In every plane there is one special line called the line at infinity. The point where any other line in the plane cuts the line at infinity is called the point at infinity in that line. All other points in the line are called finite points. Two lines in the plane which meet the line at infinity in the same point are said to have the same direction, and two lines which meet it in different points to have different directions. Two lines which have the same direction cannot meet in any finite point [I. Axiom x.], and are parallel. Two lines which have different directions must intersect in some finite point, since, if produced, they meet the line at infinity in different points. This is a fundamental conception in Geometry, it is self-evident, and may be assumed as an Axiom (see Observations on the Axioms, Book I.). Hence we may infer the following general proposition:—“Any two lines in the same plane must meet in some point in that plane; that is—(1) at infinity, when the lines have the same direction; (2) in some finite point, when they have different directions.”—See Poncelet, Propriétés Projectives, page 52. ________________ NOTE B. legendre’s and hamilton’s proofs of euclid, I. xxxii. The discovery of the Proposition that “the sum of the three angles of a triangle is equal to two right angles” is attributed to Pythagoras. Until modern times no proof of it, independent of the theory of parallels, was known. We shall give here two demonstrations, each independent of that theory. These are due to two of the greatest mathematicians of modern times—one, the founder of the Theory of Elliptic Functions; the other, the discoverer of the Calculus of Quaternions. Legendre’s Proof.—Let ABC be a triangle, of which the side AC is the greatest. Bisect BC in D. Join AD. Then AD is less than AC [I. xix. Ex. 5]. Now, construct a new triangle AB′C′, having the side AC′ = 2AD, and AB′ = AC. Again, bisect B′C′ in D′, and form another triangle AB′′C′′, having AC′′ = 2AD′, and AB′′ = AC′, &c. (1) The sum of the angles of the triangle ABC = the sum of the angles of AB′C′ [I. xvi. Cor. 1] = the sum of the angles of AB′′C′′ = the sum of the angles of AB′′′C′′′, &c. (2) The angle B′AC′ is less than half BAC; the angle B′′AC′′ is less than half B′AC′, and so on; hence the angle B(n)A(n) will ultimately become infinitely small. (3) The sum of the base angles of any triangle of the series is equal to the angle of the preceding triangle (see Dem. I. xvi.). Hence, if the annexed diagram represent the triangle AB(n+1)C(n+1), the sum of the base angles A and C(n+1) is