the Column titled Parts of a Perch and right against it you will find 5 Feet So I say that 500 Feet is 30 Perches 5 Feet Again I would know how many Chains and Links there are in 15045 Feet First seek for 10000 and write down the Chains Links and Parts of a Link contained therein Do the like by 5000 also by 40 and 5. Lastly adding them together you have your desire Feet Chain Link Parts 10000 151 51 515 5000 75 75 757 40 0 60 606 5 0 7 575 Added make 227 95 453 Answer 227 Chains 95 Links are contained in 15045 Feet One Example more and I have done with this Table How many Perches do 10573 Feet make Feet Perches Parts 10000 606 060 500 30 303 70 4 242 3 0 181 Add 640 786 The Answer is 640 Perches and 786 / 1000 of a Perch or 13 Feet I had forgot to tell you what a Furlong is it is 40 Perches in length 8 Furlongs make 1 Mile And so much of Long Measure I shall now proceed to Square Measure Planometry or the measuring the Superficies or Planes of things as Sir Jonas Moore says is done with the Squares of such Measures as a Square Foot a Square Perch or Chain that is to say by Squares whose Sides are a Foot a Perch or Chain and the Content of any Superficies is said to be found when we know how many such Squares it containeth But before we go any farther take this Table following of Square Measure A TABLE of SQUARE MEASURE  Inch          Inch 1 Links         Links 62.726 1 Feet        Feet 144 2.295 1 Yards       Yards 1296 20.755 9 1 Pace      Pace 3600 57.381 25 2.778 1 Perch     Perch 39204 625 272.25 30.25 10.89 1 Chain    Chain 627264 10000 4356 484 174.24 16 1 Acre   Acre 6272640 100000 43560 4840 1742.4 160 10 1 Mile  Mile 4014489600 64000000 27878400 3097600 1115136 102400 6400 640 1 Mile This Table is like the former of Long Measure and the use of it is the same Example If you would know how many Square Feet are contained in one Chain look for Feet at Top and Chain on the Side and in the common Angle of meeting stands 4356 so many Square Feet are contained in one Square Chain The common Measure for Land is the Acre which by Statute is appointed to contain 160 Square Perches and it matters not in what form the Acre lye in so it contains just 160 Square Perches as in a Parallelogram 10 Perches one way and 16 another contain an Acre So does 8 one way and 20 another and 4 one way and 40 the other If then having one Side given in Perches you would know how far you must go on the Perpendicular to cut off an Acre you must divide 160 the number of Square Perches in an Acre by the given Side the Quotient is your desire As for Example the given Side is 20 Perches divide 160 by 20 the Quotient is 8 By that I know That 20 Perches one way and 8 another including a Right Angle will be the two Sides of an Acre the other two Sides must be parallel to these And here I think it convenient to insert this necessary Table shewing the Length and Bredth of an Acre in Perches Feet and Parts of a Foot But if your given Side had been in any other sort of Measure As for Instance in Yards You must then have seen how many Square Yards had been in an Acre and that Summ you must have divided by the number of your given Yards the Quotient would have answered the Question EXAMPLE If 44 Yards be given for the Bredth how many Yards shall there be in Length of the Acre Bredth Length of an Acre Perches Perches Feet 10 16 0 11 14 9 12 13 5 ½ 13 12 5 1 / 12 14 11 7 1 / 12 15 10 11 16 10 0 17 9 6 9 / 12 18 8 14 8 / 12 19 8 6 11 / 12 20 8 0 21 7 10 2 / 12 22 7 4 ½ 23 6 15 ¾ 24 6 11 25 6 6 7 / 12 26 6 2 15 / 25 27 5 15 ½ 28 5 11 ¾ 29 5 8 13 / 14 30 5 5 ½ 31 5 2 ⅔ 32 5 0 33 4 14 34 4 11 ⅔ 35 4 9 5 / 12 36 4 5 ⅔ 37 4 5 ⅔ 38 4 3 ½ 39 4 1 ⅔ 40 4 0 41 3 14 22 / 24 42 3 13 ⅓ 43 3 11 21 / 24 44 3 10 ½ 45 3 9 ⅙ First I find that an Acre contains 4840 Square Yards which I divide by 44 the Quotient is 110 for the Length of the Acre And thus knowing well how to take the Length and Bredth of one Acre you may also by the same way know how to lay down any number of Acres together of which more anon Reducing of one sort of Square Measure to another is done as before taught in Long Measure by Multiplication and Division And because Mr. Gunter's Chain is chiefly used by Surveyors I shall only instance in that and shew you how to turn any number of Chains and Links into Acres Roods and Perches Note that a Rood is the fourth part of an Acre And first mark well that 10 Square Chains make one Acre that is to say 1 Chain in Bredth and 10 in Length or 2 in Bredth and 5 in Length is an Acre as you may see by this small Table Chains Chains Links Parts of a Link Length of an Acre 1 Breadth of an Acre 10 00  2 5 00  3 3 33 333 4 2 50  5 2 00  6 1 66 666 7 1 42 285 8 1 25  9 1 11 111 And thus well weighing that 10 Chains make one Acre if any number of Chains be given you to turn into Acres you must divide them by 10 and the Quotient will be the number of Acres contained in so many Chains But this Division is abbreviated by only cutting off the last Figure as if 1590 Chains were given to turn into Acres by cutting off the last Figure 1590 there is left 159 acres which is all one as if you had divided 1590 by 10. But if Chains and Links be given you together to turn into Acres Roods and Perches first from the given Summ cut off three Figures which is two Figures for the Links and one for the Chains what 's left shall be Acres And to know how many Roods and Perches are contained in the Figures cut off multiply them by 4 from the Product cutting off the three last Figures you will have the Roods And then to know the Perches multiply the Figures cut off from the Roods by 40 from which Product cutting off again three Figures you have the Perches and the Figures cut off are thousandth Parts of a Perch
