but no thickness whose Bounds are Lines as A is a Superficies or Plain contained in these Lines BC DE BD CE which hath length from B to C and Breadth from B to D but no Thickness When these bounding Lines are measured and the Content of the Superficies cast up the result is called the Area or Superficial Content of that Figure EXAMPLE Suppose the Line BC to be twelve foot in Length and the Line BD to be four Foot long they multiplyed together make 48 therefore I say 48 Square Feet is the Area or Superficial Content of that Figure A Diagonal Line is a Line running through a Square Figure dividing it into two Triangles beginning at one Angle of the Square and proceeding to the Opposite Angle In the Square ABCD AD is the Diagonal Line CHAP. III. Geometrical Problems PROB. I. How to make a Line Perpendicular to a Line Given THe Line given is AB and at the Point C it is required to erect a Line which shall be Perpendicular to AB Open your Compasses to any convenient wideness and setting one Foot of them in the Point C with the other make a Mark upon the Line at E and also at D then taking off your Compasses open them a little wider than before and setting one Foot in the Point D with the other describe the Arch FF then without altering your Compasses set one Foot in the Point E and with the other describe the Arch GG Lastly Lay your Ruler to the Point C and the Intersection of the two Arches GG and FF which is at H and drawing the Line HC you have your desire HC being Perpendicular to AB See it here done again after the very same manner but may perhaps be plainer for your Understanding PROB. ii How to raise a Perpendicular upon the End of a Line AB is the Line given and at B it is required to erect the Perpendicular BC. If you have room you may extend the Line AB to what length you please and work as above but if not then thus you may do it Open your Compasses to an ordinary extent and setting one Foot in the Point B let the other fall at adventure no matter where in Reason as at the Point ☉ then without altering the extent of the Compasses set one Foot in the Point ☉ and with the other cross the Line AB as at D Also on the other side describe the Arch E then laying your Ruler to D and ☉ draw the prickt Line D ☉ F. Lastly from the Point B you began at through the Interjection at g draw the Line B g C which is perpendicular to AB Another way to do the same I think more easie though indeed almost the same Let AB be the given Line BI the Perpendicular required Set one Foot of your Compasses in B and with the other at any ordinary extent describe the Arch CEFD then keeping your Compasses at the same extent set one Foot in C and make a Mark upon the Arch at E also setting one Foot in E make another Mark at F then opening your Compasses or else with the same Extent which you please set one Foot in E and with the other describe the Arch GG also setting one Point in F make the Arch HH then drawing a Line through the intersection of the Arches G and H to the Point first proposed B you have the Perpendicular Line IB PROB. iii. How from a Point assigned to let fall a Perpendicular upon a Line given The Line given is AB the Point is at C from which it is desired to draw a Line down to AB that may be Perpendicular to it First setting one Foot of your Compasses in the Point C with the other make a Mark upon the Line AB as at D and also at E then opening your Compasses wider or shutting them closer either will do set one Foot in the Point of Intersection at D and with the other describe the Arch gg the like do at E for the Arch hh Lastly from the Point assigned through the Point of Intersection of the two Arches gg and hh draw the Perpendicular Line CF. This is no more but the First Problem reversed The same you may do by the second Problem viz. let fall a Perpendicular nigh the end of a given Line PROB. iv How to divide a Line into any Number of Equal Parts AB is a Line given and it is required to divide it into 6 equal Parts Make at the Point B a Line Perpendicular to AB as BC do the same at A the contrary way as you see here open your Compasses to any convenient Wideness and upon the Lines BC and AD mark out five Equal Parts for it must be always one less than the Number you intend to divide the Line into which parts you may number as you see here those upon one Line one way and the other the contrary way the laying your Ruler from No. 1. on the Line BC to No. 1. on the Line AD it will intersect the Line AB at E which you may mark with your Pen and the Distance between B and E is one sixth part of the Line so proceed on 'till you come to No. 5. and then you will find that you have divided the give Line into six Equal Parts as required PROB. v. How to make an Angle Equal to any other Angle given The Angle given is A and you are desired to make one Equal to it Draw the Right Line BC then going to the Angle A set one Foot of your Compasses in the Point h and with the other at what Distance you please describe the Arch IK then without altering the extent of the Compasses set one Foot in B and draw the like Arch as fg after that measure with your Compasses how far it is from K to I and the same distance set down upon the Arch from g towards f which will fall at E after draw the Line BED and you have done PROB. vi How to make Lines Parallel to each other AB is a Line given and it is required to make a Line parallel unto it Set one foot of your Compasses at or near the end of the given line as at C and with the other describe the Arch ab do the same near the other end of the same line and through the utmost convex of those two Arches draw the Parallel line C. D. PROB. vii How to make a Line Parallel to another Line which must also pass through a Point assigned Let AB be the given line C the point through which the required Parallel line must pass Set one foot of your Compasses in C and closing them so that they will just touch and no more the Line AB describe the Arch aa with the same extent in any part of the given Line set one Foot and describe another Arch as at D then through the assigned Point and the utmost Convex of the last Arch draw the required Line CD
