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 multiplyed in it self, 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 mul­tiplyed in it self, 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 multiplyed in it self, it makes neither more nor less, but just the Number pro­pounded, viz. 81.

Again, suppose 16 be the number given, I say the Root of it is 4, because 4 mul­tiplyed in it self 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.

Or, suppose a plain Square Figure be given you as this in the Margent, and it be requi­red of you to divide it into 9 small

Squares: Your Business is to know in­to how many Parts to divide any one of the Side Lines, which here must be into 3, and that is the Root required. But now how to do this readily is the thing I am going to teach you. The Roots of all Square Numbers under 100, you have in your Multiplica­tion Table, however since it is good for you to keep them in your Mind, take this small Table of them.

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, then put a Point over the first Figure on the Right-hand, which is the place of Unites, and so proceeding to the Left-hand, 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 Left-hand that has a Point over it, which will always be the first or second Figure, and search out the Root [Page 4]of that one Figure, or both joyned 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 Left-hand, 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 Margent, and set down 1 in the Quotient, then if 1 admitted of any Multiplication, I should multiply it by it self, but since once 1 is but 1, I substract it out of the first prick'd Figure on the Left-hand, 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 write 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, then that 22 which stands under the 44 must be multiplyed by the [...] in the Quotient, whose Product is 44, which substracted out of 44, there remains 0: But you may multiply and substract to­gether thus, twice 2 is 4, which I take out of 4, and there remains 0, then I cancel the first 4 and 2 to the Left-hand, 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, [Page 5]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 summ 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 substract, 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 write 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 Tens or Hundreds, in order before him, as you do in Division, as you may see here: Then proceed, and say, how many times 46 [Page 6] [...] 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, remembring the 4 that must be put down under the pointed Figure, and when you find you can have it four times, write 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 write 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 Cyphers, if you desire to know but to the tenth part of an Unite; but if to an hundredth part add four Cyphers, if to a thousandth part of an Unite, add six Cyphers, 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 [Page 7]to know the Decimal Fraction as near as to the hun­dredth part of an Unite; I write it down as before, annexing four Cyphers 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 Left-hand 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 easie and ready, or to be wrought with as few Figures. To do it indeed by the Loga­rithms or Artificial Numbers, is very easie 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 shew 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 [Page 8]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.

APoint 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 joyned together in length, of which there are two sorts, viz. Streight and Crooked. As, AB is a Streight Line, BC two Crooked Lines.

An Angle is the meeting of two Lines in a Point; provided the two Lines so meeting, do not make one Streight Line, as the Line AB, and the Line AC, meeting together in the Point A, make the Angle BAC.

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.

All Figures contained under three Sides are called Triangles, as A, B, C.

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.

Other Figures that are contained under 5, 6, 7, or more Sides, I call Irregular, as FG, &c. Except

such as are made by dividing the Circumference of a Circle into any number of Parts; for then 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, &c. 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.

A Circle is a Figure determined with one Endless

Line, as A. Which Line is called the Cir­cumference of the Circle, in the Mid­dle whereof is a Prick or Point, by which the Circle is descri­bed, which is called the Center, from which Point or Cen­ter all Streight Lines drawn to the Cir­cumference are Equal, or of the same Length, as AB, AC, AD.

The Diameter of a Circle, is a Line which passing through the Center, cuts the Circle into two Equal Parts, or the longest Streight Line that can be made in any Circle; as BC.

The Semi-Diameter, is the half of the above-menti­oned 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.

A Quadrant is the fourth part of a Circle, made by two Diameters perpendi­cularly intersecting each o­ther,

as ABD, ADC, ABE, AEC, either of which is a Quadrant, or the fourth part of a Circle.

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.

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.

ANd first of Long Measures; which are either Inches, Feet, Yards, Perches, Chains, &c. Note that twelve Inches make one Foot, three Feet one Yard, five Yards and a half one Pole or Perch, four Perches one Chain of Gunter's, eighty Chains one Mile. But if you would bring one sort of Mea­sure into another, you must work by Multiplication or Division. As for example, Suppose you would know how many Inches are contained in twenty Yards: First reduce the Yards into Feet, by multi­plying them by 3, because 3 Feet make one Yard, the Product is 60, which multiplyed by 12, the num­ber of Inches in one Foot, gives 720, and so many Inches are contained in 20 Yards Length.

On the contrary, if you would have known how many Yards there are in 720 Inches, you must first divide 720 by 12, the Quotient is 60 Feet; that again divided by 3, the Quotient is 20 Yards. The like you must do with any other Measure, as Perches, Chains, &c. of which more by and by.

See this Table of Long Measure annexed, the use whereof is very easie: If you would know how ma­ny Feet in Length go to make one Chain; look for Chain at Top, and at the Left-hand for Feet, against which, in the common Angle of meeting, is 66, so many Feet are contained in one Chain.

But because Mr. Gunters Chain is most in use among Surveyors for measuring of Lines, I shall chiefly 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 multiplyed 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 multiplyed 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 Left-hand the Feet required.

Square Measure.

Planometry, or the measuring the Superficies or Planes of things (as Sir Jonas Moore says) is done with the Squares of such Measures, as a Square Foot, a Square Perch, or Chain, that is to say, by Squares whose Sides are a Foot, a Perch, or Chain; and the Content of any Superficies is said to be found, when we know how many such Squares it containeth.

