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 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
you think the easiest To conclude this Chapter or Section I shall in the next place shew you How to take the Plot of a Field by the Chain only using no other Instrument in the Field and that after a better manner than hitherto has been taught First therefore I shall shew you how to take the quantity of an Angle by the Chain which well understood there need be no more required For the Business of a Surveyor in the Field is no more but to measure Lines and take Angles I mean for telling the quantity of any Field or Piece of Land as how many Acres it contains or the like How by the Chain only to take an Angle in the Field But the more easie and speedy way is to take but one Chain only along the Hedges as in the foregoing Figure I set a strong Stick in the very Angle A and putting the Ring at one End of the Chain over it I take the other End in my Hand and stretch out the Chain along the First Hedge AB and where it ends as at 5 I stick down a Stick then I stretch the Chain also along the Hedge AC and at the end thereof set another Stick as at 4 then loosing my Chain from A I measure the distance 4 5 which is 74 Links which is all I need notedown in my Field-Book for that Angle and now coming to plot that Angle I take first from my Scale the distance of one Chain and placing one Foot of the Compasses in any part of the Paper as at A I describe the Arch 4 5 then I take from the same Scale 74 Links and set it off upon that Arch making Marks where the Ends of the Compasses fall as at 4 5. Lastly from A through these Marks I draw the Lines AB and AC which constitute the former Angle Remember to plot your Angles with a very large Scale and you may set off your Lines with a smaller I will give you two Examples of this way of measuring and then leave you to your own practice First How by the Chain only to Survey a Field by going round the same Let ABCDEF be the Field and beginning at A in the very Angle stick down a Staff through the great Ring at one of the Ends of your Chain and taking the other End in your Hand stretch out the Chain in length and see in what part of the Hedge AF the other End falls as suppose at a there set up a Stick and do the like by the Hedge AB and say there the Chain ends at a also measure the nearest distance between a and a which let be 1 Chain 60 Links this note down in your Field-Book measure next the length of the Hedge AB which is 12 Chains 50 Links note this down also in your Field-Book Nextly coming to B take that Angle in like manner as you did the Angle A and measure the distance BC after this manner you must take all the Angles and measure all the Sides round the Field But lest you be at a Nonplus at D because that is an outward Angle thus you must do stick a Staff down with the ring of the Chain round it in the very Angle D then taking the other end of the Chain in your Hand and stretching it at length move your self to and Fro 'till you perceive your self in a direct Line with the Hedge DC which will be at G where stick down an Arrow or one of your Surveying-Sticks then move round 'till you find your self in a direct Line with the Hedge DC and there the Chain stretched out at length plant another Stick as at H then measure the nearest Distance HG which let be 1 Chain 43 Links which note down in your Field-Book and proceed on to measure the Line DE but in your Field-Book make some some Mark against D to signifie it is an outward Angle as › or the like And when you come to plot this you must plot the same Angle outward that you took inward for the Angle GDH is the same as the Angle d D d. I made this outward Angle here on purpose to shew you how you must Survey a Wood by going round it on the Outside where you must take most of the Angles as here you do D. Having thus taken all the Angles and measured all the Sides the next thing to be done Is to lay down upon Paper according to your Field-Book Which you will find to stand thus  Cross Lines or Chords    Angles Chains Links Lines of the Field Chains Links A 1 60 AB 12 50 B 1 84 BC 23 37 C 1 06 CD 19 30 D 1 43 › DE 20 00 E 0 80 EF 29 00 F 1 52 FA 31 50 Forasmuch now as it is convenient that the Angles be made by a greater Scale than the Lines are laid down with I have therefore in this Figure made the Angles by a Scale of one Chain in an Inch and laid down the Lines by a Scale of ten Chains in one Inch. But to begin to plot take from your Scale one Chain and with that Distance in any convenient place of your Paper as at A sweep the Arch aa then from the same large Scale take off 1 Chain 60 Links and set it upon that Arch as from a to a and from A draw Lines through a and a as the Lines AB AF Then repairing to your shorter Scale take from thence the first distance viz. 12 Chains 50 Links and set it from A to B drawing the Line AB Secondly repairing to B take from your large Scale 1 Chain and setting one Foot of the Compasses in B with the other make the Arch bb also from the same Scale take your Chord Line viz. 1 Chain 84 Links and set it upon the Arch bb one Foot of the Compasses standing where the Arch intersects AB the other will fall at b then through b draw the Line BC and from your smaller Scale set off the Distance BC 23 Chains 37 Links which will fall at C where the next Angle must be made After this manner proceed on according to your Field-Book 'till you have done And here mark that you need neither in the Field nor upon the Paper take notice of the Angle F nor yet measure the Lines EF and AF for if you draw those two Lines through they will intersect each other at the true Angle F However for the Proof of your Work it is good to measure them and also to take the Angle in the Field I must not let slip in this place the usual way taught by Surveyors for the measuring a Field by the Chain only as true indeed as the former but more tedious which take as followeth The common way taught by Surveyors for taking the Plot of the foregoing Field Because I will not confound your Understanding with many Lines in one Figure I have here again placed the same First they bid you measure round the Field
