LOG 



LOG 



Here the fluxion of the area A, or y x, is y x — =ay; 



which coiTeAed •rives A — a (AF — y) = a (AP— NO) 

 = a X P V = A T X P V. That is, tlie area A P O N, 

 between any two ordinatcs, is equal to the rectangle of the 

 conlla.': fubtangent, and the difference of the ordinatcs. 

 And hence, when the abfcifTa is infinite, or the Idfl ordinate 

 equal to zero ; then the iniinitcly long area APZ isequal 

 to A T X A P, ur double the triangle A P T. 



Tojini the content of the foliil formed by the revolution of the 

 curve about its avis A Z. — The fluxion of the folid S = 



fy''x—fy'x—=f>ayy, where /> = 3.14159, &c. ; 



y 



and the correft fluent is S = i p a x (AP^ — V ) = 

 :i/> X AT X (A P' — N O ), which is half the difference 

 between two cylinders of the common altitude tr, or A T, 

 and the radii of their bafes A P, NO. And hence, fup- 

 pofinj; the axis infiinte toivards Z, and confequently the 

 ordii.ate at its extremity zero, the content of the infinitely 

 long folid will be equal to Ip a y A P^ = i/i x A T x 

 A P , or Half tiie cylinder on the fame bafe and its altitude 

 A T. 



This curve erroatly facilitates the conception of loga- 

 rithms, and affords a very obvious proof of the very impor- 

 tant propcrry of their fluxions, or very fmall increments ; 

 namely, that the fluxion of a number, is to the fluxion of its 

 logarithm as the number is to the fubtangent. As alfo 

 this property, that if their numbers be taken very nearly 

 equal, fo that their ratios may differ but a little from a ratio 

 of equality, then- difference will be very nearly proportional 

 to the logarithm of the ratio of thefe numbers to each other ; 

 which follows from the logarithmic arcs being very little dif- 

 ferent trom their chords when they are taken vei-y fmall. 

 The conilant fubtangent of this cui-\'e is, what Cotes calls, 

 the modulus of the fyflem of logarithms. This curve has 

 been treated of by a great number of very eminent mathema- 

 ticians, as Huy gens, Le Seur, Keil, Bevnouilli, Emerfon, &c. 

 See the latter author's Treatife on Curve Lines, page 19. 



LoGAKiTiTMic, Hyperlolic. See Yi\PEHTiOLic Logarithms. 



LtiGAUiTHMlc, or Loganihniical, relating to logarithms. 

 Thus we fay, logarithmic.7r;/ATOf//o-, curve, lit:e,fca/e.fpiral. 



^LOGARITHMS, formed from the Greek Xoyo?, ratio, 

 and af,ifj.c:, number ; q. d. ratio of numbers ; the indices of 

 the ratios of numberfi one to another ; or a feries of arti- 

 ficial numbers proceeding in arithmetical proportion, cor- 

 reiponding to as many others proceeding in geometrical 

 proportion ; contrived for the ealing and expediting of cal- 

 culation. 



Logarithms have been ufually defined niimcrorum propor- 

 tlonalium xquidifferentes comitts ; but this definition Dr. Hal- 

 ley and Stifelius think deficient, and more accurately de- 

 fine ti'.em, the indices or exponents of the radios of numbers ; 

 ratio being confidered as a quantity fui generis, beginning 

 from the ratio 01 equality, or i to I = o, and being af- 

 firmative when the ratio is increafmg, and negative when it 

 is decreafing. But a more limple idea of thefe numbers 

 may be formed from the following definition, iv's. The lo- 

 garithm of a number is that exponent of fome other num- 

 ber, which renders the power of the latter equal to the 

 former : thus if r' = a, r' = b, r' — c, &c. tlien is x the 

 logarithm ui a ; y the logarithm of i ; s the logarithm of 

 e, &c. Alfo r is tlien called the radix of the fyfteni, which 

 may be allumed at pkafure j but in the common tables the 

 radix is always 10. 



