I s o 



TSOLACCIO, a town of the idand of Corfica ; jS 

 miles N. of Porto Veccliio. 



ISOLEPIS, in Botany, from wo,-, equal or uniform. End 

 ^■-;,-, Tifcak. a s^enus feparated from Sclrpus by Mr. Brown, 

 in his Prodr. Nov. Holl. v. i. 23 1, on account of the want 

 of briftles at the bafe of the germen or feed, by which alone 

 it is diftinguilhed ; a mark, if conitant, certainly fuflicient 

 in fo difficult a tribe. Twelve Now Holland fpccies are 

 defined by this author, amongit which are Stirpus Jlu'itans, 

 fupinus, fdaccus, and cap'iUaris of Liimsus ; nodofut and pro- 

 lifer of Rottboll ; with fix never before defcribed. 



ISOLETTA, iH Geography, a town of Italy, in the 

 department of the Mela ; 15 miles S. of Drefcia. 



ISOMERIA, formed of ij-o;, equal, and pcfi-r, part, in 

 Algebra, a method of freeing an equation from fraftions, by 

 reducing all the fractions to one common denominator, and 

 then multiplying each member of the equation by that com- 

 mon denominator. 



Tins amounts to the fame with what is otherwife 

 called converfion of equations. See CoNVEnsiON of Equa- 

 tions. 



ISONA, in Geography, a town of Spain, in Catalonia; 

 24 miles N. of Bala^ucr. 



ISONEM A, in Botany, from tm.-, equal, and tr.yn, a threaA 

 or Jlamen. Brown. Mem. of the Wernerian Society, v. r. 



63 Clafs and order, Pentandria Monogyr.ia. Nat. Ord. 



Contorts, Linn. Apoc'mex, Brown. 



Ell'. Ch. Corolla falver-fliaped ; its mouth and tube with- 

 out foales ; limb in five deep fegrnents. Stamens prominent ; 

 filaments inferted into the mouth, fimple at the top ; anthers 

 arrow-(haped, adhering to the iiigma by their middle part. 

 Germens two ; llyle one, thread-fnaped ; ftigma thick, ob- 

 tufe. No fcales beneath the germen. Pouches . . . 



The above charaAers are taken from a (hrub gathered on 

 the African coaft, near Sierra Leone, by Smeathman, and 

 prcferved in the Bankfian herbarium. It is hairy, appa- 

 rently erect, with oppofite lea'oe:. Pvnicle terminal, oppo- 

 fitely divided, corymbofe. Leaves of the calyx with a 

 double fcale at their bafe on the infide. Tube of the corolla 

 half an inch long, cylindrical, bearded in the middle within 

 Brown. 



ISOPERIMETRY, in Malhemaiics, is a branch of the 

 higher geometry, which treats of the properties of ifoperi- 

 metrical figures, vh. of furfaccs contained under equal peri- 

 meters ; of folids under equal furfaces ; curves of equal 

 lengths, &c. Of the foregoing heads, the two firll may 

 be confidered as containing the elements of the fcience, which 

 relate principally to the maxima et minima of different fur. 

 faces and folids, when bounded by figures of equal peri- 

 meters, but of a greater or lefs number of fides, and pofited 

 in a different order. The other part, which relates to the 

 maxima et minima of curves, treats of problems of another 

 kind, and of the moil difficult nature which have engaged the 

 attention, and cxercifed the talents, of many of the greatelt 

 mathematicians of modern times ; as Newton, Leibnitz, 

 the Bernouillis, Euler, Lagrange, &c. and gave rife to 

 many warm and even rancorous difputes, particularly be- 

 tween the brothers John and James Bernoiiilli, which we 

 rtiall mention more particularly in the fubfequeni part of 

 this article, after having given a llight hiftorical iketch and 

 view of the more elementary parts of this interellmg branch 

 of mathematical inquiry. 



