LUNE

, or Lunula, or little moon, is a geometri-| cal figure, in form of a crescent, terminated by the arcs of two circles that intersect each other within.

Though the quadrature of the whole circle has never been effected, yet many of its parts have been squared. The first of these partial quadratures was that of the Lunula, given by Hippocrates of Scio, or Chios; who, from being a shipwrecked merchant, commenced geometrician. But although the quadrature of the Lune be generally ascribed to Hippocrates, yet Proclus expressly says it was found out by Oenopidas of the same place. See Heinius in Mem. de l'Acad. de Berlin, tom. ii. pa. 410, where he gives a dissertation concerning this Oenopidas. See also Circle, and Quadrature.

The Lune of Hippocrates is this: Let ABC be a semicircle, having its centre E, and ADC a quadrant, having its centre F; then the Figure ABCDA, contained between the arcs of the semicircle and quadrant, is his Lune; and it is equal to the right-angled triangle ACF, as is thus easily proved. Since , that is, the square of the radius of the quadrant equal to double the square of the radius of the semicircle; therefore the quadrantal area ADCFA is = the semicircle ABCEA; from each of these take away the common space ADCEA, and there remains the triangle ACF = the Lune ABCDA.

Another property of this Lune, which is the more general one of the former, is, that if FG be any line drawn from the point F, and AH perpendicular to it; then is the intercepted part of the Lune AGIA = the triangle AGH cut off by the chord line AG; or in general, that the small segment AKGA is equal to the trilineal AIHA. For, the angle AFG being at the centre of the one circle, and at the circumference of the other, the arcs cut off AG, AI are similar to the wholes ABC, ADC, therefore the small seg. AKGA is to the semisegment AIH, as the whole semicircle ABCA to the semisegment or quadrant ADCF, that is in a ratio of equality.

Again, if ABC (fig. 2) be a triangle, right angled at C, and if semicircles be described on the three sides as diameters; then the triangle T (ABC) is equal to the sum of the two Lunes L1, L2. For, the greatest semicircle is equal to the sum of both the other two; from the greatest semicircle take away the segments S1 and S2, and there remains the triangle T; also from the two less semicircles take away the same two segments S1 and S2, and there remains the two Lunes L1 and L2; therefore the triangle the two Lunes.