EXAMPLE 1599 Square Chains and 55 Square Links how many Acres Roods and Perches  Acres 159955   4 Answer 159 Acres 3 Rood 32 8 / 10.    Roods 3620   40  Perches 24800 On the contrary if to any number of Acres given you add a Cypher they will be turned into Chains thus 99 Acres are 990 Chains 100 Acres 1000 Chains c. The same as if you had multiplyed the Acres by 10. And if you would turn Square Chains into Square Links add four Cyphers to the end of the Chains so will 990 Chains be 9900000 Links 1000 Chains 10000000 Links all one as if you had multiplyed 990 by 10000 the number of Square Links contained in one Chain And now whereas in casting up the content of a piece of Land measured by Mr. Gunter's Chain viz. multiplying Chains and Links by Chains and Links the Product will be Square Links you must therefore from that Product cut off five Figures to find the Acres which is the same as if you divided the Product by 100000 the number of Square Links contained in one Acre then multiply the five Figures cut off by 4 and from that Product cutting off five Figures you will have the Roods Lastly multiply by 40 and take away as before 5 Figures the rest are Perches EXAMPLE Admit a Parallelogram or Long Square to be one way 5 Chains 55 Links and the other way 4 Chains 35 Links I demand the content in Acres Roods and Perches  Multiplicand 555  Multiplicator 435   2775   1665   2220 Answer 2 Acres Acres 241425   4 1 Rood Roods 165700   40 26 Perches Perches 2628000 And 28 / 100 Parts of a Perch   Lastly Because some Men chuse rather to cast up the Content of Land in Perches I will here briefly shew you how it is done which is only by dividing by 160 the number of Square Perches contained in One Acre the number of Perches given EXAMPLE Admit a Parallelogram to be in length 55 Perches and in breadth 45 Perches these two multiplied together make 2475 Perches which to turn into Acres divide by 160 the Quotient is 15 Acres and 75 Perches remaining which to turn into Roods divide by 40 the Quotient is 1 Rood and 35 Perches remaining So much is the Content of such a piece of Land viz. 15 Acres 1 Rood and 35 Perches Here follows a Table to turn Perches into Acres Roods and Perches Perches Acres Roods Perch 40 0 1 00 50 0 1 10 60 0 1 20 70 0 1 30 80 0 2 00 90 0 2 10 100 0 2 20 200 1 1 00 300 1 3 20 400 2 2 00 500 3 0 20 600 3 3 00 700 4 1 20 800 5 0 00 900 5 2 20 1000 6 1 00 2000 12 2 00 3000 18 3 00 4000 25 0 00 5000 31 1 00 6000 37 2 00 7000 43 3 00 8000 50 0 00 9000 56 1 00 10000 62 2 00 20000 125 0 00 30000 187 2 00 40000 250 0 00 50000 312 2 00 60000 375 0 00 70000 437 2 00 80000 500 0 00 90000 562 2 00 100000 625 0 00 The Use of this Table In 2475 Perches how many Acres Roods and Perches Perch Acres Rood Perch 2000 12 2 00 400 2 2 00 70 0 1 30 To which add the odd 5 Perches 0 0 05 Answer 15 1 35 CHAP. V. Of Instruments and their Use And first of the Chain THere are several sorts of Chains as Mr. Rathborne's of two Perch long Others of one Perch long some have had them 100 Feet in length But that which is most in use among Surveyors as being indeed the best is Mr. Gunter's which is 4 Pole long containing 100 Links each Link being 7 92 / 100 Inches The Description of which Chain and how to reduce it into any other Measure you have at large in the foregoing Chapter of Measures In this place I shall only give you some few Directions for the use of it in Measuring Lines Take care that they which carry the Chain deviate not from a streight Line which you may do by standing at your Instrument and looking through the Sights If you see them between you and the Mark observed they are in a streight Line otherwise not But without all this trouble they may carry the Chain true enough if he that follows the Chain always causeth him that goeth before to be in a direct line between himself and the place they are going to so as that the Foreman may always cover the Mark from him that goes behind If they swerve from the Line they will make it longer than really it is a streight Line being the nearest distance that can be between any two places Besure that they which carry the Chain mistake not a Chain either over or under in their Account for if they should the Error would be very considerable as suppose you was to measure a Field that you knew to be exactly Square and therefore need measure but one Side of it if the Chain-Carriers should mistake but one Chain and tell you the Side was but 9 Chains when it was really 10 you would make of the Field but 8 Acres and 16 Perches when it should be 10 Acres just And if in so small a Line such a great Error may arise what may be in a greater you may easily imagine But the usual way to prevent such Mistakes is to be provided with 10 small Sticks sharp at one End to stick into the Ground and let him that goes before take all into his Hand at setting out and at the End of every Chain stick down one which let him that follows take up when the 10 Sticks are done be sure they have gone 10 Chains then if the Line be longer let them change the Sticks and proceed as before keeping in Memory how often they change They may either Change at the end of 10 Chains then the hindmost Man must give the foremost all his Sticks or which is better at the end of 11 Chains and then the last Man must give the first but 9 Sticks keeping one to himself At every Change count the Sticks for fear lest you have dropt one which sometimes happens If you find the Chain too long for your use as for some Lands it is especially in America you may then take the half of the Chain and measure as before remembring still when you put down the Lines in your Field Book that you set down but the half of the Chains and the odd Links as if a Line measured by the little Chain be 11 Chains 25 Links you must set down 5 Chains 75 Links and then in plotting and casting up it will be the same as if you had measured by the whole Chain At the end of every 10 Links you may if you find it convenient have a Ring a piece of Brass or a Ragg for your more ready reckoning the odd Links When you put down
