THE ELECTRONIC JOURNAL OF COMBINATORICS (ed. March 2001), DS #5.

Venn Diagram Survey
Euler Diagrams

Euler diagrams predate Venn diagrams, but are distinct. They were introduced by Euler to aid in the understanding of syllogisms. In that context they have certain deficiencies, some of which Venn tried to rectify with Venn diagrams.

Our formal working definition of an Euler diagram is found below. Let C = { C₁, C₂, ..., C_n } be a collection of finitely intersecting simple closed curves drawn in the plane. The collection C is said to be an Euler diagram if the intersection of X₁, X₂, ..., X_n is connected, where each X_i is either int(C_i ) (the interior of C_i ) or is ext(C_i ) (the exterior of C_i ). If infinite intersections are allowed we refer to the diagrams as iEuler diagrams.

In other words, an Euler diagram is like a Venn diagram except that intersections of curve interiors are allowed to be empty. From this point of view a Venn diagram is a particular type of Euler diagram. As used in the analysis of syllogisms, each region of a Venn diagram is shaded (in red in the table below) according to whether the region can contain any members. In this way any set system can be represented by a Venn diagram. In an Euler diagram a region is present if and only if it contains at least one member; there is no shading. As a consequence not all set systems may be represented by Euler diagrams.

Euler wrote four letters in February of 1761 which contain Euler diagrams as defined above (from the Brewster translation) [Eu].

LETTER CII. Of the Perfections of a Language. Judgements and Nature of Propositions, affirmative and negative; universal or particular.
LETTER CIII. Of Syllogisms, and their different Forms, when the first Proposition is universal.
LETTER CIV. Different Forms of Syllogisms, whose first Proposition is particular.
LETTER CV. Analysis of some Syllogisms.

Euler always drew his diagrams as collections of circles, but remarks in Letter CIII, These circles, or rather these spaces, for it is of no importance of what figure they are of,.... It is to be noted that Euler never made use of the familiar 3 circle Venn diagram.

Chapter V of John Venn's book Symbolic Logic contains an explanation and comparison of Venn and Euler diagrams [Ve81]. The second part of his historical notes contains much information about the history of diagrammic reasoning.

In the table below we show the corresponding set system, Venn diagram, and Euler diagram for 2-sets. Up to a relabelling of sets, all possibilities are shown that include the empty set.

Set system	Venn diagram	Euler diagram	Comments
{Ø,A,B,AB}
{Ø,B,AB}
{Ø,A,B}
{Ø,AB}			This is an iEuler diagram.
{Ø,B}
{Ø}		.

THE ELECTRONIC JOURNAL OF COMBINATORICS (ed. March 2001), DS #5.