equall 2. If any right line fall upon two parallel right lines it maketh the outward angles on the one equall to the inward angles on the other and the two inward opposite angles on contrary sides of the falling line also equall 3. If any side of a Triangle be produced the outward angle is equall to the two inward opposite angles and all the three angles of any Triangle are equall to two right angles 4. In equiangled Triangles all their sides are proportionall as well such as contain the equall angles as also the subtendent sides 5. If any four Quantities be proportionall the first multiplied in the fourth produceth a Quantity equall to that which is made by multiplication of the second in the third 6. In all right angled Triangles the square of the side subtending the right angle is equall to both the squares of the containing sides 7. All parallelograms are double to the triangles that are described upon their bases their altitudes being equall 8. All triangles that have one and the same Base and lie between two parallel lines are equall one to the other GEOMETRICALL PROBLEMES PROBLEME I. Vpon a right line given how to erect another right line which shall be perpendicular to the right line given THe right line given is AB upon which from the point E it is required to erect the perpendicular EH Opening your Compasses at pleasure to any convenient distance place one foot in the assigned point E and with the other make the marks C and D equidistant on each side the given point E. Then opening your Compasses again to any other convenient distance wider then the former place one foot in C and with the other describe the arch GG also the Compasses remaining at the same distance place one foot in the point D and with the other describe the arch FF then from the point where these two arches intersect or cut each other which is at H draw the right line HE which shall be perpendicular to the given right line AB which was the thing required to be done PROB. II. How to erect a Perpendicular on the end of a right line given LEt OR be a line given and let it be required to erect the perpendicular RS. First upon the line OR with your Compasses opened to any small distance make five small divisions beginning at R noted with 1 2 3 4 5. Then take with your Compasses the distance from R to 4 and placing one foot in R with the other describe the arch PP Then take the distance R 5 and placing one foot of the Compasses in 3 with the other foot describe the arch BB cutting the former arch in the point S. Lastly from the point S draw the line RS which shall be perpendicular to the given line OR PROB. III. How to let fall a perpendicular from any point assigned upon a right line given THE point given is C from which point it is required to draw a right line which shall be perpendicular to the given right line AB First from the given point C to the line AB draw a line by chance as CE which divide into two equall parts in the point D then placing one foot of the Compasses in the point D with the distance DC describe the Semicircle CFE cutting the given line AB in the point F. Lastly if from the point C you draw the right line CF it shall be a perpendicular to the given line AB which was required PROB. IV. How to make an angle equall to an angle given LEt the angle given be ACB and let it be required to make another angle equall thereunto First draw the line EF at pleasure then upon the given angle at C the Compasses opened to any distance describe the ark AB also upon the point F the Compasses un-altered describe the arke DE then take with your Compasses the distance AB and set the same distance from E to D. Lastly draw the line DF so shall the angle DFE be equall to the given angle ACB PROB. V. A right line being given how to draw another right line which shall be parallel to the former at any distance required THe line given is AB unto which it is required to draw another right line parallel thereunto at the distance AC or BD. First Open your Compasses to the distance AC or AD then placing one foot in A with the other describe the arke C also place one foot in B and with the other describe the arch D. Lastly Draw the line CD so that it may only touch the arks C and D so shall the line CD be parallel to the line AB and at the distance required PROB. VI. To divide a right line given into any number of equall parts LEt AB be a line given and let it be required to divide the same into four equall parts First From the end of the given line A draw the line AC making any angle then from the other end of the given line which is at the point B draw the line BD parallel to AC or make the angle ABD equall to the angle CAB then upon the lines AC and BD set off any three equall parts which is one lesse then the number of parts into which the line AB is to be divided on â—ace line as 1 2 3 then draw lines from 1 to 3 from 2 to 2 and from 3 to 1 which lines crossing the given line AB shall divide it into four equall parts as was required PROB. VII A right line being given how to draw another right line parallel thereunto which shall also passe through a point assigned LEt AB be a line given and let it be required to draw another line parallel thereunto which shall passe through the given point C. First Take with your compasses the distance from A to C and placeing one foote thereof in B with the other describe the ark DE then take in your compasses the whole line AB and placing one foot in the point C with the other describe the arke FG crossing the former arke DE in the point H. Lastly if you draw the line CH it shall be parallel to AB PROB. VIII Having any three points given which are not situate in a right line how to finde the center of an arch of a Circle which shall passe directly through the three given points THe three points given are A B and C now it is required to finde the center of a Circle whose circumference shall passe through the three points given First Opening your Compasses to any distance greater then halfe BC place one foot in the point B and with the other describe the arch FG then the Compasses remaining at the same distance place one foot in C and with the other turned about make the marks F and G in the former arch and draw the line FG at length if need be Again opening the Compasses to any distance greater then halfe AB place one foot in
