8 



OF SPACE. 



50. Theorem. When the logarithm of 

 a quantity varies equably, the quantity varies 

 proportionally. 



Or, if 1. x—y, — =: — For xZZ.V f^a), and when v 

 ax 



becomesy+7/',3:+a.'=i'^ —V.y'—x.b'', andx'zix.i'— 



aizi. [y — l) ; but y' being constant by the supposition, 



i>' — 1 is constant, and may be called :—, and .r' ^ — ; 



a a 



xu i y 



therefore inr — , and — n — . 

 a X a 



Scholium. Numericallogarithms do not, strictly speak- 

 ing, vary by evanescent increments ; but other quantities 

 may flow continually, and be always proportionate to lo- 

 garithms : in either case the proposition is true. In Briggs's 

 logarithms, commonly used, 2i is 10, and a, the modulus, is 

 .43429448)9 ; dividing all the system by a, or multiplying 

 by Q.302585093, we have Napier's original hyperbolical lo- 



j," 

 garithms, where ^ becomes zz — , and a:z:i. 



X 



61. Theorem. The fluxion of any power 

 of a quantity, of which the exponent], is va- 

 riable, is equiil to the fluxion of the same 

 power considered as constant, together with 

 the fluxion of the exponent multiplied by 

 the power and by the hyperbolical logarithm 

 of the quantity. ^ 



If .x'-=:z, i:z.yx''-Ki + (h.l.x). x'i/ ; for h. 1. Jzri/. 



• ^ > 



(h. 1. .t), (42); now (h. 1. ») — -, (50);and izri. (h.l.^^J' 



—X. (y. (h. 1. .r)'-s,. (yr_+(h. 1. x).i/), (48, 50)z= 



X 



j'x'~'i-f(h. 1. x) xi/. I 



52. Theorem. When a variable quan- 

 tity is greatest or least, its fluxion vanishes. 



For a quantity is greatest when it ceases to increase, and 

 before it begins to decrease ; that is, when it has neither 

 increment nor decrement ; and it is least when it has 

 ceased to have a decrement and has not yet an increment. 



53. Problem. To solve a numerical 

 equation by approximation. 



The most general and useful mode of solving all nume- 

 rical equations is by approximation. Substitute for the 

 unknown quantity a number, found by trial, which nearly 

 answers to the conditions; then the error will be a finite 

 difTerence of the whole equation ; which, when small, will 



be to the error of the quantity substituted, nearly in the 

 ratio of the evanescent differences, or of the fluxions ; and 

 this ratio may be easily determined. 



Thus, if I*— 6x'+4x=6699, call 0699, 3/, then Zx'i— 



y ._._/_ y' 



I2xjr-J.4x:=y,and irz- 



and x'— - 



3x' — 12X-)-4> Sx''— 12x4-4 



nearly; now assume x:=2o, then j/z:5680, and y'— 1019, 

 whence x'zz 1.05, and x corrected is a 1.05; by repeating 

 the operation we may approach still nearer to the true 

 value 21. 



Ifx"=J(,r=- 



-' whence the common rule for the 



extraction of roots is derived. In order to find the nearest 

 integer root, the digits must be divided, beginning with 

 the units, into parcels of as many as there are units in the 

 index, and the nearest root of the last or highest parcel 

 being found, and its power subtracted, the remainder 

 must be divided by its next inferior power multiplied by 

 the given index, in order to find the next figure, adding 

 the next parcel to the remainder before the division. 

 There are also in particular cases other more compendious 

 methods. 



SECTION III. OF SPACE. 



54. Definition. A solid is a portion of 

 space limited in magnitude on all sides. 



Scholium. Space is a mode of existence incapable of 

 definition, and supposed to be understood by tradition. 



55. Definition. A surface is the limit 

 of a solid. 



56. Definition. A line is the hmit of 

 a surface. 



57. Definition. A point is the limit of 



a line. 



Scholium. The paper 

 of which this figure covers 

 a part, is an example of a 

 solid, the shaded portion represents a portion of surface: 

 the boundaries of that surface are lines, and the three ter- 

 minations or intersections of those lines are points. In 

 conformity with this more correct conception, these defi- 

 nitions are illustrated by representations of the respective 

 portions of space of which the limits are considered ; and 

 also by the more usual method of denoting a line by a 

 narrow surface, and a surface by such a line surround- 

 ing it. 



