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A48331 The compleat surveyor containing the whole art of surveying of land by the plain table, theodolite, circumferentor, and peractor ... : together with the taking of all manner of heights and distances, either by William Leybourn. Leybourn, William, 1626-1716. 1653 (1653) Wing L1907; ESTC R20856 115,157 173

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equall 2. If any right line fall upon two parallel right lines it maketh the outward angles on the one equall to the inward angles on the other and the two inward opposite angles on contrary sides of the falling line also equall 3. If any side of a Triangle be produced the outward angle is equall to the two inward opposite angles and all the three angles of any Triangle are equall to two right angles 4. In equiangled Triangles all their sides are proportionall as well such as contain the equall angles as also the subtendent sides 5. If any four Quantities be proportionall the first multiplied in the fourth produceth a Quantity equall 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 equall to both the squares of the containing sides 7. All parallelograms are double to the triangles that are described upon their bases their altitudes being equall 8. All triangles that have one and the same Base and lie between two parallel lines are equall one to the other GEOMETRICALL PROBLEMES PROBLEME I. Vpon 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 then 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 equall 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. IV. How to make an angle equall to an angle given LEt the angle given be ACB and let it be required to make another angle equall thereunto First draw the line EF 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 arke DE then 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 equall 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 arke 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 equall parts LEt AB be a line given and let it be required to divide the same into four equall 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 ABD equall to the angle CAB then upon the lines AC and BD set off any three equall parts which is one lesse then 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 equall parts as was required PROB. VII A right line being given how to draw another right line parallel thereunto which shall also passe through a point assigned LEt AB be a line given and let it be required to draw another line parallel thereunto which shall passe through the given point C. First Take with your compasses the distance from A to C and placeing one foote 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 arke FG crossing the former arke DE in the point H. Lastly if you draw the line CH it shall be parallel to AB PROB. VIII Having any three points given which are not situate in a right line how to finde the center of an arch of a Circle which shall passe directly through the three given points THe three points given are A B and C now it is required to finde the center of a Circle whose circumference shall passe through the three points given First Opening your Compasses to any distance greater then halfe 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 then halfe AB place one foot in
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 Satute 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 Hils and Mountains how to finde the lengths of the horizontall lines on which they stand severall wayes 257 24. Of mountanous 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 severall 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 inlarging 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 severall kindes and that the exact and carefull making and dividing of all such Instruments is chiefely 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 Mathematicall Instrument whatsoever is exactly made by Mr. Anthony Thompson in Hosier lane neer Smithfield London THE COMPLEAT SURVEYOR The First Book THE ARGVMENT THis first Book consisteth of divers Definitions Problemes Geometricall extracted out of the Writings of divers ancient and modern Geometricians as Euclid Ramus Clavius c. and are here so methodically disposed that any man may gradually proceed from Probleme to Probleme without interruption or being referred to any other Author for the Practicall performance of any of them Onely 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 Theoremes which had the ensuing Problemes been demonstrated must of necessity have been inserted Also the figures would have been so incumbred with multiplicity of lines that the intended Problemes 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 onely made choice of such Problems as I conceived most usefull for my present purpose and come most in use in the practice of Surveying and ought of necessity to be known by every man that intendeth to exercise himselfe in the Practice thereof and those are chiefly such as concern the reducing of Plots from one forme to another and to inlarge or diminish them according to any assigned Proportion also divers of the Problemes in this Book will abundantly help the Surveyor in the division and seperation of Land and in the laying out of any assigned quantity whereby large parcels may be readily divided into divers severals and those again sub-divided if need be Also for the better satisfaction of the Reader I have performed divers of the following Problemes both Arithmetically and Geometrically GEOMETRICALL DEFINITIONS 1. A Point is that which cannot be divided A Point or Signe is that which is void of all Magnitude and is the least thing that by minde and understanding can be imagined and conceived than which there can be nothing lesse 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 thicknesse 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 streight or crooked And of the three kindes of Magnitudes in Geometry viz. Length Breadth and Thicknesse 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 lyeth streight and equall 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 kinde of Magnitude so the motion of a Line produceth a Superficies which is the second kinde 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 extreams of a Superficies are Lines As the extreams or ends of a Line are points so the extreams or bounds of a Superficies are Lines and so the extreams 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 joyned 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 then 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 sheweth
