Nomogram

Nomogram . abacus or nomograph . It is a graphical calculation instrument, a two-dimensional diagram that allows the graphical and approximate computation of a function of any number of variables. In its most general conception, the nomogram simultaneously represents the set of equations that define a given problem and the total range of its solutions.

Table of Contents

Summary

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1 Definition
2 The tables
3 Your Use
4 Nomogram examples
5 History
6 Sources

Definition

It is a model that uses an algorithm or a mathematical formula to predict the probability of an outcome, optimized for greater predictive security. Nomograms are a tool of daily use. One of its advantages is its usefulness to carry out the synthesis of complex tables at the time of satisfying all the concerns of patients.

For example, in daily practice, patients who require a greater volume of information about the disease that motivates their consultation are assisted. Many times they demonstrate that they have read enough on the subject, express their doubts regarding the prognosis and demand certainty about the treatment that the professional suggests.

This obliges all doctors to be permanently updated, either through magazines, textbooks, courses, conferences or through the enormous amount of medical information found on the Internet. And all this information must be summarized so that it is understandable. This can be done with nomograms.

It is also an analogical calculation instrument, as is the calculation rule, by using continuous line segments to represent the discrete numerical values that the variables can assume. Nomograms used to be used in cases where obtaining an exact answer was impossible or very inconvenient, while obtaining an approximate solution was sufficient and highly desirable.

The tables

Nomograms are closely related to another traditional instrument for problem solving and succinct presentation of scientific information, tables. As the Spanish military engineer Ricardo Seco used to say in 1911 , “if it were possible to put together in a small volume a collection of tables containing the results that give the most frequently applied formulas for all the values that in practice can be taken by the different variables they contain, we would have reached the desideratum that every practical manual should try to fill. ”

But, he added, “such a collection of tables is unfeasible because, discounting the excessive work, long time necessary for their construction and the large volume they would occupy,” if there were more than three variables in the formula “there is no practical means of constructing them.” On the other hand, as it has been said, the naming techniques allow to construct nomograms of practically any number of variables.

D’Ocagne stated in Le calcul simplifié that “nomograms can be considered as complete calculation tables”, adding to the advantages already mentioned that they have over them the ease of visual interpolation, while acknowledging the drawback that the precision they can achieve the data in the tables is in principle as large as desired, while that of the nomograms is essentially very limited.

Its use

Being a nomogram the graphic representation of an equation of several variables, it must consist of as many graphic elements as there are variables in the equation. These elements will be points or lines, straight or curved, depending on the case. Given the values of all the variables except one, the one of the latter can be found by means of some immediate geometric resource (which is generally the tracing of another line that passes through that point).

Therefore, the nomogram of a two-variable equation (y = f (x)) will have two graphical elements, usually two graduated lines, or scales, arranged in such a way that the determination of the value of one of the variables (setting a point of the line) specify the value of the other, the unknown or function. The nomogram of a three-variable equation (z = f (x, y)) will normally consist of three scales, and so on.

The art of nomography consists precisely in elaborating these scales and arranging them in the plane in such a way that the drawing of straight lines that cross them determines the existing collinear points in each of the scales, points that will represent the different values related by the function in each specific case.

The relative disposition between these elements, on the other hand, cannot be predicted, since it will be determined by the nature of the problem in question or by other considerations. For example, the nomogram of the two-variable function that relates degrees Celsius of temperature to Fahrenheit may consist of two properly located parallel scales. To use it, simply place a ruler perpendicular to the scale that contains the known data; the other will be at the point where the ruler cuts its corresponding scale.

But it is evident that the space that separates the two scales does not perform any special function, so it can be progressively reduced to the point of making it disappear and that both are confused into one, which will thus be labeled on both sides, being then immediate reading the result of the conversion.

Examples of nomogram

Spiral nomogram – the Cornu spiral.
Smith’s letter, reproduced at the beginning of the article, used in electronics and systems analysis.
reticulated – semilogarithmic, double logarithmic, probabilistic paper, all intended to represent various nonlinear functions as straight lines.

A useful graph To reduce the air volume measured in ATPS, the expression Air volume STPD – (Air volume ATPS), (Factor STPD) will be used. Note: STPD factor values start at 0.570 and so in progressive order up to 1,020 (example: 0.750)

Nomogram to calculate the skin surface

Nomogram to determine the STPD factor

History

The astrolabes , quadrants, and sectors of the late Middle Ages and the Renaissance (collectively called mathematical instruments) were already intended to solve practical mathematical problems graphically and mechanically. The invention of the Gunter scale in the 17th century was the first graphical representation of a function using a graduated scale and was essential to all subsequent developments. Another decisive step was Descartes’ invention of Analytical Geometry , which allows the graphical representation of any mathematical function by means of a curve.

It was especially the military engineers and other public officials, in charge of regularly solving repetitive quantitative problems, who naturally showed greater interest in obtaining help for their task. Thus L. Pouchet published in 1797 a work titled Métrologie terrestre, which contains an appendix named Arithmétique linéaire containing the first systematic attempt to construct double-entry graphical tables.

In 1842 Léon Lalanne proposed the use of devices of this kind for the calculation of cuttings and embankments and in 1843 he formulated the principle of anamorphosis , which greatly facilitated the construction of these double-entry graphics, which he called abacus , by replacing the most of the lines curved by straight lines. J. Massau generalized this principle around 1880 .

Maurice d’Ocagne replaced in 1884 the squares and systems of curves by simple graduated scales, straight or curved, thanks to his conception of the isotope points. Later he systematized all these scattered methods into a definitive body of doctrine, which he called nomography .

Nomograms had a great development in the first three quarters of the 20th century , both in civil engineering and in the fields of chemistry , electrical, electronics and aeronautics . They were included in the manuals of the disciplines and in addition separate collections of them were published. Its limited precision, of two or three significant figures, instead restricts its use in fields such as astronomy or financial calculation, where accuracy is of paramount importance.

The refinement and popularization of calculators and electronic computers in the last quarter of the 20th century meant the practical disappearance of nomograms, greatly facilitating the complete carrying out of exact calculations that the operator would not even know how to propose on his own. Nomograms, however, still have the same usefulness as ever and present some specific advantages that were not unknown to their inventors, such as the synoptic capture of the range of values that the solution of a problem can adopt, the presentation of the structure of relations that occur between its parameters or the possibility of being used in almost any imaginable circumstance.