A MORE EXACT WAY TO Delineate the Plot of any spacious parcel of Land, as Baronies, Parishes, and Town-Lands, as alsoof Rivers, Harbours and Loughs, etc. than is as yet in practice.  
ALSO A METHOD OR FORM OF Keeping the Field-Book, AND How to cast up the superficial Content of a PLOT most Exactly.  
DUBLIN, Printed by WILLIAM BLADEN, Anno Dom. 1654.   

To the Surveyors of Lands.  
Gentlemen,  
I Think myself obliged to render an account to you, what hath occasioned the Publishing of this short tract at this time, That I may not be thought either to affect an idle glory to myself by putting a thing in Print, or to be desirous if imposing any thing herein upon other Men.  
Truly I must confess, that after several Experiments made for my own private satisfaction in the business of Survey, I finding the ordinary way of keeping a Book, and of Protraction, seldom to answer that exactness I expected, it made me to consider how those defects might (at least in some measure) be supplied; which brought into mind something I had read of that kind, (in Mr. Norwoods' Seaman's Practice) which after a more serious review and proof of, perceiving it to answer to my end, I form some short Rules out of it for my own particular use.  
Which casually communicating to the Surveyer General, upon a desire expressed by him: That the field Books might in the future Surveys, be so kept and so returned that it might be possible out of them to find how near each parcel of Land (surrounded by the Instrument) did close upon drawing up of the Plot of it; Or in case any loss or disaster should come to any of the Plots, that new ones might be framed, by having the fair and true Copies of the said field Books, respectively belonging to them.  
He, upon further Examination of it, and advice had with some other persons concerning it, seeming also much to approve of this before the ordinary way of Protraction, and commanding me to take the pains to set the whole method of it down; to the end, That it might the better be examined, and that if no inconvenience could be demonstrated in it, It might be the more generally embraced. I, in order to the said commands, and that I might not decline any service enjoined me by the public, have endeavoured to lay down the practical part, with all the plainness I possibly could.  
Wherefore premising only this caution to such as shall go out into the Field, that great care be had their Needle do play freely, and that it seat itself always in the same position, in regard that upon the goodness of the Needle depends (upon the matter) the whole of Survey. The commodities humbly conceived to be in the way here propounded above the usual, are principally these following:  
First, that (after the summing up of the Columns of the North, South, East, and West) any one may know at the first sight how much the Plot wants of closing even to a Link; and this before protraction is made, which cannot be done the usual way, until the whole be first actually laid down; besides that, in the usual way after a man hath never so carefully plotted and laid down his work, if it close not, it leaves him still doubtful whether the Error be in the Field, or in the Plot, or in both.  
Secondly, It is a most absolute way for laying down any Plot, for by the usual way of Protracting, an error once committed is carried on through the whole plot: (if not augmented) whereas in this, if an Error should be committed in one line, in the next line shall fall in his true place, and consequently that Error goes no farther.  
Thirdly, that for the casting up the Area of any Figure, it will do it most tightly, (and that without plotting if you will) Nay, a Plot cast up the usual way although a thousand times as large, yet shall not the Content be cast up so exactly as it may this way.  
Lastly, To find that there is an Error in the Field work, and yet not to know how readily to find it, will something perplex: yet you have in this way some advantage more than in the usual: for you shall see whether the Error be on the North and South sides of the Plot, or on the East and West, or both; and by this means shall be able to know where you should run a line cross your Work for the discovery of that Error.  
Whatsoever is here done with the Sine, and Sine Compliment of Arches, in the plotting of Land etc. the same may be performed with the same Certainty by Tangents, and Secants of Arches; but that being besides this business I shall say no more thereof, but remain ready to serve the Lovers of Mathematical Learning.  
DUBLIN April 20. 1654. Henry Osborne.   

A More Exact Way to Delineate the Plot of any spacious parcel of Land, as Baronies, Parishes and Town-Lands: as also of Rivers, Harbours and Loughs, etc. than is as yet in Practice  

Sect. 1. How to keep the Field Book.  
