• LOG
is called by Mr. Cotes ratio /nodularis. This ratio is that of om to oe, if op by increafing proportionally, from being
equal to oe, becomes equal to om, in the fame time that P,
by its uniform motion, defcribcs a line equal to the modulus
o e. Ibid. Art. 175, 753.
This ratio nodularis is found to be that of 2,718281828459,
fsV. to I ; or of 1 to 0.367879441171, tjfc. Cotes, Harm.
Menfur. p. 7. •
If a logarithm z be given, the number correfponding,
if z be a natural logarithm, will be expreffed by this
feries, 1 + 2 + i z> -)- J zi 4. J, 2* + m z ! +
T»o z 6 +, ^ or generally, if the modulus be M,
by this, 1+^+ ^+ 1_ + , J*, which when a = M is
chang'd into I + l-K-K+A+Tfc+s&'C'Ori-f-i- +i— +
I 1. 2
-+-
•+-
-+:
L £f\. =
1.2. 3 ' I.2.3.4 ' 1.2. 3.4. 5 " 1.2. 3.4. 5.6 TS
2.7 1 828 1 8, He. the ratio of which to I is the ratio modu- lans. See Mac Laurin, Flux. Art. 753. Cotes, Harm. Mcnf. p. 6, 7.
We have faid, that, in Lord Napier's firft fyftem, the con- ftant velocity of P, by which the logarithm A P is generated, is equal to the velocity with which p fets out from a. There- fore, if we fuppofe A P and ap to be diminifhed continually,
AP being always the logarithm of op, their ratio (hall con- tinually approach to a ratio of equality as its limit. (See Limit.) And, oa being fuppofed to reprefent unity, if ap be very fmall compared with oa, it may be fuppofed in approximations to be the logarithm of op, in this fyftem ; and any very fmall fraaion may be fuppofed to be the logarithm of the fum of unity and that fraction added together. From this it follows, that if a feries of mean proportionals be interpofed between oa and any given line oh, and the number of all the terms, exclufive of oa, be x; and on being the fecond term of the feries, an be to ab as unity is to the number q : then, by increafing continually the number of mean proportionals betwixt a and 1, the ratio of x to q fhall approach continually to the ratio of. A B (the logarithm of b) to ah, as its limit. For the number x is to unity as A B is to AN (the logarithm of on;) and unity is to $ as a n (which approaches continually to AN) is to ab. Therefore theratioof x to q approaches continually to that of A B to a b. For inftance, if oh be double of a, the ratio of x to q approaches continually to the ratio of the natural logarithm of 2 to unity, which is nearly that of 7 to 10. Mac Laurin, lib. cit. Art. 170 and De Moivre Doftr. of Chances, Probl. 5. Hyperbolic Logarithms are the fame as natural logarithms, or thofe of Napier's firft fcheme.
The reafon of the appellation is the analogy hyperbolic fpaccs have to logarithms. Thus,
It is known, that if on the afymptote O S of an hyperbola LMN, the abfeiffe O A, OP, OG, OR, £sV. beta- ken in geometric progreflion, the correfponding feclors LOU,hON,L'OQ.,a r . or the fegments LAPM, , LAG N, LARQ_,cJ'i:. will Ci R S be in arithmetical progref- fion. Hence, the fegments or feflors will be the logarithms of their correfponding ab- fciftx, OP, OG, OR, He. And the feaor L O M, or the fegment LAPM meafures the ratio O P to O A. The modulus of this fyftem of logarithms is the parallelograms O A L B, which being fuppofed unity, thefe logarithm coin- cide with Napier's.
Hyperbolic lectors meafure ratios as circular feflors or arches meafure angles. See Mac Laurin, 1. c. Art. 177. L'Hopttal$e8t.'Cofiiq, Art. 221. LOGARITHMIC (Cycl.)— Logarithmic curve. This curve was called logarithmic or logiflic by Huygcns : though others had conhdered it before him, none had named it. He has given us feveral curious properties relating to it, at the end of his treatifc of the caufe of gravity, but without demon- Orations. Thefe have been fince fupplied by Guido Grandi. See Hi/yg. Oper. Vol. 2. Amftel. 1728. Thefe properties are,
x. The fpaces comprehended between two ordinates, are as the difference of thofe ordinates. Thus let AVD be a loga- rithmic, and its ordinates A B, V C, D Q_; let the two laft continued meet A K, a parallel to the afymptote B O, in E and
Q.A
LOG
K, then will the fpaces A B C V, A B Q_D be to each other as E V to K D.
2. If A O be a L tangent in A, in- terfering CE and KQ_inl,G; the
E. fpaces AVE, AD K, will be to each other as VI and DG.
3. The fpace A K S QJ3 ; is to the
infinite fpace be- yond Q_D, and lying between the curve and its a- fymptote, as K D toDQ;.
The accurate author obferves, that when he fays the infinite fpace has a certain proportion to a finite one, he means that this infinite, or indefinite fpace, may approach fo near to a given fpace, having that certain proportion to another, that the difference may become lefs than any ailigned quantity.
4. The fubtangent is conftant.
5. This fubtangent may be found by approximation, and is to the portion of the afymptote intercepted between ordi- nates in the ratio of 2 to 1, as 43429, (Sc. to 30103, or nearly as 13 to 9. That is, as the modulus of Briggs's fyf- tem, to the logarithm of 2. See Logarithm.
6. If there be three ordinates, as AD, GH, BF, and if from the extremity B of the leaft, there be drawn a parallel to the afymptote B K, cutting the other two ordinates in R, K ; as alfo a tangent B Q_, cutting them in N, Q_; the tri-
jy[ linear fpaces ABK, H B R, will be to each other as the portions of the ordinates intercept- ed between the curve and the tangent, that is, as AQ_, HN. 7. The infinite fpace Y between the curve and its afymptote, lying be- yond any ordinate BF is double of the triangle
B F O, formed by that ordinate, the fubtangent and tangent.
In which fenfe this is to be undcrftood. See above N' 3.
8. The fpace comprehended between two ordinates is equal to the re&angle under the fubtangent and the difference of the ordinates. Thus the fpace A B F D is equal to the rec- tangle under F O, A K, or to B M X Y.
9. The folid formed by the revolution cf the infhi'e fpace lying beyond any ordinate, about the afymptote, is to a cone of the height of the fubtangent, with a bafe equal to the circle defcribed with that ordinate as radius, as 3 to 2. Thus the folid formed by the revolution of the indefinite fpace FBTZ, about F Z is to the cone formed by the revolu- tion of the triangle F B O, as 3 to 2.
jo. The folid formed by the revolution of the fame infinite fpace about the ordinate F B, is fextuple of the cone formed by the triangle F B O, revolving about F B.
11. The diftance of the center of gravity of this infinite plain fpace from the ordinate B F, is the length of the fub- tangent F O.
12. The diftance of this center from the afymptote is one fourth of the ordinate.
13. The diftance of the center of gravity of the firft of the before-mentioned folids, from its bafe, is one half of the fub- tangent.
14. The diftance of the center of gravity of the other folid, from its infinite bafe, is one eighth of its axis.
15. If there be two fegments of an hyperbola, comprized
between erdi-
D A nates to one of
its afymptotes, and if the ordi- nates of the one be to each other, as the ordinates AD, HG of the logarithmic ; and it the ordi- nates of the o- ther, be as B F, to C E ; then will the hyper- bolic fpaces be to each other, as DG to FE. And thefe hy- perbolic
