right Lines or more and are either Triangles or Triangulate that is such as are compounded of and resolvable into Triangles XIV Triangles are Figures comprehended under three right Lines and as Ramus thinks for a Reason that he gives lib. 6. pr. 6. might be better called Trilaterals but the name Triangle from the number of the Angles hath obtained Also from the nature and Quantity of their Angles these Triangles are distinguished into three sorts 1. Rectangled having one right Angle 2 Obtuse-angled having one obtuse Angle and 3. Acute-angled having all acute Angles for no Triangle can have more right or obtuse Angles than one because by by an old Rule easie to be demonstrated no Triangle upon a plain Superficies can consist of three greater Angles than such as being jointly taken are equal to two Right These three sorts of Triangles may according to the length and proportion of their sides be subdistinguished into seven for each of them may have either two equal Sides or none and the Acute-angled may have all three Sides or lines equal To all which kinds learned Men give distinct Greek Names which if mine English Reader have a mind to see they are to be found in 〈…〉 Practices Book 1. page 6. for my present purpose the above-mentioned â—rimembred distinction will abundantly suffice for be Triangles of what name or kind soever they are all capable of being exactly measured by one plain Rule as hereafter shall fully appear XV. Triangulare Figures are such as have more Angles and consequently more Sides or Lines than three and these are either Quadrangular or Multangular XVI Quadrangular Figures are such as have foâ—â— Angles and as many Side and these are either Parallellograms or Trapezia's XVII Parallellograms are Figures that are bounded with parallel Lines that is such lines as are every where of the same distance one from another so as if they were infinitely extended they would never meet like the upright lines of he Roman H. These Parallellograms are either Rectangular or Obliquangular XVIII Rectangular Parallellograms are such as have four right Angles viz. the Square or Quadrat and the long Square otherwise called the Oblong XIX The Square is that Figure that hath four right Angles and four equal Sides like any of the six Faces of a Die XX. The long Square hath also four right Angles and the oposite Sides are equal but the adjoyning Sides meeting at each Angle differ in length Of this Figure is a well printed Page in a Book and the Superficies of a well cut Sheet of Paper or an ordinary Pane of Glass XXI Obliquangled Parallellograms are such as have oblique Angles viz. two acute and two obtuse Of these there are two kinds the Rhombus and the Rhomboides XXII The Rhombus is a Figure that hath equal Sides but no right Angles like the form of a Diamond on the Cards or the most ordinary Cut of Glass in Windows whose oposite Angles are equal XXIII The Rhomboides is as it were a defective Rbombus for if from any side of a Rhombus we cut off a part with a parallel Line the Remainder will be a Rhomboides which hath neither equal Sides nor Angles but yet the opposite Sides and Angles are equal XXIV The Trapezium is a Figure that is neither parallellogram nor consequently hath equal Sides or Angles but is irregularly quadrangular as if drawn at adventure Of this shape most Fields prove that seem to the Eye to be Squares or Oblongs XXV Multangular Figures are such as contain more Sides and Angles than four and they are either regular or irregular XXVI Regular Multangulars take their names from their Number of Angles so a Pentagon Hexagon Heptagon Octogon Encagon Decagon signifie a multangular Figure of five six seven eight nine ten Angles and consequently Sides XXVII An irregular Polygon or multangular Figure is that which hath more Angles and Sides than four the Sides and Angles being unequal to one another CHAP. II. Of Geometrical Problems I. To draw a Line parallel to another at any Distance assigned Fig. 1 OPen your Compasses to the Distance given and chusing two Points conveniently distant in the Line given as here at A and B describe the Arches C and D to whose convixity if you apply a Rule the parallel Line is easily drawn II. To raise a Perpendicular upon a Line given or to cross that Line at right Angles in a Point assigned Fig. 2 Suppose the point C in the Line AB were assigned for the Perpendicular open the Compasses to a convenient distance and mark out the two points E and F in the line AB then opening them some what wider you may by setting one Foot in E and F severally describe the two Arches cutting one another at the point D from which if you draw a Line to the point C the work is done for the raising of a Perpendicular but if you be to cross the lines at right Angles you may continue the line from D through C at pleasure But if the said line AB had been given to be divided in the precise middle by another Line crossing it at right Angles the way were to set one Point of the Compasses in A and B severally and having described two Arches above the line intersecting one another as at D do the like below the line AB from the same points and with the same extent of your Compasses then through the several intersections a Rule being laid upon them a line may be drawn cutting the given line exactly in the middle at right Angles Note That when one point of your Compasses stand in A you may make both the Arches belonging to that Center above and below the line and then removing the Compasses to B you may cross them both III. To raise a Perpendicular at the End of a Line Fig. 3 Let OR be the line given then to raise a Perpendicular at R make five little equal divisions and taking four of them with your Compasses set one foot of your Compasses in R and with the other describe the Arch PP then take the distance from R to 5 and placing one Foot in 3 with the other describe the Arch BB intersecting the former in the point S then shall the line SR being drawn by a straight Rule be a Perpendicular to the line OR IV. To let fall a Perpendicular upon a given Line from any Point assigned Open your Compasses so as one Foot being set in the assigned point the other may go clear over the line given and thereby describe an Arch cutting the line at two points then shall the half distance between those two points be the point to which the Perpendicular may be drawn from the point assigned But if you think it too much pains to find the point of half distance by trial you may help your self by the second Problem For if you describe two Arches intersecting one another on the farther side of the line from the assigned point placing to
that purpose the Foot of your Compasses first in one of the Intersections of the given line and then in the other you may by laying a Rule upon the assigned Point and the Intersection of the two Arches draw a Perpendicular from the said assigned Point cutting the given Line at right Angles Note that all these Problems touching perpendiculars aim at no greater matter than what may be performed in a Mechanical way with exactness enough and much more neatly by avoiding unhandsome Pricks and Arches by the help of a small Square exactly made or for want thereof a Plate Quadrant or broad Rule having a right Angle and true Sides for if you apply one Leg of such a Square to any Line so as the Angle of the Square may touch the end of the said Line or any other Point where the Perpendicular is to be raised you may by the other Leg draw the Perpendicular In like sort to let fall a perpendicular from a point assigned you need only to apply one Leg of the Square to the Line so as the other may touch at the same time the assigned point whence you may draw the perpendicular by that Leg that toucheth the Point If the Angle of your Square be a little blunt either through ill making or long using you must allow for it when you apply it to the point in a Line And when you are drawing a Perpendicular you must stop before you reach the given line and then by applying the Leg of your Square to that part of the perpendicular already drawn so as part of that Leg may pass clearly over the given Line you may draw the rest of your perpendicular as exactly as if the Angle had been true The like course is to be taken when a line is to be crossed by another drawn quite through it at right Angles V. An Angle being given to make another equal to it Fig. 4 The Angle XAD being given and a Line drawn at pleasure as is the lowest from the point E open your Compasses to any convenient distance and setting one foot in A describe the Arch BC. Then with the same extent setting one foot in E with the other describe the Arch GH long enough to equal or exceed the other Then taking the distance BC between the points of your Compasses set one in G and with the other mark the point H in the Arch GH through which point H a line being drawn from the point E will make an Angle with the Line EG equal to the Angle given Note when we speak of the quantity of Angles their equality or unequality we never regard the length of the Lines for if you extend or contract them at pleasure the Angle is still the same But that is the greatest Angle whose lines are farthest distant from one another at the same distance from the Angular Point or the place where its lines meet VI. Any three Lines being given equal or unequal so as no one of them be longer than the other two joyned together to make a Triangle of them Fig. 5 The Lines A B C being given set the line A from D to E then with your Compasses take the length of the Line B and setting one Foot in D describe the Arch PO. This being done take with your Compasses the length of the line C and setting one foot in E with the other cross the former Arch at F from which Intersection drawing Lines by a Rule to D and E the Triangle is finished Note that if all the Sides or two of them be equal the method is the same but the labour less because we need not to take the same length twice over with the Compasses VII To find the Perpendicular of the Triangle in order to the measuring of it Fig. 6 Let the Line AB be accounted the Base and from the Angle C let fall a perpendicular as was taught Probl. 