Was there life before computer?

calculating instruments before the digital era

Download the history of the nomography with all the graphic needed to calculate

Abaques and nomograms

In 1844, Leon Lalanne created the first log-log plot in history. The product of x and y is found from their intersection with the 45° lines. Below there is a semplificated example but the original plot is able to perform also squares, cubes, roots and various engineering and chemical formulas. He called his table "abaque".

Lalanne's abaque

Download the abaque in high resolution

Lalanne envisioned many copies of his Universal Calculator posted in public squares and business meeting places for popular use but today is very rare: I know only three copies, one in the Bibliothèque de l'Ecole des mines of Paris, one in the Library of Congress of Washington and one is mine.

The original Lalanne's abaque

The original "abaque"

This system never had the success hoped for and was gradually abandoned for the more practical nomography, but the proportions between the numbers are harmonious and some abaques are really beautiful.

Abaco di Lallemand

Lallemand's abaque, 1885

The nomography was invented, as an easier replacement of the a abaques, in 1884 by Maurice d'Ocagne who replaced the Cartesian coordinates of the first calculating tables with a system of parallel scales.

The nomogram or nomograph, in its simplest form consists of three scales: the two external identify the values of the problem to solve, and joining them with a ruler you can read the result of the intersection with the central staircase. The scales may be linear or logarithmic, simple calculations are shown on straight lines but it is sometimes necessary to draw them in a circular shape.

The nomography allowed everyone to perform calculations with ease, it is sufficient to draw one or more lines without even having to know the equation that is being solved. A great help before the advent of the electronic calculators.

The nomograms are still widely used for military uses in medicine and aviation: they are quick to use and the results are sufficiently precise, and for the solution of specific problems are unsurpassed. This represented below is extremely intuitive: simply combine with a ruler the values of our weight and our height to know if we should just put on a diet. The red line is my situation: I must start to care ...

Height and weight nomogram

As slide rules nomograms are analog instruments whose accuracy is limited by the accuracy with which you can print and read the scales but instead of slide rules, can only solve the problems for which they are programmed. Often are placed in sliding tables.

How to calculate with the nomography

To multiply connect with a ruler the two factors A and B of the outer scales and read the result in the central scale, to divide reverse the process. You can also square and cube a number (or do the square and cubic root). In the examples the red line is 2x5=10 or 10/2=5 or 10/5=2; the blue line is 6x3=10 or 18/3=6 or 18/6=3. Here you can download a printable one; graphic by Alvaro Gonzales: ARC - Amigos de las Reglas de Calculo.

Download the printable nomogram

Example: 2.3 x 3.4

  • connect with a ruler 2.3 of the A scale with 3.4 of the B scale;
  • read the answer (ca. 7.81) in the AxB scale. The correct answer is 7.82.

Example: 4.5 / 7.8

  • connect with a ruler 4.5 of the AxB scale with 7.8 of the B scale;
  • read the answer (ca. 5.76) on the A scale. We know that the result of 4/8 is near 0.5, so we adjust the decimal place to get 0.576. The correct answer is 0.576

Now a division where the numerator is a square root:

  • to the left of 3.5 of the A2 scale we find on the A scale the square root of 350: 18.7;

  • now we connect 18.7 on the AxB scale with 1.51 of the A scale: on the B scale we can read the answer: ca. 12.39. A calculator would have been just a little more precise, finding 12.3896.

The slide rules work on the same principle, but this slight approximation has not prevented Von Braun to design space stations and send men on the Moon: calculating with this system is in fact less difficult than it sounds and the secret is just to be accurate and to practice.

Download the history of the nomography with all the graphic needed to calculate

Nicola Marras 2008


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