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-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
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
upon measuring a Piece of Land Whereupon I took four or five of my Scholars to the Heath with me that had only been exercised within the Walls of the School and never saw that I know of so much as a Chain laid on the ground and to the admiration of the Spectators and especially of a skilful Surveyor then living in the Town they went about their work as regularly and dispatched it with as much expedition and exactness as if they had been old Land-meters CHAP. X. 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 make IF you can procure a Chain and find it is not divided as before hath been shewed but into Feet or quarters of Yards or any such vulgar divisions make no reckoning of the divisions at all but measure it as exactly as you can to find out the true length of the whole Chain and if it fit none of those lengths mentioned in the 8th Chapter nor any of their halfs make it to fit by taking off a Link or two or piecing it out with a string then dividing the length of that Chain by 100 or the half of it by 50 find the true length of a Link according to our artificial division and having got a long stick or rod set as many of those link-lengths upon it as it will hold Then may you measure all the whole Chains by your regulated Chain and the odd links of every line by your divided stick or rod as is manifest in this Example following Being far from mine Instruments and requested by a Friend to measure him a Close I procure a pair of Compasses an ordinary Carpenters Rule of two foot divided into Inches and quarters and meeting also with a piece of an old Chain seemingly divided into feet I measure it by the Rule and finding it to be 45 feet long and some odd measure I piece it out with a pretty strong Cord that will not stretch much to 48 feet exactly then it will serve me for half a Chain of 24 feet to the Pole This 48 I multiply by 12 the number of Inches in a Foot and that product being 576 I divide by 50 the number of links in half a decimal Chain and the Quotient is 1166-50 Inches or 11 Inches and an half and an trifle over So then dividing a long stick throughout into such parts each containing 11 Inches and an half besides the breadth of the nicks I am provided of Tools to measure Lines to a Link with exactness enough In like manner would I proceed with a Cord or Rope having fitted them to some known length or other And then for protraction it were easie with the Compasses to make a plain Scale of a large sort either upon Paper or an even piece of Wood this for once may serve a mans turn well enough Besides there is a way of measuring the Perpendiculars of Triangles and Trapezia`s upon the ground it self so as to prevent the necessity of a Scale for if you have a little Square with an hole in it to turn upon the head of a little stick which you may fix where you please as you are measuring the Base of a Triangle or the Diagonal of a Trapezium you may by a very few trials find the place where the one Leg will be just in the Line which you are measuring and the other point at the Angle from which the Perpendicular falls on it and then the space between your Stick and that Angle truly measured is the Perpendicular If you have not such a Square a square Trencher or any end of a Board that hath one right Angle and two true sided will supply the want of it And now that I am mentioning this way of measuring I shall make bold to add that this is a good way and as such ordinarily used by that general Scholar and reverend Minister Mr. Samuel Langley of Tamworth in Staffordshire whose ancient acquaintance I have long esteemed both mine happiness and honour to measure a Trapezium thus though it be protracted afterwards for by measuring the Perpendiculars as aforesaid and observing at how many Chains and Links end the said Perpendiculars meet the common Base the whole Trapezium may be truly protracted without going about it this little Square competently supplying the place of an Instrument which is usually called a Cross or Square made up as it were of two small Indices like those for a Plain-Table but much less with fore-sights and back-sights and cutting one another at right Angles put together and having an hole at the Center like those things which here in Cheshire we call Yarndles being used by Country Housewives in winding of their Yarn CHAP. XI Concerning dividing of Land Artificially and Mechanically WEre it sutable to my Design or Humour to be copious or curious I had here a fait opportunity for four or five modern Survey-Books of the best Accompt lying open before me would tempt me to transcribe abundance of ingenious things but for reasons often hinted before I shall confine my self to a few plain things that will competently do this business 1. To divide a Triangle into any parts required