HAving taken and set down your notes (of Proprietors names and Lands, bounders, wind, etc.) in the Field on the left hand Pages of your Book, you may divide the right hand Page into six Columns, (as may be seen hereafter) in the first may be put the quantities of your lines measured in Chains and hundred parts, in the second Column how many degrees and parts; the Lines you have traversed, are distant from the North or South part of the Meridian towards the East or West; the other four Columns are distinguished by the names of South, North, East and West: in the which are set down the Easting, Westing, Northing and Southing of the Degrees and Distances observed by a Table (the making and use whereof shall be shown) and for the easy understanding of which, we shall here affix a Table so divided and noted as aforesaid, answering to the Scheme adjoining in which suppose the line from A. unto B. to be 73. Degrees to the South towards the West, and the length of the line A. B. to be 14. Chains and 70. links; again the line B. C. to be 75. Degrees and 36. from the South towards to West, and the length of the line B. C. to be 20. Chains: 40. links, and also the line C. D. to be 87. Degrees from the South toward the West, and length C. D. 10. Chains: 50. links; the line D. E. 9 Chains: 70 links, and the Angle 73. Degrees 30, minutes from the South to the West, and this Order is to be observed in all the rest till you have finished your surround, and may more plainly appear by the subsequent Table.  
Distances. Angles. South. North, East. West. X Z 14: 70 73: 00 S W 4 29   14: 06 4: 29 14: 06 20: 40 75: 30 S W 5 10   19: 79 9: 39 33: 85 10: 50 87: 00 S W 0 55   10: 49 9: 94 44: 34 9: 70 73: 30 S W 2 76   9: 30 12: 70 53: 64 29: 00 75: 30 S W 7 25   28: 11 19: 95 81: 75 7: 40 74: 30 S W 1 98   7: 12 21: 93 88: 87 10: 60 South. 10 60   00 00 32: 53 88: 87 14: 60 28: 30 S W 12 84   6 97 45: 37 95: 84 33: 95 82: 00 N E  4: 74 33: 61  40: 63 62: 23 25: 00 72: 00 N E  7: 74 25: 75  32: 89 38: 48 21: 40 78: 30 N E  4: 28 20: 97  28: 61 17: 51 21: 00 65: 30 N E  8: 71 19: 11  19: 90 1: 60 20: 00 4: 30 N W  19 90  1 60 00: 00 00: 00 238. 25  45. 37 45. 37 97. 44 97: 44 X Z   

Sect. II. Of the Construction and use of a Table for the more ready and exact laying down of any right Lined Figure.  
THe making of this Table is very easy, out of the Tables of natural Sins for each Degree and half Degree of the Quadrant; (or nearer if one please) for if you multiply any distance run, by the Sine and Sine Compliment answering that Degree of Azimuth you measured on, the Products shall be the number to be put in the North or South Column, and East or West; for the reason is evident from the first Axiom of plain Triangles, that if the Hypotenusa (in a right angled Triangle) be put as Radius the base, and Cathetus shall be Sins of their opposite Angles: Now in this case, the lines measured are put as the Hypote●usa, and supposed Radius and therefore the sides shall be in proportion to the Sins of their opposite Angles, and the Ratio will be as Radius, is to the Distance of two places measured in their Rumbe: so is the Sine Compliment of that Rumbe to a fourth number, to be put in the Meridian either Northward or Southward, according as the Rumbe lieth North or South. Again as Radius is to the distance of two places measured in their Rumbe: so is the Sine of the Rumbe to a fourth number, to be put in the line of East or West, according as the Rumbe lieth Eastward or Westward; and this may suffice for the construction. Now to the use, supposing our Tables already made, now this Table is made for every Degree and half Degree of the Quadrant from the North or South towards the East or West, and is only the Sine Compliment of every of these Degrees and parts, made into a Table by Duplication, Triplication, Reduplication and Bipartition, etc. unto ten; by the help whereof, you may make the Numbers only by addition; which Numbers multiplied unto ten I use whither my Number of Chains be under Ten, a Hundred, or a Thousand: only observe, that the Numbers in the Table stand for any Number of Chains under Ten; but if they shall amount unto Twenty Chains or above, and less than two Hundred, you are to adjoin a Cipher to the Number found: but if your Number of Chains shall exceed two Hundred Chains, and less than two Thousand, you are to adjoin two cyphers to the Number found, etc. But if you shall have any odd Links, you are to seek them as if they were whole Chains, only with this caution, that if your Links exceed Ten, you cut off one Figures towards the right hand: but if under Ten, than you cut off two Figures towards the right hand; the residue you place under the Number of Chains, and then add them up into one Sum, and cut of the two last Figures toward the right hand, the remainder shall show the Number of Chains and those two Numbers cut off, shall be the Number of Links. An example or two will make all easy, and show how the Numbers in the Columns of North, South, East and West in the little Table are made, which observe well.   surveying diagram 
