19. 5403295 7. 6794279 11. 8609016 be divided by IK, you must take the Distance CS= IA, and then ST will be the Quotient, whofe Loga rithm is OA+OC-OI. Let CD=0. 347, IK =0.00478. Then add the Logarithm of Unity to the Logarithm of CD; that is, put 1 or 10 before the Index thereof, and from that fubtract the Logarithm of the Divisor, and the Remainder will be the Logarithm of the Quotient, whofe Index 11. fhews that the Quotient is between the Numbers 10 and 100; and I feek the Number answering the Logarithm, which I find to be 72, 542. If the Logarithm of a Vulgar Fraction, for Example, required, the Logarithm of Unity must be added to the Logarithm of the Numerator 7, or which is all one, you must put 10 or 100 before the Index thereof, and fubduct from it the Logarithm of the Denominator 8, and there will remain the Logarithm of the Vulgar Fraction, or the Decimal .875. be 10. 8450980 ༠.༡༠༣༠༡༠༠ 9. 9420080 If the Powers of any Fraction DC be required, you muft affume 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, AE2 AC=2 AO-2 OC, whence OEOA-AE 2OC-OA, that is, the Logarithm of the Square is the Double of the Logarithm of the Root, lefs the Logarithm of Unity. In like manner, fince AG= 3 AC 3 OA-3 OC, we fhall have O GO A— AG3OC-2 OA the Logarithm of the Cube triple the Logarithm of the Root, the Double of the Logarithm of Unity. For the fame Reafon, because AI4 AC=4 OA-4 OC, we have OI=4OC-3 OA, which is the Logarithm of the Biquadrate. And univerfally, if the Power of a Fraction be n, and the Logarithm L, then fhall the Logarithm of the Power nn L-n OA+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 2-1, the Logarithm of the Power n of that Fraction will be had. 3. For For Example, if it is required to find the 6th Power of the Fraction, 05 the Logarithm of this Fraction is 8.6989700, which being multiplied by 6, gives the Number 52. 1938200; and if from 52 the Number 50, which is the Index of the Logarithm of Unity drawn into 5, be taken away, the Remainder will be the Logarithm of the 6th Power, viz. 2.1938200, to which the Number ,0000000 15625 anfwers. For the Index 2 fhews that 7 Cyphers must be put before the firft Figure. If the 8th Power of the Fraction,05 be required, by multiplying the Logarithm by 8, there will be produced 69.5917600; and fince 70, which is feven times the Index of the Logarithm of Unity, cannot be taken from 69, unless we run into negative Numbers, the Index of the Logarithm of Unity muft be fuppofed 100, and then the Index of the Logarithm of the Fraction will be 98. Now this Logarithm drawn into 8, gives 789. 5917600, and if 700, which is 7 times the Index of the Logarithm of Unity, be taken from 789, there will remain 89. 5917600, the Logarithm of the 8th Power of the Fraction 2, whofe correfpondent Number is,0000000000 39062; for fince the Index is 89, and the Difference thereof from 100 is 11; the firft fignificative Figure of the Fraction fhall be in the 11th Place from Unity; and fo there must be 10 Cyphers placed before it. If the Roots of the Powers of Fractions be defired, for Example, the Square Root of the Fraction E F, because the Root is a mean Proportional between the Fraction and Unity, you muft bifect AE in C, and then CD will be the fquare Root of the Fraction EF. But ACAE= QA-OE and fo the Loga 2 rithm of the Root OA-AC if the Cube Root of the Fraction G H be fought, this if this be taken from O A, there will OA+OG=OC=Logarithm of the Cube 3 Root of the Fraction GH. So likewife the biquadrate Root of the Fraction IK will be had, by dividing AI into four equal Parts, for the Root is the firft of three mean Proportionals between Unity and the Fraction, and confequently, if AC4 A I, then will CD be the biquadrate Root of OA-OI and fo OC the Fraction IK. But&AI= —OA—AC_3OA+01 4 4 And univerfally, if the Root of any Power n of the Fraction LM be required, the Logarithm of the Root thereof will be" OA-OA+OL that is, if the η Number - be prefix'd to the Index of the Logarithm, and the Logarithm thus augmented be divided by n, the Quotient will give the Logarithm of the Root fought. So if the Cube Root of the Fraction or .5 be fought, you must place z=n-i (fince the Cube Root is required) before