which, by the Table, I find to be 8424, which is the product of the two Numbers propos'd. Prob. 2. Let it be requir'd to find the Quotient that arifes by dividing one Number by another, fuppofe 828 by 23. To folve this by the Logarithms, I first look in the Tables for the Logarithm of 828, the Dividend, which I find to be 2.91803; then for the Logarithm of 23 the Divifor, which is 1.36173, and this laft taken from the former Logarithm, there remains 1.55630 the Logarithm of the Quotient, which answers to the Number 36 the Quotient re quired. Again, let it be required to divide 3055 by 47; by proceeding according to the last Example, the Operation will be as follows: 3.48501 the Logarithm of 3055 the Dividend, 1.67210 the Logarithm of 47 the Divifor, 1.81291 the Logarithm of the Quotient. which answers to the Number 65 the Quotient required. Prob. 3. Three Numbers being given to find a fourth proportional to them, viz. Such as fhall have the fame proportion to the third as the fecond has to the first. Rule. Take from the Tables the Logarithm of each of the propos'd Numbers, then add the Logarithms of the fecond and third together, and from the Sum take the Logarithm of the first, and the Remainder fhall be the Logarithm of the fourth number requir'd. Example. Let the three propos'd Numbers be 36, 48, 66, to which we are to find a fourth proportional; by the preceeding Rule, the Operation will ftand as follows; 1.68124 the Logarithm of 48 the 2d Term, 1.81954 the Logarithm of 66 the 3d Term, 3.50078 the Logarithm of their Product, 1.55630 the Logarithm of the 1ft Term, 36. 1.94448 the Log. of the 4th Term requir❜d. which, by looking into the Table, I find answers to the natural Number 88, which is the 4th proportional to the three propos'd Numbers. Again, let it be required to find a fourth propor tional to the three Numbers 24, 144, 123; by proceeding according to the foregoing Rule, the Operation will stand as follows: 2.15836 the Logarithm of the 2d Term 144. 2.08991 the Logarithm of the 3d Term 123. 4.24827 the Logarithm of their Product, 1.38021 the Logarithm of the 1ft Term 24. 2.86806 the Log. of 738, the 4th number requir'd. Prob. 4. To find the Square of any Number by Logarithms. Rule. Multiply the Logarithm of the given Number by 2, and the product is the Logarithm of the Square fought. Example. Required to find the Square of 36. First I look in the Table for the Logarithm of 36, and find it to be 1.55630, which doubled gives 3.11260 the Logarithm of the Square fought, which by Inspection I find answers to the natural Number 1296 the Square of 36, viz. the product of 36 mul tiply'd into itself. Prob. 5. To extract the fquare Root of any pro pos'd Number, i. e. to find a Number which mul tiply'd into itself, fhall produce the given Number, Rule Rule. Divide the Logarithm of the propos'd Number by 2, and the Quotient will be the Logarithm of the fquare Root required. 2 Example. Required to find the fquare Root of 1296. First I look in the Tables for the Logarithm of 1296, and find it to be 3.11261, which divided by 2 gives 1.55630 for the Logarithm of the fquare Root, and the natural Number answering thereto is 36 the Root required. If for the Sine, Tangent, &c. of every Degree and Minute in the Quadrant, in the natural Tables, we take the Logarithm agreeing to each, we fhall have a Table of Logarithmic Sines, Logarithmic Tangents, &c. as it is in the fecond Table at the end of this Book. In which you may observe, that each Page is divided into eight Columns, the first and last of which is Minutes, and the intermediate ones contain the Sines, Tangents, and Secants; the upper and lower Columns contain Degrees; the Column of Minutes on the left hand of each Page, anfwers to the Degrees in the top Column; and the Sines, Tangents, and Secants, belonging to these Degrees, and Minutes are in the Columns mark'd at the top with the Words, Sine, Tangent, Secant; the Column of Minutes on the right hand of each Page, anfwers