4. Upon that line at D which is ready for taking off with Compasses and measuring on a Scale of which hereafter in the Chapters of measuring the Content of Figures But if we have no occasion to draw the perpendicular but only to know the length of it as it most frequently falls out in measuring no more is needful but to set one foot of the Compasses in the Angular point C and extend the other to the Base AB so as it may touch it but not go beyond it then have we the perpendicular between the points of the Compasses VIII One Side being given how to make a Square Fig. 7 The Line CD being given raise a perpendicular at C of the length at the least of the given line then taking the line CD between the feet of your Compasses set it upon the perpendicular from the Angular Point C to A With the same distance setting one foot in D describe the Arch OP Lastly with the same distance or extent set one foot in A and with the other describe the Arch crossing the Arch OP in N from which intersection a line drawn by a Rule to A and another to D finish the Geometrical Square or Quadrant ACDN. IX To make a long Square the length and bredth being given This is so like the former that a particular Figure is not necessary to conceive of it Suppose each side of the Square in the last Problem to consist of 8 small equal parts and you were to make a long Square whose length must be equal to a side thereof viz. 8. and the bredth half so much given in a line thus 4 then when you had drawn the line CD for the length and raised the Perpendicular at C you must take the shorter Line given for the bredth and set it upon the Perpendicular from C upwards to a Point which for distinction we shall call the Point E imagining it so marked With the same extent of the Compasses describe the Arch placing to that purpose one foot in D. Lastly extending your Compasses to the length of the line CD set one foot in E and with the other cross the Arch aforesaid Then a right Line drawn from that Intersection to E and another from the same to D complete the long Square X. To make a Rbombus the Sides being given Fig. 8 If the Angles be not limited draw any oblique Angle at pleasure either Acute or Obtuse as here the Angle BAC which is Acute Then let the Line OP be the length of a Side which being taken with your Compasses set it from the angular point A in both Lines to D and E in which two points place a foot of your Compasses successively without altering them viz. in D to describe the Arch FG and in E to describe the Arch HI crossing one another in the Point K from which right lines drawn to D and E finish the Rhombus DAEK Note if any Angle be given together with the Side to limit the shape and content begin with that and proceed as before For you must
imply that the Figure I speak of is in that very Page and so it was in my Copy but the Printer and Gravers have otherwise contrived them for convenience in Copper Cuts by themselves And to give them their due they are generally done with great accuracy and none of them having any such error as is like to beget trouble or mistake to the Reader saving only that fig. 19 hath D instead of O at the Center and the Line OL in the Margin of Fig. 14. should be of the length from L to the uppermost o in the Scale and the Figures on the side should be made 1 less than they are viz. 2 should be 1 3 made 2 c. And lastly as to the Errata though I have not been so anxiously careful as to correct every literal mistake I have very diligently perused all from p. 1 to p. 224 inclusive and hope I have sufficiently restored the Sense to the places wronged when thou hast done them right by the Pen according to the Directions of the Errata following next after the contents and that you continue the Line in the Margin of p. 34. to the length of the Line OL in fig. 14. THE CONTENTS OF THE CHAPTERS Chapt. 1. OF Geometrical Definitions Divisions and Remarks p. 1. Chapt. 2. Of Geometrical Problems p. 6. Chapt. 3. To find the Superficial Content of any right lined Figure the lines being given p. 17. Chapt. 4. Concerning Chains Compasses and Scales p. 26. Chapt. 5. How to cast up the Content of a Figure the lines being given in Chains and Links p. 35. Chapt. 6. How to measure a Close or parcel of Land and to protract it and give up the Content p. 41. Chapt. 7. Concerning the measuring of Circles and their parts p. 48. Chapt. 8. Concerning Customary measure and how it may be reduced to Statute measure e Contra either by the Rule of Three or a more compendious may by Multiplication only p. 52. Chapt. 9. How a Man may become a ready Measurer by Practice in his private Study without any ones assistance or observation till he design to practise abroad p. 65. Chapt. 10. How to measure a piece of Land with any Chain of what length soever and howsoever divided yea with a Cord or Cart-rope being a good Expedient when Instruments are not at hand of a more Artificial â—ake p. 67. Chapt. 11. Concerning dividing of Land Artificially and â—echanically p. 70 Chapt. 12. Concerning the Boundaries of Land where the â—ines to be measured must begin and end p. 80. Chapt. 13. Containing a Description of the Plain-Table the â—rotractor and Lines of Chords p. 82. Chapt. 14. How to take the true Plot of a Field by the â—lain-Table upon the Paper that covers it at one or â—ore Stations p. 85. Chapt. 15. Concerning the plotting of many Closes together â—hether the ground be even or uneven p. 99. Chapt. 16. Concerning shifting of Paper p. 102. Chapt. 17. Concerning the plotting of a Town Field where 〈◊〉 several Lands Buts or Doles are very crooked â—â—th a Note concerning Hypothenusual or sloping â—â—undaries common to this and the fifteenth Chapter â— 104. Chapt. 18. Concerning taking the plot of a piece of ground 〈◊〉 the Degrees upon the Frame of the Plain-Table seâ—â—â—al ways and protracting the same p. 108. Chapt. 19. Concerning taking inaccessible Distances by the â—â—ain-Table and accessible Altitudes by the Protractor â— 121. Chapt. 20. Of casting up the Content of Land by a Table â— 193. ERRATA PAge 3 Line 16 Read Trilaterals p. 4 〈◊〉 Geodates p. 5 l. 29 Eneagon p. 10 l. 6 belong as the other two p. 16 l. 6 Centers at right Aâ—â—gles p. 28 l. 1 forefinger p. 36 l. 20 Poles or Râ—â— p. 47 l. 15 fourth Diagonal and the sixth side p. 〈◊〉 l. 28 as in this figure is ABC p. 56 l. 23 〈◊〉 and l. 24 â— 22 ½ p. 58 l. 11 28. p. 69 l. 5 side 74 l. 25 FG. p. 77 l. 27 138562 and l. 28 〈◊〉 33 r. 242030. p. 79 l. 33 triangulate p. 80 l former p. 82 l. 24 fitted p. 86 l. 2 Stationary â—â—â—stances p. 90 l. 15 Chart or Card. p. 95 l. â— Park Pond p. 103 l. 22 Line p. 104. l. 19 〈◊〉 Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9. Fig. 10. Fig. 11. Fig. 12. Fig. 13. Fig. 14 Fig. 15. Fig. 16. Fig. 17. Fig. 18. Fig. 19. Fig. 20. Fig. 21. Fig. 22. Fig. 23. Fig. 24. Fig. 25. Fig. 26. Fig. 27. Fig. 28. Fig. 29. Fig. 30. Fig. 31. Fig. 32. Fig. 33. Fig. 34. Fig. 35. Fig. 36. Fig. 37. Fig. 38. Fig. 39. THE Country Survey-Book OR LAND-METER's VADE-MECVM CHAP. I. Of Geometrical Definitions Divisions and Remarks I. A Point is that which hath no parts either of longitude or latitude but is indivisible ordinarily expressed with a small prick like a period at the end of a sentence II. A Line hath length but no bredth nor depth whose limits or extremities are Points This is either right or crooked III. A right Line lies straight and equal between its extreme points being the shortest extension between them the crocked or circular not so IV. A Superficies hath length and bredth but no depth of this Lines are the limits V. A plain Superficies is that which lieth equally or evenly between its Lines VI. An Angle is the Meeting or two Lines in one point so as not to make one straight Line and if drawn on past that point they will intersect or cross one another This is vulgar English may be called a Corner of which there be two sorts one right the other oblique VII A right Angle is that which is made by two right lines crossing or touching one another perpendicularly or squarely like an ordinary Cross or Carpenters Square VIII An oblique Angle is that which is either greater or less than a right Angle and this is of two sorts obtuse and acute IX An obtuse Angle is greater than a right Angle like the left and right Corners of a Roman X. X. An acute Angle is less than a right Angle like the highest and lowest Corners of the same Letter XI A Figure is that which is comprehended under one line or many Of this there are two kinds a Circle and a right-lined Figure XII A Circle is a perfect round Figure such as is drawn with a pair of Compasses the one Foot being turned round in a point and the other wheeled about it The point in the precise middle is called the Center the round line the Circumference or Peripheri a line going through the Center and divide the Circle into two equal parts is called the Diameter half of that line is a Somidiameter or Radius half the Circle is stiled a Semicircle the quarter a Quadrant any portion of it cut off by a right Line not touching the Center is called a Segment XIII Right-lined Figures are such as are limited by three
