When, then, the angle is greater than 45°, the distance from radius to the angle, though measured backwards upon the scale, must be reckoned a forward distance, and vice versa. The lines of logarithmic sine rhumbs, marked S.R., and tangent rhumbs, marked T.R., are formed in the same way as the lines of logarithmic sines and tangents, but are set off for the angles corresponding to the points and quarter points of the compass, instead of for degrees and minutes. We shall now proceed to explain the uses of Gunter's lines. 1. The Line of Logarithmic Numbers.-The primary divisions upon this line, as explained in its construction, represent the logarithms of all the integers from 1 to 100, while the extents to the first subdivisions will indicate tenths of an unit from the beginning of the scale to 1 in the middle, and units from 1 in the middle to 10 at the end, where the figures 2, 3, &c., stand for 20, 30, &c., as has been explained in the construction. If any of the subdivisions be further subdivided into ten parts, each of these last divisions will indicate hundredths of an unit from 1 at the beginning to 1 in the middle, and tenths of an unit from 1 in the middle to 10 at the end. Upon pocket sectors (p. 34), however, upon which Gunter's lines are now usually placed, affording a greater extent for the purpose than the six-inch plain scale (p. 33), only the part from 1 in the middle to 2 towards the right is a second time divided, and that but into five parts instead of ten, every one of which must be accounted as two-tenths. By this line the multiplication and division of numbers of any denomination. either whole or fractional may be readily accomplished, questions in proportion solved, and all operations approximatively performed with great rapidity, which can be performed by the common table of logarithms; but the numbers sought must always be supposed to be divided or multiplied by 10 as many times as will reduce them to the numbers, the logarithms of which are actually set off upon the line of numbers, and these tens must be mentally accounted for in the result. Multiplication is performed by extending from 1 on the left to the multiplier; and this extent will reach forwards from the multiplicand to the product. Thus, if 125 were given to be multiplied by 250, extend the compasses from 1 at the left hand to midway between the second and third subdivision, in the first primary division from 1 to 2, for the 125. This extent is really the logarithm of 1.25. Set off this extent towards the right from the fifth subdivision after the primary division marked 2, which is taken to represent the log. of 250, but is really the log. of 2.5, and the compasses will reach to a quarter of the next subdivision beyond the first subdivision after the primary division marked 3. The extent to this point is really the logarithm of 3.125; but in this case it represents the number 31250, because two powers of ten have been cast out from both the multiplier and multiplicand, and therefore the product must be multiplied by the product of four tens, or ten thousand; or, in other words, the first figure of the product must be reckoned as so many tens of thousands. Division, being the reverse of multiplication, is performed by extending from 1 on the left to the divisor; and this extent will reach backwards from the dividend to the quotient. Thus, if 31250 were to be divided by 250, extend the compasses from 1 on the left to 2.5, and this extent will reach backwards from 3.125 to 1.25. Then, since the divisor contained 2 powers of ten and the dividend 4, the quotient must contain 2, and therefore the result is 125. Proportion being performed by multiplication and division, extend the compasses from the first term to the second, and this extent will reach from the third to the fourth, taking care to measure in the same direction, so that, if the first be greater than the second, the third may be greater than the fourth, and vice versa. Example.-If the diameter of a circle be 7 inches, and the circumference 22, what is the circumference of another circle, the diameter of which is 10? Ex tend the compasses from 7 to 10, and this extent will reach from 22 to 314, or nearly 31 inches, the circumference required. The same thing may also be performed by extending from the first term to the third, and this extent will reach from the third term to the fourth (Euc. v. prop. 16). Thus, the extent from 7 to 22 will reach from 10 to 31.4, as before. To measure a Superficies, extend from 1 to either the breadth or length, both being reduced to the same denomination, and this extent will reach forwards from the length or breadth to the superficial content. Example.-Required the superficial content of a plank 27 feet long by 15 inches broad Extend from 1 to 1.25, for 15 inches equals 1.25 feet, and this extent will reach from 27 feet to 33.75 feet, the superficial content required Second Method.-Extend from 12 to the number of inches in the breadth, and this extent will reach in the same direction from the number of feet in the length to the number of square feet in the superficial content. Thus the extent for wards from 12 to 15 will reach forwards from 27 to 33.75, as before; while the extent backwards from 12 to 9 will reach backwards from 27 to 20-25 or 204, showing the superficial content of a plank 27 feet long by 9 inches broad to be 20-25 or 20 feet. To measure a Solid Content.-The breadth, depth, and length being all reduced to the same denomination, extend from 1 to either the breadth or depth, and this extent will reach from the depth or breadth forwards to a fourth number, which will represent the superficial content of the section at the place measured: then, if the breadth and depth be the same throughout the entire length, the extent from 1 to the superficial content thus found will reach forwards from the length to the solid content. Example.