As for Example: Suppose ABCD was a Piece of

Land, and the Length of the Line AB or CD was 4 Perches; also the Length of the Line AC or BD was 5 Perches; I say that Piece of Land contains 20 Square Perches, as you may see it here divi­ded; every little Square be­ing a Perch, having a Perch in Length for its side. If you lay down a Square Fi­gure, whose side is 1 Foot, [Page 47]and at the end of every Inch you draw Lines crossing one another, as these here, you will divide that Square Foot into 144 little Squares, or Square Inches.

Or thus, The Line ab is a Perch long or 16 Feet ½, so is the Line bd, and the,

other 2 Lines: The whole Figure abcd is called a Square Perch.

But before we go any farther, take this Table fol­lowing of Square Measure.

And first of the Chain.

THere are several sorts of Chains, as Mr. Rath­borne's of two Perch long: Others, of one Perch long, some have had them 100 Feet in length: But that which is most in use among Surveyors (as being indeed the best) is Mr. Gunter's, which is 4 Pole long, containing 100 Links, each Link being 7 92/100 Inches: The Description of which Chain, and how to reduce it into any other Measure, you have at large in the foregoing Chapter of Measures. In this place I shall only give you some few Directions for the use of it in Measuring Lines.

Take care that they which carry the Chain, de­viate not from a streight Line; which you may do by standing at your Instrument, and looking through the Sights: If you see them between you and the Mark observed, they are in a streight Line, other­wise not. But without all this trouble, they may carry the Chain true enough, if he that follows the Chain always causeth him that goeth before to be in a direct line between himself, and the place they are going to, so as that the Foreman may always cover the Mark from him that goes behind. If they swerve from the Line, they will make it longer than really [Page 55]it is; a streight Line being the nearest distance that can be between any two places.

Besure that they which carry the Chain, mistake not a Chain either over or under in their Account, for if they should, the Error would be very considerable; as suppose you was to measure a Field that you knew to be exactly Square, and therefore need measure but one Side of it; if the Chain-Carriers should mistake but one Chain, and tell you the Side was but 9 Chains, when it was really 10, you would make of the Field but 8 Acres and 16 Perches, when it should be 10 Acres just. And if in so small a Line such a great Er­ror may arise, what may be in a greater, you may easily imagine. But the usual way to prevent such Mistakes is, to be provided with 10 small Sticks sharp at one End, to stick into the Ground; and let him that goes before take all into his Hand at setting out, and at the End of every Chain stick down one, which let him that follows take up; when the 10 Sticks are done, be sure they have gone 10 Chains; then if the Line be longer, let them change the Sticks, and pro­ceed as before, keeping in Memory how often they change: They may either Change at the end of 10 Chains, then the hindmost Man must give the fore­most all his Sticks; or which is better, at the end of 11 Chains, and then the last Man must give the first but 9 Sticks, keeping one to himself. At every Change count the Sticks, for fear lest you have dropt one, which sometimes happens.

If you find the Chain too long for your use, as for some Lands it is, especially in America, you may then take the half of the Chain, and measure as before, remembring still when you put down the Lines in your Field Book, that you set down but the half of [Page 56]the Chains, and the odd Links, as if a Line measured by the little Chain be 11 Chains 25 Links, you must set down 5 Chains 75 Links, and then in plotting and casting up it will be the same as if you had mea­sured by the whole Chain.

At the end of every 10 Links, you may, if you find it convenient, have a Ring, a piece of Brass, or a Ragg, for your more ready reckoning the odd Links.

When you put down 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 shewed 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 Inclosures, the Semi­circle for Champaign Grounds, The Circumferentor, the Theodolite, &c. To describe these to you, their Parts, how to put them together, take them asun­der, &c. 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 [Page 57]Paper concerning them. Let it suffice that the only use of them all is no more (or chiefly at most) but this; viz.

To take the Quantity of an Angle.

As suppose A B and A C are two Hedges or other Fences of a Field, the Chain serves to measure the

length of the Sides AB or AC, and these Instru­ments we are speaking of are to take the Angle A. And first by the

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 [Page 58]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 move­able 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, substract 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 it self 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 shewed 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, &c. 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 publick; therefore as I have before shewed you, How by the Semicircle to take an Angle without the help of the Needle; I shall here direct you

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 Pra­ctise; 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 [Page 62]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,

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, then 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 chuse whether you will have any Lines or not, if you can write streight, and in good order, the Figures directly one under another. For this I leave you chiefly 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 shewing the Figure of such a one as are commonly sold in Shops, and teaching how to use it.

Those Lines that are numbred at top with 11, 12, 16, &c. 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,

and draw the Line AB, which shall contain in length 10 Chains, 50 Links, by the Scale of 11 in an Inch. The back-side 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.

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.

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 take the Plot of a Field at one Station in any place thereof, from whence you may see all the Angles by the Semicircle.

ADmit ABCDEF to be a Field, of which you are to take the Plot: First set your Se­micircle upon the Staff in any convenient place there­of, as at ☉, and cause Marks to be set up in every Angle: Direct your Instrument, the Flower de Luce from you to any one Angle: As for Example, to A, and espying the Mark at A through the fixed Sights, there screw fast the Instrument; then turn the move­able

[Page 72]Index about (the Semicircle remaining immove­able) 'till through the Sights thereof you espy the Mark at B. See what Degrees on the Brass Limb are cut by the Index, which let be 80, write that down in your Field-Book, so turn the Index round to every one of the other Angles, putting down in your Field-Book what Degrees the Index points to, as for Exam­ple, at C 107 Degrees, at D 185, mark that at D, the End of the Index will go off the Brass Limb, and the other End will come on; you must therefore look for what Degrees the Index cuts in the innermost Circle of the Limb at E 260, at F 315 Degrees.

