Mining Engineer V. Sergeev.

Toward a Theory of the Vertical Angle

The vagueness and confusion in the printed literature on the question of deriving formulas for the vertical angle \(h\) and for the place of the horizon \(C_v\), the practice of teaching mine-surveying and geodesy, the supervision of practical exercises, observations of workers in production, and the variety of existing types of graduations on vertical circles—all this, taken together, leads to the conviction that the question of the vertical angle theoretically still remains not yet exhausted, and that in the field of computing the vertical angle \(h\) and the place of the horizon \(C_v\), not only in practice but also in theory, wide scope is afforded for inventiveness and imagination. The need for more definite and simple formulas for \(h\) and \(C_v\) is certainly felt; evidence of this is furnished by: 1) the variety of types of graduations on vertical circles and also 2) the variety of formulas for \(h\) and \(C_v\).

The question of the theoretical derivation of the above-mentioned formulas in general, in connection with the various types of graduations on vertical circles, is nowhere touched upon in the printed literature, unfortunately. The first to take up this question was P. K. Sobolevsky, now professor at the Ural Polytechnic Institute, who in 1908, for students of the Tomsk Technological Institute, issued lithographed notes, “Toward a Theory of Geodetic Instruments,” where the question of the theory of the vertical angle and of the place of the horizon is examined in detail, in particular with respect to the type of graduations counterclockwise from \(0^\circ\) to \(360^\circ\).

In this brief note I shall touch only upon what I consider more expedient and useful with regard to: 1) the theoretical derivation of formulas for the vertical angle \(h\) and the place of the horizon \(C_v\), and 2) the type of graduations on the vertical circles of all angle-measuring geodetic instruments.

First, however, let us define the angle being measured in general, and, adhering to the definition of a horizontal angle proposed by P. K. Sobolevsky, give it in the following formulation: a measured angle in general is the difference of two directions (readings), the older minus the younger. Let us also define the concepts of the younger and older directions. The younger direction is that from which, adhering necessarily to the increase of readings on the vernier, we must begin measuring the angle; and the older is that at which, covering the measured angle by the motion of the sighting axis, the measurement must be ended. In this connection one should not confuse the older direction (reading) with the larger reading, nor the younger direction with the smaller reading, since the immediately larger reading is not always the older.

Indeed, if the alidade, moving from the younger direction to the older, passes through the zero of the circle, it is evident that the older reading, losing \(360^\circ\), becomes numerically smaller than the younger.

It may be said that, for a given angle, the seniority of directions depends only on the direction in which the graduations increase.

The dependence of the seniority of directions on the direction of increase of the graduations on the circle imposes the obligation to draw, in the field book or field journal, the type of graduations on the horizontal and vertical circles.

In proceeding to set forth the derivation of the formulas for \(h\) and \(C_v\), as a basis for this derivation we shall likewise assume the priority of directions, and we shall introduce, for the derivation of \(h\) and \(C_v\), the concept of the responsible vernier. By the responsible vernier, in our particular case, we shall mean, for all sightings, the objective vernier, i.e. the one lying at all times on the objective side. In the given type of graduation of the vertical limb (see Drawing 1), the eyepiece vernier may also be taken as the responsible vernier, since for it too the cases of the alidade passing through the zero of the limb drop out.

Let us also give the definitions of the vertical angle “\(h\)” and of the horizon position “\(C_v\)”: the vertical angle is the angle between the given direction and its projection on the plane of the horizon; the horizon position *) is the reading on the vertical circle when the sighting axis occupies a horizontal position in space.

Let us now turn to Drawings Nos. 1, 2, 3, and 4 and proceed to the actual derivation of the formulas for “\(h\)” and “\(C_v\).”

Drawing No. 1: the sighting axis \(K\) is made to coincide with the horizon with circle right, reading \({}^{R}C_v\). Drawing No. 2: the sighting axis is directed along the given direction with circle right, \(Cr.\ R.\), to the point “\(P\),” reading \(R^b\). The priority direction to the given point is \(R^b\); consequently the angle

Drawing No. 1.

