Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	draw_v	line_n	perpendicular_a	3,095	5	14.0786	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A52120	The country-survey-book: or Land-meters vade-mecum Wherein the principles and practical rules for surveying of land, are so plainly (though briefly) delivered, that any one of ordinary parts (understanding how to add, substract, multiply and divide,) may by the help of this small treatise alone and a few cheap instruments easy to be procured, measure a parcel of land, and with judgment and expedition plot it, and give up the content thereof. With an appendix, containing twelve problems touching compound interest and annuities; and a method to contract the work of fellowship and alligation alternate, very considerably in many cases. Illustrated with copper plates. By Adam Martindale, a friend to mathematical learning.	Martindale, Adam, 1623-1686.; Collins, John, 1625-1683.	1692 (1692)	Wing M854A; ESTC R217468	73,611	191

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 ●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 fo●● 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

10 11 12 16 20 24 30 c. in an Inch. Fi. 13 That short one which I give you the Figure of at A is of 10 in an Inch so noted at the top according as is usual upon Rules and Indices of plain Tables The Line marked OQ separates Unites and Tents Unites being taken upward from that Line and Tens downward mixt Numbers both ways As for Example 7 is the extent of the Compasses upon the Scale A from the Line OQ to K 30 is their extent from the Line so marked to OQ and 27 is their extent from the Line 20 to the short Line K aforesaid Here note that you must not expect to find the Letters OQ or K upon the Scales which you buy being only marks used at pleasure to make my meaning plain and likewise that this Scale of 10 in an Inch and others that are smaller all being composed after the same manner are usually made for more convenient use so long as to contain nine or ten the more the better of the great Divisions signifying Tens though the Figure at A being designed for no other use than to help your conceptions extends but a little beyond 30 that length being sufficient for my purpose in this place Fi. 14 These plain Scales especially the smaller sorts of them such as 24 or 30 in an Inch are very proper for drawing Figures upon paper where the Numbers represented by the Lines are not above 100 for then every Division may be counted as it is upon the Scale or above upon a long Scale Also in the surveying of Forests Chases and great Commons where the Lines are vastly long and the mistake of a few Links yea of half a Pole is not considerable they may be conveniently used accounting the Tens and Unites to signifie so many whole Chains and so estimating the parts of a Chain with the Compasses upon the small Divisions which a sagacious Man may do very near upon one of the larger Scales But it were much better in my opinion for ordinary measuring if the grand Divisions on the Scale were two Inches a piece as I have one upon the Index of my plain Table for then the smaller Divisions being of five in an Inch would be so large as to be subdivided into five apiece which represents 20 Links and then the half of one of those smaller Divisions signifying 10 Links and the quarter 5 a very ordinary judgment may come very near to the truth by estimation 2. But the Diagonal Scale is so well known to every Mathematical Instrument-maker so easie to be procured and every ways so fitted to Gunter's Chain and our Countryman's use that I cannot but highly commend it Of these Diagonal Scales there are two sorts the Old and New Fi. 14 By the Old one I mean such as is to be found in Mr. Leybourn's Book whereof I shall present you with a fragment with such a description as may enable you to understand the whole 1. It is made as appears by the Figure B upon eleven parallel Lines equidistant so as to include ten equal spaces which are all cut at right Angles by Transwerse Lines dividing them all into four equal parts 2. One of these Transverse Lines viz PR where it toucheth the first and last Lines separates between the Hundreds or whole Chains and the Tens representing 10 Links apiece the Chains being numbred downwards on the left hand from P only to 3 but on the Instrument it self they may go on to 9 or 10 the Rule being a Foot long but the Tens or Decads upward from P to 10. 3. From the Points of Division into Tens upon the first Line beginning at P to the like Points beginning at R in the last Line are nine Diagonal Lines drawn the first beginning at P and ending at the first Division above R. The second beginning at the first Division above P and ending at the second above R. In a word they are all drawn from one Division less from P to one more from R by which it comes to pass that every Diagonal by that time it hath passed from the first Line to the eleventh is a whole tenth part of an Inch which answers to ten Links of the Chain farther distant from the Line PR than at the Point upon the first Line whence it was drawn Fi. 14 4. Every one of these Diagonals is divided into ten equal parts by the long parallel Lines running through the whole Scale and numbred on the top from 1 to 9. Whereby it is evident that the Intersection of any of the nine parallel Lines that are numbred at the head with any Diagonal must be farther distant from the Line PR than the Intersection of the Line next before it with the same Diagonal by of that is by which answereth to a single Link of your Chain From what hath been said and inspection of the Figure B these things plainly follow which as so many clear instances will help you to understand it fully 1. The distance from PR to the second Division below it answereth to two Chains 2. The distance from PR to the eighth Division upward being taken with Compasses upon the first Line of the eleven from P to 8 answereth to 80 Links 3. Consequently the extent of the Compasses from the second grand Division below P to the eighth of the less Divisions upward is proportionable to 2 Chains 80 Links 4. The distance from PR to the first Diagonal being taken upon the parallel Line noted with 9 above answereth to 9 Links Where note that the first Diagonal is not that which is noted with 1 but that which is drawn from the point P. 5. The distance upon the same Line from PR to the Diagonal that is marked with 7 is answerable to 79 Links Fi. 14 6. The extent of the Compasses from the bottom of the Figure B upon the same Line to the same Diagonal answereth to 3 Chains 79 Links Briefly whole Chains may by Analogy be measured upon any Line from PR to the grand Division noted with the given Number Decads alone or Chains and Decads upon the first Line of the eleven where the Diagonals begin Links alone Decads with Links and Chains and De●ads with Links always upon that Line upon which the number ofodd Links stands at the head of the Scale And know that these Directions mutatis mutandis will as well fit if half an Inch be only allowed for a Chain and consequently all the Diagonals drawn within that extent as it is usual and very commodious for longer Lines upon the other end of the same Rule the grand Divisions for Chains going the contrary way ●nd noted with Numeral Figures in order It 〈◊〉 good therefore when you furnish your self with Scales to have Diagonal Scales of both ●imensions on the fore-side of your Rule and upon the back-side many plain Scales of equal parts with a Line of Chords all which you may have by enquiring only for

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 Poin● 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 ●hose Lines oftener than once and lay ever● part of ev●●y Close in its due place and that I be su●● 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 Fift●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 Ang●● A directly under the Sights of the Instrume●● to the Mark at F is as it were the pri●● Diameter whence the Degrees of the Angles 〈◊〉 to be numbred and accordingly I mark th● Angle A in my Table hereafter to be exempl●●fied with 360 Degrees But to proceed turning my Index with the fiducial edge upon 〈◊〉 Center till I see the Thread cutting the Ma●●● 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 ●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 ●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. 1● 81. 20 67. 4● 53. 3● 81. 1● 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 rend●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