All which you may note down in your Field-Book thus.

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 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 signifie 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, &c. making marks where, upon the several Lines, the distances fall.

Lastly, Between those Marks draw Lines, as AB, BC, CD, &c. and then have you the true Plot of the Field ready protracted to your hand. This Instrument is so plain and easie 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 Semi-circle, 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 shewed you how to take the Plot of a Field at one Station, by [Page 75]the Simi-circle, 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 shew 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, &c. 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 numbred from the North Eastwards 10, 20, 30, &c. to 360 deg. Some from the North Westard, which are best for this use, Pro­tractors being made accordingly: For when you [Page 76]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 Semi-circle 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

at which you would take your Observations. Ha­uing placed your Semi-circle at F, turn it about the [Page 77]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:

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 Moveable Index, to every several Angle round the whole Field, [Page 80]

and see what Degrees are cut thereby at every Angle, which note down in your Field-Book as followeth: [Page 81]

Secondly, measure the Distance between the two Stations, which let be 20 Chains, and set it down in the Field-Book.

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.

Let CDEFGHIKLMNO be a Field in which from no one Place thereof all the Angles may be seen; chuse 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.

The Distance between the two Station 31 Ch. 60 L.

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 shew 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.

ABCDE is the Field, and A the Angle appoin­ted for the Station; place your Semicircle in A, and direct the Diameter thereof 'till through the fixed Sights you see the Mark at B, then screw it fast, and turn the Index to C, observing what Degrees are there cut upon the Limb; which let be 68 Degrees; turn it further, 'till you espy D, and note

[Page 87]down the Degrees there cut, viz. 76 Degrees; do the like at E, and the Index will cut 124 Degrees: This done, measure round the Field, noting down the length of the Side Lines between Angle and Angle, as from A to B 14 Chains 00 Links, from B to C 15 Chains, 00 Links, from C to D 7 Chains 00 Links, from D to E 14 Chains 40 Links, and from E to A 14 Chains 05 Links:

Then will your Field-Book be as hereunder.

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, AE, 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 [Page 88]one Foot of the Compasses in DE, with the other cross the Line AE, 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 streight 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.

To protract this you must work as you were taught concerning the foregoing Figure, untill you have drawn the Lines AB, AC, AD, AE, 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; lastly 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 AE 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 [Page 90]place for B, there make an Angle of 51 Degrees, drawing the Line 'till it intersect AC, &c.

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.

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.

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 AG, 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 lyes 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, lye 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 lye 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 lye 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. lye upon the line EF; against the end of the Diameter make a mark, through which draw the line FG, which will intersect the line AG 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 lye 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 chiefly to shew 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 South-side of the Instrument to G, and with the Index had taken B, and so of the rest, the work would have been the same, remem­bring 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 [Page 94]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, chuse 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 [Page 95]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 substract 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 it self,) 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 [Page 96]as the Line of your Hedg, your work will be the same. Thus if AB was a bushy Hedge, to which

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

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 [Page 97]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, &c. 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.

Let A, B, C, D, be a Wood or Field, to be thus measured. Cause your Assistants to set up Marks in

every Angle thereof, not regarding the distance from the Hedges, so much as the convenience for planting [Page 98]the Instrument, so as you may see from one Mark to another. Then beginning at ☉ 1, take the quantity of that Angle, and measure the distance 1, 2. But before you begin to measure the Line, take the Off­set to the Hedge, viz. the distance ☉ e; and in taking of it, you must make that little Line ☉ e perpendicular to 1, 2; which is easily done, when your Instru­ment stands with the Fixed Sights towards 2, by turning the Moveable Index till it lye upon 90 deg. which then will direct to what place of the Hedge to measure to, as e, that little Line ☉ e: Set down in your Field-Book under title Off-set. So likewise when you come to 2, measure there the Off-set again, viz. ☉ f. Then taking the Angle at 2, measure the Line 2, 3, and the Off-sets 2 g, 3 h. The like do by all the rest of the Lines and Angles in the Field, how many soever they be. And when you come to lay this down upon Paper; first, as you have been taught before, Protract the Figure 1, 2, 3, 4. That done, set off your Off-sets as you find them in the Field-Book, viz. ☉ e, and ☉ f, perpendicular to the Line 1, 2; also ☉ g, ☉ h, perpendicular to the Line 2, 3, making Marks at e, f, g, h, and the rest; through which draw Lines, which shall intersect each other at the true Angles, and describe the true Bound-Lines of the Field or Wood.

In working after this manner, observe these two things. First, if the Wood be so thick, that you cannot go within-side thereof, you may after the same manner as well perform the Work, by going on the out-side 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 [Page 99]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.