Drawing No. 2.

\[ h = R^b - {}^{R}C_v. \ . . . . . . . . . . . . . . . . \tag{1} \]

Drawing No. 3: the sighting axis is made to coincide with the horizon with circle left, reading \({}^{L}C_v\). Drawing No. 4: the sighting axis is directed to the given point “\(P\)”

Drawing No. 3.

Drawing No. 4.

*) The horizon position is often called the zero position; it is such only when the zero of the vertical limb is correspondingly arranged.

with circle left, the reading \(L_a\). The older direction is toward the horizon; then the angle

\[ h = {}_L C_v - L_a \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (2) \]

By solving equations (1) and (2) we obtain the desired formulas for \(h\) and \(C_v\)

\[ h_0 = \frac{R_b - L_a}{2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (3) \]

\[ C_v = \frac{R_b + L_a}{2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (4) \]

Let us now use drawings 5 and 6 for a direct derivation of the same formulas. We note that \({}_R C_v = {}_L C_v\). Drawing No. 5 represents the combined depiction of the moments of measuring the angle \(h\) with circle right and circle left, the geometrical elements of the moment with circle left having been transferred into a symmetrical position with respect to the elements with circle right, downward to the right, where the sighting axis \(K\) is indicated by a dash-dotted line over both its visible and invisible parts.

Drawing No. 5.                                                                             Drawing No. 6.

The two symmetrically situated positions of the sighting axis “\(K\),” with circle right and reduced with circle left, are determined by arcs with respect to the corresponding vernier “\(b\)” with circle right; here the seniority of directions for the double angle \(h\) \((2h)\) is clear, namely: the younger direction—with circle left—is marked by the arc \(L_a\), \((L_a = L_b)\); the older direction—with circle right—is marked by the arc \(R_b\), and hence the angle

\[ 2h = R_b - L_a \quad \text{and the angle} \quad h_0 = \frac{R_b - L_a}{2} \ldots \ldots \ldots \ldots \ldots (3) \]

Drawing No. 6 also represents the moments of sighting with circle right and circle left to the horizon and to the given point, which sightings are marked by the arcs \({}_R C_v\), \({}_L C_v\), \(R_b\), and \(L_a\). For the derivation we shall rotate the arc \({}_L C_a\) by the magnitude of the arc \((m+h)\) to the right until its beginning coincides with the end of the arc \({}_R C_v\); likewise we shall rotate the arc \(L_a\) to the right by the magnitude of the arc “\(m\)” until its head coincides with the end of the arc \(R_b\); then the ends of the arcs \(({}_R C_v + {}_L C_v)\) and \((R_b + L_a)\) will lie on the same radii, and since \({}_R C_v = {}_L C_v\), it follows that

\[ 2C_v = R_b + L_a \quad \text{and} \quad C_v = \frac{R_b + L_a}{2} \ldots \ldots \ldots \ldots \ldots (4) \]

Thus, from the geometry of the drawing, the formula for the horizon \(C_v\) is obtained, already derived earlier by the analytic method, by solving

equations (1) and (2). Let us give our readings the particular values \(R_b = 127^\circ\) and \(L_a = 59^\circ\), and substitute into the formulas; then
\(h_o = \dfrac{R_b - L_a}{2} = \dfrac{127^\circ - 59^\circ}{2} = +34^\circ\),
\(C_v = \dfrac{R_b + L_a}{2} = \dfrac{127^\circ + 59^\circ}{2} = 93^\circ\); as a check,
\(h = R_b - C_v = 127^\circ - 93^\circ = +34^\circ\),
\(h = C_v - L_a = 93^\circ - 59^\circ = +34^\circ\). Thus we have obtained not only the magnitude but also the sign of the angle.

To confirm the generality of the formulas derived, let us take for observation a negative angle “\(h\).” In Fig. No. 7, the sighting axis \(K\) is brought into coincidence with the horizon at Cr. Right, reading \({}_R C_v\). In Fig. No. 8 the sighting axis is brought into coincidence with the given direction to point “\(H\)” at Cr. Right, reading \(R_b\). In this case the larger direction is the direction to the horizon, and the angle is \(-h = {}_R C_v - R_b\), whence we obtain \(h = R_b - C_v\)—the former formula (1).

Fig. No. 7.          Fig. No. 8.

