GEODAESIA: OR, THE ART OF SURVEYING AND Measuring of Land, Made EASIE. SHOWING, By Plain and Practical Rules, How to Survey, Protract, Cast up, Reduce or Divide any Piece of Land whatsoever; with New Tables for the ease of the Surveyor in Reducing the Measures of Land. MOREOVER, A more Facile and Sure Way of Surveying by the Chain, than has hitherto been Taught. AS ALSO, How to Layout New Lands in America, or elsewhere: And how to make a Perfect Map of a River's Mouth or Harbour; with several other Things never yet Published in our Language. By JOHN LOVE, Philomath. Oculus mentis excoecatus & defossus, per sola Mathematica studia instauratur & excitatur, ut res ipsas cernere queat, & a rerum nudis simulackris ad veritatem, à tenebris ad lucem, à materiae spelunca & vinculis, ad incorporeas, & invisibiles essentias sese erigere. Plato de Repub. LONDON: Printed for JOHN TAYLOR, at the Ship in S. Paul's Churchyard, MDCLXXXVIII. TO THE HONOURABLE ROBERT BOIL, Esq A MOST WORTHY PROMOTER OF ALL Truly Ingenious Knowledge, And one of the MEMBERS OF THE Royal Society: This Small TREATISE of GEODAESIA, Is humbly Dedicated, by the Meanest of his Servants, the Author, J. L. Licenced, Feb. 16. 1687/8 ROB. MIDGLEY. THE PREFACE TO THE READER. WHat would be more ridiculous, than for me to go about to Praise an Art that all Mankind know they cannot live Peaceably without? It is near hand as ancient (no doubt on't) as the World: For how could Men set down to Plant, without knowing some Distinction and Bounds of their Land? But (Necessity being the Mother of Invention) we find the Egyptians, by reason of the Nyles overflowing, which either washed away all their Bound-Marks, or covered them over with Mud, brought this Measuring of Land first into an Art, and Honoured much the Professors of it. The great Usefulness, as well as the pleasant and delightful Study, and wholesome Exercise of which, tempted so many to apply themselves thereto, that at length in Egypt (as in Bermudas now) every Rustic could Measure his own Land. From Egypt, this Art was brought into Greece, by Thales, and was for a long time called Geometry; but that being too comprehensive a Name for the Mensuration of a Superficies only, it was afterwards called Geodaesia; and what Honour it still continued to have among the Ancients, needs no better Proof than Plato's 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉. And not only Plato, but most, if not all the Learned Men of those times, refused to admit any into their Schools, that had not been first entered in the Mathematics, especially Geometry and Arithmetic. And we may see, the great Monuments of Learning built on these Foundations, continuing unshaken to this day, sufficiently demonstrate the Wisdom of the Designers, in choosing Geometry for their Ground-Plot. Since which the Romans have had such an Opinion of this sort of Learning, that they concluded that Man to be incapable of Commanding a Legion, that had not at least so much Geometry in him, as to know how to Measure a Field. Nor did they indeed either respect Priest or Physician, that had not some Insight in the Mathematics. Nor can we complain of any failure of Respect given to this Excellent Science, by our Modern Worthies, many Noblemen, Clergymen, and Gentlemen affecting the Study thereof: So that we may safely say, none but Unadvised Men ever did, or do now speak evil of it. Besides the many Profits this Art brings to Man, it is a Study so pleasant, and affords such Wholesome and Innocent Exercise, that we seldom find a Man that has once entered himself into the Study of Geometry or Geodaesia, can ever after wholly lay it aside; so natural it is to the Minds of Men, so pleasingly insinuating, that the Pythagoreans thought the Mathematics to be only a Reminiscience, or calling again to mind things formerly learned. But no longer to light Candles to see the Sun by, let me come to my business, which is to speak something concerning the following Book; and if you ask, why I writ a Book of this nature, since we have so many very good ones already in our own Language? I answer, because I cannot find in those Books, many things, of great consequence, to be understood by the Surveyor. I have seen Young men, in America, often nonplused so, that their Books would not help them forward, particularly in Carolina, about Laying out Lands, when a certain quantity of Acres has been given to be laid out five or six times as broad as long. This I know is to be laughed at by a Mathematician; yet to such as have no more of this Learning, than to know how to Measure a Field, it seems a Difficult Question: And to what Book already Printed of Surveying shall they repair to, to be resolved? Also concerning the Extraction of the Square Root; I wonder that it has been so much neglected by the Teachers of this Art, it being a Rule of such absolute necessity for the Surveyor to be acquainted with. I have taught it here as plainly as I could devise, and that according to the Old way, verily believing it to be the Best, using fewer Figures, and once well learned, charging less the Memory than the other way. Moreover, the Sounding the Entrance of a River, or Harbour, is a Matter of great Import, not only to Seamen, but to all such as Seamen live by; I have therefore done my endeavour to teach the Young Artist how to do it, and draw a fair Draught thereof. Many more things have I added, such as I thought to be New, and Wanting; for which I refer you to the Book itself. As for the Method, I have chose that which I thought to be the easiest for a Learner; advising him first to learn some Arithmetic, and after teaching him how to Extract the Square-Root. But I would not have any Neophyte discouraged, if he find the First Chapter too hard for him; for let him rather skip it, and go to the Second and Third Chapters, which he will find so easy and delightful, that I am persuaded he will be encouraged to conquer the Difficulty of learning that one Rule in the First Chapter. From Arithmetic, I have proceeded on to teach so much Geometry as the Art of Surveying requires. In the next place I have showed by what Measures Land is Surveyed, and made several Tables for the Reducing one sort of Measure into another. From which I come to the Description of Instruments, and how to Use them; wherein I have chief insisted on the Semicircle, it being the best that I know of. The Sixth Chapter teacheth how to apply all the foregoing Matters together, in the Practical Surveying of any Field, Wood, etc. divers Ways, by divers Instruments; and how to lay down the same upon Paper. Also at the end of this Chapter I have largely insisted on, and by new and easy ways, taught Surveying by the Chain only. The Seventh, Eighth, Ninth, Tenth and Eleventh Chapters, teach how to cast up the Contents of any Plot of Land; How to lay out New Lands; How to Survey a Manor, 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 Springhead, 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. holwel 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 Arithmetic in general Page 1 How to Extract the Square-Root, by Vulgar Arithmetic Page 2 How to Extract the Square-Root, by The Logarithms Page 7 CHAP. II. Geometrical Definitions. Showing 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, etc. 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, etc. 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, showing 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 showing 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 showing 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, showing what it is; and how to Reduce one sort into another Page 46 A General Table of Square Measure Page 47 A Table, showing the Length and Breadth 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 Semicircle Page 58 To take the quantity of an Angle in the Field by Circumferentor, etc. several ways ibid. Of the Field-Book Page 61 Of the Scale, with several Uses thereof; and how to make a Line of Chords Page 62, etc. 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 Semicircle; 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 Semicircle, either with the help of the Needle and Limb both together, or by the help of the Needle only ibid. How, by the Semicircle, 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, etc. 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 Angel's 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 Semicircle 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 itself: And also in such case, how to take your Angel's 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 numbered with four 90 s. or Quadrants; and two ways how to Protract the same, and examine the Work Page 103, etc. 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, etc. 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, etc. 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 Manor, 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 straight 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 Concentric with the Circumference Page 158 CHAP. XII. Trigonometry 159, etc. THis Chapter shows first the Use of the Tables of Sines and Tangents. And Secondly, contains Ten Cases for the Mensuration of Right-lined Triangles, very necessary to be understood by the Surveyor. CHAP. XIII. Of Heights and Distances. HOw to take the Height of a Tower, Steeple, Tree, or any such thing. Page 180 How to take the Height of a Tower, etc. when you cannot come nigh the foot thereof Page 183 How to take the Height of a Tower, etc. 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 Springhead, to any appointed Place Page 194 A Table of Northing or Southing, Easting or Westing. A Table of Logarithms. A Table of Artificial Sins and Tangents. A Catalogue of Books Printed for and Sold by John Tailor at the Ship in S. Paul's Churchyard. 1. THe Travels of Monsieur de Thevinot into the Levant; in Three Parts, viz. I. Into Turkey, II. Persia, III. The East-Indies; New done out of French, in Folio. 2. A Free Enquiry into the Vulgarly Received Notion of Nature; made in an Essay, Addressed to a Friend. By the Honourable Robert boil, 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 Schoolmasters 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 Easy Introduction to the Knowledge 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. Public Devotion, and the Common-Service of the Church of England Justified, and Recommended to all Honest and Well-meaning (however Prejudiced) 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 Deceased, left behind for her Children. By Mrs. Dorothy Leigh. 10. The Enchanted 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 Inquiries, 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 Europe's 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, etc. CHAP. I. Of Arithmetic. IT is very necessary for him that intends to be an Artist in the Measuring of Land, to begin with Arithmetic, as the Groundwork and Foundation of all Arts and Sciences Mathematical: and at least not to be ignorant of the five first and Principal Rules thereof, viz. Numeration, Addition, Substraction, Multiplication and Division: Which supposing every Person, that applies himself to the Study of this Art to be skilled in; or if not, referring him to Books or Masters, every where to be found, to learn: I shall name a sixth Rule, as necessary, (if not more) to be understood by the Learner; which is the Extraction of the Square Root; without which (though seldom mentioned by Surveyors in their Writings) a Man can never attain to a competent Knowledge in the Art: I shall not therefore think it unworthy my Pains (though perhaps other Men have better done it before me) to show you easily and briefly how to do it. How to Extract the Square Root. In the first place it is convenient to tell you what this Square Root is: It is to find out of any Number propounded a lesser Number, which lesser Number being multiplied in itself, may produce the Number propounded. As for Example, suppose 81 be a Number given me, I say 9 is the Root of it, because 9 multiplied in itself, viz. 9 times 9, is 81. Now 8 could not be the root, for 8 times 8 is but 64: nor could 10, for 10 times 10 is 100, therefore I say 9 must needs be the Root, because multiplied in itself, it makes neither more nor less, but just the Number propounded, viz. 81. mathematical figure Again, suppose 16 be the number given, I say the Root of it is 4, because 4 multiplied in itself makes 16. For your better understanding see this Figure, which is a great Square, containing 16 little Squares; any side of which great Square contains 4 little Squares: which is called the Square Root. mathematical figure Roots 1 2 3 4 5 6 7 8 9 Squares 1 4 9 16 25 36 49 64 81 Here you see the Root of 25 is 5, the Root of 64 is 8, and so of the rest. So far as 100 in whole Numbers, your Memory will serve you to find the Root; but if the Number propounded, whose Root you are to search out, exceed 100, than put a Point over the first Figure on the Right-hand, which is the place of Unites, and so proceeding to the Lefthand, miss the second Figure, and put a Point over the third, then missing the fourth, Point the fifth; and so (if there be never so many Figures in the Number) proceed on to the end, pointing every other Figure, as you may see here, and so many Points as there are, of so many Figures your Root will consist, which is very material to remember: Then begin at the first Figure on the Lefthand that has a Point over it, which will always be the first or second Figure, and search out the Root of that one Figure, or both joined together if there be two, and when you have found it, or the nighest less to it, which you may easily do by the Table above, or your own memory, draw a little crooked Line, as in Division, and there set it down. For Example, Let 144 be the Number whose Root I am to find; I set it down, and prick the Figures thus: Then going to the first Figure on the Lefthand, that has a Price over it, which is 1, and see what the Root of it is, which is 1 also; I therefore draw a crooked Line, as in the Margin, and set down 1 in the Quotient, then if 1 admitted of any Multiplication, I should multiply it by itself, but since once 1 is but 1, I subtract it out of the first pricked Figure on the Lefthand, and there remains 0, so that I cancel that first Figure, as having wholly done with it: If any thing had remained after the Substraction, I should have put the remainder over it. The next thing to be done, is to double what is already in the Quotient, which makes 2, which 2 I writ down under the next Figure, viz. 4, which has no Point over it, and then see how oft I can have 2 in 4: Answer, twice; I therefore set down 2 in the Quotient, and 2 likewise under the next pointed Figure, which in this Example is 4, than that 22 which stands under the 44 must be multiplied by the● in the Quotient, whose Product is 44, which substracted out of 44, there remains 0: But you may multiply and subtract together thus, twice 2 is 4, which I take out of 4, and there remains 0, than I cancel the first 4 and 2 to the Lefthand, as having done with them; then again, twice 2 is 4, which taken out of 4 leaves 0, and then I cancel the last 4 and 2, and the Question is answered, for there is 12 in the Quotient, which is the Root of 144, which may easily be proved by multiplying 12 by 12. Take another Example: Let the sum be First see what the Root of 5 is, which is 2, and place it in the Quotient, and under the first pointed Figure both, as you see here, then say two times 2 is 4, which taken out of 5, there remains one, and so have you done with the first Point. Next double the Quotient, which makes 4, and place it as you see here, under the Figure void of a Point, then see how many times 4 you can have in 14, answer 3 times, which 3 place both in the Quotient, and under the next pointed Figure, which is 7; then multiply and subtract, saying three times 4 is 12, which taken out of 14 leaves 2, which 2 write over the 4, and cancel both the 4 and the 1, as you do in Division: And three times 3 is 9, which taken out of 27, rests 18; which writ over head, and cancel what Figures you have done with, no otherwise than in Division, and so have you done with the first two Points. Now for the third pointed Figure, or if there were never so many more of them, they are done altogether as the second: viz. Double again your Quotient, it makes 46, which put down as you see here, always observing this Rule, That the last Figure of the doubled Quotient, I mean that in the place of Unites, stand under the next, void of Points: And those of your Left hand of him, viz in the places of Ten or Hundreds, in order before him, as you do in Division, as you may see here: Then proceed, and say, how many times 46 can I have in 185, or rather how many times 4 in 18: here Essay, as you do in Division, and see if you can have it four times, remembering the 4 that must be put down under the pointed Figure, and when you find you can have it four times, writ it down in the Quotient, and also under your last pointed Figure; then say four times 4 is 16, out of 18, there rests 2, which writ down, and cancel the 18 and 4. Again, four times 6 is 24, out of 25, rests 1; which put down, and cancel the 2, 5, and 6. Again, four times 4 is 16, out of 16, rests 0: and so have you done, and find the Root to be 234. I'll add but one Example more for your practice: Let the Number, whose Root is required be , see the working of it. But in this you see there is a Fraction remains, and so there will be in most Numbers, for we seldom happen upon a Number exactly Square: the Fractional Part must therefore thus be taken: before you begin to extract, add to your Number given two Ciphers, if you desire to know but to the tenth part of an Unite; but if to an hundredth part add four Ciphers, if to a thousandth part of an Unite, add six Ciphers, and then work, as before, as if it was all one entire Number, and look how many Points were placed over the Number first given, so many places of Integers will be in the Root; the rest of the Root towards the Right-hand, will be the Numerator of a Decimal Fraction. For Example, let 143 be the Number given to be extracted, and to know the Decimal Fraction as near as to the hundredth part of an Unite; I writ it down as before, annexing four Ciphers to the end of it, as you see hereunder; and after having wrought it, there comes out in the Quotient 1195, but because I had but two Points over the first Number given, viz. , I therefore at the end of two Figures in the Quotient put a Point, which parts the whole Number from the Fraction; that 11 on the Lefthand being Integers, and the 95 on the Right Centesms of an Unite, which you may either write as above, or thus, 11 95/100 if you please. There are other ways taught by Arithmeticians for finding out the Square Root of any Number; but I know no way so concise as this, and after a little practice, so easy and ready, or to be wrought with as few Figures. To do it indeed by the Logarithms or Artificial Numbers, is very easy and pleasant, but Surveyors have not always Books of Logarithms about them, when they have occasion to extract the Square Root: However I will briefly show you how to do it, and give you one Example thereof. When you have any Number given whose Square Root you desire, seek for the given Number in the Tables of Logarithms under the Title Numbers, and right against it, under the Title Logarithms, you will find the Logarithm of the said Number, the half of which is the Logarithm of the Root desired: Which half seek for under the Title Logarithm, and right against it under the Title Number, you will find the Root. EXAMPLE. Let 625 be the Number whose Root is desired: First I seek for it under the Title Numbers, and right against it I find this which I divide by 2, or take the half of it as you see: Log. 2,795880, Half. 1,397940, And finding that half under the Title Log. right against it is 25, the Root desired. See the same done by the former way with less trouble. CHAP. II. Geometrical Definitions. Appoint is that which hath neither Length nor Breadth, the least thing which can be imagined, and which cannot be divided, commonly marked as a full Stop in Writings thus (.) A Line has Length, but no Breadth nor thickness, and is made by many Points joined together in length, of which there are two sorts, viz. Streight and Crooked. As, AB is a Straight Line, BC two Crooked Lines. mathematical figure An Angle is the meeting of two Lines in a Point; provided the two Lines so meeting, do not make one Straight Line, as the Line AB, and the Line AC, meeting together in the Point A, make the Angle BAC. mathematical figure Of which Right-lined Angles there are three sorts, viz. Right Angled, Acute, Obtuse. When a Line falleth perpendicularly upon another Line, it maketh two Right Angles. mathematical figure EXAMPLE. Let CAB be a Right Line, DAMN a Line Perpendicular to it, that is to say, neither leaning towards B or C, but exactly upright; then are both the Angles at A, viz. DAB, and DAC, Right Angles; and contain each just 90 Degrees, or the fourth part of a Circle; but if the Line DA had not been Perpendicular, but had leaned towards B, then had DAC been an Obtuse Angle, or greater than a Right Angle, and DAB an Acute Angle, or lesser than a Right Angle, as you see hereunder. mathematical figure All Figures contained under three Sides are called Triangles, as A, B, C. mathematical figure Where note, The Triangle A hath three equal sides, and is called an Equilateral Triangle. The Triangle B hath two Sides equal, and the third unequal, and is called an Isosceles Triangle. The Triangle C hath three unequal Sides, and is called a Scalenum. Of four Sided Figures there are these Sorts: First, a Square, whose Sides are all equal, and Angles Right, as A. Secondly, A Long Square, or Parallelogram, whose Opposite Sides are equal, and Angles Right, as B. Thirdly, A Rhombus, whose Sides are all Equal, but no Angle Right, as C. Fourthly, A Rhomboides, whose Opposite Sides only are Equal, and no Right Angles, as D. All other four Sided Figures are called Trapezia, as E. mathematical figure Other Figures that are contained under 5, 6, 7, or more Sides, I call Irregular, as FG, etc. Except mathematical figure such as are made by dividing the Circumference of a Circle into any number of Parts; for than they are Regular Figures; having all their Sides and Angles Equal; and are called according to the number of Right Lines the Circle is divided into, or more properly according to the Number of Angles they contain, as a Pentagon, Hexagon, Heptagon, Octogon, etc. Which in plain English is no more than a Figure of Five, Six, Seven or Eight Angles; which Angles are all equal one to another, and their Sides consequently all of the same length. And thus (though I mention no more than 8,) the Circumference of the Circle may be divided into as many Parts as you please; and the Regular Figures arising out of such divisions, are called according to the number of Parts the Circle is divided into; see for your better understanding these two or three following. Pentagon Pentagon Hexagon Hexagon Heptagon Heptagon mathematical figure The Diameter of a Circle, is a Line which passing through the Centre, cuts the Circle into two Equal Parts, or the longest Straight Line that can be made in any Circle; as BC. The Semi-Diameter, is the half of the Line, as AB, AC, or AD, either of which is called a Semi-Diameter. A Chord, is any Line shorter than the Diameter, which passeth from one part of the Circumference to another, as EF. A Semicircle is the half of a Circle, as BDC, or BEC. mathematical figure A Section, Segment, or part of a Circle is a piece of the Circle cut off by a Chord Line, and is greater or less than a Semicircle, as ECFG is a Segment of the Circle EBDCG, likewise EBDCF is the greater Segment of the same Circle. A Superficies is that which hath both length and breadth, but no thickness: whose Bounds are Lines, as A is a Superficies or Plain contained in these Lines BC, DE, BD, CE, which hath length from B to C, and Breadth from B to D, but no Thickness. mathematical figure When these bounding Lines are measured, and the Content of the Superficies cast up, the result is called the Area, or Superficial Content of that Figure. EXAMPLE. Suppose the Line BC to be twelve foot in Length, and the Line BD, to be four Foot long, they multiplied together make 48; therefore I say 48 Square Feet is the Area or Superficial Content of that Figure. mathematical figure mathematical figure A Diagonal Line is a Line running through a Square Figure, dividing it into two Triangles, beginning at one Angle of the Square, and proceeding to the Opposite Angle. In the Square ABCD, AD is the Diagonal Line. CHAP. III. Geometrical Problems. PROB. I. How to make a Line Perpendicular to a Line Given. THe Line given is AB, and at the Point C it is required to erect a Line which shall be Perpendicular to AB. mathematical figure Open your Compasses to any convenient wideness, and setting one Foot of them in the Point C, with the other make a Mark upon the Line at E, and also at D; then taking off your Compasses, open them a little wider than before, and setting one Foot in the Point D, with the other describe the Arch FF, then without altering your Compasses, set one Foot in the Point E, and with the other describe the Arch GG. Lastly, Lay your Ruler to the Point C, and the Intersection of the two Arches GG and FF, which is at H, and drawing the Line HC, you have your desire, HC being Perpendicular to AB. See it here done again after the very same manner, but may perhaps be plainer for your Understanding. mathematical figure PROB. two. How to raise a Perpendicular upon the End of a Line. mathematical figure AB is the Line given, and at B it is required to erect the Perpendicular BC. If you have room you may extend the Line AB to what length you please, and work as above; but if not, then thus you may do it: Open your Compasses to an ordinary extent, and setting one Foot in the Point B, let the other fall at adventure, no matter where in Reason, as at the Point ☉, then without altering the extent of the Compasses, set one Foot in the Point ☉, and with the other cross the Line AB as at D: Also on the other side describe the Arch E; then laying your Ruler to D and ☉ draw the pricked Line D ☉ F. Lastly, from the Point B, you began at, through the Interjection at g draw the Line B g C, which is perpendicular to AB. Another way to do the same, I think more easy, though indeed almost the same. Let AB be the given Line, BY the Perpendicular required. mathematical figure Set one Foot of your Compasses in B, and with the other at any ordinary extent describe the Arch CEFD, then keeping your Compasses at the same extent, set one Foot in C, and make a Mark upon the Arch at E; also setting one Foot in E, make another Mark at