ED by c, we have in the isoceles triangle, Cos C

and in the obliqueangled-triangle DCE

.........

=

(by substituting the value of Cos

b2-2 ab Cos C

=

ready found,) a2 + b2 —2 ab

b -d

2 α

b-d

2 a,

=

a2 +

Cal

a2—bd — a2 (1 + bd), and therefore C=√ a2 (1 +

=

bd

bd

a2

a2

"In like manner may the diagonals of the remaining quadrilaterals be determined; and when reduced, with the other parts, to the scale of the map, will give at once in inches, the sides a, b, d, and diagonal c for every latitude.

"In the present supposition, the ratio of the scale is 33110; by which therefore dividing the foregoing quantities we shall have

"If so far, has been thoroughly understood, there will be no difficulty in using the following Tables, which give the parts

of every quadrilateral of 1 latitudinal degree high, and half a longitudinal degree broad, between the parallels of 8° and 36°.

"But the scale of 4 miles to 1 inch is not the only ratio which may be required; and therefore other sets of Tables are added for the scale of 8 miles, and of its several multiples by 2, 3, 4 and 6 to one inch.

"As however the scale decreases, it will be proper that the numbers of degrees contained in the sides and bases of the quadrilateral should increase. The following law will therefore be observed, which is calculated to maintain always a regular curvature, without marking by too sensible an angle, the junction of one quadrilateral with another.

*NOTE.-The divisions here directed, have for object the reduction of the Thus, quadrilaterals to the areas proposed to be comprized in the graticule. in the 1st scale, there will be an intersection at every half degree; in the 2nd, 3rd and 4th, at every degree, and in the 5th and 6th at every second degree.

Table of Quadrilateral parts for a Scale of 4 Miles to 1 Inch.

a = 1° b 0°.30',

d = 0°-30'

Table of Quadrilateral parts for a Scale of 8 Miles to 1 Inch.-N. B. a

Latitude.

Table of Quadrilateral parts for a Scale of 16 Miles to 1 Inch.-N. B. a

Table of Quadrilateral parts for a Scale of 24 Miles to 1 Inch.-N. B. a

= 3o. b = 1o. d = 1o.

Table of Quadrilateral parts for a Scale of 32 Miles to 1 Inch.-N. B. a

Table of Quadrilateral parts for a Scale of 48 Miles to 1 Inch.-N. B. a =

Although in the example which has been given a very small area has been employed, this method is nevertheless susceptible of considerable extension. At the same time it has its limits, beyond which the meridians and parallels will lose their proper curvatures. Perhaps 100 superficial degrees will be found to be the utmost extent to which it can be practised, without introducing any sensible distortion: and this area may be indifferently the product of 10 into 10, 11 into 9, 12 into 8, 13 into 7, 14 into 7, &c. whilst it signifies little, which factors express the latitude or longitude.

CHAPTER XXVII.

ON THE CONNECTION BETWEEN THE GREAT TRIGONOMETRICAL AND REVENUE SURVEYS-RATIO OF ERROR— EXTENT OF COUNTRY FOR SURVEY, AND AVERAGE COST PER SQUARE MILE.

No Revenue Survey can be considered complete and satisfactory, unless the errors committed, and the degree of accuracy attained, are distinctly reported on. Without this is scrupulously observed, no confidence can be placed in its results, and as no work whatever, executed by human means, can be free from error, all that is reasonably expected is, that unavoidable errors should not exceed the lowest practicable limits. Every great National survey has a limit of error assigned, and all work exceeding this limit is rejected.

Ratio of Error.

In the large triangulation of the Great Trigonometrical Survey of India, where of course the greatest refinement and most scrupulous care is observed, an error of one inch per mile or in part, amounts to 500 inches or 42 feet, or nearly half a second of latitude in 500 miles, which is the distance between some of the bases. The work is reckoned liable to half this error when executed with the great theodolite, and equal to the degree of correctness indicated by an inch per mile, when performed by 18-inch instruments for the subordinate series of triangles; with inferior instruments, or a less careful system, the accumulation of error would be a foot per mile, which is equal to a ratio of 20 in linear dimensions

or in area, or per cent. or 6 seconds in the above dis

tance.

But men trained to this degree of exactness are out of their place in the detail measurements of the Revenue Survey, the valuable qualities of the Trigonometrical Surveyor, which he is obliged to maintain with such care, viz., delicacy of eye and touch, and rigorous modes of thinking, would be injurious in the details, and there would only be a loss of time in changing from one work to the other. They require in fact, people differently trained, with instruments of different calibre, and of different constitutions of mind. The Trigonometrical, from being freed from details, has been able to progress more rapidly, and the great object has been to get the large triangulation out of hand, without it nothing systematic or really trustworthy can be done. Indeed the principal, and the detail work, never could have gone on pari passu, and if the latter is based on the triangulation, it cannot be liable to any great accumulation of error, and may be taken up and organized according to the scale required.

The maximum error allowed, in lineal measurement on the Revenue Survey, according to the test it is submitted to by traverse proof, is 10 links in 100 chains equal to 5.28 feet per mile of a mile, but in the actual prosecution of the extensive surveys of the season 1847-48 covering an area of about 16,000 (sixteen thousand) square miles the average ratio of correction employed for the closing of the traverses, is found to be only 2 feet per mile, or rather more than one-third of the allowed correction; per cent. therefore for the pergunnah or main circuit measurement is fully within practicability, per cent. also may be allowed for the area of the district,per cent. for the village survey area, and one per cent. for the interior detail measurement of cultivation and waste.

These are exclusively the errors of measurement. As the computation of the Revenue Survey is based upon the wrong assumption of the earth's surface being a plane, certain dis