which is Parallel to AB and passeth through the Point C. PROB. viii How to make a Triangle three Lines being given you Let the three lines given be 1 2 3 The Question is how to make a Triangle of them Take with your Compasses the length of either of the three in this Example let it be that No. 1. viz. the longest and lay it down as hereunder from A to B then taking with your Compasses the Length of the Line 2 set one Foot in B and make the Arch C also taking the length of the last Line 3. place your Compasses at A and make the Arch D which will intersect the Arch Cat the Point E from which Point of Intersection draw Lines to AB which shall constitute the Triangle AEB The Line AB being equal to the line No. 1 BE to No. 2 AE to No. 3. PROB. ix How to make a Triangle equal to a Triangle given and every way in the same Proportion First make an Angle Equal to the Angle at A as you were taught in PROB. v. Then making the Lines AD and AE equal to AB and AC draw the Line DE. Or otherwise you may do it as you were taught in PROB. viii PROB. x. How to make a Square Figure Let A be a Line given and it is required to make a square Figure each side of which shall just be the length of the Line A. First lay down the length of your Line A as AB Secondly raise a Perpendicular of the same length at B. Thirdly take the length of either of the aforementioned Lines with your Compasses and setting one Foot in C describe the Arch ee do the like at A and describe the Arch ff Fourthly draw Lines from A and C into the Point of Intersection and the Square is finished PROB. xi How to make a Parallelogram or long Square This is much like the former Admit two Lines be given you as 1 2 and it is required to make a Parallelogram of them What a Parallelogram is you may see in the Second Chapter of Definitions First lay down your longest Line as AB upon the End of which erect a Perpendicular Line equal in Length to your shortest Line and so proceed as you were taught in the foregoing Problem PROB. xii How to make a Rhombus First make an Angle suppose ACB no matter how great or small but be sure let the two Lines be of equal length then taking with your Compasses the length of one of those two Lines set one Foot in A and describe the Arch bb also set one Foot in B and describe the Arch cc. Lastly draw Lines and it is finished Two Equilateral Triangles is a Rhombus A Rhomboides differs just so much and no more from a Rhombus as a Parallelogram does from a true Square it is needless therefore I presume to shew you how to make it PROB. xiii How to divide a Circle into any number of Equal Parts not exceeding ten or otherwise how to make the Figures called Pentagon Hexagon Haptagon Octogon c. Let ABCD be a Circle in which is required to be made a Triangle the greatest that can be made in that Circle Keeping your Compasses at the same extent they were at when you made the Circle set one Point of them in any part of the Circle as at A and with the other make a Mark at E and f and draw a Line between E and f which will be one Side of the Triangle I need not tell you how to make the other two Sides for it is an Equilateral Triangle all three Sides being of Equal Length To make a Pentagon or Five-sided Figure Draw first an obscure Circle as ABCD then draw a Diameter from A to B make another Diameter Perpendicular to the first as CD then taking with your Compasses the Length of the Semi-Diameter set one Point in A and make the Marks EF drawing a Line between them as you did to make the Triangle Next set one Point of your Compasses in the Intersection at g and extend the other to C draw the Arch CH The nearest Distance between C and H viz. the Line CIH is the Side of a Pentagon and the greatest that can be made within that Circle Which with the same extent of your Compasses you may mark out round the Circle and drawing Lines the Figure will be finished To make a Hexagon or Six-sided Figure Draw an obscure Circle as you see here and then without altering the extent of the Compasses mark out the Hexagon required round the Circle for the Semidiameter of any Circle is the side of the greatest Hexagon that can be made within the same Circle This is the way Coopers use to make Heads for their Casks To make a Heptagon or Figure of Seven equal Sides and Angles You must begin and proceed as if you were going to inscribe a Triangle in a Circle till you have drawn the Line EF then taking with your Compasses the half of that Line viz. from ☉ to E or from ☉ to F mark out round the Circle your Heptagon for the half of the Line EF is one side of it To make an Octogon commonly called an Eight-square Figure First make a Circle Secondly divide it into four equal Parts by two Diameters the one perpendicular to the other as AB and CD Thirdly Set one Foot of the Compasses in A and make the Arch E E also with the same extent set one foot in C and make the Arch ff then through the Intersection of the two Arches draw a Line to the Center viz. gh Lastly Draw the Line IC or IA either of which is the side of an Octagon To make a Nonagon First make a Circle and a Triangle in it as you were taught at the beginning of this Problem then divide one third part of the Circle As for Example that A 1 2 3 B into three equal Parts Lastly draw the lines A 1 1 2 2 B c. each of these Lines is the side of a Nonagon To make a Decagon You must work altogether as you did in making a Pentagon See the Pentagon above where the distance from the Centre K to the Point at H is the side of a Decagon or Ten-sided Figure PROB. xiv Three Points being given How to make a Circle whose Circumference shall pass through the three given Points provided the three Points are not in a streight Line Let A B C be the three Points given first setting one foot of your Compasses in A open them to any convenient wideness more than half the distance between A and B and describe the Arch dd then without altering the extent set one point in B and cross the first Arch at E and E through those two Intersections draw the Line EE The very same you must do between B and C and draw the Line ff where these two Lines intersect each other as at g there is the Centre of the Circle required therefore setting one foot