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 Angles and if the Product be equal to the Sum of the quantity of all the Angles then have you wrought true There were seven Angles in that Wood therefore I multiply 180 by 5 and the Product is 900. If you Survey by taking the quantity of every Angle and if all be inward Angles you must work as before But if one or more be outward Angles you must substract them out of 180 deg and add the Remainder only to the rest of the Angles And when you multiply 180 by a Sum less by 2 than the number of your Angles you are not to account the outward Angles into the number Thus in the precedent Example I find one outward Angle viz. C 205 the quantity of which if it had been taken would have been but 155 deg That taken from 180 deg there remains 25 which I add to the other Angles and they make then in all 720. Now because C was an outward Angle I take no notice of it but see how many other Angles I have and I find 6 a number less by 2 than 6 is 4 by which I multiply 180 and the Product is 720 as before 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 It is impossible for you when you have a Hedge to measure to go at top of the Hedge itself but if you go Parallel thereto either within side or without and make your Parallel-line of the same length as the Line of your Hedg your work will be the same Thus if AB was a bushy Hedge to which you could not conveniently come nigher to plant your Instrument than ☉ let him that goes to set up your mark at B take before he goes the Distance A ☉ which he may do readily with a Wand or Rod and at B let him set off the same distance again as to ✚ where let the mark be placed for your Observation and when the Chain bears measure the distance ☉ ✚ be sure they have respect to the Hedge AB so as that they make ☉ ✚ equal to AB or of the same length But to make this more plain Suppose ABC to be a Field and for the Bushes you cannot come nigher than ☉ to plant your Instrument Let him that sets up the Marks take the distance between the Instrument ☉ and the Hedge AB which distance let him set off again nigh B and set up his Mark at D likewise let him take the distance between ☉ and the Hedge A C and accordingly set up his Mark at E. Then taking the Angle d ☉ E it will be the same as the Angle BAC So do for the rest of the Angles But when the Lines are measured they must be measured of the same length as the outside Lines as the Line ☉ d measured from G to F c. the best way therefore is for them that measure the Lines to go round the Field on the outside thereof although the Angles be taken within 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 Off-sets to the Hedge In working after this manner observe these two things First if the Wood be so thick that you cannot go within-side thereof you may after the same manner as well perform the Work by going on the out-side round the Wood. Secondly if the Lines are so long that you cannot see from Angle to Angle cause your Assistant to set up a Mark so far from you as you can conveniently see it as at N Measure the distance ☉ 1 N and take the Off-set from N to the Hedge Then at N turn the Fixed-Sights of the Instrument to ☉ 1 and and by that Direction proceed on the Line till you come to an Angle This way of Surveying is much easier done though I cannot say truer by taking only a great Square in the Field from the Sides of which the Off-sets are taken And when you have thus laid out your Square and taken all your Off-sets you will find in your Field-Book such Memorandums as these to help you Protract The Angles 4 Right-Angles The Sides 12 Chains 00 Links each I went round cum Sole or the Hedges being on my Left-hand  C. L.  C. L. In the first Line at 1 50 Off-set to a Side-Line 5 40 8 30 Off-set to an Angle 6 00  C. L.  C. L. In the second Line at 3 50 Off-set to an Angle 6 00 10 70 Off-set to an Angle 5 50  C. L.  C. L. In the third Line at 10 00 Off-set to an Angle 5 30  C. L.  C. L. In the fourth Line at 4 30 Off-set to an Angle 4 40 6 70 Off-set to an Angle 1 50 10 80 Off-set to an Angle 2 20 Now to lay down upon Paper the foregoing Work make first a Square Figure whose Side may be 12 Chains as 1 2 3 4. Then considering you went with the Sun take 1 2 for the first Line and taking from your Scale 1 Chain 50 Links set it upon the Line from 1 to 7 at 7 raise a Perpendicular as 7 6 making it according to your Field-Book 5 Chains 40 Links long Also for the second Off-set upon the same Line take from your Scale of Equal Parts 8 Chains 30 Links which set upon the line from 1 to 8 and upon 8 make the Perpendicular-line 8 B 6 Chains in length For the Off-sets of the second Line take 3 Chains 50 Links from the Scale and set it from 2 to 9 at 9 make a Perpendicular-line 6 Chains long viz. 9 C Also for the second Off-set of the same Line take 10 Chains 70 Links and set it from 2 to 10 at 10 make the Perpendicular 10 D 5 Chains 50 Links in length For the Off-sets of the third Line take from your Scale 10 Chains and set it from 3 to 11 and at 11 make the Perpendicular 11 E 5 Chains 30 Links long For the Off-sets of the fourth Line take from your Scale 4 Chains 30 Links and set it from 4 to 12 and at 12 make the Perpendicular 12 F 4 Chains 40 Links long Also take 6 Chains 70 Links and let it from 4 to 13 and at 13 make the Perpendicular 13 G 1 Chain 50 Links long Lastly take 10 Chains 80 Links and set it from 4 to 1 and at I make the Perpendicular 1 5 2 Chains 20 Links long Then have you no more to do but through the ends of these Perpendiculars to draw the bounding-Bounding-lines remembring to make Angles where the Field-Book mentions Angles and where it mentions Side-lines there to continue such Side-lines till they meet in an Angle Although I mention a Square yet you are not bound to that Figure for you may with the same success use a Parallelogram Triangle or any other Figure Nor are you bound to take
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
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
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
EXAMPLE 1599 Square Chains and 55 Square Links how many Acres Roods and Perches  Acres 159955   4 Answer 159 Acres 3 Rood 32 8 / 10.    Roods 3620   40  Perches 24800 On the contrary if to any number of Acres given you add a Cypher they will be turned into Chains thus 99 Acres are 990 Chains 100 Acres 1000 Chains c. The same as if you had multiplyed the Acres by 10. And if you would turn Square Chains into Square Links add four Cyphers to the end of the Chains so will 990 Chains be 9900000 Links 1000 Chains 10000000 Links all one as if you had multiplyed 990 by 10000 the number of Square Links contained in one Chain And now whereas in casting up the content of a piece of Land measured by Mr. Gunter's Chain viz. multiplying Chains and Links by Chains and Links the Product will be Square Links you must therefore from that Product cut off five Figures to find the Acres which is the same as if you divided the Product by 100000 the number of Square Links contained in one Acre then multiply the five Figures cut off by 4 and from that Product cutting off five Figures you will have the Roods Lastly multiply by 40 and take away as before 5 Figures the rest are Perches EXAMPLE Admit a Parallelogram or Long Square to be one way 5 Chains 55 Links and the other way 4 Chains 35 Links I demand the content in Acres Roods and Perches  Multiplicand 555  Multiplicator 435   2775   1665   2220 Answer 2 Acres Acres 241425   4 1 Rood Roods 165700   40 26 Perches Perches 2628000 And 28 / 100 Parts of a Perch   Lastly Because some Men chuse rather to cast up the Content of Land in Perches