U N G 



Note 2. — If V exceed \ J, tliat is, if llie bafc exceed the 

 femicircle, then c is negative, and b c mull be added. 



This theorem may be demonftrated in the following man- 

 ner. The fluxion of the folid is =: the A H I K drawn 

 into the fluxion of L I, which fluxion will, therefore, be 



XX — X H I • ( ufmg the fame charafters as in the de- 



monftration of the lafl; problem) =: — x [y — cV =■ — X 



2<V 2V 



h 



2cy + cc) = — X 



2V 



(■5 J J — .\- .V — 2 c y -\- c c) =. 



X {^dd— XX — dc + 2cx A L 



^ hx 

 cc) = — X 

 2V 



( J J- — XX- 



he 



hx 



2c X FL); whofe fluent, — x {ss — i;xx) 

 2 V 



X area F A I H, when I coincides with G, is — x 



V 2-V 



(tJ'' — ic), the double of which 



iis^-lc) 



the content of tlie folid D E A G D required. 



Cor. I. — If F fall in the centre of the bafe, then r rr o, 

 and J = 1) z= i ^, and the rule will be ^ ddh. 



Cor. 2. — If A F exceed ¥ B, c wiU be negative, and then 

 — ic will bfcome + lie. 



Cor. 3. — If F fall in B, s = O, and c = — 



and 



then the theorem becomes ^bh ^ half the cylinder. 



Ex. I. — If the diameter A B be 50, the height AD 

 120, and the vcrfed fine A F lo ; what is the folidity of 

 the hoof? 



Or, fuppofnig a cylindric vefTel A BCD, containing a 

 fluid, to be placed in fuch a pofition that the furface of the 

 fluid, difpofing itfelf parallel to the horizon, may cut the 

 bafc in G E, leaving 40 inches of the diameter dry, and 

 the fide of the cylinder in D, 120 inches dill ant from the 

 bafe ; to find how many ale gallons are in it, the diameter 

 of the bafe being 50 inches. 



Here i = 120, d — 50, and v = 10. Then ^d — v 



and \' \dd — c c = V 2^"- — 15' 



= 25 



15 



•^40 X lo = 20 = 



And, to find the bafe by the table of fegments, — t- = 



d 



4S = .2 ; this being found in the column of verfed fines, 



oppofite to it is the area .1118238: hence 50 x 50 x 



.1118238 = 279.5595 = ^ is the fegment or bafe. 



T'L ' i' — b c 



Ihen 6=12 X (J X 8000— 15 X 279-5595) 



= '2 X (5333 ; - 4193.3925) = 12 X 1 139.9408 = 

 13679.2896 = the folidity in inches; which, divided by 

 282, the inches in a gallon, give 48.50939 ale gallons for 

 the content. 



Ex. 2 — Suppofe the cylinder fo placed, that the furface 

 of the liqucr may biftft the bafe, and rife up the iidc to the 

 fame diftance of 120 inches from the bafe: to find the 

 content. 



Here, by note I, we have iddh = 50 x 50 X 20 = 

 50000 folid inches = 177.3049645 gallons, for the content 

 in this cafe. 



Ex. 3 — Suppofe, now, the fame vcfftl fo placed, as that 

 the furface of the liquor may leave only 10 inches of the 

 diameter dry, dill rifing to the fame diftancc of 1 20 inches 

 along the fide ; to find the content. 

 9 



U N I 



Here the part of the cylinder's bafe left dry, is equal to 

 the bafe in the firll example, viz. 279.5595, "'I'lch, there- 

 fore, taken from 50 x 50 x .78539816= 1963.4954, the 

 whole circle, leaves 1683. 9359 = b, the bafe of the ungula 

 in this example. 



Now v — 40, c = — 15, and s — 20. 



Whence /. = ( 5 x 8coo + 25259.0385) v." 



= 30592-3718 X 3 = 917-7.1154 folid inches = 325.45076 

 gallons, the content in this cafe. 



For the method of finding the folidity of the unguis or 

 hoofs of other fohds, we mull refer to Hutton's Menfura- 

 tion, part iii. § 1. 



Ungula, in Natural Hijlory, the claw, or hoof, of a 

 quadruped. 



Ungula Aids, the elk's claw. See Elk. 



Ungula, a technical name formerly applied to an abfcefs 

 of the cornea, when the difeafe was fancied to referable a 

 hoof in its fhape. 



Ungula, or Hamus, among Surgeons, a fort of hooked 

 inftrument, with which to extraft a dead fcetus out of the 

 womb. 



UNGULUS, in Antiquity, ?L remarkable kind of bracelet. 



UNHACA, in Geography, a fmall ifland m the Indian 

 fea, at the entrance of the bay of Lorenzo Marques. N. 

 lat. 26° 5'. 



UNH ALTER, in the Manege. A horfe is faid to 

 unhalter himfelf that turns off his halter. 



UNHOST, or AuNHOST, in Geography, a town of 

 Bohemia, in the circle of Schlan ; 8 miles S. of Schlan. N. 

 lat. 50° 6'. E. long. 14° 15'. 



UNIA, a fmall ifland in the Adriatic, W. of Ofero. 

 N. lat. 44° 52'. E. long. 14° 26'. 



UNIAK, or Unimak. See Oonemak. 



UNIA R A, a town of Hindooftan, in the circar of 

 Rantampour ; 18 miles S. of Rantampour. 



UNICORN, in AJlronomy. See MoNOCEROS. 



Unicorn, in Geography, a poit-town of Pennfylvania, in 

 Lancafter county ; 124 miles from Wafliington. 



Unicorn, in Natural Hijlory, an animal famous among 

 the Greek authors, under the name of j/ovoxs^i'; ; and among 

 the Latins, under that of unicorn. 



Both thefe names it takes from its diftinguifliing charac- 

 terillic, the having one horn only. Sec RHlNOCEnos. 



The firll author who wrote of the unicorn, wa8 one 

 Ctcfius, whom Ariftolle mentions as a very fufpicious 

 author. vElian fpcaks of it in very doubtful terms. The 

 other writers on the fubjeft are Philoftratus and Solinus ; 

 iEueas Sylvius, who is pope Pius II ; Marcus Paulus, 

 Aleofius, Gefiier, Garcias ab Orta, And. Marinus, &c. 

 Of thefe, fome fay it refenibles a horfe, others an afs, others 

 a goat, by its beard ; others an elephant, others a rhino- 

 ceros, others a greyhound, &c. 



Munfter and Thcvet will have it an amphibious animal, 

 and its horn to be moveable at pleafure. Others make -•dl 

 its (Irength to confift in its horn ; and add, that whea 

 purfued by the hunters, it precipitates itfelf from the lops 

 of the highell rocks, and pitches upon its horn ; which fuf- 

 tains the whole effort of its f.ill, fo that it receives no 

 damage thereby. In reality, the feveral authors do all give 

 different accounts of the figure and colour, both of the 

 animal and of its horn, and all its parts. And hence many 

 .tmoiig the moderns have fuppofed it to be a merely fabulous 

 animal. 



The legend adds, that it is wonderfully fond of challe 

 perfonsj and therefore, iu order to tike it, a virgin is placed 



