L U N E. 



two fides AD, D B, become equal, as in Jig. 3, the two 

 luncs are each equal to half that triangle, and confequently 

 the q'.iadratiire of them is determined, being each equal to 

 a given redilineai figure ; and this is what is properly called 

 the lune of Hippocrates, and it was the only one of which he 

 could determine the area, for though he, in all cafes, had the 

 nieafure of the fpace of botli together, yet it was only in the 

 cafe of equality that he could fnid the area of the fingle lune, 

 though he could r.Kvays determine a lune that fhould be equal 

 to any given reftilineal fpace. For in_/f 1^.-5, the arc DEB 

 is a quarter of a circle to the radius C B, andD H B is a 

 femicircle. If, therefore, we conftrudl the il ofceles right- 

 angled triangle B C A, ifg-'i-) equal to any given fpace, and 

 on A B defcribe the femicircle B D A, and from C as a 

 centre, and with C A as a radius, defcribe the quadrant 

 B E A, we (hall have the lune B D E A equal to the given 

 fpace as required. This, as we have obferved abo/e, was 

 the firll inilance of the quadrature of a curvilineal fpace, 

 that is, of its being fhewn equal to a reftilineal figure ; for, 

 properly fpeaking, it i-; not abfolutcly a quadrature, as was 

 that of Archimedes, when he demonilrated that every para- 

 bola was two-thirds of its circumfcribing rcdtangle ; Hip- 

 pocrates arriving at his refult only flep by (lep, by fubtradl- 

 ing equal quantities from equal fpaces, and hence finally, as 

 by chance, coming to a cafe in which a curvilineal area is equal 

 to a retlilineal one. 



This dilcovery of Hippocrates, it feems, ir.fpired him 

 with great confidence of being able to find the meafure of 

 the circle itfelf ; and the reafoning which has been attributed 

 to him on this fubjeft, though very erroneous, is Itill ex- 

 tremely plaufible. Hippocrates fuppofed a femicircle 

 A D E B {fig. 4. ) in which he drew the three chords or radii 

 A D, D E, E B, and on each of thefe chords he defcribed 

 a femicircle and a fourth, as F, equal to them. Then the four 

 femicircles A G D, D E H, EI B, and F, being each 

 equal to a quarter of the femicircle A D E B, they are 

 therefore together equal to it, and taking away from each 

 the faiall fegments A G D, D H E, E I B, we (hall have 

 on one fide the rectilineal figure A D E B, equal to the 

 three hines, together with the ftmicircle F. If, therefore, 

 the area of the lunes be taken away from the redlilineal 

 A D E B, there will remain the area of the femicir- 

 cle F, equal to a given reflilineal fpace. This reafoning, 

 however, though ingenious, is iWl very dcfedive, in confe- 

 quence of the lunes em ployed in this cale bemg dilTcrent from 

 thofe of vi-hieh Hippocrates had found the quadrature, for 

 that, as we have feeu, is bounded by a quadrant of one cir- 

 cle, and the half of another, whereas thofe in the above 

 figure are bounded by a femicircle, and the fixth part of 

 another circle, which is very different from the former, and 

 the quadrature of it equally as difficult as that of the circle 

 itfelf. All, therefore, that Hippocrates could draw from his 

 i".veilii;ation, was merely this, that if any geometer (hould be 

 able to find the area of thofe lunes, the quadrature of the 

 crcle would neceffctrily follow, and as this problem was not 

 at that time thought fo difficult as it is now known to be, it 

 is not improbable that confiderablc hopes of fuccefs were en- 

 tertained after the difcovcry which this able geometer had 

 made of the polTibility of fquaring what is indeed apparently 

 a more complex figure than the circle. In faift the quadra-' 

 ture of the circle might be accomplilhed, if we only knew 

 the ratio of the two luncs, defcribed as in Jig. ?. ; for then 

 knowing the fum of the two, and their ratio, it is obvious 

 that we fhould have the real area of each, and confequently, 

 by taking A D equal to the radius B C, the area of the cir- 

 cle would follow, as we have (liewn above. 