I will here briefly shew you how it is done which is only by dividing by 160 the number of Square Perches contained in One Acre the number of Perches given EXAMPLE Admit a Parallelogram to be in length 55 Perches and in breadth 45 Perches these two multiplied together make 2475 Perches which to turn into Acres divide by 160 the Quotient is 15 Acres and 75 Perches remaining which to turn into Roods divide by 40 the Quotient is 1 Rood and 35 Perches remaining So much is the Content of such a piece of Land viz. 15 Acres 1 Rood and 35 Perches Here follows a Table to turn Perches into Acres Roods and Perches Perches Acres Roods Perch 40 0 1 00 50 0 1 10 60 0 1 20 70 0 1 30 80 0 2 00 90 0 2 10 100 0 2 20 200 1 1 00 300 1 3 20 400 2 2 00 500 3 0 20 600 3 3 00 700 4 1 20 800 5 0 00 900 5 2 20 1000 6 1 00 2000 12 2 00 3000 18 3 00 4000 25 0 00 5000 31 1 00 6000 37 2 00 7000 43 3 00 8000 50 0 00 9000 56 1 00 10000 62 2 00 20000 125 0 00 30000 187 2 00 40000 250 0 00 50000 312 2 00 60000 375 0 00 70000 437 2 00 80000 500 0 00 90000 562 2 00 100000 625 0 00 The Use of this Table In 2475 Perches how many Acres Roods and Perches Perch Acres Rood Perch 2000 12 2 00 400 2 2 00 70 0 1 30 To which add the odd 5 Perches 0 0 05 Answer 15 1 35 CHAP. V. Of Instruments and their Use And first of the Chain THere are several sorts of Chains as Mr. Rathborne's of two Perch long Others of one Perch long some have had them 100 Feet in length But that which is most in use among Surveyors as being indeed the best is Mr. Gunter's which is 4 Pole long containing 100 Links each Link being 7 92 / 100 Inches The Description of which Chain and how to reduce it into any other Measure you have at large in the foregoing Chapter of Measures In this place I shall only give you some few Directions for the use of it in Measuring Lines Take care that they which carry the Chain deviate not from a streight Line which you may do by standing at your Instrument and looking through the Sights If you see them between you and the Mark observed they are in a streight Line otherwise not But without all this trouble they may carry the Chain true enough if he that follows the Chain always causeth him that goeth before to be in a direct line between himself and the place they are going to so as that the Foreman may always cover the Mark from him that goes behind If they swerve from the Line they will make it longer than really it is a streight Line being the nearest distance that can be between any two places Besure that they which carry the Chain mistake not a Chain either over or under in their Account for if they should the Error would be very considerable as suppose you was to measure a Field that you knew to be exactly Square and therefore need measure but one Side of it if the Chain-Carriers should mistake but one Chain and tell you the Side was but 9 Chains when it was really 10 you would make of the Field but 8 Acres and 16 Perches when it should be 10 Acres just And if in so small a Line such a great Error may arise what may be in a greater you may easily imagine But the usual way to prevent such Mistakes is to be provided with 10 small Sticks sharp at one End to stick into the Ground and let him that goes before take all into his Hand at setting out and at the End of every Chain stick down one which let him that follows take up when the 10 Sticks are done be sure they have gone 10 Chains then if the Line be longer let them change the Sticks and proceed as before keeping in Memory how often they change They may either Change at the end of 10 Chains then the hindmost Man must give the foremost all his Sticks or which is better at the end of 11 Chains and then the last Man must give the first but 9 Sticks keeping one to himself At every Change count the Sticks for fear lest you have dropt one which sometimes happens If you find the Chain too long for your use as for some Lands it is especially in America you may then take the half of the Chain and measure as before remembring still when you put down the Lines in your Field Book that you set down but the half of the Chains and the odd Links as if a Line measured by the little Chain be 11 Chains 25 Links you must set down 5 Chains 75 Links and then in plotting and casting up it will be the same as if you had measured by the whole Chain At the end of every 10 Links you may if you find it convenient have a Ring a piece of Brass or a Ragg for your more ready reckoning the odd Links When you put down
the Angles and consequently as well have performed the Work. 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 Let CDEFGHIKLMNO be a Field in which from no one Place thereof all the Angles may be seen chuse therefore two Places for your Stations as A and B and setting the Semicircle in A direct the Diameter to the Second Station B there making the Instrument fast with the Index take all the Angles at that end of the Field as CDEFGHIK and measure the Distance between your Instrument and each Angle measure also the Distance between the two Stations A and B. Secondly remove your Instrument to the Second Station at B and having made it fast so as that throug the Back Sights you may see the First Station A take the Angles at that End of the Field as NOCKLM and measure their Distances also as before all which done your Field-Book will stand thus First Station Angles Degrees Minutes Chains Links C 25 00 20 75 D 31 00 8 10 E 67 00 9 85 F 101 00 10 80 G 137 00 7 00 H 262 00 6 70 I 316 00 13 70 K 354 00 24 50 The Distance between the two Station 31 Ch. 60 L. Second Station Angles Deg. Min. Chain Link N 3 30 4 20 O 111 00 7 00 C 145 00 15 60 K 205 00 7 48 L 220 00 15 00 M 274 00 11 20 To lay this down upon Paper draw at adventure the Line PBAP then taking in with the Compasses the Distance between the two Stations viz. 31 Ch. 60 Links set it upon the Line making Marks with the Compasses as A and B A being the First Station B the Second lay the Protractor to A the North End of the Diameter towards B and mark out the several Angles observed at your First Station drawing Lines and setting off the Distances as you were taught in the beginning of this Chapter Fig. I. Do the same at B the Second Station and when you have marked out all the Distances between those Marks draw the Bound-Lines I am the briefer in this because it is the same as was taught concerning Fig I for if you conceive a Line to be drawn from C to K then would there be two distinct Fields to be measured at one Station apiece If a Field be very irregular you may after the same manner make three four or five Stations if you please but I think it better to go round such a Field and measure the bounding Lines thereof Which by and by I shall shew you how to do Note in the foregoing Figure you might as well have had your Stations in two convenient Angles as D and K and have wrought as you were taught concerning Fig. 2. the Work would have been the same 