divide the Base as the Demand imports then shall Lines drawn from the Points of Division to the opposite Angle finish the Division of the Triangle Example Fi. 20 AC the Base of the Triangle ABC being divided into 12 equal parts a Line drawn from the Angular point B to the point 6 divides the Triangle into two equal parts 2. Lines drawn to 4 and 8 divide it into three equal parts 3. Lines drawn to the Points noted with 3 6 and 9 divide it into four equal parts and so Lines drawn to 2 4 6 8 10 divide it into six equal parts Also it is very obvious that if the same Triangle were so to be divided that the one part should be double to the other a Line drawn from B to 4 or 8 doth the work Or if it be required to divide it into two parts so as the one shall be triple to the other a Line drawn from B to 3 or 9 compleats the Work So also a Line from B to 2 or 10 divides it into two parts whereof the one is quintuple or five-fold to the other and a Line from B to 1 or 11 divides it into two parts whereof the one is 11 times as large as the other Further yet if it were required this Triangle should be so divided that the two parts should in quantity bear proportion as 5 and 7 a Line from B to 5 or 7 doth that feat But to deal plainly with you I must confess that sometimes the Division will be a little more intricate than thus yet not such but that the seeming difficulty may be easily overcome by observing the method wherein I shall satisfie the following demand
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
length measured down the ridge or middle the Product gives the Content But both in this case and that mentioned in the 15th Chapter the Figure of a Plot may be somewhat disordered not only by the unevenness of the ground within for which I have given due caution already that being both the more common and more considerable case but also by the great diclivity of the Ground where the boundary-lines go either of the whole Plot or particular parcels For whereas in Plotting every Line is presumed to be Horizontal or level that it may pass from Angle to Angle the shortest way and that every part may be duly situated and none thrust another out of its right place If it be not level but falling down towards a Valley or rising up Hill or compounded of both a Line over such Ground though true for the measure and for giving up the Content will be false as to the Plot and therefore must be reduced to a level and so taken off the Scale and protracted For the doing of this there are several Instruments very proper especially Mr. Rathbourne's Quadrant upon the head of his Peractor though it were better to have a Semicircle than a Quadrant so placed and divers others But supposing my Country friend to have no other but such as I have already described I shall shew him a plain easie way much used by practical Surveyors especially in Ireland as some of themselves have told me being the very same that he may meet with in Mr. Leybourn's Book Intituled The Compleat Surveyor I mean the second way by him discovered Fig. 36 Suppose ABC to be part of an Hill falling within my plot my boundary-Boundary-line going crookedly from A to B following the Surface of the Ground To find the Horizontal Line equal to AC I cause one to stand at the point A the foot of the Hill and to hold up the end of the Chain to a convenient height and gently ascending the Hill I draw it level and make a mark where it toucheth the Hill observing the number of Links betwixt mine assistant's hand and that place where he must take his second standing and hold it up as before and so I draw it out level again till it touch the Place where he must take his third standing noting the Links as before and so proceed till at last from his fifth standing I draw the Chain level to the highest point within my plot viz. the point B. And now as the pricked Lines of this Figure put together and evidently equal to the Line AC So are the Links noted down at every Station when summed up equal to the Horizontal Line of that part of the Hill In the very same manner only inverting the Order you may find the Horizontal Line going down-hill where that is most convenient And if there be both Ascents and Descents in one Line betwixt two Angles the Horizontal Lines of both must be found and joyned together in Protraction All this concerning Declivities of rising or falling Ground is to be understood when they are considerable and a very exact plot required for small ones especially when much exactness is not expected are not regardable CHAP. XVIII Concerning Plotting a piece of Ground by the Degrees upon the Frame of the Plain-Table several ways and Protracting the same HItherto I have shewed the use of the Plain-Table as such and I think my Directions have been near as plain as the Instrument it self At which some quarrel for its over-plainness exposing the Art to proud ignorant people who judging the rest of the Surveyors work