In Fig. No. 9 the sighting axis \(K\) is directed to the horizon at Cr. Left, reading \({}_L C_v\). In Fig. No. 10, the sighting axis \(K\) is directed to the given point “\(H\)” at Cr. Left, reading \(L_a\). Here the larger direction is to the given point, and the angle is \(-h = L_a - {}_L C_v\), whence we obtain \(h = {}_L C_v - L_a\), again the former formula (2); and, solving these particular equations for \(h\), we obtain as before

\[ h_o = \frac{R_b - L_a}{2}, \qquad C_v = \frac{R_b + L_a}{2}. \]

Let us take particular values from the observa-

observations for \(R_b\) and \(L_a\), namely: \(R_b = 59^\circ\), \(L_a = 127^\circ\), then

\[ h_o=\frac{R_b-L_a}{2}=\frac{59^\circ-127^\circ}{2}=-34^\circ,\quad C_A=\frac{R_b+L_a}{2}=\frac{59^\circ+127^\circ}{2}=+93^\circ, \]

for checking,

\[ h=R_b-C_v=59^\circ-93^\circ=-34^\circ,\quad h=C_v-L_a=93^\circ-127^\circ=-34^\circ. \]

From this particular example one may be convinced that the general formulas derived do indeed give both the magnitude and the sign for the angle sought.

It should be noted that in the formulas, everywhere, the direct readings are introduced in place of the letters.

At the same time it turns out that there is absolutely no need to watch afterward “where the telescope is directed—upward or downward,” in order to give the measured angle its sign. This advice must be forgotten, since it cannot always be carried out. Our formula itself gives the sign.

For comparison let us examine the type of divisions on the Kipregel; any nonius may be taken as the corresponding one; let us take, for example, the objective one. From drawings Nos. 11–14 the seniority of the directions is clear: with Kr. Right the senior direction to the given point is \(R_b=21^\circ\); with Kr. Left the senior direction is to the horizon \(C_v\), the reading to the given point with Kr. Left is \(L_a=323^\circ\). We compose the formulas:

Drawing No. 11.

Drawing No. 12.

\[ h=R_b-{}_{R}C_v,\quad h={}_{L}C_v-L_a,\quad h_o=\frac{R_b-L_a}{2},\quad C_v=\frac{R_b+L_a}{2} \]

Drawing No. 13.

Drawing No. 14.

If we were to observe a negative angle, then (we do not give drawings) we would obtain the same general formulas. However, when substituting

when substituting these into the formulas of observed readings, one must remember the seniority of directions, and therefore the senior, but numerically smaller, reading should be increased by \(360^\circ\), after which the actual sought partial values of the angle can be obtained, for example in our case:

\[ h_o=\frac{(R_b+360^\circ)-L_a}{2} =\frac{R_b-L_a}{2} =\frac{(21^\circ+360^\circ)-323^\circ}{2} =+29^\circ, \]

\[ C_v=\frac{R_b+L_a}{2} =\frac{381^\circ+323^\circ}{2} =352^\circ,\quad \text{check: } h=381^\circ-352^\circ=+29^\circ, \]

\[ h=352^\circ-323^\circ=+29. \]

Let us suppose that we have observed a negative angle and have \(R_b=325^\circ\), \(L_a=55^\circ\). In this case, with the Circle Right the senior direction will be the direction to the horizon, and with the Circle Left the senior direction will be the direction to the given point; consequently \(360^\circ\) should be added to the senior, but numerically smaller, readings; then we obtain

\[ h_o=\frac{325-415^\circ}{2}=-45^\circ,\quad C_v=\frac{325+415}{2}=370^\circ \]

check

\[ h=R_b-C_v=325^\circ-370^\circ=-45^\circ,\quad h=370^\circ-415^\circ=-45^\circ. \]

Thus, the principle of senior and junior directions, applied to the derivation of the value \(h\) and \(C_v\) also for the type of graduations on the vertical limb as in the Kipregel, gives the former formulas, which determine for the sought angle both magnitude and sign. However, it must be stipulated that this type of graduation, in comparison with the one I propose, subdivided from \(0^\circ\) to \(360^\circ\) with the above-depicted locking of the vertical circle, is less convenient for deriving the basic formulas and, in practice, appears complicated, since in particular cases one has to add \(360^\circ\) to the directly senior readings, something the observer must remember all the time in order not to fall into error. In addition, this type of graduation is entirely unsuitable for observing zenith distances, which at times cannot be avoided, because both surveyors and mine surveyors in practice have to engage in astronomical observations to determine the azimuth of a direction, the latitude of a place, and clock correction.

