LOGARITHMS. 



In the fame manner it appear?, that the logaritlim of the index of the power of which the root !• to he found. But 



«ube is triple ; of the biquadratc. quadruple; of the fifih each of thcfe rules will require a more particular illuftration, 



power, quintuple ; of the fixth, fextuple, &c. of tha loga- which will be found in the fuhftquent part of this article, 

 ritlim of the root. 'fhe properties of logarithms hitherto mentioned, and 



Unity, therefore, is to the exponent of the power, as ths their various ufcs, are taken not.ce of by Stiftlius : but they 



logarithm of the root to the logarithm of the power. come all far (hort of the uff of logarithms in trigonometry. 



So that the logarithm of the power is had, if the loga- firll difcovered by John Napier, haron of Merehiilon, in 



rithnt of the root be multiplied by its exponent; and the 

 logarithm of the root h had, if the logarithm of the power be 

 divided by its exponent. 



And hence we derive one of the 2reat ufes of logarithms, 

 which is to expedite and faAlitate the bufmefs of multiplica- 

 tion, involution of powers, and txtradlion of roots ; the 

 former of which is here performed by mere addition, and the 

 two latter by multiplication and divifion. Thus ?, the fum 

 of the logarithms I and 2, is the logarithm of S, the pro- 

 duft of 2 and 4. In like manner 7, the fum of the loga- 

 rithms 2 and 5, is the logarithm of 128, the produdl of 4 

 snd ^2. Again, 6, the logarithm of 64, which is the tiiird 

 power of 4, or 4", is equal to j K 2. And 8, the loga- 

 rithm of 256, which is the fourth power of 4, or 4-', is equal 

 to 4 X 2. Moreover, ^, the log-irilhm of the fquare root 

 S, is half the logarithm 6, of the Square 64 ; and 2, the lo 



Scotland, and iirll publiflii-d at Edinburgh in 1614, in his 

 Mirilici Logarithmorum Canonis Defcnptio. This work 

 was trandated bv Mr. Edward Wright, and publiilied by his 

 for., with the aflillance of Mr. Driggs, in the year 1616 or 

 1618. The method of conflnn^ting the table was rcferved 

 by the ingenious author, till the fenie of the learned upon 

 his invention (hould be known ; neverthelefs Kepler, in hii 

 Chilias Logarithmorum ad totidem Numeros rotnndos, 

 pubhihed at Marpurg in 1724 ; Speidell in his New Lo.-^a- 

 rithms, publifhed in i6ig, and republiflied with conlider- 

 able additions, in a iixth impredion in 1624 ; Benj. Urhnius, 

 in his Table of Logarithms, printed at Cologne in 1625, 

 and others, at home and abroad, laboured at the computation 

 of logarithms, and conllruAcd Imall tables, conformable to 

 the plan of lord Najuer But of all thofe who aflided in 

 the conftrudion of logarithmic tables, Briggs is moll 



garithm of the cube root 4, is one-third the logarithm 6 of the confpicuous ; it was he who fu-ll fuggcllod our prefent fyf- 



cube 64. 



2. If the logarithm of unity be o, ihe logarithm of the quo- 

 tient 'will be equal to the dijference of the logarithms of the 

 di-uifor and dividend. —For as the divifor is to the dividend, 

 fo is unity to the quotient ; therefore the logarithm of the 

 quotient is a fourth equidiffeivnt number to tlic loganthinsof 

 the divifor, the dividend, and the logarithm of unity. The 

 logarithm of unity, therefore, being o, the difference of the 

 logarithm of the divifor, and that of the dividend, is the lo- 

 garithm of the quotient. Q. E. D. 



Hence appears another great advantage of log-irithnis ; 

 via. their expediting the bufmefs of divifion, and p rforn- 

 jng it by a bare fubtraftion. E. gr. 2, the differt- iice be- 

 tween 7 and J, is the logarithm of the quotient 4, obtained 

 by dividing 128 by 32. In like manner, j, the difTcrcnce 

 between 8 and j, is the logarithm of the quotient 32, ob- 

 tained by dividing 256 by 8. 



