Footnotes
The more general notion of anharmonic ratio, which includes the harmonic ratio as a special case, was also known to the ancients. While we have not found it necessary to make use of the anharmonic ratio in building up our theory, it is so frequently met with in treatises on geometry that some account of it should be given.
Consider any four points, A, B, C, D, on a line, and join them to any point S not on that line. Then the triangles ASB, GSD, ASD, CSB, having all the same altitude, are to each other as their bases. Also, since the area of any triangle is one half the product of any two of its sides by the sine of the angle included between them, we have
Now the fraction on the right would be unchanged if instead of the points A, B, C, D we should take any other four points A', B', C', D' lying on any other line cutting across SA, SB, SC, SD. In other words, the fraction on the left is unaltered in value if the points A, B, C, D are replaced by any other four points perspective to them. Again, the fraction on the left is unchanged if some other point were taken instead of S. In other words, the fraction on the right is unaltered if we replace the four lines SA, SB, SC, SD by any other four lines perspective to them. The fraction on the left is called the anharmonic ratio of the four points A, B, C, D; the fraction on the right is called the anharmonic ratio of the four lines SA, SB, SC, SD. The anharmonic ratio of four points is sometimes written (ABCD), so that
If we take the points in different order, the value of the anharmonic ratio will not necessarily remain the same. The twenty-four different ways of writing them will, however, give not more than six different values for the anharmonic ratio, for by writing out the fractions which define them we can find that (ABCD) = (BADC) = (CDAB) = (DCBA). If we write (ABCD) = a, it is not difficult to show that the six values are
The proof of this we leave to the student.
If A, B, C, D are four harmonic points (see Fig. 6, p. *22), and a quadrilateral KLMN is constructed such that KL and MN pass through A, KN and LM through C, LN through B, and KM through D, then, projecting A, B, C, D from L upon KM, we have (ABCD) = (KOMD), where O is the intersection of KM with LN. But, projecting again the points K, O, M, D from N back upon the line AB, we have (KOMD) = (CBAD). From this we have
(ABCD) = (CBAD),
or
whence a = 0 or a = 2. But it is easy to see that a = 0 implies that two of the four points coincide. For four harmonic points, therefore, the six values of the anharmonic ratio reduce to three, namely, 2,
, and -1. Incidentally we see that if an interchange of any two points in an anharmonic ratio does not change its value, then the four points are harmonic.
Fig. 49
Many theorems of projective geometry are succinctly stated in terms of anharmonic ratios. Thus, the anharmonic ratio of any four elements of a form is equal to the anharmonic ratio of the corresponding four elements in any form projectively related to it. The anharmonic ratio of the lines joining any four fixed points on a conic to a variable fifthpoint on the conic is constant. The locus of points from which four points in a plane are seen along four rays of constant anharmonic ratio is a conic through the four points. We leave these theorems for the student, who may also justify the following solution of the problem: Given three points and a certain anharmonic ratio, to find a fourth point which shall have with the given three the given anharmonic ratio. Let A, B, D be the three given points (Fig. 49). On any convenient line through A take two points B' and D' such that AB'/AD' is equal to the given anharmonic ratio. Join BB' and DD' and let the two lines meet in S. Draw through S a parallel to AB'. This line will meet AB in the required point C.
Pappus, Mathematicae Collectiones, vii, 129.
J. Verneri, Libellus super vigintiduobus elementis conicis, etc. 1522.
Kepler, Ad Vitellionem paralipomena quibus astronomiae pars optica traditur. 1604.
Desargues, Bruillon-project d'une atteinte aux événements des rencontres d'un cône avec un plan. 1639. Edited and analyzed by Poudra, 1864.
The term 'pole' was first introduced, in the sense in which we have used it, in 1810, by a French mathematician named Servois (Gergonne, Annales des Mathéématiques, I, 337), and the corresponding term 'polar' by the editor, Gergonne, of this same journal three years later.
Euler, Introductio in analysin infinitorum, Appendix, cap. V. 1748.
Œuvres de Desargues, t. II, 132.
Œuvres de Desargues, t. II, 370.
Œuvres de Descartes, t. II, 499.
Œuvres de Pascal, par Brunsehvig et Boutroux, t. I, 252.
Chasles, Histoire de la Géométrie, 70.
Œuvres de Desargues, t. I, 231.
See Ball, History of Mathematics, French edition, t. II, 233.
Newton, Principia, lib. i, lemma XXI.
Maclaurin, Philosophical Transactions of the Royal Society of London, 1735.
Monge, Géométrie Descriptive. 1800.
Poncelet, Traité des Propriétés Projectives des Figures. 1822. (See p. 357, Vol. II, of the edition of 1866.)
Gergonne, Annales de Mathématiques, XVI, 209. 1826.
Steiner, Systematische Ehtwickelung der Abhängigkeit geometrischer Gestalten von einander. 1832.
Von Staudt, Geometrie der Lage. 1847.
Reye, Geometrie der Lage. Translated by Holgate, 1897.
Ball, loc. cit. p. 261.