Geometrical Researches on the Theory of Parallels"Lobachevsky believed that another form of geometry existed, a non-Euclidean geometry, and this 1840 treatise is his argument on its behalf. Line by line in this classic work he carefully presents a new and revolutionary theory of parallels, one that allows for all of Euclids axioms, except for the last. This 1891 translation includes a bibliography and translator George B. Halsteds essay on elliptic geometry. Russian mathematician NICHOLAS LOBACHEVSKY (17921856) is best remembered as the founder (along with Janos Bolyai) of non-Euclidean geometry. He is also the author of New Foundations of Geometry (18351838) and Pangeometry (1855)." |
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acute angle angles equal angles opposite assume axes AA axioms axis BF BC the perpendicular Bolyai boundary line boundary-surface CD somewhere circle congruent consequently cos 7(a COSIMO curvature cut the line dicular draw end-points equal angles equations erected Euclid Euclidean geometry Gauss Greek Hence hypothenuse imaginary geometry intersection point isosceles triangle Kasan let fall line AA line AB line CD line DC Lobachevski mid-point Non-Euclidean opposite angles opposite triangles Parallel lines parallel to CD parallels AA pendicular perpen perpendicu perpendicular HK perpendicular sides pertain plane geometry point F prolongation quadrilateral random rectilineal triangle right angles right-angled triangles small soever soever the angle somewhere in G space sphere spherical triangle spherical trigonometry straight line Theo Theorem 16 Theorem 23 Theorem 9 Theory of Parallels three angles triangle ABC Fig triangle equal triangle the sum whence follows
Popular passages
Page 6 - If a straight line meets two straight lines, so as to " make the two interior angles on the same side of it taken " together less than two right angles...
Page 13 - DAK = fl (p) will lie also a line AK, parallel to the prolongation DB of the line DC, so that under this assumption we must also make a distinction of sides in parallelism.
Page 39 - In any triangle, the sum of the three angles is equal to two right angles, or 180°.
Page 14 - K'AE, H'AE', to the nonintersecting. In accordance with this, for the assumption H(p) = ^ir, the lines can be only intersecting or parallel; but if we assume that !!(/>) < ^JT, then we must allow two parallels, one on the one and one on the other side; in addition, we must distinguish the remaining lines into nonintersecting and intersecting. For both assumptions, it serves as the mark of parallelism that the line becomes intersecting for the smallest deviation toward the side where lies the parallel,...
Page 9 - ... without the lines meeting. A perfectly consistent and elegant geometry then follows, in which the sum of the angles of a triangle is always less than two right angles, and not every triangle has its vertices concyclic.
Page 13 - All straight lines which in a plane go out from a point can, with reference to a given straight line in the same plane, be divided into two classes — into cutting and not-cutting. The boundary lines of the one and the other class of those lines will be called parallel to the given line.
Page 13 - AF, to the not-cutting lines, as AG, we must come upon a line AH, parallel to DC, a boundary line, upon one side of which all lines AG are such as do not meet the line DC, while upon the other side every straight line AF cuts the line DC. "The angle HAD between the parallel HA and the perpendicular AD is called the parallel angle (angle of parallelism), which we will here designate by H(p) for AD = p. "If n(/>) is a right angle, so will the prolongation AE...
Page 48 - At once the supposed fact that our space does not interfere to squeeze as or stretch us whea we move, is envisaged as a peculiar property of our space. But is it not absurd to speak of space as interfering with anything? If you think so, take a knife and a raw potato, and try to cut it into a seven-edged solid...
Page 11 - In geometry I find certain imperfections which I hold to be the reason why this science, apart from transition into analytics, can as yet make no advance from that state in which it has come to us from Euclid.
Page 3 - It is well known that Euclid's twelfth axiom, even in Playfair's form of it, has been considered as needing demonstration, and that Lobatschewsky constructed a perfectly consistent theory, wherein this axiom was assumed not to hold good, or, say, a system of non-Euclidian plane geometry.
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Bibliographic information
| Title | Geometrical Researches on the Theory of Parallels |
| Author | Nicholas Lobachevsky |
| Translated by | George B. Halsted |
| Edition | reprint |
| Publisher | Cosimo, Inc., 2007 |
| ISBN | 1602064679, 9781602064676 |
| Length | 52 pages |
| Export Citation | BiBTeX EndNote RefMan |