know that to make a Rhombus or Rhomboides like to another for Figure or equal to it in Content it is not sufficient to have the same Sides for the more oblique the Angles 〈◊〉 the farther will the Rhombus differ from a 〈◊〉 and the Rhomboides from a long 〈◊〉 are and the less will be the Content But 〈◊〉 must have an Angle given which will proâ—â—â—e all the rest or else a Diagonal Line which 〈◊〉 right Line passing through the Rhombus or â—omboides from one opposite Angle to another and dividing the Figure into two equal â—â—â—angles If the former viz. an Angle be â—â—n I have shewed what use is to be made 〈◊〉 If the latter i. e. a Diagonal togetâ—â—â— with the length of the Sides you may by â—â—â—ng the length of the Sides with your Comâ—â—es and setting a Foot in the ends of the â—â—gonal Line make a Triangle on the one side â—he Diagonal by Probl. 6. and then another on the other side by the same problem the â—â—gonal being a common Base to them both this will give the Figure exactly To make a Rbomboides the Sides being given Fi. 9 〈◊〉 neither Angle nor Diagonal be given 〈◊〉 if either of them be limited the case is â—â—en to in the last problem make any Angle â—dventures as here ABC Then supposing 〈◊〉 Lines given to be OP and QR set the 〈◊〉 of the longer upon the Line BC from B 〈◊〉 and the shorter on the Line BA to E 〈◊〉 with the Compasses extended from B to E 〈◊〉 one Foot in D and describe the Arch FG. â—â—ewise with the Compasses extended to the 〈◊〉 of the Line OP setting one Foot in E 〈◊〉 the other describe the Arch HI intersecting Fig. 9 the former Arch at K from which Intersection Lines drawn to D and E finish the Rhomboides XII To make a Trapezium the Diagonal and Lines in order being given Fi. 10 Let the Line HL be the Diagonal of a Trapezium whose Sides are the Lines A B C D the Side A being counted the first as that which takes its beginning from the point H and the rest in the order as they are marked Alphabetically Then with your Compasses set to the length of the Line A place one Foot in H and with the other describe the Arch EF. Next taking the length of the Line B with the one Foot oâ— your Compasses placed in L with the other make the Arch GI intersecting the former at K from which Point of Intersection Lines drawn to H and L make the Triangle HKL Then with the extent of the Line C set one of the Feet of your Compasses at L and describe the Arch OP Lastly setting them to the length of the Line D and placing one Foot of your Compasses in H with the other make the Arch SR intersecting the former at Q 〈◊〉 shall Lines drawn from Q to L and H make uâ— the Triangle LQH and finish the Trapezium HKLQ I could have been much briefer in this problem by referring to the sixth but this being of very great and frequent use I desired to be very plain XIII To make a regular Polygon otherwise called a regular multangular or multilateral Figure consisting of many equal Sides and Angles viz. above four apiece Being satisfied what shall be the distance between the Center and every Angle with that distance describe a Circle which being equally divided into as many Parts as the Figure must have Angles or Sides for they are equal in number and Lines drawn from the Points of Division within the Circle from Point to Point ordinarily called Chords the Polygon is finished as in this Diagram Fig. 11 Suppose an Heptagon or multangular Figure of seven Sides and as many Angles be to be described every Angle being designed to be distant from the Center A seven Eighths or three quarters and an half of an Inch with that distance describe the Circle BCDEFGH which being divided into seven equal parts and Lines drawn from Point to Point the Heptagon BCDEFGH will be therein included I shall rather leave my unlearned Reader to find out the Points of Division by many tryals than to puzzle him with the Geometrical way for finding out Chords to that purpose nor shall I busie my self to tell him at large how he may divide 360 by the number of his Angles or Sides and then finding in his Quotient the Degrees and Parts belonging to every Division set them readily out by a Protractor or for want thereof by a Line of Chords for I suppose him yet ignorant of such things I shall therefore only tell him thus much A Line drawn through the Circle at the Center divides it into two equal parts which being crossed in the Center by another Line the Circle will be parted into four equal Parts or Quadrants and those by halving them into eight Parts The extent of the Compasses whereby the Circle is drawn usually called the Radius or Semidiameter will divide it into six equal parts two whereof must be a third part and half of one a twelfth part and these still easily capable of farther Division XIV Having the Sides of the Triangles whereof it consisteth orderly given to make an irregular multangler or multilateral Figure This will be more fully handled hereafter when I come to shew the method of drawing plots of Ground In the interim I will give you a Specimen of an irregular Pentagon Fi. 12 Having the Lines of three Triangles given which by a Rule hereafter to be mentioned are necessary to make up a five Figure lay down the greatest of the first viz. 20 from A to B for a Base and by Probl. 6. make a Triangle of it and the other Lines 16 and 10. viz. the Triangle ABO Secondly you find by the number o over the first Line of the second Triangle that it is the common Base to them both and therefore by the same Probl. 6. make the Triangle A B P of the Lines 20 14 18. Fi. 12 Lastly finding the Base of the third Triangle to be the same with 18 one of the Sides of the second make the Triangle PBQ of the Lines 18 11 12 So is the quinquangular Figure finished How every Line is to be found in its due order in this or any other sort of multangular Figures so as to give a true and exact account not only of the superficial Content but also of the Figure or shape and situation is to be taught hereafter in the Doctrine and Practice of protraction CHAP. III. How to find the Superficial Content of any Right-lined Figure the Lines being given AS a Foundation to what I shall say upon this Subject there are some few Geometrical principles or Theorems out of Enclid and Ramus which I desire may be remembred and because understanding is a mighty help to memory I design for my Country Reader a kind of ocular Demonstration which though not so strict and artificial as that which is to be found in
the Scales described â—n Mr. Leybourn's Book of Mr. Wynne aforesaid â—s likewise all other Mathematical Instruments Having been so large for my plain Country-man's sake I shall not proceed to the description of the new Diagonal Scale of which you may have the Figure and Description in Mr. Wing's Book For though it be an excellent good one Fi. 14 as I know by experience Mr. Hayes having at my desire furnished my noble Friend Siâ— Charles Hoghton with an artificial one of that sort when I had the honour of assisting him in Mathematical Studies yet because 't is pretty costâ—â— if well made and that before described will very well answer its end I shall at present say no more of it But my Reader may perhaps object to me thâ—â— though I have instructed him how he may make a Line of an exact length to answer to any number of Chains and Links given or found by measure upon the Diagonal Scales I have not yet shewed him how to measure a Line as suppose a Perpendicular whose length is unknown upon them To give him therefore all satisfaction though what I have writ already might help him to find this out let us suppose that in some Figure made according to the Diagonal Scale B of 10â— in an Inch we meet in measuring with an unknown Perpendicular equal to the Line in the Margin Taking it between the Points of my Compasses I first try whether it be even Chains and finding upon the first view that it is not 〈◊〉 make a second trial whether it will prove to be even Decads or Tens of Links to which purpose I set one Foot at 3 Chains in the bottom oâ— my Scale in the first Line where the Diagonals begin and the other Foot rests in the same line betwixt 6 and 7 whereby I am assured the odd Links above 3 Chains are more than 60 and less than 70. And to find how many above 60 I remove the Compasses from parallel to Fi. 14 parallel in order till one Foot in the lowest Line resting in the end of a parallel the other will touch some Diagonal at the Intersection with that line which falls out to be at L and O in the line marked with 7 shewing the whole line being measured by that Scale to signifie 3 Chains and 67 Links CHAP. V. How to cast up the Content of a Figure the Lines being given in Chains and Links HAving described these plain Instruments and in some measure shewed the use of them in severals it were very proper in the next place to teach their joynt use in measuring and protracting but because I would have my young Surveyor before I take him into a Close able to perform his whole work together I intend to shew him 1. How he ought to make his Computations 2. The Grounds or Principles that will justifie him in so doing For the first take these Rules 1. Put down your length and bredth of Squares and Oblongs and your Base and half Perpendicular of Triangles directly under one another expressed by chains and links with a prick betwixt them as was taught before Chap. 4. 