tB4 



PHILOSOPHICAL TRANSACTIONS. 



[anno 1777. 



eonsideration of molion or velocity, applicable to every thing to which fluxions 

 have been applied; and shall now proceed to the subject of this paper, after pre- 

 mising the two following detinitions. 



Defin. \ . Magnitude is that which admits of increase or decrease. 



Defin. 2. Quantity is the degree of magnitude. 



By magnitude, besides extension, I mean every thing which admits of more 

 or less, or what can be increased or diminished, sucli as ratios, velocities, powers, 

 &:c. As I shall frequently, for the sake of conciseness and conveniency, be 

 obliged to make use of particular modes of expressing geometrical magnitudes, 

 I here observe, once for all, that by such expressions as these 



A. -, A. ^ ~ , A. , A. — , he. I mean respectively a 3d proportional to 



E and a; a 4th proportional to b, a, and the difference of a and b; a 4th pro- 

 portional to D, A, and the difference of c and d ; a 4th proportional to b, a. 



anc 



B, 8iC. 



K M L 



M 



N 



To proceed then in the order in which I 

 first investigated these theorems; let a, b, c, 

 D, E, F, G, H, &c. be any number of magni- 

 tudes of the same kind, taken two and two 

 from the first; and let mn, no, nr, op, res- 

 pectively represent a, b, c, d. Let nr, op, 

 be drawn perpendicularly to vo, or otherwise 

 if in the same angle, and let the rectangles 

 or parallelograms mr, np, be completed. Let lm be a 4th proportional to op, 

 MN and NR — OP ; and let the rectangle or parallelogram La be completed. 



Then (14 e 6) lt is equal to tr, and consequently lq to mr. But (23 e 6) 

 MR has to NP the ratio compounded of the ratios of mn to no and nr to op. 

 Therefore (l e 6) ln has to no the ratio compounded of the ratios of mn to no 



and NR to OP. But ln is equal to mn -|- mn. , or a -|- a. , by con- 

 struction. Whence it appears, that a magnitude of the same kind with a and 

 B, which has to b the ratio compounded of the ratios of a to b and c to d, is 



expressed by a -f a. . 



In like manner let e, f be represented by rn, op, respectively, and let lk be a 

 4th proportional to op, ln, and qr. Then (14 e. 6) kx is equal to xr or tk 

 and xs together. But since ln has already been shown to be equal to a -f- a. 

 - ~ , LK is a 4th proportional to f, e — f, and a -I- a. ; that is equal to 



A. ^— -^ + A.^^^. ^~, by construction. Therefore kn, being equal to lk 



— IJ , 1 - F , C — D F, — F . 1 • 



U A. h A. . . And smcc kq is 



-f- LN, is equal to a -\- a. 