F, then opening your Compasses, or else with the same Extent, which you please, set one Foot in E, and with the other describe the Arch GG, also setting one Point in F, make the Arch HH, then drawing a Line through the intersection of the Arches G and H, to the Point first proposed B, you have the Perpendicular Line IB. PROB. iii. How from a Point assigned, to let fall a Perpendicular upon a Line given. The Line given is AB, the Point is at C, from which it is desired to draw a Line down to AB, that may be Perpendicular to it; mathematical figure First, setting one Foot of your Compasses in the Point C, with the other make a Mark upon the Line AB, as at D, and also at E, then opening your Compasses wider, or shutting them closer, either will do; set one Foot in the Point of Intersection at D, and with the other describe the Arch gg, the like do at E, for the Arch hh: Lastly, from the Point assigned, through the Point of Intersection of the two Arches gg, and hh, draw the Perpendicular Line CF. This is no more but the First Problem reversed: The same you may do by the second Problem, viz. let fall a Perpendicular nigh the end of a given Line. PROB. iv. How to divide a Line into any Number of Equal Parts. mathematical figure AB is a Line given, and it is required to divide it into 6 equal Parts. Make at the Point B a Line Perpendicular to AB, as BC; do the same at A the contrary way, as you see here; open your Compasses to any convenient Wideness, and upon the Lines BC, and AD, mark out five Equal Parts; for it must be always one less than the Number you intent to divide the Line into: which parts you may number, as you see here, those upon one Line one way, and the other the contrary way; the laying your Ruler from No. 1. on the Line BC, to No. 1. on the Line AD, it will intersect the Line AB at E, which you may mark with your Pen, and the Distance between B and E, is one sixth part of the Line; so proceed on till you come to No. 5. and then you will find that you have divided the give Line into six Equal Parts, as required. PROB. v. How to make an Angle Equal to any other Angle given. The Angle given is A, and you are desired to make one Equal to it. mathematical figure Draw the Right Line BC, then going to the Angle A, set one Foot of your Compasses in the Point h, and with the other at what Distance you please describe the Arch IK, then without altering the extent of the Compasses, set one Foot in B, and draw the like Arch, as fg; after that measure with your Compasses how far it is from K to I, and the same distance set down upon the Arch from g towards f, which will fall at E, after draw the Line BED, and you have done. PROB. vi. How to make Lines Parallel to each other. AB is a Line given, and it is required to make a Line parallel unto it. mathematical figure Set one foot of your Compasses at or near the end of the given line as at C, and with the other describe the Arch ab; do the same near the other end of the same line, and through the utmost convex of those two Arches draw the Parallel line C. D. PROB. seven. How to make a Line Parallel to another Line, which must also pass through a Point assigned. Let AB be the given line, C the point through which the required Parallel line must pass. mathematical figure Set one foot of your Compasses in C, and closing them so that they will just touch, (and no more) the Line AB: describe the Arch aa; with the same extent in any part of the given Line set one Foot, and describe another Arch as at D: then through the assigned Point, and the utmost Convex of the last Arch, draw the required Line CD, which is Parallel to AB, and passeth through the Point C. PROB. viij. How to make a Triangle, three Lines being given you. Let the three lines given be 1, 2, 3, The Question is how to make a Triangle of them. Take with your Compasses the length of either of the three, in this Example; mathematical figure let it be that No. 1. viz. the longest, and lay it down as hereunder from A to B; then taking with your Compasses the Length of the Line 2, set one Foot in B, and make the Arch C; also taking the length of the last Line 3. place your Compasses at A, and make the Arch D, which will intersect the Arch Cat the Point E; from which Point of Intersection draw Lines to AB, which shall constitute the Triangle AEB; The Line AB being equal to the line No. 1, BE to No. 2, A to No. 3. PROB. ix. How to make a Triangle equal to a Triangle given, and every way in the same Proportion. First make an Angle Equal to the Angle at A, as you were taught in mathematical figure PROB. v. Then making the Lines AD and A equal to AB and AC, draw the Line DE. Or otherwise you may do it as you were taught in PROB. viij. PROB. x. How to make a Square Figure. mathematical figure Let A be a Line given, and it is required to make a square Figure, each side of which shall just be the length of the Line A. First lay down the length of your Line A, as AB. Secondly, raise a Perpendicular of the same length at B. Thirdly, take the length of either of the aforementioned Lines with your Compasses, and setting one Foot in C describe the Arch ee; do the like at A, and describe the Arch ff. Fourthly, draw Lines from A and C into the Point of Intersection, and the Square is finished. PROB. xi. How to make a Parallelogram, or long Square. mathematical figure This is much like the former. Admit two Lines be given you, as 1, 2, and it is required to make a Parallelogram of them: What a Parallelogram is, you may see in the Second Chapter of Definitions. First, lay down your longest Line, as AB, upon the End of which erect a Perpendicular Line, equal in Length to your shortest Line, and so proceed, as you were taught in the foregoing Problem. PROB. xii. How to make a Rhombus. First make an Angle, suppose ACB, no matter how great or small; but be sure mathematical figure let the two Lines be of equal length; then taking with your Compasses the length of one of those two Lines, set one Foot in A, and describe the Arch bb; also set one Foot in B, and describe the Arch cc. Lastly, draw Lines, and it is finished. Two Equilateral Triangles is a Rhombus. A Rhomboides differs just so much, and no more from a Rhombus, as a Parallelogram does from a true Square; it is needless therefore, I presume, to show you how to make it. PROB. xiii. How to divide a Circle into any number of Equal Parts, not exceeding ten, or otherwise how to make the Figures called, Pentagon, Hexagon, Haptagon, Octogon, etc. Let ABCD be a Circle, in which is required to be made a Triangle, the greatest that can be made in that Circle. mathematical figure Keeping your Compasses at the same extent they were at when you made the Circle, set one Point of them in any part of the Circle, as at A, and with the other make a Mark at E and f, and draw a Line between E and f, which will be one Side of the Triangle. I need not tell you how to make the other two Sides, for it is an Equilateral Triangle, all three Sides being of Equal Length. To make a Pentagon or Five-sided Figure. Draw first an obscure Circle, as ABCD; then mathematical figure draw a Diameter from A to B; make another Diameter Perpendicular to the first, as CD; then taking with your Compasses the Length of the Semi-Diameter, set one Point in A, and make the Marks OF, drawing a Line between them, as you did to make the Triangle. Next, set one Point of your Compasses in the Intersection at g, and extend the other to C, draw the Arch CH: The nearest Distance between C and H, viz. the Line CIH, is the Side of a Pentagon, and the greatest that can be made within that Circle: Which with the same extent of your Compasses you may mark out round the Circle, and drawing Lines, the Figure will be finished. To make a Hexagon or Six-sided Figure. mathematical figure Draw an obscure Circle, as you see here, and then without altering the extent of the Compasses, mark out the Hexagon required round the Circle; for the Semidiameter of any Circle is the side of the greatest Hexagon that can be made within the same Circle. This is the way Cooper's use, to make Heads for their Casks. To make a Heptagon, or Figure of Seven, equal Sides and Angles. mathematical figure You must begin and proceed as if you were going to inscribe a Triangle in a Circle, till you have drawn the Line OF; then taking with your Compasses the half of that Line, viz. from ☉ to E, or from ☉ to F, mark out round the Circle your Heptagon, for the half of the Line OF is one side of it. To make an Octogon, commonly called an Eight-square Figure. mathematical figure First make a Circle. Secondly, divide it into four equal Parts by two Diameters, the one perpendicular to the other, as AB and CD. Thirdly, Set one Foot of the Compasses in A, and make the Arch E E; also with the same extent set one foot in C, and make the Arch ff; then through the Intersection of the two Arches draw a Line to the Centre, viz. gh. Lastly, Draw the Line IC or IA, either of which is the side of an Octagon. To make a Nonagon. mathematical figure First make a Circle, and a Triangle in it, as you were taught at the beginning of this Problem. then divide one third part of the Circle. As for Example, that A, 1, 2, 3, B, into three equal Parts. Lastly, draw the lines A 1, 1, 2, 2 B, etc. each of these Lines is the side of a Nonagon. To make a Decagon. mathematical figure You must work altogether as you did in making a Pentagon: See the Pentagon above, where the distance from the Centre K to the Point at H is the side of a Decagon or Ten-sided Figure. PROB. xiv. Three Points being given: How to make a Circle, whose Circumference shall pass through the three given Points, provided the three Points are not in a straight Line. Let A, B, C, be the three Points given; first setting one foot of your Compasses in A, open them to any convenient wideness, more than half the distance mathematical figure between A and B, and describe the Arch dd; then without altering the extent, set one point in B, and cross the first Arch at E and E, through those two Intersections draw the Line EE. The very same you must do between B and C, and draw the Line ff; where these two Lines intersect each other, as at g, there is the Centre of the Circle required; therefore setting one foot of your Compasses in g, extend the other to either of the Points given, and describe the Circle A B C. Note the Centre of a Triangle is found the same way. PROB. xv. How to make an Ellipsis, or Oval several ways. mathematical figure Fig. 1. Make three Circles whose Diameters may be in a straight 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 Centres, 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 Semidiameters, 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. A, 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. xuj. How to divide a given Line into two Equal Parts, which may be in such Proportion to each other, as two given Lines. mathematical figure 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 A; 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 EBB; and from F make a line parallel to ebb, 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 Arithmetic. 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. mathematical figure And here for a while I shall leave these Problems, till I come to show 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 show you what is necessary to be known. CHAP. IU. Of Measures. ANd first of Long Measures; which are either Inches, Feet, Yards, Perches, Chains, etc. 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 multiplied 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, etc. of which more by and by. Long Link Foot Yard Perch Chain Mile Inches 7.92 12 36 198 792 63360 Links 1.515 4.56 25 100 8000 Feet 3 16.5 66 5280 Yards 5.5 22 1760 Perch 4 320 Chain 80 See this Table of Long Measure annexed, the use whereof is very easy: If you would know how many Feet in Length go to make one Chain; look for Chain at Top, and at the Lefthand for Feet, against which, in the common Angle of meeting, is 66, so many Feet are contained in one Chain. But because Mr. Gunter's Chain is most in use among Surveyors for measuring of Lines, I shall chief insist on that measure, it being the best in use for Lands. This Chain contains in Length 4 Pole or 66 Feet, and is divided into 100 Links, each Link is therefore in length 7 92/105 Inches: If you would turn any number of Chains into Feet, you must multiply them by 66, as 100 Chains multiplied by 66, makes 6600 Feet; but if you have Links to your Chains to be turned into Feet and Parts of Feet, you must set down the Chains and Links, as if they were one whole Number, and after having multiplied that Number by 66, cut off from the Product the two last Figures to the Right-hand, which will be the Hundreth Parts of a Foot, and those on the Lefthand the Feet required. EXAMPLE. Let it be required to know how many Feet there are in 15 Inches, 25 Links. I set down thus the Multiplicand 1525 The num. of Feet in 1 Chain, Multiplicat. 66 9150 9150 Product 100650 Feet. The Product is 1006 50/100. This is so plain, it needs no other Example. But now on the other hand, if One thousand and six Feet and an half was given you to reduce into Chains and Links; you must divide 100650 by 66, the Quotient will be 1525, viz. 15 Chains, 25 Links. But for those that do not well understand Decimal Arithmetic, and may perhaps meet with harder Questions of this nature, I have here inserted A Table, showing 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 One hundred Chains. Links Feet Parts of a Foot Perches Part of a Perch Chains. Feet Perches 1 00. 66 0. 04 1 66 4 2 01. 32 0. 08 2 132 8 3 01. 98 0. 12 3 198 12 4 02. 64 0. 16 4 264 16 5 03. 30 0. 20 5 330 20 6 03. 96 0. 24 6 396 24 7 04. 62 0. 28 7 462 28 8 05. 28 0. 32 8 528 32 9 05. 94 0. 36 9 594 36 10 06. 60 0. 40 10 660 40 20 13. 20 0. 80 20 1320 80 30 19. 80 1. 20 30 1980 120 40 26. 40 1. 60 40 2640 160 50 33. 00 2. 00 50 3300 200 60 39. 60 2. 40 60 3960 240 70 46. 20 2. 80 70 4620 280 80 52. 80 3. 20 80 5280 320 90 59. 40 3. 60 90 5940 360 100 66. 00 4. 00 100 6600 400 The Explanation of the Table. If you would know how many Feet are contained in Twenty of Mr. Gunter's Chains. First, under Title Chains, seek for 20; and right against it, under Title Feet, stands 1320, the number of Feet contained in Twenty Chains. Also under Title Perches, stands 80, the number of Perches contained in Twenty Chains. Again, If you would know how many Feet are contained in Eight Links only of the Chain, seek 8 under Title Links, and right against it stands 05. 28, which is five Feet 28/100 of a Foot, something more than five Feet and a quarter. Also under Title Perches and Parts of a Perch, stands 0. 32, which signifies that 8 Links contain 0 Perch 32/100 of a Perch. But to know how many Feet are contained in any number of Chains and Links together. First seek the Feet answering to the whole Chains, and write them down next the first answering to the Links; and adding them to the other, you will have your desire. Example; In 15 Chains, 25 Links, how many Feet? First, by the Table I find 10 Chains to contain 660 Feet, which I writ down thus And when you have added them together, you find the Sum to be 1006 Feet, and 50/100 of a Foot, that is contained in 15 Chains, 25 Links. Chains, Feet, Parts, 10 660 5 330 Links 20 13 20 5 3 30 Added 1006 50 In like manner, if it had been asked, how many Perches had been contained in 15 Chains, 25 Links? In the Table against 10 Perch, Parts, Chains stands 40 5 20 20 Links 00 80 5 Links 00 20 Answer, 61 Perches 61 00 Mark, that the foregoing Table is as big again as it need to be; for you see both the Columns are alike in Figures, and only differenced by Points. I made it so for your clearer understanding of it; which when you well do, you need use no more but one Column; and that if you please, you may have placed on a Scale, or any other Instrument. But now to bring a Lesser Measure into a Greater, is so much harder than to bring a Greater into a Less, as Division is harder than Multiplication. I have therefore, for your ease, hereto annexed a large Table, with which by Inspection only, or at most by a little easy Addition, as in the former, you may change any number of Feet into Chains, Links, and Parts of a Link (remembering all this while I mean Mr. Gunter's Chain); also into Perches and Parts of a Perch. A Table, showing 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 to 10000 Feet Chain Link P. of L. Perch P. of Per. 1 0 1 515 0 060 2 0 3 030 0 121 3 0 4 545 0 181 4 0 6 060 0 242 5 0 7 575 0 303 6 0 9 090 0 363 7 0 10 606 0 424 8 0 12 121 0 484 9 0 13 636 0 545 10 0 15 151 0 606 20 0 30 303 1 212 30 0 45 454 1 818 40 0 60 606 2 424 50 0 75 757 3 030 60 0 90 909 3 636 70 1 06 060 4 242 80 1 21 212 4 848 90 1 36 363 5 454 100 1 51 515 6. 060 200 3 03 030 12 121 300 4 54 545 18 181 400 6 06 060 24 242 500 7 57 575 30 303 600 9 09 090 36 363 700 10 60 606 42 424 800 12 12 121 48 484 900 13 63 636 54 545 1000 15 15 151 60 606 2000 30 30 303 121 212 3000 45 45 454 181 818 4000 60 60 606 242 424 5000 75 75 757 303 030 6000 90 90 909 363 636 7000 106 06 060 424 242 8000 121 21 212 484 848 9000 136 36 363 545 454 10000 151 51 515 606 060 This Table is like the former, and needs not much Explanation. However I will give you an Example or two. Admit I would know how many Chains in length are contained in 500 Feet. First, in the lefthand Column, under Title Feet, I look out 500, and right against it I find 7 Chains, 57 Links, 575 Parts of 1000 of a Link, or 7 Chains, 57 575/1000. So likewise under Title Perches, I find 30 303/1000 Perches. But if you would know how many odd Feet that 303/1000 is, you must seek for 303 in 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 Ionas Moor 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. mathematical figure mathematical figure 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 Place 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 lie 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, showing the Length, and Breadth 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 Sum 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 Breadth, how many Yards shall there be in Length of the Acre? Breadth 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 Breadth 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 chief used by Surveyors, I shall only instance in that, and show 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 Breadth, and 10 in Length; or 2 in Breadth 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 Sum 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 cipher, they will be turned into Chains, thus 99 Acres are 990 Chains, 100 Acres 1000 Chains, etc. The same as if you had multiplied the Acres by 10. And if you would turn Square Chains into Square Links, add four Ciphers to the end of the Chains so will 990 Chains be 9900000 Links, 1000 Chains 10000000 Links, all one as if you had multiplied 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 last, Because some Men choose rather to cast up the Content of Land in Perches, I will here briefly show 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 straight 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 straight 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 straight 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, than 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 dropped 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, remembering 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 Rag, for your more ready reckoning the odd Links. When you put down in your Field-Book the length of any Line, you may set it thus, if you please, with a Stop between the Chains and Links, as 15 Chains 15 Links 15.15. or without, as thus 1515, it will be all one in the casting up. Of Instruments for the taking of an Angle in the Field. There are but two material things (towards the measuring of a piece of Land) to be done in the Field; the one is to measure the Lines (which I have showed you how to do by the Chain) and the other to take the quantity of an Angle included by these Lines; for which there are almost as many Instruments as there are Surveyors. Such among the rest as have got the greatest esteem in the World, are, the Plain Table for small Enclosures, the Semicircle for Champaign Grounds, The Circumferentor, the Theodolite, etc. To describe these to you, their Parts, how to put them together, take them asunder, etc. is like teaching the Art of Fencing by Book, one Hours use of them, or but looking on them in the Instrument-maker's Shop, will better describe them to you than the reading one hundred Sheets of Paper concerning them. Let it suffice that the only use of them all is no more (or chief at most) but this; viz. To take the Quantity of an Angle. mathematical figure Plain Table. Place the Table (already fitted for the Work, with a Sheet of Paper upon it) as nigh to the Angle A as you can, the North End of the Needle hanging directly over the Flower de Luce; then make a Mark upon the Sheet of Paper at any convenient place for the Angle A, and lay the Edge of the Index to the Mark, turning it about, till through the Sights you espy B, then draw the Line AB by the Edge of the Index, Do the same for the Line AC, keeping the Index still upon the first Mark, then will you have upon your Table an Angle equal to the Angle in the Field. To take the Quantity of the same Angle by the Semicircle. Place your Semicircle in the Angle A, as near the very Angle as possibly you can, and cause Marks to be set up near B and C, so far off the Hedges, as your Instrument at A stands, then turn the Instrument about till through the fixed Sights you see the Mark at B, there screw it fast; next turn the movable Index, till through the Sights thereof you see the Mark at C, then see what Degrees upon the Limb are cut by the Index; which let be 45, so much is the Angle BAC. How to take the same Angle by the Circumferentor. Place your Instrument, as before, at A, with the Flower de Luce towards you, direct your Sights to the Mark at B, and see what Degrees are then cut by the South End of the Needle, which let be 55; do the same to the Mark at C, and let the South End of the Needle there cut 100, subtract the lesser out of the greater, the remainder is 45, the Angle required. If the remainder had been more than 180 degrees, you must then have substracted it out of 360, the last remainder would have been the Angle desired. This last Instrument depends wholly upon the Needle for taking of Angles, which often proves erroneous; the Needle yearly of itself varying from the true North, if there be no Iron Mines in the Earth, or other Accidents to draw it aside, which in Mountainous Lands are often found: It is therefore the best way for the Surveyor, where he possibly can, to take his Angles without the help of the Needle, as is before showed by the Semicircle: But in all Lands it cannot be done, but we must sometimes make use of the Needle, without exceeding great trouble, as in the thick Woods of Jamaica, Carolina, etc. It is good therefore to have such an Instrument, with which an Angle in the Field may be taken either with or without the Needle, as is the Semicircle, than which I know no better Instrument for the Surveyors use yet made public; therefore as I have before shown you, How by the Semicircle to take an Angle without the help of the Needle; I shall here direct you How with the Semicircle to take the Quantity of an Angle in the Field by the Needle. Screw fast the Instrument, the North End of the Needle hanging directly over the Flower de Luce in the Chard, turn the Index about, till through the Sights you espy the Mark at B; and note what Degrees the Index cuts, which let be 40; move again the Index to the Mark at C, and note the Degrees cut, viz. 85. Subtract the Less from the greater, remains 45, the Quantity of the Angle. Or thus; Turn the whole Instrument till through the Fixed Sights you espy the Mark at B, then see what Degrees upon the Chard are cut by the Needle; which for Example are 315, turn also the Instrument till through the same Sights you espy C, and note the Degrees upon the Chard then cut by the Needle, which let be 270; subtract the Less from the Greater, (as before in working by the Circumferentor) remains 45 for the Angle. Mark if you turn the Flower de Luce towards the Marks, you must look at the Norh end of the Needle for your Degrees. Besides the Division of the Chard of the Semicircle into 360 Equal Parts or Degrees: It is also divided into four Quadrants, each containing 90 Degrees, beginning at the North and South Points, and proceeding both ways till they end in 90 Degrees at the East and West Points; which Points are marked contrary, viz. East with a W. and West with an E, because when you turn your Instrument to the Eastward, the End of the Needle will hang upon the West Side, etc. If by this way of division of the Chard, you would take the aforesaid Angle, direct the Instrument so (the Flower de Luce from you) till through the fixed Sights you espy the Mark at B; then see what Degrees are cut by the North End of the Needle, which let be NE 44; next direct the Instrument to C, and the North End of the Needle will cut NE 89; subtract the one from the other, and there will remain 45 for the Angle. But if at the first sight the Needle had hung over NE 55, and at the second SE 80, then take 55 from 90, remains 35, take 80 from 90, remains 10, which added to 35, makes 45, the Quantity of the Angle: Moreover, if at the first Sight, the North End of the Needle had pointed to NW 22, and at the second NE 23, these two must have been added together, and they would have made 45 the Angle as before. Mark, if you had turned the South part of your Instrument to the Marks, than you must have had respect to the South End of your Needle. Although I have been so long showing you how to take an Angle by the Needle, yet when we come to Survey Land by the Needle, as you shall see by and by, we need take but half the Pains; for we take not the Quantity of the Angle included by two Lines, but the Quantity of the Angle each Line makes with the Meridian; then drawing Meridian Lines upon Paper, which represent the Needle of the Instrument, by the help of a Protractor, which represents the Instrument, we readily lay down the Lines and Angles in such proportion as they are in the Field. This way of dividing the Chard into four 90, is in my Opinion, for any Work the best; but there is a greater use yet to be made of it, which shall hereafter be showed in its proper place. Of the Field-Book. You must always have in readiness in the Field, a little Book, in which fairly to insert your angles and Lines; which Book you may divide by Lines into Columns, as you shall think convenient in your Practice; leaving always a large Column to the right hand, to put down what remarkable things you meet with in your way, as