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 equall 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 equall namely the angle ABD on the one side equall to the angle ABC on the other side then are either of the two angles ABC and ABD 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 lesse than a right angle So the angle EBD is an acute angle for it is lesse than the right angle ABD 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 unequall 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 equall 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 equall as the lines AB AC AF AD and this point A is called the center of the Circle 16. A Diameter of a Circle is a right line drawn by the Center thereof and ending at the Circumference on either side dividing the Circle into two equall parts So the line BAD 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 center of the Circle And moreover it divideth the Circle into two equall parts namely BCD being on one side of the Diameter equall 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 center of any Circle do not divide it equally all the lines drawn from the center of that Circle to the Circumference cannot be equall 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 lesse then 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 lesse 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 then 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 equall sides but not right angles As the Figure C. 27. A Rhomboides is a Figure whose opposite sides are equall and whose opposite angles are also equall but it hath neither equall 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 equall sides nor equall 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 Geometricall Theoremes 1. ANy two right lines crossing one another make the contrary or verticall angles
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 equall 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 then by the 31 Probleme reduce the five sided figure ABCFG into the Triangle HAK the base whereof HK divide into two equall parts in O and draw the line OA which shall divide the five sided figure ABCFG into two equall parts Then by the 30 Probleme reduce the Trapezia FCDE into the Triangle OLM and divide the base thereof LM into two equall parts in the point P and draw the line OP which will divide the Trapezia FCDE into two equall parts and so is the whole Plot divided into two equall parts by the lines AO and OP but to performe the Probleme 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 equall parts The end of the First Book THE COMPLEAT SURVEYOR The Second Book THE ARGVMENT IN this Book is contained both a generall 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 Staffe Sockets Screws Index Label and other necessaries Now whereas these three Instruments are the most convenient for all manner of practises 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 shewed how that Instrument may be ordered to performe 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 onely as it is there contrived with the same ease dispatch and exactnesse and in many respects better as in Chap. 1. doth plainly appear so that this Instrument onely is sufficient for all manner of practises whatsoever And besides the fore-mentioned 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 INSTRVMENTS CHAP. I. Of Instruments in generall THe particular description of the severall Instruments that have from time to time been invented for the practise of Surveying would make a Treatise of it self and in this place is not so necessary to be insisted on every of the inventors in their severall Books of the uses of them having been already large enough in their construction To omit therefore the description of the Topographicall Instrument of Master Leonard Diggs the Familiar Staffe of Master John Blagrave the Geodeticall Staffe and Topographicall Glasse 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 chiefely 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 himselfe 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 kinde 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 businesse I would advise him to use the Circumferentor or Theodolite and for Townships and small Inclosure the Plain Table so altering his Instrument according at the nature or quality of the ground he is to measure doth require These three speciall 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 shew you a new metamorphosis of that Instrument making it the most absolute and universall 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 exactnesse and whatsoever may be done by the limbe 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 Staffe as in the description thereof it is so conveniently ordered will be an absolute Circumferentor and in some respects better then the ordinary one hereafter described because the Sights thereof stand at a greater distance so that thereby the visuall 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 chiefely 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 practise will appear of themselves CHAP. II. Of the Theodolite the description thereof and the detection of an errour 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 equall parts called degrees and each degree sub-divided into as many other equall parts as the largenesse of the Instrument will best permit For the diameter of this Circle it may be of any length but those usually made in brasse are about twelve or fourteen inches and the limb thereof divided as aforesaid into 360 degrees and sub-divided into other parts by diagonall 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 Geometricall Square which is described within the Circle and the sides thereof divided into certain equall 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 sheweth 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 center of the Circle upon which center also the Index of the Instrument must turn about and somtimes 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 staffe 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 c. 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 sheweth is Trigonometricall and sufficiently shewn by him I will passe that by and shew 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 equall parts by Diagonals and let it be required to finde where the concentrique Circles E F and G must be drawn so that lines drawn from the center A through the points E F and G shall divide the arch BC into four equall 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 concentricall that is upon the same center A but not equidistant therefore by the ● Probleme of the 1. Book draw the arch of a Circle which shall passe through the points B D A then divide the part of that arch which lies between B and D into four equall 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 crosse your Diagonals to divide the limb into four equall parts whereas if the Circles had been equidistant the arch would have been unequally divided and this errour is frequently practised for in the making of any Instrument they commonly divide the distance BH or CD into four equall parts and through them draw the concentrique Circles whereas by the figure you see that the farther the Circles are from the center the closer they come together but let this suffice for the correction of this Errour 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 bredth four inches and in