The problems in which it is required to find, among 

 figures of the fame or different kinds, thofe which, within 

 equal perimeters, ("hall comprehend the greate'.l furfaces, 

 and thofe folids which, under equal furfaces, iball contain 

 the greateft volume, had long engaged the i^ttentign of 



Vol. XIX. 



I s o 



mathematicians before the iiiTcntion of fluxions end dif- 

 ferent methods liad been devifed for the folution of them 

 by Dcs C.irtes, Fermat, Sluze, Iludde, md others; but thefe 

 were all fupplantcd by the fimplicity and generality of the 

 new analyfis ; after which time (lie elements of the fcience 

 fcem to have been loll fight of by mathematicians, who 

 were all engaged in the lolution of the higher order of 

 ifopcrimetrical problems. 



Simpfon was tlie firfi who condcfcendcd to treat of ihc 

 more elementary parts of this fcience, by giving, in his 

 Geometry, a very intcrcfting chapter on the maxima and 

 minima of geometrical quantities, and feme of the fimpleft 

 problems concerning ifoperimeters. The next who treated 

 the fubjeft in an elementary manner was Simon L'Huillicr 

 of Geneva, who, in 178;, publifiied his trcatife " Do Re- 

 latione mutua Cap.-\citatis et Terminorum Figurarum," f.:c. 

 His principal objeft in the compofition of that work wa3 

 to fupply the deficiency in this rcfpeft, which h- found 

 in mod of the elementary courfes, and to determine, with 

 regard to both the mofl ufual furfaces and folids, thofe 

 wiiich poflefr. the minimum of contour witli the fame ca- 

 pacity ; and reciprocally, the maximum of capacity witli iJjc 

 fame bonnd.^ry. Legcndre has alfo confidered the fame 

 fubjeft, in his " Elemcns de Gconietrie ;" Dr. Hutton, 

 in his " Courfe of Mathematics;" and Dr. Horficy, \a 

 the Pbilofophical Traufaflions, vol. Ikkv. for 1775. 



Eiivrli tf Ifoperimclry. 



Proposition I. — Of all triangles that can be contained 

 under any two given right lines, and any other line join- 

 ing their extremities, that will be the greateft that has the 

 two given lines perpendicular to each other. Fig. i, 

 Ifoperimttry. Plate IX. Geometry. 



Let A B and B D be the given lines, then will the tri- 

 angle A B D, in which they are perpendicular to each 

 other, be the greateft : for let B C = B D, and the angle 

 ABC cither greater or lefs than the right angle A B D ; 

 aiid let alfo C F be drawn parallel to A B, and meeting 

 B D in F ; and join A F, A C, A C 



Then the < B F C being a right angle, it is evident that 

 B C, or B D, is greater than B F, and therefore the 

 triangle A B D, being greater than the triangle A B F, is 

 aho greater than its equal ABC. Q. E. D. 



The fame may be otherwife demonltratcd, thus: — Affume 

 either of the two ^iven fides for the bafe of the triangle ; 

 then the area being direftly as the perpendicular let fall 

 upon that fide from the oppofite extremity of the other 

 given fide, the furface will be the greatell w hen that, per- 

 pendicular is the greateft, that is, when the other fide is 

 not inclined to that perpendicular, but coirc-des with it ; 

 hence the area is a maximum when the two given Cd^5 ar? 

 perpendicular to each other. Q. E. D. 



Prop. II. 



Of all triangles of the fame bafe, and whofg vertices fall 

 in a right line given in pofition, the one whofe perimeter 

 is a minimum, is that whofe fides make equal angles wiia 

 the given line. Fig. 2. 



Let A B be the common bafe of a ferics of t.-iangi*- 

 A B C, A B C, whofe vertices C, C, fall in the right Jjr* 

 L M, given in pofition ; then is tl\e triangle of lea!^ per;, 

 meter, that whofe fides AC, B C, make equal angles witli 

 the line L M. 



For, let B L be drawn from B perpendicular to L M, 

 and produce it to D, till D M =-- B M, and join D A ; 

 and from the point C, where this line interlcfb L M, 