Ponds, Brooks, Mills, Trees, or the like. Thus for Example, if you had taken the Angle A, and found it to contain 45 Degrees; and measured the Line AB, and found it to be 12 Chain's, 55 Links, set it down in your Field-Book thus, A degrees 45 Min. 00 Chain 12 Link 55 Or if at A you had only turned your fixed Sights to B, and the Needle had then cut 315; in the place of 45 you must have put down 315. If you Survey by Mr. Norwood's way, than there must be four Columns more for E. W. N. and Southing. You may also make two Columns more, if you please, for Off-sets, to the right and left. Lastly, You may choose whether you will have any Lines or not, if you can write straight, and in good order, the Figures directly one under another. For this I leave you chief to your own fancy; for I believe there are not two Surveyors in England, that have exactly the same Method for their Field-Notes. Of the Scale. Having by the Instruments before spoken of, measured the Angles and Lines in the Field; the next thing to be done, is to lay down the same upon Paper; for which Use the Scale serves. There are several sorts of Scales, some large, some small, according as Men have occasion to use them; but all do principally consist of no more but two sorts of Lines; the first, of Equal Parts, for the laying down Chains and Links; the second, of Chords, for laying down or measuring Angles. I cannot better explain the Scale to you, than by showing the Figure of such a one as are commonly sold in Shops, and teaching how to use it. mathematical figure Those Lines that are numbered at top with 11, 12, 16, etc. are Lines of Equal Parts, containing 11, 12, or 16 Equal Parts in an Inch. If now by the Line of 11 in an Inch, you would lay down 10 Chains, 50 Links; look down the Line under 11, and setting one foot of your Compasses in 10; close the other till it just touch 50 Links, or half a Chain, in the small Divisions. Then laying your Ruler upon the Paper; by the side thereof make two small Pricks, with the same extent of the Compasses, mathematical figure and draw the Line AB, which shall contain in length 10 Chains, 50 Links, by the Scale of 11 in an Inch. The backside of the Scale, is only a Scale of 10 in an Inch; but divided with Diagonal Lines, more nicely than the other Scales of Equal Parts. How to lay down an Angle by the Line of Chords. If it were required to make an Angle that should contain 45 Degrees. mathematical figure Draw a Line at pleasure, as AB; then setting one Foot of your Compasses at the beginning of the Line of Chords, see that the other fall just upon 60 Degrees: With that extent set one foot in A, and describe the Arch CD. Then take from your Line of Chords 45 Degrees, and setting one foot in D, make a mark upon the Arch, as at C, through which draw the Line A So shall the Angle EAB be 45 Degrees. If by the Line of Chords you would erect a Perpendicular Line, it is no more but to make an Angle that shall contain 90 Degrees. The reason why I bid you take 60 from the Line of Chords to make your Arch by, is because 60 is the Semi-diameter of a Circle whose circumference is 360. How to make a Regular Polygon, or a Figure of 5, 6, 7, 8, or more Sides, by the Line of Chords. Divide 360, the number of Degrees contained in a Circle, by 5, 6, or 7, the number of Sides you would have your Figure to contain; the Quotient taken from the Line of Chords shall be one Side of such a Figure. EXAMPLE. For to make a Pentagon, or Figure of live Sides: Divide 360 by 5, the Quotient is 72, one Side of a Pentagon. mathematical figure As for Example in a Heptagon: Divide 360 by 7, the Quotient will be 51 Deg. 25 Min. which if you take from the Line of Chords, and set off round the Circle, you will make a Heptagon, as DE, OF, FG, etc. are the Sides thereof. To make a Triangle in a Circle by the Line of Chords. First, Take the whole length of your Line of Chords, or the Chord of 90 Degrees, with your Compasses; which distance upon the Circle, set off from C to *. Then take 30 Degrees from the Line of Chords, and set that from * to H. Draw the Line CH, which is one side of the greatest Triangle that can be made in that Circle. Or you may make it, by setting off twice the Semidiameter of the Circle for 60, and 60, is 120, as well as 90, and 30. How to make a Line of Chords. First, make a Quadrant, or the fourth part of a mathematical figure Circle, as ABC; divide the Arch thereof, viz. AC, into 90 Equal Parts; which you may do, by dividing it first into three Equal Parts, and every of those Divisions into three Equal Parts more, and every of the last Divisions into ten Equal Parts. Secondly, Continue the Semi-diameter BC to any convenient length, as to D. Then setting one foot of your Compasses in C, let the other fall on 90, and de scribe the Arch 90. So likewise 80, 80; 70, 70; and the rest. CD is the Line of Chords, and these Arches cutting of it into Unequal Parts, constitute the true Divisions thereof, as you may see by the Figure: You may, if you please draw Lines Parallel to DC, as I have done here, for the better distinguishing every Tenth and Fifth Figure. Of the Protractor. The Protractor is an Instrument with which, with more ease and expedition you may lay down an Angle, than you can by the Line of Chords: also when you have Surveyed by the Needle, by placing the Diameter of the Protractor upon a Meridian Line made upon your Paper, you readily with a Needle upon the Arch of the Protractor prick off the true situation of any Line from the Meridian, without scratching the Paper, as you must do in the use of the Line of Chords. It is made almost like, and graduated altogether like the Brass Limb of a Semicircle, performing the same upon Paper, as your Instrument did in the Field: See here the Figure of it. mathematical figure For the use of the Protractor, you must have a fine Needle, such as Women sew withal, put into a small Handle of Wood, Ivory, or the like, which is to put through the Centre of the Protractor to any Point assigned upon the Paper, that the Protractor may turn round upon it. How to lay down an Angle with the Protractor. mathematical figure If you Survey according to Mr. Norwood's way before spoken of, it will be good to have the Arch of your Protractor divided accordingly, viz. into two Quadrants, or twice 90 Degrees. I need say no more of a Protractor, any ingenuous Man may easily find the several uses thereof, it being as it were, but only an Epitome of Instruments. 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 Semicircle. mathematical figure All which you may note down in your Field-Book thus. Angel's Degrees Minutes Chains Links ☉ A. 00. 00. 8. 70 ☉ B. 080. 00. 10. 00 ☉ C. 107. 00. 11. 40 ☉ D. 185. 00. 10. 50 ☉ E. 260. 00. 12. 00 ☉ F. 315. 00. 8. 78 Secondly, cause the Distance between your Instrument, and every Angle to be measured, thus from ☉ to A will be found to be 8 Chains 70 Links; from ☉ to B 10 Chains 00. all which set down in order in your Field-Book, as you see here above; and then have you done what is necessary to be done in that Field towards measuring of it. Your next work is to Protract or lay it down upon Paper. How to Protract the Former Observations taken. First draw a Line at adventure as A a, then take from your Scale, with your Compasses, the first Distance measured, viz. from ☉ to A 8 Chain 70 Links, and setting one Foot in any convenient place of the Line, which may represent the place where the Instrument stood, with the other make a Mark upon the Line as at A; so shall A be the first Angle, and ☉ the place where the Instrument stood. Secondly, Take a Protractor, and having laid the Centre thereof exactly upon ☉, and the Diameter or Meridian upon the Line A a, the Semicircle of the Protracture lying upwards. There hold it first, and with your Protracting Pin, make a mark upon the Paper against 80 deg. 107 deg. etc. as you find them out of your Field-Book. Then for those Degrees that exceed 180, you must turn the Protractor downward, keeping still the Centre upon ☉, and placing again the Diameter upon a A. Mark out by the Innermost Circle of Divisions the rest of your Observations 185, 260, 315. Then applying a Scale to ☉, and every one of the Marks, draw the pricked Lines ☉ B, ☉ C, ☉ D, ☉ E, ☉ F. Thirdly, Take in with your Compasses the length of the Line ☉ B, which you find by the Field-Book to be 10 Chains, which from ☉ set off to B. The like do for ☉ C, ☉ D, and the rest. Lastly, Draw the Lines AB, BC, CD, etc. which will enclose a Figure exactly proportionable to the Field before Surveyed. How to take the Plot of the same Field at one Station by the Plain Table. Place your Table with a sheet of Paper upon it at ☉, and making a mark upon the Paper, that shall signify where the Instrument stands, lay your Index to the mark, turning it about till you see through the Sights the mark at A; there holding it fast, draw the Line A ☉. Turn the Index to B, keeping it still upon the first mark at ☉; and when you see through the Sights the mark at B, draw the Line B ☉. Do the same by all the rest of the Angles, and having measured the distance between the Instrument, and each Angle, set it off with your Scale and Compasses, from ☉ to A, from ☉ to B, etc. making marks where, upon the several Lines, the distances fall. Lastly, Between those Marks draw Lines, as AB, BC, CD, etc. and then have you the true Plot of the Field ready protracted to your hand. This Instrument is so plain and easy to be understood, I shall give no more Examples of the Use of it. The greatest Inconveniency that attends it, is, that when never so little Rain or Dew falls, the Paper will be wet, and the Instrument useless. How to take the Plot of the same Field at one Station by the Semicircle, either with the help of the Needle and Limb both together, or by the help of the Needle only. In the beginning of this Chapter, I shown you how to take the Plot of a Field at one Station, by the Simicircle, without respect to the Needle, which is the best way: But that I may not leave you ignorant of any thing belongin to your Instrument, I shall here show how to perform the same with the help of the Needle two ways. And first with the Needle and Limb together. Fix the Instrument, as before, in ☉, making the North-Point of the Needle hang directly over the Flower-de-Luce of the Card; there screw fast the Instrument. Then turn the Index to all the Angles, noting down what Degrees are cut thereby at every Angle, as at A let be 25, at B 105, at C 132, and so of the rest round the Field. And when you have measured the Distances, and are come to Protraction, you must first draw a Line cross your Paper, calling it a North and South-Line, which represents the Meridian-Line of the Instrument. Then applying the Protractor to that Line, mark round the Degrees as they were observed, viz. 25, 105, 132, etc. and having set off the Distances, and drawn the outward Lines altogether, like what you were taught at the beginning of this Chapter, you will find the Figure to be the very same as there. Now to perform this by the Needle only, is in a manner the same as the former: For instead of turning the Index about the Limb, and seeing what Degrees are cut thereby, here you must turn the whole Instrument about, and observe at every Angle what Degrees upon the Card the Needle hangs over; which set down, and Protract as before. But here mind some Cards are numbered from the North Eastwards 10, 20, 30, etc. to 360 deg. Some from the North Westard, which are best for this use, Protractors being made accordingly: For when you turn your Instrument to the Eastward, the Needle will hang over the Westward Division, and the contrary. As for the Use of the Division of the Card into four Quadrants, I shall speak largely of by and by, therefore for the present beg your patience. How by the Semicircle to take the Plot of a Field, at one Station, in any Angle thereof, from whence the other Angles may be seen. Let ABCDEFG be the Field, and F the Angle mathematical figure at which you would take your Observations. Having placed your Semicircle at F, turn it about the North-Point of the Card from you, till through the Fixed-Sights, (Note that I call them the Fiexed-Sights which are on the Fixed-Diameter) you espy the mark at G. Then screw fast the Instrument; which done, move the Index, till through the Sights thereof you see the mark at A; and the Degrees on 〈◊〉 ●●●b there cut by it, will be 20. Move again the Index to the mark at B, where you will find it to cut 40 deg. Do the same at C, and it cuts 60 deg. likewise at D 77, and at E 100 deg. Note down all these Angles in your Field-Book; next measure all the Lines, as from F to G 14 Chain, 60 Links; from F to A 18 Chain, 20 Links; from F to B 16 Chain, 80 Links; from F to C 21 Chain, 20 Links; from F to D 16 Chain, 95 Links; from F to E 8 Chain, 50 Links; and then will your Field-Book stand thus: Angles Degrees Minutes Chains Links G 00 00 14 60 A 20 00 18 20 B 40 00 16 80 C 60 00 21 20 D 77 00 16 95 E 10 00 8 50 To Protract the former Observations. Draw a Line at adventure as G, g, upon any convenient place, on which lay the Centre of your Protractor, as at F, keeping the Diameter thereof right upon the Line G, g. Then make marks round the Protractor at every Angle, as you find them in the Field-Book, viz. against 20, 40, 60, 77, and 100; which done, take away the Protractor, and applying the Scale or Ruler to F, and each of the marks, draw the Lines FAVORINA, FB, FC, FD, and FE. Then setting off upon these Lines the true distances as you find them in the Field-Book; as for the first Line F 〈◊〉 Chain, 60 Links; for the second FAVORINA 18 Chain, 20 Links, etc. make marks where the ends of these distances fall, which let be at G, A, B, C, etc. Lastly, Between these Marks, drawing the Lines GA', AB, BC, CD, DE, OF, FG, you will have completed the Work. When you Survey thus without the help of the Needle, you must remember before you come out of the Field to take a Meridian. Line, that you may be able to make a Compass showing the true Situation of the Land, in respect of the four Quarters of the Heavens, I mean East, West, North and South; which thus you may do: The Instrument still standing at F, turn it about till the Needle lies directly over the Flower-de-Luce of the Card, there screw it fast. Then turn the movable Index, till through the Sights you espy any one Angle. As for Example. Let be D: Note then what Degrees upon the Limb are cut by the Index, which let be 10 deg. Mark this down in your Field-Book, and when you have Protracted as before directed, lay the Centre of your Protractor upon any place of the Line FD, as at ☉, turning the Protractor about till 10 deg. thereof lie directly upon the Line FD. Then against the end of the Diameter of the Protractor, make a mark, as at N, and draw the Line N ☉, which is a Meridian, or North and South Line, by which you may make a Compass. Note that you may as well take the Plot of a Field at one Station, standing in any Side thereof, as in an Angle: For if you had set your Instrument in a, the Work would be the same. I shall forbear therefore (as much as I may) Tautologies. How to take the Plot of a Field at two Stations, provided from either Station you may see every Angle, and measuring only the Stationary Distance. Let CDEFGH, be supposed a Field, to be measured at two Stations; first when you come into the Field, make choice of two Places for your Stations, which let be as far asunder as the Field will conveniently admit of; also take care that if the Stationary Distance were continued, it would not touch an Angle of the Field; then setting the Semicircle at A, the first Station, turn it about, the North Point from you, till through the Fixed Sights you espy the Mark at your second Station, which admit to be at B, there screw fast the Instrument; then turn the Movable Index, to every several Angle round the whole Field, mathematical figure and see what Degrees are cut thereby at every Angle, which note down in your Field-Book as followeth: Angles Degrees Minute's C 24 30 D 97 00 E 225 00 First Station. F 283 30 G 325 00 H 346 00 Secondly, measure the Distance between the two Stations, which let be 20 Chains, and set it down in the Field-Book. Stationary Distance 20 Chains, 00 Links. Thirdly, placing the Instrument at B, the Second Station, look backwards through the fixed Sights to the First Station at A, (I mean by looking backward, that the South Part of the Instrument be towards A) and having espied the Mark at A, make fast the Instrument, and moving the Index, as you did at the First Station to each Angle, see what Degrees are cut by the Index, and note them down as followeth; and then have you done, unless you will take a Meridian Line before you move the Instrument; which you were taught to do a little before. Angel's Degrees Minute's C 84 00 D 149 00 E 194 00 The Second Station. F 215 00 G 270 00 H 322 00 How to Protract or lay down upon Paper these foregoing Observations. First, draw a Line cross your Paper at pleasure, as the Line IK, then take from off the Scale the Stationary Distance 20 Chains, and set it upon that Line, as from A to B, so will A represent the First Station, B the Second. Secondly, apply your Protractor, the Centre thereof to the Point A, and the Diameter lying straight upon the Line BK; mark out round it the Angles, as you find them in the Field-Book, and through those Marks from A, draw Lines of a convenient Length. Thirdly, move your Protractor to the Second Station B; and there mark out your Angles, and draw Lines, as before at the First Station. Lastly, the places where the Lines of the First Station, and the Lines of the Second intersect each other, are the Angles of the Field: As for Example; At the First Station the Angle C was 24 Degrees 30 Minutes, through those Degrees I drew the Line A1. At the Second Station C was 84 Degrees: Accordingly from the Second Station I drew the Line B2; now, I say, where these two Lines cut each other, as they do at C, there is one Angle of the Field. So likewise of DE, and the rest of the Angles; if therefore between these Intersections you draw straight Lines, as CD, DE, OF, etc. you will have a true Figure of the Field. This may as well be done by taking two Angles for your Stations, and measuring the Line between them, as C and D, from whence you might as well have seen all the Angles, and consequently as well have performed the Work. How to take the Plot of a Field at two Stations, when the Field is so Irregular, that from one Station you cannot see all the Angles. mathematical figure Let CDEFGHIKLMNO be a Field in which from no one Place thereof all the Angles may be seen; choose therefore two Places for your Stations, as A and B, and setting the Semicircle in A, direct the Diameter to the Second Station B; there making the Instrument fast, with the Index take all the Angles at that end of the Field, as CDEFGHIK, and measure the Distance between your Instrument and each Angle; measure also the Distance between the two Stations A and B. Secondly, remove your Instrument to the Second Station at B; and having made it fast so, as that throug the Back Sights you may see the First Station A; take the Angles at that End of the Field, as NOCKLM, and measure their Distances also as before; all which done, your Field-Book will stand thus. First Station. Angel's Degrees Minutes Chains Links C 25 00 20: 75 D 31 00 8: 10 E 67 00 9: 85 F 101 00 10: 80 G 137 00 7: 00 H 262 00 6: 70 I 316 00 13: 70 K 354 00 24: 50 The Distance between the two Station 31 Ch. 60 L. Second Station. Angles Deg. Min. Chain. Link. N. 3. 30 4: 20 O. 111. 00 7: 00 C. 145. 00 15: 60 K. 205. 00 7: 48 L. 220. 00 15: 00 M. 274. 00 11: 20 To lay this down upon Paper, draw at adventure the Line PBAP; then taking in with the Compasses the Distance between the two Stations, viz. 31 Ch. 60 Links; set it upon the Line, making Marks with the Compasses as A and B, A being the First Station, B the Second, lay the Protractor to A the North End of the Diameter towards B, and mark out the several Angles observed at your First Station, drawing Lines, and setting off the Distances as you were taught in the beginning of this Chapter, Fig. I. Do the same at B, the Second Station; and when you have marked out all the Distances, between those Marks draw the Bound-Lines. I am the briefer in this, because it is the same as was taught concerning Fig I; for if you conceive a Line to be drawn from C to K; then would there be two distinct Fields to be measured, at one Station apiece. If a Field be very irregular, you may after the same manner make three, four or five Stations, if you please; but I think it better to go round such a Field and measure the bounding Lines thereof: Which by and by, I shall show you how to do. Note, in the foregoing Figure you might as well have had your Stations in two convenient Angles, as D and K, and have wrought as you were taught concerning Fig. 2. the Work would have been the same. How to take the Plot of a Field at one Station in an Angle (so that from that Angle you may see all the other Angles) by measuring round about the said Field. mathematical figure Then will your Field-Book be as hereunder. Angel's Degrees Minutes C 68. 00 D 76. 00 E 124. 00 Lines Chains Links AB 14. 00 BC 15. 00 CD 07. 00 DE 14. 40 EA 14. 05 To protract which draw the Line AB at adventure, and applying the Centre of the Protractor to A, (the Diameter lying upon the Line AB, and the Semicircle of it upwards) prick off the Angles, as against 68: 76: and 124: make Marks, through which Marks draw the Lines AC, AD, A, long enough be sure; then taking in with your Compasses, from off the Scale, the length of the Line AB, viz. 14 Chains, and setting one Foot of the Compasses in A, with the other cross the Line, as at B; also for BC take in 15 Chains, and setting one Foot in B, with the other cross the Line AC, which will fall to be at C; for the Line CD take in 7 Chains, and setting one Foot in C, cross the Line AD, viz. at D; then for DE, take in 14 Chains 40 Links, and setting one Foot of the Compasses in DE, with the other cross the Line A, which will fall at E: Lastly for EA take 14 Chains 5 Links with your Compasses, and setting one Point in E, see if the other fall exactly upon A, if it does, you have done the Work true, if not, you have erred; between the Crosses or intersections, draw straight Lines, which shall be the bounds of the Field, viz. AB, BC, CD, DE, EA. How to take the Plot of the foregoing Field, by measuring one Line only, and taking Observations at every Angle. Begin as you have been just before taught, till you have taken the Angles C, D, E, viz. 68, 76, and 124 Degrees; then leaving a good Mark at A, which may be seen all round the Field, go to B, measuring as you go the Distance from A to B, which is all the Lines you need to measure; and planting your Semicircle at B, direct the South Part thereof toward A, until through the back fixed Sights you see the Mark at A, there making it fast, turn the Index about till you espy C, and note down the Degrees there cut, which let be 129 Degrees; move your Instrument to C, and still keeping the South Part of the Diameter to A, turn the Index to D, where it will cut 20 Degrees; then remove to D, and espying A through the Back Sights, turn the Index to E, where it will cut 135 Degrees. Note all this in your Field-Book. Angles taken at the First Station. Angles round the Field. C 68 Degrees B. 129 Degrees D 76 C. 20 E 124 D. 135 Line AB: 14 Chains. To protract this you must work as you were taught concerning the foregoing Figure, until you have drawn the Lines AB, AC, AD, A, and set off the Line AB 14 Chains; then laying the Centre of your Protractor to B, and the South End of the Diameter, (or that marked with 180 Degrees) towards A, make a Mark against 129 Degrees, and through that mark from B, draw the Line BC, till it intersect the Line AC, which it will do at C: Lay also the Centre of the Protractor upon C, the Diameter thereof upon AC, and against 20 Degrees make a Mark, through which from C, draw the Line CD till it intersect the Line AD, which it will do at D; last place your Protractor at D, the Diameter thereof upon the Line DA, and make a Mark against 135 Degrees, through which Mark draw the Line DE, until it intersect the Line A at E, also drawing the Line EA you have done. This may be done otherwise thus, after you have, standing at A, taken the several Angles, and measured the Distance AB, you may only take the quantity of the bounding Angles, without respect to A: As the Angle at B is 51 Degrees, at C (an outward Angle, which in your Field-Book you should distinguish with a Mark ›) 138; and so of the rest. And when you come to plot, having found the place for B, there make an Angle of 51 Degrees, drawing the Line till it intersect AC, etc. You may also survey a Field after this manner, by setting up a Mark in the middle thereof, and measuring from that to any one Angle, also in the Observations round the Field, having respect to that Mark, as you had here to the Angle A. It is too tedious to give Examples of all the Varieties; besides it would rather puzzle than instruct a Neophyte. How to take the Plot of a Large Field or Wood, by measuring round the same, and taking Observations at every Angle thereof, by the Semicircle. mathematical figure Suppose ABCDEFG to be a Wood, through which you cannot see to take the Angles, as before directed, but must be forced to go round the same; first plant the Semicircle at A, and turn the North End of the Diameter about, till through the fixed Sights you see the Mark at B, then move round the Index, till through the Sights thereof you espy G, the Index there cutting upon the Limb 146 Degrees. 2. Remove to B, and as you go measure the Distance AB, viz. 23 Chains 40 Links, and planting the Instrument at B, direct the North End of the Diameter to C, and turn the Index round to A, it then pointing to 76 Degrees. 3. Remove to C, measuring the Line as you go, and setting your Instrument at C, direct the North End of the fixed Diameter to D, and turn the Index till you espy B, and the Index then cutting 205 Degrees; which, because it is an outward Angle, you may mark thus › in your Field-Book. 4. Remove to D, and measure as you go; then placing the Instrument at D, turn the North End of the Diameter to E, and the Index to C, the Quantity of that Angle will be 84 Degrees. And thus you must do at every Angle round the Field as at E, you will find the quantity of that Angle to be 142 Degrees, F 137, G 110, but there is no need for your taking the last Angle, nor yet measuring the two last Sides, unless it be to prove the Truth of your Work; which is indeed convenient: When you have thus gone round the Field; you will find your Field-Book to be as followeth. Angel's Lines Deg. Min. Ch. Lin. A. 146. 00 AB. 23. 40 B. 76. 00 BC. 15. 20 C. 205. 00 › CD. 17. 90 D. 84. 00 DE. 20. 60 E. 142. 00 OF. 18. 85 F. 137. 00 FG. 13. 60 G. 110. 00 GA'. 19. 