RESISTANCE. 



r. were a cone, its centre of gravity would not be in 

 the middle of its axis, as in tlie cylinder ; and, if it were a 

 femi-parabolical folid, neither would its centre of gravity be 

 in the middle of its length or axis, nor the centre of gravity 

 of its bafe in the middle of the axis of its bafe. But itill, 

 wherefoever thefe centres fall in the feveral figures, the two 

 arms of the lever are eltimated accordingly. 



It may be here obferved, that if the bafe, by which the 

 body is fattened into the wall, be not circular, but, e. gr. 

 parabolical, and the vertex or the parabola be at the top, 

 the motion of the frafture will not be on an immoveable 

 point, but on a whole immoveable line ; which may be 

 called tile axis of equilibrium ; and it is with regard to 

 this that the diftances of the centres of gravity are to be de- 

 termined. 



Now, a body horizontally fufpended, being fuppofed 

 fuch as that the fmalleft addition of weight would break it, 

 there is an equilibrium between its pofitive and relative 

 weight ; and of confequence thole two oppofite powers are 

 to each other reciprocally as the arms of the lever to which 

 they are applied. On the other hand, the refiftance of a 

 body is always equal to the greateft weight which it will 

 fuftaiu in a vertical fituation, without breaking, i.e. is equal 

 to its abfolute weight. Therefore, fubftituting the abfo- 

 lute weight for the refiftance, it appears that the abfolute 

 weight of a body, fufpended horizontally, is to its relative 

 weight as the diltance of its centre of gravity from the axis 

 of equilibrium is to the diltance of the centre of gravity of 

 its bafe from the fame axis. 



The difcovery of this important truth, at leaft an equi- 

 valent to it, and to which this is reducible, we owe to Ga- 

 lileo. From this fundamental proportion are eafily deduced 

 feveral confequences ; as, for inftance, that, if the diftance 

 of the centre' of gravity of the bafe from the axis of equi- 

 librium be half the diftance of the centre of gravity of the 

 body, the relative weight will only be half the abfolute 

 weight ; and that a cylinder of copper, horizontally fuf- 

 pended, whofe length is double the diameter, will break, 

 provided it weigh half what a cylinder of the fame bafe, 

 4801 fathoms long, weighs. 



On this theory of refiftance, which we owe to Galileo, 

 M. Mariotte made a very ingenious remark, which gave 

 birth to a new fyftcm. Galileo fuppofes, that, where the 

 body breaks, all the fibres break at once ; fo that the body 

 always rcfifts with its whole abfolute force, or with the 

 whole force that all its fibres have in the place where it is 

 to be broke. But M. Mariotte, finding that all bodii , 

 even glafs itfelf, bend before they broke, (hews that fibri 

 are to be conlidered as fo many little bent fprings, which 

 r.erer exert their whole force till ftretched to a certain point , 

 and never break till entirely unbent. Hence, thofe nearelt 

 the axis of equilibrium, which is an immoveable line, ai 

 ftretched lels than thole farther oft ; and, of confequence, 

 employ a lefs part of their force. 



This confideration only takes place in the horizontal fitu- 

 ation of the bodv : in the vertical, the fibres of the bafe all 

 break at once ; lo that the abfolute weight of the body n 



id the united refiftance ol all its fibres ; a greater weight 

 is, therefore, required here than in the horizontal fituation ; 

 that is, a greater weight i. requil d to overcome then- 

 united refiftance, than to overcome their feveral refifta 

 one after another. The differi nee between the two litua- 

 tions arifes hence, that, in the horizontal, there is an im- 

 moveable point or line, as a centre of motion, which is not 

 in the vertical. 



Varignon has improved on the fyftcm of M. Mariotte, 

 and (hewn, that, to Galileo's fyftem, it adds the confidera- 

 tion of the centre of percuffion. The companion of the 



2 



centres of gravity with the centres ot percuffion allord a 

 fine view, and let the whole doftrine in a moft aineeable 

 light. 



In each fyftem, the bafe, by which the body breaks, 

 moves on the axis of equilibrium, which is an immoveable 

 line in the fame bafe ; but in the fecond, the fibres of this 

 bafe are continually ftretching more and more, and that in 

 the fame ratio as they recede farther and farther from the- 

 axis of equilibrium ; and, of confequence, arc itill exerting 

 a greater and greater part of their whole force. 



Thefe unequal extenlions, like all other forces, muft have 

 fome common centre where they all meet, and with regard 

 to which they make equal efforts on each fide ; and as they 

 are precifely in the fame proportion as the velocities which 

 the feveral points of a rod moved circularly would h.i 

 one another, the centre of extenfion of the bafe, by which 

 the body breaks, or tend, to break, mull be the fame with 

 its centre of percuffion. Galileo's hypothefis, according to 

 which the fibres are fuppofed to ttretch equally, and break 

 all at once, correfponds to the calf of a rod moving paral- 

 lel to itfelf, where tin: centre of extenfion or percuffion 

 does not appear, as being confounded with the centre of 

 gravity. 



The bale of fraction being a furface, whofe particular 

 nature determines its centre of percuffion, it is neceffary 

 that this (Tiould be firft known, to find on what point of the 

 vertical axis of that bafe it is placed, and how far it is from 

 the axis of equilibrium. Indeed, we know in the general, 

 that it always a£ts with fo much the more advantage as it is 

 farther from it ; becaufe it acts by a longer arm of a lever ; 

 and of confequence it is the unequal refiftance of the fibres 

 in M. Mariotte's hypothefis, which produces the centre of 

 percuffion ; but this unequal refiftance is greater or lefs, ac- 

 cording as the centre of percuffion is placed more or lefs 

 high on the vertical axis of the bafe, in the different fur- 

 of the bafe of the fr aft lire. 



To exprefs this unequal refiftance, accompanied with all 

 the variation il is capable of, regard mult be had to the 

 ratio between the diltance of the centre of percuffion from 

 the axis ot equilibrium, and the length of the vertical axis 

 of the bafe. In which ratio, the firfl term, or the nume- 

 rator, is always lefs than the fecond, or the denominator ; 

 lo that the ratio is always a fraction lefs than unity \ and 

 the unequal refiftance of the fibres in M. Mariotte's hypo- 

 thecs is fo much the greater, or, which amounts to the 

 , approaches fo much nearer to the equal refiftance in 

 Galileo's hypothefis, as the two terms of the ratio arc- 

 nearer to an equality. 



Hence it follows, that the refiftance of bodies in M. Ma- 

 riotte's fyftem is to that in Galileo's, as the le ill of the 



terms in the ratio is to the greateft. Hence, alfo, the re- 

 fiftance being lels than what Galileo imagined, the relative 

 weight mull alio be 1 the proportion already 



ied betweei th ! ol d relative weight cannot 



fublift in the new fyftem, without an augmentation of the 

 relative weight, or a diminution ol the abfolute weight; 

 which dinniiiit: by multiplying thi weight by the 



ratio, which is alwayi lefs than unity. This done, we find 



that the abfolute weight, multiplied b\ I > the 



relative weight, as the diltance of the . ..I 



the body from the axis of equilibrium, is to the diftano 



thee- jravity of tl oi th< fraftun from the 



fame axis : which is precifeh the fame thing with the s 



formula given by M. Varignon for the fyftem of M. 



Manotte. 1:. eflfeft, alter conceiving the relative weight <>t 



I ' ' and i [ual to its abfolute weight, as 



two I ontrary power; applied lo the two arms of a lever, in 

 Hi" hypothefi of Galileo, there needs nothing to convert it 



into 