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 Then will your Field-Book be as hereunder Angles Degrees Minutes C 68 00 D 76 00 E 124 00 Lines Chains Links AB 14 00 BC 15 00 CD 07 00 DE 14 40 EA 14 05 To protract which draw the Line AB at adventure and applying the Centre of the Protractor to A the Diameter lying upon the Line AB and the Semicircle of it upwards prick off the Angles as against 68 76 and 124 make Marks through which Marks draw the Lines AC AD AE long enough be sure then taking in with your Compasses from off the Scale the length of the Line AB viz. 14 Chains and setting one Foot of the Compasses in A with the other cross the Line as at B also for BC take in 15 Chains and setting one Foot in B with the other cross the Line AC which will fall to be at C for the Line CD take in 7 Chains and setting one Foot in C cross the Line AD viz. at D then for DE take in 14 Chains 40 Links and setting one Foot of the Compasses in DE with the other cross the Line AE which will fall at E Lastly for EA take 14 Chains 5 Links with your Compasses and setting one Point in E see if the other fall exactly upon A if it does you have done the Work true if not you have erred between the Crosses or intersections draw streight Lines which shall be the bounds of the Field viz. AB BC CD DE EA How to take the Plot of the foregoing Field by measuring one Line only and taking Observations at every Angle Begin as you have been just before taught 'till you have taken the Angles C D E viz. 68 76 and 124 Degrees then leaving a good Mark at A which may be seen all round the Field go to B measuring as you go the Distance from A to B which is all the Lines you need to measure and planting your Semicircle at B direct the South Part thereof toward A until through the back fixed Sights you see the Mark at A there making it fast turn the Index about 'till you espy C and note down the Degrees there cut which let be 129 Degrees move your Instrument to C and still keeping the South Part of the Diameter to A turn the Index to D where it will cut 20 Degrees then remove to D and espying A through the Back Sights turn the Index to E where it will cut 135 Degrees Note all this in your Field-Book Angles taken at the First Station Angles round the Field C 68 Degrees B 129 Degrees D 76 C 20 E 124 D 135 Line AB 14 Chains To protract this you must work as you were taught concerning the foregoing Figure untill you have drawn the Lines AB AC AD AE and set off the Line AB 14 Chains then laying the Centre of your Protractor to B and the South End of the Diameter or that marked with 180 Degrees towards A make a Mark against 129 Degrees and through that mark from B draw the Line BC 'till it intersect the Line AC which it will do at C Lay also the Centre of the Protractor upon C the Diameter thereof upon AC and against 20 Degrees make a Mark through which from C draw the Line CD 'till it intersect the Line AD which it will do at D lastly place your Protractor at D the Diameter thereof upon the Line DA and make a Mark against 135 Degrees through which Mark draw the Line DE until it intersect the Line AE at E also drawing the Line EA you have done This may be done otherwise thus after you have standing at A taken the several Angles and measured the Distance AB you may only take the quantity of the bounding Angles without respect to A As the Angle at B is 51 Degrees at C an outward Angle which in your Field-Book you should distinguish with a Mark › 138 and so of the rest And when you come to plot
and note down in your Field-Book every Line thereof as in this Field has been before done Secondly they bid you turn all the Field into Triangles as beginning at A to measure the Diagonal AC AD AE and note them down then is your Field turned into four Triangles and the Diagonals are  Chains Links AC 33 70 AD 25 70 AE 45 40 To plot which they advise you first to draw a Line at adventure as the Line AC and to set off thereon 33 Chains 70 Links according to your Field-Book for the Diagonals then taking with your Compasses the Length of the Line AB viz. 12 Chains 50 Links set one Foot in A and with the other describe the Arch aa also take the Line BC viz. 23 Chains 37 Links and setting one Foot in C with the other describe the Arch cc cutting the Arch aa in the Point B then draw the Lines AB CB which shall be two bound Lines of the Field Secondly take with your Compasses the Length of the Diagonal AD viz. 25 Chains 70 Links and setting one Foot of the Compasses in A with the other describe the Arch as dd also taking the Line CD viz. 19 Chains 30 Links set one Foot in C and with the other describe the Arch ee cutting the Arch dd in the point D to which Intersection draw the Line CD Thirdly take with your Compasses the Length of of the Diagonal AE viz. 45 Chains 40 Links and setting one Foot in A with the other describe an Arch as ff also take the Line DE 20 Chains and therewith cross the former Arch in the Point E to which draw the Line DE. Lastly take with your Compasses the length of the Line AF viz. 31 Chains 50 Links and setting one foot in A describe an Arch as II. Also take the length of the Line EF viz. 29 Chains 00 Links and therewith describe the Arch hh which cuts the Arch II in the Point F to which Point draw the Lines AF and EF and so will you have a true Figure of the Field I have shewed you both ways that you may take your choice And now I proceed to my Second Example promised How to take the Plot of a Field at one Station near the Middle thereof by the Chain only Let ABCDE be the Field ☉ the appointed place from whence by the Chain to take the Plot thereof Stick a Stake up at ☉ through one ring of the Chain and make your Assistant take the other end and stretch it out Then cause him to move up and down till you espy him exactly in a Line between the Stake and the Angle A there let him set down a stick as at a and be sure that the stick a be in a direct Line between ☉ and A which you may easily perceive by standing at ☉ and looking to A. This done cause him to move round towards B and at the Chains end let him there stick down another stick exactly in the Line between ☉ and B as at b. Afterwards let him do the same at c at d and at e and if there were more Angles let him plant a stick at the end of the Chain in a right Line between ☉ and every Angle In the next place measure the nighest distance between stick and stick as ab 1 Chain 26 Links bc 1 Chain 06 Links cd 1 Chain 00 Links de 1 Chain 20 Links and put them down in your Field-Book accordingly Measure also the Distances between ☉ and every Angle as ☉ A 18 Chains 10 Links ☉ B 15 Chains 00 Links c. all which put down your Field-Bok will appear thus   Chains Links Subtendent or Chord-Lines ab 1 26 bc 1 06 cd 1 00 de 1 20   Chains Links Diagonal or Centre-Lines ☉ A 18 10 ☉ B 15 00 ☉ C 17 00 ☉ D 15 00 ☉ E 16 00 How to plot the former Observations Take from a large Scale 1 Chain and setting one foot