28 To protract this, draw a dark Line at adventure, as AB; upon which set off the Distance, as you see it in your Field-Book, 23 Chains 40 Links, from A to B; then laying the Centre of your Protractor upon A, and the Diameter upon the Line AB, the North End, or that of 00 Degrees towards B; on the outside of the Limb make a Mark against 146 Degrees, through which Mark from A draw the Line AGNOSTUS, so have you the first Angle and first Distance. 2. Place the Centre of the Protractor upon B, and turn it about until 76 Degrees lies upon the Line AB; there hold it fast, and against the North End of the Diameter make a Mark, through which draw a Line, and set off the Distance BC 15 Chains 20 Links. 3. Apply the Centre of the Protractor to C, (the Semicircle thereof outward, because you see by the Field-Book it is an outward Angle) and turn it about till 205 Degrees, lie upon the Line CB; then against the Upper or South End of the Diameter make a Mark, through which draw a Line, and set off 17 Chains 90 Links from C to D. 4. Put the Centre of the Protractor to D, and make 84 deg. thereof lie upon the line CD; then making a mark at the end of the Diameter or 0 deg. Through that mark draw a line, and set off 20 Chains, 60 Links, viz. DE. 5. Move the Protractor to E, and make 142 deg. to lie upon the line ED. Then at the end of the Protractor, make a mark as before, and setting off the distance 18 Chains, 85 Links, draw the line EF. 6. Lay the Centre of the Protractor upon F, and making 137 deg. lie upon the line OF; against the end of the Diameter make a mark, through which draw the line FG, which will intersect the line AGNOSTUS at G: So have you a true Copy of the Field or Wood: But you may, if you think fit to prove your Work, set off the distance from F to G; and at G apply your Protractor, making 110 deg. thereof to lie upon the line FG. Then if the end of the Diameter point directly to A, and the distance be 90 Chain, 28 Links, you may be sure you have done your Work true. Whereas I bid you put the North end of the Instrument and of the Protractor towards B, it was chief to show you the variety of Work by one Instrument; for in the Figure before this, I directed you to do it the contrary way; and in this Figure, if you had turned the Southside of the Instrument to G, and with the Index had taken B, and so of the rest, the work would have been the same, remembering still to use the Protractor the same way as you did your Instrument in the Field. Also, if you had been to have Surveyed this Field or Wood by the help of the Needle; after you had planted the Semicircle at A, and posited it, so that the Needle might hang directly over the Flower-de-Luce in the Card, you should have turned the Index to B, and put down in your Field-Book what Degrees upon the Brass Limb had then been cut thereby, which let be 20. Then moving your Instrument to B, make the Needle hang over the Flower-de-Luce, and turn the Index to C, and note down what Degrees are there cut. So do by all the rest of the Angles. And when you come to Protract, you must draw Lines Parallel to one another cross the Paper, not farther distant asunder than the breadth of the Parallelogram of your Protractor; which shall be Meridianlines, marking one of them at one end N, for North; and at the other S, for South. This done, choose any place which you shall think most convenient upon one of the Meridian lines for your first Angle at A; and laying the Diameter of your Protractor upon that Line, against 20 deg. make a mark; through which draw a line, and upon it set off the distance from A to B. In like manner proceed with the other Angles and Lines, at every Angle laying your Protractor Parallel to a North and South Line, which you may do by the Figures gratuated thereon, at either end alike. When you have Surveyed after this manner, how to know before you go out of the Field whether you have wrought true or not. Add the Sum of all your angles together, as in the Example of the precedent Wood, and they make 900. Multiply 180 by a number less by 2 than the number of Angles; and if the Product be equal to the Sum of the quantity of all the Angles, then have you wrought true. There were seven Angles in that Wood, therefore I multiply 180 by 5, and the Product is 900. If you Survey, by taking the quantity of every Angle, and if all be inward Angles, you must work as before. But if one or more be outward Angles, you must subtract them out of 180 deg. and add the Remainder only to the rest of the Angles. And when you multiply 180 by a Sum less by 2 than the number of your Angles, you are not to account the outward Angles into the number. Thus in the precedent Example I find one outward Angle, viz. C 205; the quantity of which, if it had been taken, would have been but 155 deg. That taken from 180 deg. there remains 25; which I add to the other Angles, and they make then in all 720. Now because C was an outward Angle, I take no notice of it, but see how many other Angles I have, and I find 6; a number less by 2 than 6, is 4; by which I multiply 180, and the Product is 720, as before. Directions how to Measure Parallel to a Hedge (when you cannot go in the Hedge itself,) and also in such case, how to take your Angles. It is impossible for you when you have a Hedge to measure, to go at top of the Hedge itself; but if you go Parallel thereto, either within side or without, and make your Parallel-line of the same length as the Line of your Hedge, your work will be the same. Thus if AB was a bushy Hedge, to which mathematical figure you could not conveniently come nigher to plant your Instrument than ☉; let him that goes to set up your mark at B, take before he goes the Distance A ☉, which he may do readily with a Wand or Rod; and at B let him set off the same distance again, as to ✚, where let the mark be placed for your Observation; and when the Chain bears measure the distance ☉ ✚, be sure they have respect to the Hedge AB, so as that they make ☉ ✚ equal to AB, or of the same length. But to make this more plain. Suppose ABC to be a Field; and for the Bushes, you cannot come nigher than ☉ to plant your Instrument. Let him that sets mathematical figure up the Marks, take the distance between the Instrument ☉ and the Hedge AB; which distance let him set off again nigh B, and set up his Mark at D; likewise let him take the distance between ☉ and the Hedge A C, and accordingly set up his Mark at E. Then taking the Angle d ☉ E, it will be the same as the Angle BAC: So do for the rest of the Angles. But when the Lines are measured, they must be measured of the same length as the outside Lines, as the Line ☉ d, measured from G to F, etc. the best way therefore is for them that measure the Lines, to go round the Field on the outside thereof, although the Angles be taken within. How to take the Plot of a Field or Wood, by observing near every Angle, and measuring the Distance between the Marks of Observation, by taking, in every Line, two Off-sets to the Hedge. mathematical figure In working after this manner, observe these two things. First, if the Wood be so thick, that you cannot go withinside thereof, you may after the same manner as well perform the Work, by going on the outside round the Wood Secondly, if the Lines are so long, that you cannot see from Angle to Angle, cause your Assistant to set up a Mark so far from you as you can conveniently see it, as at N: Measure the distance ☉ 1 N, and take the Off-set from N to the Hedge. Then at N turn the Fixed-Sights of the Instrument to ☉ 1, and and by that Direction, proceed on the Line till you come to an Angle. This way of Surveying is much easier done (though I cannot say truer) by taking only a great Square in the Field; from the Sides of which, the Off-sets are taken. mathematical figure And when you have thus laid out your Square, and taken all your Off-sets, you will find in your Field-Book such Memorandums as these, to help you Protract. The Angles 4 Right-Angles. The Sides 12 Chains, 00 Links each. I went round cum Sole, or the Hedges being on my Lefthand. C. L. C. L. In the first Line, at 1 50 Off-set to a Side-Line 5 40 8 30 Off-set to an Angle 6 00 C. L. C. L. In the second Line, at 3 50 Off-set to an Angle 6 00 10 70 Off-set to an Angle 5 50 C. L. C. L. In the third Line, at 10 00 Off-set to an Angle 5 30 C. L. C. L. In the fourth Line, at 4 30 Off-set to an Angle 4 40 6 70 Off-set to an Angle 1 50 10 80 Off-set to an Angle 2 20 Now to lay down upon Paper the foregoing Work, make first a Square Figure, whose Side may be 12 Chains, as 1, 2, 3, 4. Then considering you went with the Sun, take 1, 2, for the first Line; and taking from your Scale 1 Chain, 50 Links, set it upon the Line from 1 to 7: at 7 raise a Perpendicular, as 7, 6, making it according to your Field-Book 5 Chains, 40 Links long. Also for the second Off-set upon the same Line, take from your Scale of Equal Parts 8 Chains, 30 Links, which set upon the line from 1 to 8, and upon 8 make the Perpendicular-line 8 B, 6 Chains in length. For the Off-sets of the second Line, take 3 Chains, 50 Links, from the Scale, and set it from 2 to 9; at 9 make a Perpendicular-line 6 Chains long, viz. 9 C: Also for the second Off-set of the same Line, take 10 Chains, 70 Links, and set it from 2 to 10; at 10 make the Perpendicular 10 D, 5 Chains, 50 Links in length. For the Off-sets of the third Line, take from your Scale 10 Chains, and set it from 3 to 11; and at 11 make the Perpendicular 11 E, 5 Chains, 30 Links long. For the Off-sets of the fourth Line, take from your Scale 4 Chains, 30 Links, and set it from 4 to 12; and at 12 make the Perpendicular 12 F, 4 Chains, 40 Links long. Also take 6 Chains, 70 Links, and let it from 4 to 13; and at 13 make the Perpendicular 13 G, 1 Chain, 50 Links long. Lastly, take 10 Chains, 80 Links, and set it from 4 to 1; and at I make the Perpendicular 1, 5, 2 Chains, 20 Links long. Then have you no more to do, but through the ends of these Perpendiculars to draw the Bounding-lines, remembering to make Angles where the Field-Book mentions Angles; and where it mentions Side-lines, there to continue such Side-lines till they meet in an Angle. Although I mention a Square, yet you are not bound to that Figure; for you may with the same success use a Parallelogram, Triangle, or any other Figure. Nor are you bound to take the Off-sets in Perpendicular-lines, although it be the best way; for you may take the Angles with the Index, from any part of the Line. This way was chief intended for such as were not provided with Instruments; for instead of the Semicircle with a plain Cross only, you may lay out a Square, the rest of the Work being done with a Chain. How by the help of the Needle to take the Plot of a large Wood by going round the same, and making use of that Division of the Card that is numbered with four 90s or Quadrants. Let ABCDE represent a Wood; set your Instrument at A. and turn it about till through the Fixed Sights you espy B, then see what Degrees in the Division before spoken of, the Needle cuts, which let be N. W. 7, measure AB 27 Chains 70 Links; then setting the Instrument at B, direct the Sights to C, and see what then the Needle cuts, which let be N. E. 74; measure BC 39 Chains 50 Links; in like manner measure every Line, and take every Angle, and then your Field-Book will stand thus; as followeth hereunder. mathematical figure Lines Degrees Minutes Chains Links AB: N. W.: 7: 00: 28: 20 BC: N. E.: 74: 00: 39: 50 CD: S. E.: 9: 00: 38: 00 DE: N. W.: 63: 20: 14: 55 EA: S. W.: 74: 80: 28: 60 To lay down which upon Paper, draw Parallel Lines through your Paper, which shall represent Meridian, or North and South Lines, as the Lines NS, NS; then applying the Protractor (which should be gratuated accordingly with twice 90 Degrees, beginning at each End of the Diameter, and meeting in the middle of the Arch) to any convenient place of one of the Lines as to A, lay the Meridian Line of the Protractor to the Meridian Line on the Paper; and against 7 Degrees make a Mark, through which draw a Line, and set off thereon the Distance AB 28 Chains 20 Links. Secondly, apply the Centre of the Protractor to B, and (turning the Semicircle thereof the other way, because you see the Course tends to the Eastward) make the Diameter thereof lie parallel to the Meridian Lines on the Paper, (which you may do by the Figures at the Ends of the Parallelogram) and against 74 Degrees make a Mark, and set off 39 Chains 50 Links, and draw the Line BC; the like do by the other Lines and Angles, until you come round to the place where you began. This is the most usual way of plotting Observations taken after this manner, and used by most Surveyors in America, where they lay out very large Tracts of Land: but there is another way, though more tedious, yet surer; (I think first made Public by Mr. Norwood) whereby you may know before you come out of the Field, Whether you have taken your Angles, and measured the Lines truly or not, and is as followeth. mathematical figure As Radius or Sine of 90 Degrees, viz. the Right Angle C is to the Logarithm of the Line AB 20 Chains; So is the Sine of the Angle CAB 20 Degrees to the Difference of Longitude CB 6 Chains 80 Links. Secondly, to find the difference of Latitudes, or the Line AC, say, As Radius is to the Logarthm Line AB 20 Chain, so is the Sine Compliment of the Angle at A to the Logarithm of the Line AC 18 Chains 80 odd Links. Example of the foregoing Figure. In the precedent Figure, I find in my Field-Book, the first Line to run NWS 7 Degrees 28 Chain, 20 Links; now to find what Northing, and what Westing is here made, I say thus, As Radius 10,000000 Is to the Logarithm of the Line 28 Chains 20 Links, 1,450249 So is the Sine of the Angle from the Meridian, viz. 7 Degrees 9,085894 To the Logarithm of the Westing 3 Chains 43 Links Again, As Radius 10,000000 Is to the Logarithm 28 Chains 20 Links 1,450249 So is the Sine Compliment of 7 Degrees 9,996750 To the Log of the Northing 27 Ch. 99 Lin. And having thus found the Northing and Westing of that Line: I put it down in the Field-Book against the Line under the proper Titles NWS, in like manner I find the Latitude and Longitude of all the rest, and having set them down, the Field-Book will appear thus. Lines Degrees: Minutes Chains: Links N S E W AB. NW 7: 00 28: 20 27: 99 ..:.. ..:.. 03: 43 BC. NE 74: 00 39: 50 10: 89 ..:.. 37: 97 ..:.. CD. SE 9: 00 38: 00 ..:.. 37: 53 05: 95 ..:.. DE. NW 63: 20 14: 55 06: 53 ..:.. ..:.. 13: 00 EA. sweet 74: 00 28: 60 ..:.. 07: 88 ..:.. 27: 49 45: 41 45: 41 43: 92 43: 92 This done, add all the Northings together, also all the Southings, and see if they agree; also all the Easting and Westing; and if they agree likewise, than you may be sure you have wrought truly, otherwise not. Thus in this Example the sum of the Northings is 45 Chains 41 Links; so likewise is the sum of the Southings; also the sum of the Easting is 43 Chains 92 Links, so is the sum of the Westing: Therefore I say I have surveyed that Piece of Land true. But because this way of casting up the Northing, Southing, Easting or Westing, of every Line may seem tedious and troublesome to you; I have at the End of this Book, made a Table, wherein by Inspection only, you may find the Longitude and Latitude of every Line, what quantity of Degrees soever it is situated from the Meridian. Moreover, I am also obliged to show you another way of plotting the foregoing Piece of Ground according to the Table in the Field-Book of NS, EWE, as hereunder. mathematical figure Then through B draw another North and South Line parallel to the first, as NBS is parallel to NAS; and taking with your Compasses the Northing of the second Line, viz. 10 Chains 89 Links, set it upon the Line from B to ☉ 2, take also the Easting of the same Line viz. 37 Chains 97 Links, and setting one Foot of the Compasses in ☉ 2, with the other sweep the Arch cc; also take with your Compasses the length of the second Line, viz. 39 Chains 50 Links, and setting one Foot in B cross the former Arch with another dd; and that intersection is your third Angle, viz. C. It would be but tautology in me to go round thus with all the Lines; for by these two first you may easily conceive how all the rest are done: But let me put you in mind when you sweep the Arches for the Easting and Westing, to turn your Compasses the right way, and not take East for West, and West for East. Nor can I commend to you this way of plotting, the former being as true, and far easier; yet when you plot by the former way, it is very good for you to prove your Work by the Table of difference of Latitude and Longitude before you begin to protract; and when you find your Field Work true, you may lay it down upon Paper, which way you think the easiest. To conclude this Chapter or Section, I shall in the next place show you, How to take the Plot of a Field by the Chain only, using no other Instrument in the Field; and that after a better manner than hitherto has been taught. First therefore, I shall show you how to take the quantity of an Angle by the Chain; (which well understood) there need be no more required: For the Business of a Surveyor in the Field, is no more but to measure Lines and take Angles: I mean for telling the quantity of any Field or Piece of Land, as how many Acres it contains, or the like. How by the Chain only, to take an Angle in the Field. mathematical figure But the more easy and speedy way is to take but one Chain only along the Hedges; as in the foregoing Figure, I set a strong Stick in the very Angle A, and putting the Ring at one End of the Chain over it, I take the other End in my Hand, and stretch out the Chain along the First Hedge AB, and where it ends, as at 5, I stick down a Stick, than I stretch the Chain also along the Hedge AC, and at the end thereof set another Stick as at 4, then losing my Chain from A, I measure the distance 4, 5, which is 74 Links, which is all I need notedown in my Field-Book for that Angle; and now coming to plot that Angle, I take first from my Scale the distance of one Chain, and placing one Foot of the Compasses in any part of the Paper, as at A, I describe the Arch 4, 5; then I take from the same Scale 74 Links, and set it off upon that Arch, making Marks where the Ends of the Compasses fall, as at 4, 5. Lastly, from A, through these Marks I draw the Lines AB, and AC, which constitute the former Angle: Remember to plot your Angles with a very large Scale; and you may set off your Lines with a smaller. I will give you two Examples of this way of measuring, and then leave you to your own practice First, How by the Chain only to Survey a Field by going round the same. mathematical figure Let ABCDEF be the Field; and beginning at A in the very Angle, stick down a Staff through the great Ring at one of the Ends of your Chain, and taking the other End in your Hand, stretch out the Chain in length, and see in what part of the Hedge OF the other End falls: as suppose at a, there set up a Stick; and do the like by the Hedge AB, and say, there the Chain ends at (a) also; measure the nearest distance between a and a, which let be 1 Chain 60 Links, this note down in your Field-Book; measure next the length of the Hedge AB, which is 12 Chains 50 Links; note this down also in your Field-Book. Nextly, coming to B, take that Angle in like manner as you did the Angle A, and measure the distance BC: after this manner you must take all the Angles, and measure all the Sides round the Field. But lest you be at a Nonplus at D, because that is an outward Angle, thus you must do; stick a Staff down with the ring of the Chain round it in the very Angle D, then taking the other end of the Chain in your Hand, and stretching it at length, move yourself to and Fro till you perceive yourself in a direct Line with the Hedge DC, which will be at G, where stick down an Arrow, or one of your Surveying-Sticks; then move round till you find yourself in a direct Line with the Hedge DC, and there the Chain stretched out at length, plant another Stick, as at H, then measure the nearest Distance HG, which let be 1 Chain 43 Links; which note down in your Field-Book, and proceed on to measure the Line DE; but in your Field-Book make some some Mark against D, to signify it is an outward Angle, as ›, or the like: And when you come to plot this, you must plot the same Angle outward that you took inward; for the Angle GDH, is the same, as the Angle d D d. I made this outward Angle here on purpose to show you how you must Survey a Wood, by going round it on the Outside, where you must take most of the Angles, as here you do D. Having thus taken all the Angles, and measured all the Sides; the next thing to be done, Is to lay down upon Paper, according to your Field-Book: Which you will find to stand thus. Cross Lines or Chords Angles Chains Links Lines of the Field Chains Links A. 1. 60 AB. 12. 50 B. 1. 84 BC. 23. 37 C. 1. 06 CD. 19. 30 D. 1. 43 › DE. 20. 00 E. 0. 80 OF. 29. 00 F. 1. 52 FAVORINA. 31. 50 Forasmuch now as it is convenient that the Angles be made by a greater Scale than the Lines are laid down with: I have therefore in this Figure made the Angles by a Scale of one Chain in an Inch, and laid down the Lines by a Scale of ten Chains in one Inch. But to begin to plot, take from your Scale one Chain, and with that Distance, in any convenient place of your Paper, as at A, sweep the Arch aa; then from the same large Scale take off 1 Chain 60 Links, and set it upon that Arch, as from a to a; and from A draw Lines through a and a, as the Lines AB, OF: Then repairing to your shorter Scale, take from thence the first distance, viz. 12 Chains 50 Links, and set it from A to B, drawing the Line AB. Secondly, repairing to B, take from your large Scale 1 Chain, and setting one Foot of the Compasses in B, with the other make the Arch bb; also from the same Scale take your Chord Line, viz. 1 Chain 84 Links, and set it upon the Arch bb, one Foot of the Compasses standing where the Arch intersects AB, the other will fall at b; then through b draw the Line BC; and from your smaller Scale set off the Distance BC 23 Chains 37 Links, which will fall at C, where the next Angle must be made. After this manner proceed on according to your Field-Book, till you have done. And here mark that you need neither in the Field, nor upon the Paper, take notice of the Angle F, nor yet measure the Lines OF and OF, for if you draw those two Lines through, they will intersect each other at the true Angle F: However, for the Proof of your Work, it is good to measure them, and also to take the Angle in the Field. I must not let slip in this place the usual way taught by Surveyors, for the measuring a Field by the Chain only, as true indeed as the former, but more tedious, which take as followeth. The common way taught by Surveyors, for taking the Plot of the foregoing Field. Because I will not confound your Understanding with many Lines in one Figure, I have here again placed the same. First they bid you measure round the Field, and note down in your Field-Book every Line thereof, as in this Field has been before done. mathematical figure Secondly, they bid you turn all the Field into Triangles, as beginning at A, to measure the Diagonal AC, AD, A, and note them down; then is your Field turned into four Triangles, and the Diagonals are, Chains Links AC: 33. 70 AD: 25. 70 A 45. 40 To plot which, they advise you first to draw a Line at adventure, as the Line AC, and to set off thereon 33 Chains 70 Links, according to your Field-Book for the Diagonals; then taking with your Compasses the Length of the Line AB, viz. 12 Chains 50 Links, set one Foot in A, and with the other describe the Arch aa; also take the Line BC, viz. 23 Chains 37 Links, and setting one Foot in C, with the other describe the Arch cc, cutting the Arch aa in the Point B, then draw the Lines AB, CB, which shall be two bound Lines of the Field. Secondly, take with your Compasses the Length of the Diagonal AD, viz. 25 Chains 70 Links, and setting one Foot of the Compasses in A, with the other describe the Arch, as dd, also taking the Line CD, viz. 19 Chains 30 Links, set one Foot in C, and with the other describe the Arch ee, cutting the Arch dd in the point D, to which Intersection draw the Line CD. Thirdly, take with your Compasses the Length of of the Diagonal A, viz. 45 Chains 40 Links, and setting one Foot in A, with the other describe an Arch, as ff, also take the Line DE 20 Chains, and therewith cross the former Arch in the Point E, to which draw the Line DE. Lastly, take with your Compasses the length of the Line OF, viz. 31 Chains, 50 Links; and setting one foot in A, describe an Arch, as II. Also take the length of the Line OF, viz. 29 Chains, 00 Links, and therewith describe the Arch hh, which cuts the Arch TWO, in the Point F, to which Point draw the Lines OF and OF, and so will you have a true Figure of the Field. I have showed you both ways, that you may take your choice. And now I proceed to my Second Example promised. How to take the Plot of a Field at one Station, near the Middle thereof, by the Chain only. Let ABCDE be the Field, ☉ the appointed place, from whence by the Chain to take the Plot thereof. Stick a Stake up at ☉ through one ring of the Chain, and make your Assistant take the other end, and stretch it out. Then cause him to move up and down, till you espy him exactly in a Line between the Stake and the Angle A; there let him set down a stick, as at a, and be sure that the stick a be in a direct Line between ☉ and A; which you may easily perceive, by standing at ☉, and looking to A. This done, cause him to move round towards B; and at the Chains end, let him there stick down another stick exactly in the Line between ☉ and B, as at b. Afterwards let him do the same at c, at d, and at e; and if there were more Angles, let him plant a stick at the end of the Chain in a right Line between ☉ and every Angle. In the next place measure the nighest distance between stick and stick, as ab, 1 Chain 26 Links, bc 1 Chain 06 Links, mathematical figure cd 1 Chain 00 Links, the 1 Chain 20 Links, and put them down in your Field-Book accordingly. Measure also the Distances between ☉ and every Angle, as ☉ A 18 Chains 10 Links, ☉ B 15 Chains 00 Links, etc. all which put down, your Field-Bok will appear thus; Chains Links Subtendent or Chord-Lines ab 1. 26 bc 1. 06 cd 1. 00 de 1. 20 Chains Links Diagonal or Centre-Lines ☉ A. 18. 10 ☉ B. 15. 00 ☉ C. 17. 00 ☉ D. 15. 00 ☉ E. 16. 