LOG
What is here faid of the times of the motion of p when op increafes proportionally, is to be applied to tire fpaces de- scribed by P, in thofe times i with its uniform motion. Hence the chief properties of logarithms are deduced. , They are the meafures of ratios. The excefs of the logarithm of the antecedent above the logarithm of the confcquent mea- fures the ratio of thofe terms. The meafure of the ratio ot a greater quantity to a leffer is pofitive, as this ratio com- pounded with any other ratio increafes it. The ratio of equality compounded with any other ratio, neither increafes nor diminifhes it ; and its meafure is nothing. The mea- fure of the ratio of a leffer quantity to a greater is negative, as this ratio compounded with any other ratio diminifhes it. The ratio of any quantity A to unity, compounded with the ratio of unity, to A, produces the ratio of A to A, or the tatio of equality ; and the meafures of thofe two ratios deftroy each other when added together ; fo that when the one is confidered as pofitive, the other is to be confidered as ne- gative.
When op increafes proportionally, the motion of p is perpe- tually accelerated ; and, on the contrary, when op de- crcafes proportionally, the motion of p is perpetually re- tarded.
If the velocity of the point p be always as the diftance op, then will this line increafe or decreafe in the manner fup- pofed by Lord Napier ; and the velocity of the point p being the fluxion of the line op, will always vary in the fame ra- tio as this quantity itfelf. See Mac Laurin's Fluxions, Art. 151— 160.
The fluxion of any quantity is to the fluxion of its loga- rithm, as the quantity itfelf is to unity.
Hence, the fluxion of the logarithm of x will be — ■ For
x : 1 : : x : — = the fluxion of the logarithm required.
.v When op increafes proportionally, the increments generated in any equal times, are accurately in the fame ratio as the velocities of p, or the fluxions of op, at the beginning, end, or at any fimilar terms of thofe times.
When op increafes or decreafes proportionally, the fluxions of this line, of all the higher orders, increafe or decreafe in the fame proportion as the line itfelf increafes or decreafes ; fo that one rule ferves for comparing together thofe of any kind at different terms of the time. Aid in this cafe we never arrive at any conftant or invariable fluxion. If the logarithms of two quantities be always to each other in any invariable ratio, the fluxions of thofe quantities fhall be in a ratio that is compounded of the ratio of the quantities themfelves, and of the invariable ratio of their logarithms. Let p be greater than a ; ad : ap :: oa : op ; and let oa,
q a d e f g h k p x
a d, de, of, fg, &c. be continued in proportion : then by adding together a d, i de, \ef, 4 fg, C?V, we approximate continually to the value of A P, the logarithm of op. And we approximate continually to the logarithm of d, by fum- ming up the differences betwixt ad and i dc, § ef and if Si ? sgb ail d ! hi, &c.
Thefe theorems are demonftrated geometrically by Mr. Mac Laurin, lib. cit. art. 171, 172. From what has been faid, it follows, that if a : d :: op : ox, then the logarithm of x will be equal to the fum of the logarithms of op and d, that is, to the fum of a d-\- i de -f ' f ef+ i/f-f I g h+i hi, &c. mdad-—-ide+ \ ef— ±f g + >, gh —\hi, &c. which fum is 2 a d -f- * ef + * gh-^hz. Let aq zz: ad; then the logarithm of oa- will meafure the ratio of od to q. But odand oq have half their fum equal to oa, and half their difference equal to a d, which are the two firft terms of the geometric progrdflon oa, ad, de, ef, fg' lb, hi, &c. Hence, if sa— 1, and ad=x, de, 'f\fii &c - W 'M he refpeflively, x z , xi, x*, &c. and the ratioof l-\-x to I — x will be equal to that of d to q. But the logarithm of this ratio is 2 ad^-\ ef-\-\gh+, &c. there- fore the logarithm of ^~= 2 x x+ ^ *> + s a-' + § *5 -f &c.
agreeably to what has been fhewn by Dr. Halley and others. To find, for example, the logarithm of 2. From this ex-
preflion of the logarithm of ' . Then '~^~* = 2, or
1 — x 1 — x
i+* = 2 — 2x, and 3*= 1 or xz= ;, that is in decimals.
x — -33333333 t
tnd x — 0.33333333
x% = 37°3/°3
3 x3 — 1234568
X s = 411522
1 X s = 82304
- l = 45725
f x? ss 6532
- » = 5081
$>x9 = 720
x " = 564
Tr X V = 51
"',] = 6 3
r'l *' 3 = 5
- !l = j
Sum 0.34657513 Multiply by 2
0.69315026 Napier's.
LOG.
