LUNATION, the period or time he, tween one new moon and another : it is also called the synodical month, consist ing of 29d .12h 44' 3" 11-3ds; exceeding the periodical month by 2d 5h 0' 55". LUNE, in mathematics, is a geometri ral figure, in form of a crescent, termi mted by the arcs of two circles that in terstct each other within. Though the quadsature of the whole circle has never been aected, yet many of its parts have been sqtiqed. 'The first of these partial quadraturcs 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 ex pressly says, r; was found out by Oenopi das of the same 'Mace. The lune of Hip pocrates is this : let A B C, Plate IX. Miscel. fig. 7, be a s•:mi-circle, having its centre E; and ADC a quadrant, having its centre P ; then the gore ABCD A. contained between the arcs of the semi circle and quadrant, is his lune ; and it is equal to the right-angled triangle A C F, as is thus easily proved. Since A P.= 2 A E', that is, the square oc the radius of the quadrant equal to double the square of the radius of the senui•circle ; therefore the quadrant-area, ALCF A, is = the semicircle of ABGE A; from each of these take away the common space ADCE A, and there remains the triangle A C F = the lune A BCD A.
Another property of this lune, which is the more general one of the former, is, that if F G be any.line drawn from the point F, and A II perpendicular to it ; , then is the intercepted part of the lune A GI A == the triangle A G II, cut off by the chord line A G; or, in general, that the small segment, A K G A, is equal to the tri-lineal A I ft A. For, the angle A F G being at the centre of the one cir cle, and at the circumference of the other, the arcs cut off A G, A I are similar to the wholes A li C, A D C, therefore the small segment AK G A is to the semi segment A I H, as the whole semi-circle A B C A to the semi-segment or quadrant A D C F, that is, in a ratio of equality. Again, if A B C (fig. 8) be a triangle, right-angled at C, and it semi-circles be described on the three sides as diameters ; then the triangle T (A B C) is equal to the sum of the two limes L 1, L 2. For the greatest semi-circle is equal to the sum of both the other two ; from the greatest semi-circle take away the seg ments S 1, and S 2, and there remains the triangle T; also from the two less semi circles take away the same two segments S 1 and S 2, and there remains the two tunes L 1, and L 2; therefore the trian gle T = L 1 + L 2, the two tunes.