Demonjlrations of the Fundumenial Property of the Lever enumerated. 541 



interefting to afcertain preci'fely whether the colouring matter has not more analogy witli 

 earths than has hitherto been thought, and whether it would no: be advantageous, particu. 

 larly m red dies, to employ the foaps of tin and of cochineal. 



IV. 



Obfervaliom on the Fundamental Property of the Lever .- -with a Proof of the Principle ajfumed 

 by Archimedei in his Demonjlration. By the Rev. S. Fl^'CE * A.M. F.R S 



J. HE want of a demonftration of the property of the lever upon clear and felf-evident 

 prmciples has juftiy been confidered as a great defideratum (defed) in the fcienceof mechanic . 

 as the moft important parts of that branch of natural philofophy are founded upon it 

 Arclnmedes was, I believe, the firfl who attempted it. He fuppofes, that if two equal bo- 

 dies be placed upon a lever, their efted to turn it about any point is the fame as if they 

 were placed in the middle point between them. This propofitlon is byjio means felf-evi- 

 dent, and therefore the inveftigation which is founded upon it has been rejeded as imper-' 

 te«. Huygens obferves, that fome mathematicians, not fatisfied with the principle here 

 taken for granted, have, by altering the form of the demonftration, endeavoured to render 

 Its defefls lefs fenf.ble, but without fuccefs. He then attempts a demonftration of his 

 own, in which he takes for granted, that if the fame weight be removed to a greater 

 diftance from the fulcrum, the effect to turn about the ievdr will be greater : tliis is a prin 

 Ciple by nomeans to be admitted, when we are fuppofed to be totally ignorant of the 

 effeas of weights upon a lever at different diftances from the fulcrum. Moreover if it 

 were felf-evident his demonftration only holds when the lengths of the arms are commen- 

 lurable. Sir I.Newton has given a demonftration, in which it is fuppofed, that if a given 

 weight aa m any direftion, and any radii be drawn from the fulcrum to the line of di 

 reaion, the effea to turn the lever will be the fome, on whichever of the radii it -.ds 

 But fome of the moft eminent mathematicians fince his time have objeded to- this prin" 

 ople as being far from felf-evident; and, in confequence thereof, have attempted to de- 

 monftrate the propofition upon more clear and fatisfadory principles. The demonftr.nion 

 by M Laur.n, as far as it goes, is certainly very fotisfaaory ; but as he colleds the truth of 

 the propofition only from induaion, and has not extended it to the cafe where the arms 

 arc incommenfurable, his demonftration is imperfed. The demonftration given by'Dr" 

 Hamilton, ,n his Effays, depends upon this propofition, that wJien a body is at reft and 

 aaed upon by three forces, they will be as the tiiree fides of a triangle parallel to the'di 

 reaions of the forces. Now this is true when the three forces ad at any point of a body ; 

 whereas, confidcring the lever as the body, the three forces .id at different points, and 



rrcfumcd «.. known .0 chc ancient mailers. On this fubjca. c.nfult rhc jd c.hi.r of ,hc Journal of .i,c 



I'olyttchnic School, p 426 and 427. "" 01 me 

 • Phil. Tranf. m.ucc.xciv. 33. 



'■ therefore 