colours you must alwayes work with one end of the Chain from you This Chain being thus divided and marked you have every whole Pole equall to ten Primes or 100 Seconds every three quarters of a Pole equall to seven Primes and a halfe or 75 Seconds every halfe Pole equall to five Primes or 50 Seconds and lastly every quarter of a Pole equall to two Primes and a halfe or 25 Seconds And here is to be noted that in the ordinary use of this Chain for measuring and platting you need take notice only of Unites and Primes which is exact enough for ordinary use but in case that separation or division of Lands into severall parts you may make use of Seconds Of Mr. GUNTERS Chain AS every Pole of Master Rathborns Chain was divided into 100 Links so Master Gunters whole Chain which is alwayes made to contain four Poles is divided into 100 Links one of these Links being four times the length of the other Now if this Chain be made according to the Statute each Perch to contain 16½ Feet then each Link of this Chain will contain 7 Inches and 92 100 of an Inch and the whole Chain 729 Inches or 66 Foot In measuring with this Chain you are to take notice only of Chains and Links as saying such a line measured by the Chain contains 72 Chains 48 Links which you may expresse more briefely thus 72,48 and these are all the Denominations which are necessary to be taken notice of in Surveying of Land For the ready counting of the Links of this Chain there ought to be these distinctions namely In the middle thereof which is at two Poles end let there be hanged a large Ring or rather a plate of brasse like a Rhombus so is the whole Chain by this plate divided into two equall parts Secondly Let each of these two parts be divided into two other equall parts by smaller Rings or Circular plates of brasse so shall the whole Chain be divided into four equall parts or Perches each Perch containing 25 Links Thirdly At every ten Links let be fastened a lesser Ring then the former or else a Plate of some other fashion as a Semicircle or the like And lastly at every fift link if you please may be fastened other marks so by this means you shall most easily and exactly count the Links of your Chain without any trouble The Chain being thus distinguished it mattereth not which end thereof be carryed forward because the notes of distinction proceed alike on both sides from the middle of the Chain ¶ Here note that in all the examples in this Book the lines are supposed to be measured by this four Pole Chain of Master Gunter it being the best of any other the manner how to cast up the content of any plot measured therewith shall be hereafter taught in its due place Cautions to be observed in the use of any Chain IN measuring a large distance with your Chain you may casually mistake or misse a Chain or two in keeping your account from whence will ensue a considerable errour Also in measuring of distances when you go not along by a hedge side you can hardly keepe your Instrument Chain and Mark in a right line which if you do not you must necessarily make your measured distance greater then in reality it is For the avoyding of either of these mistakes you ought to provide ten small sticks or Arrows which let him that leadeth the Chain carry in his hand before and at the end of every Chain stick one of these Arrows into the ground which let him that followeth the Chain take up so going on till the whole number of Arrows be spent and then you may conclude that you have measured ten Chains without any further trouble and these ten Chains if the distance you are to measure be large you may call a Change and so you may denominate every large distance by Changes Chains and Links Or you may at the end of every ten Chains set up another kinde of stick by which standing at the Instrument you may see whether your eye the stick and the Mark to which you are to measure be in a right line or not and accordingly guide those that carry the Chain with the more exactnesse to direct it to the Mark intended How to reduce any number of Chains and Links into Feet IN the practise of many Geometricall Conclusions as in the taking of Heights and Distances hereafter taught it is requisite to give your measure in such cases in Feet or Yards and not in Poles or Perches yet because your Chain is the most necessary Instrument to measure withall I thought it convenient in this place to shew you how to reduce any number of Chains and Links into Feet which is thus Multiply your number of Chains and Links together as one whole number by 66 cutting off from the product the two last figures towards the right hand so shall the rest of the product be Feet and the two figures cut off shall be hundred parts of a Foot EXAMPLE Let it be required to know how many Feet are contained in 5 Chains 32 Links First Set down your 5 Chains 32 Links as is before taught and as you see in the first Example with a Comma between the Chains and Links then multiplying this 5 Chains 32 Links by 66 the product will be 35112 from which cut off the two last figures toward the right hand with a Comma then will the number be 351,12 which is 351 Feet and 12 100 parts of a foot and so many Feet are contained in 5 Chains 32 Links Example I. 5,32 66 3192 3192 351,12 Example II. 9,05 66 5430 5430 597,30 But let the number of Chains be what they will if the number of Links be lesse then 10 as in the second Example it is 9 Chains 5 Links you must place a Cypher before the five Links as there you see and then multiplying that number viz. 9,05 by 66 the product will be 59730 from which taking the two last figures there will remain 597 Feet and â—… 100 parts of a Foot The like may be done for any other number of Chains and Links whatsoever According to these Examples is made the Table following which sheweth how many Feet are contained in any number of Chains and Links from 5 Links to 10 Chains for every fift Link which is sufficient for ordinary use by which Table you may see that in 6 Chains 40 Links is contained 422 Feet and 40 100 of a Foot Also in 5 Chains 55 Links is contained 366 Feet and 30 100 parts of a Foot and so of any other A TABLE shewing how many Feet and parts of a Foot are contained in any number of Chains and Links between five Links and eight Chains  0 1 2 3 4 5 6 7 0  66,00 132,00 198,00 264,00 330,00 396,00 462,00 5 3,30 69,30 135,30 201,30 267,30 333,30 399,30 465,30 10 6,60 72,60 138,60 204,60 270,60 336,60 402,60
severall parcels in this kinde you will have use only of your Chain then when you begin your work you must first write the name of the field and in the first columne of your Booke or paper you must write the Tenants name and the tenour by which he holds the same from what place you begin to measure and upon what point of the compasse you passe from thence and observing this direction in all the rest you may if need require bound every parcell This being noted in your Book observe the species or shape of the Furlong whether it be all of one length or not if of one length then you need take the length thereof but once for all but if it be irregular that is in some places shorter and in others longer then you must take the length thereof at every second or third breadth and expresse the same in your Book under the title of length As for the expressing of the severall breadths you need but to crosse over the whole Furlong taking every mans breadth by the middle thereof and entering the same as you passe along but in case there be a considerable difference at either end then I would advise you to take the breadth at either end and adde them together into one sum then take the half of that summe for your mean or true breadth and enter it in your book or paper under the title of breadth In this manner you may proceed from one Furlong to another till you have gone through the whole field which when you have done and noted down the severall lengths and breadths in your book you may multiply the length and breadth of every parcell together as is taught before and so shall you have the quantity of every parcell by it selfe which quantity must be noted downe in the three last columns of your Book as in the following example