-What is the solid content of a pillar 1 foot 3 inches square, and 21 feet 9 inches long? The extent from 1 to 1.25 reaches forward from 1.25 to 1.56, the superficial content of a section of the pillar; and the extent from 1 to 1.56 reaches from 21.75 to 34, or more accurately to 33.93, the solid content in feet*. 2. The Lines of Logarithmic Sines and Tangents.—These lines are generally used, in connection with the line of numbers, for solving all proportions in which any of the terms are functions of angles, as sines, tangents, &c., and, in fact, all questions in which such quantities appear as factors or divisors. We will exemplify their use by giving the solution, by their aid, of the several cases of right-angled trigonometry. Case 14. The hypothenuse and angles being given, to find the perpendicular and base. * Our limits forbid us from entering further upon the uses of the line of logarithmic numbers; but the student will, we hope, from what he sees here, be easily enabled to apply it to every case of mensuration, and, in short, to almost every arithmetical operation. Additions and subtractions, however, cannot be performed by it. These cases are, in fact, the solutions, by the aid of Gunter's lines, of the following proportions, which will be obvious to the student upon inspec- Rad. : sin. ▲ :: AB BO Case 1. Sin. B rad. :: AC: AB Sin. B sin. A:: AC: BO Rad. tan. A :: AC BO rad. : sin. B Case 2. AB AC Case 3. Case 4. Note.-One acute angle of a right-angled triangle being the complement of the other, or the sum of the two acute angles being equal to 90, when one of the acute angles is given, the other is also given. Solution.-Extend the compass from 90°, or radius, on the line of A sines to the number of degrees in B either of the acute angles, and that extent will reach backwards, on the line of numbers, from the hypothenuse to the side opposite this angle. Example.-Given the hypothenuse AB=250, and the angle A= 35° 30′ Extend from 90° to 35° 30′ on the line of sines, and this extent will reach from 250 to 145 on the line of numbers. Extend from 90° to 54° 30′ on the line of sines, and this extent will reach from 250 to 203.5 on the line of numbers Case 2. The angles, and one side being given, to find the hypothenuse, and the other side. Solution-Extend from the angle opposite the given side to 90°, or radius, on the line of sines, and this extent will reach forwards from the given side to the hypothenuse on the line of numbers. Again, extend from the angle opposite the given side to the angle opposite the required side, and this extent will reach in the same direction on the line of numbers, from the given side to the required side. Or, extend from radius, or 45°, on the line of tangents, to the angle opposite the required side, and the extent will reach, in the same direction on the line of numbers, from the given side to the required side; recollecting that, when the angle is greater than 45°, the extent is to be taken on the scale backwards from rad. or 45° to the complement of the angle, but is to be reckoned a forward distance, the logarithmic tangents of angles greater than 45° exceeding the logarithmic tangents of 45°, or radius, by as much as the logarithmic tangents of their complements fall short of it. Example. Given the angle a=35° 30′ and side Ac=203·5. Extend from 54° 30' to 90°, or rad., upon the line of sines, 90° 0' BAC 35 30 ABC 54 30 .. AB=250 .. BC=145 Or extend backwards from 45°, rad., to 35° 30' on the line of tangents. and this extent will reach backwards from 203.5 to 145 on the line of numbers, as before*. Case 3. The hypothenuse and one side being given, to find the angles and the other side. Solution.-Extend from the hypothenuse to the given side on the line of numbers, and this extent will reach from 90 or rad. to the angle opposite the given side upon the line of sines. The other angle is the complement of this. Extend upon the line of sines from the rad. to the angle last found, which is opposite the required side, and this extent will reach from the hypothenuse to the required side. Example.-Given the hypothenuse AB=250, and the side AC=203·5. Extend backwards from 250 to 203.5 on the line of numbers, and this extent will reach from 90° to 54° 30′ on the line of sines 90° 0' ... ABO=54 30 Extend from 90 to 35° 30′ on the line of sines, and this extent will reach backwards from 250 to 145 on the line of numbers BAC=35 30 .. BO=145 Case 4. The two sides being given, to find the angles and the hypothenuse. Solution.-Extend from one side to the other upon the line of numbers, and this extent will reach backwards upon the line of tangents from rad. to the least angle, and to the same point, considered as a forward distance, representing the greatest angle, which is the complement of the least. Again, extend on the line of sines from one of the angles just found to rad., and this extent will reach from the side opposite the angle taken to the hypothenuse. Example.-Given AC=203.5 and BC145. Extend backwards upon the line of numbers from 203.5 to 145, and this extent will reach backwards from 45° to 35° 30′ on the line of tangents, which is the angle opposite the side 145 If we measure forwards from 145 to 203-5, then from rad. to 35° 30' is to be considered a forward distance, and the angle to be taken as the complement of 35° 30′, that is, 54° 30', which is the angle opposite the side 203.5. Again, extend from 33° 38′ to 90° on the line of sines, and this extent will reach from 145 to 250 upon the line of numbers 90° 0' .. BAC 35 30 ... ABO=54 30 .. A B = 250 *The property that tan.: rad. : sine cosine, may be made a test of the accuracy of the scale, since the distance from 45 to any angle upon the line of tangents ought to be the same as the distance from the angle to its complement upon the line of sines. |