2. If the odd links were under ten put a Cypher before the numeral Figure expressing them as there also was shewed and if therâ— be no odd links but all even chains put twâ— Cyphers after the prick 3. Multiply length by bredth and Base 〈◊〉 the half Perpendicular according to the Rulâ— for finding the Content of Figures Chap. 3. 4. From their Product cut off 5 Figures accounâ—ing Cyphers for such reckoned from thâ— right hand backward with a dash of your 〈◊〉 so shall those to the left hand signifie Acres 5. If those five cut off were not all Cypherâ— multiply them by 4 and cutting off fivâ— towarâ—â— the right hand again the rest will be Roods 〈◊〉 Quarters 6. If amongst these five Figures towards the right hand that were cut off at the second Multiplication there be any Figures besides Cyphers multiply all the five by 40 and cutting off fivâ— again by a dash of your Pen those on the left hand signifie square Perches Poles or Roods A few Examples will make all plain Quest 1. What is the content of a Square 〈◊〉 Sides are every one of them 7 Chains 25 Links Length 7.25 Bredth 7.25 3625 1450 5075 525625 525625 4 102500 40 100000 Answ 5 Acres 1 Rood and 1 Perch as here appears Quest 2. In a long Square whose length is 14 Chains and the bredth 6 Chains 5 Links what is contained Length 14.00 Bredth 6.05 7000 84000 847000 4 188000 40 3520000 Answ 8 Acres 1 Rood and 35 Perches as the Work makes it evident Quest 3. In a Triangle whose Base is 3 Chains and half the Perpendicular 98 Links what is the Content The Base 3.00 Half Perpend 0.98 2400 2700 29400 4 117600 40 704000 Answ 0 Acres 1 Rood 7 Perches as here is plain There be other ways of Computation by Scales Tables c. but that this is sound and demonstrative I come now to shew by these following Steps 1. It is evident that in this way of Multiplication the Product is square Links for every Chain being 100 Links it is all one to multiply 7.25 by 7.25 or 725 by 725 without pricks for the pricks signifie something as to Conceptions but nothing at all in Operation The Product therefore of the first Example was really 525625 Links 2. Every Chain being 4 Perches long it follows that 5 Chains or 20 Perches in length and 2 Chains or 8 Perches in bredth make an Acre or 160 square Perches for 20 being multiplied by 8 gives 160. 3. From hence it plainly followeth farther that there are exactly 100000 square Links in an Acre for 5 Chains multiplied by 2 is the same with 500 Links by 200 which makes 100000. And he deserveth not the name of an Arithmetician that is ignorant of this old plain Rule When the Devisor consists of 1 and Cyphers as 10 100 1000 10000 100000 c. cut off from the right hand so many Figures of the Dividend as the Devisor hath Cyphers accounting them the Remain so shall the rest on the left side be the Quotient It is plain then that 525625 square Links make 5 Acres and 25625 square Links over Thus I have made it clear to a very ordinary capacity that as far as concern Acres the Rules for Computation are good Now for Roods and Perches though I might turn off my Reader with that known Rule in Decimal Arithmetick Multiplying Decimal Fractions by known Parts gives those known Parts in Integers due regard being had to the separation I shall proceed in my plain way thus If 25625 square Links which remain above an Acre do contain any quarter or quarters of an Acre then if they be multiplned by 4 and divided by 100000 that is five cut off from the Product they will contain so many Acres as now they do quarters or Roods for any number of quarters multiplied by 4 must
Suppose a large Triangle of common Land be to be divided amongst three Tenants A B and C according to the quantity of their Tenements A having 19 Acres of Land to his Tenement B13 and C7 the Base of the Triangle being found by measure to be 17 Chains and 27 Links and the Dem and is where the Points of Division must be placed in the Base so as Lines drawn from thence to the opposite Angle shall truly limit each mans part To answer this let us add 13 and 07 to 19 as in the Margin and they give 39 So is the work plainly reduced to the Rule of Fellowship and therefore to find every mans distinct portion we need only to multiply the Base by his 19 13 07 39 particular number and divide that product by 39 the sum of all their numbers as here is plain A 39 17. 27 ∷ 19 19 15543 1727 39 32813 841 14 39 312 161 156 53 39 14 B 39 17. 27 ∷ 13 13 5181 1727 39 22451 575 2◠3◠195 295 273 221 195 26 C 39 17. 27 ∷ 7 7 39 12089 309 38 39 117 389 351 38 From these Operations it is plain that if we set off from the Angular point where the Base begins 8 Chains 41 Links and a little above the third part of a Link upon the Base for A and where that ends 5 Chains and 75 Links and 2 3 of a Link for B and consequently leave between this second division and the other end of the Base 3 Chains and almost 10 Links for C Lines drawn from those points of division to the opposite Angle will give each man his due What I have said touching the division of Triangles upon their Bases will with a little variation serve for the dividing of all sorts of Parallellograms whether Square Long-squares Rhombus's or Rhomboides's all the difference is that in stead of drawing Lines from Points in the Base to the opposite Angle you must draw parallel Lines from Points in one opposite side to another as will be sufficiently plain by this one Instance Fig. 7 Suppose the square Figure in the 8th Prop. of the second Chapter to represent a Close of six Acres and I am to cut off an Acre at the side AC having set off the 6th part of the Line CD from C towards D and also from A towards N a Parallel drawn between those Points takes off exactly a 6th part or an Acre If it be not thought convenient as in some cases it is not to cut off a piece so long and narrow you may by the Rule of Three find what other length of any greater breadth will limit an equal quantity to it Or you may multiply the breadth by 2 3 or any other and divide the length by the same number that you multiplied the breadth by Or lastly if you set out a double proportion that is 2 6 or 1 3 from C towards D and from the Point where it falleth draw a Line to the Angle A you will have a Triangle equal to 1 6 of the Square ACDN. But to return to Triangles the most simple and primitive of all Rectilinears and therefore the most considerable in this case of partition as giving Laws often to the rest It may fall out that a Triangle must be divided convenience so requiring by a Line from some Point in a side so as that Line may either be parallel to some other side or not parallel to any For the former case take this Example following out of Mr. Wing Lib. 5. Prob. 5. Fig. 21 Let ABC be a Triangle given and it is required to cut off 3 5 by a Line parallel to AB First on the Line AC describe the Semi-circle AEC whose Diameter CA divide into five equal parts according to the greater term and upon three of those parts the lesser term erect the Perpendicular DE which cutteth the arch Line in E then set the Line CE from C to F and from thence draw the Line EG parallel to AB so will the Triangle CGF contain 3 5 of the Triangle ABC as was required Fig. 22 Now for the latter case when the line of partition goes not parallel with any side take this Example Let ABC be a Triangle given to be divided into two parts which shall bear proportion to one another as 3 and 2 by a Line drawn from the point D in the Base or Line AC From the limited point D draw a Line to the Angle B then divide the Base AC into five equal parts and from the third point of Division draw the Line to E parallel to BD. Lastly from E draw the Line ED. So shall the Trapezium ABED be in content as 3 to 2 to the new Triangle DEC I have now done with the Division of Triangles when I have added these three Advertisements 1. You must be sure to take very exactly the distance of every point where