to be as easie as looking through sights at a Mark and drawing lines by a Rule are apt to undertake to use it or slight the skill of such as do Others say and that truly that for vast things such as Forests Chases c. the Circumferentor is more proper And every one must grant that in wet Weather either that a Peractor a Theodolite or Semicircle must needs be better than the Plain-Table covered with the paper which cannot endure wet Hence it is that some Artists have to good purpose shewed how the Box screwed to the Index and that made to turn on the head of the three-legged Staff become a Circumferentor And if these thus fixed be turned about upon the Center of the Table they will say some with good reason Mr. Leybourn for one perform the work of the Peractor much better than the Peractor it self Others shew as I shall briefly that taking the Instrument as it is without the charge of further fitting it or trouble of removing the Box the Index turned upon the Center will by help of the Degrees on the Frame perform the work of the Theodolite to which the Semicircle is near of kin And though I might easily answer all these Objections by saying the first is frivolous such foolish Arrogance being easily contemned or checkt if worth the while by putting such conceited Fools upon the harder part of the work The second impertinent to our purpose who design not to plot such vast parcels of Land And the third concerning only an extraordinary case and that well provided for otherways for sure no man that hath not a Body of the same Metal with his Instruments will ordinarily measure Land in continual Rain a sudden shower may be fenced against by a cover and if any be so eager upon his work I have shewed â—how it may be done in the former Chapters of this Book without planting any Instrument at all by Chain Scale and Compasses alone Yet I shall shew how the plot of a Field may be taâ—en by the Degrees on the Frame not every way that I could imagine nor that I could transcribe for that would be tedious but two ways only whereof the one is proper for an ordinary Close where all the Angles may be seen from 〈◊〉 Station within it the other fitting any parâ—â—l of Land though much larger whatever be the Figure of it For the former take this Example Fig. 37 Let ABCDE represent the Figure of Field to be plotted by the Plain-Table in rain Weather I put on the Frame without a Papâ—â— the graduated side upwards and plant it iâ— some convenient place whence I can see aâ—â— the Angles as at O then placing the Index upon thâ— Table so as the fiducial edge do 〈◊〉 the same time go through the Center upon the Table and the Lines upon the Frame of thâ— Table cutting it perpendicularly at 360 wheâ—â— the Degrees begin and end and 180 the 〈◊〉 act half I turn about the Table upon the Staâ—â—â—head till through the Sights the Side markeâ— with 180 being next mine eye I see thâ—â— Angle A and then screw it fast observinâ— where my Needle cutteth and by back-sighâ— causing a mark to be set up in the Line CD 〈◊〉 the point F that the Instrument may be keâ—â— firm from moving or be rectified if it be moved during the Work And now the Liâ—â— AOF
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
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 Deâ—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 â—nd noted with Numeral Figures in order It 〈◊〉 good therefore when you furnish your self with Scales to have Diagonal Scales of both â—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 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
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.
matter is not weighty 't is so easie to be retrieved for if you lay it on a clean paper and prick through every Angle lines drawn between those points will give you the Plot neat and perfect One thing more before I close this long Chapter the Artist sometimes loseth his labour of Protraction through some error in the Field so as his Plot will not close 't is therefore good to know before we begin that work whether it will or no for which purpose if we take a Number less by two than the number of Angles in the Plot and thereby-multiply 180 that Product being found to be equal to all the Angles the Plot will close and so it appears by our Plot in this present work the Multiplier being 3 because the Angles are 5 and the Multiplier must be two less than the number of Angles 180 3 540 Deg. Min. 89. 10 116. 30 137. 30 67. 15 129. 