00 How to plot the former Observations. Take from a large Scale 1 Chain, and setting one foot of the Compasses in any convenient place of the Paper, as at ☉, make the Circle abcde. Then taking for your first Subtendent, or Chord-line, 1 Chain, 26 Links; set it upon the Circle, as from a to b. From ☉ through a and b, draw Lines, as ☉ A, ☉ B, which be sure let be long enough. Then take your second Subtendent from the same large Scale, viz. 1 Chain, 6 Links, and set it upon the Circle from b to c, and through c draw the Line ☉ C. When thus you have set off all your Subtendents, and drawn Lines through their several Marks, repair to a smaller Scale; and upon the Lines drawn, set off your Diagonal or Centre Lines, as you find them in the Field-Book: So upon the Line ☉ a you must set off 18 Chains, 10 Links, making a Mark where it falls, as at A: Upon the Line ☉ b 15 Chains, 00 Links, which falls at B; and so by all the rest. Lastly, draw the Lines AB, BC, CD, etc. and the Work will be finished. It would be but running things over again, to show you how, after this manner, to Survey a Field at two or three Stations, or in any Angle thereof, etc. For if you well understand this, you cannot be ignorant of the rest. CHAP. VII. How to cast up the Contents of a Plot of Land. HAving by this time sufficiently shown you how to Survey a Field, and lay down a true Figure thereof upon Paper; I come in the next place to teach you how to cast up the Contents thereof; that is to say, to find out how many Acres, Roods and Perches it containeth. And first Of the Square, and Parallelogram. mathematical figure To cast up either of which, multiply one Side by the other, the Product will be the Content. EXAMPLE. Let A be a true Square, each side being 10 Chains; multiply 10 Chains 00 Links by 10 Chains 00 Links, facit 1000000. from which I cut off the five last Figures, and there remains just 10 Acres for the Square A. Again, In the Parallelogram B, let the side A b or c D be 20 Chains, 50 Links; and the side ac or b D 10 Chains, 00 Links: Multiply ab, 20 Chains, 50 Links, by ac 10 Chains, 00 Links, facit 2050000. from which cutting off the last five Figures, remains 20 Acres. Then if you multiply the Figures cut off, viz. 50000 by 4, facit 200000; from which cutting off five Figures, remains 2 Roods; and if any thing but 100000 had been left, you must have multiplied again by 40; and then cutting off again five Figures, you would have had the odd Perches: See it done hereunder. I need not have multiplied 00 by 40; for I know 40 times Nothing is Nothing; but only to show you in what order the Figures will stand when you have odd Perches, as presently we shall light on. So much is the Content of the long Square B, viz. 20 Acres, 2 Roods, 00 Perch. 20.50 10.00 Acres 2050000 4 Roods 200000 40 Perches 000000 Of Triangles. The Content of all Triangles are found, by multiplying half the Base by the whole Perpendicular; or the whole Base by half the Perpendicular; or otherwise, by multiplying the whole Base and whole Perpendicular together, and taking half that Product for the Content. Either of these three ways will do, take which you please. EXAMPLE. mathematical figure 6,85 10,00 Acres 685000 4 Roods 340000 40 Perches 1600000 So likewise in the Triangle B, the Perpendicular ab is 13 Chains, 70 Links; which multiplied by half the Base, will give the same Content. Also in the Triangle C, if you multiply half the Base E d, by the Perpendicular c F, the Product will be the Content of that Triangle. And here Note, that you are not confined to any Angle, but you may let fall your Perpendicular from what Angle you please, taking the Line on which it falls for the Base. Thus in the Triangle A, if from b you let fall a Perpendicular, take bd, and the half of ac for finding the Content. Also in the Triangle C, you may from E let fall your Perpendicular, although it falls without the Triangle; and the half of EGLANTINE, and the whole of cd, shall be the true Content of the Triangle C; but than you must remember to extend the Base-line cd. Remember this, all Triangles having the same Base, and lying between Parallel-lines, are of the same Content; so the Triangles ABC have the same Base, and lie between the Lines E c and G b, and are therefore of the same Content. To find the Content of a Trapezia. Draw between two opposite Angles a straight Line, as AB; then is the Trapezia reduced into two Triangles, viz. ABC and ABDELLA, which you may measure as before taught, and adding their Products together, you will have the true content of the Trapezia. Or a Little shorter, thus: mathematical figure Half the common Base AB 18,50 The Sum of the two Perpendiculars 12,20 37000 3700 1850 Acres 2257000 4 Roods 228000 40 Perches 1120000 How to find the Content of an Irregular Plot, consisting of many Sides and Angles. To do this, you must first by drawing Lines from Angle to Angle, reduce the Plot all into Trapeziaes' and Triangles; after which measure every Trapezia and Triangle severally, and adding their Contents altogether, you will have the true Content of the whole Plot. EXAMPLE. mathematical figure In the annexed Figure ABCDEFGHI, I draw the Line AD, which cuts off the Trapezia K; also the Line AGNOSTUS, which cuts off the Trapezia L: And lastly the Line GE, which makes the Trapezia M, and the Triangle N, so is the whole Plot reduced into the three Trapeziaes' K, L, M, and the Triangle N; all which I measure as before taught, and put them down as hereunder. Acres Roods Perches The Trapezia K contains 21: 2: 12 The Trapezia L contains 26: 3: 18 The Trapezia M contains 30: 2: 16 The Triangle N contains 6: 2: 24 The Content of the Plot 85: 2: 30 By which you find the whole Plot to contain 85 Acres, 2 Rood, 30 Perches. If the Sides of the Plot had been given in Perches, Yards, Feet, or any other Measure, you must still cast up the Content after this manner, and then your Product will be Perches, Yards, etc. To turn which into Acres, Roods and Perches, I have largely treated of in the beginning of this Book. How to find the Content of a Circle, or any Portion thereof. To find the Content of the whole Circle, it is convenient, That first you know the Diameter and Circumference thereof; one of which being known, the other is easily found; for as 7 is to 22, so is the the Diameter to the Circumference: And as 22 is to 7, so is the Circumference to the Diameter. In this annexed Figure, the Diameter AB is 2 Chains, or 200● Links, which multiplied by 22, and mathematical figure the Product divided by 7, giveth 6 Chains 28 Links, and something more for the Circumference. Now, to know the Superficial Content multiply half the Circumference by half the Diameter, the Product will be the Content: Half the Circumference is 3 Chains 14 Links; half the Diameter 1 Chain 00 Links; which multiplied together, the Product is 3,1400 Square Links, or 1 Rood 10 Perch, the Content of the Circle. Again, By the Diameter only to find the Content. As 14 is to 11, so is the Square of the Diameter to the Content. The Square of the Diameter is 40000, which multiplied by 11, makes 440000, which divided by 14 gives 31428, or 1 Rood 10 Perch, and something more for the Content. How to measure the Superficial Content of the Section of a Circle. Multiply half the Compass thereof by the Semidiameter of the Circle, the Product will answer your desire. In the foregoing Circle, I would know the Content of that little piece DCB; the Arch DB is 78 Links ½; the half of it 39 ¼, which multiplied by 1 Chain, 00 Links, the Semidiameter gives 3925 Square Links, or 6 Perches ¼. How to find the Content of a Segment of a Circle without knowing the Diameter. Let EFG be the Segment, the Chord OF is 1 Chain 70 Links, or 170 Links, the Perpendicular GH 50 Links; now multiply ⅔ of the one by the whole of the other, the Product will be the Content, the two thirds of 170 is nearest 113, which multiplied by 50 produces 5650 Square Links or 9 Perches. How to find the Superficial Content of an Oval. The common way is to multiply the long Diameter by the shorter, and from that Product extract the Square Root, which you may call a mean Diameter; then as if you were measuring a Circle, say, As 14 to 11, so the mean Diameter to the Content of the Oval; but this is not exact: A better way is; As 1, 27/100 is to the length of the Oval; so is the breadth to the Content, or nearer, as 1,27324 to the length; so the breadth to the Content. How to find the Superficial Content of Regular Polygons; as Pentagons, Hexagons, Heptagons, etc. Multiply half the sum of the Sides, by a Perpendicular, let fall from the Centre upon one of the Sides, the Product will be the Area or Superficial Content of the Polygon. In the following Pentagon the Side BC is 84 Links, the whole sum of the five Sides, mathematical figure therefore must be 420, the half of which is 210, which multiplied by the Perpendicular AD 56 Links, gives 11760 Square Links for the Content, or 18 Perches 8/10 of a Perch, almost 19 Perches. I have been shorter about these three last Figures than my usual Method, because they very rarely fall in the Surveyors way to measure them in Land, though indeed in Broad Measure, Paving, etc. often. CHAP. VIII. Of laying out New Lands, very useful for the Surveyors, in his Majesty's Plantations in America. A certain quantity of Acres being given, how to lay out the same in a Square Figure. ANnex, to the Number of Acres given, 5 Ciphers, which will turn the Acres into Links; then 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 join to the 100 5 Ciphers, 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 Cornfield one Square Acre: I add to one five Ciphers, 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 Ciphers, 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 Ciphers, 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, etc. times longer than it is broad. In Carolina, all Lands lying by the Sides of Rivers, except Seignories or Baronies, are (or aught, 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 Ciphers; which sum divide by the number given for the Proportion between the length and breadth, as 4, 5, 6, 7, etc. the Root of the Quotient will show 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 Ciphers, and it makes 100,00000, which sum 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, shows 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 cyphers,) 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 cyphers 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 OF have been the Perpendicular mathematical figure 50, and by drawing the Lines FAVORINA, 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 Ciphers to the 100, and it makes 100,00000 Links, which multiply by 14, facit 140000000; which divided by 11, giveth 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 merely for Speculation, and seldom or never come in Practice, 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 plate 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. mathematical figure There are several other ways taught by Surveyors for reducing Plots or Maps, as Mr. Rathboxn, and after him Mr. holwel, 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 numbered 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 glue 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, etc. of the same 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 numbered with Equal Parts from the Centre-hole to the end; and ⅓ part thereof numbered the other way to the end with the same number of Equal Parts, though 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, glue 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 Centres 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 practice. How to change Customary-Measure into Statute, and the contrary. In some Parts of England, for Wood-Lands; and in most Parts of Ireland, for all sorts of Lands; they account 18 Foot to a Perch, and 160 such Perches to make an Acre, which is called Customary-Measure: Whereas our true Measure for Land, by Act of Parliament, is but 160 Perches for one Acre, at 16 Foot ½ to the Perch. Therefore to reduce the one into the other, the Rule is, As the Square of one sort of Measure, is to the Square of the other; So is the Content of the one, to the Content of the other. Thus if a Field measured by a Perch of 18 Feet, accounting 160 Perches to the Acre, contain 100 Acres; How many Acres shall the same Field contain by a Perch of 16 Feet ½? Say, if the Square of 16 Feet ½, viz. 272. 25. give the Square of 18 Feet, viz. 324. What shall 100 Acres Customary give? Answer 119 9/10 of an Acre Statute. Knowing the Content of a piece of Land, to find out what Scale it was plotted by. First, by any Scale measure the Content of the Plot; which done, argue thus: As the Content found, is to the Square of the Scale I tried by; So is the true Content, to the Square of the true Scale it was plotted by. Admit there is a Plot of a piece of Land containing 10 Acres, and I measuring it by the Scale of 11 in an Inch, find it to contain 12 Acres 1/10 of an Acre. Then I say, If 12 2/10 give for its Scale 11: What shall 100 give? Answer 10. Therefore I conclude that Plot to be made by a Scale of 10 in the Inch. And so much concerning Reducing Lands. CHAP. X. Instructions for Surveying a Manor, County, or whole Country. To Survey a Manor observe these following Rules. 1. WAlk or ride over the Manor 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 Manor 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 bond Hedges in this manner. mathematical figure 3. Take a true Draught of all the Roads and By-Lanes in the Manor, putting down also the true Butting 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 Manor, 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 Manor, though not complete, which to make so, go to all the Butting 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 Manor. 5. When thus you have plotted all the Fields, according to the Butting 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 lie so within others, that they are not bounded on either Side by a Road, Lane nor River; than 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 Manor-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 Manor be fairly drawn, and a Compass in some waste part of the Paper; also a Scale, the same by which it was plotted: You must also beautify such a Draught with Colours and Cuts according as you shall see convenient. Writ down also in every Field the true Content thereof; and if it be required, the Names of the present Possessors, and their Tenors: by which they hold it of the Lord of the Manor. The Quality also of the Land, you may take notice of as you pass over it, if you have Judgement 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, and plotted by a large Scale, Survey next all the Rocks, Sands or other Obstacles that lie at the entrance of every River, Harbour, Bay or Road upon the Coast of that County or Country; which plot down accordingly, as I shall teach you in this Book by and by. 3. Survey all the Roads, taking notice as you go along of all Towns, Villages, great Houses, Rivers, Bridges, Mills, Cross Ways, etc. Also take the bearing at two Stations of all such Remarks, as you see out of the Road, or by the Side thereof. 4. Also Survey all the Rivers, taking notice how far they are Navigable, what (and where the) Branches runs into them, what Fords they have, Bridges, etc. 5. All this being exactly plotted, will give you a truer Map of the County than any that I know of hath been yet made in England: However you may look upon old Maps, and if you find therein any thing worth the Notice that you have not yet put down, you may go and Survey it; and thus by degrees you may so finish a County, that you need not so much as leave out one Gentleman's House; for hardly will it scape but every remarkable thing will come into your View, either from the Roads, the Rivers or Sea-Coast. 6. Lastly, with a large Quadrant take the true Latitude of the Place, in three or four Places of the County, which put down upon the Edge of your Map accordingly. CHAP. XI. Of dividing Lands. How to divide a Triangle several ways. SUppose ABC to be a Triangular Piece of Land, containing 60 Acres, to be divided between two Men, the one to have 40 Acres cut off towards A, mathematical figure and the other 20 Acres towards C; and the Line of Division to proceed from the Angle B. First Measure the Base AC, viz. 50 Chains 00 Links; then say by the Rule of Three, If the whole Content 60 Acres give 50 Chain for its Base, what shall 40 Acres give? Multiply and Divide, the Quotient will be 33 Chains 33 Links; which set off upon the Base from A to D, and draw the Line BD, which shall divide the Triangle as was required. If it had been required to have divided the same into 3, 4, 5, or more unequal Parts; you must, in the like manner, by the Rule of Three have found the length of each several Base; much after the same manner as Merchants part their Gains, By the Rule of Fellowship. There are several ways of doing this by Geometry, without the help of Arithmetic, but my Business is not to show you what maybe done, but to show you how to do it, the most easy and practicable way. How to divide a Triangular Piece of Land into any Number of Equal or Unequal Parts, by Lines proceeding from any Point assigned in any Side thereof. Let ABC be the Triangular Piece of Land, containing 60 Acres to be divided between three Men, the first to have 15 Acres, the second 20, and the third 25 Acres, and the Lines of Division to proceed from D: First measure the Base, which is 50 Chains; then divide the Base into three Parts, as you have been before taught, by saying, If 60 give 50, what shall 15 give? Answer, 12 Chains 50 Links for the mathematical figure first Man's Base; which set off from A to E. Again, Say if 60 give 50, what shall 20 give? Answer, 16 Chains 66 Links for the second Man's Base; which set off from OF, then consequently the third Man's Base, viz. from F to C must be 20 Chains 84 Links: This done, draw an obscure Line from the Point assigned D, to the opposite Angle B, and from E and F draw the Lines EH and FG, parallel to BD. Lastly, from D, draw the Lines DH, DG, which shall divide the Triangle into three such Parts as were required. How to divide a Triangular Piece of Land, according to any Proportion given, by a Line Parallel to one of the Sides. mathematical figure First, divide the Base, as has been before taught, and the point of Division will fall in D, AD being 33 Chains 33 Links, and DC 16 Chains 67 Links. Secondly, find a mean Proportion between AD and AC; by multiplying the whole Base 50 by AD 33, 33, the Product is 16665000, of which sum extract the Root, which is 40 Chains 82 Links, which set off from A to E. Lastly from E draw a Line parallel to BC, as is the Line OF; which divides the Triangle, as demanded. Of dividing Four-Sided Figures or Trapeziaes. Before I begin to teach you how to divide Pieces of Land of four Sides, it is convenient first to show you how to change any Four-Sided Figure into a Triangle; which done, the Work will be the same as in dividing Triangles. How to reduce a Trapezia into a Triangle, by Lines drawn from any Angle thereof. Let ABCD be the Trapezia to be reduced into a Triangle, and B the Angle assigned: Draw the mathematical figure Dark Line BD, and from C make a Line Parallel thereto, as CE; extend also the Base AD, till it meet CE in E; then draw the Line BE, which shall make the Triangle BAE equal to the Trapezia ABCD. Now to divide this Trapezia according to any assigned Proportion is no more but to divide the Triangle ABE; as before taught, which will also divide the Trapezia. EXAMPLE. Suppose the Trapezia ABCD containing 124 Acres 3 Roods and 8 Perches, is to be divided between two Men, the first to have 50 Acres, 2 Rood and 3 Perches; the other 74 Acres, 1 Rood and 5 Perches, and the Line of Division to proceed from B. First, Reduce all the Acres and Roods into Perches, then will the Content of the Trapezia be 19968 Perches; the first Man's Share 8083 Perches; the second 11885. Secondly, Measure the Base of the Triangle, viz. A 78 Chains 00 Links; Then say, If 19968 the whole Content give for its Base 78 Chains 00 Links, What shall 8083, the first Man's part give? Answer 31 Chains 52 Links; which set off from A to F, and drawing the Line FB, you divide the Trapezia as desired; the Triangle ABF being the First Man's Portion, and the Trapezia BCFD, the second's. How to reduce a Trapezia into a Triangle, by Lines drawn from a Point assigned in any Side thereof. ABCD is the Trapezia, E the Point assigned from whence to reduce it into a Triangle, and run the division Line; the Trapezia is of the same Content mathematical figure as the former, viz. 19968 Perches, and it is to be divided as before, viz. one Man to have 8083 Perches, and the other 11885. First for to reduce it into a Triangle, draw the Lines ED, EC, and from A and B make Lines parallel to them, as OF, BG; then draw the Lines EGLANTINE, OF, and the Triangle EFG will be equal to the Trapezia ABCD; which is divided as before; for when you have found by the Rule of Proportion, What the first Man's Base must be, viz. 31 Chains 52 Links, set it from F to H, and draw the Line HE, which shall divide the Trapezia according to 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, mathematical figure AD; and parallel thereto BF, EGLANTINE extending the Base from C to F, and from D to G; then draw the Lines OF, AGNOSTUS, 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. mathematical figure 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 BASILIUS. 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 straight Line through it. mathematical figure ABCDEFGHI is a Field to be divided between two Men in equal Halfs, by a straight 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 ARE, also QT parallel thereto. Draw also AT, and then have you turned two of the Lines into one. Fifthly, From T draw the Line 'tis; and parallel thereto, the Line RV. Draw also TU. Then is your Figure divided into two Equal Parts, by the two Lines AT and TU. Lastly, Draw the Line AV, and parallel thereto TW. Draw also AW, which will cut the Figure into two Equal Parts by a straight 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 straight one, as above. This above is a good way of Dividing Lands, but Surveyors seldom take so much pains about it. I shall therefore show you how commonly they abbreviate their Work, and is indeed An easy 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. mathematical figure 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 straight 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. mathematical figure From ☉ to any Angle draw a Line for the first Division-line, as ☉ A. Then consider that the first Angle A ☉ B 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 Concentric 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. mathematical figure 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 showed 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 Lefthand under M for the 10 Minutes, and right against under the Title Sin. stands the Sine required, viz. 9,659517; also in the same Line under the Title Tang. stands the Tangent of 25°: 10′, viz. 9,710282: But if the Degrees exceed 45, then look at the Foot of the Tables for the Degrees, and up the Right-hand Column for the Minutes; and right against you will find the Sine and Tangent above the Title Sine Tang. thus the Sine of 64° Degrees 50′ Minutes is 9,956684, the Tangent thereof is 10,328037. How to find the Cousin or Sine Compliment; the Cotangent or Tangent Compliment of any given Degrees and Minutes. The Cousin or Cotangent is nothing more but the Sine and Tangent of the remaining Degrees and Minues after substraction from 90, thus, take 25 Degrees 10 Minutes from 90 Degrees, 00 Minutes, there will remain 64 Degrees 50 Minutes, the Sine of which, is as before 9,956684, and that is the Sine Compliment of 25 Degrees 10 Minutes. But the more ready way to find the Cousin or Co-tangent of any number of Degrees given, is to look for the Degrees and Minutes, as before taught, for Sines and Tangents, and right against, under the Titles Cousin and Cotangent; or above, if the Degrees exceed 45, you will find the Cousin or Cotangent require: Thus the Cousin of 30 Degrees 15 Minutes is 9,702236; the Cotangent of 58 Degrees 10 Minutes is 9,792974. Any Sine or Tangent, Co-sine or Co-tangent being given, to find the Degrees and Minutes belonging thereto. This is only the converse of the former, for you must seek in the Tables for the Sine, etc. given, or the nighest that can be be found thereto; and right against it you will find the Minutes and Degrees overhead. Let the Sine 8,742259 be given, right against it stands 3 Degrees 10 Minutes. Remember well that Multiplication is performed with these Logarithm Tables by Addition, and Division by Substraction. If I were to multiply 5 by 4, first I look for the Logarithm of 5, which is 0,698970 The Logarithm of 4 is 0,602060 Added together, they make 1,301030 which 1,301030 I seek for in the Logarithm Tables, and right against, under Title Num. stands 20, the Product of 5 multiplied by 4. If I were to divide 20 by 5, first I look for the Logarithm of 20, which as above, is 1,301030 The Logarithm of 5 is 0,698970 After Substraction remains 0,602060 and the Number answering to that Logarithm, you will find to be 4. And thus by Addition and Substraction the Rule of Three, is performed with the Logarithms, viz. by adding the two last together, and out of their Product substracting the First. EXAMPLE. If 15 give 32, what shall 45 give? The Logarithm of 15 is 1,176091 The Logarithm of 45 is 1,653212 The Logarithm of 32 is 1,505150 The two last added together, make 3,158362 Out of which I subtract the first, and there remains 1,982271 Against which 1,982271, I find the Number 96. I answer therefore, If 15 gives 32, 45 shall give 96. This you must observe to do in the following Cases of Triangles, always to add the second and third numbers together, and from their Product to Subtract the first, the remainder will be the Logarithm Number, Sine or Tangent, of your required Line or Angle. Certain Theorems for the better understanding Right-Lined Triangles. 1. A Right-Lined Triangle is a Figure comprehended within three Straight Lines. 2. Which is either Right-Angled as A, having one Right Angle, which contains just 90 Degrees, viz. that at b; or else Obliqne as B, which consists of three Acute Angles, neither of them so great as 90 Degrees; or which consists of two Acute Angles and one Obtuse, viz. at that D. mathematical figure 3. All the three Angles of any Triangle are equal to two Right Angles, or 180 Degrees; so that one Angle being known, the other two together are known also; or two being known, the third is also known by Substracting the two known Angles out of 180 Degrees, the remainder is the third Angle. 