appears Mordon Field The Tenants names and tenour Length Breadth Content C. L. C. L. A. R. P. Abel Johnson from the pond S. E. free 32 76 3 45 11 1 12 Nicholas Somes for three lives 30 12 2 63 7 3 30 Robert Dorton for Life 28 60 8 12 23 0 36 James Norden at Will 25 11 12 35 31 0 2 CHAP. XLXI. To finde the horizontall line of any hill or mountain THis proposition differeth nothing from those formerly taught in the taking of Altitudes Wherefore suppose you should meet with a hill or mountain as ABD the thing required is to finde the length of the line BD on which the mountain standeth First place your Instrument at the very foot of the Hill exactly levell then let one go to the top of the hill at A and there place a mark which must be so much above the top of the hill as the top of the Instrument is from the ground then move the Label up and down till through the sights thereof you see the top of the mark at A and note the degrees cut by the Label on the Tangent line for that is the quantity of the angle ABC which suppose 47 degrees then by consequence the angle BAC must be 43 degrees the complement of the former to 90 degrees then measure the side of the hill AB which suppose to contain 71 Feet then in the Triangle ABC there is given the side AB 71 foot and the angle BAC 43 degrees together with the right angle ACB 90 degrees and you are to finde the side BC which to perform say As the Sine of the angle ACB 90 degrees Is to the side AB 71 feet So is the Sine of the angle BAC 43 degrees To the side BC 48½ feet Then because the hill descends on the other side you must place your Instrument at D observing the angle ADC to contain 41 degrees and the angle DAC 49 degrees and the side AD 80 feet now to finde the side CD the proportion will be As the Sine of the angle ACD 90 degrees Is to the side AD 80 feet So is the Sine of the angle CAD 49 degrees To the side CD 60½ feet Which added to the line BC giveth 109 feet which you may reduce into Chains by dividing it by 66 and this line must be protracted instead of the hypothenusall lines AB and AD. Another way There is another way also used by some for the measuring of horizontall lines which is without the taking of the Hils altitude or using of any Arithmeticall proportion but by measuring with the Chain only the manner whereof is thus Suppose ABC were a hill or mountain and that it were required to finde the length of the Horizontall line thereof AC At the foot of the hill or mountain as at A let one hold the Chain up then let another take the end thereof and carry it up the hil holding it levell so shall the Chain meet with the hill at D the length AD being 60 Links then at D let the Chain be held up again and let another carry it along levell till it meet with the side of the hill at E the length being 54 Links then again let one stand at E and hold up the Chain another going before to the top of the hill at B the length being 48 Links these three numbers being added together make 162 Links or 1 Chain 62 Links which is the length of the horizontall line AC This way of measuring is by some practised but the other in my opinion is far to be preferred before it only when you are destitute of better helps you may make use hereof ¶ But if the hill or mountain should have a descent back again on the other side you must then use the same way of working as before and adde all together for the horizontall line CHAP. XLVII How to plot Mountanous and uneven grounds with the best way to finde the content thereof FOr the plotting of any mountanous or uneven piece of ground as ABCDEFG you must first place your Instrument at A and direct the sights to B measuring the line AB then in regard that from B to C there is an ascent or hill you must finde the horizontall line thereof and draw that upon your Table accounting thereon the length of the hypothenusall line then measure round the field according to former directions and having the figure thereof upon your Table reduce it into Trapezias as into the Trapezias ABEG BCDE and the Triangle GEF then from the angles A C E and F let fall the perpendiculars AK CH EI and FM Now in regard there are many hils and valleys all over the field you must measure with your Chain in the field over hill and dale from B to D and to the line BD set the number of Chains and links as you finde them by measuring which will be much longer then the streight line BD measured on your Scale then by help of your Instrument finde the point H in the line BD and measure with your Chain from C to H over hill
from the foot of the farthermost Sight all along the Ruler to the foot of the nethermost Sight and up the side thereof and is numbred from 1 to 90 by 10 20 30 40 50 c. ending at the foot of the furthermost Sight from whence the line proceeded The use of this line of Tangents in taking of Heights is shewed in the fourth Book is used with the Tables of Sines and Logarithms treated of in the third Book without which Tables or something equivalent thereunto this line of Tangents will be of little use therefore it will be convenient to have upon the Index of your Table the lines of Artificiall Numbers Sines and Tangents by which you may work any proportion required very speedily and exactly so that if you be destitute of your Tables these Lines will sufficiently help you There is yet another way by which you may take any altitude or reduce Hypothenusall to Horizontall lines only by Vulgar Arithmetick without the help of Tables by having a line of equall parts divided on the edge of the Index and another line of the same equall parts on the Label by which lines and Vulgar Arithmetick an Altitude may very well be taken Now because I intend only to shew in generall the use of these equall parts I will therefore do it in this place because I shall have occasion to speak no more thereof hereafter The use thereof briefely is thus Now for the reducing of Hypothenusall to Horizontall lines having measured the Hypothenusall line with your Chain the proportion will be As the equall parts cut on the Label Are to the equall parts cut on the Index So is the length of the Hypothenusall line measured To the length of the Horizontall line required I thought good to give the Reader a view of the severall wayes there are to perform these conclusions leaving every man at liberty to use that which he best liketh or all if he please for all the lines may very well be put upon one Instrument without any confusion of lines but the way which I shall chiefly insist upon in the prosecuting of this Work shall be by the line of Tangents as being in my opinion the best of all Now when I come to shew you the use of this line of Tangents with the Tables of Sines and Logarithms in the resolving of Triangles I will also shew you how to perform the same Propositions by the lines of Artificiall Numbers Sines and Tangents and therefore I would advise every man to have these so necessary lines upon his Index Fourthly Unto this Instrument also belongeth a Box and Needle which is to be fastned to the side of the Table by help of two screws so that it may be taken off and put on at pleasure In the bottome of this Box must be placed a Card divided into 360 degrees numbered if you please after the usuall manner from the North