I have drawn this following Figure so, that at once you may see all the variety of this way of Working. The best way, indeed, is to contrive your Square

so, that, if possible, you may from the Sides thereof go upon a Perpendicular-line to any of the Angles. But if that cannot be, then Perpendicular-lines to the Sides may do as well, as you see here 1, 5, 7, 6, [Page 100]to be. To begin therefore, plant your Semi-circle in any convenient place of the Field, for taking a large Square, as at 1; and laying the Moveable Index upon 90 deg. look through the Sights, and cause a Mark to be set up in that Line, as at 4: Looking also through the Fixed-Sights, cause another Mark to be set up, as at 2. Measure out from your In­strument, towards either of these Marks, any num­ber Chains, as 1, 2, 12 Chains; 1, 4, 12 Chains. But as you measure, remember to take the Off-sets in a Perpendicular-line to every Angle or Side, if there be occasion, as here at 7, which is 1 Chain, 50 Links from my Station I take an Off-set to a side of the Hedge, and put it down accordingly 5 Chains, 40 Links. So at 8 I take an Off-set to an Angle, viz. 8 B, 6 Chains; which Off set is at the end of 8 Chains, 30 Links in my first Line. Then seeing in that Line there is no more occasion of Off-sets, I measure on to 2, making the Line 1, 2, 12 Chains. Then planting my Instrument at 2, I direct the Fixed-Sights to my first Station, and laying the Index upon 90 deg. I cause a Mark to be set up, so as that I may see it through the Sights; and upon that Line, as I measure out 12 Chains, I take the Off-sets C 9, D 10. In like manner you must do for the other Angle, Lines and Off-sets.

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 Left-hand.

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 [Page 102]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 Bounding­lines, remembring 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 [Page 103]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 chiefly intended for such as were not provided with Instruments; for instead of the Semi-circle 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 numbred with four 90 ^s 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.

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 lye 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 Publick 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.

When you have Surveyed the Ground as above di­rected, and find your Field-Book to stand as before; cast up what Northing, Southing, Easting or West­ing [Page 106]every Line makes; that is to say, How far at the End of every Line you have altered your Meridian, and what Distance upon a Meridian-Line you have made: As for Example, suppose AB was the Side of a Field measured to be 20 Chains, NS a Meridian-Line,

the Angle CABNE 20 Degrees. The business is to find the Length of the Line AC, which is called the Northing, or the difference of Latitude; also the length of the Line CB, which is called the Easting, or Difference of Longitude, which you may do indif­ferently truly by laying them down thus upon Paper: But passing this and the Gunter's Scale, the only way is by the Tables of Sines and Logarithms, where the Proportion is this.

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 Complement of the Angle at A to the Logarithm of the Line AC 18 Chains 80 odd Links.

How by the Chain only, to take an Angle in the Field.

First measure along the Hedge AB, any small di­stance, as A2 two Chains; also measure along the

[Page 112]Hedg AC what number of Chains you please, no matter whether they be equal to the former or not; as A3 two Chains; next measure the distance 2, 3, viz. 1 Chain 68 Links; and then have you done in the Field. To plot which, draw the Line AB at adventure, and set off 2 Chains from A to 2; then take with your Compasses the distance A3, 2 Chains, and setting one Foot in A, describe the Arch 2, 3; take also with your Compasses the di­stance 2, 3, viz. 1 Chain 68 Links; and setting one Foot in 2, with the other cross the former Arch; through which Cross draw the Line AC; which with AB will make an Angle equal to the Angle in the Field.

But the more easie 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, then I stretch the Chain also along the Hedge AC, and at the end thereof set another Stick as at 4, then loosing 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 [Page 113]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.

Let ABCDEF be the Field; and beginning at A in the very Angle, stick down a Staff through the [Page 114]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 AF 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 your self to and Fro 'till you perceive your self 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 your self 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 signifie 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 [Page 115]Angle here on purpose to shew 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.

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, AF: [Page 116]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 EF and AF, 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 [Page 117]the Field, and note down in your Field-Book every Line thereof, as in this Field has been before done.

Secondly, they bid you turn all the Field into Tri­angles, as beginning at A, to measure the Diagonal AC, AD, AE, and note them down; then is your Field turned into four Triangles, and the Diagonals are, [Page 118]

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 AE, 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 AF, viz. 31 Chains, 50 Links; and setting one foot in A, describe an Arch, as II. Also take the length of the Line EF, viz. 29 Chains, 00 Links, and therewith describe the Arch hh, which cuts the Arch II, in the Point F, to which Point draw the Lines AF and EF, and so will you have a true Figure of the Field.

I have shewed 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 [Page 120]☉ 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,

cd 1 Chain 00 Links, de 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, &c. all which put down, your Field-Bok will appear thus; [Page 121]

Of the Square, and Parallelogram.

To cast up either of which, multiply one Side by the other, the Product will be the Content.

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.

To find the Content of a Trapezia.

Draw between two opposite Angles a streight Line, as AB; then is the Trapezia reduced into two Triangles, viz. ABC and ABD, 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:

Take the length of the Line AB, which let be 37 Chain 00 Links; take also the length of the Per­pendicular

D e, which let be 7 Chains 40 Links; also C d 4 Chains 80 Links: add the two Perpendi­culars together, and they make 12 Chains 20 Links, which multiply by half the common Base AB 18 Chains 50 Links, and the Product is 22 Acres, 2 Rood, 11 Perch, as appears by the Operation hereunder.

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.

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, [Page 129]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 multiplyed by 22, and

the Product divided by 7, gives 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 multiplyed together, the Product is 3,1400 Square Links, or 1 Rood 10 Perch, the Content of the Circle. Again,

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 [Page 131]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 bredth to the Content, or nearer, as 1,27324 to the length; so the bredth to the Content.

How to find the Superficial Content of Regular Polygons; as Pentagons, Hexagons, Hep­tagons, &c.

Multiply half the summ 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 summ of the five Sides,

therefore must be 420, the half of which is 210, which multiplyed 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, &c. often.

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 Cy­phers, 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.

