LOG 



LOG 



from Lagauia, lie may rot be at a lofs what to call them. 

 Thele are 



L. ferpyllijolia. Br. n. 9 — " Somewhat (lirubby. Leaves 

 ovate. .Siipiilas w'thin the footilalks, fringed like the calyx. 

 Flowers terminal, fomewhat corymbofo." — Gathered by 

 Mr. Brown in the foiith part of New Holland. 



L.piifiila. Br. n. 10.—" Herbaceous. Leaves clliptT- 

 cal. Stip'.ilas triangular, within the footftalks. Flowers 

 axillary, folitary." — Native of Port Jackfon. 



L. campmiu/atn. Br. n. II "Herbaceous. Leaves 



linear, v.-ithuut liipulas. Flowers terminal. Flower-ftalks 

 and calyx downy." — From the fouth part of New FIol- 

 land. 



LOG.ARITHMIC, Atmo.si'iierical, is a curve 

 {Phue XL Anahfis, Jig. 2.) dcfcribed in the following 

 manner: let the point C reprefent the centre of the earth, 

 C A the earth's femidiamctcr, and A B any height above 

 the furfacc ; at A , place a right line A D, of any finite 

 length, at right angles with AC. In, the right line AC, 

 towards C, take A p fuch, tliat C A may be to A ^ in the 

 proportion of C B to B A. In a right line drawn through 

 ,3, at right angles with A C, lake |3 E, of fuch length, as 

 ito be to A D in the proportion of the denfuy of the air at 

 B to the denlity at A, the earth's fnrface. The curve, 

 which the point E always touches, is a logarithmic, of 

 which A C is the afymptote ; and is called by Dr. Horflcy 

 the atmofpherico'I logarithmic. 



Imagine this curve defcribed, and take another height 



C A X A ^ 



A b, and take Ate = , and draw €e parallel to 



/3E, meeting the curve in e. Then foS is tlie logarithm of 

 the ration of f3 E to to f, or of the denlity at B to the denfity 

 at b. But if the greater of the two heights, A B and 

 A b, bear but a very fmall proportion to the femidiameter 

 of the earth, their difference B b will be very nearly equal 

 topb. 



For, becaufe C B ; B A = C A ; A/S by conltruftion. 

 Therefore, by converfion, CB : CA = CA : C;3. 



In like manner, and by inverfion, CA; Cb = C? :CA, 

 by equi-dillance perturbatc, CB : Cb = C? : C^, 



and converting, CB : Jib =: C^ : f(3, 



by permutation, Bi : ,3^ =CB:Cb. 



But when A B is infinitely diminifhed, C B = C A ulti- 

 iiialely. Alfo A i being infinitely diniinidicd, C ? = C A 

 ultimately. Therefore C B = C te ultimately, and B ^ = 

 ^ ? ultimately. Q. E. D. 



Now A B and A b will always be fo fmall, with refpeft 

 to C A, if B and i be fuppofed to reprefent any acccffible 

 places, that C B, C S, anu B b, fiS, may always, in this 

 iafe, be confidered as in their ultimate proportion of 

 equality. 



It is ftill therefore to be admitted, as a principle, in prac- 

 tice, that the difference of elevation of any two places is as 

 the difference of the tabular logarithms of the heights of the 

 ijaickfilver in the barometer at the fame time at both places ; 

 that is, it is the logarithm of the ratio of thofe heights in 

 fome fyftem of logarithms. And the heights of the quick- 

 lilver being given by obfenation, the difference of elevation 

 will be known, if that particular fyllem. can be determined ; 

 that is, if the 7>wdulus of thi fyfti-m, or the length of the 

 lubtangent of the curve D Ef of that fyftem, can be afcer- 

 tained, in fome known meafure, as Enghfh fathoms, or 

 Paris toifes. 



