VOL. XXI.] PHILOSOPHICAL TRANSACTIONS. 457 



As I undertook, to demonstrate some theorems to geometricians, I did not think 

 it necessary to pursue every thing very minutely. But that I may oblige the 

 animadvertor, I will now demonstrate that lemma, Prop. I, because I cannot 

 express it more fully than I have already done, in the following words : — 

 " Three powers, constituted in equilibrio, have the same ratio as three right 

 lines, which are parallel to the directions of the powers, or are inclined in a 

 given angle, and terminated by their mutual concourse." 



As suppose three powers are in equilibrio, that either draw, press, or any 

 how act according to three right lines pa, pb, pc, fig. 6, pi. 11; and let the 

 three right lines ef, fd, de, be inclined to these directions in any given angle ; 

 that is, let the angles eaf, fbp, dcp, be equal : I say the powers a, b, c, are to 

 one another, as the right lines fe, fd, de. Let the right lines ap, bp, cp, be 

 produced to g, h, k. In the quadrilateral faep, because by the hypotheses the 

 external angle eap is equal to the internal and opposite angle pbf, the two in- 

 ternal opposite angles pap and fbp will be equal to two right angles; and since 

 all the four internal angles are equal to four right angles, the other two angles 

 F and apb, opposite in the same quadrilateral, will also be equal to two right 

 angles. But apb and bpg make two right ones ; therefore the angle f is equal 

 to BPG. In like manner d and e may be shown equal to bpk and apk. 



Now because the three powers are in equilibrio, they are immoveable, and 

 therefore any one of them, in respect of the two others that remain in equilibrio, 

 may be considered as a fulcrum. If b is the fulcrum, by a welt known theorem 

 in mechanics, the power a is to the power c, as the sine of the angle bpk to 

 the sine of the angle bpg, that is, as the sine of the angle d to the sine of the 

 angle p ; that is, as the right line fe to the right line de. Again, supposing 

 c the fulcrum, the power a is to the power b, as the sine of the angle cph to 

 the sine of the angle cpg, or the sine of the angle bpk to the sine of the angle 

 APK ; that is, the sign of the angle d to the sine of the angle e, or as the right 

 line FE to FD. Therefore the three powers a, b, and c, are as the right lines 

 E, FD, and de. q. e. d. 



We must now say something about the application of this mechanical lemma. 

 If the absolute gravity of the little line do, fig. 7, expounded by do, as said 

 above in Prop. 1 , is conceived to be collected in its centre of gravity m, and 

 this heavy line, by virtue of its gravity, endeavours to descend according to the 

 direction MF perpendicular to dD ; the power drawing according to md, which 

 is in equilibrio with the said heavy line by the foregoing lemma, is to its mo- 

 mentum or power drawing according to mf, as So is to ^d. For the angle Sv>A, 

 in which d^ is inclined to md, is equal to the angle dsF, in which AS \s, inclined 

 to MF, for each is the complement of the angle d to a right angle. And this 



VOL. IV. 3 N 