of the Compasses in any convenient place of the Paper as at ☉ make the Circle abcde Then taking for your first Subtendent or Chord-line 1 Chain 26 Links set it upon the Circle as from a to b. From ☉ through a and b draw Lines as ☉ A ☉ B which be sure let be long enough Then take your second Subtendent from the same large Scale viz. 1 Chain 6 Links and set it upon the Circle from b to c and through c draw the Line ☉ C. When thus you have set off all your Subtendents and drawn Lines through their several Marks repair to a smaller Scale and upon the Lines drawn set off your Diagonal or Centre Lines as you find them in the Field-Book So upon the Line ☉ a you must set off 18 Chains 10 Links making a Mark where it falls as at A Upon the Line ☉ b 15 Chains 00 Links which falls at B and so by all the rest Lastly draw the Lines AB BC CD c. and the Work will be finished It would be but running things over again to shew you how after this manner to Survey a Field at two or three Stations or in any Angle thereof c. For if you well understand this you cannot be ignorant of the rest CHAP. VII How to cast up the Contents of a Plot of Land. HAving by this time sufficiently shewed you how to Survey a Field and lay down a true Figure thereof upon Paper I come in the next place to teach you how to cast up the Contents thereof that is to say to find out how many Acres Roods and Perches it containeth And first Of the Square and Parallelogram To cast up either of which multiply one Side by the other the Product will be the Content EXAMPLE Let A be a true Square each side being 10 Chains multiply 10 Chains 00 Links by 10 Chains 00 Links facit 1000000. from which I cut off the five last Figures and there remains just 10 Acres for the Square A. Again In the Parallelogram B let the side A b or c D be 20 Chains 50 Links and the side ac or b D 10 Chains 00 Links Multiply ab 20 Chains 50 Links by ac 10 Chains 00 Links facit 2050000 from which cutting off the last five Figures remains 20 Acres Then if you multiply the Figures cut off viz. 50000 by 4 facit 200000 from which cutting off five Figures remains 2 Roods and if any thing but 000s had been left you must have multiplied again by 40 and then cutting off again five Figures you would have had the odd Perches See it done hereunder I need not have multiplied 00 by 40 for I know 40 times Nothing is Nothing but only to shew you in what order the Figures will stand when you have odd Perches as presently we shall light on So much is the Content of the long Square B viz. 20 Acres 2 Roods 00 Perch  20.50  10.00 Acres 2050000  4 Roods 200000  40 Perches 000000 Of
from the Number thus increased extract the Root which shall be the Side of the proposed Square EXAMPLE Suppose the Number given be 100 Acres which I am to lay out in a Square Figure I joyn to the 100 5 Cyphers and then it is Square Links the Root of which is 3162 nearest or 31 Chains 62 Links the length of one Side of the Square Again If I were to cut out of a Corn-Field one Square Acre I add to one five Cyphers and then is it the Root of which is 3 Chains 16 Links and something more for the Side of that Acre How to lay out any given Quantity of Acres in a Parallelogram whereof one Side is given Turn first the Acres into Links by adding as before 5 Cyphers that number thus increased divide by the given Side the Quotient will be the other Side EXAMPLE It is required to lay out 100 Acres in a Parallelogram one Side of which shall be 20 Chains 00 Links first to the 100 Acres I add 5 Cyphers and it is 100,00000 which I divide by 20 Chains 00 Links the Quotient is 50 Chains 00 Links for the other Side of the Parallelogram How to lay out a Parallelogram that shall be 4 5 6 or 7 c. times longer than it is broad In Carolina all Lands lying by the Sides of Rivers except Seignories or Baronies are or ought by Order of the Lord's Proprietors to be thus laid out To do which first as above taught turn the given quantity of Acres into Links by annexing 5 Cyphers which summ divide by the number given for the Proportion between the length and bredth as 4 5 6 7 c. the Root of the Quotient will shew the shortest Side of such a Parallelogram EXAMPLE Admit it were required of me to lay out 100 Acres in a Parallelogram that should be five times as long as broad First to the 100 Acres I add 5 Cyphers and it makes 100,00000 which summ I divide by 5 the Quotient is 2000000 the Root of which is nearest 14 Chains 14 Links and that I say shall be the short Side of such a Parallelogram and by multiplying that 1414 by 5 shews me the longest Side thereof to be 70 Chains 70 Links How to make a Triangle that shall contain any number of Acres being confined to a certain Base Double the given number of Acres to which annexing first five Ciphers divide by the Base the Quotient will be the length of the Perpendicular EXAMPLE Upon a Base given that is in length 40 Chains 00 Links I am to make a Triangle that shall contain 100 Acres First I double the 100 Acres and annexing five Ciphers thereto it makes 200,00000 which I divide by 40 Chains 00 Links the limited Base the Quotient is 50 Chains 00 Links for the height of the Perpendicular As in this Figure AB is the given Base 40 upon any part of which Base I set the Perpendicular 50 as at C then the Perpendicular is CD Therefore I draw the Lines DA DB which makes the Triangle DAB to contain just 100 Acres as required Or if I had set the Perpendicular at E then would EF have been the Perpendicular 50 and by drawing the Lines FA FB I should have made the Triangle FAB containing 100 Acres the same as DAB If you consider this well when you are laying out a new piece of Land of any given Content in America or elsewhere although you meet in your way with 100 Lines and Angles yet you may by making a Triangle to the first Station you began at cut off any quantity required How to find the Length of the Diameter of a Circle which shall contain any number of Acres required Say as 11 is to 14 so will the number of Acres given be to the Square of the Diameter of the Circle required EXAMPLE What is the Length of the Diameter of a Circle whose Superficial Content shall be 100 Acres Add five Cyphers to the 100 and it makes 100,00000 Links which multiply by 14 facit 140000000 which divided by 11 gives for Quotient 12727272 the Root of which is 35 Chains 67 Links and better almost 68 Links And so much shall be the Diameter of the required Circle I might add many more Examples of this nature as how to make Ovals Regular Polygons and the like that should contain any assigned quantity of Land. But because such things are meerly for Speculation and seldom or never come in Practise I at present omit them CHAP. IX Of Reduction How to Reduce a large Plot of Land or Map into a lesser compass according to any given Proportion or e contra how to Enlarge one THe best way to do this is if your Plot be not over-large to plat it over again by a