IM 



AK 



i 



i i i i 



1*14*1' """7 

 Q IAT t *. . C i. 



c = 4 



= 



and in the hyperbola (as will readily appear from 

 FLI-IIONS, $ 150, Ex 5.) 



(t *\ . * *\ 



,+, > '=*,_< > 



ratio of magnitude to be the numerical ratio between the Imaginary 

 magnitudes of the two lines and the ratio of potit ion, ^ 

 the inclination of the one line to the other, or the angle '*""V" > 

 they contain. Again he lays it down, that four straight 

 lines are in projjortion of magnitude and position, win n 

 between the two last there is the<Mme ratio of magni- 

 tude iind position as between the two first. That is, 

 a, b, c, d, to be the magnitudes, we must 



i art perfectly analogous in their form, 

 'in the first set */+ 1 be put instead of the 

 /I, it will be immnrlistfly trans- 

 i into the second. By the first act of formula, the 

 theory of the Arithmetic of Sines may be invea- 

 d. and by the second, a corresponding theory re- 

 bong to the co-ordinates of an hyperbola, many of the 

 ' ; i of the two curves will be i 



I., ll.r .,tl,, : r 



tieal. and some will d.ffrr only in the sign, of the terms. 

 In both, the rewiha will be alike free from the ima- 

 ginary sign, although in the on* cos* the steps by 

 which they have been found are unintelligible, and in 

 the other they art perfectly significant. This agree- 

 ment of two method* so very different at the discovery 

 of truth, the ingenious writer *ttiibmas to the analogy 

 that take* place between the subjects of investigation, 

 which is w dose, that every property of the one m*y. 



Ktion*. be transferred to i" 

 it hapjH-nv th.t ;r,- .,,* r-li. ;;< ,.,-ri, n 

 v character*, although destitute of 



i u reference to others 



rrctlv s method of 

 i property of the hyperbola, and 

 then leave us to conclude from analogy that the same 

 H uys.il i belong* slao to the circle. All that we art as- 



lUfnl of ' ) '. .* !'! k " ' f * : " ^ r !_.!'. tl I Till :t t >-n- 



II ' nTb*' proved of the hyperbek ; hut if from 



ooU trenafcr that n*rlm.*i to the cnxle. 



ksswst be m tunnqinaui of the 



have -7- = -T, also the angle contained by a and i equal 

 6 a 



to the angle contained by e and d, 



\\ hen the consequent of the first ratio is the antece- 

 dent of the second, the proportion of magnitude and 

 position is said to be continued, and the middle term is 

 a mean proportional of magnitude and position between 

 the other two. From this it follows, that the middle 

 term bisects the angle made by the two extremes. 



These observations being premised, he gives the fol- 

 lowing theorem as the foundation of his theory. Ima. 

 ftnart, quaxtutn of the n,rm + a+S 1 represent in the 

 ft- Hieitif of potiliun perpendiculari to the a fit of the at- 

 citur, ana tectpr.ifally, i*r/**dictilarj to the Of it of the 

 abtituc are tmngtnanet of tfiii form. For, putting -|-o 

 and a for straight lines lying in opposite directions, 

 according to M Fran<jai, the quantity r: a v/ 1 is a 

 mean prouattjonal of magnitude and position between 

 Now it has been premised, that a line, which 



yet nfltes of re 



1 '.. \. i. t '..,' 



TW 



is prowl of the by- 



1 nW to I I tru. '..f 



of theewves, w,il 



boa bren hromrht forward bv M Beiee. m lha 

 PHL 2Vmw IMJsl, also by M. Arnmi, in a w. 



ism * fmpm*imt*nty i so that the 



. y^-l. histomlrf btms| the sign o an a 



siaTUt m*tk*aV mMMa 



MM WOm immw flO 



i that ia, if a line to tnr ngot be 



a* esmal tee to the left by - 

 sjenmtakw to and eual to eit 



equal to either of these will be 



is a msan proportional in magnitude and position be- 

 tween two lines, ought to bisect the angle they con- 

 lore, in the present case, the mean roust be 

 perpendicular to the axis of the abciasse, and will lie 

 hove or below the axis according as it i +">/ 1, or 

 a+/ I. Reciprocally, every perpendicular to the 

 axis of the abciisoj moat, according to the same princi- 

 ple, be men proportional between +m and a ; it is 

 therefi*! an imaginary quantity of the form rfca^/ 1. 

 Sch is the substance of M Francais* demomtration ; 

 bwt to M it seem* to be by no mean* setisnctory We 



1 ami 



br-n jut ow 



r erj c* e moiv 



y. end sAer the Hikmst rxasmstauoa, will be 



i. have 1,0 othar cmhv to the evidence of <h> 



ia seeking the 

 it +a and 



t, h.. 



.I'niajf- 

 wouW 



here seiight the mean of meanitiids independently of 

 ppekiim. and then the mean of position independently 

 Thee* would have required different 

 first would have given +/< X = for 

 lid, putting o for a right an. 

 gW. the second would have given 4(9+9)=? ' or to 

 mean of position. W* cannot, however, we any useful 

 emathssson deducible from the remit. The author of 

 the theory, bv calling the one* 4-o and a, seems to 

 h*v* inverted them at once with magnitude and post, 

 tion while at the seme time be seeks the mean by a 

 which applies to them as thing* having only 

 tie. 



i proof offered by Mr. Buee in support of the 

 of the proposition, that ^ I is the sign of per- 

 |iss>Mi nlei ilj . i* not more conclusive. He suppose* 

 three equal straight lines to meet in a point, two of 

 them to be in one strsight line, and the third to be at 

 right angles to them both He calls the line taken to 

 the right -f I , then that taken to the left, he says, must 

 be I . and the third, which must be a mean propor- 

 tional between them, must be v/ (I*), or more sim- 

 ply v/ I. Hence he infers, that v/ I is the sign of 

 ptrpntdtcularity. The inconcluiiveness of this reason- 

 ing, ha* been well exposed by an able critic in the 

 lifclllilah Review, vol. xii. July IftOH, where it is ob- 

 served, that any imaginable conclusion might have been 

 derived in the tame manner. For example, the third 

 line, instead of being at right angles, may be supposed 