thicknesse about three quarters of an inch one side whereof is divided into divers equall parts most fitly of ten or twelve in an inch so that it may be used as the Scale of a Protractor the Instrument it selfe 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 halfe Diameter and about half an inch deep in which is placed a Card divided commonly into 120 equall 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 Diall drawn to finde the hour of the day and Azimuth of the Sun within the Box is hanged a Needle touched with a Load-stone and covered over with a cleer glasse to preserve it from the weather On the upper part of this Instrument is also described a Table of naturall Sines collected answerable to the Card in the box that is to say if the Card be divided but into 120 parts the Sines must be so also but if into 360 the Sines 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 Center of a supposed Circle the Semidiameter whereof is the distance from the Wyer to the edge of the Instrument underneath the same which parts is imaginarily divided into sixty equall 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 tens There is also belonging to these divisions a little Ruler at one end whereof is a little hole to put it upon the wyer 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
a line into such equall parts so that your stationarie 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 shew 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 exactnesse attein the true distance and scituation of any Mark or Marks far remote without approaching neer 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 alwayes 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 severall 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 Compasse 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 severall angles thereof make choise 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 severall 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 applyed to your Table from L to C shall give you the point C representing your third mark In this manner you must deale 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 alwayes 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 severall 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 alwayes 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
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 horizontall And this short Ruler with the divisions thereof and those on the edge of the Instrument serve for taking of altitudes chiefly and for the reducing of hypothenusall to horizontal lines CHAP. IV. 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 exactnesse THe Table it selfe is a Parallelogram containing in length about fourteen inches and a halfe and in bredth eleven inches it is composed of three severall 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 joynted 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 inclosure by the Table This frame must have upon it neer the inward edge Scales of equall 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 bredth and one third part of an inch in thicknesse 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 brasse pin and also a thred and plummet to place your Instrument horizontall 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 Wyer on the top of the shorter Sight and the slit on the longer Sight stand precisely over the fiduciall edge of the Index The space or distance of these two Sights one from the other is to be equall to the divided part of the longer Sight Upon the longer Sight is to be placed a Vane of brasse 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 Geometricall 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 bredth thereof there is drawn a line called the line of Level dividing the side of the projected Square into two equall parts also the same side is on this Sight divided into a hundred equall parts which are numbered upwards and downwards from the line of Levell by fives and tens to fifty on either side which divisions are called the Scale There is also on the same Sight another sort of division representing the hypothenusall Lines of the same Square as they increase by Unites and are likewise numbered upwards and downwards from the line of Levell from one to twelve by 1 2 3 c. sometimes signifying 101 102 103 c. these divisions shew how much any hypothenusall or slope line drawn over the same Square exceedeth the direct horizontall 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 Levell upward and downward by fives and tens to 25 which divisions are called the Quadrant Unto this Instrument as unto all others belong these necessary parts as the Socket the Staffe the Box and Needle c. ¶ According to this description have Plain Tables formerly been made but if unto it be added these additionall parts and alterations which make it lesse cumbersome then before it will be the most exact absolute and universall 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 equall parts as in the description the other side may have projected upon it the 180 degrees of a Semicircle from a Center noted in the superficies of the Table which degrees must be numbered from the left hand towards the right when the Center is next to you by fives and tens 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 winde 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 threed above and the other a slit above and a threed 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 practise 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 Hypothenusall to Horizontall 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 Brasse 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 horizontall if you look through this small Sight and by the Pin on which the Label hangeth moving the Label too and fro till you espie the mark you look at then will the Labell shew 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 lesse cumbersome then 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
