several Furlongs in Common-fields unto divers Tenants. HAving plotted the whole Field, Common, or other Enclosure, with its particular bounds, as you observe them in the survey of the whole Manor, or if you only survey that particular, you must take special notice of all the bounds thereof, then provide a Book or paper which must be ruled or divided into 8 Columns, in the first whereof towards the left hand is to be written the Tenant's name, and the tenor by which he holds the same Land, the two next Columns are to contain the length of every man's Furlong in Chains and Links. In the two next Columns is expressed the breadth of every man's Furlong in Chains and Links, as by the Letters over the head of each column doth appear. In the three last Columns is to be expressed the quantity of each tenant's Furlong in Acres, Roods and Perches. In the laying out of several parcels in this kind, you will have use only of your Chain; then when you begin your work, you must first write the name of the field, and in the first column of your Book or paper, you must write the Tenant's name, and the tenor by which he holds the same, from what place you begin to measure, and upon what point of the compass you pass from thence, and observing this direction in all the rest, you may (if need require) bound every parcel. This being noted in your Book, observe the species or shape of the Furlong, whether it be all of one length or not, if of one length, than you need take the length thereof but once for all, but if it be irregular, that is, in some places shorter and in others longer, than you must take the length thereof at every second or third breadth, and express the same in your Book under the title of length. As for the expressing of the several breadths, you need but to cross over the whole Furlong, taking every man's breadth by the middle thereof, and entering the same as you pass along, but in case there be a considerable difference at either end, than I would advise you to take the breadth at either end, and add them together into one sum, then take the half of that sum for your mean or true breadth, and enter it in your book or paper under the title of breadth. In this manner you may proceed from one Furlong to another till you have gone through the whole field, which when you have done and noted down the several lengths and breadths in your book, you may multiply the length and breadth of every parcel together, as is taught before, and so shall you have the quantity of every parcel by itself, which quantity must be noted down in the three last columns of your Book as in the following example appears. Mordon Field. The Tenants names and tenor. Length. Breadth. Content. C. L. C. L. A. R. P. Abel Johnson, from the pond S. E. free. 32 76 3 45 11 1 12 Nicholas Somes, for three lives. 30 12 2 63 7 3 30 Robert Dorton, for Life. 28 60 8 12 23 0 36 James Norden, at Will. 25 11 12 35 31 0 2 CHAP. XLXI. To find the horizontal line of any hill or mountain. THis proposition differeth nothing from those formerly taught in the taking of Altitudes. Wherefore, suppose you should meet with a hill or mountain as ABDELLA, the thing required is to find the length of the line BD on which the mountain standeth. First, place your Instrument at the very foot of the Hill, exactly level, then let one go to the top of the hill at A, and there place a mark, which must be so much above the top of the hill; as the top of the Instrument is from the ground; then move the Label up and down till through the sights thereof you see the top of the mark at A, and note the degrees cut by the Label on the Tangent line, for that is the quantity of the angle ABC, which suppose 47 degrees, then by consequence the angle BAC must be 43 degrees, the compliment of the former to 90 degrees, then measure the side of the hill AB, which suppose to contain 71 Feet, then in the Triangle ABC there is given the side AB 71 foot and the angle BAC 43 degrees, together with the right angle ACB 90 degrees, and you are to find the side BC, which to perform, say, As the Sine of the angle ACB, 90 degrees, Is to the side AB 71 feet; So is the Sine of the angle BAC, 43 degrees, To the side BC: 48½ feet. Then (because the hill descends on the other side) you must place your Instrument at D, observing the angle ADC to contain 41 degrees, and the angle DAC 49 degrees, and the side AD 80 feet: now to find the side CD the proportion will be, As the Sine of the angle ACD, 90 degrees, Is to the side AD, 80 feet; So is the Sine of the angle GOD, 49 degrees, To the side CD 60½ feet. Which added to the line BC, giveth 109 feet, which you may reduce into Chains, by dividing it by 66, and this line must be protracted instead of the hypothenusal lines AB and AD. Another way. There is another way also used by some for the measuring of horizontal lines, which is without the taking of the Hills altitude, or using of any Arithmetical proportion, but by measuring with the Chain only, the manner whereof is thus. Suppose ABC were a hill or mountain, and that it were required to find the length of the horizontal line thereof AC. At the foot of the hill or mountain, as at A, let one hold the Chain up, then let another take the end thereof and carry it up the hill, holding it level, so shall the Chain meet with the hill at D, the length AD being 60 Links, then at D let the Chain be held up again, and let another carry it along level till it meet with the side of the hill at E, the length being 54 Links: then again let one stand at E and hold up the Chain, another going before to the top of the hill at B, the length being 48 Links, these three numbers being added together make 162 Links, or 1 Chain 62 Links, which is the length of the horizontal line AC. This way of measuring is by some practised, but the other (in my opinion) is far to be preferred before it, only when you are destitute of