Eastward but the Card by which all the Examples in this Book were framed was numbered from the North Westward by 10 20 30 c. to 360 contrary to the common custome There belongeth also to this Instrument a Socket of Brasse to be screwed on the back side of the Table into which must be put the head of the three legg'd Staffe this Staffe ought to be joynted in the middle so that it may be the more portable For the Socket it may be a plain one but a Ball and Socket with an endlesse screw is the best of all for by help thereof you may place the Table or any other Instrument either Horizontall Verticall or in any other position ¶ Note that this Instrument if made according to these directions is the most absolute Instrument for a Surveyor to use CHAP. V. Of Chains the severall sorts thereof OF Chains there are divers sorts as namely Foot Chains each link containing a Foot or 12 Inches and so the whole Pole or Perch will contain 16½ Links or Feet answering to the Statute denomination Some Chains have each Pole divided into 10 equall parts and these are called Decimall Chains and this grosse division may be convenient in some practises The Chains now used and most esteemed amongst Surveyors are especially two namely that generally used by Master Rathborne which hath every Perch divided into 100 Links and that of Master Gunter which hath four Poles divided into 100 Links so that each Link of Master Gunters Chain is as long as four of Master Rathborns Now because these Chains are most esteemed of and used by Surveyors I will therefore make a generall description of them both leaving every man at liberty to take his choise Of Mr. RATHBORNS Chain THe Chain which Master Rathborne ordinarily used as himselfe saith contained in length two Statute Poles or Perches each Pole containing in length 16½ feet which is 198 Inches then each Pole was divided into 10 equall parts called Primes every of which contained in length 19◠Inches again every of those Primes was sub-divided into 10 other equall parts called Seconds so that every of these Seconds contained in length 1 49 50 Inch so that the whole Pole Perch Unite or Commencement as he calleth it was divided into 130 equall parts or Links called Seconds The Chain or one Pole thereof being thus divided at the end of every 50 Links or halfe Pole let a large Curtain ring be fastned so shall you have in a whole Chain of two Perches long three of these Rings the middlemost being the division of the two Poles Then at the end of every Prime that is at the end of every ten Links let a smaller Curtain Ring be fastened By this distinction of Rings the Chain is divided into these three denominations Unites Primes and Seconds whose Characters are these ◯ · · so that if you would expresse 40 Unites 8 Primes and 7 Seconds they are thus to be written 408̇7̇ by which you may perceive that those Figures which have no pricks over them are Unites or Intigers and the figure under the first point Primes and under the next Seconds so also three Unites seven Primes and two Seconds will stand thus 37̇2̇ Besides these divisions Master Rathborn for his own use sewed at the end of every two Primes and a halfe which is a quarter of a Pole a small red cloth and at every seven Primes and a halfe being three quarters of a Pole the like of yellow or other discernable colour which much helped him in the ready reckoning of the several Rings upon the Chain remembring this Rule That if it be the next Ring short of the Red it is two Primes if the next over three if the next short of the yellow seven Primes if the next over eight if the next short of the great halfe Ring it is four the next over six and if the next short of the middle great Ring it is nine and if the next over one ¶ But here is to be noted that if you use this distinction by
side QS 303 and the side RQ 176 and the angle comprehended by them namely the angle RQS 110 degrees 30 minutes and it were required to finde either of the other angles First Take the summe and difference of the two given sides their summe is 479 and their difference is 127. Then knowing that the three angles of all right lined Triangles are equall to two right angles or 180 degrees by the 17. Theor. of Chap. 3. therefore the angle RQS being 110 degrees 30 minutes if you substract this angle from 180 degrees the remainder will be 69 deg 30 min. which is the summe of the two unknown angles at R and S the halfe whereof is 34 deg 45 min. The side QS 303 The side QR 176 The summe of the sides 479 The difference of the sides 127 The halfe sum of the two unknown angles 34 deg 45 min. The summe and difference of the sides being thus found and also the halfe summe of the two unknown angles the proportion by which you must finde the angles severally is As the Logarithm of the summe of the sides 479 2,680335 Is to the Logarithm of the difference of the sides 127 2,103804 So is the Tangent of the halfe summe of the two unknown angles 34 degrees 45 minutes 9,841187 the summe of the second and third numbers 11,944991 the first number substracted 2,680335 To the Tangent of 10 degrees 25 minutes 9,264656 These 10 degrees 25 minutes being added to the halfe summe of the two unknown angles namely to 34 degrees 45 minutes the summe will be 45 degrees 10 minutes the quantity of the angle QRS which is the greater angle of the two Also these 10 degrees 25 minutes being substracted from the same halfe sum there remaineth 24 degrees 20 minutes for the angle QSR which is the lesser of the unknown angles and thus are either of the enquired angles easily found By the lines of Tangents and Numbers Extend the Compasses from the summe of the sides 479 to the difference of the sides 127 the same extent upon the line of Tangents will reach from the Tangent of 34 degrees 45 minutes which is the halfe summe of the two unknown angles to the Tangent of 10 degrees 25 minutes and these 10 degrees 25 minutes added to and substracted from the halfe summe as before is shewed will give the quantity of either of the two unknown angles CASE X. The three sides of a right lined plain Triangle being given how to finde the Area or the superficiall content thereof EXAMPLE Let the Triangle given be ABC the sides thereof being 20 13 11 how much is the superficiall content thereof The summe of the sides is 44 the halfe summe is 22 the differences betwixt each side and that halfe are 2 9 11 which numbers rank in this order following The halfe summe 22 1,342423 The differences 2 0,301030 9 0,954243 11 1,041393 The summe of the Logarithms 3,639089 The Area or Content required 66. 