4. To know well what the Quantity of an Angle is, take this following Demonstration. mathematical figure An Angle that cuts off less than 90 Degrees, is called an Acute Angle, as HEF, which takes but 45 Degrees from the Circle. mathematical figure 5. Every Triangle hath six Parts, viz. three Sides and three Angles; the Sides are sometimes called Legs, but most commonly in Right-Angled Triangles, the Bottom Line, as BC is called the Base, AC the Perpendicular, and the longest Line AB is called the Hypothenuse. The Sides are all in proportion to the Sins of their opposite Angles; so that any three parts of the six being known, the rest may easily be searched out. 6. When an Angle exceeds 90 Degrees, subtract it out of 180, and work by the remainder. CASE i. In Right-Angled Triangles, the Base being given, and the Acute Angle at the Base; how to find the Hypothenusal Line, and the Perpendicular. mathematical figure As the Sine Compliment of the Angle at A is to the Logarithm of the Base 26, So is Radius or the Sine of 90° to the Logarithm of the Hypothenuse AC 30. The Sine Compliment of 30 Degrees is 9,937531 The Logarithm of 26 is 1,414973 The Radius, or Sine of 90° 10,000000 The two last added together 11,414973 Remains, after Substracting the first Number 1,477442 Which if you look for in your Logarithm Tables, you will find the Number answering thereto to be 30, and so long is the Hypothenusal-line required. Note in your Tables, when you cannot find exactly the Logarithm you look for, you must take the nearest thereto, as in this Example I find 1,477121 to be the nearest to 1477442. Mark also, that whereas I say, as the Sine-complement of the Angle at A, etc. you may as well say, as the Sine of the Angle at C is to the Log. etc. for the Angle at A being given in a Right-angled Triangle, you cannot be ignorant of the Angle at C. If you mind the Rule above, that all the three Angles of a Triangle are equal to two right Angles, or 180 Degrees; for if you take the Right-Angle at B 90°, and that at A 30° both known, and subtract them out 180°, there remains only 60° for the Angle at C. But in pursuance of our Question. How to find the Perpendicular. As the Sine of the Angle ACB 60° is to the Log. of the Base 26 AB; So the Sine of the Angle CAB 30° to the Log. of the Perpendicular CB 15. Note, when I put three Letters to express an Angle, the Middlemost Letter denotes the Angular-Point. The Sine of 60 deg. is 9,937531 The Log. of the Base 26 AB, is 1,414973 The Sine of 30 deg. is 9,698970 The two last added 11,113943 From which subtract the first, and remains 1,176412 The nearest number answering to which is 15, which is the Length of the Perpendicular-line CB. Or otherwise; the Hypothenusal-line being first found, viz. AC 30. you may find the Perpendicular thus: As the Sine of the Right-Ang. CBA or Rad. 10,000000 is to the Log. of the Hypoth. AC 30 1,477121 So is the Sine of the Angle CAB 30 deg. 9,698970 to the Log. of the Perpendicular 15 1,176091 CASE two. The Perpendicular and Angle ACB being given to find the Base and Hypothenusal. mathematical figure As the Co-sine of the Angle ACB is to the Logarith. of the Perpendicular BC 15; So is the Sine of the Angle ACB to the Logarith. of the Base AB 26. The Co-sine of the Angle ACB 60°, is 9,698970 The Log. of CB 15, is 1,176091 The Sine of the Angle ACB 60, is 9,937531 11,113622 The nearest Log. answering to 26, is 1,414652 For the Hypothenusal. As the Sine-complement of the Angle ACB 60° is to the Log. of the Perpendicular CB 15 So is the Sine of the Angle ABC, or Radius 90° to the Log. of the Hypothenusal 30° The Co-sine of the Angle ACB, is 9,698970 The Log. of the Perpend. CB 15, is 1,176091 The Radius 10,000000 The Log. of the Hypothenusal 30 1,477121 Or otherwise thus; the Base being first found, to find the Hypothenusal. As the Sine of the Angle ACB 60° 9,937531 is to the Log. of the Base 26 1,414973 So is Radius 10,000000 to the Log. of the Hypothenusal (30) 1,477442 CASE three The Hypothenusal, and either of the Acute Angles given, to find the Base and Perpendicular. mathematical figure Let the Hypothenusal be AC 30 The Angle CAB 30° To find the Base AB, work thus: As the Sine of the Right-Angle CBA 90°, or Radius 10,000000 is to the Log. of the Hypoth. AC 30 1,477121 So is the Co-sine of the Angle CAB 30 9,937531 to the Log. of the Base AB (26) 1,414652 To find the Perpendicular CB, work thus. As the Sine of the Right-Angle CBA 90°, or Radius 10,000000 is to the Log. of the Hypoth. AC 30 1,477121 So is the Sine of the Angle CAB 30 9,698970 to the Log. of the Perpend. (15) 1,176091 Or otherwise; the Base being found, to find the Perpendicular thus: As the Co-sine of the Angle CAB 30° 9,937531 is to the Log. of the Base AB 26 1,414973 So is the Sine of the Angle CAB (30°) 1,698970 11,113943 to the nearest Log. of the Perpend. (15) 1,176412 CASE iv. The Hypothenusal and Base being given, to find the two Acute Angles, viz. ACB, and CAB. Let AC, the Hypothenusal, be 30° AB the Base 26. and the Angle ACB required. mathematical figure As the Logarithm of the Hypothenusal AC 30 is to Radius, or the Sine of the Angle CBA 90; So is the Logarithm of the Base AB 26 to the Sine of the Angle ACB 60. The Operation. The Logar. of the Hypothenusal AC 30 is 1,477121 The Radius 10,000000 The Logarithm of the Base AB 26 1,414973 The Sine of ACB, the Angle required, 60° 9,937852 For the Angle CAB, work thus. As the Logar. of the Hypothenuse AC 30 1,477121 is to the Radius 90 10,000000 So is the Logarithm of the Base AB 26 1,414973 to the Cousin of the Angle required 30 9,937852 CASE v. The Hypothenusal and Perpendicular being given, to find the Angles and Base. mathematical figure The Hypothenusal is 30 The Perpendicular 15 ABC a Right Angle. Now to find the Angle at A work thus. As the Logar. of the Hypothenusal AC 30 1,477121 to the Radius 10,000000 So is the Logar. of the Perpendicular 15 CB 1,176091 to the Sine of the Angle at A 30° 9,698970 To find the Angle at C work thus. As the Logarithm of the Hypothenusal AC 30 is to the Radius 90 Degrees, So is the Logarithm of the Perpendicular CB 15 to the Co-sine of the Angle at A 30, viz. 60 Deg. Lastly to find the Base, work as you were taught in Case 2. Here note that any two Sides of a Right Angled Triangle being given: the third Side may be found by extraction of the Square Root. EXAMPLE. mathematical figure In the Right Angled Triangle A, let the given Base be 20, the Perpendicular 15, and the Hypothenusal required. Square the Base 20, or multiply it by itself, and it makes 400; Square also the Perpendicular 15, and it makes 225, add the two Squares together, and they make 625, from which Sum extract the Square Root, which Root is the length of the Hypothenusal, viz. 25; but if the Hypothenusal, and either of the other Sides be given to find the third, you must Subtract the Lesser Square out of the Greater, and the Root of the remainder is the Side required: As for Example, the Hypothenusal 25 is given, and the Base 20, to find the Perpendicular multiply the Hypothenusal in itself, and it makes 625 Multiply the Base in itself and it makes 400 which 400 Subtract from 625, there remains 225 the Root of which is 15, the Perpendicular required. CASE vi. Of Obliqne Angled Plain Triangles. Two Sides of an Obliqne Triangle being given, and an Angle opposite to either of the Sides, how to find the other two Angles and the third Side. mathematical figure In the Triangle ABC there is given the Side AB 40, the Side BC 32, the Angle at A 40 Degrees, and the Angle at C is required. Note that in Obliqne Triangles, the same Rule holds good as in Right-Angled Triangles; viz. That the Sides are in such proportion one to another, as the Sins of their opposite Angles. As the Logarithm of the Side BC 32 1,505150 is to the Sine of the Angle A 40 9,808067 So is the Logarithm of the Side AB 40 1,602060 11,410127 to the Sine of the Angle at C 53°: 28′ 9.904977 To find the Angle at B, Add the two known Angles together, viz. that at A 40, and that at C 53.28, and they make 93 Degrees 28 Minutes; which substracted from 180 Degrees, leaves 86 Degrees 32 Minutes, which is the Angle at B. Lastly, to find the Line AC, say, As the Sine of the Angle A 40 9,808067 is to the Logarithm of the Side BC 32 1,505150 So is the Sine of the Angle B 86°: 32 9,999204 11,504354 to the Log. of the Side AC required 50 1,696287 Mark, that though the nearest whole number answering to the Logarithm 1,696287 be 50; yet if you go to Fractions, the length of the Line AC is but 49 69/100. CASE seven. Two Angles being given, and a Side opposite to one of them, to find the other opposite Side. In the foregoing Triangle there is given the Angle A 40 Degrees, the Angle C 53 Degrees 28 Minutes; also the Side AB 40: To find the Side BC work thus. As the Sine of the Angle C 53°: 28′ 9,904992 is to the Logarithm of the Side AB 40 1,602060 So is the Sine of the Angle A 40 9,808067 11,410127 To the Log. of the Side BC, nearest 32 1,505135 CASE viij. Two Sides of a Triangle being given, with the Angle contained by them, to find either of the other Angles. mathematical figure In the Triangle ABC there is given the Side AB 197 The Side AC 500 The Angle at A 40 Degrees; Now to find either of the other Angles work thus. As the Log. of the Sum of the 2 Sides 697 2,843233 is to the Logar. of their Difference 303 2,481443 So is the Tangle of the half Sum of the two Opposite Angles 70 Degrees 10,438934 12,920377 to the Tangent of 50 Degrees 4 Min. 10,077144 which 50° 4′ added to the half Sum of the two unknown Angles, viz. 70° makes 120° 4′, which is the Quantity of the Angle at B, also taken from 70, leaves 19 deg. 56′, which is the Angle at C. CASE ix. Three Sides of an Obliqne Triangle being given, to find the Angles. mathematical figure You must first Divide your Obliqne Triangle into two Right Angled Triangles thus. In the Triangle ABC The Side AC is 50 The Side AB 36 The Side BC 20 The Sum of the two Lesser Sides 56 The Difference of the two Lesser Sides 16 As the Log. of the greatest Side AC 50 1,698970 is to the Logar. of the Sum of the two Lesser Sides 56 1,748188 So is the Differ. of the two Lesser Sides 16 1,204120 2,952308 to the Log. of a fourth Number 18 1,253338 Subtract this 18 out of the greatest Side AC 50, and there remains 32, the half of which, viz. 16, is the Base of the Lesser Right-Angled Triangle, and the remainder of the Line AC, viz. AD 34 is the Base of the Greater Right-Angled Triangle, with which this Obliqne Triangle is divided. And now of either Right-Angled Triangle BDC, or BDA, you have the Base and Hypothenuse given to find the Angles; which you must do as you were before taught, Case iv. Note that you may better and easier find the fourth Number, for dividing an Oblique-angled Triangle into two Right-Angled Triangles by Vulgar Arithmetic, than by the Tables of Logarithms, for in the above Triangle, if you say, If 50 give 56, what shall 16 give? Multiply and Divide, the Answer is 17 46/50. There is another way used by Arithmeticians, in my Opinion better than the former, which is this. Square the three given Sides, add the two greater Squares together; and from that Sum Subtract the Lesser; half the remainder divide by the greatest Side; the Quotient will be the Base of the Greater Right-Angled Triangle. EXAMPLE. In the foregoing Triangle, the Square of the greatest Side AC 50, is 2500 The Square of the Side AB 36, is 1296 Added together, make 3796 From which subtract the Square of the least Side 400 Remains 3396 The Half 1698 Which 1698 divide by 50 the longest Side; the Quotient is 33 42/50, the Base of the greater Right-Angled Triangle, viz. AD; and that being substracted out of 50, leaves 16 2/50, for the Base of the smaller Right-Angled Triangle, viz. DC. CASE x. The three Sides of an Obliqne Triangle being given, how to find the Superficial Content without knowing the Perpendicular. From half the Sum of the three Sides, subtract each particular Side. Add the Logarithms of the three Differences, also the Log. of half the Sum of the three Sides together. Half the Total is the Log. of the Content required. In the foregoing Triangle, the Sides are 50, 36, 20, their Sum is 106: The half Sum 53. The differences between the half Sum and each particular Side, are 3 Log. 0.477121 17 1.230449 33 1.518514 The half Sum 53 1.724276 Total added 4.950360 The Half 2.475180 The Number answering to that Log. is 298 which is the Content of the Triangle required. By Vulgar Arithmetic, thus. Multiply the First Difference by the Second; that Product by the Third; that Product by the Half Sum. Lasty, Extract the Square-Root, and you have the Superficial Content. So 3 multiplied by 17. makes 51; which multiplied by 33, makes 1683. that multiplied by 53, the half Sum makes 89199. the Square-Root of which is 298, the Content required. CHAP. XIII. Of Heights and Distances. How to take the Height of a Tower, Steeple, Tree, or any such thing. LEt AB be a Tower, whose Height you would know. mathematical figure To this 40 Yards you must add the height of your Instrument from the Ground; or which is better, look through your Fixed-Sight to the Tower, and mark where your Sight falls upon the Tower, and measure from that place to the ground, which add to the former Height found. In this way of taking heights, the Ground ought to be very level, or you may make great Mistakes. Also the Tower or Tree should stand perpendicular: Or else you must measure to such a place, where a Perpendicular would fall, if let down; as AB is not a Perpendicular, but A d, therefore measure the Distance C d, for you Base. This you may plainly understand by the foregoing Figure; for if standing at C, you were to take the Height of the Tower and Steeple to E: The Angle ECB is the same as the Angle ACB; and if you measure only CB or CD, you will make the Height FE the same as DA; which by the Figure you plainly perceive to be a great Error: Therefore to take the Height FE, you should measure from C to F. How to take the Height of a Tower, etc. when you cannot come nigh the Foot thereof. In the foregoing Figure, let AB be the Tower, and suppose CB to be a Moat, or some other hindrance, that you cannot come nigher than C to take the Height. Therefore at C plant your Instrument, and take (as before) the Angle ACB 58 deg. Then go backwards any convenient distance, as to G, there also take the Angle AGB 38 deg. This done, subtract 58 from 180, so have you 122 deg. the Angle ACG. Then 122 and 38 being taken from 180, remains 20 for the Angle GAC. The Distance GC measured, is 26. Now by Trigonometry, say, As the Sine of the Angle A 20 9534052 is to the Log. of the Distance GC 26 1414973 So is the Sine of the Angle G 38 9789342 11204315 to the Log of the Line AC 47 1,670263 Again, As Radius the Right-Angle B 10,000000 is to the Log. of the Line AC 47 1672098 So is the Sine of the Angle C 58 9928420 To the Log. Height of the Tower 40 Yards 1,600518 But still, as I told you before, the Ground is understood to be level. However, if it be not, I will show you, How to take the Height of a Tower, etc. when the Ground either riseth or falls. mathematical figure To take this at two Stations, without approaching the Foot of the Tower, is no more than what has been said before; for if you take your Angles at C, and then measure to F, and there in like manner, as before, take your Angles again, thereby you may find all the Angles, and the Line OF, then say, As the Sine of the Angle ABF is to the Logarithm of the Line FAVORINA, So is the Sine of the Angle AFB To the Logarithm of the Height of the Tower AB. Of Distances. Although I have before shown how to take Distances by Surveying a Field at two Stations, yet since it seems naturally to come in here again, I will give you one Example thereof: Suppose this following Figure to be a Piece of a River, and you measuring along one Side of it, would as well know the Breadth of it, as also make a true Plot thereof, by putting down what remarkable things are seen on the other Side. mathematical figure To Protract this, draw the Line NS for a Meridian, and laying your Protractor upon it, the Centre thereof to ☉ 1; against NW 6 make a Mark for the Line that goes to ☉ 2. Also against NW 17 make a Mark for the Tree, and against 40 and 52, for the Windmill and House. Then from ☉ 1 through these Marks draw the Lines ☉ A, ☉ B, ☉ C, ☉ 2. Secondly, Take from your Scale 18 Ch. 20 Lin. and set it off upon the Line ☉ 2, which will reach to ☉ 2. There lay again the Centre of your Protractor, the Diameter thereof parallel to the Line NS, and make Marks, as you see in the Field-Book, against NE 15. NW 77. sweet 20. sweet 50. NW 28. NW 4°. and through these Marks draw Lines. The first Line directs to your third Station; the second Line NW 77. directs you to the Tree C upon the River's bank; for that Line cutting the Line ☉ 1 C, shows you by the Intersection where the Tree stood, and also the Breadth of the River. Also the Line sweet 20 cuts the Line from the first Station NW 52, in the place where the House A stands upon the Bank of the River. If therefore you draw a Line from A to C, it will represent the farther Bank of the River. And so you may proceed on Plotting, according to the Notes in your Field-Book; and you will not only have a true Plot of the River, but also know how far the Windmill B, and the House D, stand from the Waterside. How to take the Horizontal-line of a Hill. When you measure a Hill, you must measure the Superficies thereof, and accordingly cast up the Contents. But when you Plot it down, because you cannot make a Convex Superficies upon the Paper, you must only plot the Horizontal or Base thereof; which you must shadow over with the resemblance of a Hill, that other Surveyors, when they apply your Scale thereto, may not say you was Mistaken. And you may find this Horizontal or Base-line, after the same manner as you have been taught to take heights. mathematical figure But if you have occasion to measure the whole Hill, plant again your Instrument at B, and take the Angle CBD, which let be 46 deg. Measure also the Distance BC 21 Ch. Then say, As Radius 10000000 is to the Line BC 21 Ch. 00 Lin. (Log.) 1322219 So is the Line of the Angle CBD 46 9856934 to the part of the Base DC 15 Ch. 12 Lin. 1,179153 Which 15. 12. added to 8.90, makes 24 Chains, 2. Links, for the whole Base AGNOSTUS; which is to be plotted, and not AB and BC; although they are to be measured to find the Content of the Land. I mentioned this way, for your better understanding how to take the Base of part of a Hill; for many times your Survey ends upon the side of a Hill. But if you find you are to take in the whole Hill, you need not take altogether so much pains as by the former way. As thus: Take, as before, the Angle A 58 deg. Measure also AB. Then at B take the whole Angle ABC 78 deg. Subtract these two from 180 deg. remains 44 for the Angle at C. Then say, As the Sine of the Angle C 44 is to the Log. of the Side AB; So is the Sine of the Angle ABC to the Log. of the Base AC. How to take the Shoals of a River's Mouth, and Plot the same. Measure first the Sea coast on both Sides of the River Mouth, as far as you think you shall have occasion to make use thereof; and make a fair Draught thereof, putting down every remarkable thing in its true Situation, as Trees, Houses, Towns, Wind-mills, etc. Then going out in a Boat to such Sands or Rocks as make the Entrance difficult, at every considerable bend of the Sands, take with a Sea-Compass the bearing thereof to two known Marks upon the Shore, and having so gone round all the Sands and Rocks, you may easily upon the Plot before taken, draw Lines which shall intersect each other at every considerable Point of the Sands, whereby you may truly prick out the Sands, and give good Directions either for laying Buyos, or making Marks upon the Shore for the Direction of Shipping. EXAMPLE mathematical figure It would be too tedious for you, and troublesome for me, to give you all the Observations, I having already in this Treatise so often described the same thing before; therefore I will mention only one place of Observation more; and if by that you do not understand the whole, I know not how to make you. In the Sand C, I find the bend (2,) and there, as I should do at all the rest, I take two Observations to such things on the Shore, as are most conspicuous unto me, viz. First, to the Beacon, which bears from me S. W. 25 deg. Secondly, to the Windmill, which bears from me N. W. 40 deg. Now after I have taken the other Angles or Bends of that Sand, and am come Home, I draw a Line from the Beacon-opposite to my Observation S. W. 25 deg. viz. N. E. 25 deg. Also from the Wind mill I draw a Line S. E. 40 deg. Now where these two Lines intersect each other, as they do at 2, I mark for one Point of the Sand C. In like manner as I did this, I observe, and protract every Line of the Sand C, and of all the other Sands and Rocks, be there never so many; and so will you have a fair Map, fitting for Seamens Use, better done, I think, than in any place of the World yet, except for the Harbours of Utopia. Now to give Direction for Seamens coming in here, draw a Line through the middle of the South Channel, which Line will cut both the Church and Windmill; so that if a Ship coming from the Southward, brings the Church and Windmill both into one, and keep them so, she may boldly run in, till she brings the River's mouth fair open, and then sail up the River. Likewise coming from the Northward, must first bring the Tree and Beacon both into one, and keep them so till the River's mouth is fair open. But lest they should mistake, and run upon the ends of the Sands A or B, it would be necessary that a Mark was set up behind the Red-House, in a straight Line with the middle of the River, as Then a Ship coming from the Southward, or Northward, let her keep her former Marks both in one, till she bring the Red House and both in one; and then keeping them so, run boldly up the River, till all Danger is past. I have put down this Windmill and Beacon, not as if such good Marks would always happen; but to show you how to place Marks, if it be required; or to lay Buoys. You must mind after you have taken all the Sands, to take the Soundings also, quite cross the Channels, all up and down, and to put them down accordingly; the best time for doing which, is at Low-Water, in Spring-Tides. How to know whether Water may be made to run from a Springhead to any appointed Place. For this Work, the Diameter of the Semicircle is a little too short; however an indifferent shift may be made therewith, but it is better to get a Water-level, such as you may buy at the Instrument-makers'; with which being provided, as also with two Assistants, and each of them with a Staff divided into Feet, Inches, and Parts of an Inch, go to the Springhead; and causing your first Assistant to stand there with his Staff perpendicular, make the other go in a Right-line towards the place designed for bringing the Water, any convenient distance, as 100, 150, or 200 Yards, and there let him stand, and hold his Staff perpendicular also. Then set your Instrument nigh the Midway between them, making it stand Level, or Horizontal; and look through the Sights thereof to your first Assistant's Staff, he moving a white piece of Paper up or down the Staff, according to the Signs you make to him, till through the Sights you espy the very Edge of the Paper. Then by a Sign make him to understand that you have done with him; and let him write down how many Feet, Inches and Parts the Paper rested upon. Also going to the other end of your Level, do the same by the second Assistant, and let him write down also what number of Feet, etc. the Paper was from the Ground. This done, let your first Assistant come to the second Assistant's place, and there let him again stand with his Staff; and let the second Assistant go forward 100 or 200 Yards, as before; and placing yourself and Instrument in the midst, between them, take your Observations altogether, as before, and let them put them down in like manner: And so must you do till you come to the place whereto the Water is to be conveyed. Then examine the Notes of both your Assistants, and if the Notes of the second Assistant exceed that of the first, you may be sure the Place is lower than the Springhead, and that therefore Water may be well conveyed. But if the first's Notes exceed the seconds, you may conclude it impossible, without Engines, or the like. The first Assistant's Note Stat. Feet Inch. Pts. ☉ 1 4 3 5 ☉ 2 12 4 2 ☉ 3 3 5 1 20 0 8 The second Assistants Note Stat. Feet Inch. Pts. ☉ 1 14 5 1 ☉ 2 4 6 3 ☉ 3 9 2 4 28 1 8 Here you see the second Assistant's Note exceeds the first, 8 Feet, 1 Inch; which is enough to bring the Water with a strong current, and to make it also rise up 6 or 7 Feet in the House, if occasion be; for such as have written of this Matter, allow but 4 Inches and ½ Fall in a Mile to make the Water run. A TABLE OF THE Northing or Southing, Easting or Westing of every Degree from the Meridian, according to the Number of Chains run upon any Degree. Distance, 1 Deg. Distance, 2 Deg. Distance, 3 Deg. N. S. E. W. N. S. E. W. N. S. E. W. 1 1.0 .0 1 1.0 .0 1 1.0 .1 2 2.0 .0 2 2.0 .1 2 2.0 .1 3 3.0 .0 3 3.0 .1 3 3.0 .1 4 4.0 .1 4 4.0 .1 4 4.0 .2 5 5.0 .1 5 5.0 .2 5 5.0 .2 6 6.0 .1 6 6.0 .2 6 6.0 .3 7 7.0 .1 7 7.0 .2 7 7.0 .4 8 8.0 .1 8 8.0 .3 8 8.0 .4 9 9.0 .2 9 9.0 .3 9 9.0 .5 10 10.0 .2 10 10.0 .3 10 10.0 .5 20 20.0 .4 20 20.0 .7 20 20.0 1.0 30 30.0 .5 30 30.0 1.0 30 30.0 1.6 40 40.0 .7 40 40.0 1.4 40 40.0 2.1 50 50.0 .9 50 50.0 1.7 50 50.0 2.6 60 60.0 1.1 60 60.0 2.1 60 59.9 3.1 70 70.0 1.2 70 70.0 2.4 70 69.9 3.7 80 80.0 1.4 80 80.0 2.8 80 79.9 4.2 90 90.0 1.6 90 89.9 3.1 90 89.9 4.7 100 100.0 1.8 100 99.9 3.5 100 99.9 5.2 Dist. E. W. N. S. Dist. E. W. N. S. Dist. E. W. N. S. 89 Deg. 88 Deg. 87 Deg. Distance, 4 Deg. Distance, 5 Deg. Distance, 6 Deg. N. S. E. W. N. S. E. W. N. S. E. W. 1 1.0 .1 1 1.0 .1 1 1.0 .1 2 2.0 .1 2 2.0 .2 2 2.0 .2 3 3.0 .2 3 3.0 .3 3 3.0 .3 4 4.0 .3 4 4.0 .3 4 4.0 .4 5 5.0 .3 5 5.0 .4 5 5.0 .5 6 6.0 .4 6 6.0 .5 6 6.0 .6 7 7.0 .5 7 7.0 .6 7 7.0 .7 8 8.0 .6 8 8.0 .7 8 8.0 .8 9 9.0 .6 9 9.0 .8 9 8.9 .9 10 10.0 .7 10 10.0 .9 10 9.9 1.0 20 20.0 1.4 20 19.9 1.7 20 19.9 2.1 30 29.9 2.1 30 29.9 2.6 30 29.8 3.1 40 39.9 2.8 40 39.8 3.5 40 39.8 4.2 50 49.9 3.5 50 49.8 4.4 50 49.7 5.2 60 59.9 4.2 60 59.8 5.3 60 59.7 6.3 70 69.8 4.9 70 69.7 6.1 70 69.6 7.3 80 79.8 5.7 80 79.7 7.1 80 79.6 8.3 90 89.8 6.3 90 89.7 7.9 90 89.5 9.4 100 99.8 7.0 100 99.6 8.7 100 99.5 10.4 Dist. E. W. N. S. Dist. E. W. N. S. Dist. E. W. N. S. 86 Deg. 85 Deg. 84 Deg. Distance, 7 Deg. Distance, 8 Deg. Distance, 9 Deg. N. S. E. W. N. S. E. W. N. S. E. W. 1 1.0 .1 1 1.0 .1 1 1.0 .2 2 2.0 .2 2 2.0 .3 2 2.0 .3 3 3.0 .4 3 3.0 .4 3 3.0 .5 4 4.0 .5 4 4.0 .6 4 4.0 .6 5 5.0 .6 5 5.0 .7 5 4.9 .8 6 6.0 .7 6 5.9 .8 6 5.9 .9 7 6.9 .8 7 6.9 1.0 7 6.9 1.1 8 7.9 1.0 8 7.9 1.1 8 7.9 1.3 9 8.9 1.1 9 8.9 1.3 9 8.9 1.4 10 9.9 1.2 10 9.9 1.4 10 9.9 1.6 20 19.9 2.4 20 19.8 2.8 20 19.8 3.1 30 29.8 3.7 30 29.7 4.2 30 29.6 4.7 40 39.7 4.9 40 39.6 5.6 40 39.5 6.3 50 49.6 6.1 50 49.5 7.0 50 49.4 7.8 60 59.6 7.3 60 59.4 8.3 60 59.3 9.4 70 69.5 8.5 70 69.3 9.7 70 69.1 10.9 80 79.4 9.8 80 79.2 11.1 80 79.0 12.5 90 89.3 11.0 90 89.1 12.5 90 88.9 14.1 100 99.3 12.2 100 99.0 13.9 100 98.8 15.6 Dist. E. W. N. S. Dist. E. W. N. S. Dist. E. W. N. S. 83 Deg. 82 Deg. 81 Deg. Distance, 10 