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f’ and * as the centres of gravity are points in which all8a es > ar >sing from the weights of the several parts ofHv e th 6 ^ od * e s, are conceived to be united, one may con-| Stavif We '»ht of the whole cylinder applied in the centrei y °f its mass, i. e. in a point in the middle of its axis:

| Cavity o J e ® Is tance of the cylinder applied in the centre ofi base , its base, i. e. in the centre of the base; it beingw nich resists the fracture.

ion • l cylinder breaks by its own weight, all the;, l j^. on an immovable extremity of a diameter of the

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Ho Se * rila extremity, therefore, is the fixed point of a lever,of 0 arm s are the radius of the base, and half the axis;5, ‘Vn, C r °, nSequence - the two opposite forces do not only acti? 4t 'v e p Ves > an< l by their absolute force, but also by the‘be <: ‘orce tVio„ ,i— e —™ *v, „,;.r

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fix r| IOrce Hey derive from their distance with regard to^ ea ce °fi the lever.

ip 'H, vp U evidentl y follows, that a cylinder, e. gr. of copper,v e8s th lI 2 a by suspended, will not break by its own weight,I H a j atl fi°ur hundred and eighty fathoms long, will breakio s tj! en g th in an horizontal situation ; because the*H; v , ttll s latter case, contributes two ways to the frac-of a , as it makes it of such a weight, and as it is anIq Hall e ^ Ver to w bich the weight is applied. Hence, also,h° ^ r e;ilj l be base is, the less length or weight will suffice> e it ’ * 3ot * 1 because the resistance is really less, and. jf t\Yo ac , t ? by a less arm of a lever,it • et>I> th ■ n ders of the same matter, having their bases| ls evj<j ' n He same proportion, be suspended horizontally;jHf, both* 1 ^ Hut the greater has more weight than the| a less r -° n a °count of its length, and of its base. But itt a ? e r a rrn Sl ® ta nce on account of its length, considered as atv * ts bas e / a ' ever > and bas only more resistance on accountb^’Sbt ^ ’ therefore it exceeds the lesser in its bulk andH jj, e than in resistance, and, consequently, it must

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'all - ® tve |

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s tr a fi> peonf See w by, upon making models and machines inat| gth 0 |. e are apt to be mistaken as to the resistance and

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ilg . . horizontal pieces, when they come to' r designs in large, by observing the same propor-

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W,’ * s no i.fi stna H* Galileo’s doctrine of resistance, there-a nd e s P ec ulation, but becomes applicable in arclii-a other :

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sj t a 8 al Wa ° u required to break a body placed horizontally,to j ess than that required to break it in a vertical

if) >e ratio p t b' s weight being greater or less, accordingifiisiHs °| He two arms of the lever, the whole theory'be « 16 WeV| C '^ e to this : viz., to find what part of thefiftUr e l,gbt the relative weight is to be, supposingfitav 6 't i e body known; which indeed is necessary,Hp '^> or t) ” le figure that determines the two centres ofof a Con e * e tw ° ar ms of the lever. For if the body, e. gr.a Xig a ^ a ce ntre of gravity would not be in the middle

^bd^Vv,He cylinder; and, if it were a semi-para-

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of 1>e 'ther would its centre of gravity be in the

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or ax ' s * nor ^ ie centre of gravity of its» 1 of the axis of its base. But still, where-

levet Cen ^ res fall in the several figures, the two arms1 T est * m ated accordingly.

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hp a . ls _ fasten'^ 6 - °b serv ed, that if the base, by which theX °bfia] ) into the wall, be not circular, but, e. gr.V a a of' tb d pbe vertex of the parabola be at the top, thea a ‘ ra cture will not be on an immovable point,e quilif. ® ‘ rQ movable line; which may be called theH &S °f tl rturn > an d it is with regard to this, that theH a bod* 6 , centres of gravity are to be determined.

He snf ii ° r ' zon tally suspended, being supposed sucha lest addition of weight would break it, there

is an equilibrium between its positive and relative weight;and, of consequence, those two opposite powers are to eachother reciprocally as the arms of the lever to which they areapplied. On the other hand, the resistance of a body isalways equal to the greatest weight which it will sustain ina vertical situation without breaking, i. e. is equal to itsabsolute weight. Therefore, substituting the absolute weightfor the resistance, it appears that the absolute weight of a body,suspended horizontally, is to its relative weight as the distanceof the centre of gravity from the axis of equilibrium is tothe distance of the centre of gravity of its base from thesame axis.

The discovery of this important truth, at least an equiva-lent to it, and to which this is reducible, we owe to Galileo.From this fundamental proposition are easily deduced severalconsequences : as, for instance, that if the distance of thecentre of gravity of the base from the axis of equilibrium behalf the distance of the centre of gravity of the body, therelative weight will only be half the absolute weight; andthat a cylinder of copper, horizontally suspended, whoselength is double the diameter, will break, provided it weighhalf what a cylinder of the same base, 4801 fathoms long,weighs.

On this theory of resistance of the illustrious Florentine,M. Mariotte made a very ingenious remark, which gave birthto a new system. Galileo supposes, that where the bodybreaks, all the fibres break at once ; so that the body alwaysresists with its whole absolute force, or with the whole forcethat all its fibres have in the place where it is to be broken.But M. Mariotte, finding that all bodies, even glass itself,bend before they break, shows that fibres are to be consideredas so many little bent springs, which never exert their wholeforce till stretched to a certain point, and never break tillentirely unbent. Hence, those nearest the axis of equili-brium, which is an immovable line, are stretched less thanthose farther off; and, of consequence, employ a less part oftheir force.

This consideration only takes place in the horizontal situa-tion of the body : in the vertical, the fibres of the base allbreak at once ; so that the absolute weight of the body mustexceed the united resistance of all its fibres ; a greater weightis therefore required here than in the horizontal situation ;that is, a greater weight is required to overcome their unitedresistance than to overcome their several resistances one afteranother. The difference between the two situations ariseshence, that in the horizontal there is an immoveable point,or line, as a centre of motion, which is not in the vertical.

Yarignon has improved on the system of M. Mariotte, andshown, that to Galileo’s system it adds the consideration ofthe centre of percussion. The comparison of the centresof gravity with the centres of percussion afford a fine view,and set the whole doctrine in a most agreeable light.

In each system, the base, by which the body breaks, moveson the axis of equilibrium, which is an immoveable line inthe same base ; but in the second, the fibres of this base arecontinually stretching more and more, and that in the sameratio as they recede farther and farther from the axis of equi-librium ; and, of consequence, are still exerting a greater andgreater part of their whole force.

These unequal extensions, like all other forces, must havesome common centre where they all meet, and, with regardto which, they make equal efforts on each side ; and, as theyare precisely in the same proportion as the velocities whichthe several points of a rod moved circularly would have toeach other, the centre of extension of the base, by whichthe body breaks, or tends to break, must be the same withthe centre of percussion. Galileo’s hypothesis, according to

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