smaller Scale But if it be large as a Map of a County or the like the only way is to compass in the Plot first with one great Square and afterwards to divide that into as many little Squares as you shall see convenient Also make the same number of little Squares upon a fair piece of Paper by a lesser Scale according to the Proportion given This done see in what Square and part of the same Square any remarkable accident falls and accordingly put it down in your lesser Squares and that you may not mistake it is a good way to number your Squares I cannot make it plainer than by giving you the following Example where the Plot ABCD made by a Scale of 10 Chains in an Inch is reduced into the Plot EFGH of 30 Chains in an Inch. There are several other ways taught by Surveyors for reducing Plots or Maps as Mr. Rathboxn and after him Mr. Holwell adviseth to make use of a Scale or Ruler having a Centre-hole at one end through which to fasten it down on a Table so that it may play freely round and numbred from the centre-Centre-end to the other with Lines of Equal Parts The Use of which is thus Lay down upon a smooth Table the Map or Plot that you would reduce and glew it with Mouth-glew fast to the Table at the four corners thereof Then taking a fair piece of Paper about the bigness that you would have your reduced Plot to be of and lay that down upon the other the middle of the last about the middle of the first This done lay the Centre of your Reducing Scale near the Centre of the white Paper and there with a Needle through the Centre make it fast yet so that it may play easily round the Needle Then moving your Scale to any remarkable thing of the first Plot as an Angle a House the bent of a River or the like See against how many Equal Parts of the Scale it stands as suppose 100 then taking the ⅓ the ¼ the ⅕ or any other number thereof according to the Proportion you would have the reduced Plot to bear and make a mark upon the white Paper against 50 25 33 c. of the same
the former Proportion How to reduce an Irregular Five-Sided Figure into a Triangle and to divide the same Let ABCDE be the Five-Sided Figure to reduce which into a Triangle draw the Lines AC AD and parallel thereto BF EG extending the Base from C to F and from D to G then draw the Lines AF AG which will make the Triangle AFG equal to the Five Sided-Figure If this was to be divided into two equal Parts take the half of the Base of the Triangle which is FH and from H draw the Line HA which divides the Figure ABCDE into two equal Parts The like you may do for any other Proportion If in dividing the Plot of a Field there be Outward Angles you may change them after the following manner Suppose ABCDE be the Plot of a Field and B the outward Angle Draw the Line CA and parallel thereto the Line BF Lastly The Line CF shall be of as much force as the Lines CB and BA So is that five-sided Figure having one outward Angle reduced into a four-sided Figure or Trapezia which you may again reduce into a Triangle as has been before taught How to Divide an Irregular Plot of any number of Sides according to any given Proportion by a streight Line through it ABCDEFGHI is a Field to be divided between two Men in equal Halfs by a streight Line proceeding from A. First consider how to divide the Field into five-sided Figures and Trapezias that you may the better reduce it into Triangles As by drawing the Line KL you cut off the five-sided Figure ABCHI which reduce into the Triangle AKL and measuring half the Base thereof which will fall at Q draw the Line QA Secondly Draw the Line MN and from the Point Q reduce the Trapezia CDGH into the Triangle MNQ which again divide into Halfs and draw the Line QR Thirdly From the Point R reduce the Trapezia DEFG into the Triangle ROP and taking half the Base thereof draw the Line RS and then have you divided this Irregular Figure into two Equal Parts by the three Lines AQ QR RS. Fourthly Draw the Line AR also QT parallel thereto Draw also AT and then have you turned two of the Lines into one Fifthly From T draw the Line T S and parallel thereto the Line RV Draw also TV. Then is your Figure divided into two Equal Parts by the two Lines AT and TV. Lastly Draw the Line AV and parallel thereto TW Draw also AW which will cut the Figure into two Equal Parts by a streight Line as was required You may if you please divide such a Figure all into Triangles and then divide each Triangle from the Point where the Division of the last fell and then will your Figure be divided by a crooked Line which you may bring into a streight one as above This above is a good way of Dividing Lands but Surveyors seldom take so much pains about it I shall therefore shew you how commonly they abbreviate their Work and is indeed An easie way of Dividing Lands Admit the following Figure ABCDE contain 46 Acres to be divided in Halfs between two Men by a Line proceeding from A. Draw first a Line by guess through the Figure as the Line AF. Then cast up the Content of either Half and see what it wants or what it is more than the true Half should be As for Example I cast up the Content of AEG and find it to be but 15 Acres whereas the true Half is 23 Acres 8 Acres being in the part ABCDG more than AEG Therefore I make a Triangle containing 8 Acres and add it to AEG as the Triangle AGI then the Line AI parts the Figure into equal Halfs If it had been required to have set off the Perpendicular the other way you must still have made the end of it but just touch the Line ED as LK does For the Triangle AKG is equal to the Triangle AGI each 8 Acres And thus you may divide any piece of Land of never so many Sides and Angles according to any Proportion by streight Lines through it with as much certainty and more ease than the former way Mark you might also have drawn the Line AD and measured the Triangle AGD and afterwards have divided the Base GD according to Proportion in the Point I which I will make more plain in this following Example Suppose the following Field containing 27 Acres is to be divided between three Men each to have Nine Acres and the Lines of Division to run from a Pond in the Field so that every one may have the benefit of the Water without going over one another's Land. From ☉ to any Angle draw a Line for the first Division-line as ☉ A. Then consider that the first Angle A ☉ B is but 674 Perches and the second B ☉ C 390 both together but 1064 Perches less by 376 than 1440 one Man's Portion You must therefore cut off from the third Angle C ☉ D 376 Perches for the first Man's Dividing-line which thus you may do The Base DC is 18 Chains the Content of the Triangle 1238 Perches Say then if 1238 Perches give Base 18 Chains 00 Links What shall 376 Perches give Answer 5 Chains 45 Links which set from C to F and drawing the Line ☉ F you have the first Man's part viz. A ☉ F. Secondly See what remains of the Triangle C ☉ D 376 being taken out and you will find it