Scale And thus turning the Scale about you may first reduce all the outermost parts of the Plot. Which done you must double the lesser Plot first ½ thereof and then the other by which you may see to reduce the innermost part near the Centre But I advise rather to have a long Scale made with the Centre-hole for fixing it to the Table in about one third part of the Scale so that ⅔ of the Scale may be one way numbred with Equal Parts from the Centre-hole to the end and ⅓ part thereof numbred the other way to the end with the same number of Equal Parts tho lesser Upon this Scale may be several Lines of Equal Parts the lesser to the greater according to several Proportions Being thus provided with a Scale glew down upon a smooth Table your greater Plot to be reduced and close to it upon the same Table a Paper about the bigness whereof you would have your smaller Plot. Fix with a strong Needle the Centre of your Scale between both then turning the longer end of your Scale to any remarkable thing of your to be reduced Plot see what number of Equal Parts it cuts as suppose 100 there holding fast the Scale against 100 upon the smaller end of your Scale make a mark upon the white Paper so do round all the Plot drawing Lines and putting down all other accidents as you proceed for fear of confusion through many Marks in the end and when you have done although at first the reduced Plot will seem to be quite contrary to the other yet when you have unglewed it from the Table and turned it about you will find it to be an exact Epitome of the first You may have for this Work divers Centers made in one Scale with Equal Parts proceeding from them accordingly or you may have divers Scales according to several Proportions which is better What has been hitherto said concerning the Reducing of a Plot from a greater volume to a lesser the same is to be understood vice versa of Enlarging a Plot from a lesser to a greater But this last seldom comes in practise How to change Customary-Measure into Statute and the contrary In some Parts of England for wood-Wood-Lands and in most Parts of Ireland for all sorts of Lands they account 18 Foot to a Perch and 160 such Perches to make an Acre which is called Customary-Measure Whereas our true Measure for Land by Act of Parliament is but 160 Perches for one Acre at 16 Foot ½ to the Perch Therefore to reduce the one into the other the Rule is As the Square of one sort of Measure is to the Square of the other So is the Content of the one to the Content of the other Thus if a Field measured by a Perch of 18 Feet accounting 160 Perches to the Acre contain 100 Acres How many Acres shall the same Field contain by a Perch of 16 Feet ½ Say if the Square of 16 Feet ½ viz. 272. 25. give the Square of 18 Feet viz. 324. What shall 100 Acres Customary give Answer 119 9 / 10 of an Acre Statute Knowing the Content of a piece of Land to find out what Scale it was plotted by First by any Scale measure the Content of the Plot which done argue thus As the Content found is to the Square of the Scale I tried by So is the true Content to the Square of the true Scale it was plotted by Admit there is a Plot of a piece of Land containing 10 Acres and I measuring it by the Scale of 11 in an Inch find it to contain 12 Acres 1 / 10 of an Acre Then I say If 12 2 / 10 give for its Scale 11 What shall 100 give Answer 10. Therefore I conclude that Plot to be made by a Scale of 10 in the Inch. And so much concerning Reducing Lands CHAP. X. Instructions for Surveying a Mannor County or whole Country To Survey a Mannor observe these following Rules 1. WAlk or ride over the Mannor once or twice that you may have as it were a Map of it in your Head by which means you may the better know where to begin and proceed on with your Work. 2. If you can conveniently run round the whole Mannor with your Chain and Instrument taking all the Angles and measuring all the Lines thereof taking notice of Roads Lanes or Commons as you cross them Also minding well the Ends of all dividing Hedges where they butt upon your bound Hedges in this manner 3. Take a true Draught of all the Roads and By-Lanes in the Mannor putting down also the true Buttings of all the Field-Fences to the Road. If the Road be broad or goes through some Common or Wast Ground the best way is to measure and take the Angles on both Sides thereof but if it be a narrow Lane you may only measure along the midst thereof taking the Angles and Off-sets to the Hedges and measuring your Distances truly Also if there be any considerable River either bounds or runs through the Mannor survey that also truly as is hereafter taught 4. Make a true Plot upon Paper of all the foregoing Work and then will you have a Resemblance of the Mannor though not compleat which to make so go to all the Buttings of the Hedges and there Survey every Field distinctly plotting it accordingly every Night or rather twice a Day till you have perfected the whole Mannor 5. When thus you have plotted all the Fields according to the Buttings of the Hedges found in your first Surveys you will find that you have very nigh if not quite done the whole Work But if there be any Fields lye so within others that they are not bounded on either Side by a Road Lane nor River then you must also Survey them and place them in your Plot accordingly as they are bounded by other Fields 6. Draw a fair Draught of the whole putting down therein the Mannor-House and every other considerable House Wind-mill Water-mill Bridg Wood Coppice Cross-paths Rills Runs of Water Ponds and any other Matter Notable therein Also in the fair Draught let the Arms of the Lord of the Mannor be fairly drawn and a Compass in some wast part of the Paper also a Scale the same by which it was plotted You must also beautifie such a Draught with Colours and Cuts according as you shall see convenient Write down also in every Field the true Content thereof and if it be required the Names of the present Possessors and their Tenures by which they hold it of the Lord of the Mannor The Quality also of the Land you may take notice of as you pass over it if you have Judgment therein and it be required of you How to take the Draught of a County or Country 1. If the County or Country is in any place thereof bounded with the Sea Survey first the Seacoast thereof measuring it all along with the Chain and taking all the Angles thereof truly 2. Which done