Thefe properties of logarithms, however, are more obvious 

 according to our latter definition. For in that cafe, if 

 r' = a, and r' ^ b, » and y being the logarithms of a and b, 

 we have immediately from the firlt principles of algebra, 



j^ = f' 



= ab 



r" ~ r> = 





r ' = 





= C/' 



Multiplication. 



Divifion. 



Involution. 

 Evolution. 



From which formnlx it is evident, that the logarithm 

 «f the product of a multiplied by b is equal to the fum 

 •f the logarithms of a and b. The logarithm of the quo- 

 tient of a divided by b, is equal to the difference of the 

 logarithms of a and b. The logarithm of the «th power 

 of a is equal to n times the logarithm of a. And the 

 logarithm of the nth root of a, is equal to the logarithm 

 cf a divided by n. Therefore, univerlally, to multiply two 

 numbers together, we mutl take the fum of their loga- 

 rithms : to divide one number by another, we lubtract the 

 logarithm of the latter from the logarithm of the former. 

 To involve a number to any power, we muft multiply its 

 logarithm by the index of the power. And to extract the 

 joot of any number, we mull divide its logarithm by the 



tern, and laboured more than any one in the computation ot 

 the numbers it contains. In the prefent Hate of analylis 

 many comparatively fliort and f afy methods may be employed 

 for this purpole, that were unknown to the early writers ; 

 and for want of which the labour attending the firlt com. 

 putation was exceedingly great ; fome idea of which may 

 be formed from the following illiillralion. 



To find the logarithm of any number, according to Briggs' s 

 method. — I. Becaufe, I. 10. 1^0. 1000. joooo, &c. conlli- 

 tuie a geometrical progreffion, their logarithms maybe taken 

 at pleafure : to be able, then, to cxprcls the logarithms of the 

 intermed.ate numbers by decimal fractions, take 0.00000000, 

 i.oooooooo, 2.00000000, 3.00000000, 4.00000000. &c. 

 2. It is manifeft, that for thofe numbers which are not con- 

 tained in the fcale of geometrical progreffion, the juit lo- 

 gar;tlims cannot be had : yet they may be had fo near the 

 truth, that, as to matters of ufe, they fhall be altogether as 

 good as if ilriftly jull. To make this appear, fuppofe 

 the I'garithm of the number 9 were required ; between 

 1.0000000 and 10.0000000, find a mean proportional, and 

 belvecn their logarithms 0.00000000, and i.oooooooo an 

 equidiffereut mean, which will be the logarithm thereof ; 

 that IS, of a number exceeding three by liV-ViyVcJ ^nd 

 therefore far remote from nine. Between 3 and 10, there- 

 fore, lind another mean proportional, which may come fome- 

 what nearer 9 ; and between 10 and this mean another ilili ; 

 and fo on between the numbers next greater and next lef* 

 than 9, till at lall you arrive at 9 \ o o o ■:, o . o - ; which 

 not being one millionth part from 9, its logarithm may, 

 without any lenlibie error, be taken for that of 9 itfelf. 

 Seeking then in each cafe for the logarithms of the mean 

 proportionals, yoH will at laft have 0.954251, which is ex- 

 ceedingly near the true logarithm of 9. 3, If in like man- 

 ner you lind mean proportionals between 1.0000000 and 

 3 1622777, and alhgn the proper logarithms to each, you 

 will at length have the logarithm of the number 2, and 

 fo of the rell. 



Such was the method employed by the «arly computors 

 of logarithms : and though they had certain means of 

 abridging the operations in particular cafes, yet it is evi- 

 dent that the computation of them was not effefted with- 

 out immenfe labour ; a particular and interefting account of 

 which, with an explanation of the feveral modifications of 



tb« 