better helps you may make use hereof. ¶ But if the hill or mountain should have a descent back again on the other side, you must then use the same way of working as before, and add all together for the horizontal line. CHAP. XLVII. How to plot Mountainous and uneven grounds, with the best way to find the content thereof. FOr the plotting of any mountainous or uneven piece of ground, as ABCDEFG, you must first place your Instrument at A, and direct the sights to B, measuring the line AB, then in regard that from B to C there is an ascent or hill, you must find the horizontal line thereof, and draw that upon your Table, accounting thereon the length of the hypothenusal line, then measure round the field according to former directions, and having the figure thereof upon your Table reduce it into Trapezias, as into the Trapezias ABEG, BCDE, and the Triangle GEF; then from the angel's A C E and F let fall the perpendiculars AK, CH, EI, and FM. Now in regard there are many hills and valleys all over the field, you must measure with your Chain in the field over hill and dale from B to D, and to the line BD set the number of Chains and links as you find them by measuring, which will be much longer than the straight line BD measured on your Scale; then by help of your Instrument find the point H in the line BD, and measure with your Chain from C to H, over hill and dale as before, and to this perpendicular CHANGED set the number as you find it by the Chain: then find the perpendicular IE, and measure that with your Chain also, all which lines (in respect of the hills and valleys) will be found much longer than if they were measured by your Scale: then by the measured lines BD, CHANGED and IE, cast up the content of the Trapezia BCDE. In this manner you must cast up the content of the Trapezia ABEG, and the Triangle GEF, and this is the exactest way I can prescribe for the mensuration of uneven grounds, which being well and carefully performed, will not vary much of the true content: For it is apparent that if such mountainous grounds were plotted truly according to their area in plano, the figure thereof would not be contained within its proper limits, and being laid down amongst other grounds would swell beyond the bounds, and force the adjoining grounds out of their places; now for distinction in your Plot you may shadow them off with hills as in this figure, lest any man seeing your plot should measure by your Scale, and find your work to differ. CHAP. XLVIII. How to take the Plot of a whole Manor, or of divers parcels of Land lying together, whether Wood-lands or Champion plains, by the Plain Table. ALthough practice, in the performance hereof, be better than many words, and that the rules already delivered are of sufficient extent to perform the work of this Chapter, yet (for farther satisfaction in this particular) I will herein deliver the most sure and compendious way I can imagine. Suppose therefore that the following figure ALMNPQSTYXGH and K were part of a Manor, or divers parcels of land lying together, and that it were required to take the plot thereof upon your Plain Table. Now the best way (in my opinion) is first to go round about the whole quantity to be measured, and draw upon your Table a perfect plot thereof, as if it were one entire field (which you may do by the 31 Chap. of this Book) and then to make separation and division thereof in an orderly way, as is taught in this Chapter: But before you begin your work, it will be very necessary to ride or walk about the whole Manor, or at least so much as you are to survey, that you may be the better acquainted with the several bounders, and in your passage you ought to take special notice of all eminent things lying in your way, as Churches, Houses, Mills, Highways, Rivers, etc. which will much help you, also in this your passage it were necessary to take notice of some convenient place to begin your work as followeth. Having made choice of some convenient place in the periphery or outward part of the Manor, as at A, place there your Table, turning it about till the Needle hang over the Meridian line in the Card, and there fix it, then upon the Table (with most convenience) assign any point at pleasure, as A, unto which point lay the Index, and turn it about till through the sights you see a mark set up at the next angle at L, then by the side of the Index draw the line ALL, which suppose to contain 8 Chains 68 links, take these 8 Chains 68 links from any Scale, and place that length upon your Table from A to L. 3. Remove your Table to M, and lay the Index upon the line ML, turning the Table about till through the sights you espy a mark set up at the angle L, where your Table last stood, and there fixing it, you shall still find the Needle to hang directly over the Meridian line, if you proceed truly in your work: then laying the Index to the point M, turn it about till through the sights you espy some mark set up at the next angle at N, and draw a line by the side of the Index, then measuring with your Chain from M to N, you shall find it to contain 7 Chains 27 links, which take from the same Scale as before, and place the length thereof upon your Table from M unto N. 4. Place your Instrument at N, laying the Index upon the line NM, and turn the Table about till through the sights you see a mark set up at your former station at M, and there fix the Table, so will the needle hang over the meridian line as before, then turn the Index about upon the point N, till through the sights you espy the next angle at P, and draw a line by the side thereof, then measure the distance NP 9 Chains 32 links, which take from the Scale, and set it upon your Table from N unto P. In this manner must you go round about the whole Manor, making observation at every angle thereof, as at P Q S T Y X G H and K, and setting down the length of every line upon your