how to cast up the Contents of any Plot of Land How to lay out New Lands How to Survey a Mannor County or Country Also how to Reduce Divide Lands Cum multis aliis The Twelfth Chapter consists wholly of Trigonometry The Thirteenth Chapter is of Heights and Distances including amongst other things how to make a Map of a River or Harbour Also how to convey Water from a Spring-head to any appointed Place or the like Lastly At the end of the Book I have a Table of Northing or Southing Easting or Westing or if you please to call it so A Table of Difference of Latitude and departure from the Meridian with Directions for the Use thereof Also a Table of Sines and Tangents and a Table of Logarithms I have taken Example from Mr. Holwell to make the Table of Sines and Tangents but to every Fifth Minute that being nigh enough in all sense and reason for the Surveyor's Use for there is no Man with the best Instrument that was ever yet made can take an Angle in the Field nigher if so nigh as to Five Minutes All which I commend to the Ingenious Reader wishing he may find Benefit thereby and desiring his favourable Reception thereof accordingly I conclude READER Your Humble Servant J. L. ADVERTISEMENT SUch Persons as have occasion for the Instruments mentioned in this Book or any other Mathematical Instruments whatsoever may be furnished with the same at Reasonable Rates by John Worgan Instrument-Maker at his Shop under the Dial of St. Dunstan's Church in Fleestreet London THE CONTENTS CHAP. I. OF Arithmetick in general Page 1 How to Extract the Square-Root by Vulgar Arithmetick Page 2 How to Extract the Square-Root by The Logarithms Page 7 CHAP. II. Geometrical Definitions Shewing what is meant by A Point Page 9 A Line ibid. An Angle ibid. A Perpendicular Page 10 A Triangle Page 11 A Square Page 12 A Parallelogram ibid. A Rhombus and Rhomboides ibid. A Trapezia ibid. An Irregular Figure Page 13 A Regular Polygon as Pentagon Hexagon c. Page 14 A Circle with what thereto belongs ibid. A Superficies Page 15 Parallel-Lines Page 16 Diagonal-Lines ibid. CHAP. III. Geometrical Problems 1. How to make a Line Perpendicular to another two ways Page 17 2. How to Raise a Perpendicular upon the end of a Line two ways Page 18 3. How from a Point assigned to let fall a Perpendicular upon a Line given Page 20 4. How to Divide a Line into any Number of Equal Parts Page 21 5. How to make an Angle equal to any other Angle given Page 22 6. How to makes Lines Parallel to each other Page 23 7. How to make a Line Parallel to another Line which must also pass through a Point assigned Page 24 8. Three Lines being given how to make thereof a Triangle ibid. 9. How to make a Triangle equal to a Triangle given Page 25 10. How to make a Square Figure Page 26 11. How to make a Long Square or Parallelogram ibid. 12. How to make a Rhomubs or Rhomboides Page 27 13. To make Regular Polygons as Pentagons Hexagons Heptagons c. Page 28 14. Three Points being given how to make a Circle whose circumference shall pass through the three given Points Page 32 15. How to make an Ellipsis or Oval several ways Page 33 16. How to Divide a given Line into two Parts which shall be in such Proportion to each other as two given Lines Page 36 17. Three Lines being given to find a Fourth in Proportion to them Page 37 CHAP. IV. Of Measures in general I. OF Long Measure shewing by what kind of Measures Land is Surveyed and also how to Reduce one sort of Long Measure into another Page 39 A General Table of Long Measure ibid. A Table shewing how many Feet and Parts of a Foot also how many Perches and Parts of a Perch are contained in any number of Chains and Links from one Link to an hundred Chains Page 41 A Table shewing how many Chains Links and Parts of a Link also how many Perches and Parts of a Perch are contained in any number of Feet from 1 Foot to 10000 Page 44 II. Of Square Measure shewing what it is and how to Reduce one sort into another Page 46 A General Table of Square Measure Page 47 A Table shewing the Length and Bredth of an Acre in Perches Feet and Parts of a Foot Page 49 A Table to turn Perches into Acres Roods and Perches Page 53 CHAP. V. Of Instruments and their Use OF the Chain Page 54 Of Instruments for the taking of an Angle in the Field Page 56 To take the quantity of an Angle in the Field by Plain Table Page 57 To take the quantity of an Angle in the Field by Semi-circle Page 58 To take the quantity of an Angle in the Field by Circumferentor c. several ways ibid. Of the Field-Book Page 61 Of the Scale with several Vses thereof and how to make a Line of Chords Page 62 c. Of the Protractor Page 68 CHAP. VI. HOw to take the Plot of a Field at one Station in any place thereof from whence you may see all the Angles by the Semi-circle and to Protract the same Page 71 How to take the Plot of the same Field at one Station by the Plain Table Page 74 How to take the Plot of the same Field at one Station by the Semi-circle either with the help of the Needle and Limb both together or by the help of the Needle only ibid. How by the Semi-circle to take the Plot of a Field at one Station in any Angle thereof from whence the other Angles may be seen and to Protract the same Page 76 How to take the Plot of a Field at two Stations provided from either Station you may see every Angle and measuring only the Stationary Distance Also to Protract the same Page 79 82 c. How to take the Plot of a Field at two Stations when the Field is so Irregular that from one Station you cannot see all the Angles Page 83 How to take the Plot of a Field at one Station in an Angle so that from that Angle you may see all the other Angles by measuring round about the said Field Page 86 How to take the Plot of the foregoing Field by measuring one Line only and taking Observations at every Angle Page 88 How to take the Plot of a large Field or Wood by measuring round the same and taking Observations at every Angle by the Semi-circle Page 90 When you have Surveyed after this manner how to know before you go out of the Field whether you have wrought true or not Page 94 Directions how to Measure Parallel to a Hedge when you cannot go in the Hedge it self And also in such case how to take your Angles Page 95 How to take the Plot of a Field or Wood by observing near every Angle and Measuring the Distance between the Marks of Observation by taking in every Line two