Let the Line A. B. contain in length 14 Chains, 70 Links, and the Angle observed is 73 Degrees from the South part of the Meridian towards the West, under 73 Degrees in the Table, and against 10 in the side I find 292; also under 17 degrees (the Compliment of 73 Degrees to 90.) and against 10 in the side I find 956 these two Numbers I place thus, then in the same Table under 73 Degrees, I seek my four, Chains against 4 in the side, and I find 117 to be added to 292 afore found; also in the next Column under 17 Degrees, and against 4 in the side, I find 383 to be added to 956 found in the said Table as before. 292 956 117 383 20 67 4: 29 14: 06 Now here are 70 Links, which 70 I seek under 73 Degrees, and against 7 in the side, and find 205; also against 7 in the side, and under 17 Degrees I find 669; now according to the rule, because above ten Links I cut off the last Figure to the right hand, and they stand thus 20: 67 to be added to the aforesaid Numbers, this done I sum up these Numbers, and they will be 429 and 14 06 but when I have cut off the two last Figures to the right hand, than they will be 4: 29 and 14: 06, that is 4 Chains 29 Links to be put in the South Column, and 14 Chains 6 Links to be put in the West; and always observe to put the Number arising from the Degree you measured on, into the Column of South or North, according to his Title: and the Number arising from the Compliment in the Column of East or West, with regard to his Title. The second line B. C. is 20 Chains 40 Links, and the Angle 75: 30 first I seek under 75: 30 and against 2 in the side (because my Table gives not 20) I find 50 also under 14: 30 the Compliment of 75: 30 in the Column adjoining I find 194 to both these, I adjoin a Cipher according to the rule, and they will stand thus also in the same Column, and against 4 I seek 40 Links and I find 100 and 387 which when I have cut off the last Figure to the right hand, they will stand thus 11 and 29 which added to the former Numbers and the two last Figures cut off it will be 5 Chains 10 Links for the South, and 19 Chains 79 Links to be put into the West, and so I proceed with all the rest until I have finished; 500 1940 10 39 5: 10 19: 79 and being thus returned unto my first station, I sum up severally these four Columns of South, North, East & West, and finding that the sum of the South Column is equal to that of the North, and the sum of the East is equal to the sum of the West, I conclude the work to be truly performed; whereas i● there had been any difference it shows an Error, and if that difference had been great, it is necessary to examine the work again, and so to correct it; and thus much for our business in the Field, we come now to the plotting. And for our more quick dispatch, we add severally the several Numbers in each Column together, as in the Example following; and here I begin with the South Column, and put them into the Table noted with X thus, 4: 29 4: 29 & more by 5: 10 is equal to 9: 39 and 9: 39 added to 0: 55 is equal to 9: 94 and 9: 94 added to 2: 76 is equal to 12: 70. and 12: 70 added to 7: 25 is equal to 19: 95. and 19: 95 added to 1: 98 is equal to 21: 93. & 21: 93 added to 10: 60 is equal to 32: 53. & 32: 53 added to 12: 84 is equal to 45: 37. And thus is the South Column summed up, and the several Numbers in the Northern Column, are to be subducted out of the South Column; Thus 4: 74 substracted from 45: 37 there rests 40: 63. and 7: 74 subducted from 40: 63 there rests 32: 89. and 4: 28 substracted from 32: 89 there remains 28: 61. and 8: 71 subducted from 28: 61 there rests 19: 90. and 19: 90 subducted from 19: 90 there rests 00: 00. And thus you see I have drawn the North and South Columns into the Column X. I shall proceed in the same manner with the East and West Columns, as to draw them into the Column Z. first 14: 06 and 14: 06 added to 19: 79 is equal to 33: 85. and 33: 85 added to 10: 49 is equal to 44: 34 & 44: 34 added to 9: 30 is equal to 33: 64. and 53: 64 added to 28: 11 is equal to 81: 75. & 81: 75 added to 7: 12 is equal to 88: 87. and 88: 87 added to 00: 00 is equal to 88: 87. (and this is the reason of twice 88: 87 in the Table Z) & 88: 87. added to 6: 97 is equal to 95: 84. and so far is the West Column summed up: the East Column follows, and is to be subducted out of the West thus, 33: 61 substracted from 95: 84 there rests 62: 23. and 23: 75 subducted from 62: 23 there rests 38: 48. and 20: 97 subducted from 38 48 there rests 17: 51. and 19: 11 substracted from 17: 51, which because I cannot do, I subduct 17: 51 out of 19: 11 there rests 1: 60. East, from which I subduct 1: 60 in the West, and there remains 00: 00. And thus is my two Columns of X and Z made, by which I lay down the Scheme.  