Table as you find it by measuring with your Chain, you shall have upon your Table the figure of one large plain; which must include all the rest of the work, and in thus going about you shall (if you have truly wrought all the way) find your plot to close exactly in the point A, where you began, but if it do not, go over your work again, for otherwise, all that you do afterwards within the same will be false. ¶ Here note, that if one sheet of paper will not contain your whole plot, you must then shift your paper in this manner: when any line falleth off of your Table, draw two lines at right angles cross your paper, which the equal divisions on the frame will help you to do; then lay another clean sheet of paper upon your Table, and by the same parallel divisions at the contrary end of the Table, draw two other lines at right angles, and upon them note what part of your Plot crossed the two other lines before drawn, and at those points begin to go forward with the rest of your work: and thus may you shift divers papers one after another, if need be. Having thus drawn the true plot of the outward bounds or periphery of the whole Manor upon your Table, as the figure ALMNPQSTYXGH and K; and exactly closed your plot at A where you began, you may proceed now to lay out the several Closes therein contained, in this manner. 1. Place your Table at A, laying the Index and sights upon the line ALL before drawn, and turn it about till through the sights you espy the angle L, and there fixing it, the needle will hang directly over the Meridian line in the Card: then turn the Index about upon the point A, till through the sights you espy a mark set up at the angle B, and by the side of the Index draw the line AB containing 6 Chains 43 Links. 2. Remove the Table to B, laying the Index on the line BASILIUS, and turn the Table about till through the sights you see the angle A, then fix it, and turn the Index about upon B, till you see the next angle at C, drawing the line BC by the side of the Index, which suppose to contain 8 Chains 5 Links. 3. Place the Table at C, laying the Index upon the line CB, and turn it about till through the sights you see your former station at B, and there fixing it, turn the Index about upon the point C, till through the sights you see the angle at E, and draw the line CE containing 10 Chains 22 Links which set from C to E, and again (before you move your Table) direct the sights to O and draw the line OC containing 6 Chains 64 links, which take from your Scale and set from C to O, and (because O is the next angle to the bounder) you may (without placing your Instrument at O, or measuring the distance ON) draw the line ON upon your Table, which (if the rest of the work be true) will contain 4 Chains 45 links. 5. Remove your Instrument to T, laying the Index upon the line 'tis, and turn it about till through the sights you espy the angle at S, & there fixing it, turn the Index about upon the point T, till through the sights you espy the next angle at V, and by the side of the Index draw the line TV containing 6 Chains 15 Links, which set upon the Table from T to V: now (because V is the angle next the bounder) you may only draw the line VG, without placing your Instrument at V, or measuring the distance VG upon the ground, which (if the rest of the work be true) will contain 6 Chains 38 Links. 6. Bring your Instrument to Q, and lay the Index upon the line PQ, turning the Table about till through the sights you see the angle at P, then fixing the Table there, turn the Index about upon the point Q, till through the sights you espy the angle at R, and by the side of the Index draw the line QR containing 10 Chains 75 Links, which set from Q to R. Lastly, Bring your Table to R, and laying the Index on the line QR, turn the Table about till through the sights you see the angle at Q, and there fix it, then turn the Index about upon the point R, till through the sights you espy the angle at D, and draw the line RD, which (if the rest of the work be true) will contain 5 Chains 3 Links. Thus have you an exact and perfect draught of the whole Manor, or of several enclosures, in the performance whereof I have been something large, because I would show the most natural way first: but the same thing may be performed with more brevity as followeth, wherein (if you mark it well) you shall plainly perceive that half the work will be abreviated, and the same thing effected with almost half the measuring. Having made choice of the angle A to begin your work, place your Table there, turning it about till the Needle hang directly over the Meridian line in the Card, and there fix it, then assign any point upon the Table, for your beginning station, as the point A, and laying the Index to this point, turn it about till through the sights you espy the next angle at L, then draw the line ALL containing 8 Chains 68 Links, which take from your Scale and set from A to L: and also (before you move your Table) direct the sights to B, and by the side of the Index draw the line AB, but you need not measure the length thereof. 2: Then go forward with your work as in the former part of this Chapter, placing your Table at the angles L M and N, and when you come to N, and have drawn the line NP, you may (before you move your Table) draw the line NO, but not measure it. 3. Also when you come to the angle Q, and have drawn the line QS, you may draw the line QR also, at once placing of the Table. 