How to lay out any given Quantity of Acres in a Parallelogram; whereof one Side is given.

Turn first the Acres into Links, by adding as be­fore 5 Cyphers, that number thus increased, divide by the given Side, the Quotient will be the other Side.

How to lay out a Parallelogram that shall be 4, 5, 6, or 7, &c. times longer than it is broad.

In Carolina, all Lands lying by the Sides of Rivers, except Seignories or Baronies, are (or ought, by Order of the Lord's Proprietors to be) thus laid out. To do which, first as above taught, turn the given quantity of Acres into Links, by annexing 5 Cyphers; which summ divide by the number given for the Pro­portion between the length and bredth, as 4, 5, 6, 7, &c. the Root of the Quotient will shew the shortest Side of such a Parallelogram.

How to make a Triangle that shall contain any number of Acres, being confined to a certain Base.

Double the given number of Acres, (to which annexing first five Ciphers,) divide by the Base; the Quotient will be the length of the Perpendicular.

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.

How to Reduce a large Plot of Land or Map into a lesser compass, according to any given Proportion; or e contra, how to Enlarge one.

THe best way to do this, is, if your Plot be not over-large, to plat it over again by a smaller Scale: But if it be large, as a Map of a County, or the like, the only way is to compass in the Plot first with one great Square; and afterwards to divide that into as many little Squares, as you shall see conve­nient. Also make the same number of little Squares upon a fair piece of Paper, by a lesser Scale, accord­ing to the Proportion given. This done, see in what Square, and part of the same Square, any re­markable accident falls, and accordingly put it down in your lesser Squares; and that you may not mi­stake, it is a good way to number your Squares. I cannot make it plainer, than by giving you the fol­lowing Example, where the Plot ABCD, made by a Scale of 10 Chains in an Inch, is reduced into the Plot EFGH, of 30 Chains in an Inch.

There are several other ways taught by Surveyors for reducing Plots or Maps, as Mr. Rathboxn, and after him Mr. Holwell, adviseth to make use of a Scale or Ruler; having a Centre-hole at one end, through which to fasten it down on a Table, so that it may play freely round; and numbred from the Cen­tre-end to the other, with Lines of Equal Parts: The Use of which is thus. Lay down upon a smooth Table, the Map or Plot that you would reduce, and glew it with Mouth-glew fast to the Table at the four corners thereof. Then taking a fair piece of Paper about the bigness that you would have your reduced Plot to be of, and lay that down upon the other; the middle of the last about the middle of the first. This done, lay the Centre of your Reducing Scale near the Centre of the white Paper, and there with a Needle through the Centre make it fast; yet so, that it may play easily round the Needle. Then moving your Scale to any remarkable thing of the first Plot, as an Angle, a House, the bent of a River, or the like: See against how many Equal Parts of the Scale it stands, as suppose 100; then taking the ⅓, the ¼, the ⅕, or any other number thereof, accord­ing to the Proportion you would have the reduced Plot to bear; and make a mark upon the white Paper against 50, 25, 33, &c. of the same 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 [Page 140]Scale may be one way numbred with Equal Parts from the Centre-hole to the end; and ⅓ part thereof numbred the other way to the end with the same number of Equal Parts, tho lesser. Upon this Scale may be several Lines of Equal Parts, the lesser to the greater, according to several Proportions. Being thus provided with a Scale, glew down upon a smooth Table your greater Plot to be reduced; and close to it upon the same Table, a Paper about the bigness whereof you would have your smaller Plot. Fix with a strong Needle the Centre of your Scale between both; then turning the longer end of your Scale to any remarkable thing of your to be reduced Plot, see what number of Equal Parts it cuts, as sup­pose 100; there holding fast the Scale, against 100 upon the smaller end of your Scale, make a mark upon the white Paper; so do round all the Plot, drawing Lines, and putting down all other accidents as you proceed, for fear of confusion, through many Marks in the end; and when you have done, al­though at first the reduced Plot will seem to be quite contrary to the other; yet when you have unglewed it from the Table, and turned it about, you will find it to be an exact Epitome of the first. You may have for this Work divers Centers made in one Scale, with Equal Parts proceeding from them accordingly; or you may have divers Scales, according to several Proportions, which is better.

What has been hitherto said concerning the Re­ducing of a Plot from a greater volume to a lesser, the same is to be understood vice versa, of Enlarging a Plot, from a lesser to a greater. But this last seldom comes in practise.

How to change Customary-Measure into Sta­tute, and the contrary.

In some Parts of England, for Wood-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.

To Survey a Mannor observe these following Rules.

1. WAlk or ride over the Mannor once or twice, that you may have as it were a Map of it in your Head, by which means, you may the better know where to begin, and proceed on with your Work.

2. If you can conveniently run round the whole Mannor with your Chain and Instrument, taking all the Angles, and measuring all the Lines thereof; ta­king notice of Roads, Lanes or Commons as you [Page 143]cross them: Also minding well the Ends of all divid­ing Hedges, where they butt upon your bound Hedges in this manner.

3. Take a true Draught of all the Roads and By-Lanes in the Mannor, putting down also the true Buttings of all the Field-Fences to the Road. If the Road be broad, or goes through some Common or Wast Ground, the best way is to measure, and take the Angles on both Sides thereof; but if it be a nar­row Lane, you may only measure along the midst thereof, taking the Angles and Off-sets to the Hedges, and measuring your Distances truly: Also if there be any considerable River either bounds or runs through the Mannor, survey that also truly, as is hereafter taught.