Bill though Hippocrates and the ancient geometers were 



unable to fquare any other lune, except that above-men- 

 tioned, yet the moderns have found feveral other cafes in 

 which the quadrature may be obtained, as alfo certain por- 

 tions of them cut off by right lines, drawn in certain direc- 

 tions. In the lune of Hippocrates, the radii of the bound- 

 ing circles are to each other in the ratio of two to one ; but if 

 the two circles are to each other as three to one, or as 

 three to two, or as five to one, or five to three, they may 

 alfo be fquared, or may be conlfrufted equal to given fpaces, 

 by means of the fimplc elements of geometry ; but other ra- 

 tios, as four to one, fix to one, feven to one,' Sec. require 

 the affiftance of the higher geometry, being of a fimilar 

 dcfcription of problems to thofe of trifeftiiig an angle, 

 doubling a cube, &c. ; and can only be folved by the fame 

 means. 



We fliall take this opportunity of giving a fummary of 

 fome of the mod curious obfervations, added by modern 

 geometers to the difcovery of Hippocrates. 



1. If from the centre F, {Jig. J.) there he drawn any 

 ftraight line whatever F E, cutting off the portion of the 

 lune AEG i\, th^t portion will be qnadrable, and equal 

 to the reftilineal triangle A H E. For it may be readily 

 demonilrated, that the fegment A E will be equal to the fc- 

 mi-fegment A G H. 



2. From the point E, if EI be let fall perpendicularly on 

 A C, and F I and E F be drawn, the fame portion of 

 the lune A E G A will be equal to the triangle A F I. 

 For it may be eafily demonilrated, that the triangle A F I is 

 equal to the triangle A H E. 



3. The lune, therefore, may be divided in a given ratio, by 

 a line drawn from the centre F ; nothing more being necel- 

 fary than to divide the diameter A C in fuch a manner, that 

 A I fhall beto C I in that ratio ; to raife E 1 perpendicular 

 to A C, and to draw the line E F : then the two fegments 

 of the lune AGE and G E C will be in the ratio of A I 

 to CI. 



All thefe remarks were firfl made by M. Artus de Lionnc, 

 bifliop of Gap, who publifhed them in a work, entitled 

 '•■ Curvilincorum Amsenios Contemplatio," 1654, 410., and 

 afterwards the following' were added by other geometers. 



. 4. If two circles, forming the lune of Hippocrates, be 

 completed, the refult will be another lune, which may be 

 called the conjugate to the for.mer, and in whicli mixlilineal 

 fpaces may be found, which may be fquared as in the pre- 

 ceding cafes. 



From the point F, if there be drawn any radius F M, in- 

 terfedling the two circles in R and M, we fhall have the 

 mixtilineal fpace R A M R equal to the reftilineal triangle 

 L A R ; which can be eafily demonilrated ; for it may be 

 readily feen that the fegment A R, of the fmall circle, is 

 equal to the femi-fcgment L A M of the greater. 



6. Hence, if the diameter in O touch the fmall circle in 

 F, it follows that the mixtilineal fpace A R F n; A will be 

 equal to the triangle A S F, right-angled at S, or to half 

 the lune A G C B A. We might have added here various 

 other properties relating to lunes and their fegments, but 

 our limits will not admit of it ; we muil therefore refer the 

 curious reader to Ozamam's " Mathematical Recreations," 

 where the fuhjeft is amply ilhiilrated See alfo the remarks 

 of David Gregory, Calwell, and Wailis, on the quadra- 

 ture of the lunula, in Phil. Tranf. N' 2^^, or vol. iv. 

 p. 452, New Abridgment ; and for " the Umieiifions of the 

 folids generated by theconverfion of the lunula of Plippo- 

 crates, and of its parts about feveral axes, with the iuc- 

 faces generated by that converfion," fee DeMoivre's paper 

 in the Philofophical Tranfadlions, N'' 205, or vol. iv. p. 505, 

 New Abridgment. 



LUNELLE- 



