considered Euclidean, all characterized by apparently rigorous, stylized deduction from first
principles, are discussed here.
Data (“Things Given”) is mentioned first by Pappos (Collection 7.3, p. 636.18–19 H.) in his
list of works useful for analysis, that is, the finding of solutions to problems and of proofs of
propositions, by supposing that what has to be done is accomplished or that what is to be
proved is true, and asking what else must be accomplished or true as a result: the idea is that
when one reaches things one knows how to accomplish or prove, one will be able to reverse
the steps and produce a solution or proof for what is sought.
Optics is essentially a treatise on monocular perspective. It is assumed that vision is a
matter of the emission of rectilinear rays from the eye which strike an object and form a
cone with vertex in the eye and base a plane figure determined by the shape of the object
seen, and that the relative apparent size of an object is determined by the size of the angle
“under which” it is seen and its relative apparent position by the relative position of the rays
under which it is seen; the rays are treated as discrete straight lines, so that an object will not
be seen if it falls between rays.
Catoptrics takes the same approach to mirror vision, treating plane, convex, and concave
mirrors.
Phenomena is an essay in very elementary geometric astronomy, the main point of which
seems to be showing that certain astronomical appearances can be represented and under-
stood geometrically. In the prologue simple astronomical data are invoked to justify the
claim that the sphere of the fixed stars rotates uniformly about a fixed axis and that the eye
of an observer is at the center of the sphere, and geometrical definitions are given of such
astronomical terms as “horizon,” and “meridian.” Among the theorems proved are the
assertion that if two stars lie on a great circle which has no point in common with the arctic
circle (the circle including all stars that are never seen to set), the one which rises earlier sets
earlier (prop. 4).
The name of “Euclid” is associated first and foremost with
the Elements, apparently a single treatise in which propositions
are derived from principles labeled as “definitions,” “postu-
lates,” and “common notions” (the last frequently called
axioms). Careful scholarship of the last century has made
clear that the work is a compilation based on several sources.
The subject of book 1 is the geometry of plane rectilinear
figures. The book is noteworthy for avoiding the use of pro-
portions and for postponing the use of the parallel postulate
until it is required. Book 2 introduces what is now frequently
called geometric algebra in a series of geometric propositions
corresponding to what we know as algebraic equations; for
example, proposition 2, which corresponds to “(x+y)^2 = x^2 + y^2
- 2xy,” says that if AGB is a straight line, the square with side
equal to AB [SQ(AB)] is equal to SQ(AG) plus SQ(BG) plus two times the rectangle with
sides equal to AG and BG.
Book 3 treats circles and their relations to straight lines and angles, Book 4 the inscription
in circles and circumscription about circles of rectilinear figures. Book 5 brings in pro-
portionality, developing a theory based on a definition which says of four magnitudes A, B,
C, D that A:B :: C:D if and only if for any multiples m·A, n·B, m·C, n·D of those magnitudes,
if m·A is greater than, equal to, or less than n·B, m·C is accordingly greater than, equal to, or
Euclid’s geometric alge-
bra (1. prop. 2) © Mueller
EUCLID OF ALEXANDRIA