a dividing line cutteth any side to one of the ends of the same side as in this last Figure the distances BE and AD which distances being applied to the Scale by which the Triangle was protracted will shew at how many Chains and links-Links-end you are to make your dividing Line on the Field it self 2. The proportions by which you are to divide are not always so formally given as in the former examples but are sometimes to be found out by Arithmetical working as in this case Suppose a Triangular Field of 6 Acres 2 Roods and 31 Perches must be divided so as the one of the two parts shall be 4 Acres 3 Roods and 5 Perches and the other consequently 1 Acre 3 Roods and 26 Perches reduce both measures into Perches and the one will be 765 and the other 306. Their Sum is 1071 which by their common measure being reduced into their lowest terms of proportion in whole Numbers will be 5 2 and 7 which shews that the Triangle being divided into 7 equal parts the one must have 5 of those 7 parts and the other 2. And observe that it will be sufficient to find the common measure between the Sum of the terms and either of the terms the method whereof is shewed in every Arithmetick Book for reducing Fractions into their lowest terms But if my unlearned Reader cannot skill of that work he may multiply either of the parts as suppose 765 by the length of the Base which we will suppose to be 8 Chains and 75 Links or 875 Links and that product divided by 1071 the Content of the whole Close in Perches gives by the Rule of Three direct 625 Links or 6 Chains and 1 Pole the true distance from either end of the Base that his mind or occasions may direct him to begin with to the point of Division for the Division must be not only for proportion or quantity but also as to position or situation of parts upon the Paper as it is required to be on the Ground 3. In these and all other divisions
the slit of the back-sight I see the thread cutting the mark at A and then screw it fast so will my Needle if a good one hang directly over the same point that it did at the first station but however that be fore-sight and back-sight will do the business for which purpose it is good to take back-marks as well as fore-marks at every station as was taught in the Example of a single station only taking notice that the back-mark when the Instrument is planted in an Angle must needs be out of the Field as suppose here at O. But to proceed Having measured the distance between the first and second station and finding it to be 7. 10 I set it upon the Line OP from A to Q where I make another point to represent the second station and turning about my Index with the fiducial edge upon that point and so looking through the sights at the Angles GHIK I draw Lines towards them on my Paper and having measured between every one of those four Angles and the Instrument I set those Measures as I did the other with my Scale and Compasses from Q towards every Angle upon his proper Line and then having drawn the black bounding-lines from A to B from B to C and so round about the Close the Protraction is finished But here to make this Figure yet more advantageous let me according to my usual method add some Advertisements 1. Sometimes a Station is so taken that you may measure towards two Angles at once as here from Q to G and H in which case you are to set down the Chains and Links where the first Angle falleth but still be proceeding to the further Angle causing the remainder of the Chains at the fore-end to advance beyond the former Angle so going on with whole Chains so far as you can to which the odd Links at the end are to be added 2. If at any of your Stations as suppose A you can see an Angle for example E to which you cannot measure in a direct Line without passing the boundaries of your parcel of Land given to be measured you may notwithstanding take in that Angle by a strait measured Line as I have done provided it may be lawfully done without trespass and conveniently without troublesom passing of Fences otherwise it must be taken from another station 3. I here took one of my Stations at an Angle and the other within the body of the Field to shew the variety of working taught by other Authors and that 't is no great matter where you make your Stations so you can see the Angles else it had been full as convenient to have taken my first Station also within the body of the Field as suppose at R. 4. Though this Figure representeth to your eye only two Stations A and Q your fancy may multiply them at pleasure for suppose the Angle H could not have been seen from A or Q how easie had it been to have set up a Mark at S and then to have removed the Instrument thither observing the same directions â—hat were given at the removal from A to Q. II. In the second Method the Instrument is to be planted twice or oftener as occasion is the Rules for removal of the Instrument fore-sight and back-sight and measuring the distance of Stations being the same as formerly was â—aught but instead of measuring to and from â—very Angle we only view each Angle through â—he Sights from two Stations having applied â—he fiducial edge to the Points representing â—hose Stations and having drawn Lines with the point of the Compasses or a protracting Needle the Interfections represent the Angles from which the boundary-lines may be drawn so is the Field protracted Which that my Reader may understand let him note these three Figures Fig. 26 Here in these three Figures the Angles are marked Alphabetically ABCDEF c. and â—he Stations by a point in a small Circle numbred 1 2 or 1 2 3 according to their number and order Fig. 26 The first of these Figures represents the Plotâ—ing of a Field at two Stations within it from both which all the Angles may be seen Fig. 27 The second performs the same work by two Stations taken without the Field by which 〈◊〉 a Close may be measured though the present Possessor will not give us leave to come â—nto it Fig. 28 The third shews how the Work may be performed at three Stations or more when two such places cannot be found whence to view alâ— the Angles which last having more of difficulty than the two former though indeed noâ— very much and the Explanation of that will sufficiently help to the understanding of them I shall a little explicate the meaning of it in these particulars 1. From the first Station taken acording to former directions I see the Angles ABCD FGK and acordingly draw Lines upon my Paper towards them from the point representing that Station by the fiducial edge of my Indeâ— with the point of my Compasses 2. Having removed my Instrument to the second Station and in so doing observed the Rules before given touching such removals I thence see the Angles ABDEFGHIK and draw Lines upon my Paper towards them from the point representing the second Station And now viewing my Work I find upon my Paper Interfections for the Angles ABDF GK but only single Lines toward the Angles CEHI therefore 3. Removing the Instrument regularly as before to a third Station I thence see those four Angles CEHI and drawing Lines towards them I have interfections for them also so that having drawn the Lines AB BC c. from one Interfection to another I have the Field perfectly protracted For these bounding-lines from Angle to Angle do not only signifie the Boundaries of a piece of Land given to be measured limiting the Figure or shape thereof and are to that purpose given in this and all other Survey-books but also are the true distance by a Scale from Angle to Angle for the Plot upon the Paper I mean by the same Scale by which the stationary distances were laid down upon their own lines And this holds true in all kind of true plotting whether in this Method or any other III. The third Method is that of Circuition and this hath several varieties according to these three following Cases 1. When the distance from Angle to Angle without any exception is measured quite round the Plot either within or without 2. When the distance is taken only between some more notable Angles and the Perpendiculars of the rest measured as you pass along their Bases within the Plot proper for plain solid ground 3. When the like is done without the Plot as in the case of Plotting thick Woods Meres Pools Bogs c. The first of these is very easie consisting in nothing but planting the Instrument at every Angle either within or without as necessity and convenience determine it observing the former directions for planting and removing
