Talk:Geometry

Edit to Notes

Note 3 full citation is Greek and Vedic Geometry Frits Staal Journal of Indian Philosophy 27 (1/2):105-127 (1999)

Geometry is, along with arithmetic, one of the oldest branches of mathematics

Can someone point out what this "arithmetic" branch actually contains? Arithmetic is maybe a term used 2000 years ago for something we'd classify now as closer to computer science than mathematics (counting, computing), but now it means some system of operations that is a part of common algebraic structures as rings or fields 83.24.88.130 (talk) 06:32, 13 August 2025 (UTC)

In this context "arithmetic" more or less means "doing calculations with numbers". –jacobolus (t) 06:33, 13 August 2025 (UTC)

In Euclid's Elements § Books VII to X: Number theory, arithmetic refers to number theory. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 08:12, 13 August 2025 (UTC)

When this article says "Geometry is, along with arithmetic, one of the oldest branches of mathematics", it does not mean "number theory". If we wanted to say number theory, we would just use that name. –jacobolus (t) 18:22, 13 August 2025 (UTC)

5th Postulate

@D.Lazard and 47k500and8edits for christmas: Edit permalink/1323405246 reverted the text which says: if angles m∠α + m∠β < 180°, then the two lines across from each other will intersect in § Axioms with the comment Wrong statement of the parallel postulate. The Fifth Postulate is

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

The only thing I see wrong is not mentioning that they are on the same side.

Also, Euclid introduced certain axioms, or postulates, should mention common notions; the modern term axiom encompasses Euclid's common notions and postulates. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:02, 21 November 2025 (UTC)

This formulation does not implies that if the two angles sums to two right angles, then the lines do not intersect. This fact is fundamental in the parallel postulate, as the formulation you quote remains valid in spherical geometry, where the intersection exists even if the sum of the two angles equal 180° (take the equator and two meridians). In fact, a correct formulations of the postulate would be

If a line segment intersects two straight lines, the two interior angles on the same side sum to less than two right angles if and only if the two lines, if extended indefinitely, meet on that side.

I guess that the wrong formulation results from a mistranslation due to the fact that the distinction between "if" and "if and only if" was not always clear at that time.

In any case, Euclid postulated the existence of parallel lines, and without the "if and only if" of my formulation, this existence cannot be proved, as shown by spherical geometry. D.Lazard (talk) 18:05, 21 November 2025 (UTC)

References

Organization

Currently, § Non-Euclidean geometry is under § Differential geometry, even though it is more closely related to § Euclidean geometry. Meanwhile, several important geometries are missing or only referred to incidentally. I suggest adding some or all of

Linear geometries

Overview

Absolute geometry

Affine geometry

Elliptic geometry

Hyperbolic geometry

Minkowski space

Projective geometry

Spherical geometry

Differential geometry

Symplectic geometry

I'm not sure whether to keep Euclidean geometry as a full section, or make it a subsection of the new Linear geometries. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:57, 20 March 2026 (UTC)