1,819544 And this Area or superficiall Content thus found is alwayes of the same nature with the sides of the Triangle that is to say if the sides of the Triangle be given in feet then is the content found in feet also if the sides be Perches you shall have the content in perches and so of any other measure whatsoever I might add hereunto divers other Cases but in this place at present let these suffice The end of the Third Book THE COMPLEAT SURVEYOR The Fourth Book THE ARGVMENT OVr businesse hitherto hath been to provide necessary Instruments and to learn such things which of necessity ought to be known before we enter the Fields to Survey Being thus provided we come now to apply them severall wayes First in taking of Heights and Distances whether accessible or in-accessible and then in Surveying of Land In this Book every kinde of work is performed three severall wayes by three severall Instruments viz. the Plain Table the Theodolite and Circumferentor by which the congruity and harmony of the severall Instruments may be easily discerned and the truth of every Example may the better appear Here is also divers wayes of Surveying by one and the same Instrument that is to take the Plot of a Field severall wayes and to measure all kinde of Grounds whatsoever whether Woodland or other Here is also shewn how to take the Plot of a whole Mannor and to keepe your account in your Field-Book after the best and most easiest manner with divers Rules Cautions and Directions throughout the whole Book inserted THE APPLICATION AND VSE of the severall Instruments before described in the practise of SVRVEYING CHAP. I. Of the use of the Scale HAving before described the severall Instruments belonging to Surveying I will now shew the use of them and first of the Scale The Scale is principally intended for the laying out of lines for which purpose the severall Scales of equal parts are there divided some of greater and some of lesser quantities the uses of all the lines being the same for each line is divided into 11 equall parts representing 11 Chains and these grand divisions are numbered with Arithmeticall Figures by 1 2 3 c. to 10 then the uppermost large division is again divided into ten other smaller parts each part containing 10 links of your Chain each of which smaller parts you may suppose to be again divided into ten other lesser parts representing single Links of your Chain 1. Any length being measured by your Chain how to lay down the same distance upon paper Suppose that measuring along a hedge with your Chain you finde the length thereof to contain 5 Chains 60 Links Now to take this distance from your Scale and lay it down upon paper do thus First Draw a line as AB then place one foot of your Compasses upon your Scale at the figure 5 for your five Chains and extend the other foot to six of the small divisions which represents the 60 Links then set this distance upon the line drawn from A to B so shall the line AB contain 5 Chains 60 Links if you take the distance from the Scale of 10 in an Inch. But if you would have your line shorter and yet to contain 5 Chains 60 Links then take your distance from a smaller Scale as of 12 16 20 or 24 in an Inch so shall the 5 Chains 60 Links end at C if taken from the Scale of 12 in an inch or at D by the Scale of 16 or at E by the Scale of 24 either of which lines will contain 5 Chains 60 Links and be in proportion one to the other as the Scales from whence they were taken And in this manner may any number of Chains and Links be taken from any of the Scales 2. A right line being given to finde how many Chains and Links are therein contained according to any Scale assigned Suppose AB were a line given and it were required to finde how many Chains and Links are contained therein
on the frame of the Table which supplies the use thereof Thirdly When I mention or make use of the Circumferentor I mean the Index with the Box and Needle screwed to the Staffe ¶ Having thus given you a sufficient description of the severall Instruments and their parts I come now to the use of them shewing how any angle in the field may be measured by any of them And 1. How to observe an angle in the Field by the Plain Table Suppose EK and KG to be two hedges or two sides of a field including the angle EKG and that it were required to draw upon your Table an angle equall thereunto First place your Instrument as neer the angular point K as conveniencie will permit turning it about till the North end of the Needle hang directly over the Flower-de-luce in the Box and then screw the Table fast Then upon your Table with your protracting pin or Compasse point assigne any point at pleasure upon the Table and to that point apply the edge of the Index turning the Index about upon that point till through the sights thereof you espie a mark set up at E or parallel to the line EK and then with your protracting pin or Compasse point or Black-lead draw a line by the side of the Index to the assigned point upon the Table Then the Table remaining immoveable turn the Index about upon the same point and direct the sights to a mark set up at G or parallel thereto that is so far distant from G as your Instrument is placed from K and then by the side of the Index draw another line to the assigned point so shall you have drawn upon your Table two lines which shall represent the two hedges EK and KG and those lines shall include an angle equall to the angle EKG and although you know not the quantity of this angle yet you may by the 1 or 2 Chapters of this Book finde the quantity thereof if there were any need for in working by this Instrument it is sufficient only to give the symetry or proportion of angles and not their quantities as in working by the Theodolite or Circumferentor it is Also in working by the Plain Table there needeth no protraction at all for you shall have upon your Table the true figure of any angle or angles which you observe in the field in their true positions without any farther trouble 2. How to finde the quantity of an angle in the field by the Theodolite Let it be required to finde the quantity of the angle EKG by the Theodolite place your Instrument at K laying the Index on the diameter thereof then turn the whole Instrument about the Index still resting on the Diameter till through the sights you espie the mark at E then screwing the Instrument fast there turn the Index about upon the center till through the sights you espie the mark at G then note what degrees on the frame of the Table are cut by the Index which you will finde to be 114 degrees and that is the quantity of the angle EKG 3. How to finde the quantity of any angle in the field by the Circumferentor If it were required to finde the quantity of the former angle EKG by the Circumferentor First place your Instrument as before at K with the Flower-de-luce in the Card towards you then direct your sights to E and observe what degrees in the Card are cut by the South end of the Needle which let be 296 then turning the Instrument about the staffe the Flower-de-luce alwayes towards you direct the sights to G noting then also what degrees are cut by the South end of the Needle which suppose 182 this done alwayes substract the lesser number of degrees out of the greater as in this Example 182 from 296 and the remainder is 114 degrees which is the true quantity of the angle EKG Again the Instrument standing at K and the sights being directed to E as before suppose that the South end of the Needle had cut 79 degrees and then directing the sights to G the same end of the needle had cut 325 degrees now if from 325 you substract 79 the remainder is 246 but because this remainder 246 is greater then 180 you must therefore substract 246 the remainder from 360 and there will remain 114 the true quantity of the inquired angle and thus you must alwayes do when the remainder exceedeth 180 degrees ¶ This adding and substracting for the finding of