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first Place to the Left-hand; for they are lefser than the Logarithm of the Number 10, whose Beginning is Unity, and the Logarithms of the Numbers between 10 and 100 begin with Unity; for they are greater than 1,0000000, and less than 2,0000000. Also the Logarithms between 100 and 1000 begin with 2; for they are greater than the Logarithm of 100 which begins with 2, and less than the Logarithm of 1000 that begins with 3. In the same manner it is demonstrated,
that the first Figure to the Left-hand of the Loga: rithms between 1000 and 10000 muft be 3; and he
first Figure to the Left-hand of the Logarithms between 10000 and 100000 will be 4; and so on.
The first Figure of every Logarithm to the Lefthand is called the Characteristic, or Index, because it thews the highest or most remote Place of the Number from the Place of Units. For Example, if the Index of a Logarithm be 1, then the highest or most remote Place from Unity of the correspondent Number, to the Left-hand, will be the Place of Tens. If the Index be 2, the most remote Figure of the correspondent Number shall be in the second Place from Unity, that is, it shall be in the Place of Hundreds ; and if the Index of a Logarithm be 3, the last Figure of the Number answering to it, Ihall be in the Place of Thousands. The Logarithms of all Numbers that are in decuple or fubdecuple Progreffion, only differ in their Characteristics, or Indices, they being written in all other places with the fame Figures. For Example, the Logarithms of the Numbers 17, 170, 176, 17000, are che lame, unless in the.r Indices; for since I is to 17, as 10 to 170, and as 100 to 1700, and as 1000 to 17000; therefore the Distances between 1 and
between 10 and 170, beiween ico and 1700, and be tween 1000 and 17000, shall be all equal. And so, since the Distance between 1 and 17, or the Logarithm of the Number 17 is 1.2304489, the Logarithm of the Number 170 will be=2.2304489, and the Logarithm of the Number 1700 Thall be 3.2304489, because the Logarithm of the Number 2.00000oo. In like manner, since the Logarithm of the Number 1000=3.0000000, the Logarithm of the Number 19@co fhall be 4.2304489.
So also the Numbers, 6748. 674,8. 67,48. 6,748. 0,6748. 0,06748, are continual Proportions in the Ratio of 10 to 1 ; and so their Distances from each 6 7 4 8 3,8291751 other shall be equal to the 6 7 4,8 2,8291751 Diftance or Logarithm of 6 724 8 1,8291751 the Number 10, or equal 6,7 4 8 0,8291751 to 1,00000co. And so, 0,6 7481-1,8291751 since the Logarithm of 0,06 7481-2,8291751 the Number 6748 is 3,8291751, the Logarithms of the other Numbers shall be as in the Margin; where you may observe, that the Indices of the last two Logarithms are only negative, and the other Figures positive; and so, when those other Figures are
to be added, the Indices must be fybtracted, and contrariwise.
Of the Arithmetic of Logarithms in whole
Numbers, or whole Numbers adjoined to
BEcause, in Multiplication, Unity is to the Multi
plier, as the Multiplicand is to the Product; the Distance between Unity and the Multiplier shall be equal to the Distance between the Multiplicand and the Product. If therefore the Number G H be to be multiplied by the Number EF, the Distance between GH and the Product must be equal to the Distance, AE, or to the Logarithm of the Multiplier ; and fo, if G L be taken equal to A E, the Number L M hall be the. Product ; that is, if the Logarithm of the Multiplicand AG be added to the Logarithm of the Multiplier A E, the Sum shall be the Logarithm of the Product.
In Division, the Divisor is to Unity, as the Dividend is to the Quotient; ard so the Diftance between the Divifor and Unity shall be equal to the Distance between the Dividend and the Quotient. So if L M be to be divided by EE, the Diftance E A shall be equal to the Distance between L M and the Quotient;
and so, if LG be taken equal to EA, the Quotient will be at G, that is, if from AL, the Logarithm of the Dividend, be taken GL, or AE, the Logarithm of the Divisor, there will remain AG, the Logarithm of the Quotient.
And from hence it appears, that whatsoever Operations in common Arithmetic are performed by multiplying or dividing of great Numbers, may be done much eafier, and more expeditiously, by the Addition or Subtraction of Logarithms.
For Example, Let the Number 7589 be to be multiplied by 6757. Now, if the Logarithms of those Numbers be Log. 3.8801846 added together, as in the Margin, Log. 3.8297539 their Sum will be the Logarithm Log. 7.7099385 of the Product, whose Index 7 thews, that there are seven Places of Figures, besides Unity, in the Product; and in seeking this Logarithm in Tables, or the nearest equal to it, I find that the Number answering thereto, which is lesser than the Product, is 51278000; and the Number greater than the Product is 51279000; and if the adjoined Differences, and proportional Parts, be taken, the Numbers that muft be added to the Place of Hundreds and Tens in the Product are 87; and that which must be added in the Place of Unity, will neceffarily be 3, since feven Times 9=63; and to the true Product shall be 51278873. If the Index of the Logarithm had been 8 or 9, then the Numbers to be added in the Place of Tens or Hundreds could not be had from those Tables of Logarithms which consist of but 7 Places of Figures, besides the Characteristic; and To, in this Case, the Valacquian or Briggian Tables should be used; in the former of which, the Logarithms are all to ten Places of Figures, and in the latter to fourteen.