4. Make a true Plot upon Paper of all the forego­ing Work; and then will you have a Resemblance of the Mannor, though not compleat, which to make so, go to all the Buttings of the Hedges, and there Survey every Field distinctly, plotting it accordingly every Night, or rather twice a Day, till you have perfected the whole Mannor.

5. When thus you have plotted all the Fields, ac­cording to the Buttings of the Hedges found in your first Surveys, you will find that you have very nigh, if not quite done the whole Work: But if there be any Fields lye so within others, that they are not bounded on either Side by a Road, Lane nor Ri­ver; then you must also Survey them, and place them in your Plot, accordingly as they are boun­ded by other Fields.

6. Draw a fair Draught of the whole, putting down therein the Mannor-House, and every other considerable House, Wind-mill, Water-mill, Bridg, Wood, Coppice, Cross-paths, Rills, Runs of Wa­ter, Ponds, and any other Matter Notable therein. Also in the fair Draught, let the Arms of the Lord of the Mannor be fairly drawn, and a Compass in some wast part of the Paper; also a Scale, the same by which it was plotted: You must also beautifie such a Draught with Colours and Cuts according as you shall see con­venient.

Write down also in every Field the true Content thereof; and if it be required, the Names of the present Possessors, and their Tenures: by which they hold it of the Lord of the Mannor.

The Quality also of the Land, you may take notice of as you pass over it, if you have Judgment therein, and it be required of you.

How to take the Draught of a County or Country.

1. If the County or Country is in any place there­of bounded with the Sea, Survey first the Seacoast thereof, measuring it all along with the Chain, and taking all the Angles thereof truly.

2. Which done, and plotted by a large Scale, Survey next all the Rocks, Sands or other Obstacles that lye at the entrance of every River, Harbor, 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, &c. 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, &c.

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.

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,

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 maner, 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 Arithmetick, but my Business is [Page 147]not to shew you what maybe done, but to shew you how to do it, the most easie and practicable way.

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 shew you how to change any Four-Sided Figure into a Triangle; [Page 149]which done, the Work will be the same as in divi­ding Triangles.

How to reduce an Irregular Five-Sided Figure into a Triangle, and to divide the same.

Let ABCDE be the Five-Sided Figure; to re­duce which into a Triangle, draw the Lines AC,

AD; and parallel thereto BF, EG extending the Base from C to F, and from D to G; then draw the Lines AF, AG, which will make the Triangle AFG equal to the Five Sided-Figure. If this was [Page 152]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.

How to Divide an Irregular Plot of any number of Sides, according to any given Proportion, by a streight Line through it.

ABCDEFGHI is a Field to be divided be­tween two Men in equal Halfs, by a streight Line proceeding from A.

First, consider how to divide the Field into five-sided Figures and Trapezias, that you may the better re­duce it into Triangles: As by drawing the Line KL, you cut off the five-sided Figure ABCHI; which reduce into the Triangle AKL, and measuring half the Base thereof, which will fall at Q, draw the Line QA.

Secondly, Draw the Line MN, and from the Point Q reduce the Trapezia CDGH into the Tri­angle MNQ; which again divide into Halfs, and draw the Line QR.

Thirdly, From the Point R, reduce the Trapezia DEFG into the Triangle ROP; and taking half the Base thereof, draw the Line RS; and then have you divided this Irregular Figure into two Equal Parts by the three Lines AQ, QR, RS.

Fourthly, Draw the Line AR, also QT parallel thereto. Draw also AT, and then have you turned two of the Lines into one.

Fifthly, From T draw the Line TS; and parallel thereto, the Line RV. Draw also TV. Then is your Figure divided into two Equal Parts, by the two Lines AT and TV.

Lastly, Draw the Line AV, and parallel thereto TW. Draw also AW, which will cut the Figure into two Equal Parts by a streight Line, as was required.

You may, if you please, divide such a Figure all into Triangles; and then divide each Triangle from the Point where the Division of the last fell, and then will your Figure be divided by a crooked Line, which you may bring into a streight one, as above.

This above is a good way of Dividing Lands, but Surveyors seldom take so much pains about it. I shall therefore shew you how commonly they abbre­viate their Work, and is indeed

An easie way of Dividing Lands.

Admit the following Figure ABCDE contain 46 Acres, to be divided in Halfs between two Men, by a Line proceeding from A.

Draw first a Line by guess, through the Figure, as the Line AF. Then cast up the Content of either Half, and see what it wants, or what it is more than the true Half should be.

As for Example. I cast up the Content of AEG, and find it to be but 15 Acres; whereas the true Half is 23 Acres; 8 Acres being in the part ABCDG, more than AEG. Therefore I make a Triangle containing 8 Acres, and add it to AEG, as the Triangle AGI; then the Line AI parts the Figure into equal Halfs.

But more plainly how to make this Triangle: Measure first the Line AG, which is 23 Chains, 60 Links. Double the 8 Acres, they make 16; to which add five Cyphers to turn them into Chains and Links, and then they make 1600000; which divide by AG 2360, the Quotient is 6 Chains, 77 Links; for the Perpendicular HI, take from your Scale 6 Chains, 77 Links, and set it so from the Base AGF, that the end of the perpendicular may just touch the Line ED, which will be at I. Then draw the Line AI, which makes the Triangle AGI [Page 156]

just 8 Acres, and divides the whole Figure, as de­sired.

