This seems more reasonable than any of the various plans suggested, and it is found in so many practical geometries of the first century of printing that it seems to have long been a common expedient.
Practical geometries became very popular, and the reaction against the idea of mental discipline threatened to abolish the old style of text.
In the eighteenth century some excellent textbooks on geometry appeared in France, among the best being that of Legendre (1794), which influenced in such a marked degree the geometries of America.
Most geometries of any importance written since his time have been based upon Euclid, improving the sequence, symbols, and wording as occasion demanded.
In some elementary and in most higher geometries the perpendicular is called a normal to the plane.
Non-Euclidean Geometry: a discussion of geometries other than that of the space of experience.
The existence of such Euclidean quasi-geometries was first pointed out by Clifford.
But previously a description of the chief characteristic properties of elliptic and of hyperbolic geometries will be given, assuming the standpoint arrived at below under VII.
Thus analytical, though not actual, geometries exist for four and more dimensions.
These geometries will be called "Projective Geometry" and "Descriptive Geometry.
It begins with Arthur Cayley, who showed that metrical properties are projective properties relative to a certain fundamental quadric, and that different geometries arise according as this quadric is real, imaginary or degenerate.
Now the ordinary meaning of distance is required in non-Euclidean as in Euclidean Geometries--indeed, it is only in metrical properties that these Geometries differ.
It is evident, however, that the ordinary notion of distance has been presupposed in setting up the coordinate system, so that we do not really get alternative Geometries on one and the same plane.
Nevertheless, theGeometries of different surfaces of equal curvature are liable to important differences.
The Geometries of the plane and the cylinder, therefore, have much in common.
Geometries of the plane and the cylinder--are differences in projective properties[117].
Hence our Euclidean straight line, though it may serve to illustrate other Geometries than Euclid's, can only be dealt with correctly by Euclid.
It remains to discuss the manner in which non-Euclidean Geometries result from the projective definition of distance, as also the true interpretation to be given to this view of Metageometry.
Finally, the differences which appear between the Geometriesof different spaces of the same curvature--e.
The number of geometries compatible with these premises will be limited.
It may first be asked whether this reduction is possible, whether the number of necessary axioms and that of imaginable geometries are not infinite.
So vanishes the objection so far as two-dimensional geometries are concerned.
Since several geometries are possible, is it certain ours is the true one?
The two-dimensional geometries of Riemann and Lobachevski are thus correlated to the Euclidean geometry.
All the geometries I considered had thus a common basis, that tridimensional continuum which was the same for all and which differentiated itself only by the figures one drew in it or when one aspired to measure it.
These geometries of Riemann, in many ways so interesting, could never therefore be other than purely analytic and would not lend themselves to demonstrations analogous to those of Euclid.
But the difference between the two procedures is the difference between Euclidean and non-Euclidean geometries or the difference between perceptual space notions and conceptual space notions.
The real criterion then of all geometries is neither truth, conformability nor necessity, but consistency and convenience.
This celebrated postulate has proven to be the most fruitful ever devised; for it embodies in itself the possibility of threegeometries based respectively upon the following assumptions, namely: I.
In thesegeometries the sum of the angles of a triangle is not two right angles, and the departure from two right angles increases as the size of the triangle increases.
In other words there are alternative metrical geometries which all exist by an equal right so far as the intrinsic theory of space is concerned.
The above list will hopefully give you a few useful examples demonstrating the appropriate usage of "geometries" in a variety of sentences. We hope that you will now be able to make sentences using this word.