4. When you come to observe at the angle T, and have drawn the line TIE, you may at the same time also draw the line TV, but need not measure it. 5. When you come to the angle G, and have drawn the line GH, you may also draw the line GV, which will cut the line TV in the point V; and at the same time also you may draw the line GF containing 6 Chains 68 Links. Having thus gone round the whole Manor, and made a plot of the outward part or periphery thereof, and also drawn the lines AB, NO, QR, TV, GV and GF, as you went along the bounder, the remainder of the work will (by this means) be much abreviated, for you have no more to do, but 1. To place your Table at F, laying the Index upon the line FG, and to turn it about till through the sights you espy the angle at G, and fixing it there direct the sights to E, and draw the line OF containing 5 Chains 50 Links. 2. Place the Table at E, and lay the Index on the line OF, turning the Table about, till you see through the sights the angle F, then fix it, and turn the Index about upon the point E till through the sights you espy the angle at C, and by the side of the Index draw the line EDC, which containeth 10 Chains 22 Links. Then because that from C to D there is 4 Chains, set 4 Chains from C to D and draw the line DR, which will cut the line QR in the point R, leaving the line DR to contain 5 Chains 3 Links. Lastly, place the Table at C, laying the Index on the line CE, turning it about till through the sights you see the angle at E, and there fixing it, turn the Index about upon the point C, and direct the sights to B and O, drawing the lines CB and CO. And thus have you upon your Table an exact plot of your Manor with great ease and celerity. There is yet another way to perform this work: when you have taken the true plot of the outward bounds or periphery of the whole Manor upon a sheet or more of paper; if you will take the pains to go over every particular enclosure again, and draw particular plots of every parcel by the same Scale wherewith you laid down the plot of the periphery; then over the plot of every particular Enclosure, draw parallel Meridian's, and when you have thus plotted every particular, if you cut them off by their bounders, and lay them one by another according to their situation within the plot of the whole periphery, you shall find that those plots (if your work be true) will justly fill the plot of the whole, leaving no vacuity. CHAP. XLIX. How to take the plot of a whole Manor, or of divers severals whether Woodland or Champion plains, by the Theodolite, Circumferentor, or Peractor. BY what hath been hitherto delivered concerning the harmony between the Theodolite, Circumferentor and Peractor, you may perceive that the working by any one of them being rightly understood, the application thereof to any of the other will be apprehended at the first sight, I will therefore instance in the Circumferentor as being most general. Let the example of the last Chapter serve, where the figure ALMNPQSTUXGHK represented part of a Manor. Then having provided your Field-book ready ruled, you must at the head of one of the leaves thereof writ the Title of the Manor, the County in which it is, and who is Lord thereof, As, The Manor of Elsmore in the County of S. for the Honourable R. B. Lord thereof. Then beginning with your first Close write over the head of your Field-book the Tenant's name, the name of the Close, and the tenor by which he holds the same, so for the first Close. Henry Grey, Casbey Close, Pasture, Free. Under this draw a line quite through your Book, then beginning to survey this Close, place your Instrument at A, and direct your sights to L, noting the degrees there cut, which let be 160 degrees 45 minutes, which 160 degrees 45 minutes must be noted in the first and second Columns of your Field-book, then measure the distance ALL 8 Chains 68 Links, which place in the third & fourth Columns. 2. Remove your Instrument to L, and direct the sights to M, the needle cutting 181 degrees 30 min. and the line LM containing 6 Chains 55 Links, which note down in your Field-book. 3. Place your Instrument at M, and direct the sights to N, the needle cutting 233 degrees, and the line MN 7 Chains 27 Links, which note in your Field-book. And in regard you are to leave the hedge or bounder ALMN, adjoining to Wisby Common, (which appertaineth to another Manor, and therefore only the name inserted for your remembrance when you come to protraction) you must draw a line quite through your Field Book, and in the last Column thereof write Wisby Common, which denotes unto you that you are to leave the bounder of Wisby Common. 4. Place your Instrument at N and direct the sights to O the needle cutting 355 deg. 40 min. and the distance NO being 4 Chains 45 Links, which note in your Field-book as before. 5. Place your Instrument at O, and direct the sights to C, the needle cutting 309 degrees 30 minutes, and the line OC containing 6 Chains 64 Links, which note in your Field-book. Now because at these two observations you went against the hedge or bounder of Banton plain, you must against them write in your Field-bok Banton plain, and because you are now to leave the hedge or bounder of Banton plain, draw a line quite through your Field-book, 6. Place your Instrument at C, and direct the sights to B, the needle cutting 54 degrees 00 minutes, and the distance CB being 8 Chains 5 Links, the degrees and minutes must be noted in the first and second columns of your Field-book, and the Chains and Links in the third and fourth. 