colours you must alwayes work with one end of the Chain from you This Chain being thus divided and marked you have every whole Pole equall to ten Primes or 100 Seconds every three quarters of a Pole equall to seven Primes and a halfe or 75 Seconds every halfe Pole equall to five Primes or 50 Seconds and lastly every quarter of a Pole equall to two Primes and a halfe 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 severall parts you may make use of Seconds Of Mr. GUNTERS Chain AS every Pole of Master Rathborns Chain was divided into 100 Links so Master Gunters whole Chain which is alwayes 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 then 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 expresse more briefely 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 brasse like a Rhombus so is the whole Chain by this plate divided into two equall parts Secondly Let each of these two parts be divided into two other equall parts by smaller Rings or Circular plates of brasse so shall the whole Chain be divided into four equall parts or Perches each Perch containing 25 Links Thirdly At every ten Links let be fastened a lesser Ring then 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 carryed 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 misse a Chain or two in keeping your account from whence will ensue a considerable errour Also in measuring of distances when you go not along by a hedge side you can hardly keepe your Instrument Chain and Mark in a right line which if you do not you must necessarily make your measured distance greater then in reality it is For the avoyding 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 kinde 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 exactnesse to direct it to the Mark intended How to reduce any number of Chains and Links into Feet IN the practise of many Geometricall 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 withall I thought it convenient in this place to shew 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 lesse then 10 as in the second Example it is 9 Chains 5 Links you must place a Cypher 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 sheweth 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 shewing 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
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 finde 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 severall Scales of equall parts a Line or Scale of Chords which is numbered by 10 20 30 c. 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 alwayes 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 then 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 finde what number of degrees and minutes are contained therein Suppose CAB were an angle given and that it were required to finde 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 finde 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 chiefe uses of the Protractor may be performed by the line of Chords last spoken of yet for avoyding 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 center of the Protractor in which point also fix your protracting pin and turn the Protractor about upon the center till the Meridian line of the Protractor noted in the description thereof with EF 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 finde the quantity thereof by the Protractor Suppose DCB were an angle given and that it were required to finde the quantity thereof by the Protractor First you must apply the center of the Protractor to the point C and the Meridian line thereof directly upon the line DC then shall you finde 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 togeth● 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 bindes both the paper to the boards and the boards 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 center and so to the end of the degrees Then put the Socket upon the head of the Staffe 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 center in any part of the Table which you shall think most convenient for the bringing on of the work which you intend But if you use your Instrument as a Theodolite then the Index must be turned about upon the Center of the Table for which purpose there is a piece of wier which goes through a small hole of brasse fastened to the Index and so into the center by which means the Index keepes his constant place only moving upon the center 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 Staffe with a brasse screw-pin fitted for that purpose so that the Staffe being fixed in any place the Index and fights may turn about at pleasure without moving of the Staffe and now is your Instrument a good Circumferentor nay better then that before described in the second Book Also when you have occasion to measure any Altitude hang the Labell upon the farther Sight and thus are you exactly fitted for all occasions CHAP. IV. 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 it selfe but are not denominated by their quantities but by their symetry 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
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 severall observations into one sum and multiply 180 degrees by a number lesse by two then the number of angles in the field and if the product of this multiplication be equall to the totall summe of your angles then is your work true otherwise not EXAMPLE In the work of the 32 Chap. the angles found were as in the margine the summe 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 lesse 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 summe 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 inclosure 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 EF 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 gon round the field in this manner and collected the degrees cut and the lines measured in the severall columns of your Field book according to former directions you shall finde 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 wonderfull 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 looke 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 losse of time So the Instrument being placed at A if you direct the sights to B you shall finde 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 exactnesse 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 largenesse 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 somwhat lesse then the breadth of the Scale of your Protractor These parallel lines thus drawn do represent Meridians 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 EF must alwayes be laid when you protract any observations taken by the Circumferentor as in the Chapter before going Your paper or parchment being thus prepared assigne any point upon any of the Meridians as A upon which point place the center 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 looke in your Field-book what degrees the needle cut at A which were 191 degrees now because the degrees were more then 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 center of the Protractor on the point B with the meridian line thereof parallel to one of the pricked Meridians drawn upon the paper and seeing the degrees cut at B were more then 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 center of the Protractor on the point C the Meridian line thereof lying parallel to one of the pricked Meridians drawn on the paper then 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 center 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 Meridians drawn upon your paper and because the degrees cut by the Needle at this angle were lesse then 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 EF containing 9 Chains 71 Links 6. Lay the center 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 alwayes observing this for a generall rule to lay the meridian line of the Protractor upon or parallel to one of the Meridians 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 lesse then 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 kinde of practises fully intimated so that you may perceive by what hath been hitherto delivered that for some