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 â—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
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
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
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 â—… 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
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 Minutes 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 severall 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 alwayes 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 finde 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 wil 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 finde the South end of the Needle alwayes 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 center of your Protractor so that the Meridian line EF 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 then 180 degrees you must therefore turn the Semicircle of the Protractor downwards yet the line EF 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 deale with all the rest of the lines as LD LE and LF Lastly draw the lines AB BC CD DE EF and FA 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 seldome 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 assigne 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 severall 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 EF FG and GH which lines
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
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
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 â— 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
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 Staffe ¶ Having thus given you a sufficient description of the severall Instruments and their parts I come now to the use of them shewing 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 EK 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 equall thereunto First place your Instrument as neer the angular point K as conveniencie 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 Compasse point assigne 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 espie a mark set up at E or parallel to the line EK and then with your protracting pin or Compasse point or Black-lead draw a line by the side of the Index to the assigned point upon the Table Then the Table remaining immoveable 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 EK and KG and those lines shall include an angle equall 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 finde the quantity thereof if there were any need for in working by this Instrument it is sufficient only to give the symetry 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 finde the quantity of an angle in the field by the Theodolite Let it be required to finde 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 espie the mark at E then screwing the Instrument fast there turn the Index about upon the center till through the sights you espie the mark at G then note what degrees on the frame of the Table are cut by the Index which you will finde to be 114 degrees and that is the quantity of the angle EKG 3. How to finde the quantity of any angle in the field by the Circumferentor If it were required to finde 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 staffe the Flower-de-luce alwayes 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 alwayes substract 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 substract 79 the remainder is 246 but because this remainder 246 is greater then 180 you must therefore substract 246 the remainder from 360 and there will remain 114 the true quantity of the inquired angle and thus you must alwayes do when the remainder exceedeth 180 degrees ¶ This adding and substracting for the finding of angles may seeme 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 Mannor 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 shewed how to finde 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 Labell Horizontall upon the Staffe When you have screwed the Index and sights to the Staffe as a Circumferentor before you put the Labell upon the brasse pin or wier 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 thred and plummet hang directly upon a line which is gaged from under the pin all along the Sight and then doth the Instrument stand horizontall or levell which it must alwayes 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 usefull in taking of heights and in surveying of mountanous 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 levell direct the sights thereof towards CA and there fix it hanging the Labell on the farthermost fight upon a pin for that purpose then move the Labell 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 espie the very top of the object to be measured at C then note what degree of the Tangent line is cut by the Labell which suppose 30 and that is the quantity of the angle of Altitude it being equall to the angle CBA Thus by the Rules in this Chapter
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
apprehended at the first sight I will therefore instance in the Circumferentor as being most generall Let the example of the last Chapter serve where the figure ALMNPQSTVXGHK represented part of a Mannor Then having provided your Field-book ready ruled you must at the head of one of the leaves thereof write the Title of the Mannor the County in which it is and who is Lord thereof As The Mannor 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 Tenants name the name of the Close and the tenour 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 AL 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 adjoyning to Wisby Common which appertaineth to another Mannor 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 BA 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 parcell is next fittest to be taken in hand which let be Bay Wood and withall 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 expresse in the title of this second Close at what angle you begin the same unlesse you had begun it where you ended the last at A and then it is not materiall wherefore seeing you are best to begin at C looke 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 margine place 2 for the number of your second parsell and then proceed in your work of surveying this parcell 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 finde 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 write 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 looke 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 finde to be 355 degrees 40 minutes and 4 Chains 45 Links therefore at the end of that line where you finde 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 parcell 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 parcell also till you come to close at D and having finished draw a double line Then considering that Church field is next fitest to be surveyed and that it is most convenient to begin the same at Q therefore looke what degrees the needle cut at Q in the surveying of Banton plain which were 15 degrees 40 minutes â—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 â—rite in your Field-book as
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 shew how by your severall Instruments yo may take all manner of heights and distances whether accessible or inaccessible severell wayes 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 severall Instruments before mentioned as the Plain Table Theodolite and Circumferentor and also an angle of Altitude by the Index and the Labell thereunto annexed I conceive it now convenient to shew how all manner of heights and distances may be readily and exactly measured severall wayes 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 finde the distance between R and S. First place your Table at R and thereon assigne any point at pleasure unto which point apply the edge of your Index turning it about upon that point till through the sights you espie the mark at S and draw a line by the side of â—he Index as RS. Then in some other convenient place of the field as at Q let a staffe or other mark be erected and the Table remaining as before turn the Index about till through the sights you espie the mark at Q drawing a line by the side thereof as RQ so have you described upon your Table an angle equall 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 staffe 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 espie 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 espie 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 equall to the Triangle QRS the correspondent sides and angles thereof being proportionally equall 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 tooke the side QR you shall finde 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 finde 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 scituated that by reason of water or other impediments you cannot approach neere unto them And here note rhat 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 finde 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 espie your second station at Q and there screw it fast then turn the Index about upon the center till through the sights you espie 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 espie your mark at S and fixing the Instrument there turn the Index about til 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 Oblique 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 finde the two other sides RS and QS which you may finde 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 complement of the other two to 180 degrees by the 17 of Chap. 3 Lib. 3. Then to finde 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 â—… 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 then 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 finde 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 substract 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 substract 91 degrees 00 minutes from 340 degrees 30 minutes the remainder is 249 degrees 30 minutes which because it exceedeth 180 degrees substract 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 complement of the other two to 180 degrees and 4 the stationary distance QR 176 foot whereby you may finde 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 stationarie distance 176 foot Having these things given if you resolve the Triangle QRS you shall finde 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 severall 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 usuall way Suppose it were required to draw upon paper or pastboard the true symetry 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 Labell and Tangent line HAving in the 5 Section of the 4 Chapter of this Book shewn how to observe an angle of Altitude by the Labell 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 center 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 BA lie just under 30 degrees then 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 BA was 400 foot take 400 from any of your Scales of equall 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 finde 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 Labell 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 finde 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 stationarie distance AD was 200 foot you may immediately finde the Altitude BC by protraction thus First draw a line as AC in which line let A represent your first station whereon lay the center of your Protractor and make the angle BAC to contein 50 degrees as hath been severall times before shewn 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 center 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 tooke 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 scituation 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 scituate 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 intend to describe in your Map Then at one of these places as at K place your Table and neere one of the sides thereof draw a line parallel to the edge of the Table In this line assigne any point as K for your first station and laying the Index upon this line turn the Table about till through