how to cast up the Contents of any Plot of Land How to lay out New Lands How to Survey a Mannor County or Country Also how to Reduce Divide Lands Cum multis aliis The Twelfth Chapter consists wholly of Trigonometry The Thirteenth Chapter is of Heights and Distances including amongst other things how to make a Map of a River or Harbour Also how to convey Water from a Spring-head to any appointed Place or the like Lastly At the end of the Book I have a Table of Northing or Southing Easting or Westing or if you please to call it so A Table of Difference of Latitude and departure from the Meridian with Directions for the Use thereof Also a Table of Sines and Tangents and a Table of Logarithms I have taken Example from Mr. Holwell to make the Table of Sines and Tangents but to every Fifth Minute that being nigh enough in all sense and reason for the Surveyor's Use for there is no Man with the best Instrument that was ever yet made can take an Angle in the Field nigher if so nigh as to Five Minutes All which I commend to the Ingenious Reader wishing he may find Benefit thereby and desiring his favourable Reception thereof accordingly I conclude READER Your Humble Servant J. L. ADVERTISEMENT SUch Persons as have occasion for the Instruments mentioned in this Book or any other Mathematical Instruments whatsoever may be furnished with the same at Reasonable Rates by John Worgan Instrument-Maker at his Shop under the Dial of St. Dunstan's Church in Fleestreet London THE CONTENTS CHAP. I. OF Arithmetick in general Page 1 How to Extract the Square-Root by Vulgar Arithmetick Page 2 How to Extract the Square-Root by The Logarithms Page 7 CHAP. II. Geometrical Definitions Shewing what is meant by A Point Page 9 A Line ibid. An Angle ibid. A Perpendicular Page 10 A Triangle Page 11 A Square Page 12 A Parallelogram ibid. A Rhombus and Rhomboides ibid. A Trapezia ibid. An Irregular Figure Page 13 A Regular Polygon as Pentagon Hexagon c. Page 14 A Circle with what thereto belongs ibid. A Superficies Page 15 Parallel-Lines Page 16 Diagonal-Lines ibid. CHAP. III. Geometrical Problems 1. How to make a Line Perpendicular to another two ways Page 17 2. How to Raise a Perpendicular upon the end of a Line two ways Page 18 3. How from a Point assigned to let fall a Perpendicular upon a Line given Page 20 4. How to Divide a Line into any Number of Equal Parts Page 21 5. How to make an Angle equal to any other Angle given Page 22 6. How to makes Lines Parallel to each other Page 23 7. How to make a Line Parallel to another Line which must also pass through a Point assigned Page 24 8. Three Lines being given how to make thereof a Triangle ibid. 9. How to make a Triangle equal to a Triangle given Page 25 10. How to make a Square Figure Page 26 11. How to make a Long Square or Parallelogram ibid. 12. How to make a Rhomubs or Rhomboides Page 27 13. To make Regular Polygons as Pentagons Hexagons Heptagons c. Page 28 14. Three Points being given how to make a Circle whose circumference shall pass through the three given Points Page 32 15. How to make an Ellipsis or Oval several ways Page 33 16. How to Divide a given Line into two Parts which shall be in such Proportion to each other as two given Lines Page 36 17. Three Lines being given to find a Fourth in Proportion to them Page 37 CHAP. IV. Of Measures in general I. OF Long Measure shewing by what kind of Measures Land is Surveyed and also how to Reduce one sort of Long Measure into another Page 39 A General Table of Long Measure ibid. A Table shewing how many Feet and Parts of a Foot also how many Perches and Parts of a Perch are contained in any number of Chains and Links from one Link to an hundred Chains Page 41 A Table shewing how many Chains Links and Parts of a Link also how many Perches and Parts of a Perch are contained in any number of Feet from 1 Foot to 10000 Page 44 II. Of Square Measure shewing what it is and how to Reduce one sort into another Page 46 A General Table of Square Measure Page 47 A Table shewing the Length and Bredth of an Acre in Perches Feet and Parts of a Foot Page 49 A Table to turn Perches into Acres Roods and Perches Page 53 CHAP. V. Of Instruments and their Use OF the Chain Page 54 Of Instruments for the taking of an Angle in the Field Page 56 To take the quantity of an Angle in the Field by Plain Table Page 57 To take the quantity of an Angle in the Field by Semi-circle Page 58 To take the quantity of an Angle in the Field by Circumferentor c. several ways ibid. Of the Field-Book Page 61 Of the Scale with several Vses thereof and how to make a Line of Chords Page 62 c. Of the Protractor Page 68 CHAP. VI. HOw to take the Plot of a Field at one Station in any place thereof from whence you may see all the Angles by the Semi-circle and to Protract the same Page 71 How to take the Plot of the same Field at one Station by the Plain Table Page 74 How to take the Plot of the same Field at one Station by the Semi-circle either with the help of the Needle and Limb both together or by the help of the Needle only ibid. How by the Semi-circle to take the Plot of a Field at one Station in any Angle thereof from whence the other Angles may be seen and to Protract the same Page 76 How to take the Plot of a Field at two Stations provided from either Station you may see every Angle and measuring only the Stationary Distance Also to Protract the same Page 79 82 c. How to take the Plot of a Field at two Stations when the Field is so Irregular that from one Station you cannot see all the Angles Page 83 How to take the Plot of a Field at one Station in an Angle so that from that Angle you may see all the other Angles by measuring round about the said Field Page 86 How to take the Plot of the foregoing Field by measuring one Line only and taking Observations at every Angle Page 88 How to take the Plot of a large Field or Wood by measuring round the same and taking Observations at every Angle by the Semi-circle Page 90 When you have Surveyed after this manner how to know before you go out of the Field whether you have wrought true or not Page 94 Directions how to Measure Parallel to a Hedge when you cannot go in the Hedge it self And also in such case how to take your Angles Page 95 How to take the Plot of a Field or Wood by observing near every Angle and Measuring the Distance between the Marks of Observation by taking in every Line two