the instrument and also for measuring the stationary-lines on the ground and protracting them on the paper as is manifest in this Example Fig. 29 Let ABCDEF be a Park-Pond or Close to be protracted I first plant my Instrument at A and direct the sights to a Mark in the Angle B drawing a dry Line from a convenient Poinâ— on my Paper towards B on the ground and having measured by my Chain the distance AB I set it by a Scale upon the Correspondent Line from A to B drawing a black-line between them with Ink or a Lead-pen the Extremities whereof are the Points A and B on my Paper and the little pricked Line that goes beyond B represents the remainder of the dry Line drawn at random as to length with the point of the Compasses Then setting up a Mark at A if there was none before I remove the Instrument to B and laying the fiducial edge to the Line AB I turn about the Instrument upon the Saff till through the sight I perceive the thread cutting the Mark at A and my Needle if I have one directly over the same point that it was when it was planted at A and so screw it fast My next work is to lay the fiducial edge to the point B and direct the sights to C drawing a dry Line towards it and setting the distance BC measured by the Chain from B to C. In this manner I proceed surrounding the Close till I come at last to A where I began by planting the Instrument at every Angle using the help of Back-sight and my Needle as I did at B and then from the point representing my present station directing the sight to the next Angle as I did from B to C. In the second Case we do not plant the Instrument at every Angle but at the more considerable taking in the smaller by their Perpendiculars from the Base as we pass along of which this following Figure may be an Instance Fig. 30 Let ABCDEFGHIKL be a Close to be measured by planting the Instrument only at ACF and K I have the main substance of the Close in the Trapezium ACFK and for the five small Triangles which must be added to the Trapezium they may be easily protracted by the help of such a little Square as was mentioned towards the latter end of the Tenth Chapter for thereby finding at how many Chains and Links distance from A upon the ground the Perpendicular B b falleth upon the Line AC and having measured the length of that Perpendicular and taken it between my Compass-points off my Scale I erect a Perpendicular of that length at b which is the point upon the Paper where so many Chains and Links determine as were measured upon the Ground from the Angles A to the place where the Perpendicular fell on Ac viz. at b. Just in the same manner I raise the other Perpendiculars Dd Gg Ii and Ll and then by the help of the Perpendiculars I draw from and to the proper Angles the Boundary-lines AB BC CD DE FG GH HI IK K L LA which together with the Line EF between the Angles E and F give the true Plot of the Field in one large Trapezium and five small Triangles ready for casting up The third case is so like the second that there needeth no new Direction concerning it but to annex one plain Diagram all the difference consisting in this that because we cannot go within it being supposed to be some Pool Bog or Thicket we must of necessity go on the out-side and consequently all the Triangles made by inward Angles and their Lines upon the measured Bases must be excluded by the boundary-lines from being any parts of the Plot as here is manifest Fig. 31 Supposing ABCDEFGHIKLMN OP to be a great Pool though here be fifteen Angles I plant my Table only five times viz. at AEFH and M and upon the dry Lines AE FH HM and MA I raise their Perpendiculars in due places according to measure and also of a right height by which and my five stationary Angles I draw the bounding-lines of the Plot excluding all the Triangles as foreign to it they being no resemblances of any part of the Pool but of Land adjacent Where note 1. That both in this and the former Case such a little Square as I mentioned in the Second Chapter will be very useful for speedy raising of Perdendiculars but where the Triangles are very small it needs not be used not the other mentioned Chap. 10. 2. That if by reason of troublesom Brush-wood Gorse or Bogs c. I could not have measured close to the sides EF HI or LM it would be the same thing if I went parallel to them And this is a shift that the practical Surveyor will oft be put to make use of in other cases as well as this CHAP. XV. Concerning the Plotting of many Closes together whether the Ground be even or uneven THough I design not so high in this Manual as to make my Reader able to Survey Lordships and Forests much less to draw Maps of Countries but to measure a parcel of Land with truth and judgment yet I would have him so expert as not to be puzzled if any should desire him to draw as it were a true Map of a Tenement or small Demesn consisting of several Closes for which purpose let him that knows no better observe this Method Fig. 32 Suppose ABCDEFGHIKLM to be a Tenement or small Demesn divided into fourteen Closes to be measured and protracted according to their several shapes and situation I first draw the Plot of the whole by the Method of Circuition planting mine Instrument either at every Angle or only the most considerable either within or without as I find most convenient This being done a Line from B to M gives the Triangle ABM from the first Close In the next place I go round the second Close beginning at M then to B and so about cum scle to M again And then for the third Close I plant my Table at C and go round to B the Line BC being protracted already and so of all the rest still observing which are common Lines belonging to several Closes representing the Fences that I may avoid the trouble of measuring â—hose Lines oftener than once and lay everâ— part of evâ—â—y Close in its due place and that I be suâ—â— to keep the Instrument throughout the whole Work to its true Position by Needle fore-sight and back-sight There are I confess divers other ways of doing this Work but none more sure or plain especially if the ground be uneven for in that case if you protract according to the length of Lines measured from your Station to the Angles you will put your Closes into unproportionable shapes except you reduce Hypothenusal Lines to Horizontal by instruments or otherwise which is somewhat troublesom and the like may be said when you Plot with the Chain only Indeed the Method
of measuring only the stationary distances were very proper for setting out the Figure of each particular Close provided the distance of the stations be large and taken if possible upon pretty even ground which sometimes may be done though most of the Close be uneven and the Work so ordered as not to make too acute Angles but because this requireth skill and care I rather advise my young Artist to use the circling way as ordinarily most commodious in my poor judgment but not prejudicing other men's that may differ from me in opinion and where need requires let him observe the directions in the 17 Chapter But which way soever you go to work there is one very necessary Rule to be observed If the Ground be uneven considerably you must not give up the Content by measuring the Bases and Perpendiculars of the Triangles on the Paper by your Scale but you must measure the Lines correspondent to them on the Ground and cast up the Content according to that measure And if it be desired that you should adjoyn to your Plot as is usual a Scale of Chains to measure distances by you must either by making the Forms of Hills erect and reverse or some other Note in writing mark out your uneven Ground lest those that try it by the Scale judge your work erroneous for though you make that Scale exactly correspondent to that you protracted by as you ought to do the Hills and Dales in the ground truly measured may make a considerable alteration It is convenient when you plant your Table that the Needle hang just over the North-point of the Compass under it in the Box then may you by the Lines overthwart the Frame of the Table easily draw two Lines quite through the Plot cutting one another at right Angles the one pointing at North and South and the other at East and West And if your skill serve you to make the Two and thirty Points of the Compass upon the place where they intersect and to draw the Forms of the Houses Woods and other remarkable things upon the Demesn and the course of Brooks and Rivers running through it it will add to your commendation And so it will also if you take in such parcels of Land bounding it whether common or peculiar to other men as will make your Plot to look handsomly like a perfect Square or Oblong But however that be you must be sure to protract truly all Lanes going into it or through it and all Closes of other mens mixed with it and also all considerable Ponds Ways and Outlets with the Names of the Closes and quality of the Ground whether Meadow Pasture Arable c. CHAP. XVI Concerning shifting of Paper IN such work as that of the last Chapter it may sometimes fall out through the multitude and largeness of Fields that one sheet will not hold your whole Plot in which case you may help your self by shifting Paper as we call it thus Fig. 33 Let ABCD represent our sheet of paper that covereth the Table upon which the Plot of the large piece of Land EFGHIK should be drawn having made my first station at E and the second at F I find my paper will not receive the Line FG but however I draw so far as it will go to the edge of the paper and planting my Table again at E proceed in my Circuition the contrary way to K and I where I find my self again at a loss for my Line IH but draw it also to the edge of the Paper Then with the Point of my Compasses striking the Line PO parallel to the edge of the Paper