Theeaficft method of doing this, that theory fuggefls, is 

 to compare barometers at two ilations, fuppofe B and b, 

 .each of a kuown elevation A B and A^ above the level of 



the fea. For the logarithms of any given ratio, in different 

 fyflems, are proportional to the fubtangents ; and the dif- 

 ference of elevation, B b, diminifhed in the proportion of 

 C B, (the diitance of the higher ilation from the earth's 

 centre,) to C ^, (a third proportional to C i, the dillance 

 of the lower ftation from the earth's centre, and C A, the 

 earth's femidiameter,) is tlie logarithm of the ratio of the 

 denfity at B, to the denfity at "h, (that is, of the columns of 

 quickfilver fuftained in the barometer at B and b,) in the at- 

 inofpherical fyllem. Therefore, as the difference of the 

 tabular logarithms of tliefe columns, to the fubtangent of 

 the tabular fyftem, fo (liould Bi, diminiflied as hath been 

 iaid, (that is, fo {liould /So,) be to tlie lubtangent of the 

 atmofpherical logarithmic. The utmoll height to which we 

 can afccnd, above the level of the fea, is fo fmall, that the 

 reduttion of B i may, even, in this inveftigation, always be 

 neglcfled. For, if A B were four Englifli miles, which 

 e.'^cecds the greateft accclTible heights, even of the Peruvian 

 mountains, and A 1? three, (3t would be fcarce one part in 

 500 lefs than Bi. So that, by comparing barometers at 

 different elevations, within a mile above the level of the fea, 

 the lubtangent of the atmofpherical curve might be deter- 

 mined, as it Ihould feem, witliout fenfible error, by taking- 

 fimply the difference of elevation, without reducfion, for the 

 logarithm of the ratio of the obferved height of the quick- 

 filver in the atmofpiierical fyilem. 



The fubtangent is different in length at different times ; 

 though M. de Luc has (hewn, that it is conflant in a given 

 temperature ; fo that if the temperature of the air is + l6-J 

 of his fcale, the difference of the tabular logarithms of the 

 heights of the quickfilver in the barometer, gives the dif- 

 ference of elevation in looodthsof a Paris toife; whence the 

 number, which is the modulus of Briggs's fyftem, exprcffes 

 the length of the fubtangent of the atmofpherical curve, 

 fuch as it is in that temperature, in looodths of a Paris toife, 

 Phil. Tranf. vol. hiv. part i. p. 231, &c. 



LoftAUiTiiMic, or Logistic Curve, is a curve which 

 obtained its name from its properties and ufes in explaining 

 and conftrutling logarithms ; becaufe its ordinates are in 

 geometrical progreffion, while the correfponding abfciffas 

 are in arithmetical progreffion ; fo that the abfciffas are 

 the logarithms of the correfponding ordinates. Hence the 

 curve may be conftrudled in the following manner. Fig. 3. 

 F late XI. Jnalyfis. 



Upon any right line as an axis, take the equal parts A B, 

 B C, CD, &c. or the arithmetical progreffion A B, AC, 

 AD, &c. and at the points A, B, C, D, &c. ereft the 

 perpendicular ordinates A P, B Q, C R, D S, &c. in a 

 geometrical progreffion, and the curve line drawn through 

 the extremities of thefe ordinates P, Q, R, S, &c. is the 

 logarithmic or logiftic curve, its abfcift'as A B, A C, A D, 

 being as the logarithms of the refpetlive correfponding or- 

 dinates B g, C R, D S, S:c. 



Hence, 1? any abfciffa A N = .-<;, its ordinate N O = _y, 

 A P = I, and rt = a certain conftant quantity, or the 

 modulus of the logarithms, then the equation of the curve 

 is .V — a X log. y = log. ji" ; the fluxion of which being 



taken, it will be ;c = — ; whence the following proportion, 



y 



but in any curve v : .v ;: y : the fubtangent A T, and 

 therefore the fubtangent to this curve, is evtry where equal 

 to the fame conftast quantity a, the modulus of the lo- 

 garithms. 



'L'o'fiod the «rea contained hettuccn any t-wo cidinates. — 

 ■' Here 



