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and so if LG be taken equal to E A, the Quotient will be at G: that is, if from AL, the Logarithm of the Dividend, be taken GL, or A E, the Logarithm of the Divisor, there will remain AG, the Logarithm of the Quotient.
And from hence it appears, that whatsoeverOpet tations in common Arithmethick are performed by multiplying or dividing of great Numbers, may be much easier, and more expediently done by the Addition or Subtraction of Logarithms.
For Example, let the Number 7589 be to be mula tiplied by 6757. Now, if the Logarithms of thofe Numbers be Log. 3. 8801 846 added together, as in the Margin Log. 3. 8297539 their Sum will be the Logarithm Log. 7. 7099385 of the Product, whofe Index 7 shews that there are feven Places of Figures, besides Unity, in the Products and in feeking 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 must 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 necessarily be 3, since seven times 9=63, and so 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 Hundredths or Tenths, could not be had from those Tables of Logarithms which confift but of 7 Places of Figures, besides the Characteristick, and so in this case, the Plaquian 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 78956 be to be divided by, 278, by subtracting the Log. 4. 8954004 Logarithm of the Divisor from Log. 2. 4440448 the Logarithm of the Dividend, Log. 2, 4513556 the 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 to it is manifeft; that the Distance of the Square from Unity, is double of the Distance of its Root from the fame : Also the Diftance 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 fo 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 Logarithm of the Cube Root of the Number; and a fourth Part, the Logarithm of the Biquadrate Root of that Number.
Hence, the Extraction 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 Logarithm of V 5, to which the Number 2.23606 nearly answers.
CH A P. III.
Of the Arithmetick of Logarithms, when
the Numbers are Fractions. Fig. 3.
HEN Fractions are to be worked by Loga-
ble of adding one Part of a Logarithm, and fubtracting the other, that Logarithms do not begin from an integral Unit, but from
some Unit that is the tenth or hundredth Place of Decimal Fractions : For Example, let PO be jo , and from this let the Logarithms begin. Now this Fraction is ten times more distant from Unity to the left Hand, than the Number 10 is distant therefrom to the right; for there are no 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 Suppofition,
will not be o, but O A will be 10.000.000; for i the Distance of any Tenth from Unity is 1.000000o,
whence the Distance of the Number 10 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 whose Indices are -1, or —2,or --3, & ca 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 Index thereof. And the Logarithm of any Tens will have ion for the Index, that of any Hundreds 102, and so on; all the Indices being augmented by the Number 100.
The Logarithms of all Fractions that are greater than PO (whereat they begin) will be positive. And since the Numbers 10, 1, the tot, ToT, &c. are in a continued geometrical Progression, they will be equally distant from each other ; and accordingly their Logarithms will be equidifferent : And so when the Logarithm of 10, is 11.0000000, and the Logarithm of Unity is 10. 0000000, and the Logarithm of the Fraction will be 9. 0000000, and the Logarithm of the Fraction tha will be 8.0000000, and in like Manner, the Index of the Logarithm of doo will be
Also for the fame Reason, if the Index of the Logarithm of Unity be 100, and of 10 be 101, then will the Index of the Logarithm of the Fraction it be 99, and the Index of the Logarithm of too will be 98, and the Index of Logarithm of the Fraction too shall be 97, &c. And these Indices shew 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 Distance thereof from the Index of Unity, (which is 10) viz. 6, shews that the first Significative Figure of the Decimal, is in the sixth Place from Unity; and therefore, five Cyphers are to be prefixed z
thereto towards the left Hand. So allo if the Index of Unity be 100, and the Index of the Fraction be 8o, the first Figure thereof shall be in the 20th Place from Unity, and 19 Cyphers are to be prefixed thereto.
Now, let it be required to multiply the Fraction GH by the Fraction DC. Because Unity is to the Multiplier, 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. Therefore, if there be taken GI=AC, the Product IK shall be at I. And accordingly, if from O G, the Logarithm of the Multiplicand, there be taken GI or AC, there will remain OI, the Logarithm of the Product. But A COA-OC, which taken from OG, there will remain OG+OCOA=O1, that is, if the Logarithm of the Multiplier and Multiplicand be added together, and from the Sum be taken the Logarithm of Unity, (which is always expressed by 10 or 100 with Cyphers) the Logarithm of the Product will be had. For Example, let the decimal Fraction o, 00734 be to be multiplied by the Fraction 0, 000876. Set down 100 for the Index of the Logarithm of Unity, and then the Logarithms of the Fractions will be as in the Margin, which being added together, and the Logarithm of Unity being taken away 97, 8556961 from the Sum, the Remainder is the 96, 9425041 Logarithm of the Product, whose In- 94. 8082002 dex 94 thews that the first Figure of the Product is in the fixth Place froin Unity, and so there must be five Cyphers prefixed, and then the Product will be, o0000642984.
In Division, the Divifor is to Unity, as the Dividend is to the Quotient ; and fo the Diftance between the Divifor and Unity shall be equal to the Distance between the Dividend and the Quotient. And so if the Fraction IK be to be divided by DC, you must take IG=CA, and the Place of the Quotient shall be G. But CA=OA-OC, which being added to OI, we have O A+01-OC=OG, that is, if the Logarithm of Unity be added to the Logarithm of the Dividend, and from the Sum be taken the Logarithm of the Divisor, there will remain the Logarithm of the Quotient-s so if the Number CD be to
be divided by IK, you must take the Distance CSS IA, and then ST will be the Quotient, whose Logarithm is OA+OC-O1. Let CD=0. 347, IK =0.00478. Then add the Logarithm of Unity to the Logarithm of CD; 19. 5403295 that is, put i or 10 before the Index 7. 6794279 thereof, and from that fubtract the Lo
11. 8609016 garithm of the Divisor, and the Remainder will be the Logarithm of the Quotient, whose Index 11. fhews that the Quotient is between the Numbers 10 and 100; and I seek the Number answering the Logarithm, which I find to be 72, 542. If the Logarithm of a Vulgar Fraction, for Example, z be required, the Logarithm of Unity must be added to the Logarithm of the Nu 10. 8450980 merator 7, or which is all one, you must 0. 9030900 put to or 100 before the Index thereof,
9. 9420080 and subduct from it the Logarithm of the Denominator 8, and there will remain the Logarithm of 'the Vulgar Fraction }, or the Decimal .875.
If the Powers of any Fraction DC be required, you must assume EC, EG, GI, IL, each equal to AC; and then EF will be the Square, GH the Cube, and IK the Biquadrate of the Number DC; for they are continually proportional from Unity. Befides, A'E= 2 AC=2 A 0-2 OC, whence OE=OA - AE =20C-OA, that is, the Logarithm of the Square is the Double of the Logarithm of the Root, less the Logarithm of Unity. In like manner, since AG= 3 A CS3 OA-30 C, we shall have O G=OA AG=30C-20A = the Logarithm of the Cube =triple the Logarithm of the Root, – the Double of the Logarithm of Unity. For the fame Reafon, because Al=4 AC=4 OA-40 C, we have Ol=40C-30 A, which is the Logarithm of the Biquadrate. And universally, if the Power of a Fraction be n, and the Logarithm L, then shall the Loga. rithm of the Power n=nL nOA+OA, that is, if the Logarithm of a Fraction be multiplied by n, and from the Product be taken the Logarithm of Unity, multiplied by » -- 1, the Logarithm of the Power n of that Fraction will be had.