BC and the Line QO parallel to DC and cutting PO in O I throw aside that paper for a while covering the Instrument with a new one which I mark with the figure 2 for my second sheet Fig. 34 Upon which second sheet the leading part whereof is represented by the three Lines meeting in the Angular points A and B I draw PO parallel to AB the leading edge of the paper and crossing it at right Angles in the point O by a parallel to BC viz. the Line OR being of the same distance from BC that QO in the former sheet was from DC Then with a Rule and a sharp Pen-knife I cut off the end of the first sheet at the Line PO and applying the edge of it to the Line PO of the second sheet so as it may touch that Line all along and the Line QO of the former touch the Line OR in the latter so as to make one Line with it I draw the Lines PG being the Remainder of the Line FG and the Line OH being the remainder of the Line IH and from their extremities the Line GH And if the Plot required it you might proceed on in the second sheet and annex a third and a fourth c. as there is occasion These sheets may be pieced together with Mouth-Glew or fine Paste applying the edge of the former as you did upon the Table to the Line PO of the latter And note here once for all that when I speak of applying the edge of the paper to a Line I mean the precise edge cut by the Line PO but when I speak of drawing Lines to the edge of the Paper upon the Table I hope none will think me so absurd as to mean the edge that is couched under the Frame but that my meaning is that the Lines must be continued on the paper till they touch the Frame CHAP. XVII Concerning the Plotting of a Town-Field where the several Lands Butts or Doles are very crooked VVith a Note concerning Hypothenusal or sloaping Boundaries common to this and the Fiftâ—enth Chapter Fig. 35 SUppose ABCDE divided in the manner of a common Field into seven parts or Doles belonging to seven several men First Plot the whole as before hath been taught then measuring from A to B upon the Land set one Note down as you go along at how many Chains or Links or both the Division is between Dole and Dole and accordingly mark them out by the help of Scale and Compasses in the Line AB on the Paper plot In the very same manner you must measure and mark out the Lines OC and ED which being done take the Paper off the Instrument and laying it before you on a Table with the side AE towards you the Compasses must be so opened and placed as by a few tryals they may that one foot resting upon the Table the other may pass through the Points of Division upon all the three Lines viz. AB OC and ED as in this Figure they do If the Content of any one or more of these Parts Butts or Doles be desired without Plotting it may easily be done without your Plain-Table thus Take the breadth by your Chain at the head middle and lower end and adding these Numbers together the third part of their Sum is the equated breadth by which multiplying the
passing upon the Land from the Angâ—â— A directly under the Sights of the Instrumeâ—â— to the Mark at F is as it were the priâ—â— Diameter whence the Degrees of the Angles 〈◊〉 to be numbred and accordingly I mark thâ— Angle A in my Table hereafter to be exemplâ—â—fied with 360 Degrees But to proceed turning my Index with the fiducial edge upon 〈◊〉 Center till I see the Thread cutting the Maâ—â—â— at B the said edge cuts upon the Frame at 〈◊〉 Deg. 15. Min. which I note down for that Angle The like work I do turning the Sights to CD and D but not to F for there is no Angle but only a Mark in the Boundary and I find mine Index to cut for every Angle as I have marked them within the pricked Circle of the last Figure viz. 157 Deg. 35 Min. for C 225 Deg. 20 Min. for D and 278 Deg. and 50 Min. for E. Then I measure or cause to be strictly measured by others the Distances betwixt the Place where the Instrument stands and every Angle and find them to be as I have set them upon the pricked lines in the little Circle viz. A4 Chains â—0 Links B4 Chains 3 Links C3 Ch. 84 Li. D5 Ch. 35 Li. E5 Cha. 6 L. And now my Table both for Lines and Angles is thus perfected and the Work is ready for Protraction within Doors  D. M. C. L A 360 00 4 20 B 76 15 4 03 C 157 35 3 84 D 225 20 5 35 E 278 50 6 06 Your judgment will easily inform you that in such weather we shall hardly stand to make our Table neat and formal but any thing how rude soever that we can understand doth the feat A Welsh Slate with a sharp Style or for want thereof a Black-lead Pen and a smooth end of an hard Board like a Trencher is more convenient at such a season than Pen Ink and Paper But of all I would commend for expedition a Red-lead Pen whereby you may mark out every Angle neatly with one touch upon the Table it self just where it toucheth the Frame by help of the fiducial edge and close by it the length of the Line from the Center to that Angle All which may be easily cleared off by a wet Sponge or Cloath so soon as you have protracted Or if through the sponginess of the wood the head of the Table which we use to cover with paper were made a little reddish what great harm were that We are forced to do it more real wrong by the points of the Compasses in the ordinary way Now to protract our Observations I draw upon a paper the Line AF at adventures so it be long enough and stick a Pin in it at pleasure for the Center O upon which I place the Center of the Protractor so as the straight side or Diameter of the Protractor may just lie upon the Line AF the Limb or Arched-side being upwards towards B by help whereof I make a prick or point on the paper at 76 Deg. 15 Min. for B and at 157 Deg. 35 Min. for C according to the numbers nearest to the Limb. Then turning the Protractor about on the Pin with the Arch or Limb down towards D and E till the Diameter lie again just upon the Line AF I number downwards from the right hand towards the left by that rank of Figures that are nearer to the Center beginning 190 200 c. and over against the places where 225 Deg. 20 Min. and 278 Deg. 50 Min. fall I prick the Paper at the side of the Limb and through those four points I draw so many several Lines having laid aside the Protractor upon which and also upon the Line AO I mark out by Points the true measure of every Line by a Scale from the Center and from those points drawing the Lines AB BC CD DE and EA I have the true Plot of the Field Where note by the way that we estimate Minutes as well as we can both upon the Frame of the Plain-Table and the Protractor accounting half a Degree 30 Minutes a third part â—0 a fourth part 15 c. And though by this means it is impossible to avoid small errours 't is easie to avoid sensible ones and the like may be said when we protract by a Line of Chords of which I now come to treat Having proceeded in the Field as before and made my Table for Lines and Angles or done that which is equivolent by a Red-lead Pen I draw the Line AF and having extended my Compasses to the Radius or 60 Degrees on a Line of Chords I set one Foot towards the middle of the Line AF and with the other I describe a Circle like that in this Figure of a five-angled Field but much larger according to the length of the Radius Then extending the Compasses from the beginning of the Line to 76 Deg. 15 Min. I set one foot in the Intersection of the Circle by the Line A and with the other foot make a mark in the Circumference of the Circle upwards towards the right-hand and through it draw the dry Line BO In the next place I substract the angle 76. 15 from 157. 35 where the Index cut for the Angle C and there resteth 81 Deg. 20 Min. which I take off the Line as before and set it upon the Circumference from the Intersection by BO towards the end of the Diameter marked with F and through the Point were it falleth draw the dry line CO. In like manner I subtract 157 Deg. 35 Min. from 225 Deg. 20 Min. and the difference is 67 Deg. 45 Min. which I set from the Intersection by the Line CO downwards past the prime Diameter AF and through the point where it falleth draw the Line DO Lastly Having subtracted 225 Deg. 20 Min. from 278 Deg. 50 Min. there resteth 53 Deg. 30 Min. which must be set downward towards the left-hand from the Intersection by DO and through the point where that falleth I draw the Line EO And now when I have set the particular Measures upon every Line and drawn the Boundary lines as I must have done if I had used a Protractor the Plot is finished But for better assurance that I have done my Work well I take the measure of the remaining Angle AOE upon its proper Arch viz. from the Intersection of the Circumference by AF to the Intersection by EO and applying it to the Line of Chords I find it to be 81 Deg. 10 Min as it ought to be for it should be the Complement of 276 Deg. 50 Min. to 360 and so it is And for further satisfaction I sum up the Degrees and Minutes of all the five Angles which for plainness sake I have noted in every one of them on the outside of the Circle in the Figure so often referred to and their sum is 360 as it ought to be and as here is evident 76. 1â— 81. 20 67. 4â— 53. 3â— 81. 1â— 360.
