Investigation of the accuracy of readings on alignment
nomograms
N.D. Haasbroek
Nederlandse Commissie voor Geodesie, Delft, 1959. 172 pagina's. € 18,00 Harde kaft: € 22,50
Introduction
To my knowledge only a few data in nomographic literature are known
of the accuracy of readings on alignment nomograms. In his thesis
Nomographische Lösung von Kurs - und Rentabilitatsaufgaben (Amsterdam
1935) dr.ir. A. Gabel mentions some sources on pages 23 and 24. So R.
Soreau in his Nomographie ou Traité des Abaques says that the relative
error obtained from a reading on an alignment nomogram is 0.001, without
advancing any argument for this assertion. Maurice d'Ocagne, the founder
of modern nomography, says in his Calcul graphique et Nomographie "that
the readings of a trained observer are correct to 0.20 or 0.25 mm". Here
too the plausibility of this proposition is not shown in any way.
In the dissertation that Gabel himself gives on the accuracy of readings on
nomograms he refers to the investigations of C.Reinhertz in Zeitschrift für
Vermessungswesen. Reinhertz gives there a formula for the relation between
the size of an interval and the relative accuracy of the estimation in that
interval. If mI is the standard value of the relative estimation error in an
interval of I mm then, according to Reinhertz:
mI = K/√I (1) wherein K is a constant.
From his own observations Reinhertz found, according to Gabel, K = 0.075.
From the observations of F.J. Dorst could be derived, according to Gabel, an
amount K = 0.057.
It must be remarked that the investigation of Reinhertz related to readings
on levelling rods with the aid of the wires of levelling instruments. The
horizontal wire always intersected the rod scale at right angles. According
to Reinhertz one can regard formula (1) "als die normale (ansehen) und sie
als diejenige Beziehung hinstellen welche bei gut construierten Fernrohren
und scharfen Bildern im Algemeinen als zutreffend angenommen werden darf".
If the brightness of the images or the thickness of the wires changes, he
adds, the exponent n in
mI = K/In differs from 0.5.
It is obvious that several influences in Reinhertz' experiments will not be
of any importance in an investigation of the accuracy of readings on
nomograms. Other influences however, which did not have to be examined by
Reinhertz, can be of much importance in such an investigation.
It is therefore strange that Gabel thought formula (1) to be applicable to
the computation of relative accuracies of estimations in nomogram intervals.
It is true that he mentions the result K = 0.070 of a rough check, but the
usefulness of the formula itself does not come up for discussion.
A serious objection too is that Gabel does not mention that an oblique
intersection of the index line with the scale of the dependent variable
affects the standard error in the estimation of the final result in an
unfavourable way. His computations of standard errors in nomographic
readings are therefore contestable.
In several passages in their books the Russian authors M.W. Pentkowski and
B.A. Newski, pay attention to the influence of the inaccuracy of the
estimations on the results, read from nomograms. Both suppose that the
standard error in the estimation increases with the cosecans of the angle
between the index line and the scale on which one has to read. There is no
evidence of how they have come to this supposition and whether extensive
experiments have borne it out. Furthermore, in several other passages of
their important works they make suppositions and vague assertions on the
accuracy of the estimation which cannot be verified. An extensive
investigation therefore seemed to be necessary. An investigation of the
accuracy of readings on nomograms with point fields should be included.
If numerical results of the readings on the nomograms were also available,
eventual systematic errors as described by H. Bäckström in his paper: Über
die Dezimalgleichung beim Ablesen von Skalen could be shown.
Contents
1. Determination of standard errors 7
2. Determination of systematic errors 30
3. Practical results of the investigation 50
4. Investigation into individual performances of test persons 80
5. Readings on nomograms with point fields 102
6. Summary and conclusions 134
Appendix 144