angles may seeme tedious to some but here the Reader is desired to take notice that for quick dispatch the Circumferentor is as good an Instrument as the best for in going round a field or in surveying of a whole Mannor you are not to take notice of the quantity of any angle but only to observe what degrees the needle cutteth which in those cases is sufficient as will appear hereafter but in taking of distances by the Circumferentor it is altogether necessary as may appear by the 7 Chap. following and for that reason I have here shewed how to finde an angle by the Circumferentor and also that you might thereby perceive what congruity and harmony there is in all the three Instruments 4. How to set the Index and Labell Horizontall upon the Staffe When you have screwed the Index and sights to the Staffe as a Circumferentor before you put the Labell upon the brasse pin or wier you must hang a line and plummet upon that pin and then put on the Label then move the Index up and down till the thred and plummet hang directly upon a line which is gaged from under the pin all along the Sight and then doth the Instrument stand horizontall or levell which it must alwayes do when you take an altitude therewith 5. How to observe an angle of Altitude The Label which is to be hanged on one of the sights of the Circumferentor as was intimated in the description thereof and the Tangent line on the edge of the Index is only for the finding of angles of Altitude and is therefore only usefull in taking of heights and in surveying of mountanous and uneven grounds The manner how to observe an angle of Altitude by this Label and the Tangent line on the Index is thus Suppose CA to be a Tree Tower or Hill whose height were required Your Instrument being placed at B exactly levell direct the sights thereof towards CA and there fix it hanging the Labell on the farthermost fight upon a pin for that purpose then move the Labell too and fro along the side of the Index till through the sight at the end of the Label and by the Pin on which the Label hangeth you espie the very top of the object to be measured at C then note what degree of the Tangent line is cut by the Labell which suppose 30 and that is the quantity of the angle of Altitude it being equall to the angle CBA Thus by the Rules in this Chapter
having at each end thereof a Semicircle is inferiour to none but the Instrument being very chargeable and the use thereof very intricate and tedious I shall wholly omit to speak any more of it There is another way also which Master Rathborn used which was with a Ruler by him invented for that purpose which would indifferent well reduce a plot from one bignesse to another according to some particular proportions The making of this Ruler is so well known and the use thereof so apparent that I shall not need to say any thing concerning the description or use of it I only intimate that there is such a Ruler that those which please may have it made Another way is by one line divided into 100 or 1000 equall parts only which by the help of Arithmetick will perform this work very well but this as being very tedious I neglect To passe by these and divers others which I could name I shall say somthing of the Parallelogram which for generality exactnesse and dispatch surpasseth all the rest unto which in my opinion there is none comparable Of Parallelograms there are diverse sorts but that which I shall instance in consisteth but of four Rulers only the making whereof is well known to the Instrument maker and the manner of using it is as followeth Take the plot which you would reduce and fasten it to a Table with Mouth-glew then by it upon the same Table fasten your fair paper or parchment upon which you would have your new plot then having fitted your Parallelogram according to the proportion into which you would have your plot reduced fix the Parallelogram to the Table by a point for that purpose then put your drawing pen into some one hole on one of the sides of the Parallelogram and upon it a plummet of lead or brasse to keepe the pen down close to the paper when it is moved thereupon and here note that at any time when the Parallelogram is thus fitted the point that sticketh in the Table the Pen which is to draw and the Tracer which you must move along the lines of your old plot will lie alwayes in a right line but this by the way Your Parallelogram being fixed to the Table and the pen in its true place fitted to draw take the Tracer in your right hand and with it lightly go over all the lines of your old plot so shall the motion thereof occasion the pen to draw upon your clean paper or parchment the true and exact figure of your former Plot though of another bignesse which will be in proportion to the greater according to the situation of the sides of the Parallelogram which will better appeat by the sight of the Instrument then words can possibly explain it CHAP LII How to draw a perfect draught of a whole Mannor and to furnish it with all necessary varieties also to trick and beautifie the same in which as in a Map the Lord of the Mannor may at any time by inspection only see the symetry scituation and content of any parcell of his Land HAving protracted your plot according to your intended bignesse and written the content of each Close about the middle thereof you may about the bounds of every field or Inclosure with a small Pensill and some transparent green colour neatly go over your black lines so shall you have a transparent stroke of green on either side of your black line which will adde a great lustre and beauty to your Plot. Then in your Wood-land grounds draw diverse little Trees in the most materiall places and shadow your mountanous and uneven grounds with hils and valleys expressing all kinde of Bogs Groves High-wayes Rivers c. distinguishing them by lively colours according to their similitudes Then in some convenient place of the Plot without the Inclosures draw a Circle and therein describe the 32 points of the Mariners Compasse according to the situation of the grounds with a Flower-de-luce at the North part thereof Then in some other convenient place of your plot make a Scale equall to that by which your plot was protracted Lastly in some other convenient place towards the upper part thereof draw the Coat of Arms belonging to the Lord of the Mannor with Mantle Helme Crest and Supporters or in a Compartment but be sure you blazon the Coat in its true Colours THE Mannor of Lee. These things being well performed your plot will be a neat Ornament for the Lord of the Mannor to hang in his Study or other private place so that at pleasure he may see his Land before him and the quantity of all or every parcell thereof without any further trouble Also in your plot must be expressed the Mannor-house according to its symetry or situation with all other houses of note also all Water-mils Wind-mils and whatsoever else is necessary that may be put into your Plot without confusion For farther explanation of what hath been delivered in this Chapter I have here added the figure of a small Mannor which will be sufficient for example sake CHAP. LIII How to finde whether water way be conveyed from a Spring head to any appointed place THere is an Instrument called