SOL 



rtde AB, will, therefore, be •? — '-, of which I pub'- is 



4 

 fvidently two third parts. See Sphere and Spheroid. 



Ex.3. To find the Content of a parabohc Conoid. From 

 the equation a"' - " -i'' = y" of the generating curve, we 



obtain jr = a~^ X .r^, and s { = py'.r) —pa 

 X xx^- ; and, therefore, s = pa m 



a m— » 1 X ft 



X 



2n 



— + 1 

 m 



pa 



: m— 2 n rtl X m 



X 



py-- 



2 n + m 

 m X 



= pa' 



2 n ■}- m 

 the content of the fohd ; which, 



2 B + m 



therefore, is \o py'^ x, the content of the circiimfcribing 

 cylinder, as wj to 2 n -|- m. Whence the fohd generated 

 by the conical parabola (in which m = 2, and ?i =: 1) 

 appears to be juft half of its circumfcribing cylinder. See 

 Parabolic Conoid. 



Ex. 4. Tojindthe Content of an hyperbolic Conoid. From 



the equation y^ =^ — {ax ■\- .v x) of the generating hyper- 

 bola, we have s ipy' x) = ^-^ {axx + x' x), and con- 

 fequently t =: ^—j- (-ia.-c' + ^ .\-^) = the content of the 



conoid ; which, therefore, is to (^—^ [ax -\- .v') .t J t!)at of 



a cylinder of the fame bafe and altitude, as i a + -\ x to 

 « + K. This ratio, if .v be very fmall, will become as 

 I to 2 very nearly : whence it may be inferred, that the con- 

 tent of a very fmall part of any folid generated by a cHrve, 

 whofe radius of curvature at the vertex is a finite quantity, 

 is half that of a cylinder of the fame bafe and altitude very 

 nearly ; becaufe any fuch curve, for a fmall dillancc, will 

 differ infenfibly from an hyperbola, whofe radius of cur- 

 vature, at the vertex, is the fame. 



Ex'. 5. To Jind the Content of a parabolic Spindle, gene- 

 rated by the Rotation of a given Parabola A C B [fg. 4. ) 

 about its Ordinate A B. Put C M, tlie abfcifTe of the given 

 parabola, — a, and the femi-ordinate A M (or B M) = i ; 

 and fuppofing E N F to be any feftion of the fohd parallel 

 to D C, let its dillancc M N or E P from D C be denoted 

 by iu : then, by the property of the curve, we (hall have 



AM' [b^) : EP' {iv'-) ■.■.CM(a):C^ = ~: thtK. 



fore E N (= C M - C P) = a - f-^' = ^SLtJ^I, 



and confequently /. x E N' = '--^ [b* - zFm' + tu') 

 = the area of the feftion E F : which, multiplied by w, 

 the fluxion of M N, gives ^~ (b>-z'u - zb^w'-il + 



w'ti«) for the fluxion of the folidity, whofe fluent •^— 



b* 

 (i*w — ib'w^ + t':"'"). when w becomes = b, is 

 Apa-b\ 

 ( ~Ye~) "^"^ '^^ content of the folid, See Pvra.midoid, 



8 L 



For other examples of a fimilar kind, fee Simpfon's 

 Fluxions, vol. i. fett. 9, and other elementary treatifes on 

 that fubjeft. 



Solidity, in ArchiteSure, is applied both to the con- 

 fidence of the ground on whicli the foundation of a building 

 is laid, and to a mafs of mafonry of extraordinary thicknefs, 

 without any cavity within. The folidity of the Egyptian 

 pyramids is inconceivable. 



SOLIDS, in Anatomy, &c. denote all the continuous and 

 continent parts of the body ; thus called, in oppofition to 

 the fluids, or parts contained in them. 



Of the folid kind, are the bones, cartilages, ligaments, 

 membranes, fibres, mulcles, tendons, arteries, veins, nerves, 

 gland'!, lymphedufts, and lafteals. 



SO\AY!>\JQ,m Ancient Coinage. (SeeAiRELs.) Ac- 

 cording to Phny's account, gold was coined at Rome fixty- 

 two years after filver, i. e. 547 U. C. or B. C. 204; and 

 then the fcruple pafled, as he informs us, for 20 fellerces. 

 It was afterwards thought proper to coin 40 pieces out of 

 the pound of gold ; and, as he fays, our princes have, by 

 degrees, diminillied their vveiglit to 45 in the pound. The 

 pieces that now remain confirm Pliny's account. In the 

 firil coinage, the aurei were 4S in the pound ; afterward', 

 as Pliny fays, there were ,\o in the pound, and the aurens 

 was raifed from 106 grains, the woiglit of tlie didrachm of 

 this coinage, to 126 grains. From Pliny and the coins it 

 appears, that in the firll coinage, the fcruple of gold palled 

 for 20 lellerces ; the drachm of three fcruples was 60 fef- 

 tertii, or ij filver denarii ; and the didrachm, or aureus, 

 the common Roman gold coin, was worth 30 filver denarii, 

 .equal to j/. Iterling ; gold being ts filver as lyf to i. 

 The aureus feems to have continued at 30 filver denarii 

 till Sylla's time ; but about the year of Rome 675, B. C. 77, 

 the aureus fell to the rate of 40 in the pound, as Pliny 

 informs us, and being reduced near the fcale of the Greek m 

 X^^^oii pafled for 20 denarii, as the later for 20 drachmas, I 

 being in currency 13J. 4^/. Englifli. This is the more 

 probable, becaufe we know from Suetonius, that the great 

 Cxfar brought fo much gold from Gaul, that it fold at 

 3000 nummi a pound, that is, nine times its weight iu 

 filver ; but the Gallic gold wa» of a very bafe fort. How. 

 ever, in the reign of Claudius, the aureus pafled for too 

 fellertii, or 25 filver denarii ; at which rate it remained. 

 This was 16/. ?id. Englilh in currency ; but valuing gold 

 at 4/. an ounce, the intrinfic value of the aureus is about 

 i/. The aureus fell by degrees, as Pliny fays, to 45 in the 

 pound. From the coins it is clear, that it was in the time 

 of the civil wars of Otho and Vitellius, tliat the aureus fell 

 from 40 in the pound, or about 125 troy grains at a 

 medium, to 45 iu the pound, or about 1 10 grains of medial 

 weight each. It continued of this llandard till the time 

 of Elagabalus, when it fell to about 92 grafus at an aver- 

 age, or near 55 in the pound. That the aureus paffed for 

 25 filver denarii down to Alexander Sevcrus, is clear ; and 

 fuppofing that (tandard to remain, as we have no authority 

 for a change till the time of Conltantine I., the double 

 aureus will have borne 50 filver denarii, and the aureus 25. 

 The " triens" mull have had eight filver denarii, and two 

 denarii aurei ; and the double triens, 16 filver denarii or 

 argentei, and four denarii aurei. The denarius was not then 

 worth above 14^. EngUfli. The only change Aurelian 

 made in the money, was probably reilrifted to the gold ; 

 for it is certain that under him, and his fucceffor Probus, 

 the common gold piece, or aureus, is of 1 00 grains, a 

 fize confined to thele two emperors. There are alio halves 

 of .about 50 grains ; and double aurei, commonly of very 

 fine workmaiifiiip, of upward of 200 grains. Down to 



Conllantine 



