Of course it is entirely possible to assume that through a point more than one line can be drawn parallel to a given straight line, in which case another type of geometry can be built up, equally rigorous with Euclid's. This was done at the close of the first quarter of the nineteenth century by Lobachevsky (1793-1856) and Bolyai (1802-1860), resulting in the first of several "non-Euclidean" geometries.[50] Taking the problem to be that of stating a reasonably small number of geometric assumptions that may form a basis to supplement the general axioms, that shall cover the most important matters to which the student must refer, and that shall be so simple as easily to be understood by a beginner, the following are recommended: 1. _One straight line, and only one, can be drawn through two given points._ This should also be stated for convenience in the form, _Two points determine a straight line_. From it may also be drawn this corollary, _Two straight lines can intersect in only one point_, since two points would determine a straight line. Such obvious restatements of or corollaries to a postulate are to be commended, since a beginner is often discouraged by having to prove what is so obvious that a demonstration fails to commend itself to his mind. 2. _A straight line may be produced to any required length._ This, like Postulate 1, requires the use of a straightedge for drawing the physical figure. The required length is attained by using the compasses to measure the distance. The straightedge and the compasses are the only two drawing instruments recognized in elementary geometry.[51] While this involves more than Euclid's postulate, it is a better working assumption for beginners. 3. _A straight line is the shortest path between two points._ This is easily proved by the method of Euclid[52] for the case where the paths are broken lines, but it is needed as a postulate for the case of curve paths. It is a better statement than the common one that a straight line is the shortest _distance_ between two points; for distance is measured on a line, but it is not itself a line. Furthermore, there are scientific objections to using the word "distance" any more than is necessary. 4. _A circle may be described with any given point as a center and any given line as a radius._ This involves the use of the second of the two geometric instruments, the compasses. 5. _Any figure may be moved from one place to another without altering the size or shape._ This is the postulate of the homogeneity of space, and asserts that space is such that we may move a figure as we please without deformation of any kind. It is the basis of all cases of superposition.