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 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 fhall be equal to the Distance between the Multiplicand and the Product. Therefore, if there be taken G I-AC, the Product IK fhall be at I. And, accordingly, if from OG, the Logarithm of the Multiplicand, there be taken G I or A Č, there will remain OI, the Logarithm of the Product. But AC=OA-OC, which taken from OG, there will remain O G+OCOA=OI; 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 0,00734 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 from the Sum, the Remainder is the Logarithm of the Product, whofe Index 94 fhews, that the firft 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 0,00000642984. 97,8656961 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 Quotient. And fo, if the Fraction IK be to be divided by DC, you must take IG=CA, and the Place of the Quotient fhall be G. But CA≈OA~OC, 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 Quotient; fo if the Number CD be to be 19.5403295 7.6794279 11.8609016 be divided by I K, you must take the Distance CS= 1A, and then ST will be the Quotient, whofe Logarithm is OA+OC-OI. Let CD=0.347, IK, =0.00478. Then add the Logarithm of Unity to the Logarithm of CD; that is, pnt I or 10 before the Index thereof, and from that fubtract the Logarithm of the Divifor, 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 Legarithm, which I find to be 72,594. If the Logarithm of a Vulgar Fraction, for Example, be required, the Logarithm of Unity must be added to the Logafithm of the Numerator 7; or, which is all one, you must put 10 or 100 befor 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. 10.8450980 0 9030900 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, AE= 2 AC 2 AO-2 OC; whence O E OA-AE =2 OC-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 A G= 3 AC3OA-3 OC, we fhall have O GO AAG=3OC—2 Ŏ A≈ the Logarithm of the Cube, triple the Logarithm of the Root,-the Double of the Logarithm of Unity. For the fame Reason, because AI4 AC 4OA-4 OC, we have OI =40C-3OA, 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 O A÷O A; that is, if the Logarithm of a Fraction be multiplied by n, aud from the Product be taken the Logarithm of Unity, multiplied by n-1, the Logarithm of the Power n of that Fraction will be had. Z 4 For I 20 For Example, if it is required to find the 6th Power of the Fraction T,05, the Logarithm of this Fraction 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 ,000000015625 anfwers. For the Index 2 fhews, that 7 Cyphers must be put before the first 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, whofe correfpondent Number is ,000000000039062. 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 EF; becaufe the Root is a mean Proportional between the Fraction and Unity, you muft bifect AE in C, and then CD will be the Square Root of the Fraction EF. OA-OE But AC AE = 2 ; and fo the Loga rithm of the Root = OA-AC= OA+OE 2 And if the Cube Root of the Fraction GH be fought, this fhall be the first of two mean Proportionals between Unity and GH; and fo, if AG be divided into three equal Parts, the firft of which is AC, then CD shall be the Root fought: And becaufe AC=AG= OA-OG, if this be taken from QA, there will 3 remain ⚫ remain 2OA+OG 3 OC Logarithm of the Cube 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 A CAI, then will CD be the Biquadrate Root of the Fraction OA-OI ; and fo OC= IK. But AC÷AI= And univerfally, if the Root of any Power n of the Fraction L M be required, the Logarithm of the Root NOA OA+OL thereof will be ; that is, if the n Number 1 be prefixed 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 2=-1 (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, anfwering to this Logarithm, is the Root fought. CHA P. IV. Of the Rule of Proportion by Logarithms. THE 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 fift Term, then will the Quotient be the fourth proportional Term fought. But this fourth Term is much eafier 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 ber remaining will be the Logarithm of the fourth, fought. Or this may be done fomething eafier yet, if instead of the Logarithm of the first 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 firft 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, the Logarithm of the fourth Term is found by only one Addition of three Numbers. The reafon of this will be manifest from hence: Let there be three Numbers A, B, C, the firft of which is to be taken from the Sum of the fecond and third. Now this may not only be done by the common Way, but, likewise, if there be any other 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 15 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 the Sum will be 108, from which 100 being taken, there remains the Number 8. 85 23 108 Here follow fome Trigonometrical Examples of the Rule of Proportion folved by Logarithms. Let ABC be a Right-lined 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 Piane Trig.), as the Sine of the Angle A is to the Sine of the Angle B, fo is BC to A C. And becaufe the Logarithm Sine of the Arith. Comp. S, A. 0.2228938 Log. A C. 9.9951656 3.5413296 13.7593890 Angle A is the first Term of the Analogy, I substitute 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 |