to be 862 Perches which is less by 578 than 1440. Therefore from the Triangle D ☉ E cut off 578 Perches and the point of Division will fall in G. Draw the Line ☉ G which with ☉ A and ☉ F divides the Figure into three Equal Parts How to Divide a Circle according to any Proportion by a Line Concentrick with the first All Circles are in Proportion to one another as the Squares of their Diameters therefore if you divide the Square of Diameter or Semi-diameter and extract the Root you will have your desire EXAMPLE Let ABCD be a Circle to be equally divided between two Men. The Diameter thereof is 2 Chains The Semi-diameter 1 Chain or 100 Links The Square thereof 10100 Half the Square The Root of the Half 71 Links which take from your Scale and upon the same Centre draw the Circle GEHF which divides the Circle ABCD into Equal Parts CHAP. XII Trigonometry Or the Mensuration of Right Lined Triangles THe Use of the Table of Logarithm Numbers I have shewed you in Chap. I. concerning the Extraction of the Square Root Here follows The use of the Tables of Sines and Tangents Any Angle being given in Degrees and Minutes how to find the Sine or Tangent thereof Let 25 Degrees 10 Minutes be given to find the Sine and Tangent thereof first in the Table of Sines and Tangents at the Head thereof seek for 25 and having found it look down the first Column on the Left-hand under M for the 10 Minutes and right against
since it seems naturally to come in here again I will give you one Example thereof Suppose this following Figure to be a Piece of a River and you measuring along one Side of it would as well know the Breadth of it as also make a true Plot thereof by putting down what remarkable things are seen on the other Side To Protract this draw the Line NS for a Meridian and laying your Protractor upon it the Centre thereof to ☉ 1 against NW 6 make a Mark for the Line that goes to ☉ 2. Also against NW 17 make a Mark for the Tree and against 40 and 52 for the Wind-mill and House Then from ☉ 1 through these Marks draw the Lines ☉ A ☉ B ☉ C ☉ 2. Secondly Take from your Scale 18 Ch. 20 Lin. and set it off upon the Line ☉ 2 which will reach to ☉ 2. There lay again the Centre of your Protractor the Diameter thereof parallel to the Line NS and make Marks as you see in the Field-Book against NE 15. NW 77. SW 20. SW 50. NW 28. NW 4° and through these Marks draw Lines The first Line directs to your third Station the second Line NW 77. directs you to the Tree C upon the Rivers bank for that Line cutting the Line ☉ 1 C shews you by the Intersection where the Tree stood and also the Bredth of the River Also the Line SW 20 cuts the Line from the first Station NW 52 in the place where the House A stands upon the Bank of the River If therefore you draw a Line from A to C it will represent the farther Bank of the River And so you may proceed on Plotting according to the Notes in your Field-Book and you will not only have a true Plot of the River but also know how far the Wind-mill B and the House D stand from the Water-side How to take the Horizontal-line of a Hill. When you measure a Hill you must measure the Superficies thereof and accordingly cast up the Contents But when you Plot it down because you cannot make a Convex Superficies upon the Paper you must only plot the Horizontal or Base thereof which you must shadow over with the resemblance of a Hill that other Surveyors when they apply your Scale thereto may not say you was Mistaken And you may find this Horizontal or Base-line after the same manner as you have been taught to take Heighths But if you have occasion to measure the whole Hill plant again your Instrument at B and take the Angle CBD which let be 46 deg Measure also the Distance BC 21 Ch. Then say As Radius 10000000 is to the Line BC 21 Ch. 00 Lin. Log. 1322219 So is the Line of the Angle CBD 46 9856934 to the part of the Base DC 15 Ch. 12 Lin. 1,179153 Which 15. 12. added to 8.90 makes 24 Chains 2. Links for the whole Base AG which is to be plotted and not AB and BC although they are to be measured to find the Content of the Land. I mentioned this way for your better understanding how to take the Base of part of a Hill for many times your Survey ends upon the side of a Hill. But if you find you are to take in the whole Hill you need not take altogether so much pains as by the former way As thus Take as before the Angle A 58 deg Measure also AB Then at B take the whole Angle ABC 78 deg Substract these two from 180 deg remains 44 for the Angle at C. Then say As the Sine of the Angle C 44 is to the Log. of the Side AB So is the Sine of the Angle ABC to the Log. of the Base AC How to take the Shoals of a Rivers Mouth and Plot the same Measure first the Sea coast on both Sides of the River Mouth as far as you think you shall have occasion to make use thereof and make a fair Draught thereof putting down every remarkable thing in its true Situation as Trees Houses Towns Wind-mills c. Then going out in a Boat to such Sands or Rocks as make the Entrance difficult at every considerable bend of the Sands take with a Sea-Compass the bearing thereof to two known Marks upon the Shore and having so gon round all the Sands and Rocks you may easily upon the Plot before taken draw Lines which shall intersect each other at every considerable Point of the Sands whereby you may truly prick out the Sands and give good Directions either for laying Buyos or making Marks upon the Shore for the Direction of Shipping EXAMPLE It would be too tedious for you and troublesom for me to give you all the Observations I having already in this Treatise so often described the same thing before therefore I will mention only one place of Observation more and if by that you do not understand the whole I know not how to make you In the Sand C I find the bend 2 and there as I should do at all the rest I take two Observations to such things on the Shore as are most conspicuous unto me viz. First to the Beacon which bears from me S. W. 25 deg Secondly to the Wind-mill which bears from me N. W. 40 deg Now after I have taken the other Angles or Bends of that Sand and am come Home I draw a Line from the Beacon-opposite to my Observation S. W. 25 deg viz. N. E. 25 deg Also from the Wind mill I draw a Line S. E. 40 deg Now where these two Lines intersect each other as they do at 2 I mark for one Point of the Sand C. In like manner as I did this I observe and protract every Line of the Sand C and of all the other Sands and Rocks be there never so many and so will you have a fair Map fitting for Seamens Use better done I think than in any place of the World yet except for the Harbours of Eutopia Now to give Direction for Seamens coming in here draw a Line through the middle of the South Channel which Line will cut both the Church and Wind-mill so that if a Ship coming from the Southward brings the Church and Wind-mill both into one and keep them so she may boldly run in till she brings the Rivers mouth fair open and then sail up the River Likewise coming from the Northward must first bring the Tree and Beacon both into one and keep them so till the Rivers mouth is fair open But lest they should mistake and run upon the ends of the Sands A or B it would be necessary that a Mark was set up behind the Red-House in a streight Line with the middle of the River as Then a Ship coming from the Southward or Northward let her keep her former Marks both in one till she bring the Red House and both in one and then keeping them so run boldly up the River till all Danger is past I have put down this Wind-mill and Beacon not as if