of your Compasses in g extend the other to either of the Points given and describe the Circle A B C. Note the Centre of a Triangle is found the same way PROB. xv How to make an Ellipsis or Oval several ways Fig. 1. Make three Circles whose Diameters may be in a streight Line as AB Cross that Line with another Perpendicular to it at the Centre of the middle Circle as cd draw the Lines ce ch dg df Set one foot of the Compasses in D and extend the other to g describing the part of the Ellepsis gf with the sameextent setting foot one in c describe the other part he The two Ends are made by parts of the two outermost small Circles as you see fe gh Fig. 2. Draw two small Circles whose circumference may only touch each other Then taking the distance between their Centers or either of their Diameters set one foot of your Compasses in either of their Centres as that marked 2 and with the other make an Arch at a also at b then moving your Compasses to the Centre of the other Circle cross the said Arches at a and b which Crosses let be the Centres of two other Circles of equal bigness with the first Then through the Centres of all the Circles draw the Lines AB CD EH FG which done place one foot of the Compasses in the Centre of the Circle I and extend the other to C describing the Arch of the Ellipsis CE The same you must do at 2 to describe the part BH and then is your Ellipsis finished Fig. 3. This needs no Description it being so like the two former Figures and easier than either of them Here Note that you may make the Ovals 1 and 3 of any determined length for in the length of the first there is four Semi-diameters of the small Circles and in the last but three If therefore any Line was given you of which length an Oval was required you must take in with your Compasses the fourth part of the Line to make the the Oval Fig. 1. and the third part to make the Oval Fig. 3 and with that extent you must describe the small Circles The Breadth will be always proportional to the Length But if the Breadth be given you take in also the fourth part thereof and make the Oval Fig. 2. Fig. 4. This Ellipsis is to be made having Length and Breadth both given Let AB be the Length CD the Breadth of a required Oval First lay down the Line AB equal to the given length and cross it in the middle with the Perpendicular CD equal to the given Breadth Secondly take in half the Line AB with your Compasses viz. AE or BE set one foot in C and make two marks upon the Line AB viz. f and g also with the same extent set one foot in D and cross the former marks at f and g. Thirdly at the Points f and g fix two Pins or if it be a Garden-plat or the like two strong Sticks Then putting a Line about them make fast the two ends at such an exact length that stretching by the two Pins the bent of the Line may exactly touch A or B or C or D or h as in this Diagram it does at h so moving the Line still round it will describe an exact Oval PROB. xvi How to divide a given Line into two Equal Parts which may be in such Proportion to each other as two given Lines Let AB be the given Line to be divided in such Proportion as the line C is to the line D. First from A draw a Line at pleasure as AE then taking with your Compasses the line C set it off from A towards E which will fall at F Also take the line D and set off from F to E. Secondly draw the line EB and from F make a line parallel to eb as FG which shall intersect the given line AB in the Proportional Point required viz at G making AG and GB in like proportion to each other as CC and DD. Example by Arithmetick The line CC is 60 Feet Perches or any thing else the line DD is 40 the line AB is 50 which is required to be divided in such proportion as 60 to 40. First add the two lines C and D together and they make 100 Then say if 100 the whole give 60 for its greatest part what shall 50 the whole line AB give for its greatest Proportional part Multiply 50 by 60 it makes 3000 which divided by 100 produces 30 for the longest part which 30 taken from 50 leaves 20 for the shortest part as therefore 60 is to 40 so is 30 to 20. PROB. xvii Three Lines being given to find a Fourth in Proportion to them Let ABC be the three Lines given and it is required to find a fourth Line which may be in such proportion to C as B is to A A 14 B 18 C 21 which is no more but performing the Rule of Three in Lines As if we should say if A 14 give B 18 what shall C 21 give Answer 27. But to perform the same Geometrically work thus And here for a while I shall leave these Problems till I come to shew you how to divide any piece of Land and to lay out any piece of a given quantity of Acres into any Form or Figure required And in the mean time I shall shew you what is necessary to be known CHAP. IV. Of Measures ANd first of Long Measures which are either Inches Feet Yards Perches Chains c. Note that twelve Inches make one Foot three Feet one Yard five Yards and a half one Pole or Perch four Perches one Chain of Gunter's eighty Chains one Mile But if you would bring one sort of Measure into another you must work by Multiplication or Division As for example Suppose you would know how many Inches are contained in twenty Yards First reduce the Yards into Feet by multiplying them by 3 because 3 Feet make one Yard the Product is 60 which multiplyed by 12 the number of Inches in one Foot gives 720 and so many Inches are contained in 20 Yards Length On the contrary if you would have known how many Yards there are in 720 Inches you must first divide 720 by 12 the Quotient is 60 Feet that again divided by 3 the Quotient is 20 Yards The like you must do with any other Measure as Perches Chains c. of which more by and by Long Link Foot Yard Perch Chain Mile Inches 7.92 12 36 198 792 63360 Â Links 1.515 4.56 25 100 8000 Â Â Feet 3 16.5 66 5280 Â Â Â Yards 5.5 22 1760 Â Â Â Â Perch 4 320 Â Â Â Â Â Chain 80 See this Table of Long Measure annexed the use whereof is very easie If you would know how many Feet in Length go to make one Chain look for Chain at Top and at the Left-hand for Feet against which in the common Angle of