to the Degrees in the foot of the Page, and the Sines, Tangents, and Secants, anfwering to thefe Degrees and Minutes, are in the Columns mark'd at the bottom with the Words, Sine, Tangent, Secant; the Degrees in the top Column beginning at o, proceed to 44 where they end, and thofe at the foot of the Page begin at 89 proceed to 45 in a decreafing Series, the Degrees in the different Columns being the Com+ plement of each other, From what has been faid, we may eafily find the Sine, Tangent, or Secant, of any Arch, from our Tables, by looking for the gi ven Number of Degrees at the head or foot of the Page, according as they are lefs or greater than 45, and in the proper fide Column for the odd Minutes, if there be any; then below or above the Word, Sine, Tangent, or Secant, and on the fame line with the Minutes, we fhall have that requir❜d. Example 1. Required to find the Ŝine of 36 deg. 40 min. To find this, I look at the head of the Page for 36 deg. and in the fide Column, on the left hand, for 40 min. then below the Word Sine, and on the fame line with 40, I find 9.77609, which is that requir❜d. Example 2. Requir'd the Tangent of 54 deg. 30 min. To find this, I look at the foot of the Page (because the Degrees propos'd are greater than 45) for 54 deg. and in the right hand fide Column for 30 min. then in the Column mark'd with Tangent at it's bottom, and on the fame line with the 30 min. in the fide Column, I find 10.14673, which is the Log-Tangent requir'd. The Reverse of this, viz. The Logarithm of a Sine, Tangent, or Secant, being given to find the Arch belonging to it, is perform'd by only looking in the proper Column for the nearest Logarithm to that propos'd, and the Degrees and Minutes answering thereto is what was requir'd. In these Tables the Secants might have been wanting, because all the Proportions in which the Secants are concern'd may be wrought without them, by the Sines and Tangents only, as fhall be fhewn particularly, in the Solution of the feveral Cafes of plain Trigonometry. 72. The Chord, Sine, Tangent, &c. of any Arch in one Circle, is to the Chord, Sine, Tangent, &c. of the fame Arch in another Circle, juft as the Radius of the one is to the Radius of the other; for 'tis plain, the greater the Radius is, the greater is the Circle defcribed by that Radius, and confequently the greater any particular Arch of that Circle is, and and fo the Sine, Tangent, &c. of that Arch is alfo the greater; therefore, in general, the Chord, Sine, Tangent, &c. of any Arch is proportionable to the Radius of the Circle. 73. In all Circles the Chord of 60 is always equal in length to the Radius. Thus in the Circle AEBD, if the Arch AEB be an Arch of 60 degrees, then drawing the Chord AB, I fay AB fhall be equal to the Radius CB or AC; for in the Triangle ACB, the Angle ACB is 60 degrees, being measured by the Arch AEB; therefore the Sum of the other two Angles is 120 degrees, (by Cor. 1. of 61ft but fince AC and CB are equal the two Angles CAB, CBA will also be equal; confequently each of them half their Sum 120, viz. 60 degrees; therefore all the three Angles are equal to one another, confequently all the Legs, there D A E B fore AB is equal to CB. Cor. Hence the Radius from which the Lines on any Scale were form'd, is the Chord of 60 on the Line of Chords. 74. If in two Triangles ABC, abc all the Angles of the one be equal to all the Angles in the other, each to each refpectively, that is, the Angle BAC equal to the Angle bac, the Angle ACB equal to the Angle acb, and the Angle ABC equal to the Angle abc; then the Legs oppofite to the equal Angles are proportionable, viz. Â B: ab :: AC a c and AB: ab:: BC: bc and AC: ac:: BC be; for being infcribed in two Circles, 'tis plain, fince the Angle BAC is equal the Angle bac, the Arch BDC is equal the Arch bdc, and confequently the Chord BC is to the Chord bc, as the Radius of the Circle ABC to the Radius of the Circle abc (by the 72d); the fame way the Chord |