Unity, which is in the firft Place to the Left-hand; and then the Logarithm of the Side AC will be given, and the Number anfwering thereto is $746,306, 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, A B 24 Degrees 3', and AC=42° Degrees 8'; the Angle B is required. Let B A be produced to M, so that BM BC; then will A M, the Difference of the Sides BC, BA, be equal to 5 Degrees 57'. Now (by Cafe 11. in oblique angled Spherical Triangles) fay, As the Rectangles under the Sines of the Legs is to the Square of Radius, fo is the Rectangle under the Sines of the Arcs AC+AM AC-AM , to the Square 18 Degrees 6'; and becaufe the firft Term of the Analogy is the Rectangle under the Sines of AB, BC, and the fecond Term is the Square of Radius, the Sum of the Logarithm Sines A B, BC, must 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 2 and 2 the Logarithm Sines of each of the Arcs A B, BC, 19.7930549 ments Arith- Log. S, metical of 2 thefe Sines 2 Log. S Angle B be taken, and thofe Complements and the faid Sines be all added together, then fhall the Sum be the Logarithm of the Square of the Sine of half the Angle B. And so the Half thereof, viz. the Logarithm 9.8965274, is the Logarithm Sine of half the Angle B-51 Degrees 59' 56", and the Double of this Angle fhall be 103 Degrees 59′ 52" B, which was fought. CHAP. CHA P.. V. Of the continual Increments of proportional Quantities, and how to find by Logarithms, any Term in a Series of Proportionals, either increafing or decreafing. Fig. 3. IF any where in the Axis of the Logarithmetical Curve, there be taken any Number of equal Parts SV, VY, YQ, &c. and at the Points S, V, Y, Q&c be raifed the Perpendiculars S T, V X, Y Z, QI, &c. then, from the Nature of the Curve, fhall all thefe Perpendiculars be continually proportional; and therefore, alfo, the continual Increments Xx, Zz, Пx, fhall be proportional to their Wholes. For fince ST: VX:: VX: YZ:: YZ: Qп, it fhall be (by Divifion of Proportion) ST:X:: VX: Z≈ : : YZ : Пā; and (by Compofition of Proportion) VX: Xx:: YZ: Zz::Q:II. Hence if Xx be any Part of any Right Line ST, then will Zz be the fame Part of the Right Line V X, and alfo II the fame Part of the Right Line YZ. For Example, if Xx be the Part of ST, then will Zz VX, and П- YZ; or, which comes to the fame, we fhall have V X= ST+ST, YZ=VX+ V X, alfo, QII = % YZ+YZ. Now make, as ST is to V X, fo is Unity A B to NR; then, fhall AN SV; and fo each of the Right Lines SV, VY, YQ, &c. fhall be equal to the Logarithm of RN; and A V, the Logarithm of the Term V X, fhall be equal to AS+AN Logarithm of ST+Logarithm of NR. Alfo A Y, the Logarithm of the Term Y Z, fhall be equal to AS+ 2 AN Logarithm ST + 2 Logarithm NR; and AQ, the Logarithm of the Term QII, fhall be equal to AS +3AN Logarithm ST + 3 Logarithm NR. And univerfally, if the Logarithm of the Number N R be multiplied by a Number, expreffing the Distance of any Term from the firft, and the Product be added to to the Logarithm of the firft Term, then will the Logarithm of that Term be had: But if a Series of Proportionals be decreafing, that is, if the Terms diminish in a continual Ratio, and QII be the firft Term then the Logarithm of any other will be had, by multiplying the Logarithm of the Number NR, by a Number that exprefles the Distance of its Term from the first, and fubtracting the Product from the Logarithm of the firft. And if the faid Product be greater than the Logarithm of the firft Term, then the Logarithms muft begin from a Unit in fome Place of Decimal Fractions, as from OP, and then the Logarithm of the Number QI will be OQ. Now, let LM represent any Money, or Sum of Money, put out to Intereft, fo that the Intereft thereof be accounted but at the End of every Year, and let Kk be the Gain or Interest thereof at the End of the first Year; then will IK be the Sum of the Intereft and Principal. And again I K, becoming the Principal at the End of the first Year, Hh, which is proportional to IK, or in a conftant Ratio, will be the Gain at the End of the fecond Year; and fo HG, at the End of the fecond Year, will become the Principal; and at the End of the third Year Ff, proportional to GH, will be the Gain. Now let us fuppofe the Principal to be augmented every Year Part thereof, fo that I KLM+LM, GHIK+IK, EF GHGH, and fo on. And accordingly, the Terms LM, IK, GH, EF, &c. are continual Proportionals, and it is required to find the Amount of the Money at the End of any Number of Years, Let L M be a Farthing. Because L M is to IK as I to I, or as I to 1.05, as A B is to N R, then will NR=1.05, whofe Logarithm A N, is 0.021 1893, or, more accurately, 9.021892991, it is required to find the Amount of a Farthing, put out at Compound