which you would take your Observations Hauing placed your Semi-circle at F turn it about the North-Point of the Card from you till through the Fixed-Sights Note that I call them the Fiexed-Sights which are on the Fixed-Diameter you espy the mark at G. Then screw fast the Instrument which done move the Index till through the Sights thereof you see the mark at A and the Degrees on 〈◊〉 â—â—â—b there cut by it will be 20. Move again the Index to the mark at B where you will find it to cut 40 deg Do the same at C and it cuts 60 deg likewise at D 77 and at E 100 deg Note down all these Angles in your Field-Book next measure all the Lines as from F to G 14 Chain 60 Links from F to A 18 Chain 20 Links from F to B 16 Chain 80 Links from F to C 21 Chain 20 Links from F to D 16 Chain 95 Links from F to E 8 Chain 50 Links and then will your Field-Book stand thus Angles Degrees Minutes Chains Links G 00 00 14 60 A 20 00 18 20 B 40 00 16 80 C 60 00 21 20 D 77 00 16 95 E 10 00 8 50 To Protract the former Observations Draw a Line at adventure as G g upon any convenient place on which lay the Centre of your Protractor as at F keeping the Diameter thereof right upon the Line G g. Then make marks round the Protractor at every Angle as you find them in the Field-Book viz. against 20 40 60 77 and 100 which done take away the Protractor and applying the Scale or Ruler to F and each of the marks draw the Lines FA FB FC FD and FE Then setting off upon these Lines the true distances as you find them in the Field-Book as for the first Line F 〈◊〉 Chain 60 Links for the second FA 18 Chain 20 Links c. make marks where the ends of these distances fall which let be at G A B C c. Lastly Between these Marks drawing the Lines GA AB BC CD DE EF FG you will have compleated the Work. When you Survey thus without the help of the Needle you must remember before you come out of the Field to take a Meridian Line that you may be able to make a Compass shewing the true Situation of the Land in respect of the four Quarters of the Heavens I mean East West North and South which thus you may do The Instrument still standing at F turn it about till the Needle lies directly over the Flower-de-Luce of the Card there screw it fast Then turn the moveable Index till through the Sights you espy any one Angle As for Example Let be D Note then what Degrees upon the Limb are cut by the Index which let be 10 deg Mark this down in your Field-Book and when you have Protracted as before directed lay the Centre of your Protractor upon any place of the Line FD as at ☉ turning the Protractor about till 10 deg thereof lye directly upon the Line FD. Then against the end of the Diameter of the Protractor make a mark as at N and draw the Line N ☉ which is a Meridian or North and South Line by which you may make a Compass Note that you may as well take the Plot of a Field at one Station standing in any Side thereof as in an Angle For if you had set your Instrument in a the Work would be the same I shall forbear therefore as much as I may Tautologies How to take the Plot of a Field at two Stations provided from either Station you may see every Angle and measuring only the Stationary Distance Let CDEFGH be supposed a Field to be measured at two Stations first when you come into the Field make choice of two Places for your Stations which let be as far asunder as the Field will conveniently admit of also take care that if the Stationary Distance were continued it would not touch an Angle of the Field then setting the Semicircle at A the first Station turn it about the North Point from you till through the Fixed Sights you espy the Mark at your second Station which admit to be at B there screw fast the Instrument then turn the Moveable Index to every several Angle round the whole Field and see what Degrees are cut thereby at every Angle which note down in your Field-Book as followeth Angles Degrees Minutes  C 24 30  D 97 00  E 225 00 First Station F 283 30  G 325 00  H 346 00  Secondly measure the Distance between the two Stations which let be 20 Chains and set it down in the Field-Book Stationary Distance 20 Chains 00 Links Thirdly placing the Instrument at B the Second Station look backwards through the fixed Sights to the First Station at A I mean by looking backward that the South Part of the Instrument be towards A and having espyed the Mark at A make fast the Instrument and moving the Index as you did at the First Station to each Angle see what Degrees are cut by the Index and note them down as followeth and then have you done unless you will take a Meridian Line before you move the Instrument which you were taught to do a little before Angles Degrees Minutes  C 84 00  D 149 00  E 194 00 The Second Station F 215 00  G 270 00  H 322 00  How to Protract or lay down upon Paper these foregoing Observations First draw a Line cross your Paper at pleasure as the Line IK then take from off the Scale the Stationary Distance 20 Chains and set it upon that Line as from A to B so will A represent the First Station B the Second Secondly apply your Protractor the Centre thereof to the Point A and the Diameter lying streight upon the Line BK mark out round it the Angles as you find them in the Field-Book and through those Marks from A draw Lines of a convenient Length Thirdly move your Protractor to the Second Station B and there mark out your Angles and draw Lines as before at the First Station Lastly the places where the Lines of the First Station and the Lines of the Second intersect each other are the Angles of the Field As for Example At the First Station the Angle C was 24 Degrees 30 Minutes through those Degrees I drew the Line A1 At the Second Station C was 84 Degrees Accordingly from the Second Station I drew the Line B2 now I say where these two Lines cut each other as they do at C there is one Angle of the Field So likewise of DE and the rest of the Angles if therefore between these Intersections you draw streight Lines as CD DE EF c. you will have a true Figure of the Field This may as well be done by taking two Angles for your Stations and measuring the Line between them as C and D from whence you might as well have seen all