Logarithm or 2.
To find the kg. of 3, find firft the logarithm of | — *T A '
I — X
where a- =4. To the logarithm of | add the logarithm of 2, and the fum will be the kgaritlmi of 3. If the logarithm
of 10 were required j fuppofe \y = — » thenar = *
To the logarithm of -^r add twice the logarithm of 3, and the fum will be the logarithm of 10, which will be found to
be 2.30258509, £&■. and its reciprocal or — , —
2.3025a, &£. will be =; 0.43429448, tff<\
Now to convert any of Nap'iers's or of the natural loga- rithms into BriggSi where the logarithm of 10 is unity, fay* as Napier's logarithm of 10 is to 1, fo is any other of Na- pier's logarithms to Brigg's logarithm of the fame number. Hence, Briggs's logarithm will be equal to Napier's divided by 2.30258509, I&c. or which comes to the lame thing, multiplied by its reciprocal, 0.43429448, is\: And vice •uerfa^ Brjggs's logarithm being given, to find Napier's cor- refponding, multiply Briggs's by 2.302585C9, &c. In genera!, to find Briggs's logarithm of any number m
Let — T- — n= », and confequcntly x = — •
l—x n-\-%
x by 0.86858896, &fa (= 2 x 0.43429, &c.) and then continually multiply this product by xx, and divide the re- fulting products, refpec~Uvely, by the terms of the progref- fion x, 3,5,7,6^. The fum of thefe quotients will be Briggs's logarithm required.
The firft logarithms being once found, we may thence form the logarithm of any prime number. For inftance, let it be required to find the logarithm of 19. Take two integers, one immediately greater, the other lefs than 19, that is, 18 and 20. Thefe not being primes, their logarithms may be found from thofe of their factors, 2, 3, and 5 ; thus /. 20 = 2/. 2 -f- l.$. and /. 18 = 2/. 3 -J- /., 2. But the fquare of the mean number 19 always exceeds the rectangle of the two above and below it by 1. (Since in general, ««=i K-f-iX n — 1 -f-i = BWtAl-^i). Find therefore the loga-
Multiply
-ithm of s
- = » where x = T \ , -
To this logarithm add
= /. 18 -f- /.20, the fum will be /. 361, garithm of 19. See Coles's Harm. Menf.
the logarithm of 360 : half of which is the It Prop. 2.
As to the various compendia arifmg in the practice of con- structing a table of logarithms ', we refer to Gardinei's, and to Sherwin's Tables.
We have already obferved, that Napier's, or the natural lo- garith?n^ of any quantity, is found by fuppofing the uniform motion of the point P by which the logarithm of op is ge- nerated, to be equal to the velocity of p, at the term of time when the logarithms begin to be generated. But the uniform motion of P may be fuppofed equal to the motion of^> at any other term, as when it comes to r, in which cafe the conftant velocity of P is to the velocity with which p fets out from a^ when the logarithm begins to be generated, as oe is to oa. Thus we may have as many lo- garithms as we pleafe ; and the line e is what Mr. Cotea calls the modulus of the fyftem.
The meafures of a given ratio in the different fyftems are in the fame proportion as the lines oe t e being always the term where the velocity of p becomes equal to the conftant velocity of P. Hence, the logarithm of any quantity in Napier's firft fyftem, becomes equal to the logarithm of the fame quantity in any other fyftem wbofe modulus is oe> by multiplying by the number which expreffes the ratio of oe to o a ; and the modulus of any fyftem is to' the modulus of any other fyftem, as the logarithm of any given quantity in the firft is to its logarithm in the fecond. Thus in Napier's firit: febeme, in the fame time that op from being equal to oa becomes equal to ten times oa, the point P deicribes a line, the ratio of which to oa is that of 2.30258509, feV. to unity. But it was found convenient that the logarithm of 10 fhou!d be 1 ; and the motion of P was fuppofed to be fa far diminifhed, that the fpace defcribed by it in that time might be equal only to oa ; that is, its velocity, in this cafe, was fuppofed lefs than its velocity in the former, in the fame ratio, as I is lefs than 2.30258, &c. If oe be taken lets than oa in the fame ratio, the velocity of P fhall be equal, in this cafe, to the velocity of p at e ; and oe will be the modulus of this fyftem, which will therefore be
exprefledby 0.43429448, tff. = 2,30^09^. ° 9 bfi "?g unity. Mac Laur. Lib. cit. Art. 174.
When a ratio is given, its meafure is always as the modulus of the fyftem. It is therefore the fame invariable ratio that is always meafured by the modulus of the fyftem, which