We will conlider ihefe numbers under, each of the two 



fatter definitions. According to the firft ; if nrity be made 

 the common conftquent of all ratios, or the common (land- 

 ard to which all other numbers are to be referred, 'hen 

 every logarithm will be the numeral exponent of the ratio 

 of its natural number to tmity. E.gr. the ratio of 8l to 1 

 contains the four following ratios, viz. that of 81 to 27, 

 17 to 9, 9 to 3, and 3 to 1 , or ° ' 



i I 



2 7 



X 



T » 



but all thefe ratios are equal to one another, and 

 fZ = ^ X ^ X J- X I = 0' ; confequently the logarithm' 

 of 81, is four times as great as that of 3. In the fame 

 manner, the ratio of 24 to I, or \'^ = .j-*- x '-- X i — ^ 

 X I X f^ ; and, therefore, the logarilhm of 24 is equal to 

 the fum of the logarithms of 2, 3, and 4. And, univcr- 

 fally, the magnitude of the ratio of A to i, is to the mag- 

 nitude (if the r.itio of B to i, as the logarithm of A to 

 the logarithm of B. Hence we derive a method of mea- 

 furing all ratios whatever, let their ccnlequents be what 

 they will : e. g. the ratio of A to B, is the excefs of the 

 ratio of A to i, above the ratio of B to i ; therefore the nu- 

 meral exponent of the ratio of A to B, will be the excefs of 

 the numeral exponent of the ratio of A to I, above the nu- 

 meral exponent of the ratio of B to 1, that is, the excefs of 

 the logarithm of A above the logarilhm of B : therefore the 

 magnitude of the ratio of A to B is to the magnitude of 

 the ratio of C to D as the e.xcefs of the logarithm of A 

 above the logarithm of B, which is the meafure of the for- 

 mer ratio, is to the excefs of the logarithm of C above that 

 of D, which is the meafure of the latter ratio : and thus 

 we fee that logarithms are as true and proper meafurea o£ 

 ratios, as circular arcs are of angles > 



The nature and genius of logarithms w.ll be eafily con* 

 ceived from what follows : — -A feries of quantities increailng 

 or decreafing according to the fame ratio is called a geo- 

 metricid progrcjion ; e. gr. I. 2. 4. 8. 16 32, &c. A feric3 

 of quantities increafinj or decreafing, according to the fame 

 difference, is called an arithmetical progrejfian ; e. gr. 3. 6. 9. 

 12. 15. 18. 21. Now, if under the numbers proceeding in 

 a gedraetrical ratio, be placed as many of thuf- proceeding 

 in the arithmetical one, thefe laft are crdled the lo'ariLhnw 

 of the firfl. 



Suppofe e.gr. two progrefuons :. 

 Geomet. i. 2. 4. 8. 16. 32. 64. 1,28. 256. 5.12* 

 Ariihinet. o. 1. 2. 3. 4. 5. 6. 7. 8. 9. 

 Logarithms. 

 O will be the logarithm of the firll term ; vix. I ; c, of the 

 6th, 32 ; 7, the logarithm of the 8th, ic8, ^c. 



Thefe indices or logarithms may be adapted to any geo- 

 metric feries ; and, therefore, there may be as many kinds of. 

 indices or logarithms, as there can be taken kinds jaf geo- 

 metric feries ; but the logarithms mod co.-ivcnicnt for com- 

 mon ufe, are thole adapted to a geometrical feries increafino- 

 in a ten-fold progreilioii, as in the fequel. The doctrine and 

 ufe of logarithms may be conceived from the following 

 propofitions. 



1. If the logarithm of unity be O, the kgarithm of the futturrt: 

 or prrjduB '■mH! be eqiid to the fum of the logarithm^ uf ths 

 fiitors. — For as unity is to one of ti.e fattors, fa is the 

 other factor to the product. So that the K)^arithm of the 

 produtl is 3 fourth eqiiidiflerent term to the logariihjTi of 

 unity, and thofe of the faftors ; but the logarithm of unity 

 being o, the fum of the logarithms of the f iclors mull be 

 the logarithm of the fadtum, or produft. Q. E.D. Hence,. 

 fincethefaAors of a fqirare are equal to each other, i.e. a 

 fquare is the factum or product of its root multiplied into 

 itfelf, the logarithm of the f^uare will be littubk ihe loga- 

 rithm of the root. 



Id 