having found the place for B there make an Angle of 51 Degrees drawing the Line 'till it intersect AC c. You may also survey a Field after this manner by setting up a Mark in the middle thereof and measuring from that to any one Angle also in the Observations round the Field having respect to that Mark as you had here to the Angle A. It is too tedious to give Examples of all the Varieties besides it would rather puzzle than instruct a Neophyte How to take the Plot of a Large Field or Wood by measuring round the same and taking Observations at every Angle thereof by the Semicircle Suppose ABCDEFG to be a Wood through which you cannot see to take the Angles as before directed but must be forced to go round the same first plant the Semicircle at A and turn the North End of the Diameter about 'till through the fixed Sights you see the Mark at B then move round the Index till through the Sights thereof you espy G the Index there cutting upon the Limb 146 Degrees 2. Remove to B and as you go measure the Distance AB viz. 23 Chains 40 Links and planting the Instrument at B direct the North End of the Diameter to C and turn the Index round to A it then pointing to 76 Degrees 3. Remove to C measuring the Line as you go and setting your Instrument at C direct the North End of the fixed Diameter to D and turn the Index till you espy B and the Index then cutting 205 Degrees which because it is an outward Angle you may mark thus › in your Field-Book 4. Remove to D and measure as you go then placing the Instrument at D turn the North End of the Diameter to E and the Index to C the Quantity of that Angle will be 84 Degrees And thus you must do at every Angle round the Field as at E you will find the quantity of that Angle to be 142 Degrees F 137 G 110 but there is no need for your taking the last Angle nor yet measuring the two last Sides unless it be to prove the Truth of your Work which is indeed convenient When you have thus gone round the Field you will find your Field-Book to be as followeth Angles Lines  Deg. Min.  Ch. Lin. A 146 00 AB 23 40 B 76 00 BC 15 20 C 205 00 › CD 17 90 D 84 00 DE 20 60 E 142 00 EF 18 85 F 137 00 FG 13 60 G 110 00 GA 19 28 To protract this draw a dark Line at adventure as AB upon which set off the Distance as you see it in your Field-Book 23 Chains 40 Links from A to B then laying the Centre of your Protractor upon A and the Diameter upon the Line AB the North End or that of 00 Degrees towards B on the outside of the Limb make a Mark against 146 Degrees through which Mark from A draw the Line AG so have you the first Angle and first Distance 2. Place the Centre of the Protractor upon B and turn it about until 76 Degrees lyes upon the Line AB there hold it fast and against the North End of the Diameter make a Mark through which draw a Line and set off the Distance BC 15 Chains 20 Links 3. Apply the Centre of the Protractor to C the Semicircle thereof outward because you see by the Field-Book it is an outward Angle and turn it about 'till 205 Degrees lye upon the Line CB then against the Upper or South End of the Diameter make a Mark through which draw a Line and set off 17 Chains 90 Links from C to D. 4. Put the Centre of the Protractor to D and make 84 deg thereof lye upon the line CD then making a mark at the end of the Diameter or 0 deg Through that mark draw a line and set off 20 Chains 60 Links viz. DE. 5. Move the Protractor to E and make 142 deg to lye upon the line ED. Then at the end of the Protractor make a mark as before and setting off the distance 18 Chains 85 Links draw the line EF. 6. Lay the Centre of the Protractor upon F and making 137 deg lye upon the line EF against the end of the Diameter make a mark through which draw the line FG which will intersect the line AG at G So have you a true Copy of the Field or Wood But you may if you think fit to prove your Work set off the distance from F to G and at G apply your Protractor making 110 deg thereof to lye upon the line FG. Then if the end of the Diameter point directly to A and the distance be 90 Chain 28 Links you may be sure you have done your Work true Whereas I bid you put the North end of the Instrument and of the Protractor towards B it was chiefly to shew you the variety of Work by one Instrument for in the Figure before this I directed you to do it the contrary way and in this Figure if you had turned the South-side of the Instrument to G and with the Index had taken B and so of the rest the work would have been the same remembring still to use the Protractor the same way as you did your Instrument in the Field Also if you had been to have Surveyed this Field or Wood by the help of the Needle after you had planted the Semicircle at A and posited it so that the Needle might hang directly over the Flower-de-Luce in the Card you should have turned the Index to B and put down in your Field-Book what Degrees upon the Brass Limb had then been cut thereby which let be 20. Then moving your Instrument to B make the Needle hang over the Flower-de-Luce and turn the Index to C and note down what Degrees are there cut So do by all the rest of the Angles And when you come to Protract you must draw Lines Parallel to one another cross the Paper not farther distant asunder than the breadth of the Parallelogram of your Protractor which shall be Meridianlines marking one of them at one end N for North and at the other S for South This done chuse any place which you shall think most convenient upon one of the Meridian lines for your first Angle at A and laying the Diameter of your Protractor upon that Line against 20 deg make a mark through which draw a line and upon it set off the distance from A to B. In like manner proceed with the other Angles and Lines at every Angle laying your Protractor Parallel to a North and South Line which you may do by the Figures gratuated thereon at either end alike When you have Surveyed after this manner how to know before you go out of the Field whether you have wrought true or not Add the Sum of all your angles together as in the Example of the precedent Wood and they make 900. Multiply 180 by a number less by 2 than the number
