56 Mechanic Powers. Book II.

ance, but ſtill the Semidiameter requires to be leſs, becauſetheNicht bears or preſſes obliquely on che Axes of the Wheels B andC; hence the weight of the bell on thoſe axcs 18 according tothe Jin es A B, A C, and not according to the perpendicular A D:There fre 25 MU D to A B, ſo reciprocally the weight on A B toe weiht onA D; but the weight on either axis is as the ſub.dupie(or half) summ to the Whole weight; therefore theweight on B A is leſs than ſubduple, and the proportion con-tinuing is made leſs; For when the Semidiameters A E and BE are given, the whole B A is known; and B D, and alſo itsequal B E is known; therefore(by tlie 4th. Of 1ſt. of Euclid.)alſo A D will be known, whoſe quadrate you will find, if fromthe Square of B A you ſubtract the Square of B D.

Therefore if by Hypotheſis , B A, be 3, its Square is 9; andB D 2, its Square is 4, which ſubtracted from 9, there remainsthe quadrate 5, whoſe root is 2, 23 and is the right line DA:The weight therefore on B A to the whole weight of the Bellon both axes B and C, is as 2, 23 to 6,0: But becauſe thereſiſtance of like ſolids is in triplicate proportion of their homo-jogal ſides(and in Cylinders the proportion of their diameters ishad) if you ſeck two mean proportional numbers between 6,oo and 2, 23 Which you may do if you draw the quadrate ofone extream into the other extream, for the Cube root of tlieproduct is the term next to that number whoſe ſquare you aſ-ſumed: Therefore firſt, the Square of 65 oo being 36, oc oo,draw it into 2, 23 and the cube root of the product S oꝛ8ooοοο152 1, 3: But the quadrate of the other extream 2, 23 1849729being drawn into 6, os, gives 298 3740 whoſe cube root 3 10,is the other mean; Therefore the 4 numbers 600, 43 13,3 10, 223,are continual proportionals, the fractions being llighted;Wherefore if you make it as COO to 437, ſo the ſemidlameterA E to B̃ N, this ſemidiameter ſought will be ſuffcient to bearthe weight.

And becauſe B E is the double of A E, and AE to B N ismade as 600 to 431, BE will be to B N, 28 1 200 to 431; Andaccording to this ſame proportion will the Semiquadrants bethat are deſcribed by them. But the eighth part of the Peri.pbery from the radius B E, is equal to the quadrant ſrom the ra-dius A E; therefore the quadrant E O to the Semiquadrant ofthe radius B N, is alſo as 1200, to 431, Therefore if the rubingupon of the axis ot the bell with the fixed Prop and hole, be 1200,

the

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