Returning to the derivation of the basic general formulas for the angle \(h\) and the place of the horizon \(C_v\), let us note the following: 1) the proposed derivation of the formulas is a theoretical derivation, and not an empirical one such as is proposed in manuals; 2) the derived formulas in all particular cases give, for the sought angle, both magnitude and sign; consequently, by introducing the seniority of directions in measuring the vertical angle, an obligation is imposed to derive 1) the angle as a difference, not as a sum, and 2) the place of the horizon as a sum. Indeed, every separately measured quantity, whether it be a length or an angle, must be obtained by the method of differences of readings, and the vertical angle is a measured quantity. The place of the horizon is a reading, and only a reading (which may be whatever it likes), that determines one position of the sighting axis, namely the horizontal one, and therefore it is impossible to derive the formula for \(C_v\) by the method of difference. However, it must be noted that in manuals, even those of this year, formulas are encountered: for the vertical angle

\[ h=\frac{R+L}{2}, \]

for the place of the horizon

\[ C_v=\frac{R-L}{2},\quad C_v=\frac{L-R}{2}, \]

and no general theoretical justification is given.

In concluding this brief note: 1) on the desirable type of graduations on the vertical limb of angle-measuring, geodetic, and mine-surveying instruments, and 2) on the uniform derivation of formulas for “\(h\)” and “\(C_v\),” I express an earnest wish and appeal to everyone to propagate the idea of the senior and junior directions, as bringing order and clarity into the question of measuring the vertical angle, instead of the confusion of various contradictory formulas used in practice today.

In addition to the types of graduations considered here, there are also running types, which, for reasons of brevity, are not examined here in detail, such as: from two opposite zeros to \(90^\circ\), from one zero to \(180^\circ\) in both directions, where, with seniority of directions and the corresponding vernier, by choosing on the limb a positive and a negative direction of counting the graduations, one may obtain the same formulas

\[ h_c=\frac{R_b-L_a}{2},\quad h=R_b-{}_R C_v,\quad h={}_L C_v-L_a,\quad C_v=\frac{R_b+L_a}{2} \]

or, in the case of a positive direction of counting the graduations counterclockwise,

\[ h_c=\frac{L_a-R_b}{2},\quad h=L_a-{}_L C_v,\quad h={}_R C_v-R_b,\quad C_v=\frac{L_a+R_b}{2}, \]

which give for the required angle both magnitude and sign, without regard to where the telescope is directed, i.e., upward or downward.

However, these types of graduations also cannot be recommended, for the reason that here one must still specify positive and negative directions of graduations on the limb (which must necessarily be done with these types of graduations), and also because with these types of graduations absurd results for the sought angle are inevitable in practice, for example, in the case when the measured angle exceeds the value \((90-C)\),*) which is an inevitable consequence of transferring the origin of counting.

Taking all this into account, it would be necessary also to establish uniformity in the manufacture of the type of graduations on the vertical limbs of all angle-measuring instruments. In this, preference should be given to the type of graduations from \(0^\circ\) to \(360^\circ\) with such an inclination of the vertical limb that the line of sight approximately coincides with the diameter \(90^\circ\)—\(270^\circ\), because under this condition: 1) either the eyepiece or the objective vernier may be taken as the responsible vernier for all sightings, and 2) cases of the alidade passing through the zero of the limb are eliminated.

*) \(C_v=-5^\circ;\ R=+85^\circ20'\), \(h=R-C_v\), \(h=+85^\circ20'-(-5^\circ)=+90^\circ20'\); the vertical angle turned out to be greater than \(90^\circ\). \(C_v=-5^\circ\), \(C_v=+87^\circ\), \(h=-5^\circ-(+87^\circ)=-92^\circ\); the angle came out negative, whereas the telescope had been directed upward and the angle should have been positive.