7. Remove your Instrument to B, and direct the sights to A, the needle cutting 19 degrees 30 minutes, and the distance BASILIUS being 6 Chains 43 Links, the degrees and minutes must be noted in the first and second Columns of your Field-book, and the Chains and Links in the third and fourth. Now because at these two last observations you went against the hedge or bounder of Bay Wood, you must therefore against them write Bay Wood, and because you have now finished your first Close you must draw a double line through your Book for your remembrance. Then consider which parcel is next fittest to be taken in hand, which let be Bay Wood, and withal at what angle thereof it is most meet to begin, which suppose C; and here (for your help when you come to protraction) you must express in the title of this second Close at what angle you begin the same (unless you had begun it where you ended the last at A, and then it is not material) wherefore seeing you are best to begin at C, look in your Field-book (on the work of the last Close) what degrees and minutes the needle cut at C which were 54 degrees, and 8 Chains 5 Links, therefore against that number make this ☉ or the like mark, and write the Title for your second Close thus. Samuel White, Bay-wood, by Lease, begin at ☉. By this means you shall readily know when you come to protraction, where to begin with this prcell, and in the margin place (2) for the number of your second parsell, and then proceed in your work of surveying this parcel as before you did for the other till you have gone round about the same ending at A where you first began, noting down all your observations both of lines and angles, with the particular bounders as you go along in your Field-book, in all respects as you did those of the first Close, and in thus doing you shall find that at your first observation from C to E, that you went partly by the hedge or bounder of Banton plain, and partly by the hedge or bounder of Church-field, and therefore against the degrees of that observation writ Banton plain and Church-field, there drawing a line: then at your two next observations at E and F you went along the hedge or bounder of Church-field, and at the three last observations at G H and K you went against the hedge or bounder of Wisby Common, there finishing your second parsell, wherefore draw a double line quite through your Field-book. These two parcels being finished, consider which is next fittest to be taken in hand, and where to begin it, which suppose Banton plain, and to begin at N, wherefore look in your field-book what degrees the needle cut when you made observation at N in the surveying of Gosby Close, and left the bounder of Wisby Common, which degrees you shall find to be 355 degrees 40 minutes, and 4 Chains 45 Links, therefore at the end of that line where you find 355 degrees 40 minutes, and 4 Chains 45 Links, make this + or some other mark for a remembrance when you come to protraction, then for the next parcel write in your Field-book. George Burton, Banton plain, for two lives, begin at ✚. This being done place your Instrument at N, and direct the sights to P, the needle cutting 220 degrees 20 minutes, and the line NP containing 9 Chains 32 Links, which note in your Field-book, and because at this observation you went by the hedge or bounder of Wisby Common, and are now to leave it, therefore draw a line and write Wisby Common, and in this manner must you go about this parcel also till you come to close at D, and having finished draw a double line. Then considering that Church field is next fittest to be surveyed, and that it is most convenient to begin the same at Q, therefore look what degrees the needle cut at Q in the surveying of Banton plain which were 15 degrees 40 minutes, ●nd 10 Chains 75 Links, against which in your Fild-book make this ♓ or the like mark for your remembrance, and for your next Close ●rite in your Field-book as followeth. Thomas King, Church field, by Lease, begin at ♓. Then placing your Instrument at Q, direct the sights to S, noting the degrees cut, and the length of every line measured, with your particular bounders, as you did in the other Closes before, till you come to enclose at G, and when you have done, draw a double line quite through your Field-book, and write the title of the next Close to be surveyed in this manner. John Nichols, Odcumb Close Free, begin at—. Then placing your Instrument at T, direct the sights to Y, and note the degrees cut and the lines measured as in those before, till you have gone round the field to G. And thus, if there were never so many Enclosures you may (without confusion) easily distinguish the work of the one from the other, and be able (remembering the premises) to draw a plot thereof at any time, remembering always that those numbers in the Margin of your Book, aught to be placed severally in your Plot in those Closes they represent. The Manor of Elsmore, in the County of S. for the Honourable R. B. Lord thereof, (1) Henry Grey, Cosbey Close, Pasture, Free. 160 45 8 68 Wisby Common. 181 30 6 55 233 00 7 27 355 40 4 45 + Banton plain. 309 30 6 64 54 00 8 5 ☉ Bayliff Wood 19 30 6 43 (2) Samuel White, Bay Wood, by Lease, begin at ☉. 320 00 10 22 Banton plain, & Church field. 15 30 5 50 Church field. 337 45 6 68 87 30 6 84 Wisby Common. 113 30 6 73 153 30 6 69 (3) George Burton, Banton plain, for 2 Lives, begin at +. 220 20 9 32 Wisby Common. 299 30 10 50 The Forest. 15 40 10 75 ♓ Church field 53 30 5 3 (4) Thomas King, Church field, by Lease, begin at ♓. 316 20 13 12 The Forest. 17 15 10 83 Church Lane. 56 00 6 15 — Odcumb Close. 24 10 6 38 (5) John Nichols, Odcumb Close, Free, begin at— 334 30 7 3 Church Lane. 48 30 6 25 101 30 6 18 These Instructions being sufficient for the application and use of the Field book, I shall desire all men to make frequent trial and practise thereof, and compare the Book with the Plot, and protracting the same according to the directions hereafter given, you will find it to be most exact and facile. Here by the way, I might give directions whereby to take in divers severals at once, if the bounders be regular, which will much ease you both in surveying and protracting, but by small practise this and divers other abreviations will appear of themselves. I have here added one leaf of your Field-book as it ought to be ruled, which take for an example, it being the collections of the work of this Chapter, with the several lines, angles and bounders, as you observed them in your Survey. CHAP. L. How to protract or draw the plot of a whole Manor, or of divers enclosures, the observations of the several angles, lines and