Off-sets to the Hedge Page 97 An easier way to do the same by taking only one Square and many Off-sets Page 99 How by the help of the Needle to take the Plot of a large Wood by going round the same and making use of that division of the Card that is numbred with four 90 s. or Quadrants and two ways how to Protract the same and examin the Work Page 103 c. How by the Chain only to take an Angle in the Field Page 111 How by the Chain only to Survey a Field by going round the same Page 113 The Common Way taught by the Surveyors for taking the Plot of the foregoing Field Page 116 How to take the Plot of a Field at one Station in any part thereof from whence all the Angles may be seen by the Chain only Page 119 CHAP. VII How to cast up the Contents of a Plot of Land. OF the Square and Parallelogram Page 122 Of Triangles Page 123 To find the Content of a Trapezia Page 125 How to find the Content of an Irregular Plot consisting of many Sides and Angles Page 127 How to find the Content of a Circle or any Portion thereof Page 128 How to find the Content of an Oval Page 130 How to find the Content of Regular Polygons c. Page 131 CHAP. VIII Of Laying out New Lands A Certain quantity of Acres being given how to lay out the same in a Square Figure Page 132 How to lay out any given quantity of Acres in a Parallelogram whereof one Side is given Page 133 How to lay out a Parallelogram that shall be four five six or seven times c. longer than broad ibid. How to make a Triangle that shall contain any number of Acres being confined to a given Base Page 134 How to find the Length of the Diameter of a Circle that shall contain any number of Acres required Page 136 CHAP. IX Of Reduction HOw to Reduce a large Plot of Land or Map into a lesser compass according to any given Proportion Or e contra how to enlarge one three several ways Page 137 How to change Customary Measure into Statute contra Page 141 Knowing the Content of a piece of Land to find out what Scale it was Plotted by ibid. CHAP. X. Instructions for Surveying A Mannor County or Country Page 142 CHAP. XI Of Dividing Lands HOw to Divide a Triangular piece of Land several ways Page 146 How to Reduce a Trapezia into a Triangle by Lines drawn from any Angle thereof Also how to Reduce a Trapezia into a Triangle by Lines drawn from a Point assigned in any Side thereof Page 149 How to Reduce a Five-sided Figure into a Triangle and to Divide the same Page 151 How to Divide an Irregular Plot of any number of Sides according to any given Proportion by a streight Line through it Page 153 An easier way to do the same with two Examples Page 155 How to Divide a Circle according to any Proportion by a Line Concentrick with the Circumference Page 158 CHAP. XII Trigonometry 159 c. THis Chapter shews first the Vse of the Tables of Sines and Tangents And Secondly contains Ten Cases for the Mensuration of Right-lin'd Triangles very necessary to be understood by the Surveyor CHAP. XIII Of Heights and Distances HOw to take the Heighth of a Tower Steeple Tree or any such thing Page 180 How to take the Heighth of a Tower c. when you cannot come nigh the foot thereof Page 183 How to take the Heighth of a Tower c. when the Ground either riseth or falls Page 184 How to take Distances by an Example of a River Page 185 How to take the Horizontal Line of a Hill Page 189 How to take the Rocks or Sands at the Entrance of a River or Harbour and to Plot the same Page 191 How to know whether Water may be made to run from a Spring-Head to any appointed Place Page 194 A Table of Northing or Southing Easting or Westing A Table of Logarithms A Table of Artificial Sines and Tangents A Catalogue of Books Printed for and Sold by John Taylor at the Ship in S. Paul's Church-Yard 1. THe Travels of Monsieur de Thevinot into the Levant in Three Parts viz. I. Into Turkie II. Persia III. The East-Indies New done out of French in Folio 2. A Free Enquiry into the Vulgarly Receiv'd Notion of Nature made in an Essay Address'd to a Friend By the Honourable Robert Boyle Esq Fellow of the Royal Society The same is also in Latin for the Benefit of Foreigners 3. The Martyrdom of Theodora and of Didymus by a Person of Honour 4. The Declamations of Quintilian being an Exercitation or Praxis upon his Twelve Books concerning the Institution of an Orator Translated from the Oxford-Theatre Edition into English by a Learned and Ingenious Hand with the Approbation of several Eminent School-Masters in the City of London 5. England's Happiness in a Lineal Succession and the Deplorable Miseries which ever attended Doubtful Titles to the Crown Historically demonstrated from the Wars between the Two Houses of York and Lancaster 6. Academia Scientiarum Or The Academy of Sciences Being a Short and Easie Introduction to the Knowledg of the Liberal Arts and Sciences with the Names of those Famous Authors that have written on every particular Science In Latin and English By D. Abercromby M. D. 7. Publick Devotion and the Common-Service of the Church of England Justified and Recommended to all Honest and Well-meaning however Prejudic'd Dissenters By a Lover of his Country and the Protestant Religion 8. The Best Exercise To which is added a Letter to a Person of Quality concerning the Holy Lives of the Primitive Christians By Anthony Horneck Preacher at the Savoy 9. The Mother's Blessing Or The Godly Counsel of a Gentlewoman not long since Deceas'd left behind for her Children By Mrs. Dorothy Leigh 10. The Inchanted Lover Or The Amours of Narcissus and Aurelia a Novel By Peter Bellon Author of the Pilgrim 11. Good and Solid Reasons why a Protestant should not turn Papist in a Letter to a Romish Priest 12. Curious Enquiries being Six brief Discourses viz. I. Of Longitude II. the Tricks of Astrological Quacks III. of the Depth of the Sea IV. of Tobacco V. Of Europes being too full of People VI. The Various Opinions concerning the Time of Keeping the Sabbath 13. The Works of Dr. Thomas Comber in Four Parts Folio 14. Weekly Memorials for the Ingenious or an Account of Books lately set forth in several Languages with other Accounts relating to Arts and Sciences 15. Legrand's Historia Sacra 16. Poetical History by Gualtruchius 17. London Dispensatory by Nicholas Culpeper 18. Father Simon 's Critical History of the Eastern Nations 19. History of the Progress of Ecclesiastical Revenues 26. The Several Ways of Resolving Faith by the Controvertists of the Church of England and the Church of Reme GEODAESIA OR THE ART OF Measuring Land c. CHAP. I. Of Arithmetick IT
