¿BCDEeb, there be taken bBAGFeb, the Remainder will be the Map ABCDEFGA.

As before, Half the Sum of Bb, and Cc, multiplied by be, will be the Area of the Trapezium bBCc; after the fame Manner, Half the Sum of Cc, and Dd, multiplied by cd, will give the Area of the Trapezium cCDd; and again, Half the Sum of Dd, and Ee multiplied by de, gives the Area of the Trapézium dDE; and the Sum of these three Trapezia will be the Area of the Figure bBCDEeb.

Again, in the fame Manner, Half the Sum of bB, and a multiplied by ab, will give the Area of the Trapezium bВAa; and Half the Sum of aA, and gG, by ag, gives the Trapezium aAGg; to these add the Trapezia gGFf, andƒFEe, which are found in the like Manner, and you will have the Figure bBAGFEeb, and this taken from bBCDEeb, will leave the Map ABCDEFGA. QE, F.

It will be fufficient to protract this Kind of Work, and from the Map to determine the Area, as well as in Plate X. Fig. 3. to find the Areas of the Pieces 34563, and 6p76, from Geometrical Conftructions.

How to determine the Station where a Fault his been committed in a Field-Book, without the Trouble of going round the whole Ground a fecond Tme.

From every fourth or fifth Station, if they he not very long ones, or oftner if they are, let an I efection be taken to any Object, as to any particular

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Part

Part of a Castle, House, or Cock of Hay, &c. or if all these be wanting, to a long Staff with a white Sheet or Napkin fet thereon to render the Object more confpicuous, and let this be placed on the Summit of the Land, and let the respective Interfections fo taken be inferted on the left Hand Side of the Field-Book, oppofite to the Stations from whence they were respectively taken.

. In your Protraction as you proceed, let every Interfection be laid off from the respective Stations from whence they were taken, and let these Lines be continued; if they all converge or meet in one Point we thence conclude all is right, or fo far as they do converge; but if we find a Line of Interfection to diverge or fly off from the reft, we may be fure that either a Miftake has happened between the Station, the foregoing Interfection was taken at, and the Station from whence the Interfection Line diverges; or there must be an Error in the Interfection; but to be affured in which of these the Fault is, protract on to the next Intersection, and having fet it off, if it converges with the reft, tho' the foregoing one did not, we may conclude the Fault was committed in taking the laft Interfection but one, and none in any Station, and that fo far is true as is protracted; but if this as well as the foregoing Interfection diverge, or fly from the Point of Concourfe or converging Point of the reft, the Error muft have its Rife from fome Station or Stations, at or after that, from whence the laft converging Interfection Line was taken; fo that by going to that Station on the Ground and proceeding on to that where the next, or from whence the following diverging Interfection was taken we can readily and with little Trouble fet all to Rights.

But

But in moft Tracts of Land, one Object cannot be feen from every Station, or from perhaps one fourth of them; in this Cafe we are under the Neceffity to move the Pole after we begin to lose Sight of it to fome other Part of the Land, where it may be seen from as many more Stations as poffible; which is easily done by viewing the Mearing before it be furveyed: The Pole then being fixed in an advantageous Place, the first Interfection to it is beft to be made from the fame Station from whence the last one was taken, and then as often as may be thought convenient as before; in like Manner the whole may be done by the Removal of the Pole.

When we here speak of Stations we do not mean fuch as are ufually taken at every particular Angle of the Field: For it is to be apprehended, that every fkilful Surveyor, particularly fuch who use Calculation, will take the longest Distances poffible, not only to leffen the Number of Stations, for the Ease of either Protraction or Calculation, but with greater Certainty to account for the Land paffed by, on the right Hand or on the left, which is taken by Off-fets: And furely it will be allowed that any Measure taken on the Ground and the Content thence arithmetically computed, will be much more accurate than that which is obtained from any Geometrical Projection.

From what has been faid it is plain, that from this Method any Fault committed in a Survey can be readily determined, and therefore must be much preferable to the prefent Method of taking Dia

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gonals,

gonals, or the Bearings and Lengths of Lines across Land, to accomplish that End, which laft Method is too frequently ufed by Surveyors to approximate or arrive near the Content, which will ever remain uncertain, let thefe Diagonals be ever fo many, till the Station or Stations wherein the Error or Errors were committed, be found; and the Fault or Faults be corrected.

Where one Diagonal is taken it may perhaps close or meet with one Part of the Survey and not with the other; in this Cafe, if the Surveyor would difcover his Error, he must furvey that Part of the Land which did not clofe, and this may be half or more, of the whole. And fhould the Diagonal clofe with neither Part, but be too long, or too short, or fhould it fall on either Side of the affigned Point it was to clofe with, he ought to go over the whole, and make a new Survey of it in order to discover his Error.

A Number of Diagonals are frequently taken, the Sum of the Lengths of which very often exceeds the Circuit of the Ground, and after all they are but Approximations, and the Content remains uncertain as before; therefore he who returns a Map, made up by the Affiftance of Diagonals, where there remains a Misclosure in any one Part, runs the Rifque of being detected in an Error, and must fuffer Uneafiness in his Mind, as he cannot be certain of the Return he makes.

The frequent Mifclofures which are botched up by Diagonals, occafion the many and frequent fcanda

fcandalous Broils and Animofities between Surveyors, which tend to the Lofs of Character of the one or the other, and indeed often to the Difrepute of both, as well as to that of the Science they profefs.

But these may be easily remedied by Intersections, and the Bearing or Line be adjusted where the Fault was committed, and till this be found, nothing canbe certain.

To