the Off-sets in Perpendicular-lines although it be the best way for you may take the Angles with the Index from any part of the Line This way was chiefly intended for such as were not provided with Instruments for instead of the Semi-circle with a plain Cross only you may lay out a Square the rest of the Work being done with a Chain How by the help of the Needle to take the Plot of a large Wood by going round the same and making use of that Division of the Card that is numbred with four 90s or Quadrants Let ABCDE represent a Wood set your Instrument at A. and turn it about till through the Fixed Sights you espy B then see what Degrees in the Division before spoken of the Needle cuts which let be N. W. 7 measure AB 27 Chains 70 Links then setting the Instrument at B direct the Sights to C and see what then the Needle cuts which let be N. E. 74 measure BC 39 Chains 50 Links in like manner measure every Line and take every Angle and then your Field-Book will stand thus as followeth hereunder Lines Degrees Minutes Chains Links AB N. W. 7 00 28 20 BC N. E. 74 00 39 50 CD S. E. 9 00 38 00 DE N. W. 63 20 14 55 EA S. W. 74 80 28 60 To lay down which upon Paper draw Parallel Lines through your Paper which shall represent Meridian or North and South Lines as the Lines NS NS then applying the Protractor which should be gratuated accordingly with twice 90 Degrees beginning at each End of the Diameter and meeting in the middle of the Arch to any convenient place of one of the Lines as to A lay the Meridian Line of the Protractor to the Meridian Line on the Paper and against 7 Degrees make a Mark through which draw a Line and set off thereon the Distance AB 28 Chains 20 Links Secondly apply the Centre of the Protractor to B and turning the Semicircle thereof the other way because you see the Course tends to the Eastward make the Diameter thereof lye parallel to the Meridian Lines on the Paper which you may do by the Figures at the Ends of the Parallelogram and against 74 Degrees make a Mark and set off 39 Chains 50 Links and draw the Line BC the like do by the other Lines and Angles until you come round to the place where you began This is the most usual way of plotting Observations taken after this manner and used by most Surveyors in America where they lay out very large Tracts of Land but there is another way though more tedious yet surer I think first made Publick by Mr. Norwood whereby you may know before you come out of the Field Whether you have taken your Angles and measured the Lines truly or not and is as followeth As Radius or Sine of 90 Degrees viz. the Right Angle C is to the Logarithm of the Line AB 20 Chains So is the Sine of the Angle CAB 20 Degrees to the Difference of Longitude CB 6 Chains 80 Links Secondly to find the difference of Latitudes or the Line AC say As Radius is to the Logarthm Line AB 20 Chain so is the Sine Complement of the Angle at A to the Logarithm of the Line AC 18 Chains 80 odd Links Example of the foregoing Figure In the precedent Figure I find in my Field-Book the first Line to run NW 7 Degrees 28 Chain 20 Links now to find what Northing and what Westing is here made I say thus As Radius 10,000000 Is to the Logarithm of the Line 28 Chains 20 Links 1,450249 So is the Sine of the Angle from the Meridian viz. 7 Degrees 9,085894 To the Logarithm of the Westing 3 Chains 43 Links Again As Radius 10,000000 Is to the Logarithm 28 Chains 20 Links 1,450249 So is the Sine Complement of 7 Degrees 9,996750 To the Log of the Northing 27 Ch. 99 Lin. And having thus found the Northing and Westing of that Line I put it down in the Field-Book against the Line under the proper Titles NW in like manner I find the Latitude and Longitude of all the rest and having set them down the Field-Book will appear thus Lines Degrees Minutes Chains Links N S E W AB NW 7 00 28 20 27 99 .. .. .. .. 03 43 BC NE 74 00 39 50 10 89 .. .. 37 97 .. .. CD SE 9 00 38 00 .. .. 37 53 05 95 .. .. DE NW 63 20 14 55 06 53 .. .. .. .. 13 00 EA SW 74 00 28 60 .. .. 07 88 .. .. 27 49    45 41 45 41 43 92 43 92 This done add all the Northings together also all the Southings and see if they agree also all the Eastings and Westings and if they agree likewise then you may be sure you have wrought truly otherwise not Thus in this Example the summ of the Northings is 45 Chains 41 Links so likewise is the summ of the Southings also the summ of the Eastings is 43 Chains 92 Links so is the summ of the Westings Therefore I say I have surveyed that Piece of Land true But because this way of casting up the Northing Southing Easting or Westing of every Line may seem tedious and troublesome to you I have at the End of this Book made a Table wherein by Inspection only you may find the Longitude and Latitude of every Line what quantity of Degrees soever it is situated from the Meridian Moreover I am also obliged to shew you another way of plotting the foregoing Piece of Ground according to the Table in the Field-Book of NS EW as hereunder Then through B draw another North and South Line parallel to the first as NBS is parallel to NAS and taking with your Compasses the Northing of the second Line viz. 10 Chains 89 Links set it upon the Line from B to ☉ 2 take also the Easting of the same Line viz. 37 Chains 97 Links and setting one Foot of the Compasses in ☉ 2 with the other sweep the Arch cc also take with your Compasses the length of the second Line viz. 39 Chains 50 Links and setting one Foot in B cross the former Arch with another dd and that intersection is your third Angle viz. C. It would be but tautologie in me to go round thus with all the Lines for by these two first you may easily conceive how all the rest are done But let me put you in mind when you sweep the Arches for the Easting and Westing to turn your Compasses the right way and not take East for West and West for East Nor can I commend to you this way of plotting the former being as true and far easier yet when you plot by the former way it is very good for you to prove your Work by the Table of difference of Latitude and Longitude before you begin to protract and when you find your Field Work true you may lay it down upon Paper which way