the Logarithm thereof, and there will be had 29,6989700, a third Part of which is 9,8996566, which is equal to the Logarithm of the Cube Root of the Fraction, and the Number 7937 answering to this Logarithm, is the Root fought. CHA P. IV. Of the Rule of Proportion by Logarithms. T HE Rule of Proportion fhews how, by having three Numbers given, a fourth Proportional to them may be found, viz. if the fecond and third Terms be multiplied by one another, and the Product divided by the firft Term, then will the Quotient be the fourth proportional Term fought. But this fourth Term is much easier found by Logarithms; for if the Logarithm of the firft Term be taken from the Sum of the Logarithms of the second and third Term, the Num ber remaining will be the Logarithm of the fourth fought. Ör this may be done fomething easier yet, if instead of the Logarithm of the firft Term be taken its Complement Arithmetical, or the Difference of that Logarithm, and the Number 10. 0000000, which is done by fetting down the Difference between each Figure of the Logarithm, and the Figure 9; for then if that Arithmetical Complement be added to the Sum of the other two Logarithms, and if Unity, which is the first Figure to the left Hand, be taken from the Sum, the Remainder will be the Logarithm of the fourth Term fought; and fo by this Way, Logarithms of the fourth Term are found by only one Addition of three Numbers. The Reafon of this will be manifeft from hence: Let there be three Numbers A, B, C, from which the firft is to be taken from the Sum of the fecond and third. Now this may not only be done by the common Way, but likewife, if there be any other third Number E taken, and from this there be taken A, there will remain E-A, and if the Numbers B, C, and E-A be all added together, and from their Sum be taken E, there will remain B+C-A. So if the Number 1.5 be to be taken from 23, take the Complement of the Number 15 to 100, which is 85, and add this Number to 23, and 23 the Sum will be 108, from which 100 being 108 taken, there remains the Number 8. 85 Hence follow fome Trigonometrical Examples of the Rule of Proportion folv'd by Logarithms. Let A B C be a right-angled Triangle, wherein are given, the Angle A 36 Degrees 46', the Angle B 98 Degrees 32', and the Side BC 3478, the Side A Cis required. Say (by Cafe 1. of plain Trig.) as the Sine of the Angle A is to the Sine of the Angle B, fo is BC to A C. And because the Logarithm Sine of the Arith. Comp. S, A. Log. B C. 0.2228938 9.9951656 3.5413296 X3.7593890 Angle A is the first Term of the Analogy, I fubftitute its Complement Arithmetical for the fame, and add the Logarithm of BC, the Logarithm of S, B, and the faid Complement all three together, and reject Unity, which is in the firft Place to the left Hand, and then the Logarithm of the Side A C will be given, and the Number answering thereto is 5706,306 equal to the Side fought A C. Let there be a spherical Triangle ABC, in which are given all the Sides, viz. BC= 30 Degrees, AB 24 Degrees 4', and AC 42 Degrees 8', the Angle B is required. Let BA be produced to M, fo that BM BC, then will A M, the Difference of the Sides B C, B A, be equal to 5 Degrees 56. Now (by Cafe 11. in oblique-angled fpherical Triangles) fay, as the Rectangle under the Sines of the Legs, is to the Square of Radius, fo is the Rectangle under the AC+AM AC-AM Sines of the Arcs Square of the Sine of one half the Angle B. AC+AM But 2 2 2 ་ to the AC-AM =24 Degrees 2, andAC 2 18 Degrees 6'; and because the first Term of the Analogy is the Rectangle under the Sines of AB, BC, and fecond Term is the Square of Radius, the Sum of the Logarithm Sine of A B, BC, muft be taken from double the Logarithm of Radius, and what remains must be added to the Sum of the Logarithm S, of AC+AM AC-AM which is the fame as if and 2 the Logarithm Sines of each of the Arcs AB, BC, were fub the Complements and the faid Sines be all added together; then shall the Sum be the Logarithm of the Square of the Sine of half the Angle B. And fo the half of the Logarithm 9.8965274 is the Logarithm Sine of half the Angle B51 Degrees 59′ 56′′, and the Double of this Angle fhall be 103 Degrees 59′ 52′′B, which was fought. CHAP. Z 4 |