in your Field-Book the length of any Line you may set it thus if you please with a Stop between the Chains and Links as 15 Chains 15 Links 15.15 or without as thus 1515 it will be all one in the casting up Of Instruments for the taking of an Angle in the Field There are but two material things towards the measuring of a piece of Land to be done in the Field the one is to measure the Lines which I have shewed you how to do by the Chain and the other to take the quantity of an Angle included by these Lines for which there are almost as many Instruments as there are Surveyors Such among the rest as have got the greatest esteem in the World are the Plain Table for small Inclosures the Semicircle for Champaign Grounds The Circumferentor the Theodolite c. To describe these to you their Parts how to put them together take them asunder c. is like teaching the Art of Fencing by Book one Hours use of them or but looking on them in the Instrument-maker's Shop will better describe them to you than the reading one hundred Sheets of Paper concerning them Let it suffice that the only use of them all is no more or chiefly at most but this viz. To take the Quantity of an Angle Plain Table Place the Table already fitted for the Work with a Sheet of Paper upon it as nigh to the Angle A as you can the North End of the Needle hanging directly over the Flower de Luce then make a Mark upon the Sheet of Paper at any convenient place for the Angle A and lay the Edge of the Index to the Mark turning it about till through the Sights you espy B then draw the Line AB by the Edge of the Index Do the same for the Line AC keeping the Index still upon the first Mark then will you have upon your Table an Angle equal to the Angle in the Field To take the Quantity of the same Angle by the Semicircle Place your Semicircle in the Angle A as near the very Angle as possibly you can and cause Marks to be set up near B and C so far off the Hedges as your Instrument at A stands then turn the Instrument about 'till through the fixed Sights you see the Mark at B there screw it fast next turn the moveable Index 'till through the Sights thereof you see the Mark at C then see what Degrees upon the Limb are cut by the Index which let be 45 so much is the Angle BAC How to take the same Angle by the Circumferentor Place your Instrument as before at A with the Flower de Luce towards you direct your Sights to the Mark at B and see what Degrees are then cut by the South End of the Needle which let be 55 do the same to the Mark at C and let the South End of the Needle there cut 100 substract the lesser out of the greater the remainder is 45 the Angle required If the remainder had been more than 180 degrees you must then have substracted it out of 360 the last remainder would have been the Angle desired This last Instrument depends wholly upon the Needle for taking of Angles which often proves erroneous the Needle yearly of it self varying from the true North if there be no Iron Mines in the Earth or other Accidents to draw it aside which in Mountainous Lands are often found It is therefore the best way for the Surveyor where he possibly can to take his Angles without the help of the Needle as is before shewed by the Semicircle But in all Lands it cannot be done but we must sometimes make use of the Needle without exceeding great trouble as in the thick Woods of Jamaica Carolina c. It is good therefore to have such an Instrument with which an Angle in the Field may be taken either with or without the Needle as is the Semicircle than which I know no better Instrument for the Surveyors use yet made publick therefore as I have before shewed you How by the Semicircle to take an Angle without the help of the Needle I shall here direct you How with the Semicircle to take the Quantity of an Angle in the Field by the Needle Screw fast the Instrument the North End of the Needle hanging directly over the Flower de Luce in the Chard turn the Index about till through the Sights you espy the Mark at B and note what Degrees the Index cuts which let be 40 move again the Index to the Mark at C and note the Degrees cut viz. 85. Substract the Less from the greater remains 45 the Quantity of the Angle Or thus Turn the whole Instrument 'till through the Fixed Sights you espy the Mark at B then see what Degrees upon the Chard are cut by the Needle which for Example are 315 turn also the Instrument till through the same Sights you espy C and note the Degrees upon the Chard then cut by the Needle which let be 270 substract the Less from the Greater as before in working by the Circumferentor remains 45 for the Angle Mark if you turn the Flower de Luce towards the Marks you must look at the Norh end of the Needle for your Degrees Besides the Division of the Chard of the Semicircle into 360 Equal Parts or Degrees It is also divided into four Quadrants each containing 90 Degres beginning at the North and South Points and proceeding both ways 'till they end in 90 Degrees at the East and West Points which Points are marked contrary viz. East with a W. and West with an E because when you turn your Instrument to the Eastward the End of the Needle will hang upon the West Side c. If by this way of division of the Chard you would take the aforesaid Angle direct the Instrument so the Flower de Luce from you 'till through the fixed Sights you espy the Mark at B then see what Degrees are cut by the North End of the Needle which let be NE 44 next direct the Instrument to C and the North End of the Needle will cut NE 89 substract the one from the other and there will remain 45 for the Angle But if at the first sight the Needle had hung over NE 55 and at the second SE 80 then take 55 from 90 remains 35 take 80 from 90 remains 10 which added to 35 makes 45 the Quantity of the Angle Moreover if at the first Sight the North End of the Needle had pointed to NW 22 and at the second NE 23 these two must have been added together and they would have made 45 the Angle as before Mark if you had turned the South part of your Instrument to the Marks then you must have had respect to the South End of your Needle Although I have been so long shewing you how to take an Angle by the Needle yet when we come to Survey Land by the Needle as you shall see by and by we