from the Number thus increased extract the Root which shall be the Side of the proposed Square EXAMPLE Suppose the Number given be 100 Acres which I am to lay out in a Square Figure I joyn to the 100 5 Cyphers and then it is Square Links the Root of which is 3162 nearest or 31 Chains 62 Links the length of one Side of the Square Again If I were to cut out of a Corn-Field one Square Acre I add to one five Cyphers and then is it the Root of which is 3 Chains 16 Links and something more for the Side of that Acre How to lay out any given Quantity of Acres in a Parallelogram whereof one Side is given Turn first the Acres into Links by adding as before 5 Cyphers that number thus increased divide by the given Side the Quotient will be the other Side EXAMPLE It is required to lay out 100 Acres in a Parallelogram one Side of which shall be 20 Chains 00 Links first to the 100 Acres I add 5 Cyphers and it is 100,00000 which I divide by 20 Chains 00 Links the Quotient is 50 Chains 00 Links for the other Side of the Parallelogram How to lay out a Parallelogram that shall be 4 5 6 or 7 c. times longer than it is broad In Carolina all Lands lying by the Sides of Rivers except Seignories or Baronies are or ought by Order of the Lord's Proprietors to be thus laid out To do which first as above taught turn the given quantity of Acres into Links by annexing 5 Cyphers which summ divide by the number given for the Proportion between the length and bredth as 4 5 6 7 c. the Root of the Quotient will shew the shortest Side of such a Parallelogram EXAMPLE Admit it were required of me to lay out 100 Acres in a Parallelogram that should be five times as long as broad First to the 100 Acres I add 5 Cyphers and it makes 100,00000 which summ I divide by 5 the Quotient is 2000000 the Root of which is nearest 14 Chains 14 Links and that I say shall be the short Side of such a Parallelogram and by multiplying that 1414 by 5 shews me the longest Side thereof to be 70 Chains 70 Links How to make a Triangle that shall contain any number of Acres being confined to a certain Base Double the given number of Acres to which annexing first five Ciphers divide by the Base the Quotient will be the length of the Perpendicular EXAMPLE Upon a Base given that is in length 40 Chains 00 Links I am to make a Triangle that shall contain 100 Acres First I double the 100 Acres and annexing five Ciphers thereto it makes 200,00000 which I divide by 40 Chains 00 Links the limited Base the Quotient is 50 Chains 00 Links for the height of the Perpendicular As in this Figure AB is the given Base 40 upon any part of which Base I set the Perpendicular 50 as at C then the Perpendicular is CD Therefore I draw the Lines DA DB which makes the Triangle DAB to contain just 100 Acres as required Or if I had set the Perpendicular at E then would EF have been the Perpendicular 50 and by drawing the Lines FA FB I should have made the Triangle FAB containing 100 Acres the same as DAB If you consider this well when you are laying out a new piece of Land of any given Content in America or elsewhere although you meet in your way with 100 Lines and Angles yet you may by making a Triangle to the first Station you began at cut off any quantity required How to find the Length of the Diameter of a Circle which shall contain any number of Acres required Say as 11 is to 14 so will the number of Acres given be to the Square of the Diameter of the Circle required EXAMPLE What is the Length of the Diameter of a Circle whose Superficial Content shall be 100 Acres Add five Cyphers to the 100 and it makes 100,00000 Links which multiply by 14 facit 140000000 which divided by 11 gives for Quotient 12727272 the Root of which is 35 Chains 67 Links and better almost 68 Links And so much shall be the Diameter of the required Circle I might add many more Examples of this nature as how to make Ovals Regular Polygons and the like that should contain any assigned quantity of Land. But because such things are meerly for Speculation and seldom or never come in Practise I at present omit them CHAP. IX Of Reduction How to Reduce a large Plot of Land or Map into a lesser compass according to any given Proportion or e contra how to Enlarge one THe best way to do this is if your Plot be not over-large to plat it over again by a smaller Scale But if it be large as a Map of a County or the like the only way is to compass in the Plot first with one great Square and afterwards to divide that into as many little Squares as you shall see convenient Also make the same number of little Squares upon a fair piece of Paper by a lesser Scale according to the Proportion given This done see in what Square and part of the same Square any remarkable accident falls and accordingly put it down in your lesser Squares and that you may not mistake it is a good way to number your Squares I cannot make it plainer than by giving you the following Example where the Plot ABCD made by a Scale of 10 Chains in an Inch is reduced into the Plot EFGH of 30 Chains in an Inch. There are several other ways taught by Surveyors for reducing Plots or Maps as Mr. Rathboxn and after him Mr. Holwell adviseth to make use of a Scale or Ruler having a Centre-hole at one end through which to fasten it down on a Table so that it may play freely round and numbred from the Centre-end to the other with Lines of Equal Parts The Use of which is thus Lay down upon a smooth Table the Map or Plot that you would reduce and glew it with Mouth-glew fast to the Table at the four corners thereof Then taking a fair piece of Paper about the bigness that you would have your reduced Plot to be of and lay that down upon the other the middle of the last about the middle of the first This done lay the Centre of your Reducing Scale near the Centre of the white Paper and there with a Needle through the Centre make it fast yet so that it may play easily round the Needle Then moving your Scale to any remarkable thing of the first Plot as an Angle a House the bent of a River or the like See against how many Equal Parts of the Scale it stands as suppose 100 then taking the ⅓ the ¼ the ⅕ or any other number thereof according to the Proportion you would have the reduced Plot to bear and make a mark upon the white Paper against 50 25 33 c. of the same
