X: measure

 

content

Roots of the meaning

Various measures

Definitions for „module“, „modulus“ and „modulo“

E. g. Elasticity modulus

Module in abstract algebra

Modulor

insert: Module in architecture

 

 

 

Roots of the meaning

 

The roots of these meanings go back on the Greek word "metron" (measure, yardstick, boundary) and Latin "modus" (respectively in the reduction form: "modulus"). The primary meaning is measure in a double meaning, as unit (content) and as measuring instrument (yardstick).

 

“Modulus” occurs shortly after 40 BC with Horaz and Varro; then it was used various times by the well-known architect and military engineer Vitruvius in his "Book on Architecture" (approx. 23 BC), usually as architectural standard, namely the radius of a column. Vitruvius used also „modulatio“ for the calculation of measurements from a standard unit.

 

„Module“ has been introduced in 1547 in French and in 1583 (or 1586) in English. „Modulus“ was introduced in 1563 in English.

 

 

Various measures

 

In Randle Cotgraves Dictionary French-English (1611) we read:

„Module (m.):

that whereby a whole worke is measured, proportioned or squared;

also, the measure, bignesse, or quantitie of a thing;

also, a certaine measure in conduits or conveyances of water;

also, modulation, melodie, or measure in Musicke.”

 

For precise evidence of „Earliest Known Uses“ of „modulus“ - and „modular“ - in English see:

http://www.leidenuniv.nl/fsw/verduin/stathist/1stword.htm

http://jeff560.tripod.com/m.html

 

 

Definitions for „module“, „modulus“ and „modulo“

 

In modern dictionaries we find “module” in a general sense as well as in architecture, technology, mathematics and psychology (for further meanings see: chap. XXV: Original &/ or copy – component):

 

1. in general:

a) a model or measure

b) a standard or unit for measuring

 

2. in architecture:

a) the size of some one part taken as a unit of measure by which the proportions of an architectural composition are regulated

b) a measure of proportion among the parts of a classical order, the size of the diameter or semidiameter of the base of a column shaft usually being taken as a unit

 

3. in technology (gear wheels):

a) the effective diameter divided by the number of teeth (of metric-sized gear wheels).

b) the diameter of the pitch circle in millimeters divided by the number of teeth

 

4. in mathematics:

a) a subset of an additive group that is also a group under addition

b) a mathematical set that is a commutative group under addition and that is closed under multiplication which is distributive from the left or right or both by elements of a ring

 

5) in psychology:

one of the inherent cognitive or perceptual powers of the mind

 

For “modulus” we find:

 

1. in mathematics:

a) the logarithm of e to the base 10

b) the factor by which a logarithm of one base must be multiplied to become the logarithm of another base

c) a number by which two other numbers can be divided so that both give the same remainder

d) the absolute value of a complex number, computed by adding the squares of each element, then finding the square root of the resulting sum

e) the magnitude of a number irrespective of whether it is positive or negative

 

2. in physics:

a) a coefficient expressing the degree to which a substance exhibits a particular property

b) a number, coefficient, or quantity that measures a force, function, or effect: modulus of elasticity.

c) a constant factor relating a physical effect to the force producing it

 

 

For „modulo“ we find in the „Oxford English Dictionary“:

 

A. prep.

a. With respect to a modulus of (a specified value).

b. In extended use.

(a) With respect to an equivalence defined by (some feature), disregarding differences indicated by (some unimportant feature);

(b) taking into account (a particular consideration, aspect, assumption, etc.).

B. adj. (attrib.).

Modular; involving calculations with respect to a modulus.

 

 

E. g. Elasticity modulus

 

Aristotle, in his “Mechanica Problemata”, describes studies on the static behavior of geometrically similar bars with respect to torsion (Moritz Weber, 1919, 364).

 

The first attempts to measure the elasticity of bars were made by Galilei (1638 – for an illustration see John Desmond Bernal (1970, 405), for an extensive description see Helga Portz, 1994).

Not long after, the versatile physicist and inventor Robert Hooke discovered the law of elasticity (1660): A material body is deformed linearly with respect to the forces or load affecting it. In 1807 the versatile physicist and physician Thomas Young devised a measure for the stiffness of a material, later called “Young’s modulus”.

 

 

Module in abstract algebra

 

Around 1900 in the field of mathematics started the so-called “abstract algebra”, dealing with algebraic structures as magmas and groups, rings and fields, modules and vector spaces.

In the American Mathematical Monthly (34, 1927, 64) E. T. Bell wrote: “The class concept was introduced by Dedekind [1871] as follows. A set M of elements of I(a) which is closed under subtraction, and hence under addition and subtraction, is called a module. If M be such that, if b is any element of M, and g is any element of I(a), then bg is an element of M, the module M is called an ideal. Hence an ideal of (a) is closed under addition and subtraction and multiplication by elements of I(a).”

 

The Deutsche Bibliothek in Frankfurt has more than 150 German titles on modules and rings 1960-2000.

A good Handbook is written by Robert Wisbauer (1988; English 1991); Günter Geisler wrote a thesis: „Zur Modelltheorie von Moduln“ (1994). In English Mike Y. Prest published “Model Theory and Modules” (1988).

Among others published a “Module theory”: Thomas Scott Blyth (1977), Carl Clifton Faith and Sylvia Wiegand (1979), Alberto Facchini (1998) and Michael E. Keating (1998).

 

 

Modulor

 

Inspired by Vitruvius and Leonardo da Vinci the famous Swiss architect Le Corbusier developed in the years 1942-48 the “Modulor”, an architectural yardstick, based on a human figure of 1.83 meter (6 feeet).

 

 

Bibliography

Allgemeine Nachschlagewerke/ dictionaries and encyclopedias

model: special topics

Mathematische Modelle