kinde of works one Instrument is better then another and for large and spacious businesses the Circumferentor is the best the Needle being good and no impediment neere to hinder the playing or vertue 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 errour 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 then a Theodolite only the Box and Needle is so fitted to the center 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 immoveable 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 espie the mark or angle you would looke at then shall the Index cut the same degrees and minutes upon the Limbe 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 confesse is somthing 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 then 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 center till through the sights you espie your second angle at B then 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 severall 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 then by the Needle as here you see 10 minutes are lost in estimation which the Index giveth more precisely nay somtimes you may possibly misse halfe 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 finde 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 espie 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
severall parcels in this kinde 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 columne of your Booke or paper you must write the Tenants name and the tenour by which he holds the same from what place you begin to measure and upon what point of the compasse you passe from thence and observing this direction in all the rest you may if need require bound every parcell 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 then 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 then you must take the length thereof at every second or third breadth and expresse the same in your Book under the title of length As for the expressing of the severall breadths you need but to crosse over the whole Furlong taking every mans breadth by the middle thereof and entering the same as you passe along but in case there be a considerable difference at either end then I would advise you to take the breadth at either end and adde them together into one sum then take the half of that summe 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 severall lengths and breadths in your book you may multiply the length and breadth of every parcell together as is taught before and so shall you have the quantity of every parcell by it selfe which quantity must be noted downe in the three last columns of your Book as in the following example appears Mordon Field The Tenants names and tenour 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 finde the horizontall 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 ABD the thing required is to finde the length of the line BD on which the mountain standeth First place your Instrument at the very foot of the Hill exactly levell 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 complement 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 finde 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 finde 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 CAD 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 hypothenusall lines AB and AD. Another way There is another way also used by some for the measuring of horizontall lines which is without the taking of the Hils altitude or using of any Arithmeticall 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 finde the length of the Horizontall 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 hil holding it levell 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 levell 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 horizontall 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 adde all together for the horizontall line CHAP. XLVII How to plot Mountanous and uneven grounds with the best way to finde the content thereof FOr the plotting of any mountanous 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 finde the horizontall line thereof and draw that upon your Table accounting thereon the length of the hypothenusall 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 angles A C E and F let fall the perpendiculars AK CH EI and FM Now in regard there are many hils 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 finde them by measuring which will be much longer then the streight line BD measured on your Scale then by help of your Instrument finde 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 CH set the number as you finde it by the Chain then finde the perpendicular IE and measure that with your Chain also all which lines in respect of the hils and vallies will be found much longer then if they were measured by your Scale then by the measured lines BD CH 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 mountanous 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 adjoyning grounds out of their places now for distinction in your Plot you may shadow them off with hils as in this figure lest any man seeing your plot should measure by your Scale and finde your work to differ CHAP. XLVIII How to take the Plot of a whole Mannor or of divers parsels of Land lying together whether Wood-lands or Champion plains by the Plain Table ALthough practise in the performance hereof be better then 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 Mannor 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 walke about the whole Mannor or at least so much as you are to survey that you may be the better acquainted with the severall bounders and in your passage you ought to take speciall notice of all eminent things lying in your way as Churches Houses Mils High-wayes Rivers c. 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 peripherie or outward part of the Mannor 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 assigne 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 AL 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 espie a mark set up at the angle L where your Table last stood and there fixing it you shall still finde 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 espie 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 finde 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 espie 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 Mannor 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 finde 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 finde 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 crosse your paper which the equall 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 peripherie of the whole Mannor 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 severall Closes therein contained in this manner 1. Place your Table at A laying the Index and sights upon the line AL before drawn and turn it about till through the sights you espie 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 espie 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 BA 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