RESISTANCE. 



fame matter and length, whatever their bafes may be, have 

 an equal refiftance, when vertically fufpended. 



But if the length ef the cylinder be increafed, without 

 increafmg its bafe, its weight is increafed, while the refift- 

 ance or llrength remains the fame : confequently it is 

 weakened by its additional length, and has a greater ten- 

 dency to break. 



Hence, to find the greateft length a cylinder of any 

 matter may have, to break with its own weight j it is only 

 neceffary to know what weight is jutt fufficient to break 

 another' cylinder of the fame bafe and matter: for the 

 length of the required cylinder muft be fuch, that its weight 

 may be equal to the weight of the firft cylinder, together 

 with the additional weight that was employed in producing 

 the feparation. 



Thus let / denote the firll length of the cylinder, i its 

 weight, g the given weight the lengthened cylinder is to 

 bear, and <tu the leaft weight that breaks the cylinder /, 



alfo x the length fought: then as / : x :: c : - - = the 



weight of the longeft cylinder ; and this, together with 

 the given weight g, muft be equal to c + «; : hence 



, ' e x c + w — S ; ,i 



then, -- + g ~ c + w ; or x = -- / = the 



length fought. When the cylinder isjuft to break with its 

 own weight only, then g — o, and the expreffion is fimply 



c + IV 

 X = /. 



c 



If one end of the cylinder were fixed horizontally into a 

 wall, and the reft fufpended thence, its weight and refiftance 

 would then aft in a different manner ; and if it be broke by 

 the action of its weight, the rupture would be at the end 

 fixed into the wall. A circle or plane contiguous to the 

 wall, and parallel to the bafe, and confequently vertical, 

 would be detached from the contiguous circle within the 

 plane of the wall, and would defcend. All the motion is 

 performed on the loweft extremity of the diameter, which 

 remains immoveable, while the upper extremity defcribes a 

 quadrant of a circle, and till the circle, which before was 

 vertical, become horizontal ; i. c: till the cylinder be entirely 

 broken. 



In the fracture of the cylinder it is vifible two forces 

 have acted, and the one has overcome the other : the weight 

 of the cylinder, which arofe from its whole mafs, has over- 

 come the refiftance which arofe from the largenefs of the 

 bafe ; and as the centres of gravity are points in which all 

 the forces, ariling from the weights of the feveral parts of 

 the fame bodies, are conceived to be united, one may con- 

 ceive the weight of the whole cylinder applied in the centre 

 of gravity of its mafs, /. e. in a point in the middle of its 

 axis : and the refiftance of the cylinder applied in the centre 

 of gravity of its bate, ;'. e. in the centre of the bafe ; it being 

 the bafe which refills the fracture. 



When the cylinder breaks by its own weight, all the mo- 

 tion is on an immoveable extremity of a diameter of the 

 bafe. This extremity, therefore, is the fixed point of a 

 lever, whofe two arms are the radius of the bafe, and half 

 the axis ; and, of confequence, the two oppofite forces do 

 not only act of themfelves, and by their abfolute force, but 

 alfo by the relative force they derive from their diftance with 

 regard to the fixed point of the lever. 



Hence it evidently follows, that a cylinder, e. gr. of 

 copper, which, vertically fufpended, will not break by its 

 own weight, if lefs than four hundred and eighty fathom 

 long, will break with a lefs length in an horizontal fitua- 

 tion ; becaufe the length, in this latter cafe., contributes 



two ways to the fracture ; botli as it makes it of fuch a 

 weight, and as it is an arm of a lever to which the weight 

 is applied. Hence, alfo, the fmaller the bafe is, the lefs 

 length or weight will fuffice to break it ; both becaufe the 

 refiftance is really lets, and becaufe it acts by a lefs arm of 

 a lever. 



Hence, to find the length a prifm will bear, when fixed 

 in an horizontal pofition, before it breaks, either by its own 

 weight, or by the addition of any adventitious weight ; 

 take any length of fuch a prifm, and load it with weights 

 till it break ; then put 



/ = the length of this prifm, 



c = its weight, 



<w — the weight that breaks it, 



ii = the diftance of the weight w t 



g = any given weight to be bori 



d — its diftance, 



x — the length required to break it. 



I ' ■ 



ex 



r 



the weight of the prii 



ex 



T 



C X ' 



11 



= its momentum; alfo d ? = the 



mentum of the 



htj 



ex' 



2/ 



dg 



ntum of t! 



prifm x, and its additional weight g. 



In like manner we have | c 1 + a ii<, for the mo- 

 mentum of the fhorter prifm, together with the weight tu. 

 Confequently we obtain the following equation : 



ex" 1 



+ dg — \cl + aw; 



2/ 



from which is found x 



V* 



w + \c I — dg) 2 / 



, the 



length fought, or that by which the cylinder will break 

 with the weight g, at the diftance d. If this laft weight be 

 nothing, or the length be required when the cylinder would 

 juft break with its own weight, then we fhall have dg = c, 



/{a to + " 



and the expreflion becomes fimply 



V 



[cl)2l 



If two cylinders of the fame matter, having their bafe.-, 

 and lengths in the fame proportion, be fufpended hori- 

 zontally ; it is evident, that the greater has more weight 

 than the Idler, both on account of its length, and of its 

 bafe. But it hath lefs refiftance on account of its length, 

 considered as a longer arm of a lever, and has only more re- 

 fiftance on account of its bafe ; therefore it exceeds the leffer 

 in its bulk and weight more than in refiftance, and confe- 

 quently it muft break more eafily. 



Hence, we fee why, upon making modeL. and machines 

 in Email, people are apt to be mittaken as to the refiftance 

 and ftrength of certain horizontal pieces, when they come 

 to execute their deiigns in large, by obferving the fame pro- 

 portion as in the fmall. Galileo's doctrine of refiftance, 

 therefore, is no idle fpeculation, but becomes applicable in 

 architecture, and other arts. 



The weight required to break a body, placed horizontally, 

 being always lefs than that required to break it in a vertical 

 iituation ; and this weight being to be greater or lef:-, ac- 

 cording to the ratio of the two arms of the lever, the whole 

 theory is always reducible to this : viz. to find what part 

 of the ablolute weight the relative weight is to be, fiippofing 

 the figure of the body known ; which indeed is neceffar\ , 

 becaufe it is the figure that determines the two centres of 

 gravity, or the two arms of the lever. For if the body, 



e.gr. 