If it had been required to have set off the Perpen­dicular the other way, you must still have made the end of it but just touch the Line ED, as LK does: For the Triangle AKG is equal to the Triangle AGI, each 8 Acres.

And thus you may divide any piece of Land of never so many Sides and Angles, according to any Proportion, by streight Lines through it, with as much certainty, and more ease than the former way.

Mark, you might also have drawn the Line AD, and measured the Triangle AGD, and afterwards have divided the Base GD, according to Proportion, [Page 157]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.

First from the Pond ☉ draw Lines to every Angle, as ☉ A, ☉ B, ☉ C, ☉ D, ☉ E,; and then is the Figure

divided into five Triangles, each of which measure, and put the Contents down severally; which Con­tents reduce all into Perches, so will the Triangle.

• A ☉ B be 674 Perches,
• B ☉ C be 390 Perches,
• C ☉ D be 1238 Perches,
• D ☉ E be 911 Perches,
• E ☉ A be 1107 Perches,

the whole Content being 4320 Perches, or 27 Acres, each Man's Proportion being 1440 Perches.

From ☉ to any Angle draw a Line for the first Divi­sion-line, as ☉ A. Then consider that the first Angle A ☉ B is but 674 Perches, and the second B ☉ C 390, both together but 1064 Perches, less by 376 than 1440, one Man's Portion. You must therefore cut off from the third Angle C ☉ D 376 Perches for the first Man's Dividing-line; which thus you may do: The Base DC is 18 Chains; the Content of the Triangle 1238 Perches: Say then, if 1238 Perches give Base 18 Chains, 00 Links: What shall 376 Perches give? Answer 5 Chains, 45 Links; which set from C to F, and drawing the Line ☉ F, you have the first Man's part, viz. A ☉ F.

Secondly, See what remains of the Triangle C ☉ D 376 being taken out, and you will find it to be 862 Perches, which is less by 578 than 1440. Therefore from the Triangle D ☉ E cut off 578 Perches, and the point of Division will fall in G. Draw the Line ☉ G, which with ☉ A and ☉ F, divides the Figure into three Equal Parts.

How to Divide a Circle according to any Pro­portion, by a Line Concentrick with the first.

All Circles are in Proportion to one another as the Squares of their Diameters; therefore if you divide the Square of Diameter or Semi-diameter, and ex­tract the Root, you will have your desire.

The use of the Tables of Sines and Tangents.

Any Angle being given in Degrees and Minutes, how to find the Sine or Tangent thereof.

Let 25 Degrees 10 Minutes be given to find the Sine and Tangent thereof; first in the Table of Sines and Tangents, at the Head thereof seek for 25, and having found it, look down the first Column on the Left-hand under M for the 10 Minutes, and right against 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 Cosine or Sine Complement; the Cotangent or Tangent Complement of any given Degrees and Minutes.

The Cosine 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 Complement of 25 Degrees 10 Minutes.

But the more ready way to find the Cosine or Co­tangent 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 Cosine and Cotangent; or above, if the De­grees exceed 45, you will find the Cosine or Cotan­gent require: Thus the Cosine 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, &c. 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

which 1,301030 I seek for in the Logarithm Tables, and right against, under Title Num. stands 20, the Product of 5 multiplyed by 4.

If I were to divide 20 by 5, first I look for the Lo­garithm of 20, which as above, is

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.

Certain Theorems for the better understanding Right-Lined Triangles.

1. A Right-Lined Triangle is a Figure comprehen­ded within three Streight Lines.

2. Which is either Right-Angled as A, having one Right Angle, which con­tains just 90 Degrees, viz. that at b; or else Oblique 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.

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.

Let ABCD be a Circle, whose Circumference is divided (as all Circles you must esteem so to be) into 360 Equal Parts, which are called De­grees, and each of those Degrees into 60 Equal Parts more, which are called Minutes: Now a Right-Angled Triangle is that which cuts off one fourth [Page 164]

part of this Circle, viz. Degrees, as you see the Triangle EFG to do.

An Angle that cuts off less than 90 Degrees, is called an Acute Angle, as HEF, which takes but 45 Degrees from the Circle.

GEI is an Obtuse Angle, for the two Lines that

[Page 165]proceed from E, take in between them more than a quarter of the Circle, viz. 113 Degrees.

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 Sines 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, substract it out of 180, and work by the remainder.

How to take the Heighth of a Tower, Steeple, Tree, or any such thing.

LEt AB be a Tower, whose Height you would know.

Frist, At any convenient distance, as at C, place your Semi-circle, or what other Instrument you judge more fit for the taking an Angle of Altitude, as a large Quadrant, or the like, and there observe the Angle ACB. But to be more plain, place your Semi-circle at C; and having turn'd it down by a Plumb, make it to stand Horizontal, which it does when a Plummet-line fixt to the Centre, falls just upon 90 deg. (in some Semi-circles there is a Line on the Back-side of the Brass Limb on purpose for the setting of it Horizontal.) Then (first skrewing the Instru­ment fast) move the Index up and down, till through the Sights you espy the top of the Tower at A. See then what Deg. upon the Limb are cut by the Index, [Page 181]which let be 58, so much is your Angle of Altitude. Measure next the distance betwen your Instrument and the foot of the Tower, viz. the Line CD, which

let be 25 Yards: Then have you all the Angles given, (admitting the Angle the Tower makes with the Ground, viz. d to be a Right-Angle) and the Base to find the Perpendicular AB; which you may do, as you were taught in Case I. Of Trigonometry: For if you take 58 from 90, there remains 32 for the Angle at A. Then say, [Page 182]

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 Heighth found. In this way of taking Heighths, 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 Heighth 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 Heighth FE the same as DA; which by the Figure you plainly perceive to be a great Error: Therefore to take the Heighth FE, you should measure from C to F.