Distances. Angles. South. North. East. West. X Z 14: 70 73: 00 S W 4 29   14: 06 4: 29 14: 06 20: 40 75: 30 S W 5 10   19: 79 9: 39 33: 85 10: 50 87: 00 S W 0 55   10: 49 9: 94 44: 34 9: 70 73: 30 S W 2 76   9: 30 12: 70 53: 64 29: 00 75: 30 S W 7 25   28: 11 19: 95 81: 75 7: 40 74: 30 S W 1 98   7: 12 21: 93 88: 87 10: 60 South. 10 60   00 00 32: 53 88 .87 14: 60 28: 30 S W 12 84   6 97 45: 37 95: 84 33: 95 82: 00 N E  4: 74 33: 61  40: 63 62: 23 25: 00 72: 00 N E  7: 74 25: 75  32: 89 38: 48 21: 40 78: 30 N E  4: 28 20: 97  28: 61 17: 51 21: 00 65: 30 N E  8: 71 19: 11  19: 90 1: 60 20: 00 4: 30 N W  19 90  1 60 00: 00 00: 00 238. 25  45. 37 45. 37 97. 44 97: 44 X Z   

Sect. III. How to Delineate the Scheme.  
COnsidering which way the Ground lies, I take a point for my first Station; So as the whole may fall conveniently within the Plot, which let be the point A. By which point I draw a Meridian and Parallel, namely two right lines intersecting one another at right Angles, whereof let N S be the Meridian running North and South, and E W. the Parallel running East and West: And first I look to the South and North Columns, and there in the South Column, and against the first station I find 4: 29. and even with it in the West I find 14: 06 then with my Compasses, on a Scale of equal parts, I measure this 4: 29 on the Meridian line, from A. towards S, to 1. Also I take the length of 14: 06 of the Seal, and set it on the line of East and West, from A to 1; then keeping your Compasses at that distance, set one point in the Meridian line at 1, and with the other point describe a small Arch towards B. then set the Compasses to the first distance 4: 29 and setting one point in 1 on the line of East and West Cross the aforesaid Arch, and in the point of crossing note it with a prick, and a line drawn from this point to the point A represents my first line A B 14: 70. Then I observe against my second Station in the Column X Z. for the South 9: 39 and for the West 33: 85. First I take the 9: 39 on the Scale of equal Parts and lay it as before on the Meridian line N. S. from A. to 2; also I take the length 33: 85 and measure it on the line of East and West from A to 2: then keeping your Compasses at that distance set one point in the Meridian line at 2. and with the other foot describe an Arch toward C. then setting the Compasses at the former distance 939 set one point in the Line of East and West at 2. and with the other foot Cross the aforesaid Arch, and note the point of crossing for a line drawn from thence to the point B. is my second line 20: 40, and thus you may lay down the distances C.D.E. etc. till you have included the figure, the practice of this but once makes all perfect.  
 surveying diagram   

Sect. FOUR How to cast up the Superficial Content.  
NOw for the exact casting up the Area, the very bare Inspection will Declare it; for in the first Triangle A. B. 1, it is plain by the Table that the Base is 14: 06, the Cathetus 4: 29; the second Triangle B. C. Q. the Base 19: 79 and the Cathetus 5: 10; also the Parallelogram 1. B. 2. Q. hath for his sides the Cathetus of the first Triangle 4: 29, & Base of the second Triangle 19: 79; So it is obvious for all the rest; then having considered the length & breadth of the Rectangle Figure a b c d (within which the greatest extent of your Plot is contained) drawn into one another, and from the product you shall subduct the products of all those Triangles, and Parrallelograms comprehended within the said Rectangle Figure, and perimeter of your Plot; the remainder shall be the Superficial Content of that Figure: as in the foregoing Scheme the sides of the Rectangle Figure containing the Plot are 45: 37 and 97: 44, which drawn into one another, is 44208528; and the Products of the several Triangles and Parrallelograms are 23799003; which subducted from the former, there rests 204: 09525, which are the Number of square Links in the Plot, and if your Chain be four Pole; than are you to cut off the five last Figures towards the right hand, the residue shall be the Numbers of Acres, as thus, 204: 09525 is 204 Acres, and 9525 Links; but if your Chain be two Pole, than the fourth  of this number shall be the Number of Acres, viz. 51 Acres and 2381 Links, or almost 4 Perch; for like Plains are in a Duplicate Ratio to their Homologal sides by 20. Eucl. 6. (that is) as a Quadrat to a Quadrat; and therefore if three Lines are proportional, it shall be as the first is to the third: So is the superficies upon the first, to the superficies on the second; For example, 1. 2. 4, are proportional therefore as 1. is to 4: so is the square of 1. to the square of 2. upon this depends the reason of reduction of Plots or quantities, either bigger or lesser.   
FINIS.