Intereft, at the End of 600 Years. Multiply AN by 600, and the Product will be 12.7135794, and to this Product add the Logarithm of the Fraction, viz. 97.0177288 (for a Farthing is Part of a Pound), and the Sum 109.7313082 fhall be the Logarithm of the Number fought; and fince the Index 109 exceeds the Index of Unity by 9, there shall be nine Places of Figures above Unity in the correfpondent Num ber; ber; and that Number, being fought in the Tables, will be found greater than 5386500000, and lefs than 5386600000. And therefore a Farthing put out, at Interest upon Intereft, at 5 per Cent. per Annum, at the End of 600 Years will amount to above 5386500000 Pounds; which Sum could hardly be made up, by all the Gold and Silver that has been dug out of the Bowels of the Earth, from the Beginning of the World to this Time. Let QII expound any Sum of Money due to fome Perfon at the End of a full Year. Now it is certain, that if the Debtor fhould pay down, at prefent, the Whole Sum of Money, he would lofe the yearly Ufury or Intercit that his Money would gain him; and fo a leffer Sum, being put out to Intereft, will, at the End of one Year, together with the Interest thereof, be equal to the Sum of Money QII. Now this prefent Sum of Money, which together with the Intereft thereof,is equal to the Sum of Money QII, is called the present Worth of the Money QII. Let AN be the Logarithm of the Ratio which the Principal has to the Sum of the Principal and Interest, that is, if the Principal be twenty Times the yearly Intereft, let AN be the Logarithm of the Number 1+ or 1.05, and take QY equal to AN then will A Y be the Logarithm of the present Worth of the Money QII. For it is manifeft, that the Mo ney Y Z put out to Intereft, will, at the End of one Year, amount to the Money QII; and fo, to have the Logarithm of the prefent Worth thereof, or Y Z, the Logarithm AN muft be taken from the Logarithm AQ, and there will remain the Logarithm AY of the pretent Worth, or Y Z. But if the Sum QI be not due till the End of two Years, then the Logarithm 2 AN must be fubtracted from the Logarithm AQ, and there will remain AV, the Logarithm of the prefent Worth, or of the Sum that must be paid at prefent for the Money QII due at the End of two Years. For it is manifeft, that the Money V X being put out to Intereft, will, at the End of two Years amount to the Sum of Money QII. By the fame Realon, if the Sum QI be not due until the End of three Years, the Logarithm 3 AN must be subtracted from the Logarithm of QII, and the Remainder AS fhall be the Logarithm of the Number ST, or ST. ST fhall be the prefent Worth of the Sum QII due at the three Years End. And univerfally, if the Logarithm AN be multiplied by the Number of Years, at the End of which the Sum QII is due, and the Number produced be taken from the Logarithm AQ, then will the Logarithm of the prefent Worth of the Sum QII be had. And from hence it is manifeft, if 5386500000 Pounds be due to fome Society at the End of 600 Years, then would the prefent Worth of that vaft Sum of Money be fcarcely a Farthing. If the proportional Right Lines HG, EF, A B,CD, Fig.4. are Ordinates to the Axis of the Logarithmetical Curve, and if their Ends F H, D B, be joined by Right Lines, which, produced, meet the Axis in the Points P and K, then the Right Lines G P, A K, will be always equal. For fince GH: EF::AB: CD; it will be, as GH: FS::AB: DR. But because of the equiangular Triangles PGH, HSF, as alfo KAB, BRD, we have PG:H:: (GH:Fs::AB:DR::) KA: BR. And fince the Confequents Hs, BR, are equal, the Antecedents PG, K A, fhall be alfo equal. W.W.D. If the Right Lines CD, EF, equally accede to A B, GH, to that the Foint D at laft may coincide with B, and the Point F with H, then the Right Lines D BK, FHP, which did cut the Curve before, will be changed into the Tangents BT, HV. And the Right Lines AT, G V, will be always equal to each other; that is, the Portion of the Axis AT, or GV, intercepted between the Ordinate and the Tangent, which is called the Subtangent, will every where be of a constant and given Length. And this is one of the chief Properties of the Logarithmetical Curve; for the different Species or Forms of thofe Curves are determined by the Subtangents. The Logarithms, or the Diftances from Unity of the fame Number, in two Logarithmetick Curves of different Species, will be proportional to the Subtangents of their Curves. For let H BD, SNY, Fig. 4, 5. be Curves, whofe Subtangents are AT, MX, and let AB MN Unity; alfo, DC=QY; then shall A C, the Logarithm of the Number CD, in the Logarithmetick Curve HD, be to M Q, the Logarithm of the Number QY (or of the faid CD), in the Curve SY; as the Subtangent AT is to the Subtangent MX. |