How to take the Heighth of a Tower, &c. 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 Heighth. 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, sub­stract 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,

Again,

But still, as I told you before, the Ground is un­derstood to be level. However, if it be not, I will shew you,

How to take the Heighth of a Tower, &c. when the Ground either riseth or falls.

AB is the Tower, CB the Hill whereon you are to take the Heighth of the Tower; plant your Semi­cirle

in any place of the Hill, as at C, then turn it down, and make it stand Horizontal, as before di­rected, the Diameter then pointing to d of the Tow­er, [Page 185]turn the Moveable Index to A, and take the An­gle AC d; which let be 19 Degrees 30 Minutes. Take also the Angle d CB, which is 45 Degrees 30 Minutes; measure also the Distanee CB 56 Yards, take 19 Degrees 30 Minutes out of 90 Degrees 00 Mi­nutes, there remains 70 Degrees 30 Minutes for the Angle at A, then say

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 AF, then say,

As the Sine of the Angle ABF is to the Logarithm of the Line FA,

So is the Sine of the Angle AFB To the Logarithm of the Heighth of the Tower AB.

Of Distances.

Although I have before shewed 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 [Page 186]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.

Beginning at ☉ 1, the first Station, cause one of your Assistants to go to the next Bend of the River, as ☉ 2, and there set up a Mark for you; then see what Angle from the Meridian ☉ 1, ☉ 2 makes, which let be N. W. 6 Degrees; also seeing several Marks on the other Side of the River, take their Bearings, as the House A, which stands upon the Bank, and is a good Mark for the Bredth of the River bears N. W. 52 Degrees, the Wind-mill B up in the Land, bears N. W. 40 deg. The Tree C by the Water-side, bears N. W. 17 deg. All this note down in your Field-Book, and measure the distance ☉ 1, ☉ 2, 18 Chains, 20 Links. After this coming to ☉ 2, see how the next bend of the River bears from you, viz. ☉ 3; which is NE 15 deg. See also how the House A there bears from you, viz. S. W. 20 deg. The Wind-mill S. W. 50 deg. The Tree N. W. 77. Also as you are going forward, if you see any thing more at this second Station, take the bearing thereof, as a noted House D up in the Land, bears N. W. 28° And a Church E close by the Rivers brink N. W. 4° Measure the distance 2, 3, and placing your Instru­ment at 3, the Church bears from you N.W. 88 deg. The House up in the Land D you cannot see for the Church, therefore let it alone for the next Station. But here you may see forward a little Village F, the first House whereof bears from you N. W. 32 deg. Measure the distance 3, 4, and planting your Instru­ment in 4, the first House of the Village F bears from you S. W. 32 deg. and the House D, which you could [Page]

[Page 188]not see at the third Station, S. W. 24°. Having put down all these things in your Field-Book, it will not look much unlike this,

• ☉ 1 N. W. 6° 18 Ch. 20 Lin.
• Observation A Tree upon the brink of the River, bears N. W. 17° 00′
• Observation A Wind-mill up in the Land N. W. 40° 00′
• Observation A House upon the Rivers bank N. W. 52° 00′
• ☉ 2 N. E. 15° 18 Ch. 10 Lin.
• The Tree N. W. 77° These look back to the Observation of ☉ 1.
• The House S. W. 20° These look back to the Observation of ☉ 1.
• The Wind-mill S. W. 50° These look back to the Observation of ☉ 1.
• A noted House far up in the Land N. W. 28° Forward Observati­ons.
• A Church upon the Rivers bank N. W. 4° Forward Observati­ons.
• ☉ 3 N. W. 15° 20 Ch. 50 Lin.
• The Church bears N. W. 88° These look back to the Obser. of ☉ 2.
• The noted House cannot be seen These look back to the Obser. of ☉ 2.
• The end of a little Village N. W. 32 A forward Observation.
• ☉ 4—
• The end of the little Village S. W. 32° These respect ☉ 3 and ☉ 2.
• The House respecting ☉ 2 in the Land S. W. 24° These respect ☉ 3 and ☉ 2.

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, [Page 189]for the Wind-mill 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. SW 20. SW 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 Rivers bank; for that Line cutting the Line ☉ 1 C, shews you by the Intersection where the Tree stood, and also the Bredth of the River. Also the Line SW 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 Wind-mill B, and the House D, stand from the Water-side.

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 [Page 190]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 Heighths.

For suppose ABCD a Hill, whose Base you would know. Plant your Semi-circle at A, and cause a Mark to be set up at B, so high above the top of the

Hill, as the Instrument stands from the Ground at A; and making your Instrument Horizontal, take the Angle BAD 58 deg. Measure the Distance AB 16 Chains, 80 Links. Then say,

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, [Page 191]

Which 15. 12. added to 8.90, makes 24 Chains, 2. Links, for the whole Base AG; 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. Substract 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 Rivers 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, &c. Then going out in a Boat to such [Page 192]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 gon 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.

How to know whether Water may be made to run from a Spring-head to any appointed Place.

For this Work, the Diameter of the Semi-circle 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 Spring-head; 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 Mid-way 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 [Page 196]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, &c. 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 your self 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 Spring-head, 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.

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.