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67 4 8 3,8291751 2,8291758

67
6 74,8

So alfo the Numbers, 6748. 674, 8. 67, 48. 6, 748. 0, 6748. 0, 06748, are continual Proportionals in the Ratio of 10 to 1; and fo their Distances from each other fhall be equal to the Distance or Logarithm of the Number 10, or equal to 1,0000000. And fo fince the Logarithm of the Num- 0,0 6 7 4 82,8291751 ber 6748 is 3,8291751, the

6 7,4 8

1,8291751

6,7 4 8

0,829175€

0,674 81,8292751

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Logarithms of the other Numbers fhall be as in the Margin; where you may obferve that the Indices of the last two Logarithms are only negative, and the other Figures pofitive; and fo when those other Figures are to be added, the Indices muft be fubtracted, and contrariwife.

CHA P. II.

Of the Arithmetick of Logarithms in whole Numbers, or whole Numbers adjoined to decimal Fractions. Fig. 2.

B

Ecaufe, in Multiplication, Unity is to the Multiplier, as the Multiplicand is to the Product, the Distance between Unity and the Multiplier, fhall 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 GL be taken equal to AE, the Number L M fhall be the Product, that is, if the Logarithm of the Multiplicand AG be added, to the Logarithm of the Multiplier A E, the Sum fhall be the Logarithm of the Product.

In Divifion, the Divifor is to Unity, as the Dividend is to the Quotient; and fo the Distance between the Divifor and Unity fhall be equal to the Distance between the Dividend and the Quotient. So if LM be to be divided by EF, the Distance E Afhall be equal to the Diftance between L M and the Quotient,

and

and fo 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 AE, the Logarithm of the Divifor, there will remain AG, the Logarithm of the Quotient.

And from hence it appears, that whatsoever Ope+ 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.

Log. 3. 8801846
Log. 3. 8297539
Log. 7. 7099385

For Example, let the Number 7589 be to be multiplied by 6757. Now, if the Logarithms of thofe Numbers be added together, as in the Margin their Sum will be the Logarithm of the Product, whofe Index 7 fhews that there are feven Places of Figures, befides Unity, in the Product; and in feeking this Logarithm in Tables, or the neareft equal to it, I find that the Number answering thereto, which is leffer 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 neceffarily be 3, fince feven times 9=63, and fo the true Product fhall 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, befides the Characteristick, and fo in this Cafe, the Vaquian or Briggian Tables fhould 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 fubtracting the Logarithm of the Divifor from the Logarithm of the Dividend, the Logarithm of the Quotient, will be had. And to this Logarithm, the Number 282, 719 answers; which therefore fhall be the Quotient. Becaufe Unity, any affumed Number, the Square thereof, the Cube, the Biquadrate, &c. are all con

Log. 4. 8954004
Log. 2. 4440448
Log. 2, 4513556

tinual Proportionals, their Distances from each other fhall be equal to one another. And fo it is manifeft, that the Distance of the Square from Unity, is double of the Distance of its Root from the fame: Alfo 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 fo if the Logarithm of any Number be doubled, we fhall 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 contrariwife, if the Logarithm of any Number be bifected, we fhall have the Logarithm of the square Root thereof: Moreover, a third Part of the faid 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 eafily performed, by dividing a Logarithm into as many Parts as there are Units in the Index of the Power. So if you want the fquare 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 5, to which the Number 2.23606 nearly answers.

CHA P. III.

Of the Arithmetick of Logarithms, when the Numbers are Fractions. Fig. 3.

W

HEN Fractions are to be worked by Logarithms, it is neceffary, for avoiding the Trouble of adding one Part of a Logarithm, and fubtracting the other, that Logarithms do not begin from an integral Unit, but from fome Unit that is the tenth or hundredth Place of Decimal Fractions: For Example, let PO be, and from this let the Logarithms begin. Now this Fraction is ten times more diftant from Unity to the left Hand, than the Number 10 is diftant therefrom to the right; for there are 10 proportional Terms in the Ratio of 10 to 1, from Unity to PO. And so if AB be Unity,

the

the Logarithm thereof, according to this Suppofition, will not be o, but OA will be 10.0000000; for. the Distance of any Tenth from Unity is 1.0000000, whence the Distance of the Number 10 from PO will be 11.0000000. Alfo the Diftance of the Number 100 from PO, or its Logarithm, beginning from P O, fhall be 12.000oooo, 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 thofe 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 diftant from Unity, than 10 is distant from it; and fo the Logarithm of Unity will have 100 for the Index thereof. And the Logarithm of any Tens will have 101 for the Index, that of any Hundreds 102, and fo on; all the Indices being augmented by the Number 100.

I

The Logarithms of all Fractions that are greater than PO (whereat they begin) will be pofitive. And fince the Numbers 10, 1, , TOT, TUBE, &c. are in a continued geometrical Progreffion, they will be equally diftant from each other; and accordingly their Logarithms will be equidifferent: And fo when the Logarithm of 10, is 11.0000000, and the Logarithm of Unity is 10. ooooooo, and the Logarithm of the Fraction will be 9. ooooooo, and the Logarithm of the Fraction will be 8.0000000, and in like Manner, the Index of the Logarithm of Too will be 7. Alfo 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 be 99, and the Index of the Logarithm of will be 98, and the Index of Logarithm of the Fraction fhall be 97, &c. And these Indices fhew in what Place from Unity, the firft 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, fhews that the firft Significative Figure of the Decimal, is in the fixth Place from Unity; and therefore, five Cyphers are to be prefixed

1000

thereto

thereto towards the left Hand. So alfo if the Index of Unity be 100, and the Index of the Fraction be 80, the first Figure thereof fhall be in the zoth 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 fhall be equal to the Distance between the Multiplicand and the Product. Therefore, if there be taken GI AC, the Product IK fhall 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=QI, 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 expreffed 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 o, 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 from the Sum, the Remainder is the Logarithm of the Product, whose Index 94 fhews that the first Figure of the Product is in the fixth Place from Unity, and fo there must be five Cyphers prefixed, and then the Product will be, oooo0642984.

97, 8556961

96, 9425041

94. 8082002

In Divifion, the Divifor is to Unity, as the Dividend is to the Quotient; and fo the Distance between the Divifor and Unity fhall be equal to the Distance between the Dividend and the Quotient. And fo 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 CAOA-O C, which being added to OI, we have OA+OI-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 Divifor, there will remain the Logarithm of the Quotients fo if the Number CD be to

be

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