Deg. Distance, 11 Deg. Distance, 12 Deg N. S. E. W. N. S. E. W. N. S. E. W. 1 1.0 .2 1 1.0 .2 1 1.0 .2 2 2.0 .3 2 2.0 .4 2 2.0 .4 3 3.0 .5 3 2.9 .6 3 2.9 .6 4 3.9 .7 4 3.9 .8 4 3.9 .8 5 4.9 .9 5 4.9 .9 5 4.9 1.0 6 5.9 1.0 6 5.9 1.1 6 5.9 1.2 7 6.9 1.2 7 6.9 1.3 7 6.8 1.5 8 7.9 1.4 8 7.8 1.5 8 7.8 1.7 9 8.9 1.6 9 8.8 1.7 9 8.8 1.9 10 9.9 1.7 10 9.8 1.9 10 9.8 2.1 20 19.7 3.5 20 19.6 3.8 20 19.6 4.2 30 29.6 5.2 30 29.4 5.7 30 29.3 6.2 40 39.4 6.9 40 39.3 7.6 40 39.1 8.3 50 49.2 8.7 50 49.1 9.5 50 48.9 10.4 60 59.1 10.4 60 58.9 11.4 60 58.7 12.5 70 68.9 12.1 70 68.7 13.4 70 68.5 14.6 80 78.8 13.9 80 78.5 15.3 80 78.3 16.6 90 88.6 15.6 90 88.3 17.2 90 88.0 18.7 100 98.5 17.4 100 98.9 19.1 100 97.8 20.8 Dist. E. W. N. S. Dist. E. W. N. S. Dist. E. W. N. S. 80 Deg. 79 Deg. 78 Deg. Distance, 13 Deg. Distance, 14 Deg. Distance, 15 Deg. N. S. E. W. N. S. E. W. N. S. E. W. 1 1.0 .2 1 1.0 .2 1 1.0 .3 2 2.0 .4 2 1.9 .5 2 1.9 .5 3 2.9 .7 3 2.9 .7 3 2.9 .8 4 3.9 .9 4 3.9 1.0 4 3.9 1.0 5 4.9 1.1 5 4.8 1.2 5 4.8 1.3 6 5.9 1.3 6 5.8 1.4 6 5.8 1.6 7 6.8 1.6 7 6.8 1.7 7 6.8 1.8 8 7.8 1.8 8 7.8 1.9 8 7.7 2.1 9 8.8 2.0 9 8.7 2.2 9 8.7 2.3 10 9.8 2.2 10 9.7 2.4 10 9.7 2.6 20 19.5 4.5 20 19.4 4.8 20 19.3 5.2 30 29.2 6.7 30 29.1 7.3 30 29.0 7.8 40 39.0 9.0 40 38.8 9.7 40 38.6 10.3 50 48.7 11.2 50 48.5 12.1 50 48.3 12.9 60 58.5 13.5 60 58.2 14.5 60 58.0 15.5 70 68.2 15.7 70 67.9 16.9 70 67.6 18.1 80 78.0 18.0 80 77.6 19.4 80 77.3 20.7 90 87.7 20.2 90 87.3 21.8 90 86.9 23.3 100 97.4 22.5 100 97.0 24.2 100 96.6 25.9 Dist. E. W. N. S. Dist. E. W. N. S. Dist. E. W. N. S. 77 Deg. 76 Deg. 75 Deg. Distance, 16 Deg Distance, 17 Deg. Distance, 18 Deg. N. S. E. W. N. S. E. W. N. S. E. W. 1 1.0 .3 1 1.0 .3 1 1.0 .3 2 1.9 .6 2 1.9 .6 2 1.9 .6 3 2.9 .8 3 2.9 .9 3 2.8 .9 4 3.8 1.1 4 3.8 1.2 4 3.8 1.2 5 4.8 1.4 5 4.8 1.5 5 4.7 1.5 6 5.8 1.7 6 5.7 1.7 6 5.7 1.8 7 6.7 1.9 7 6.7 2.0 7 6.6 2.2 8 7.7 2.2 8 7.6 2.3 8 7.6 2.5 9 8.6 2.5 9 8.6 2.6 9 8.5 2.8 10 9.6 2.8 10 9.6 2.9 10 9.5 3.1 20 19.2 5.5 20 19.1 5.8 20 19.0 6.2 30 28.8 8.3 30 28.7 8.8 30 28.5 9.3 40 38.4 11.0 40 38.3 11.7 40 38.0 12.4 50 48.1 13.8 50 47.8 14.6 50 47.6 15.4 60 57.7 16.5 60 57.4 17.5 60 57.1 18.5 70 67.3 19.3 70 66.9 20.5 70 66.6 21.6 80 76.9 22.0 80 76.5 23.4 80 76.1 24.7 90 86.5 24.8 90 86.1 26.3 90 85.6 27.8 100 96.1 27.6 100 95.6 29.2 100 95.1 30.9 Dist. E. W. N. S. Dist. E. W. N. S. Dist. E. W. N. S. 74 Deg. 73 Deg. 72 Deg. Distance, 19 Deg. Distance, 20 Deg. Distance, 21 Deg. N. S. E. W. N. S. E. W. N. S. E. W. 1 .9 .3 1 .9 .3 1 .9 .4 2 1.9 .6 2 1.9 .7 2 1.9 .7 3 2.8 1.0 3 2.8 1.0 3 2.8 1.1 4 3.8 1.3 4 3.8 1.4 4 3.7 1.4 5 4.7 1.6 5 4.7 1.7 5 4.7 1.8 6 5.7 2.0 6 5.6 2.0 6 5.6 2.1 7 6.6 2.3 7 6.6 2.4 7 6.5 2.5 8 7.5 2.6 8 7.5 2.7 8 7.5 2.9 9 8.5 2.9 9 8.5 3.1 9 8.4 3.2 10 9.4 3.3 10 9.4 3.4 10 9.3 3.6 20 18.9 6.5 20 18.8 6.8 20 18.7 7.2 30 28.4 9.8 30 28.2 10.3 30 28.0 10.7 40 37.8 13.10 40 37.6 13.7 40 37.3 14.3 50 47.3 16.3 50 47.0 17.1 50 46.7 17.9 60 56.7 19.5 60 56.4 20.5 60 56.0 21.5 70 66.2 22.8 70 65.8 23.9 70 65.3 25.1 80 75.6 26.1 80 75.2 27.4 80 74.7 28.7 90 85.1 29.3 90 84.6 30.8 90 84.0 32.3 100 94.5 32.6 100 94.0 34.2 100 93.3 35.8 Dist. E. W. N. S. Dist. E. W. N. S. Dist. E. W. N. S. 71 Deg. 70 Deg. 69 Deg. Distance, 22 Deg. Distance, 23 Deg. Distance, 24 Deg. N. S. E. W. N. S. E. W. N. S. E. W. 1 .9 .4 1 .9 .4 1 .9 .4 2 1.9 .7 2 1.8 .8 2 1.8 .8 3 2.8 1.1 3 2.8 1.2 3 2.7 1.2 4 3.7 1.5 4 3.7 1.6 4 3.6 1.6 5 4.6 1.9 5 4.6 1.9 5 4.6 2.0 6 5.6 2.2 6 5.5 2.3 6 5.5 2.4 7 6.5 2.6 7 6.4 2.7 7 6.4 2.8 8 7.4 3.0 8 7.4 3.1 8 7.3 3.2 9 8.3 3.4 9 8.3 3.5 9 8.2 3.7 10 9.3 3.7 10 9.2 3.9 10 9.1 4.1 20 18.5 7.5 20 18.4 7.8 20 18.3 8.1 30 27.8 11.2 30 27.6 11.7 30 27.4 12.2 40 37.1 15.0 40 36.8 15.6 40 36.5 16.3 50 46.4 18.7 50 46.0 19.5 50 45.7 20.3 60 55.6 22.5 60 55.2 23.4 60 54.8 24.4 70 64.9 26.2 70 64.4 27.3 70 63.9 28.5 80 74.2 30.0 80 73.6 31.2 80 73.1 32.5 90 83.4 33.7 90 82.8 35.2 90 82.2 36.6 100 92.7 37.5 100 92.0 39.1 100 91.3 40.7 Dist. E. W. N. S. Dist. E. W. N. S. Dist. E. W. N. S. 68 Deg. 67 Deg. 66 Deg. Distance, 25 Deg. Distance, 26 Deg. Distance, 27 Deg. N. S. E. W. N. S. E. W. N. S. E. W. 1 .9 .4 1 .9 .4 1 .9 .5 2 1.8 .8 2 1.8 .9 2 1.8 .9 3 2.7 1.3 3 2.7 1.3 3 2.7 1.4 4 3.6 1.7 4 3.6 1.8 4 3.6 1.8 5 4.5 2.1 5 4.5 2.2 5 4.5 2.3 6 5.4 2.5 6 5.4 2.6 6 5.3 2.7 7 6.3 3.0 7 6.3 3.1 7 6.2 3.2 8 7.2 3.4 8 7.2 3.5 8 7.1 3.6 9 8.1 3.8 9 8.1 3.9 9 8.0 4.1 10 9.1 4.2 10 9.0 4.4 10 8.9 4.5 20 18.1 8.4 20 18.0 8.8 20 17.8 9.1 30 27.2 12.7 30 27.0 13.1 30 26.7 13.6 40 36.2 16.9 40 36.0 17.5 40 35.6 18.2 50 45.3 21.1 50 44.9 21.9 50 44.5 22.7 60 54.4 25.4 60 53.9 26.3 60 53.5 27.2 70 63.4 29.6 70 62.9 30.7 70 62.4 31.8 80 72.5 33.8 80 71.9 35.1 80 71.3 36.3 90 81.6 38.0 90 80.9 39.4 90 80.2 40.9 100 90.6 42.3 100 89.9 43.8 100 89.1 45.4 Dist. E. W. N. S. Dist. E. W. N. S. Dist. E. W. N. S. 65 Deg. 64 Deg. 63 Deg. Distance, 28 Deg. Distance, 29 Deg. Distance, 30 Deg. N. S. E. W. N. S. E. W. N. S. E. W. 1 .9 .5 1 .9 .5 1 .9 .5 2 1.8 .9 2 1.7 1.0 2 1.7 1.0 3 2.6 1.4 3 2.6 1.4 3 2.6 1.5 4 3.5 1.9 4 3.5 1.9 4 3.5 2.0 5 4.4 2.3 5 4.4 2.4 5 4.3 2.5 6 5.3 2.8 6 5.2 2.9 6 5.2 3.0 7 6.2 3.3 7 6.1 3.4 7 6.1 3.5 8 7.1 3.7 8 7.0 3.9 8 6.9 4.0 9 7.9 4.2 9 7.9 4.3 9 7.8 4.5 10 8.8 4.7 10 8.7 4.8 10 8.7 5.0 20 17.7 9.4 20 17.5 9.7 20 17.3 10.0 30 26.5 14.1 30 26.2 14.5 30 26.0 15.0 40 35.3 18.8 40 35.0 19.4 40 34.6 20.0 50 44.1 23.5 50 43.7 24.2 50 43.3 25.0 60 53.0 28.2 60 52.5 29.1 60 52.0 30.0 70 61.8 32.9 70 61.2 33.9 70 60.6 35.0 80 70.6 37.6 80 70.0 38.8 80 69.3 40.0 90 79.5 42.2 90 78.7 43.6 90 77.9 45.0 100 88.3 46.9 100 87.5 48.5 100 86.6 50.0 Dist. E. W. N. S. Dist. E. W. N. S. Dist. N. S. N. S. 62 Deg. 61 Deg. 60 Deg. Distance, 31 Deg. Distance, 32 Deg. Distance, 33 Deg. N. S. E. W. N. S. E. W. N. S. E. W. 1 .9 .5 1 .8 .5 1 .8 .5 2 1.7 1.0 2 1.7 1.1 2 1.7 1.1 3 2.6 1.5 3 2.5 1.6 3 2.5 1.6 4 3.5 2.1 4 3.4 2.1 4 3.4 2.2 5 4.3 2.6 5 4.2 2.6 5 4.2 2.7 6 5.1 3.1 6 5.1 3.2 6 5.0 3.3 7 6.0 3.6 7 5.9 3.7 7 5.9 3.8 8 6.9 4.1 8 6.3 4.2 8 6.7 4.4 9 7.7 4.6 9 7.6 4.8 9 7.6 4.9 10 8.6 5.1 10 8.5 5.3 10 8.4 5.4 20 17.1 10.3 20 17.0 10.6 20 16.8 10.9 30 25.7 15.4 30 25.4 15.9 30 25.2 16.3 40 34.3 20.6 40 33.9 21.2 40 33.5 21.8 50 42.9 25.7 50 42.4 26.5 50 41.9 27.2 60 51.4 30.9 60 50.9 31.8 60 50.3 32.7 70 60.0 36.0 70 59.4 37.1 70 58.7 38.1 80 68.6 41.2 80 67.8 42.4 80 67.1 43.6 90 77.1 46.3 90 76.3 47.7 90 75.5 49.0 100 85.7 51.5 100 84.8 53.0 100 83.9 54.5 Dist. E. W. N. S. Dist. E. W. N. S. Dist. E. W. N. S. 59 Deg. 58 Deg. 57 Deg. Distance, 34 Deg. Distance, 35 Deg. Distance, 36 Deg. N. S. E. W. N. S. E. W. N. S. E. W. 1 .8 .6 1 .8 .6 1 .8 .6 2 1.7 1.1 2 1.7 1.1 2 1.6 1.2 3 2.5 2.7 3 2.5 1.7 3 2.4 1.8 4 3.3 2.2 4 3.3 2.3 4 3.2 2.3 5 4.1 2.8 5 4.1 2.9 5 4.0 2.9 6 5.0 3.4 6 4.9 3.4 6 4.8 3.5 7 5.8 3.9 7 5.7 4.0 7 5.7 4.1 8 6.6 4.5 8 6.6 4.6 8 6.5 4.7 9 7.5 5.0 9 7.4 5.2 9 7.2 5.3 10 8.3 5.6 10 8.2 5.7 10 8.1 5.9 20 16.6 11.2 20 16.4 11.5 20 16.2 11.8 30 24.9 16.8 30 24.6 17.2 30 24.3 17.6 40 33.2 22.4 40 32.8 22.9 40 32.4 23.5 50 41.4 28.0 50 41.0 28.7 50 40.4 29.4 60 49.7 33.5 60 49.1 34.4 60 48.5 35.3 70 58.0 39.1 70 57.3 40.2 70 56.6 41.1 80 66.3 44.7 80 65.5 45.9 80 64.7 47.0 90 74.6 50.3 90 73.7 51.6 90 72.8 52.9 100 82.9 55.9 100 81.9 57.4 100 80.9 58.8 Dist. E. W. N. S. Dist. E. W. N. S. Dist. E. W. N. S. 56 Deg. 55 Deg. 54 Deg. Distance, 37 Deg. Distance, 38 Deg. Distance, 39 Deg. N. S. E. W. N. S. E. W. N. S. E. W. 1 .8 .6 1 .8 .6 1 .8 .6 2 1.6 1.2 2 1.6 1.2 2 1.6 1.3 3 2.4 1.8 3 2.4 1.8 3 2.3 1.9 4 3.2 2.4 4 3.1 2.5 4 3.1 2.5 5 4.0 3.0 5 3.9 3.1 5 3.9 3.1 6 4.8 3.6 6 4.7 3.7 6 4.7 3.8 7 5.6 4.2 7 5.5 4.3 7 5.4 4.4 8 6.4 4.8 8 6.3 4.9 8 6.2 5.0 9 7.2 5.4 9 7.1 5.5 9 7.0 5.7 10 8.0 6.0 10 7.9 6.2 10 7.8 6.3 20 16.0 12.0 20 15.8 12.3 20 15.5 12.6 30 24.0 18.0 30 23.6 18.5 30 23.3 18.9 40 31.9 24.1 40 31.5 24.6 40 31.1 25.2 50 39.9 30.1 50 39.4 30.8 50 38.8 31.5 60 47.9 36.1 60 47.3 36.9 60 46.6 37.8 70 55.9 42.1 70 55.2 43.1 70 54.4 44.0 80 63.9 48.1 80 63.3 49.0 80 62.2 50.3 90 71.9 54.2 90 70.9 55.4 90 69.9 56.6 100 79.9 60.2 100 78.8 61.6 100 77.7 62.9 Dist. E. W. N. S. Dist. E. W. N. S. Dist. E. W. N. S. 53 Deg. 52 Deg. 51 Deg. Distance, 40 Deg. Distance, 41 Deg. Distance, 42 Deg. N. S. E. W. N. S. E. W. N. S. E. W. 1 .8 .6 1 .8 .7 1 .7 .7 2 1.5 1.3 2 1.5 1.3 2 1.5 1.3 3 2.3 1.9 3 2.3 2.0 3 2.2 2.0 4 3.1 2.6 4 3.0 2.6 4 3.0 2.7 5 3.3 3.2 5 3.8 3.3 5 3.7 3.3 6 4.6 3.8 6 4.5 3.9 6 4.4 4.0 7 5.4 4.5 7 5.3 4.6 7 5.2 4.7 8 6.1 5.1 8 6.0 5.2 8 5.9 5.3 9 6.9 5.8 9 6.8 5.9 9 6.7 6.0 10 7.7 6.4 10 7.5 6.6 10 7.4 6.7 20 15.3 12.9 20 15.1 13.1 20 14.9 13.4 30 23.0 19.3 30 22.6 19.7 30 22.3 20.1 40 30.6 25.7 40 30.2 26.2 40 29.7 26.8 50 38.3 32.1 50 37.7 32.8 50 37.2 33.5 60 46.0 38.6 60 45.3 39.4 60 44.6 40.1 70 53.6 45.0 70 52.8 45.9 70 52.0 46.8 80 61.3 51.4 80 60.4 52.5 80 59.4 53.5 90 68.9 57.9 90 67.9 59.0 90 66.9 60.2 100 76.6 64.3 100 75.5 65.6 100 74.3 66.9 Dist. E. W. N. S. Dist. E. W. N. S. Dist. E. W. N. S. 50 Deg. 49 Deg. 48 Deg. Distance, 43 Deg. Distance, 44 Deg. Distance, 45 Deg. N. S. E. W. N. S. E. W. N. S. E. W. 1 .7 .7 1 .7 .7 1 .7 .7 2 1.5 1.4 2 1.4 1.4 2 1.4 1.4 3 2.2 2.0 3 2.2 2.1 3 2.1 2.1 4 2.9 2.7 4 2.9 2.8 4 2.8 2.8 5 3.6 3.4 5 3.6 3.5 5 3.5 3.5 6 4.4 4.1 6 4.3 4.2 6 4.2 4.2 7 5.1 4.8 7 5.0 4.9 7 4.9 4.9 8 5.8 5.4 8 5.8 5.6 8 5.6 5.6 9 6.6 6.1 9 6.5 6.2 9 6.4 6.4 10 7.3 6.8 10 7.2 6.9 10 7.1 7.1 20 14.6 13.6 20 14.4 13.9 20 14.1 14.1 30 21.9 20.5 30 21.6 20.8 30 21.2 21.2 40 29.2 27.3 40 28.8 27.8 40 28.3 28.3 50 36.6 34.1 50 36.0 34.7 50 35.3 35.3 60 43.9 40.9 60 43.2 41.7 60 42.4 42.4 70 51.2 47.7 70 50.3 48.6 70 49.5 49.5 80 58.5 54.6 80 57.5 55.6 80 56.6 56.6 90 65.8 61.4 90 64.7 62.5 90 63.6 63.6 100 73.1 68.2 100 71.9 69.5 100 70.7 70.7 Dist. E. W. N. S. Dist. E. W. N. S. Dist. E. W. N. S. 47 Deg. 46 Deg. 45 Deg. THE USE OF THE Foregoing Table, I Have already sufficiently in the 6th. Chapter of this Book Taught you the use of this Table; however, because it is made somewhat different from such of this kind as have been made by others, I will briefly byan Example, or two, Explain it to you. Admit in Surveying a Wood, or the like, you run a Line N. E. 40 Degrees, 10 Chains: or in plainer terms, a Line 10 Chains in Length, that makes an Angle with the Meridian of 40 Degrees to the Eastward; and you would put down in your Field-Book the Northing, and Easting of this Line under their proper Titles N. and E. according to Mr. Norwood's way of Surveying Taught in the 6th. Chapter. First at the Head of the Table find 40 Degrees, then in the Column of distances seek for 10 Chains: which had, you will find to stand right against it under the Title N. 7. 7, for the Northing, which is 7 Chains, 7/10 of a Chain: and for the Easting under the Title E. 6. 4, which is 6 Chains 4/10 of a Chain, as nigh as may be expressed in the Tenth part of a Chain: But if you would know to one Link, add an 0 to the distance, so will 10 be 100, which seek for in the same Page of the Table, and right against it you will find under Title N. 76. 6 or 7 Chains, 66 Links for your Northing, and under Title E, 64. 3, or 6 Chains 43 Links for your Easting: which found, put down in your Field-Book accordingly; and having done so by all your Lines, if you find the Northing, and Southing, the same, also the Easting, and Westing, you may be sure you have wrought true, otherwise not. If the distance consists of odd Chains, and Links, as most commonly it so falls out, then take them severally out of the Table, and by adding all together you will have your desire: as for Example. Suppose my distance run upon any Line be NW. 35 Degrees, 15 Chains, 20 Links: First in the Table I find the Northing of 10 Chains to be N. Ch. Ch. Lin. 10 8 19 5 4 10 20 Links 0 16 4/10 12 45 4/10 which added together makes 12 Chains 45 Links, for the Northing of that distance run: In like manner under 35 Degrees, and Title W, I find the Westing of the same Line, as here Ch. Ch. L. 10 5 74 5 2 87 20 Links 11 4/10 8 72 4/10 by which I conclude the Northing of that Line to be 12 Chains 45 Links, and the Westing 8 Chains 72 Links: which thus you may prove by the Logarithms. As Radius 10,000000 Is to the distance 15.20 3,181844 So is the Sign of the Corpse 35 Deg. 9,758772 To the Westing 8 Chains 72 Links 2,940616 And, as Radius 10,000000 To the distance 15 Chains 20 Links 3,181844 So Cousin of the Course 55 9,913364 To the Northing 12 Chains 45 Links 3,095208 Mark that if your Course had been SE, it would have been the same thing as NW: for you see in the Tables N, and S. E, and W, are joined together. If your Degrees exceed 45, then seek for them at the Foot of the Table: and over the Titles NS, EWE, find out the Northing, Southing, Easting or Westing. I think this to be as much as need be said concerning the preceding Table: As for the finding the Horizontal Line of a Hill, and such like things by the Table, before you have half well read through the Chapter of Trigonometry, your own Ingenuity will fast enough prompt you to it. A TABLE OF Sines & Tangents To every Fifty Minute OF THE QUADRANT. 0. M. SIN. Co-sine TAN. Co-Tangent. 0 0.000000 10.000000 0.000000 Infinita 60 5 7.162696 10.000000 7.162696 12.837304 55 10 7.463726 9.999998 7.463727 12.536273 50 15 7.639816 9.999996 7.639820 12.360180 45 20 7.764754 9.999993 7.764761 12.235239 40 25 7.861662 9.999989 7.861674 12.138326 35 30 7.940842 9.999983 7.940858 12.059142 30 35 8.007787 9.999977 7.007809 11.992191 25 40 8.065776 9.999971 8.065806 11.934194 20 45 8.116926 9.999963 8.116963 11.883037 15 50 8.162681 9.999954 8.162737 11.837273 10 55 8.204070 9.999944 8.204126 11.795874 5 60 8.241855 9.999934 8.241921 11.758079 0 Co-sine SIN. Co-Tang. TAN. M 89. 1. M. SIN. Co-sine TAN. Co-Tangent. 0 8.241855 9.999934 8.241921 11.758079 60 5 8.276614 9.999922 8.276691 11.723309 55 10 8.308794 9.999910 8.308884 11.691116 50 15 8.338753 9.999897 8.338856 11.661144 45 20 8.366777 9.999882 8.366895 11.633105 40 25 8.393101 9.999867 8.393234 11.606766 35 30 8.417919 9.999851 8.418068 11.581932 30 35 8.441394 9.999834 8.441560 11.558440 25 40 8.463665 9.999816 8.463849 11.536151 20 45 8.484848 9.999797 8.485050 11.514950 15 50 8.505045 9.999778 8.505267 11.494733 10 55 8.524343 9.999757 8.524586 11.475414 5 60 8.542819 9.999735 8.543084 11.456916 0 Co-sine SIN. Co-Tang. TAN. M 88 2. M. SIN. Co-sine TAN. Co-Tangent. 0 8.542819 9.999735 8.543084 11.456916 60 5 8.560540 9.999713 8.560828 11.439172 55 10 8.577566 9.999689 8.577877 11.422123 50 15 8.593948 9.999665 8.594283 11.405717 45 20 8.609734 9.999640 8.610094 11.389906 40 25 8.624965 9.999614 8.625352 11.374648 35 30 8.639680 9.999586 8.640093 11.359907 30 35 8.653911 9.999558 8.654352 11.345648 25 40 8.667689 9.999529 8.668160 11.331840 20 45 8.681043 9.999500 8.681544 11.318456 15 50 8.693998 9.999469 8.694529 11.305471 10 55 8.706577 9.999437 8.707140 11.292860 5 60 8.718800 9.999404 8.719396 11.280604 0 Co-sine SIN. Co-Tang. TAN. M 87. 3. M. SIN Co-sine TAN. Co-Tangent. 0 8.718800 9.999404 8.719396 11.280604 60 5 8.730688 9.999371 8.731317 11.268683 55 10 8.742259 9.999336 8.742922 11.257078 50 15 8.753528 9.990301 8.754227 11.245773 45 20 8.764511 9.999265 8.765246 11.234754 40 25 8.775223 9.999227 8.775995 11.224005 35 30 8.785675 9.999189 8.786486 11.213514 30 35 8.795881 9.999150 8.796731 11.203269 25 40 8.805852 9.999110 8.806742 11.103258 20 45 8.815599 9.999069 8.816529 11.183471 15 50 8.825130 9.999027 8.826103 11.173897 10 55 8.834456 9.998984 8.835471 11.164529 5 60 8.843585 9.998941 8.844644 11.155356 0 Co-sine SIN. Co-Tang. TAN. M. 86. 4 M. SIN. Co-sine TAN. Co-Tangent. 0 8.843585 9.998941 8.844644 11.155356 60 5 8.852525 9.998896 8.853628 11.146372 55 10 8.861283 9.998851 8.862433 11.137567 50 15 8.869868 9.998804 8.871064 11.128936 45 20 8.878285 9.998757 8.879529 11.120471 40 25 8.886542 9.998708 8.887833 11.112167 35 30 8.894643 9.998659 8.895984 11.104016 30 35 8.902596 9.998609 8.903987 11.096013 25 40 8.910404 9.998558 8.911846 11.088154 20 45 8.918073 9.998506 8.919568 11.080432 15 50 8.925609 9.998453 8.927156 11.072844 10 55 8.933015 9.998399 8.934616 11.065384 5 60 8.940296 9.998344 8.941952 11.058048 0 Co-sine SIN. Co-Tang. TAN. M 85 5. M. SIN. Co-sine TAN. Co-Tangent. 0 8.940296 9.998344 8.941952 11.058048 60 5 8.947456 9.998289 8.949168 11.050832 55 10 8.954499 9.998232 8.956267 11.043733 50 15 8.961429 9.998174 8.963255 11.036745 45 20 8.968249 9.998116 8.970133 11.029867 40 25 8.974962 9.998056 8.976906 11.023094 35 30 8.981573 9.997996 8.983577 11.016423 30 35 8.988083 9.997935 8.990149 11.009851 25 40 8.994497 9.997872 8.996624 11.003376 20 45 9.000816 9.997809 9.003007 10.996993 15 50 9.007044 9.997745 9.009298 10.990702 10 55 9.013182 9.997680 9.015502 10.984498 5 60 9.019235 9.997614 9.021620 10.978380 0 Co-sine SIN. Co-Tang. TAN. M 84. 6. M. SIN. Co-sine TAN. Co-Tangent. 0 9.019235 9.997614 9.021620 10.978380 60 5 9.025203 9.997547 9.027655 10.972345 55 10 9.031089 9.997480 9.033609 10.966391 50 15 9.036896 9.997411 9.039485 10.960515 45 20 9.042625 9.997341 9.045284 10.954716 40 25 9.048279 9.997271 9.051008 10.948992 35 30 9.053859 9.997199 9.056659 10.943341 30 35 9.059367 9.997127 9.062240 10.937760 25 40 9.064806 9.997053 9.067752 10.932248 20 45 9.070176 9.996979 9.073197 10.926803 15 50 9.075480 9.996904 9.078576 10.921424 10 55 9.080719 9.996828 9.083891 10.916109 5 60 9.085894 9.996751 9.089144 10.910856 0 Co-sine SIN. Co-Tang. TAN. M 83. 7. M. SIN Co-sine TAN. Co-Tangent. 0 9.085894 9.996751 9.089144 10.910850 60 5 9.091008 9.996673 9.094336 10.905664 55 10 9.096062 9.996594 9.099468 10.900532 50 15 9.101056 9.996514 9.104542 10.895458 45 20 9.105992 9.996433 9.109559 10.890441 40 25 9.110873 9.996351 9.114521 10.885479 35 30 9.115698 9.996269 9.119429 10.880571 30 35 9.120469 9.996185 9.124284 10.875716 25 40 9.125187 9.996100 9.129087 10.870913 20 45 9.129854 9.996015 9.133839 10.866161 15 50 9.134470 9.995928 9.138542 10.861458 10 55 9.139037 9.995841 9.143196 10.856804 5 60 9.143555 9.995753 9.147803 10.852197 0 Co-sine SIN. Co-Tang. TAN. M 82. 8 M. SIN. Co-sine. TAN. Co-Tangent. 0 9.143555 9.995753 9.147803 10.852197 60 5 9.148026 9.995664 9.152363 10.847637 55 10 9.152451 9.995573 9.156877 10.843123 50 15 9.156830 9.995482 9.161347 10.838653 45 20 9.161164 9.995390 9.165774 10.834226 40 25 9.165454 9.995297 9.170157 10.829843 35 30 9.169702 9.995203 9.174499 10.825501 30 35 9.173908 9.995108 9.178799 10.821201 25 40 9.178072 9.995013 9.183059 10.816941 20 45 9.182196 9.994916 9.187280 10.812720 15 50 9.186280 9.994818 9.191462 10.808538 10 55 9.190325 9.994720 9.195606 10.804394 5 60 9.194332 9.994620 9.199713 10.800287 0 Co-sine SIN. Co-Tang. TAN. M 81 9 M. SIN. Cousin. TAN. Co-Tangent. 0 9.194332 9.994620 9.199713 10.800287 60 5 9.198302 9.994159 9.203782 10.796218 55 10 9.202234 9.994418 9.207817 10.792183 50 15 9.206131 9.994316 9.211815 10.788185 45 20 9.209992 9.994212 9.215780 10.784220 40 25 9.213818 9.994108 9.219710 10.780290 35 30 9.217609 9.994003 9.223607 10.776393 30 35 9.221367 9.993897 9.227471 10.772529 25 40 9.225092 9.993789 9.231302 10.768698 20 45 9.228784 9.993681 9.235103 10.764897 15 50 9.232444 9.993572 9.238872 10.761128 10 55 9.236073 9.993462 9.242610 10.757390 5 60 9.239670 9.993351 9.246319 10.753681 0 Co-sine. SIN. Co-Tang. TAN. M. 80 10. M. SIN. Co-sine. TAN. Co-Tangent. 0 9.239670 9.993351 9.246319 10.753681 60 5 9.243237 9.993240 9.249998 10.750002 55 10 9.246775 9.993127 9.253648 10.746352 50 15 9.250282 9.993013 9.257269 10.742731 45 20 9.253761 9.992898 9.260863 10.739137 40 25 9.257211 9.992783 9.264428 10.735572 35 30 9.260633 9.992666 9.267967 10.732033 30 35 9.264027 9.992549 9.271479 10.728521 25 40 9.267395 9.992430 9.274964 10.725036 20 45 9.270735 9.992311 9.278424 10.721576 15 50 9.274049 9.092190 9.281858 10.718142 10 55 9.277337 9.992069 9.285268 10.714732 5 60 9.280599 9.991947 9.288652 10.711348 0 Co-sine. SIN. Co-Tang. TAN. M 79. 11. M SIN. Co-sine. TAN. Co-Tangent. 0 9.280599 9.991974 9.288652 10.711348 60 5 9.283836 9.991823 9.292013 10.707987 55 10 9.287048 9.991699 9.295349 10.704651 50 15 9.290236 9.991574 9.298662 10.701338 45 20 9.293399 9.991448 9.301951 10.698049 40 25 9.296539 9.991321 9.305218 10.694782 35 30 9.299655 9.991193 9.308463 10.691537 30 35 9.302748 9.991064 9.311685 10.688315 25 40 9.305819 9.990934 9.314885 10.685115 20 45 9.308867 9.990803 9.318064 10.681936 15 50 9.311893 9.990671 9.321222 10.678778 10 55 9.314897 9.990538 9.324358 10.675642 5 60 9.317879 9.990404 9.327475 10.672525 0 Co-sine. SIN. Co-Tang. TAN. M 78 12 M. SIN. Co-sine. TAN. Co-Tangent. 0 9.317879 9.990404 9.327475 10.672525 60 5 9.320840 9.990270 9.330570 10.669430 55 10 9.323780 9.990134 9.333646 10.666354 50 15 9.326700 9.989997 9.336702 10.663298 45 20 9.329599 9.989860 9.339739 10.660261 40 25 9.332478 9.989721 9.342757 10.667243 35 30 9.335337 9.989582 9.445755 10.664245 30 35 9.338176 9.989441 9.348735 10.651265 25 40 9.340996 9.989300 9.351697 10.648303 20 45 9.343797 9.989157 9.354640 10.645360 15 50 9.346579 9.989014 9.357566 10.642434 10 55 9.349343 9.988869 9.360474 10.639526 5 60 9.352088 9.988724 9.363364 10.636636 0 Co-sine. SIN. Co-Tang. TAN. M. 77 13. M. SIN. Cousin. TAN. Co-Tangent. 0 9.352088 9.988724 9.363364 10.636636 60 5 9.354815 9.988578 9.366237 10.633763 55 10 9.357524 9.988430 9.369094 10.630906 50 15 9.360215 9.988282 9.371933 10.628067 45 20 9.362889 9.988133 9.374756 10.625244 40 25 9.365546 9.987983 9.377563 10.622437 35 30 9.368185 9.987832 9.380354 10.619646 30 35 9.370808 9.987679 9.383129 10.616871 25 40 9.373414 9.987526 9.385888 10.614112 20 45 9.376003 9.987372 9.388631 10.611369 15 50 9.378577 9.087217 9.391360 10.608640 10 55 9.381134 9.987061 9.394073 10.605927 5 60 9.383675 9.986904 9.396771 10.903229 0 Co-sine. SIN. Co-Tang. TAN. M 76. 14. M. SIN. Co-sine. TAN. Co-Tangent. 0 9.383675 9.986904 9.396771 10.603229 60 5 9.386201 9.986746 9.399455 10.600545 55 10 9.388711 9.986587 9.402124 10.597876 50 15 9.391206 9.986427 9.404778 10.595222 45 20 9.393685 9.986266 9.407419 10.592581 40 25 9.396150 9.986104 9.410045 10.589955 35 30 9.398600 9.985942 9.412658 10.587342 30 35 9.401035 9.985778 9.415257 10.584743 25 40 9.403455 9.985613 9.417842 10.582158 20 45 9.405862 9.985447 9.420415 10.579585 15 50 9.408254 9.985280 9.422974 10.577026 10 55 9.410632 9.985113 9.425519 10.574481 5 60 9.412996 9.984944 9.428052 10.571948 0 Co-sine. SIN. Co-Tang. TAN. M. 75. 15. M SIN. Co-sine. TAN. Co-Tangent. 0 9.412996 9.984944 9.428052 10.571948 60 5 9.415347 9.984774 9.430573 10.569427 55 10 9.417684 9.984603 9.433080 10.566920 50 15 9.420007 9.984432 9.435576 10.564424 45 20 9.422318 9.984259 9.438059 10.561941 40 25 9.424615 9.984085 9.440529 10.559471 35 30 9.426899 9.983911 9.442988 10.557012 30 35 9.429170 9.983735 9.445435 10.554565 25 40 9.431429 9.983558 9.447870 10.552130 20 45 9.433675 9.983381 9.450294 10.549706 15 50 9.435908 9.983202 9.452706 10.547294 10 55 9.438129 9.983022 9.455107 10.544893 5 60 9.440338 9.982842 9.457496 10.542504 0 Co-sine. SIN. Co-Tang. TAN. M 74. 16. M SIN. Co-sine. TAN. Co-Tangent. 0 9.440338 9.982842 9.457496 10.542504 60 5 9.442535 9.982660 9.459875 10.540125 55 10 9.444720 9.982477 9.462242 10.537758 50 15 9.446893 9.982294 9.464599 10.535401 45 20 9.449054 9.982109 9.466945 10.533055 40 25 9.451204 9.981924 9.469280 10.530720 35 30 9.453342 9.981737 9.471605 10.528395 30 35 9.455469 9.981549 9.473919 10.526081 25 40 9.457584 9.981361 9.476223 10.523777 20 45 9.459688 9.981171 9.478517 10.521483 15 50 9.461782 9.980981 9.480801 10.519199 10 55 9.463864 9.980789 9.483075 10.516925 5 60 9.465935 9.980596 9.485339 10.514661 0 Co-sine. SIN. Co-Tang. TAN. M 73. 17. M SIN. Co-sine. TAN. Co-Tangent. 0 9.465935 9.980596 9.485339 10.514661 60 5 9.467996 9.980403 9.487593 10.512407 55 10 9.470446 9.980208 9.489838 10.510162 50 15 9.472086 9.980012 9.492073 10.507927 45 20 9.474115 9.979816 9.494299 10.505701 40 25 9.476133 9.979618 9.496515 10.503485 35 30 9.478142 9.979420 9.498722 10.501278 30 35 9.480140 9.979220 9.500920 10.499080 25 40 9.482128 9.979019 9.503109 10.496891 20 45 9.484107 9.978817 9.505289 10.494711 15 50 9.486075 9.978615 9.507460 10.492540 10 55 9.488034 9.978411 9.509622 10.490378 5 60 9.489982 9.978206 9.511776 10.488224 0 Co-sine SIN. Co-Tang. TAN. M. 72. 18. M SIN. Co-sine TAN. Co-Tangent. 0 9.489982 9.978206 9.511776 10.488224 60 5 9.491922 9.978001 9.513921 10.486079 55 10 9.493851 9.977794 9.516057 10.483943 50 15 9.495772 9.977586 9.518186 10.481814 45 20 9.497682 9.977377 9.520305 10.479695 40 25 9.499584 9.977167 9.522417 10.477583 35 30 9.501476 9.976957 9.524520 10.475480 30 35 9.503360 9.976745 9.526615 10.473385 25 40 9.505234 9.976532 9.528702 10.471298 20 45 9.507099 9.976318 9.530781 10.469219 15 50 9.508956 9.976103 9.532853 10.467147 10 55 9.510803 9.975887 9.534916 10.465084 5 60 9.512642 9.975670 9.536972 10.463028 0 Co-sine SIN. Co-Tang. TAN. M. 71. 19 M. SIN. Co-sine TAN. Co-Tangent. 0 9.512642 9.975670 9.536972 10.463028 60 5 9.514472 9.975452 9.539020 10.460980 55 10 9.516294 9.975233 9.541061 10.458939 50 15 9.518107 9.975013 9.543094 10.456906 45 20 9.519911 9.974792 9.545119 10.454881 40 25 9.521707 9.974570 9.547138 10.452862 35 30 9.523495 9.974347 9.549149 10.450851 30 35 9.525275 9.974122 9.551153 10.448847 25 40 9.527046 9.973897 9.553149 10.446851 20 45 9.528810 9.973671 9.555139 10.444861 15 50 9.530565 9.973444 9.557121 10.442879 10 55 9.532312 9.973215 9.559097 10.440903 5 60 9.534052 9.972986 9.561066 10.438934 0 Co-sine SIN. Co-Tang. TAN. M 70. 20. M SIN. Co-sine. TAN. Co-Tangent. 0 9.534052 9.972986 9.561066 10.438934 60 5 9.535783 9.972755 9.563028 10.436972 55 10 9.537507 9.972524 9.564983 10.435017 50 15 9.539223 9.972291 9.566932 10.433068 45 20 9.540931 9.972058 9.568873 10.431127 40 25 9.542632 9.971823 9.570809 10.429191 35 30 9.544325 9.971583 9.572738 10.427262 30 35 9.546011 9.971351 9.574660 10.425340 25 40 9.547689 9.971113 9.576576 10.423424 20 45 9.549360 9.970874 9.578486 10.421514 15 50 9.551024 9.970635 9.580389 10.419611 10 55 9.552680 9.970394 9.582286 10.417714 5 60 9.554329 9.970152 9.584177 10.415823 0 Co-sine SIN. Co-Tang. TAN. M. 69. 21. M SIN. Co-sine. TAN. Co-Tangent. 0 9.554329 9.970152 9.584177 10.415823 60 5 9.555971 9.969909 9.586062 10.413938 55 10 9.557606 9.969665 9.587941 10.412059 50 15 9.559234 9.969420 9.589814 10.410186 45 20 9.560855 9.969173 9.591681 10.408319 40 25 9.562468 9.968926 9.593542 10.406458 35 30 9.564075 9.968678 9.595398 10.404602 30 35 9.565676 9.968429 9.597247 10.402753 25 40 9.567269 9.968178 9.599091 10.400909 20 45 9.568856 9.967927 9.600929 10.399071 15 50 9.570435 9.967674 9.602761 10.397239 10 55 9.572009 9.967421 9.604588 10.395412 5 60 9.573575 9.967166 9.606410 10.393590 0 Co-sine SIN. Co-Tang. TAN. M. 68 22. M SIN. Co-sine. TAN. Co-Tangent. 0 9.573575 9.967166 9.606410 10.393590 60 5 9.575136 9.966910 9.608225 10.391775 55 10 9.576689 9.966653 9.610036 10.389964 50 15 9.578236 9.966395 9.611841 10.388159 45 20 9.579777 9.966136 9.613641 10.386359 40 25 9.581312 9.965876 9.615435 10.384565 35 30 9.582840 9.965615 9.617224 10.382776 30 35 9.584361 9.965353 9.619008 10.380992 25 40 9.585877 9.965090 9.620787 10.379213 20 45 9.587386 9.964826 9.622561 10.377439 15 50 9.588890 9.964560 9.624330 10.375670 10 55 9.590387 9.964294 9.626093 10.373907 5 60 9.591878 9.964026 9.627852 10.372148 0 Co-sine SIN. Co-Tang. TAN. M 67. 