SOLIDITY. 



by which it excludes every other body from that place which 

 itfelf pofTeffes. 



Solidity, in this fenfe, is a property common to all bodies, 

 whether fohd or fluid. It is ufually called impenetrability ; 

 but folidity exprefles it belt, as carrying fomewhat more of 

 pofitive with it than the other, which is a negative idea. 



The idea of folidity, Mr. Locke obferves, arifes from the 

 refiftance we find one body make to the entrance of another 

 into its own place. Solidity, he adds, feems the moft ex- 

 tenfive property of body, as being that by which we con- 

 ceive it to till fpace ; it is didinguiihed from mere /pace, by 

 this latter not being capable of refiftance or motion. 



It is diftinguifhed from hardneft, which is only a firm 

 cohefion of the folid parts, fo as they may not eafily change 

 their fituation. 



The difficulty of changing fituation gives no more foli- 

 dity to the hardeft body than to the fofteft ; nor is a diamond 

 properly a jot more folid than water.' By this we diltinguidi 

 the idea of the extenfion of body from that of the exten- 

 fion of fpace : that of body is the continuity or cohefion 

 of fohd, feparable, moveable parts ; that of fpace, the con- 

 tinuity of unfohd, infeparable, immoveable parts. 



The Cartefians, however, will, by all means, deduce foli- 

 dity, or, as they call it, impenetrability, from the nature of 

 extenfion ; they contend that the idea of the former is con- 

 tained in that of the latter ; and hence they argue againft a 

 vacuum. Thus, fay they, one cubic foot of extenfion can- 

 not be added to another, without having two cubic feet of 

 extenfion ; for each has in itfelf all that is required to con- 

 Ititute that magnitude. And hence they conclude, that 

 every part of fpace is folid, or impenetrable, inafmuch as 

 of its own nature it excludes all others. But the conclufion 

 is falfe, and the inftance they give follows from this, that 

 the parts of fpace are immoveable, not from their being im- 

 penetrable or folid. 



For an account of a late controverfy concerning the foli- 

 dity of matter, fee Matter. 



Solidity, in Geometry, the quantity of fpace contained 

 in a folid body ; called alfo the folid content, and the cube 

 of it. 



The folidity of a cube, prifm, cylinder, or parallelepiped, 

 is had by multiplying its bafis into its height. 



The folidity of a pyramid or cone is had by multiplying 

 either the whole bafe into a third part of the height, or the 

 whole height into a third part of the bafe. 



Solidity of any irregular Body, To find the. Put the 

 body in a hollow parallelepiped, and pour water or fand 

 upon it, and note the height of the water or fand A B 

 (Plate XIV. Geometry, fig. 2.) ; then taking out the body, 

 obferve at what height the water (or fand, when levelled) 

 ftands, as AC. SubtraA A C from A B, the remainder 

 will be B C. Thus is the irregular body reduced to a pa- 

 rallelepiped, whole bafe is FCGE, audits altitude BC. 

 To find the folidity of which, fee Parallelepiped. 



Suppofe, e. gr. A B to be 8, and A C, j ; then will 

 B C be 3 : fuppofe, again, D B, 12, D E, 4 ; then will the 

 folidity of the irregular body be found 144. 



If the body be fuch as that it cannot be well laid in 

 fuch a kind of channel, e. gr. if it be required to meafure 

 the folidity of a llatuc, as it ftands; a quadrangular prifm 

 or parallelepiped is to be framed over it ; the rell as before. 



Solidity of ahollorv Body, To find the. If the body be 

 not comprifed in the number of regular bodies, its fohdity 

 is found as in the preceding problem. If it be a parallel- 

 epiped, prifm, cylinder, fphere, pyramid, or cone, the 

 folidity firll of the whole body, including the cavity, then 

 that of the cavity, which is fuppofedto have the fame figure 



Vol. XXXIIL 



with the body itfelf, is to be found, according to the re- 

 fpeftive methods delivered under Pakallelepiped, Prism, 

 &c. For the latter being fubtrafted out of the former, the 

 remainder is the folidity of the hollow body required. 



For the centrobaryc metliod of meafuring the folidity ot 

 bodies, fee Centrobaryc Method. 



To find the Solidity of Bodies by the Method of Fluxions. 

 Let ABC {Plate XIY. ylnalyfts,fig. I.) reprefcnt any 

 folid conceived to be generated by a plane P Q, pafling 

 over it with a parallel motion : let H /', perpendicular to 

 PQ, be taken to exprefs the fluxion of A H (x), or the 

 velocity with which the generating plane is carried : alfo 

 let the area of the part E m F n be denoted by A : 

 then it follows, from the definition of a fluxion, that the 

 fluxion of the folid A E F will be exprefled by A x. 

 Whence, by expounding A in terms of x (according to 

 the nature of the figure), and then taking the fluent, the 

 contents of the fohd, which we may reprefeiit by s, will be 

 given. But, when the propofed folid is that arifing from 

 the revolution of any given curve A E B about A H D, 



as an axis, the fluxion (j) of the folidity may be exhibited 

 in a manner more convenient for praftice ; for, putting 

 the area (3.141592, &c.) of the circle, whoie radius is 

 unity, = p, and the ordinate 'E.H =■ y, it will be i ^ : ^■' 

 '•'• P '• Py^' ^'1^ *''^3 °f •^'"^ circle EoiFn, which being 

 fubftituted for A, we have s = py^ x. 



Ex. I. To find the Content of a Cone ABC {fg. 2.) 

 Let the given altitude A D be = a, and the femidiameter 

 of its bafe B D = ^, and the dillance A F of the circle 

 E G from the vertex A, = .r ; then we fhall have, by 



bx • 

 fimilar triangle.<i, a : ^ :: .x : EF (y) = — . Whence 



(=/^'.v) = 



pb' 



vcrfe method of fluxions, t = 



and, confequently, by the in- 

 pb' . 



( = AD), gives 



pb^ 



= P 



3^" 

 X BD' 



which, when sc = j 



A D for the 



content of the whole cone ABC; which appears, from 

 hence, to be juft ;,d of a cylinder of the fame bafe and alti- 

 tude. See Cone. 



Ex. 2. To find the Content of a Spheroid A F B H 

 {fig. 3.) and alfo of a Sphere. Let the axis A B, about 

 which the folid is generated, be = a, and the other axis 

 F H of the generating ellipfe = i ; it follows, from the 

 property of the ellipfe, that a- : 6' :: x (a — x) (AD 



X B D) : ji' (D E") = — (ax - x.v) : vrhence we ha»e 



J. LI 



s =z {pyx =)^ - {a XX — x"a), and 



— ^:cs) = the fegment A I £. 

 pb'- 



a' 

 Which, when A D (x) 



{\aKm 



pb' 



a O 



= AB (a), becomes^ (-i a' — ^ a^) — 



■i pab' = the content of the whole fphcroid. Where, if S 

 (F H) be taken — a {A B), we fhall obtain ipa^ for the 

 true content of a fpheic whofc diameter is a. Hence, a 

 fphere, or a fpheroid, is jds of its circumfcribing cylinder ; 



pb' 

 for the area of the circle FH being expreffed by — , the 



4 



content of the cylinder, whofe diameter is F H, and alti- 

 S f tudc 



