Geodetical Operations

Excerpts

• Geodetical Operations

  If the three angles of a plane triangle be observed, the fact that their sum ought to be equal to two right angles affords a test of the accuracy with which the observations are made.

• Geodetical Operations

  Now in modern observations $h$ will not exceed the circular measure of a few seconds, so that, if $C$ be not very small, $h\cot C$ is practically insensible.

• Geodetical Operations

  The degree of closeness with which the measured length agrees with the calculated length is a test of the accuracy of the survey.

• Geodetical Operations

  One of the most important applications of Trigonometry, both Plane and Spherical, is to the determination of the figure and dimensions of the Earth itself, and of any portion of its surface.

• Geodetical Operations

  An important part of any survey consists in the measurement of a horizontal line, which is called a *base*.

• Geodetical Operations

  At various points of the country suitable stations are selected and signals erected; then by supposing lines to be drawn connecting the signals, the country is divided into a series of triangles.

• Geodetical Operations

  This formula is called General Roy’s rule, as it was used by him in the Trigonometrical survey of Great Britain and Ireland.

• Geodetical Operations

  Now the area is not known *exactly* unless the elements of the spherical triangle are known *exactly*; but it is found that in such cases as occur in practice an approximate value of the area is sufficient.

• Geodetical Operations

  thus in observing three angles, we suppose that in one observation a certain error is made, in a second observation the same numerical error is made but with an opposite sign, and in the remaining observation no error is made.

• Geodetical Operations

  The three methods which we have indicated were all used by Delambre in calculating the triangles in the French survey (*Base du Systeme Metrique*, Tome iii. page 7).

• Geodetical Operations

  This process, by which we find the angle $COD$ from the angle $AOB$, is called *reducing an angle to the horizon*.

Equations

• Geodetical Operations

  s=Er^2

  The area s of a spherical triangle on the Earth's surface equals the circular measure E of its spherical excess times the square of the Earth's radius r.

• Geodetical Operations

  E=\frac{n\pi}{180\centerdot 60\centerdot 60}

  The circular measure E of an angle equals n seconds converted through 180 degrees of π radians and 60 minutes of 60 seconds each.

• Geodetical Operations

  = \frac{n}{206265}\ \text{ approximately;}

  Approximately, the circular measure E equals n divided by 206265, the number of seconds in one radian.

• Geodetical Operations

  s=\frac{nr^2}{206265}\,

  The area s of the spherical triangle, in square feet, equals n r squared divided by 206265, where n is the spherical excess in seconds and r the Earth's radius in feet.

• Geodetical Operations

  \frac{\pi r}{180}=365155

  The length of one degree of arc on the Earth's surface, πr/180, equals the measured value 365155 feet, which fixes the Earth's radius r.

• Geodetical Operations

  \log n = \log s - 9.326774

  The logarithm of the spherical excess in seconds equals the logarithm of the triangle's area in square feet minus 9.326774, so n is found once s is known.

• Geodetical Operations

  \delta A + \delta B + \delta C = 0

  The errors in the three observed angles A, B, C sum to zero, because the altered angles are supposed to sum correctly.

• Geodetical Operations

  \dfrac{a \sin (C + \delta C)}{\sin (A + \delta A)}

  Treating the triangle as approximately plane, the true side c is a sin(C + δC) divided by sin(A + δA), where a is the known side opposite A.

• Geodetical Operations

  \sin (C + \delta C) = \sin C + \delta C \cos C

  For a small error δC, sin(C + δC) is approximately sin C plus δC times cos C.

• Geodetical Operations

  \sin (A - \delta B - \delta C) = \sin A - (\delta B + \delta C) \cos A

  For small errors, sin(A − δB − δC) is approximately sin A minus (δB + δC) times cos A.

• Geodetical Operations

  \cot C + \cot A = \dfrac{\sin (A + C)}{\sin A \sin C} = \dfrac{\sin B}{\sin A \sin C}

  The sum of the cotangents of A and C equals sin B divided by sin A sin C, using A + C = 180° − B; the book states this holds approximately for the nearly plane triangle.

• Geodetical Operations

  \frac{a \sin B}{\sin^2 A} \delta C + \frac{a \sin C \cos A}{\sin^2 A)} \delta B

  The approximate error in the side c due to the angle errors δB and δC is a sin B δC over sin²A plus a sin C cos A δB over sin²A; the source prints a stray closing parenthesis in the second denominator, which is reproduced here as printed.

• Geodetical Operations

  \frac{a \sin C}{\sin^2 A} \delta B + \frac{a \sin B \cos A}{\sin^2 A} \delta C

  The approximate error in the side b due to the angle errors δB and δC is a sin C δB over sin²A plus a sin B cos A δC over sin²A.

• Geodetical Operations

  \cos (\theta + x)=\frac{\cos \theta-\sin h \,\sin k}{\cos h \,\cos k}

  The cosine of the reduced horizontal angle θ + x equals (cos θ − sin h sin k) divided by cos h cos k; this formula is exact.

• Geodetical Operations

  \cos \theta-x \sin \theta=\frac{\cos \theta-hk} {1-\frac{1}{2}(h^2+k^2)}

  To first order in the small quantities x, h and k, cos θ − x sin θ equals (cos θ − hk) divided by 1 − ½(h² + k²).

Problems

No exercises in this chapter.