bounders being noted in your Field-book. PRovide a Skin of Velom, or Parchment, or divers sheets of paper neatly fastened together with Mouth-glew, according to the magnitude or greatness you intent to have your Plot, which paper or parchment let be ruled all over with 〈◊〉 parallel lines, representing Meridian's, as is taught in the 36 Chapter of this Book, the distance of which lines one from another must not exceed the breadth of the Scale of your Protractor. Now suppose you were to protract the observations of the last Chapter, laying your Field-book before you, consider which way your plot will extend, and accordingly begin your work, as at the point A, upon which point A place the centre of your Protractor, turning it about, till the correspondent divisions at each end of the Scale of the Protractor lie directly upon one of the parallel meridians, and staying the Protractor there, look in your Field-book what degrees and minutes the needle cut at your first observation at A, which were 160 degrees 45 minutes, therefore against 160 degrees 45 minutes of your Protractor, make a mark, and through that mark and the point A, draw the line ALL, containing 8 Chains 68 Links. Then place the centre of the Protractor upon the point L, in all respects as before, and finding your next degrees and length to be 181 degrees 30 minutes, and the length 6 Chains 55 Links, therefore against 181 degrees 30 minutes of your Protractor make a mark, and through it draw the line LM containing 6 Chains 55 Links. Then place the centre of the Protractor upon the point M, and look in your Field-book what degrees were cut at M, protract those degrees (as before) and draw the line MN containing 7 Chains 27 Links. Then place the centre of the Protractor upon the point N, the degrees cut being 355 degrees 40 minutes, and the line NO containing 4 Chains 45 Links, and because against these 355 degrees 40 minutes you find in your Field-book this mark + there placed, you must therefore (with Black lead or the like) make the same mark at the point N upon your paper, to signify that you must there begin to protract some other Close. In this manner must you proceed with all the other lines and angles as you find them noted in your Field-book, till you have gone over your first Close, and closed your plot at A. Having thus finished your first enclosure, you must deal in the same manner with the second, third and fourth, and so on, were there never so many. And to know where to begin to protract your second enclosure, you must have recourse to your Field-book, where you shall find this mark ☉ at which you must begin your second enclosure, which is Bay Wood, and the like mark upon your paper at the point C, which is your remembrancer to put you in mind that at the point C you must begin to protract your second Enclosure, as you did your first Close. ¶ In this manner of protracting there is no difference nor cautions to be observed, more than those already hinted in Chap. 36 and 38 of this Book, viz. that if the degrees to be protracted be under 180, to lay the Semicircle of the Protractor upwards or from you, and if they be above 180, to lay the Semicircle downwards. CHAP. LI. The figure of any plot being given, how to enlarge or diminish the same according to any assigned proportion. IT may so fall out that when you have taken the plot of a whole Manor upon your Plain Table, in divers sheets of paper, or observed the angles, and afterwards protracted them, as in the two last Chapters, it may so fall out that your plot may be either bigger or lesser then is desired, now if at any time it be required to enlarge or diminish any plot according to any proportion, this Chapter will accomplish your desire. The Instruments for the performance hereof are divers, as was intimated in the 9 Chapter of the 2 Book. Now for generality and exactness, the two Indices there spoken of, having at each end thereof a Semicircle, is inferior to none, but the Instrument being very chargeable, and the use thereof very intricate and tedious, I shall wholly omit to speak any more of it. There is another way also which Master Rathborn used, which was with a Ruler by him invented for that purpose, which would indifferent well reduce a plot from one bigness to another according to some particular proportions. The making of this Ruler is so well known, and the use thereof so apparent, that I shall not need to say any thing concerning the description or use of it: I only intimate that there is such a Ruler, that those which please may have it made. Another way is by one line divided into 100 or 1000 equal parts only, which by the help of Arithmetic will perform this work very well, but this (as being very tedious) I neglect. To pass by these and divers others which I could name, I shall say something of the Parallelogram, which for generality, exactness, and dispatch, surpasseth all the rest, unto which (in my opinion) there is none comparable. Of Parallellograms there are divers sorts, but that which I shall instance in, consisteth but of four Rulers only, the making whereof is well known to the Instrument maker, and the manner of using it is as followeth. Take the plot which you would reduce, and fasten it to a Table with Mouth-glew, then by it, upon the same Table, fasten your fair paper or parchment, upon which you would have your new plot; then having fitted your Parallelogram according to the proportion into which you would have your plot reduced, fix the Parallelogram to the Table, by a point for that purpose: then put your drawing pen into some one hole on one of the sides of the Parallelogram, and upon it a plummet of lead or brass to keep the pen down close to the paper, when it is moved thereupon: and here note, that at any time when the Parallelogram is thus fitted, the point that sticketh in the Table, the Pen which is to draw, and the Tracer which you