35 540 Â This kind of tryal is grounded upon two principles of Euclid and Ramus mentioned in the first and third Chapters of this Book shewing that in all plain Triangles all the Angles taken together are equal to two right Angles and that the sides consequently the Angle also of every triangulate Figure are more by two than the Triangles of which it consisteth But I think it not proper to be large in such things whereof my young Artist is like to make but little use for when all is done I confess with Mr. Wing in his Art of Surveying Lib. 6. Chap. 15 that this way of Plotting by the Degrees is far more troublesome tedious and liable to errour than the other ways upon a sheet of paper and therefore not ordinarily to be used but when necessity compelleth us CHAP. XIX Concerning taking inaccessible Distances by the Plain-Table and accessible Altitudes by the Protractor Fig. 39 THe Substance of what is to be said for the first of these is gathered from the Instructions given for Plotting a Field by measuring only the Stationary-distance but to make the case more plain to an ordinary capacity Suppose the Line AC to be the unknown breadth of a River over which a Bridge of Boats is to be laid and the General that he may inform himself what store of Boats and Planks is necessary to be brought down commands me to tell him the true distance from A where he is at present to C a little Boat-house on the other side the water To satisfie his demand I plant my Table covered with a paper at A causing one to set me up a mark at B at a good distance from me along the Bank of the River the further the better if distance do not hinder sight Then having chosen a point to represent A and laid the fiducial edge upon it I direct my sights towards C and B and strike lines towards them Which done I set up a mark at A and from thence measure to B 6 Chains 32 Links and so plant my Instrument at B laying the fiducial edge to the line AB and turning about the head of the Instrument upon the Staff till through the sights I spy the Mark at A and then screw it fast In the last place I take 6 Chains 32 Links off my Scale and set it on the Line AB from A to B and laying the fiducial edge to the point B from thence direct the sights to C and draw the Line BC meeting or cutting the Line AC in C So shall the space AC measured on the Scale viz. 8 Chains 29 Links be the distance desired and because the Chain is 22 Yards long if I multiply 8. 29 by 22 the Product is 182 Yards and 38 100 of a Yard which by reduction is some little more than 13 Inches and 3 5 of an Inch. Now to take the height of a Tree Tower or Steeple by a Protractor without any Arithmetical operation hang a Plumet with a fine Silk Thread at the Center of it and hold it stedfastly with that end to your Eye where the Numbers begin then look streight along the Diameter as if you were to shoot in a Cross-bow without a Sight still removing backward and forward as there is occasion till you see the top of the Tree Tower or Steeple and the Thread at the same time fall upon 45 degrees 10 shall the distance from your Eye to the Tree Tower or Steeple measured in an horizontal or level Line together with the height of your Eye above the bottom of it be equal to the height thereof If either for convenience of sight or any other reason you think good to set the other end of the Diameter to your eye then the Thread for the tryal aforesaid must fall upon 135 Deg instead of 45. Other ways of doing this work by this Instrument or a Quadrant with the help of Trigonometry and by other Instruments I forbear at present till I write a second Part considering whose benefit is here intended CHAP. XX. Of casting up the Content of Land by a Table TO make up the number of my Chapters to an even score I shall add one at the desire of my worthy Friend Mr. S. L. before mentioned to whose Experience and Communicativeâ—ess I acknowledge my self indebted for the notion of measuring crooked Lands or Doles at the middle and both ends marking every where how the Divisions fall as is mentioned in the beginning of Chapter 17. concerning the use of a Table borrowed out of the 46th Chapter of Mr. Leybourns Compleat Surveyor second Edition Page 271 which with the use take as followeth Links R. P. 1 Â Â 00000 4 00 90000 3 24 80000 3 08 70000 2 32 60000 2 16 50000 2 00 40000 1 24 30000 1 08 20000 0 32 10000 0 16 8750 0 14 8125 0 13 7500 0 12 6875 0 11 6250 0 10 5625 0 09 5000 0 08 4375 0 07 3750 0 06 3125 0 05 2500 0 04 1875 0 03 1250 0 02 625 0 01 This Table consists of three Columns the first containing Links the second Roods or Quarters of Acres the third Perches and the use of it is thus 7. 25 5. 50 36250 3625 3625 3. 98750 Suppose a Field to be 7 Chains and 25 Links long and 5 Chains 50 Links broad these by multiplication make 398750 as here is evident whereof five Figures being cut off towards the right-hand the Figure 3 signifies Acres and the rest viz. 98750 denote parts and to reduce them into Roods and A R P 3 3 24 Â Â 14 3 3 38 Perches I first subtract from 98750 the greatest number of Links in my Table that can be subtracted from it viz. 90000 and put down for it 3 Roods 24 Perches which I find over against it in the annexed Columns and the remain being 8750 I look in the Table and find over against it 14 Perches which by addition makes 3 Roods 38 Perches So is the whole Content of the Field 3 Acres 3 Roods 38 Perches But not