of your Compasses in g extend the other to either of the Points given and describe the Circle A B C. Note the Centre of a Triangle is found the same way PROB. xv How to make an Ellipsis or Oval several ways Fig. 1. Make three Circles whose Diameters may be in a streight Line as AB Cross that Line with another Perpendicular to it at the Centre of the middle Circle as cd draw the Lines ce ch dg df Set one foot of the Compasses in D and extend the other to g describing the part of the Ellepsis gf with the sameextent setting foot one in c describe the other part he The two Ends are made by parts of the two outermost small Circles as you see fe gh Fig. 2. Draw two small Circles whose circumference may only touch each other Then taking the distance between their Centers or either of their Diameters set one foot of your Compasses in either of their Centres as that marked 2 and with the other make an Arch at a also at b then moving your Compasses to the Centre of the other Circle cross the said Arches at a and b which Crosses let be the Centres of two other Circles of equal bigness with the first Then through the Centres of all the Circles draw the Lines AB CD EH FG which done place one foot of the Compasses in the Centre of the Circle I and extend the other to C describing the Arch of the Ellipsis CE The same you must do at 2 to describe the part BH and then is your Ellipsis finished Fig. 3. This needs no Description it being so like the two former Figures and easier than either of them Here Note that you may make the Ovals 1 and 3 of any determined length for in the length of the first there is four Semi-diameters of the small Circles and in the last but three If therefore any Line was given you of which length an Oval was required you must take in with your Compasses the fourth part of the Line to make the the Oval Fig. 1. and the third part to make the Oval Fig. 3 and with that extent you must describe the small Circles The Breadth will be always proportional to the Length But if the Breadth be given you take in also the fourth part thereof and make the Oval Fig. 2. Fig. 4. This Ellipsis is to be made having Length and Breadth both given Let AB be the Length CD the Breadth of a required Oval First lay down the Line AB equal to the given length and cross it in the middle with the Perpendicular CD equal to the given Breadth Secondly take in half the Line AB with your Compasses viz. AE or BE set one foot in C and make two marks upon the Line AB viz. f and g also with the same extent set one foot in D and cross the former marks at f and g. Thirdly at the Points f and g fix two Pins or if it be a Garden-plat or the like two strong Sticks Then putting a Line about them make fast the two ends at such an exact length that stretching by the two Pins the bent of the Line may exactly touch A or B or C or D or h as in this Diagram it does at h so moving the Line still round it will describe an exact Oval PROB. xvi How to divide a given Line into two Equal Parts which may be in such Proportion to each other as two given Lines Let AB be the given Line to be divided in such Proportion as the line C is to the line D. First from A draw a Line at pleasure as AE then taking with your Compasses the line C set it off from A towards E which will fall at F Also take the line D and set off from F to E. Secondly draw the line EB and from F make a line parallel to eb as FG which shall intersect the given line AB in the Proportional Point required viz at G making AG and GB in like proportion to each other as CC and DD. Example by Arithmetick The line CC is 60 Feet Perches or any thing else the line DD is 40 the line AB is 50 which is required to be divided in such proportion as 60 to 40. First add the two lines C and D together and they make 100 Then say if 100 the whole give 60 for its greatest part what shall 50 the whole line AB give for its greatest Proportional part Multiply 50 by 60 it makes 3000 which divided by 100 produces 30 for the longest part which 30 taken from 50 leaves 20 for the shortest part as therefore 60 is to 40 so is 30 to 20. PROB. xvii Three Lines being given to find a Fourth in Proportion to them Let ABC be the three Lines given and it is required to find a fourth Line which may be in such proportion to C as B is to A A 14 B 18 C 21 which is no more but performing the Rule of Three in Lines As if we should say if A 14 give B 18 what shall C 21 give Answer 27. But to perform the same Geometrically work thus And here for a while I shall leave these Problems till I come to shew you how to divide any piece of Land and to lay out any piece of a given quantity of Acres into any Form or Figure required And in the mean time I shall shew you what is necessary to be known CHAP. IV. Of Measures ANd first of Long Measures which are either Inches Feet Yards Perches Chains c. Note that twelve Inches make one Foot three Feet one Yard five Yards and a half one Pole or Perch four Perches one Chain of Gunter's eighty Chains one Mile But if you would bring one sort of Measure into another you must work by Multiplication or Division As for example Suppose you would know how many Inches are contained in twenty Yards First reduce the Yards into Feet by multiplying them by 3 because 3 Feet make one Yard the Product is 60 which multiplyed by 12 the number of Inches in one Foot gives 720 and so many Inches are contained in 20 Yards Length On the contrary if you would have known how many Yards there are in 720 Inches you must first divide 720 by 12 the Quotient is 60 Feet that again divided by 3 the Quotient is 20 Yards The like you must do with any other Measure as Perches Chains c. of which more by and by Long Link Foot Yard Perch Chain Mile Inches 7.92 12 36 198 792 63360 Â Links 1.515 4.56 25 100 8000 Â Â Feet 3 16.5 66 5280 Â Â Â Yards 5.5 22 1760 Â Â Â Â Perch 4 320 Â Â Â Â Â Chain 80 See this Table of Long Measure annexed the use whereof is very easie If you would know how many Feet in Length go to make one Chain look for Chain at Top and at the Left-hand for Feet against which in the common Angle of