must move along the lines of your old plot, will lie always in a right line, but this by the way: Your Parallelogram being fixed to the Table, and the pen in its true place fitted to draw, take the Tracer in your right hand, and with it, lightly go over all the lines of your old plot, so shall the motion thereof occasion the pen to draw upon your clean paper or parchment, the true and exact figure of your former Plot, though of another bigness, which will be in proportion to the greater according to the situation of the sides of the Parallelogram, which will better appeat by the sight of the Instrument, than words can possibly explain it. CHAP, LII. How to draw a perfect draught of a whole Manor, and to furnish it with all necessary varieties, also to trick and beautify the same: in which, (as in a Map) the Lord of the Manor may at any time (by inspection only) see the symmetry, situation and content of any parcel of his Land. HAving protracted your plot according to your intended bigness, and written the content of each Close about the middle thereof, you may about the bounds of every field or Enclosure, with a small Pencil, and some transparent green colour, neatly go over your black lines, so shall you have a transparent stroke of green on either side of your black line, which will add a great lustre and beauty to your Plot. Then in your Wood-land grounds, draw divers little Trees in the most material places, and shadow your mountainous and uneven grounds with hills and valleys, expressing all kind of Bogs, Groves, Highways, Rivers, etc. distinguishing them by lively colours according to their similitudes. Then in some convenient place of the Plot, without the Enclosures, draw a Circle, and therein describe the 32 points of the Mariner's Compass according to the situation of the grounds, with a Flower-de-luce at the North part thereof. Then in some other convenient place of your plot, make a Scale equal to that by which your plot was protracted. Lastly, in some other convenient place towards the upper part thereof, draw the Coat of Arms belonging to the Lord of the Manor, with Mantle, Helm, Crest, and Supporters; or in a Compartment, but be sure you blazon the Coat in its true Colours. THE Manor of Lee. These things being well performed, your plot will be a neat Ornament for the Lord of the Manor to hang in his Study, or other private place, so that at pleasure he may see his Land before him, and the quantity of all or every parcel thereof without any further trouble. Also in your plot must be expressed the Manor-house according to its symmetry or situation, with all other houses of note, also all Water-mils, Windmills, and whatsoever else is necessary, that may be put into your Plot without confusion. For farther explanation of what hath been delivered in this Chapter, I have here added the figure of a small Manor, which will be sufficient for example sake. CHAP. LIII. How to find whether water way be conveyed from a Spring head, to any appointed place. THere is an Instrument called a water-level, for the performance hereof, the making whereof is sufficiently known. Now if it were required to know whether water may be conveyed in Pipes or Trenches from a Spring head to any determinate place, observe the following directions. Place your water-level at some convenient distance from the Spring head, in a right line towards the place to which the water is to be conveyed, as at 30, 40, 60, or 100 yards distant from the Springhead. Then having in a readiness two long straight poles (which you may call your station staves) divided into Feet, Inches, and parts of Inches from the bottom upwards: being thus provided, cause one (whom you may call your first assistant) to set up one of the said staves at the Spring head, and require another (which you may call your second assistant) to erect the other staff beyond your Instrument at 30, 40, 60, or 100 yards forward, towards the place to which the water should be conveyed. These station staves being erected perpendicular, and your water-level in the mid way precisely horizontal, go to the end of the Level, and looking through the sights, cause your first assistant to move a leaf of paper up and down your station staff, till through the sights you see the very edge thereof, and then by some known sign or sound, intimate to him that the paper is then in its true position, then let this first assistant note against what number of Feet, Inches, and parts of an Inch the edge of the paper resteth, which he must note down in a paper. Then your water-level remaining , go to the other end thereof, and looking through the sights towards your other station staff, cause your second assistant to move a leaf of paper along the staff, till you see the very edge thereof through the sights, and then (by some known sign or sound) cause him to take notice what number of Feet, Inches, and parts of an Inch, are cut by the said paper, which will him also to keep in mind, or note in a paper as your first assistant did. This done, require your first assistant to bring his station staff from the Spring head, and cause your second assistant to take that staff and carry it forward towards the place to which the water is to be conveyed, 30, 40, 60, or 100 yards, and there to erect it perpendicular as before, letting your first assistant stand at that staff where your second assistant before stood; then in the mid way between your two assistants, place your water-level exactly horizontal, and looking through the sights thereof, cause your first assistant to move a paper up and down, and when you give him a sign to note what number of Feet, Inches, and parts of an Inch are cut by the paper, and note them down, then going to the other end of your water-level, look through the sights, and cause your second assistant to move a paper along the Staff, and to note the Feet, Inches, and parts