If the Number 78596 be to be divided by 276, by subtracting the Log. 4.8954004 Logarithm of the Divisor from the Log. 2.4440448 Logarithm of the Dividend, the Log. 2.4513556 Logarithm of the Quotient will be had. And to this Logarithm, the Number 282, 719 answers; which therefore shall be the Quotient.
Because Unity, any assumed Number, the Square thereof; the Cube the Biquadrate, &c. are all con
tinual Proportionals, their Distances from each other shall be equal to one another. And so it is manifeft; that the Distance of the Square from Unity is double of the Distance of its Root from the same: Also the Distance of the Cube is triple of the Distance of its Root; and the Distance of the Biquadrate is quadruple of the Distance of its Root from Unity, &c. And lo, if the Logarithm of any Number be doubled, we shall have the Logarithm of its Square; if it be tripled, we shall have the Logarithm of its Cube ; and if it be quadrupled, the Logarithm of its Biquadrate. And contrariwise, if the Logarithm 'of any Number be bifected, we shall have the Logarithm of the Square Root thereof: Moreover, a third Part of the said Logarithm will be the Logarithin of the Cube Root of the Number; and a fourth Part, the Logarithm of the Biquadrate Root of that Number.
Hence, the Extractions of all Roots are easily performed, by dividing a Logarithm into as many Parts as there are Units in the Index of the Power. So if you want the Square Root of 5, the Half of 0,6989700 must be taken, and then that Half 0,3494850 will be the Logarithm of the Square Root of 5, or the Lo. garithm of w 5, to which the Number 2,236068 nearly answers.
CH A P. III.
Of the Arithmetic of Logarithms, when tbe
Numbers are Fractions. Fig. 3. WHEN Fractions are to be worked by Loga
rithms, it is neceffary, for avoiding the Trouble of adding one Part of a Logarithm, and subtracting the other, that Logarithms do not begin from an integral Unit, but from some Unit that is in the Tenth or Hundredth Place of Decimal Fractions : For Example, let PO be toortoort, and from this let the Logarithm begin. Now this Fraction is ten Times more distant from Unity to the Left-hand, than the Number so is distant therefrom to the Right; for there are 10 proportional Terms in the Ratio of 10 to 1, from Unity to PO. And so, if A B be Unity,
the Logarithm thereof, according to this Supposition, will not be o, but O A=10.0000000. Now the Distance of Ten from Unity is 1.2000000, whence the Distance of the Number ro from PO will be 11.0000000. Also the Distance of the Number 100 from PO, or its Logarithm, beginning from PO, shall be 12.0000000; and the Logarithm of 1000, or the Distance from PO, will be 13.0000000. And thus, the Indices of all Logarithms are augmented by the Number 10; and those Fractions whofe Indices are -1,or-2, or~-3, &c. are now made 9, 8, or 7, &c.
But if Logarithms begin from the Place of a Fraction, whose Numerator is Unity, and Denominator Unity with 100 Cyphers added to it (which they must do when Fractions occur that are less than PO), then that Fraction will be 100 Times more distant from Unity, than 10 is distant from it; and so the Logarithm of Unity will have 100 for the Indox thereof. And the Logarithm of any Tens will have 101 for the Index, that of any Hundreds 102, and so on; all the Indices being augmented by the Number 100.
The Logarithms of ali Fractions ebat are greater than PO (whereat they begin) will be p.:titive. And fince the Numbers 10, I, 1, töös too, &c. are in a continual Geometrical Progresiion, they will be equally diftant from each other; and accordmoly their Logarithms will be equidifferent: And fo, when the Logarithm of 10 is 11.00000oo, and ihe Logarithm of Unity is 10.0000000; then the Logarithm of the Fraction to will be 9.000.000, and the Logarithin of the Fraction will be 8.0000000; and, in like manner, the Index of the Logariihm of ooo will,
7. Also, for the same Reason, if the Index of the Logarithm of Unity he 100, and of 10 be 101, then will the Index of the Logarithm of the Fraction is be 99, and the Index of the Logarithm of will be 98, and the Index of the Logarithm of the Fraction to shall be 97, &c. And these Indices Thew in what Place from Unity the first Figure of the Fraction, not being a Cypher, must be put. For Example, if the Index be 4, the Diftance thereof from the Index of Unity (which is 10), viz. 6, Thews that the first significa- , tive Figure of the Decimal is in the sixth Place from Unity, and therefore five Cyphers are to be prefixed