Scale And thus turning the Scale about you may first reduce all the outermost parts of the Plot. Which done you must double the lesser Plot first ½ thereof and then the other by which you may see to reduce the innermost part near the Centre But I advise rather to have a long Scale made with the Centre-hole for fixing it to the Table in about one third part of the Scale so that ⅔ of the Scale may be one way numbred with Equal Parts from the Centre-hole to the end and ⅓ part thereof numbred the other way to the end with the same number of Equal Parts tho lesser Upon this Scale may be several Lines of Equal Parts the lesser to the greater according to several Proportions Being thus provided with a Scale glew down upon a smooth Table your greater Plot to be reduced and close to it upon the same Table a Paper about the bigness whereof you would have your smaller Plot. Fix with a strong Needle the Centre of your Scale between both then turning the longer end of your Scale to any remarkable thing of your to be reduced Plot see what number of Equal Parts it cuts as suppose 100 there holding fast the Scale against 100 upon the smaller end of your Scale make a mark upon the white Paper so do round all the Plot drawing Lines and putting down all other accidents as you proceed for fear of confusion through many Marks in the end and when you have done although at first the reduced Plot will seem to be quite contrary to the other yet when you have unglewed it from the Table and turned it about you will find it to be an exact Epitome of the first You may have for this Work divers Centers made in one Scale with Equal Parts proceeding from them accordingly or you may have divers Scales according to several Proportions which is better What has been hitherto said concerning the Reducing of a Plot from a greater volume to a lesser the same is to be understood vice versa of Enlarging a Plot from a lesser to a greater But this last seldom comes in practise How to change Customary-Measure into Statute and the contrary In some Parts of England for Wood-Lands and in most Parts of Ireland for all sorts of Lands they account 18 Foot to a Perch and 160 such Perches to make an Acre which is called Customary-Measure Whereas our true Measure for Land by Act of Parliament is but 160 Perches for one Acre at 16 Foot ½ to the Perch Therefore to reduce the one into the other the Rule is As the Square of one sort of Measure is to the Square of the other So is the Content of the one to the Content of the other Thus if a Field measured by a Perch of 18 Feet accounting 160 Perches to the Acre contain 100 Acres How many Acres shall the same Field contain by a Perch of 16 Feet ½ Say if the Square of 16 Feet ½ viz. 272. 25. give the Square of 18 Feet viz. 324. What shall 100 Acres Customary give Answer 119 9 / 10 of an Acre Statute Knowing the Content of a piece of Land to find out what Scale it was plotted by First by any Scale measure the Content of the Plot which done argue thus As the Content found is to the Square of the Scale I tried by So is the true Content to the Square of the true Scale it was plotted by Admit there is a Plot of a piece of Land containing 10 Acres and I measuring it by the Scale of 11 in an Inch find it to contain 12 Acres 1 / 10 of an Acre Then I say If 12 2 / 10 give for its Scale 11 What shall 100 give Answer 10. Therefore I conclude that Plot to be made by a Scale of 10 in the Inch. And so much concerning Reducing Lands CHAP. X. Instructions for Surveying a Mannor County or whole Country To Survey a Mannor observe these following Rules 1. WAlk or ride over the Mannor once or twice that you may have as it were a Map of it in your Head by which means you may the better know where to begin and proceed on with your Work. 2. If you can conveniently run round the whole Mannor with your Chain and Instrument taking all the Angles and measuring all the Lines thereof taking notice of Roads Lanes or Commons as you cross them Also minding well the Ends of all dividing Hedges where they butt upon your bound Hedges in this manner 3. Take a true Draught of all the Roads and By-Lanes in the Mannor putting down also the true Buttings of all the Field-Fences to the Road. If the Road be broad or goes through some Common or Wast Ground the best way is to measure and take the Angles on both Sides thereof but if it be a narrow Lane you may only measure along the midst thereof taking the Angles and Off-sets to the Hedges and measuring your Distances truly Also if there be any considerable River either bounds or runs through the Mannor survey that also truly as is hereafter taught 4. Make a true Plot upon Paper of all the foregoing Work and then will you have a Resemblance of the Mannor though not compleat which to make so go to all the Buttings of the Hedges and there Survey every Field distinctly plotting it accordingly every Night or rather twice a Day till you have perfected the whole Mannor 5. When thus you have plotted all the Fields according to the Buttings of the Hedges found in your first Surveys you will find that you have very nigh if not quite done the whole Work But if there be any Fields lye so within others that they are not bounded on either Side by a Road Lane nor River then you must also Survey them and place them in your Plot accordingly as they are bounded by other Fields 6. Draw a fair Draught of the whole putting down therein the Mannor-House and every other considerable House Wind-mill Water-mill Bridg Wood Coppice Cross-paths Rills Runs of Water Ponds and any other Matter Notable therein Also in the fair Draught let the Arms of the Lord of the Mannor be fairly drawn and a Compass in some wast part of the Paper also a Scale the same by which it was plotted You must also beautifie such a Draught with Colours and Cuts according as you shall see convenient Write down also in every Field the true Content thereof and if it be required the Names of the present Possessors and their Tenures by which they hold it of the Lord of the Mannor The Quality also of the Land you may take notice of as you pass over it if you have Judgment therein and it be required of you How to take the Draught of a County or Country 1. If the County or Country is in any place thereof bounded with the Sea Survey first the Seacoast thereof measuring it all along with the Chain and taking all the Angles thereof truly 2. Which done