of an Inch as before. Then cause your first assistant to bring away his station-staffe, and cause your second assistant to take it and carry it 30, 40, 60, or 100 yards forwarder towards the place to which the water is to be conveyed, and leaving your first assistant at the place where your second assistant last stood, place your water-level again in the mid way between your two Assistants, and looking through the sights as before, cause each of them to move a leaf of paper up and down their station staves, and note down in their several papers the number of Feet, Inches, and parts of an Inch cut, when you looked through the sights of your Water-levell. In this manner you must go along from the Spring head, to the place unto which you would have the water conveyed, and if there be never so many several stations, you must, in all of them, observe this manner of work precisely, so by comparing the notes of your two Assistants together, you may easily know whether the water may be conveyed from the Spring head to the desired place or not. ¶ Here note, that in your passage between the Spring head and the appointed place, from station to station, you must observe this order, otherwise great error will ensue, viz. that your first assistant must at every station, stand between the Spring head and your water-level: and your second assistant must always stand between your water-level and the place to which the water is to be conveyed, thus by observing this order in your work you shall have no confusion, neither shall one of your Assistants take more pains than the other. Having thus orderly proceeded from the Spring head to the place appointed, call both your Assistants together, and cause them to give in their notes of the observations at each station, and add them together severally: then if the note of the second assistant exceed (or be greater than) the note of the first assistant, take the lesser out of the greater, and the remainder will show you how much the appointed place to which the water is to be brought is lower than the Spring head. The First Assistants Note. Station Feet Inch parts 1 15 3 ½ 2 2 1 ¼ 3 1 6 0 Sum 18 10 ¾ The Second Assistants Note. Station Feet Inch parts 1 3 2 ¾ 2 14 0 ¼ 3 3 11 0 Sum 21 2 0 By this Table you may perceive that the notes of the first assistant collected at his several stations being added together, amounteth to 18 Feet, 10 Inches; and ¾ of an Inch: and the notes of your second assistant at his several stations being added together amounteth to 21 Feet and 2 Inches, so the number of the first assistants observations being taken from the number of the second, there will remain 2 Feet, 3 Inches and ¼ of an Inch, and so much is the place to which the water is to be brought, lower than the Spring head, according to the straight water-level, and therefore the water may easily be conveyed thither. ¶ Here note, that when you have called your two Assistants together, and examined their several Notes, and added them together, if then you shall find the sum of your first assistants Note to be greater than the Sum of your second assistants Note, that then it is impossible to bring the water from that Spring head to the intended place: but if the Sums of the Notes of your two Assistants do exactly agree; there is then a possibility of effecting it, if the distance be but short, though with more charge and difficulty. ¶ Note 2, That the most approved Authors concerning this particular do aver, that at every miles end there ought to be allowed 4½ Inches more than the straight Level, for the current of the water. ¶ Note 3, If there be any Hill lying in the way between the Spring head and the place to which the water is to be conveyed, you must then cut a Trench by the side of the Hill in which you must lay your pipes equal with the straight water level, with the former allowance. And in case there be a Valley you must then make a Trunk of strong wood well underproped with strong pieces of Timber, and well pitched or leaded, as is done in divers places between Ware and London. ¶ Note 4, That in your conveying of water to an appointed place, it is not convenient to bring it from the Spring head by the nearest distance or in a straight line, but by a crooked or winding way; and you ought also to lay the pipes one up and another down, but this is to be observed but in some cases only, where the water will have too violent a current. Another way. There is another way whereby you may know whether water will be brought to any place or not, which in very large distances ought to be considered. Take the distance between the Spring head and the place to which the water is to be brought, which multiply in itself, add the product thereof to the Square of the Earth's Semidiameter, viz. to the square of 3436 4/11 Italian miles, than out of the product thereof extract the Square Root, and then from that Square Root take 3436 4/11 miles, the remainder is the difference between the line of level, and the water or circular level. Thus have I finished my intended discourse of Surveying of Land, in which I have studied rather to make every particular therein contained plain and perspicuous to the meanest capacity, then with too much brevity to obscure that which I chiefly aimed at, namely, to instruct the ignorant: I confess I may be justly blamed by those who are Masters of the Art, or have a considerable knowledge thereof already, for using too many circumlocutions, but I answer, it was not written for their sakes, yet I hope it will not be rejected by them; and although I do not attempt to teach such more than they know already, yet (possibly) they may herein find something worth their perusal and practice, or (at least) it may be a remembrancer unto them to bring to mind what otherwise they may have forgotten: But ceasing to apologise any more for my Book, let it now speak for itself. FINIS.