a Water-Levell for the performance hereof the making whereof is sufficiently known Now if it were required to know whether water may be conveyed in Pipes or Trenches from a Spring head to any determinate place observe the following directions Place your Water-Levell at some convenient distance from the Spring head in a right line towards the place to which the water is to be conveyed as at 30 40 60 or 100 yards distant from the Spring-head Then having in a readinesse two long streight poles which you may call your station staves divided into Feet Inches and parts of Inches from the bottome upwards being thus provided cause one whom you may call your first assistant to set up one of the said staves at the Spring head and require another which you may call your second assistant to erect the other staffe beyond your Instrument at 30 40 60 or 100 yards forward towards the place to which the water should be conveyed These station staves being erected perpendicular and your Water-Levell in the mid way precisely horizontall go to the end of the Levell and looking through the sights cause your first assistant to move a leafe of paper up and down your station staffe till through the sights you see the very edge thereof and then by some known signe or sound intimate to him that the paper is then in its true position then let this first assistant note against what number of Feet Inches and parts of an Inch the edge of the paper resteth which he must note down in a paper Then your Water-Levell remaining immoveable go to the other end thereof and looking through the sights towards your other station staffe cause your second assistant to move a leafe of paper along the staffe till you see the very edge thereof
delivered may the true quantity of any angle be easily taken and this is the most convenient use to be first placed I will now shew how by your severall Instruments yo may take all manner of heights and distances whether accessible or inaccessible severell wayes with divers other necessary conclusions incident thereunto CHAP. V. How to take an inaccessible Distance at two stations by the three forementioned Instruments and first by the Plain Table YOu are taught in the last Chapter how to make observation of any angle in the field by the severall Instruments before mentioned as the Plain Table Theodolite and Circumferentor and also an angle of Altitude by the Index and the Labell thereunto annexed I conceive it now convenient to shew how all manner of heights and distances may be readily and exactly measured severall wayes whether they be accessible or inaccessible and first of distances ¶ You may remember that I formerly intimated that the measuring of a Height or Distance is only to resolve a Triangle so that when you make any observation either of Height or Distance the observation of angles which you make are the angles of some Triangle and the lines which you measure on the ground are the sides of the same Triangle and these are the given parts of the Triangle The manner how to take a distance by the Plain Table is thus Suppose you were standing in a field at R and that at S were some eminent mark as a Tree Church House or such like and that it were required to finde the distance between R and S. First place your Table at R and thereon assigne any point at pleasure unto which point apply the edge of your Index turning it about upon that point till through the sights you espie the mark at S and draw a line by the side of â—he Index as RS. Then in some other convenient place of the field as at Q let a staffe or other mark be erected and the Table remaining as before turn the Index about till through the sights you espie the mark at Q drawing a line by the side thereof as RQ so have you described upon your Table an angle equall to the angle QRS Then with your Chain measure the distance QR which let be 176 foot then take with your Compasses 176 out of any Scale and set it upon your Table from R to Q so shall this point Q upon your Table represent the mark at Q in the field This done set up a staffe a R and remove your Table to Q laying the Index upon the line QR and holding it fast there turn the whole Table about till through the sights you espie the mark set up at your former place of standing at R then screw the Table fast and lay the Index on the point Q turning it about till through the sights you espie your mark at S then draw a line by the side of the Index which will cut the line RS first drawn in the point S. By this means shall you have upon your Table a Triangle equall to the Triangle QRS the correspondent sides and angles thereof being proportionally equall with those in the field therefore if with your Compasses you take the length of the side RS and apply that distance to the same Scale from whence you tooke the side QR you shall finde it to contain 400 foot and that is the distance between R and S. Likewise if you take with your Compasses the length of the line QS and apply it to the same Scale you shall finde it to contain almost 303 and so many foot is the distance QS ¶ In this manner may the distance between any two places be measured although they be so scituated that by reason of water or other impediments you cannot approach neere unto them And here note rhat when you take your second station that you take it as large as the ground will permit so shall your work be so much the truer by now much the distance taken is the larger CHAP. VI. How to take an inaccessible distance at two stations by the Theodolite IN the former Diagram let R and Q be two stations from either of which it is required to finde the distance to S. First place your Instrument at R laying the Index and sights upon the Diameter thereof turning the whole Instrument about till through the sights you espie your second station at Q and there screw it fast then turn the Index about upon the center till through the sights you espie the mark at S noting the degrees cut by the Index which suppose 45 degrees 10 minutes Then remove your Instrument to Q laying the Index on the Diameter thereof and holding it there turn the whole Instrument about till through the sights you espie your mark at S and fixing the Instrument there turn the Index about til through the sights you see the mark set up at your former station at R noting the degrees there cut which let be 110 degrees 30 minutes This done measure the distance of your two stations Q R which let be 176 feet 10 in the Oblique angled Triangle QSR you have given 1 the angle SRQ 45 degrees 10 minutes the angle observed at your first station 2 the angle RQS 110 degrees 30 minutes which was the angle observed at your second station And 3 you have given the side RQ 176 foot which is the distance of your two stations and you are to finde the two other sides RS and QS which you may finde by the 7 Case of the 4 Chapter of the 3 Book in this manner for Having the two angles QRS and RQS given you have also the third angle RSQ given 24 degrees 20 minutes it being the complement of the other two to 180 degrees by the 17 of Chap. 3 Lib. 3. Then to finde the other two sides the proportion is I. For the side QS As the sine of the angle RSQ 24 degrees 20 minutes Is to the Logarithm of the side RQ 176 foot So is the sine of the angle QRS 45 degrees 10 minutes To the Logarithm of the side QS 303 foot ferè II. For the side RS. As the sine of the angle QRS â—… degrees 10 minutes Is to the Logarithm of the side QS 303 foot So is the sine of the angle RQS 110 deg 30 min. or 69 d. 30 m. To the Logarithm of the side RS 400 foot Which is the distance required ¶ I have been larger upon this particular then I intended having sufficiently insisted thereon before in the dimension of plain Triangles but that the Reader may fully understand these necessary conclusions I have in this example used all the perspicuity I could imagine so that in the subsequent Chapters I may be the briefer for this being well understood he may easily apprehend any of the other at the first view CHAP. VII How to take an in-accessible distance at two stations by the Circumferentor LEt it be required to finde the distance from