the former Proportion How to reduce an Irregular Five-Sided Figure into a Triangle and to divide the same Let ABCDE be the Five-Sided Figure to reduce which into a Triangle draw the Lines AC AD and parallel thereto BF EG extending the Base from C to F and from D to G then draw the Lines AF AG which will make the Triangle AFG equal to the Five Sided-Figure If this was to be divided into two equal Parts take the half of the Base of the Triangle which is FH and from H draw the Line HA which divides the Figure ABCDE into two equal Parts The like you may do for any other Proportion If in dividing the Plot of a Field there be Outward Angles you may change them after the following manner Suppose ABCDE be the Plot of a Field and B the outward Angle Draw the Line CA and parallel thereto the Line BF Lastly The Line CF shall be of as much force as the Lines CB and BA So is that five-sided Figure having one outward Angle reduced into a four-sided Figure or Trapezia which you may again reduce into a Triangle as has been before taught How to Divide an Irregular Plot of any number of Sides according to any given Proportion by a streight Line through it ABCDEFGHI is a Field to be divided between two Men in equal Halfs by a streight Line proceeding from A. First consider how to divide the Field into five-sided Figures and Trapezias that you may the better reduce it into Triangles As by drawing the Line KL you cut off the five-sided Figure ABCHI which reduce into the Triangle AKL and measuring half the Base thereof which will fall at Q draw the Line QA Secondly Draw the Line MN and from the Point Q reduce the Trapezia CDGH into the Triangle MNQ which again divide into Halfs and draw the Line QR Thirdly From the Point R reduce the Trapezia DEFG into the Triangle ROP and taking half the Base thereof draw the Line RS and then have you divided this Irregular Figure into two Equal Parts by the three Lines AQ QR RS. Fourthly Draw the Line AR also QT parallel thereto Draw also AT and then have you turned two of the Lines into one Fifthly From T draw the Line T S and parallel thereto the Line RV Draw also TV. Then is your Figure divided into two Equal Parts by the two Lines AT and TV. Lastly Draw the Line AV and parallel thereto TW Draw also AW which will cut the Figure into two Equal Parts by a streight Line as was required You may if you please divide such a Figure all into Triangles and then divide each Triangle from the Point where the Division of the last fell and then will your Figure be divided by a crooked Line which you may bring into a streight one as above This above is a good way of Dividing Lands but Surveyors seldom take so much pains about it I shall therefore shew you how commonly they abbreviate their Work and is indeed An easie way of Dividing Lands Admit the following Figure ABCDE contain 46 Acres to be divided in Halfs between two Men by a Line proceeding from A. Draw first a Line by guess through the Figure as the Line AF. Then cast up the Content of either Half and see what it wants or what it is more than the true Half should be As for Example I cast up the Content of AEG and find it to be but 15 Acres whereas the true Half is 23 Acres 8 Acres being in the part ABCDG more than AEG Therefore I make a Triangle containing 8 Acres and add it to AEG as the Triangle AGI then the Line AI parts the Figure into equal Halfs If it had been required to have set off the Perpendicular the other way you must still have made the end of it but just touch the Line ED as LK does For the Triangle AKG is equal to the Triangle AGI each 8 Acres And thus you may divide any piece of Land of never so many Sides and Angles according to any Proportion by streight Lines through it with as much certainty and more ease than the former way Mark you might also have drawn the Line AD and measured the Triangle AGD and afterwards have divided the Base GD according to Proportion in the Point I which I will make more plain in this following Example Suppose the following Field containing 27 Acres is to be divided between three Men each to have Nine Acres and the Lines of Division to run from a Pond in the Field so that every one may have the benefit of the Water without going over one another's Land. From ☉ to any Angle draw a Line for the first Division-line as ☉ A. Then consider that the first Angle A ☉ B is but 674 Perches and the second B ☉ C 390 both together but 1064 Perches less by 376 than 1440 one Man's Portion You must therefore cut off from the third Angle C ☉ D 376 Perches for the first Man's Dividing-line which thus you may do The Base DC is 18 Chains the Content of the Triangle 1238 Perches Say then if 1238 Perches give Base 18 Chains 00 Links What shall 376 Perches give Answer 5 Chains 45 Links which set from C to F and drawing the Line ☉ F you have the first Man's part viz. A ☉ F. Secondly See what remains of the Triangle C ☉ D 376 being taken out and you will find it to be 862 Perches which is less by 578 than 1440. Therefore from the Triangle D ☉ E cut off 578 Perches and the point of Division will fall in G. Draw the Line ☉ G which with ☉ A and ☉ F divides the Figure into three Equal Parts How to Divide a Circle according to any Proportion by a Line Concentrick with the first All Circles are in Proportion to one another as the Squares of their Diameters therefore if you divide the Square of Diameter or Semi-diameter and extract the Root you will have your desire EXAMPLE Let ABCD be a Circle to be equally divided between two Men. The Diameter thereof is 2 Chains The Semi-diameter 1 Chain or 100 Links The Square thereof 10100 Half the Square The Root of the Half 71 Links which take from your Scale and upon the same Centre draw the Circle GEHF which divides the Circle ABCD into Equal Parts CHAP. XII Trigonometry Or the Mensuration of Right Lined Triangles THe Use of the Table of Logarithm Numbers I have shewed you in Chap. I. concerning the Extraction of the Square Root Here follows The use of the Tables of Sines and Tangents Any Angle being given in Degrees and Minutes how to find the Sine or Tangent thereof Let 25 Degrees 10 Minutes be given to find the Sine and Tangent thereof first in the Table of Sines and Tangents at the Head thereof seek for 25 and having found it look down the first Column on the Left-hand under M for the 10 Minutes and right against