U N G 



U N G 



or poma of the (hell, which ftopped the mouth at pleafure, 

 and from under which the creature thruft out its tongue to 

 feed ; and he adds, that the fliell-fifh to which it belonged 

 was taken in the marfhes of India, when the waters were 

 dried away ; and that the Indian fpikenard growing in great 

 abundance in thefe marfhes, the creature became fweet- 

 fcented in every part, by feeding on it. However, he con- 

 cludes with telling us, that there were only two kinds 

 brought into Greece in his time, the one from the Red fea, 

 the other from Babylon. 



The truth is, that fpikenard grows neither in the Red 

 fea, nor any where about Babylon, but only in India, be- 

 yond the Ganges, or about its banks. The fpikenard alfo 

 does not grow in the water, but only in marfhy places, and 

 therefore can never be in the way of feeding fhell-fi(h. 

 Avicenna, perceiving the abfurdity and contradiftion of 

 Diofcorides's account, fays that the (hell-fi(h was found in 

 an ifland in the Indies, on which iflaiid the fpikenard alfo 

 grew in great abundance. But this account fuppofes that 

 the fliell-tifh, to which the unguis odoratus belongs, may be 

 found on dry land ; whereas it is certain, that no (hell-fi(h, 

 living in the water, can fubfift without fome means of 

 clofing up its cavity, fo as to keep out the water at plea- 

 fure ; this is done in the bivalve kinds, by clofing the two 

 valves ; but in the ftromboide ones, by drawing down this 

 operculum, which is the unguis odoratus, to the mouth of 

 the (hell. A land-(hell, therefore, can have no occafion for 

 fuch a part as the poma or operculum, and no fuch drug as 

 the unguis odoratus can be found about it. But it is to be 

 obferved, that Avicenna did not know that the unguis 

 odoratus was a covering or operculum of the mouth of a 

 (hell, but thought that it was only a fragment cut or broken 

 indeterminately from any part of the (hell. This therefore 

 might appear no abfurdity to him ; and the thin and flat 

 ungues he faw mis^ht appear fragments artificially cut from 

 fome of the thin-(lielled kind of land-fnails. See Blatta 

 Byzantma. 



UNGULA, in Geometry, is the feSion of a cylinder, cut 

 off by a plane paiTing obliquely through the plane of the 

 bafe, and part of the cylindric furface. 



Or, more generally, an ungula, or hoof, is a part cut 

 off a folid by a plane oblique to the bafe. — To find the 

 curve furface of the ungula D E A G D of a cylinder 

 (Plate XY. Geometry, Jg. 19.) put h = the height AD, 

 ■v = the verfed fine of A E, J — the diameter A B, a = 

 the arc E A G of the bafe, s = the right fine F G, and 



I K = — X (> ~ f ) ; and hence the fluxion of the fur- 



c = the cofine of the half arc ; then 



X A is the 



V 



face, or z X I K, is 

 c z) 



X {yi 



cz) = — X 



01 



h 



cz) =. (when 



convex furface : /'. e. from the produft of the diameter and 

 fine, fubtraft the produft of the arc and cofine, and multi- 

 ply the difference by the height, and divide by the verfed 

 fine. 



Note I . — When F is the centre of the bafe, then i; = j 

 =: ^ (/, and c =z o ; in which cafe the theorem becomes dh, 

 viz. the produil of the diameter and height equal to the 

 curve furface. 



Note 2. — When A F exceeds ^ A B, then a c muft be 

 added. 



For the demonftration of this theorem, draw H I, I K 

 parallel to F A and A D refpeftively, and join the points 

 H, K ; fince it is evident that the furface is generated by 

 the motion of I K along the arc A I G, K I x the fluxion 

 of I A will be the fluxion of the furface. Therefore put 

 2 := A I, » ::=z its fine I L, and y = its cofine ; then H I 

 ■^ y — c ; and, by fimilar triangles, FA : AD : : HI 



the fluent of which is =: — x [\dK 



1} 



A I = A G ) — X {\ds — \ac); the double of which is 



ds — ac) = the whole convex furface DEAGD 

 1 . — If F be the centre ; then v = s = ^ d, and 



V 



Cor. 



c := o ; and then the theorem becomes barely dh ■=■ \ times 

 the triangle FDA. 



Cor. 2. — When A F exceeds \d, f is negative, and then 



— ac becomes -\- ac. 



Cor. 3. — If F coincide with B ; then j = o, and c = 



— \-v ; and the theorem becomes \ah ^ the furface of the 

 half cyhnder. 



Example I. — Let the diameter A B (^) be 100, the height 



Then 



!</ 



AD (/>) 140, and the verfed fine A F [i;) 10. 



— 1) — 50 — lO = 40 = ir; and ^' \dd — cc = '>/ i^oo — l6oo 



= "k^ 900 = 30 = 



^"^P = 



.6 is the fine 



reduced to the radius I, to which, in a table of fines, belong 

 36' 52.268' — 36.87113 degrees. Then by the rule given 

 under Arc of a Circle, tlie length of the arc a wiU be 

 .01745329 X 36.87113 X too = 64.352252. Whence 



ds — ac , , „, 



X b= (3000 — 2574.09008) X 14 = 425.90992 



X 14 = 5962.73888 = the convex furface required. 



Ex. 2. — If the diameter and height be 1 00 and 140, as 

 before, ahd the feftion be made through the centre of the 

 bafe, or 11 = -1 (/ = 50 ; what is the convex furface ? 



Here, by note i, dh = 100 x 140 = 14000 = the con- 

 vex furface required. 



Ex. 3. — Suppofing d and h Hill the fame, and 1; = go; 

 to find the convex furface. 



Here i</— •i; = 50— 90 = — 40 = *:, j = 30, the fame 

 as before, but it is here the fine of the fupplemental arc, 

 which therefore is 180 — 36.87113 = 143.12887 degrees. 

 Hence .01745329 x 143.12887 x 100 = 249.807013 = 

 the arc a. Or, the arc may be fooncr found by only fub- 

 trafting the arc in the firft example, rra. 64.352252, from 

 314.159265, the whole circumference. 



Then, by note 2, 



ds + ac 



h = V ( 3000 + 9992.28052 ) 



= V X 12992.28052 — 20210.21414, the convex furface 

 required. 



To Jind the Solidity of ihe Hoof of a Cylinder. — From -J 

 of the cube of the right fine, fubtraft the produft of the 

 bafe and cofine of half the arc of the bafe ; then multiply 

 the difference by the height, and divide by the verfed fine, 

 the quotient will be the fohdity required. That is, putting, 

 as before, h = the height A D, "u — the verfed fine A F, 

 s — the right fine F G, <r = the cofine = 5 A B — A F, 



's^ — be 

 b = the bafe or area of the feg. G A E G ; then ^ 



X /6 = the folidity. 



Note I. — If F be the centre, that is, if the bafe be equal 

 to the femicircle, then u = j, and r =: o ; and therefore 

 ^ h s 1 = i d d h n ihe fohdity in that cafe. 



Note 