00 My Reader may now perhaps expect that I teach him how to take a Plot at two or more Stations when all the Angles cannot be seen from one But because this is so easie from the grounds already laid to any that is Ingenious and in part rendâ—red unnecessary by the Method presently following I shall content my self to give this general hint When you have from one Station taken in all the Angles you can see from thence and then are to remove to your second Station do just as you would do if the Table were covered with a Paper only it is at your choice whether you would guide your self for back-sight by a Line that may be rubbed off drawn upon the Table it self from the Center to the Degrees on the Frame along the fiducial edge or by noting only what Degrees it cuts on either side of the Center the edge passing through it that by the help thereof and the Needle the Instrument may be placed in the same Line and Situation as before for taking in the rest of the Angles if it can be if not another Station must be taken after the same manner But now to my second Method Fig. 38 Let ABCDE be the Figure of a Field to be plotted the Weather being bad I send mine Assistants to find the length of every side measuring it about cum sole beginning at A who return me such an account of every side in Chains and Links as I have noted them upon the Figure and in the Table following viz. AB 3 Chains 73 Links BC 4 Chains 91 Links c. In the mean season I make haste to find the Angles and without curiosity plant the Instrument at B and laying the Index on the Center I look at C and find the Index cutting 10 Deg. 15 Min. and looking at A it cuts 126 Deg. 45 Min. out of which if I subtract 10 Deg. 15. Min. there resteth 116 Deg 30 Min. for the Angle A but because I like not my Quarters so well as to subtract there I set them down thus B A 126. 45 C 10. 15. the meaning whereof is that B notes the Angle and CA the Lines meeting there cutting such Degrees on the Frame and the reason why I set A above is for more ready subtracting afterwards then removing to the Angle C and thence looking at B and D I find the Index to cut as here expressed C B 153. 10 C 15. 40. In like manner I find at D thus D C 96. 05 E 28. 50 At E thus E D 141. 20 A 11. 45. And lastly at A I find them thus A E 98. 30 B. 9. 20 An. D. M Sides Ch. L. A 89 10 A B 3 73 B 116 30 B C 4 91 C 137 30 C D 1 88 D 67 15 D E 6 64 E 129 35 E A 2 29 This being done I hast under Covert and by Subtraction find 116 Deg. 30 M. for the Angle B. 137 Deg. 30 M. for C. 67 Deg. 15 Min. for D. 129 Deg. 35 Min. for E. and 89. Deg. 10 Min. for A as you find them on the Figure and in this Table together with the length of the Lines Note that there is a way to find the Angles without Subtraction if at every Station you lay the fiducial edge over the Center and the Divisions 180 and 360 turning about the head of the Instrument upon the Staff till through the Sights you see one of the Neighbouring Angles for the Index turned upon the Center to the other Angle will give you the quantity of the Angle you are at but this exact planting at every Angle is more tedious than the other and therefore not so fit for wet weather But now to protract this Plot First by my Scale Rule and Compass I draw the Line AB in length 3 Chains 73 Links ending at the pont B then laying the Center of my Protractor upon the Line AB so as the Center of it be upon the Point A and that end of the Diameter from which the Numbers are reckoned on the Arch or Limb towards B I make a point for the Angle A at 89 Deg. 10 Min. by the guidance whereof and the point A I draw the Line AE which according to my Scale must be 2 Chains 29 Links In like manner placing the Diameter upon AE just as it was upon AB and the Center upon the point E I mark out by the Limb for the Angle E 129 Deg. 20 Min. by which I draw the Line ED 6 Chains 64 Links In the next place I bring the Center of the Protractor to the point D its Diameter lying on the Line ED and its Limb towards A by which I prick out 67 Deg. 15 Min. for the Angle at D and draw the Line 1 Chain 88 Links Lastly the Center being at C and the Diameter upon the Line DC in such manner as before at other Angles I prick out by the Limb or Arch 137 Deg. 30 Min. and draw the Line CB for at B my Plot should close and if rightly done the Angle at B will be 116 Deg. 30 Min. and the side BC 4 Chains 91 Links which by measure I find so to be But if I plot by a Line of Chords I am not bound to this Order but may go from A to B and so round that way if I please which I could not so well do with a Protractor without reckoning my Numbers backward yet it must be granted that a Line of Chords neither doth the work so quickly nor conveniently for this is the way When I have drawn the Line AB of a right length I set the Compasses to the Radius and placing one Foot of the Compasses in the point B and with the other describe an Arch of a competent length beginning at that side of the Line AB that is designed to be the inward-side and upon this Arch 116 Deg. 30 Min. must be set but because my Line of Chords gives me only 90 I set them first on from the Line AB and then take off the remainder 26 Deg. 30 Min. I joyn them to the 90 upon the Arch making a Point through which the Line B must be drawn of a due length In the like manner must I do at CE but the Angles at A and D need no such piecing being capable of being measured out by a Line of Chords at once Nor do your Angles only give you trouble in this kind of work but oft-times your Lines will be found too short to receive the touch of an Arch upon the Radius especially if the Line of Chords be large and your Scale little and so it may often fall out when you use the Protractor upon such short lines as AE and CD of this last Figure In which case a Rule must be applied to them and they must be extended to a due length that the Arches may meet them without the Figure And if those Extensions of lines and describing of Arches spoil the beauty of your Plot the