R and Q to S. First place your Instrument at R and direct the sights to S observing what degrees the South end of the Needle cutteth which let be 315 degrees 30 min. then turning the Instrument about direct the sights to Q observing what degrees the needle there cutteth which let be 270 degrees 20 minutes therefore from 315 degrees 30 minutes substract 270 degrees 20 minutes and there will remain 45 degrees 10 minutes which is the quantity of the angle SRQ Then remove the Instrument to Q and direct the sights to R the Needle cutting 91 degrees 00 minutes also direct the sights to S the needle cutting 340 degrees 30 minutes now if you substract 91 degrees 00 minutes from 340 degrees 30 minutes the remainder is 249 degrees 30 minutes which because it exceedeth 180 degrees substract from 360 degrees and there remains 110 degrees 30 min. the true quantity of the angle RQS Having thus obtained the two angles RQS and SRQ you must measure the stationary distance QR 176 foot so have you given in the Triangle QRS 1 the angle RQS 110 degrees 30 minutes 2 the angle QRS 45 degrees 10 minutes 3 the angle QSR 20 degrees 10 minutes the complement of the other two to 180 degrees and 4 the stationary distance QR 176 foot whereby you may finde the other sides QS and RS according to the doctrine delivered in the foregoing Chapter  dg min. First station at R degrees cut 315 30 270 20 The quantity of the angle QRS 45 10 Second station at Q degrees cut 340 30 91 00  249 30  360 00 The quantity of the angle RQS 110 30 The stationarie distance 176 foot Having these things given if you resolve the Triangle QRS you shall finde the side RS to contain 400 foot and the side QS 303 foot ferè as in the last Chapter CHAP. VIII How to protract or lay down a Distance taken according to the directions of the two last Chapters upon paper by help of your Protractor or line of Chords WHen you make any observations in the field by the Theodolite or Circumferentor you are to note down the quantities of the severall lines and angles observed in the field in a Book or paper so that they may be ready at hand when you come to protraction and this is the usuall way Suppose it were required to draw upon paper or pastboard the true symetry or proportion of the distance taken in the last Chapter CHAP. IX How to take the altitude of any Tower Tree Steeple or the like being accessible by the Labell and Tangent line HAving in the 5 Section of the 4 Chapter of this Book shewn how to observe an angle of Altitude by the Labell and Tangent line we now come to the further use thereof Suppose therefore that the line CA were a Tree Tower Steeple or other thing whose height were required This proportion being wrought according to the former directions the side CA will be found to contain almost 231 foot and that is the height of CA required CHAP. X. How to protract or lay down upon paper the observation made in the last Chapter HAving drawn a line upon your paper as BA place the center of the Protractor upon B now because when you made your observation at B the degrees cut were 30 turn the Protractor about till the line BA lie just under 30 degrees then with your protrocting pin make a mark by the edge of your Protractor against 00 degrees and draw the line BC so shall the angle CBA contain 30 degr Then because the measured distance BA was 400 foot take 400 from any of your Scales of equall parts and set that distance from B to A and from the point A erect the perpendicular AC which perpendicular being taken in your Compasses and measured upon the same Scale from whence the 400 foot was taken you shall finde it to contain almost 231 foot and so much is the altitude CA as before CHAP. XI How to take an in-accessible Altitude by the Labell and Tangent line Then must you make a second work in the Triangle BCD in which you have given 1. The angle BDC 64 degrees 2. The angle DBC 26 degrees 3. The side DB 633 foot And you are to finde the side BC the altitude required wherefore say again As the sine of the angle BCD 90 degrees Is to the Logarithm of the side DB 633 foot So is the sine of the angle BDC 64 degrees To the Logarithm of the Altitude BC Which according to the former Doctrine will be found to be 569 foot CHAP. XII How to Protract the observation taken in the last Chapter WHen you have made your observation as in the last Chapter and noted down in a Book or otherwise that the degrees cut at your first station at A were 50 and the degrees cut at the second station at D were 64 and that your stationarie distance AD was 200 foot you may immediately finde the Altitude BC by protraction thus First draw a line as AC in which line let A represent your first station whereon lay the center of your Protractor and make the angle BAC to contein 50 degrees as hath been severall times before shewn and draw the line AB Then upon the line AC set off the distance of your two stations 200 foot from A to D then bring your Protractor to D which represents your second station and placing the center of your Protractor thereupon set off an angle of 64 degrees as BDC and draw the line DB then where these two lines AB and DB intersect or meet which is in the point B from that point let fall the perpendicular BC the length whereof being measured upon the same Scale from whence you tooke the distance AD will give you 569 foot and that is the altitude of AB which was required CHAP. XIII How to take the distance of divers places one from another according to their true scituation in plano and to make as it were a Map thereof by the Plain Table THis Proposition is of good use to describe in plano the most eminent places in a Town or City and to make as it were a Map thereof Let A B C D E F G be certain eminent places scituate in some Town or City and let it be required to describe all those places upon paper by which the distance of any of them one from another may be readily found At some convenient distance from the City Town or Field make choice of two other convenient places as K and L from either of which you may plainly discern all the marks which you intend to describe in your Map Then at one of these places as at K place your Table and neere one of the sides thereof draw a line parallel to the edge of the Table In this line assigne any point as K for your first station and laying the Index upon this line turn the Table about till through