23. M SIN. Co-sine. TAN. Co-Tangent. 0 9.591878 9.964026 9.627852 10.372148 60 5 9.593363 9.963757 9.629606 10.370394 55 10 9.594842 9.963488 9.631355 10.368645 50 15 9.596315 9.963217 9.633099 10.366690 45 20 9.597783 9.962945 9.634838 10.365162 40 25 9.599244 9.962672 9.636572 10.363428 35 30 9.600700 9.962398 9.638302 10.361698 30 35 9.602150 9.962123 9.640027 10.359973 25 40 9.603594 9.961846 9.641747 10.358253 20 45 9.605032 9.961569 9.643463 10.356537 15 50 9.606465 9.961290 9.645174 10.354826 10 55 9.607892 9.961011 9.646881 10.353119 5 60 9.609313 9.960730 9.648583 10.351417 0 Co-sine SIN. Co-Tang. TAN. M. 66. 24. M. SIN. Co-sine TAN. Co-Tangent. 0 9.609313 9.960730 9.648583 10.351417 60 5 9.610729 9.960448 9.650281 10.349719 55 10 9.612140 9.960165 9.651974 10.348026 50 15 9.613545 9.959882 9.653663 10.346337 45 20 9.614944 9.959596 9.655348 10.344652 40 25 9.616338 9.959310 9.657028 10.342972 35 30 9.617727 9.959023 9.658704 10.341296 30 35 9.619110 9.958734 9.660376 10.339624 25 40 9.620488 9.958445 9.662043 10.337957 20 45 9.621861 9.958154 9.663707 10.336293 15 50 9.623229 9.957863 9.665366 10.334634 10 55 9.624591 9.957570 9.667021 10.332979 5 60 9.625948 9.957276 9.668673 10.331327 0 Co-sine SIN. Co-Tang. TAN. M. 65. 25. M. SIN. Co-sine TAN. Co-Tangent. 0 9.625948 9.957276 9.668673 10.331327 60 5 9.627300 9.956981 9.670320 10.329680 55 10 9.628647 9.956684 9.671963 10.328073 50 15 9.629989 9.956387 9.673602 10.326398 45 20 9.631326 9.956089 9.675237 10.324763 40 25 9.632658 9.955789 9.676869 10.323131 35 30 9.633984 9.955488 9.678496 10.322504 30 35 9.635306 9.955186 9.680120 10.319880 25 40 9.636623 9.954883 9.681740 10.318260 20 45 9.637935 9.954579 9.683356 10.316644 15 50 9.639242 9.954274 9.684968 10.315032 10 55 9.640544 9.953968 9.686577 10.313423 5 60 9.641842 9.953660 9.688182 10.311818 0 Co-sine SIN. Co-Tang. TAN. M 64. 26. M SIN. Co-sine TAN. Co-Tangent. 0 9.641842 9.953660 9.688182 10.311818 60 5 9.643135 9.953352 9.689783 10.310217 55 10 9.644423 9.953042 9.691381 10.308619 50 15 9.645706 9.952731 9.692975 10.307025 45 20 9.646984 9.952419 9.694566 10.305434 40 25 9.648258 9.952106 9.696153 10.303847 35 30 9.649527 9.951791 9.697736 10.302264 30 35 9.650792 9.951476 9.699316 10.300684 25 40 9.652052 9.951159 9.700893 10.299107 20 45 9.653308 9.950841 9.702781 10.297534 15 50 9.654558 9.950522 9.704036 10.295964 10 55 9.655805 9.950202 9.705603 10.294397 5 60 9.657047 9.949881 9.707166 10.292834 0 Co-sine SIN. Co-Tang. TAN. M 63. 27. M SIN Co-sine TAN. Co-Tangent. 0 9.657047 9.949881 9.707166 10.292834 60 5 9.658284 9.949558 9.708726 10.291274 55 10 9.659517 9.949235 9.710282 10.289718 50 15 9.660746 9.948910 9.711836 10.288104 45 20 9.661970 9.948584 9.713386 10.286614 40 25 9.663190 9.948257 9.714933 10.285067 35 30 9.664406 9.947929 9.716477 10.283523 30 35 9.665617 9.947600 9.718017 10.281983 25 40 9.666824 9.947269 9.719555 10.280445 20 45 9.668027 9.946937 9.721089 10.278911 15 50 9.669225 9.946604 9.722621 10.277379 10 55 9.670419 9.946270 9.724149 10.275851 5 60 9.671609 9.945935 9.725674 10.274326 0 Co-sine. SIN. Co-Tang. TAN. M. 62. 28. M. SIN. Co-sine TAN. Co-Tangent. 0 9.671609 9.945935 9.725674 10.274326 60 5 9.672795 9.945598 9.727197 10.272803 55 10 9.673977 9.945261 9.728716 10.271284 50 15 9.675155 9.944922 9.730233 10.269767 45 20 9.676328 9.944582 9.731746 10.268254 40 25 9.677498 9.944241 9.733257 10.266743 35 30 9.678663 9.943899 9.734764 10.265236 30 35 9.679824 9.943555 9.736269 10.263731 25 40 9.680982 9.943210 9.737771 10.262229 20 45 9.682135 9.942864 9.739271 10.260729 15 50 9.683284 9.942517 9.740767 10.259233 10 55 9.684430 9.942169 9.742261 10.257739 5 60 9.685571 9.941819 9.743752 10.256248 0 Co-sine SIN. Co-Tang. TAN. M 61. 29. M. SIN. Co-sine TAN. Co-Tangent. 0 9.685571 9.941819 9.743751 10.256248 60 5 9.686709 9.941469 9.745240 10.254760 55 10 9.687843 9.941117 9.746726 10.253274 50 15 9.688972 9.940763 9.748209 10.251791 45 20 9.690098 9.940409 9.749689 10.250311 40 25 9.691220 9.940054 9.751167 10.248833 35 30 9.692339 9.939697 9.752642 10.247358 30 35 9.693453 9.939339 9.754115 10.245885 25 40 9.694564 9.938980 9.755585 10.244415 20 45 9.695671 9.938619 9.757052 10.242948 15 50 9.696775 9.938258 9.758517 10.241483 10 55 9.697874 9.937895 9.759979 10.240021 5 60 9.698970 9.937531 9.761439 10.238561 0 Co-sine SIN. Co-Tang. TAN. M. 60. 30. M. SIN. Co-sine TAN. Co-Tangent. 0 9.698970 9.937531 9.761439 10.238561 60 5 9.700062 9.937165 9.762897 10.237103 55 10 9.701151 9.936799 9.764352 10.235648 50 15 9.702236 9.936431 9.765805 10.234195 45 20 9.703317 9.936062 9.767255 10.232745 40 25 9.704395 9.935692 9.768703 10.231297 35 30 9.705469 9.935320 9.770148 10.229852 30 35 9.706539 9.934948 9.771592 10.228408 25 40 9.707606 9.934574 9.773033 10.226967 20 45 9.708670 9.934199 9.774471 10.225529 15 50 9.709730 9.933822 9.775908 10.224092 10 55 9.710786 9.933445 9.777342 10.222658 5 60 9.711839 9.933066 9.778774 10.221226 0 Co-sine. SIN. Co-Tang. TAN. M 59 31. M. SIN Co-sine TAN. Co-Tangent. 0 9.711839 9.933066 9.778774 10.221226 60 5 9.712889 9.932685 9.780203 10.219797 55 10 9.713935 9.932304 9.781631 10.218369 50 15 9.714978 9.931921 9.783056 10.216944 45 20 9.716017 9.931537 9.784479 10.215521 40 25 9.717053 9.931152 9.785900 10.214100 35 30 9.718085 9.930766 9.787319 10.212681 30 35 9.719114 9.930378 9.788736 10.211264 25 40 9.720140 9.929989 9.790151 10.209849 20 45 9.721162 9.929599 9.791563 10.208437 15 50 9.722181 9.929207 9.792974 10.207026 10 55 9.723197 9.928815 9.794383 10.205617 5 60 9.724210 9.928420 9.795789 10.204211 0 Co-sine. SIN. Co-Tang. TAN. M 58. 32. M. SIN. Co-sine. TAN. Co-Tangent. 0 9.724210 9.928420 9.795789 10.204211 60 5 9.725219 9.928025 9.797194 10.202806 55 10 9.726225 9.927629 9.798596 10.201404 50 15 9.727228 9.927231 9.799997 10.200003 45 20 9.728227 9.926831 9.801396 10.198604 40 25 9.729223 9.926431 9.802792 10.197208 35 30 9.730217 9.926029 9.804187 10.195813 30 35 9.731206 9.925626 9.805580 10.194420 25 40 9.732193 9.925222 9.806971 10.193029 20 45 9.733177 9.924816 9.808361 10.191639 15 50 9.734157 9.924409 9.809748 10.190252 10 55 9.735135 9.924001 9.811134 10.188866 5 60 9.736109 9.923591 9.812517 10.187483 0 Co-sine. SIN. Co-Tang. TAN. M 57 33. M. SIN. Cousin. TAN Co-Tangent. 0 9.736109 9.923591 9.812517 10.187483 60 5 9.737080 9.923181 9.813899 10.186101 55 10 9.738048 9.922769 9.815280 10.184720 50 15 9.739013 9.922355 9.816658 10.183342 45 20 9.739975 9.921940 9.818035 10.181965 40 25 9.740934 9.921524 9.819410 10.180590 35 30 9.741889 9.921107 9.820783 10.179217 30 35 9.742842 9.920688 9.822154 10.177846 25 40 9.743792 9.920268 9.823524 10.176476 20 45 9.744739 9.919846 9.824893 10.175107 15 50 9.745683 9.919424 9.826259 10.173741 10 55 9.746624 9.919000 9.827624 10.172376 5 60 9.747562 9.918574 9.828987 10.171013 0 Co-sine. SIN. Co-Tang. TAN. M. 56 34. M. SIN. Co-sine. TAN. Co-Tangent. 0 9.747562 9.918574 9.828987 10.171013 60 5 9.748497 9.918147 9.830349 10.169651 55 10 9.749429 9.917719 9.831709 10.168291 50 15 9.750358 9.917290 9.833068 10.166932 45 20 9.751284 9.916859 9.834425 10.165575 40 25 9.752208 9.916427 9.835780 10.164220 35 30 9.753128 9.915994 9.837134 10.162866 30 35 9.754046 9.915559 9.838487 10.161513 25 40 9.754960 9.915123 9.839838 10.160162 20 45 9.755872 9.914685 9.841187 10.158813 15 50 9.756782 9.914246 9.842535 10.157405 10 55 9.757688 9.913806 9.843882 10.156118 5 60 9.758591 9.913365 9.845227 10.154773 0 Co-sine. SIN. Co-Tang. TAN. M 55. 35. M SIN. Co-sine. TAN. Co-Tangent. 0 9.758591 9.913365 9.845227 10.154773 60 5 9.759492 9.912922 9.846570 10.153430 55 10 9.760390 9.912477 9.847913 10.152087 50 15 9.761285 9.912031 9.849254 10.150746 45 20 9.762177 9.911584 9.850593 10.149407 40 25 9.763067 9.911136 9.851931 10.148069 35 30 9.763954 9.910686 9.853268 10.146732 30 35 9.764838 9.910235 9.854603 10.145397 25 40 9.765720 9.909782 9.855938 10.144062 20 45 9.766598 9.909328 9.857270 10.142730 15 50 9.767475 9.908873 9.858602 10.141398 10 55 9.768348 9.908416 9.859932 10.140068 5 60 9.769219 9.907958 9.861261 10.138739 0 Co-sine SIN. Co-Tang. TAN. M 54. 36. M. SIN. Co-sine. TAN. Co-Tangent. 0 9.769219 9.907958 9.861261 10.138739 60 5 9.770087 9.907498 9.862589 10.137411 55 10 9.770952 9.907037 9.863915 10.136085 50 15 9.771815 9.906575 9.865240 10.134760 45 20 9.772675 9.906111 9.866564 10.133436 40 25 9.773533 9.905645 9.867887 10.132133 35 30 9.774388 9.905179 9.869209 10.130791 30 35 9.775240 9.904711 9.870529 10.129471 25 40 9.776090 9.904241 9.871849 10.128151 20 45 9.776937 9.903770 9.873167 10.126833 15 50 9.777781 9.903298 9.874484 10.125516 10 55 9.778624 9.902824 9.875800 10.124200 5 60 9.779463 9.902349 9.877114 10.122886 0 Co-sine. SIN. Co-Tang. TAN. M. 53. 37. M. SIN. Cousin. TAN Co-Tangent. 0 9.779463 9.902349 9.877114 10.122886 60 5 9.780300 9.901872 9.878428 10.121572 55 10 9.781134 9.901394 9.879741 10.120259 50 15 9.781966 9.900914 9.881052 10.118948 45 20 9.782796 9.900433 9.882363 10.117637 40 25 9.783623 9.899951 9.883672 10.116328 35 30 9.784447 9.899467 9.884980 10.115020 30 35 9.785269 9.898981 9.886288 10.113712 25 40 9.786089 9.898494 9.887594 10.112406 20 45 9.786906 9.898006 9.888900 10.111100 15 50 9.787720 9.897516 9.890204 10.109796 10 55 9.788532 9.897025 9.891507 10.108493 5 60 9.789342 9.806532 9.892810 10.107190 0 Co-sine. SIN. Co-Tang. TAN. M. 52. 38. M. SIN. Co-sine. TAN. Co-Tangent. 0 9.789342 9.896532 9.892810 10.107190 60 5 9.790149 9.896038 9.894111 10.105889 55 10 9.790954 9.895542 9.895412 10.104588 50 15 9.791757 9.895045 9.896712 10.103288 45 20 9.792557 9.894546 9.898010 10.101990 40 25 9.793354 9.894046 9.899308 10.100692 35 30 9.794150 9.893544 9.900605 10.099395 30 35 9.794942 9.893041 9.901901 10.098099 25 40 9.795733 9.892536 9.903197 10.096803 20 45 9.796521 9.892030 9.904491 10.095509 15 50 9.797307 9.891523 9.905785 10.094215 10 55 9.798091 9.891013 9.907077 10.092923 5 60 9.798872 9.890503 9.908369 10.091631 0 Co-sine. SIN. Co-Tang. TAN. M. 51. 39 M. SIN. Cousin. TAN Co-Tangent. 0 9.798872 9.890503 9.908369 10.091631 60 5 9.799651 9.889990 9.909660 10.090340 55 10 9.800427 9.889477 9.910951 10.089049 50 15 9.801201 9.888961 9.912240 10.087760 45 20 9.801973 9.888444 9.913529 10.086471 40 25 9.802743 9.887926 9.914817 10.085183 35 30 9.803511 9.887406 9.916104 10.083895 30 35 9.804276 9.886885 9.917391 10.082609 25 40 9.805039 9.886362 9.918677 10.081323 20 45 9.805799 9.885837 9.919962 10.080038 15 50 9.806557 9.885311 9.921247 10.078753 10 55 9.807314 9.884783 9.922530 10.077470 5 60 9.808067 9.884254 9.923814 10.076186 0 Co-sine. SIN. Co-Tang. TAN. M. 52. 40. M. SIN. Co-sine TAN. Co-Tangent. 0 9.808067 9.884254 9.923814 10.076186 60 5 9.808819 9.883723 9.925096 10.074904 55 10 9.809569 9.883191 9.926378 10.073622 50 15 9.810316 9.882657 9.927659 10.072341 45 20 9.811061 9.882121 9.928940 10.071060 40 25 9.811804 9.881584 9.930220 10.069781 35 30 9.812544 9.881046 9.931499 10.068501 30 35 9.813283 9.880505 9.932778 10.067222 25 40 9.814019 9.879963 9.934056 10.065944 20 45 9.814753 9.879420 9.935333 10.064667 15 50 9.815485 9.878875 9.936611 10.063389 10 55 9.816215 9.878328 9.937887 10.062113 5 60 9.816943 9.877780 9.939163 10.060837 0 Co-sine SIN. Co-Tang. TAN. M 49. 41. M. SIN. Co-sine TAN. Co-Tangent. 0 9.816943 9.877780 9.939163 10.060837 60 5 9.817668 9.877230 9.940439 10.059561 55 10 9.818392 9.876678 9.941713 10.058287 50 15 9.819113 9.876125 9.942988 10.057012 45 20 9.819832 9.875571 9.944262 10.055738 40 25 9.820550 9.875014 9.945535 10.054465 35 30 9.821265 9.874456 9.946808 10.053192 30 35 9.821977 9.873896 9.948081 10.051919 25 40 9.822688 9.873335 9.949353 10.050647 20 45 9.823397 9.872772 9.950625 10.049375 15 50 9.824104 9.872208 9.951896 10.048104 10 55 9.824808 9.871641 9.953167 10.046833 5 60 9.825511 9.871073 9.954437 10.045563 0 Co-sine SIN. Co-Tang. TAN. M. 48. 42. M SIN. Co-sine. TAN. Co-Tangent. 0 9.825511 9.871073 9.954437 10.045503 60 5 9.826211 9.870504 9.955708 10.044292 55 10 9.826910 9.869933 9.956977 10. 04●●23 50 15 9.827606 9.869360 9.958247 10.041753 45 20 9.828301 9.868785 9.959516 10.040484 40 25 9.828993 9.868209 9.960784 10.039216 35 30 9.829683 9.867631 9.962052 10.037948 30 35 9.830372 9.867051 9.963320 10.036680 25 40 9.831058 9.866470 9.964588 10.035412 20 45 9.831742 9.865887 9.965855 10.034145 15 50 9.832425 9.865302 9.967123 10.032877 10 55 9.833105 9.864716 9.968389 10.031611 5 60 9.833783 9.864127 9.969656 10.030344 0 Co-sine SIN. Co-Tang. TAN. M 47. 43. M SIN. Co-sine. TAN. Co-Tangent. 0 9.833783 9.864127 9.969656 10.030344 60 5 9.834460 9.863538 9.970922 10.029078 55 10 9.835134 9.862946 9.972188 10.027812 50 15 9.835807 9.862353 9.973454 10.026546 45 20 9.836477 9.861758 9.974720 10.025280 40 25 9.837146 9.861161 9.975985 10.024015 35 30 9.837812 9.860562 9.977250 10.022750 30 35 9.838477 9.859962 9.978515 10.021485 25 40 9.839140 9.859360 9.979780 10.020220 20 45 9.839800 9.858756 9.981044 10.018956 15 50 9.840459 9.858151 9.982309 10.017691 10 55 9.841116 9.857543 9.983573 10.016427 5 60 9.841771 9.856934 9.984837 10.015163 0 Co-sine SIN. Co-Tang. TAN. M 46. 44 M. SIN. Co-sine. TAN. Co-Tangent. 0 9.841771 9.856934 9.984837 10.015162 60 5 9.842424 9.856323 9.986101 10.013899 55 10 9.843076 9.855711 9.987365 10.012635 50 15 9.843725 9.855096 9.988629 10.011371 45 20 9.844372 9.854480 9.989893 10.010107 40 25 9.845018 9.853862 9.991156 10.008844 35 30 9.845662 9.853242 9.992420 10.007580 30 35 9.846304 9.852620 9.993683 10.006317 25 40 9.846944 9.851997 9.994947 10.005053 20 45 9.847582 9.851372 9.996210 10.003790 15 50 9.848218 9.850745 9.997473 10.002527 10 55 9.848852 9.850116 9.998737 10.001263 5 60 9.849485 9.849485 10.000000 10.000000 0 Co-sine SIN. Co-Tang. TAN. M 45. A TABLE OF Logarithm Numbers. N. Log. N. Log. N. Log. N. Log. 1 0.000000 41 1.612784 81 1.908485 121 2.082785 2 0.301030 42 1.623249 82 1.913814 122 2.086359 3 0.477121 43 1.633468 83 1.919078 123 2.089905 4 0.602060 44 1.643452 84 1.924279 124 2.093422 5 0.698970 45 1.653212 85 1.929419 125 2.096910 6 0.778151 46 1.662758 86 1.934498 126 2.100371 7 0.845098 47 1.672098 87 1.939519 127 2.103804 8 0.903090 48 1.681241 88 1.944482 128 2.107209 9 0.954242 49 1.690196 89 1.949390 129 2.110589 10 1.000000 50 1.698970 90 1.954242 130 2.113943 11 1.041393 51 1.707570 91 1.959041 131 2.117271 12 1.079181 52 1.716003 92 1.963788 132 2.120574 13 1.113943 53 1.724276 93 1.968483 133 2.123852 14 1.146128 54 1.732394 94 1.973128 134 2.127105 15 1.176091 55 1.740362 95 1.977723 135 2.130334 16 1.204120 56 1.748188 96 1.982271 136 2.233539 17 1.230449 57 1.755875 97 1.986772 137 2.136721 18 1.255272 58 1.763428 98 1.991226 138 2.139879 19 1.278753 59 1.770852 99 1.995635 139 2.143015 20 1.301230 60 1.778151 100 2.000000 140 2.146128 21 1.322219 61 1.785330 101 2.004321 141 2.159219 22 1.342422 62 1.792391 102 2.008600 142 2.152288 23 1.361728 63 1.799340 103 2.012837 143 2.155336 24 1.380211 64 1.806180 104 2.017033 144 2.158362 25 1.397940 65 1.812913 105 2.021189 145 2.161368 26 1.414973 66 1.819544 106 2.025306 146 2.164353 27 1.431364 67 1.826075 107 2.029384 147 2.167317 28 1.447158 68 1.832509 108 2.033424 148 2.170262 29 1.462398 69 1.838849 109 2.037426 149 2.173186 30 1.477121 70 1.845098 110 2.041393 150 2.176091 31 1.491361 71 1.851258 111 2.045323 151 2.178977 32 1.505150 72 1.857332 112 2.049218 152 2.181844 33 1.518514 73 1.863323 113 2.053078 153 2.184691 34 1.531479 74 1.869232 114 2.056905 154 2.187521 35 1.544068 75 1.875061 115 2.060698 155 2.190332 36 1. 5●6303 76 1.880813 116 2.064458 156 2.193125 37 1.568202 77 1.886491 117 2.068186 157 2.195899 38 1.579783 78 1.892094 118 2.071882 158 2.198657 39 1.591064 79 1.897627 119 2.075547 159 2.201397 40 1.602060 80 1.903090 120 2.079181 160 2.204110 161 2.206826 201 2.303196 241 2.382017 281 2.448706 162 2.209515 202 2.305351 242 2.383815 282 2.450249 163 2.212187 203 2.307496 243 2.385606 283 2.451786 164 2.214844 204 2.309630 244 2.387389 284 2.453318 165 2.217484 205 2.311754 245 2.389166 285 2.454845 166 2.220108 206 2.313867 246 2.390935 286 2.456366 167 2.222716 207 2.315970 247 2.392697 287 2.457889 168 2.225309 208 2.318063 248 2.394452 288 2.459392 169 2.227887 209 2.320146 249 2.396199 289 2.460898 170 2.230449 210 2.322219 250 2.397940 290 2.462398 171 2.232996 211 2.324282 251 2.399674 291 2.463893 172 2.235528 212 2.326336 252 2.401401 292 2.465383 173 2.238046 213 2.328379 253 2.403121 293 2.466868 174 2.240549 214 2.330414 254 2.404834 294 2.468347 175 2.243038 215 2.332438 255 2.406540 295 2.469822 176 2.245513 216 2.334454 256 2.408239 296 2.471292 177 2.247973 217 2.336459 257 2.409933 297 2.472756 178 2.250420 218 2.338456 258 2.411619 298 2.474216 179 2.252853 219 2.340444 259 2.413299 299 2.475671 180 2.255273 220 2.342422 260 2.414973 300 2.477121 181 2.257679 221 2.344392 261 2.416641 301 2.478566 182 2.260071 222 2.346353 262 2.418301 302 2.480007 183 2.262451 223 2.348305 263 2.419956 303 2.481443 184 2.264818 224 2.350248 264 2.421604 304 2.482874 185 2.267172 225 2.352183 265 2.423246 305 2.484299 186 2.269513 226 2.354108 266 2.424882 306 2.485721 187 2.271842 227 2.356026 267 2.426511 307 2.487138 188 2.274158 228 2.357935 268 2.428135 308 2.488551 189 2.276462 229 2.359835 269 2.429752 309 2.489958 190 2.278754 230 2.361728 270 2.421364 310 2.491362 191 2.281033 231 2.363612 271 2.432969 311 2.492760 192 2.283301 232 2.365488 272 2.434569 312 2.494155 193 2.285557 233 2.367356 273 2.436163 313 2.495544 194 2.287802 234 2.369216 274 2.337751 314 2.496929 195 2.290035 235 2.371068 275 2.439333 315 2.498311 196 2.292256 236 2. 3729●2 276 2.440909 316 2.499687 197 2.294466 237 2.374748 277 2.442479 317 2.501059 198 2.296665 238 2.376577 278 2.444045 318 2.502427 199 2.298853 239 2.378398 279 2.445604 319 2.503791 200 2.301029 240 2.380211 280 2.447158 320 2.505149 321 2.506505 361 2.557507 401 2.603144 441 2.644439 322 2.507856 362 2.558709 402 2.604226 442 2.645422 323 2.509203 363 2.559907 403 2.605305 443 2.646404 324 2.510545 364 2.561101 404 2.606381 444 2.647383 325 2.511883 365 2.562293 405 2.607455 445 2.648360 326 2.513218 366 2.563481 406 2.608526 446 2.649335 327 2.514548 367 2.564666 407 2.609594 447 2.650308 328 2.515874 368 2.565848 408 2.610660 448 2.651278 329 2.517196 369 2.567026 409 2.611723 449 2.652246 330 2.518514 370 2.568202 410 2.612784 450 2.653213 331 2.519828 371 2.569374 411 2.613842 451 2.654177 332 2.521138 372 2.570543 412 2.614897 452 2.655138 333 2.522444 373 2.571709 413 2.615950 453 2.656098 334 2.523746 374 2.572872 414 2.617000 454 2.657056 335 2.525045 375 2.574031 415 2.618048 455 2.658011 336 2.526339 376 2.575188 416 2.619093 456 2.658965 337 2.527629 377 2.576341 417 2.620136 457 2.659916 338 2.528916 378 2.577492 418 2.621176 458 2.660865 339 2.530199 379 2.578639 419 2.622214 459 2.661813 340 2.531479 380 2.579784 420 2.623249 460 2.662758 341 2.532754 381 2.580925 421 2.624282 461 2.663701 342 2.534026 382 2.582063 422 2.625312 462 2.664642 343 2.535294 383 2.583199 423 2.626340 463 2.665581 344 2.536558 384 2.584331 424 2.627366 464 2.666518 345 2.537819 385 2.585461 425 2.628389 465 2.667453 346 2.539076 386 2.586587 426 2.629409 466 2.668386 347 2.540329 387 2.587711 427 2.630428 467 2.669317 348 2.541579 388 2.588832 428 2.631444 468 2.670246 349 2.542825 389 2.589949 429 2.632457 469 2.671173 350 2.544008 390 2.591065 430 2.633468 470 2.672098 351 2.545307 391 2.592177 431 2.634477 471 2.673021 352 2.546543 392 2.593286 432 2.635484 472 2.673942 353 2.547775 393 2.594393 433 2.636488 473 2.674861 354 2.549003 394 2.595496 434 2.637489 474 2.675778 355 2.550228 395 2.596597 435 3.638489 475 2.676694 356 2.551449 396 2.597695 436 2.639486 476 2.677607 357 2.552668 397 2.598790 437 2.640481 477 2.678518 358 2.553883 398 2.599883 438 2.641475 478 2.679428 359 2.555094 399 2.600973 439 2.642465 479 2.680336 360 2.556303 400 2.602059 440 2.643453 480 2.681241 481 2.682145 521 2.716838 561 2.748963 601 2.778874 482 2.683047 522 2.717671 562 2.749736 602 2.779596 483 2.683947 523 2.718502 563 2.750508 603 2.780317 484 2.684845 524 2.719331 564 2.751279 604 2.781037 485 2.685742 525 2.720159 565 2.752048 605 2.781755 486 2.686636 526 2.720986 566 2.752816 606 2.782473 487 2.687529 527 2.721811 567 2.753583 607 2.783189 488 2.688419 528 2.722634 568 2.754348 608 2.783904 489 2.689309 529 2.723456 569 2.755112 609 2.784617 490 2.690196 530 2.724276 570 2.755875 610 2.785329 491 2.691081 531 2.725095 571 2.756636 611 2.786041 492 2.691965 532 2.725912 572 2.757396 612 2.786751 493 2.692847 533 2.726727 573 2.758155 613 2.787460 494 2.693727 534 2.727541 574 2.758912 614 2.788164 495 2.694605 535 2.728354 575 2.759668 615 2.788875 496 2.695482 536 2.729165 576 2.760422 616 2.789581 497 2.696356 537 2.729974 577 2.761176 617 2.790285 498 2.697229 538 2.730782 578 2.761928 618 2.790988 499 2.698101 539 2.731589 579 2.762679 619 2.791691 500 2.698970 540 2.732394 580 2.763428 620 2.792392 501 2.699838 541 2.733197 581 2.764176 621 2.793092 502 2.700704 542 2.733999 582 2.764923 622 2.793791 503 2.701568 543 2.734799 583 2.765669 623 2.794488 504 2.702430 544 2.735599 584 2.766413 624 2.795185 505 2.703291 545 2.736397 585 2.767156 625 2.795880 506 2.704151 546 2.737192 586 2.767898 626 2.796574 507 2.705008 547 2.737987 587 2.768638 627 2.797268 508 2.705863 548 2.738781 588 2.769377 628 2.797959 509 2.706718 549 2.739572 589 2.770115 629 2.798651 510 2.707570 550 2.740363 590 2.770852 630 2.799341 511 2.708421 551 2.741152 591 2.771587 631 2.800029 512 2.709269 552 2.741939 592 2.772322 632 2.800717 513 2.710117 553 2.742725 593 2.773055 633 2.801404 514 2.710963 554 2.743509 594 2.773786 634 2.802089 515 2.711807 555 2.744293 595 2.774517 635 2.802774 516 2.712649 556 2.745075 596 2.775246 636 2.803457 517 2.713491 557 2.745855 597 2.775974 637 2.804139 518 2.714329 558 2.746634 598 2.776701 638 2.804821 519 2.715167 559 2.747412 599 2.777427 639 2.805501 520 2.716003 560 2.748188 600 2.778151 640 2.806179 641 2.806858 681 2.833147 721 2.857935 761 2.881385 642 2.807535 682 2.833784 722 2.858537 762 2.881955 643 2.808211 683 2.834421 723 2.859138 763 2.882525 644 2.808886 684 2.835056 724 2.859739 764 2.883093 645 2.809559 685 2.835691 725 2.860338 765 2.883661 646 2.810233 686 2.836324 726 2.860937 766 2.884229 647 2.810904 687 2.836957 727 2.861534 767 2.884795 648 2.811575 688 2.837588 728 2.862131 768 2.885361 649 2.812245 689 2.838219 729 2.862728 769 2.885926 650 2.812913 690 2.838849 730 2.863323 770 2.886491 651 2.813581 691 2.839478 731 2.863917 771 2.887054 652 2.814248 692 2.840106 732 2.864511 772 2.887617 653 2.814913 693 2.840733 733 2.865104 773 2.888179 654 2.815578 694 2.841359 734 2.865696 774 2.888741 655 2.816241 695 2.841985 735 2.866287 775 2.889302 656 2.816904 696 2.842609 736 2.866878 776 2.889862 657 2.817565 697 2.843233 737 2.867467 777 2.890421 658 2.818226 698 2.843855 738 2.868056 778 2.890979 659 2.818885 699 2.844477 739 2.868643 779 2.891537 660 2.819543 700 2.845098 740 2.869232 780 2.892095 661 2.820201 701 2.845718 741 2.869818 781 2.892651 662 2.820858 702 2.846337 742 2.870404 782 2.893207 663 2.821514 703 2.846955 743 2.870989 783 2.893762 664 2.822168 704 2.847573 744 2.871573 784 2.894316 665 2.822822 705 2.848189 745 2.872156 785 2.894869 666 2.823474 706 2.848805 746 2.872739 786 2.895423 667 2.824126 707 2.849419 747 2.873321 787 2.895975 668 2.824776 708 2.850033 748 2.873902 788 2.896526 669 2.825426 709 2.850646 749 2.874482 789 2.897077 670 2.826075 710 2.851258 750 2.875061 790 2.897627 671 2.826723 711 2.851869 751 2.875639 791 2.898176 672 2.827369 712 2.852479 752 2.876218 792 2.898725 673 2.828015 713 2.853089 753 2.876795 793 2.899273 674 2.828659 714 2.853698 754 2.877371 794 2.899821 675 2.829304 715 2.854306 755 2.877947 795 2.900367 676 2.829947 716 2.854913 756 2.878522 796 2.900913 677 2.830589 717 2.855519 757 2.879096 797 2.901458 678 2.830229 718 2.856124 758 2.879669 798 2.902003 679 2.832869 719 2.856729 759 2.880242 799 2.902547 680 2.832509 720 2.857332 760 2.880814 800 2.903089 801 2.903633 841 2. 92476● 881 2.944976 921 2.964259 802 2.904174 842 2.925312 882 2.945468 922 2.964731 803 2.904716 843 2.925828 883 2.945961 923 2.965202 804 2.905256 844 2.926342 884 2.946452 924 2.965672 805 2.905796 845 2.926857 885 1.946943 925 2.966142 806 2.906335 846 2.927370 886 2.947434 926 2.966611 807 2.906874 847 2.927883 887 2.947924 927 2.967079 808 2.907411 848 2.928396 888 2.948413 928 2.967548 809 2.907949 849 2.928908 889 2.948902 929 2.968016 810 2.908485 850 2.929419 890 2.949390 930 2.968483 811 2.909021 851 2.929929 891 2.940878 931 2.968949 812 2.909556 852 2.930439 892 2.950365 932 2.969416 813 2.910091 853 2.930949 893 2.950851 933 2.969882 814 2.910624 854 2.931458 894 2.951338 934 2.970347 815 2.911158 855 2.931966 895 2.951823 935 2.970812 816 2.911690 856 2.932474 896 2.952308 936 2.971276 817 2.912222 857 2.932981 897 2.952792 937 2.971739 818 2.912753 858 2.933487 898 2.953276 938 2.972203 819 2.913284 859 2.933993 899 2.953759 939 2.972666 820 2.913814 860 2.934498 900 2.954243 940 2.973128 821 2.914343 861 2.935003 901 2.954725 941 2.973589 822 2.914872 862 2.935507 902 2.955207 942 2.974050 823 2.915399 863 2.936011 903 2.955688 943 2.974512 824 2.915927 864 2.936514 904 2.956168 944 2.974972 825 2.916454 865 2.937016 905 2.956649 945 2.975432 826 2.916980 866 2.937518 906 2.957128 946 2.975891 827 2.917506 867 2.938019 907 2.957607 947 2.976349 828 2.918030 868 2.998519 908 2.958086 948 2.976808 829 2.918555 869 2.939019 909 2.958564 949 2.977266 830 2.819078 870 2.939519 910 2.959041 960 2.977724 831 2.919601 871 2.940018 911 2.959518 951 2.978181 832 2.920123 872 2.940516 912 2.959995 952 2.978637 833 2.920645 873 2.941014 913 2.960471 953 2.979093 834 2.921166 874 2.941511 914 2.960946 954 2.979548 835 2.921686 875 2.942008 915 2.961401 955 2.980003 836 2.922206 876 2.942504 916 2.961895 956 2.980458 837 2.922725 877 2.942999 917 2.962369 957 2.980912 838 2.923244 878 2.943495 918 2.962840 958 2.981366 839 2.923762 879 2.943989 919 2.963315 959 2.981819 840 2.924279 880 2.944483 920 2.963788 960 2.982271 961 2.982723 971 2.987219 981 2.991669 991 2.996074 962 2.983175 972 2.987666 982 2.992111 992 2.996512 963 2.983626 973 2.988113 983 2.992554 993 2.996949 964 2.984077 974 2.988559 984 2.992995 994 2.997386 965 2.984527 975 2.989005 985 2.993436 995 2.997823 966 2.984977 976 2.989449 986 2.993877 996 2.998259 967 2.985426 977 2.989895 987 2.994317 997 2.998695 968 2.985875 978 2.990339 988 2.994756 998 2.999133 969 2.986324 979 2.990783 989 2.995196 999 2.999565 960 2.988772 980 2.991226 990 2.995635 1000 3.000000 The use of these Tables hath been already at large showed in the First and Twelfth Chapters; therefore I shall say no more of them here. FINIS.