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transfer them to the line CE, then will the part BF be thus divided into a Line of Secants ; being placed at the distance of the radius CB, from the centre of the Sector, and beginning at B, where the radius ends.
It may be of use in many cases to observe, 1. That the Chord of 60° AG, is equal to radius AC or CG. 2. That the Sine of 60° GI, bisects the radius AC in I; and, therefore, the sine GH of 30° is equal io the radius or Cl. 3. And, therefore, the secani CL of 60°, is equal to twice the radius AC; for CI is to CG as CA to CL. 4. Therefore, Cosine is to Radius as Radius to the Secant. 5. Also the Tangent AL is to Radius AC as Radius BC is to the Co-tan
From what has been said, the reason appears why the Line of Lines, or equal parts L, terminates upon the sector at 10; the Line of Chords C at 60°; the Line of Sines at S at 90°; the larger Tangents T at 45° ; and that the lesser Tangents, and also the Secants, are of indefinite length.
From the nature of the Sector, consisting of two pairs, or legs, moveable upon a central joint, it is requisite that the lines should be laid on the Sector by pairs, viz. one of a sort on each leg, and all of them issuing from the centre, all of the same length, and every two containing the same angle. We shall now illustrate the nature of working problems Sector-wise, as follows, by the Line of Lines, or Equal Parts, LL. Let CL, CL, fig. 9, be the Two Lines of Lines upon
the Sector, opened to an angle LCL; join the divisions 4 and 4, 7 and 7, 10 and 10, by the dotted lines ab, cd, LL. Then, by the pature of similar triangles, it is CL to Cb, as LL to ab ; and CL to Cd, as LL to cd; and, therefore, al is the same part of LL, as Cb is of CL. Consequently, if LL be 10, then ab will be 4, and cd will be 7 of the same parts.
And hence, though the Lateral Scale CL be fixed, yet a Parallel Scale LL, is obtainable at pleasure ; and, therefore, though the Lateral Radius is of a determined length in the Lines of Chords, Sines, Tangents, and Secants, yet the Parallel Radius may be had of any size you want, by means of the Sector, as far as its length will admit; and all the Parallel Sines, &c. peculiar to it; as will be evident by the following Examples in each pair of lines.
1. In the Lines of Equal Parts. Having three numbers given, 4, 7, 16, to find out a fourth proportional. To do this, take the lateral extent of 16 in the line CL, and apply it parallelwise, from 4 to 4, by a proper opening of the sector; then take the parallel distance from 7 to 7 in your compasses, and applying one foot in C, the other will fall on 28 in the Line of Lines CL, and is the number required; for 4 :7:: 16: 28.
2. In the Lines of Chords. Suppove it required to lay off an angle ACB, Ag. 3, equal to 35°, then, with any convenient opening of the Sector, take the extent from 60 to 60, and with it, as radius, on the point C describe the arch AD indefinitely; then, in the same opening of the Sector, take the parallel distance from 35° 10 35°, and set it from A to B in the arch AD, and draw AB, and it makes the angle at C required.
3. In the Lines of Sines. The lines of Sines, Tangents, and Secants, are used in conjunction with the Live of Lines in the solution of all the cases of plain Trig. onometry ; thus, let there be given, in the triangle ABC, fig. 10, the side AB=230; and the angle ABC=36° 30'; to find the side AC. Here the angle at C is 53° 30'. Then take the lateral distance 230, from the Line of Lines, and make it a parallel from 53° 30' to 53° 30' in the Line of Sines; then the parallel distance between 36° 30' in the same lines, will reach laterally from the centre to 170,19 in the Line of Lines for the side AC required.
4. In the Lines of Tangenis. If, instead of making the side BC radius, as before, you make AB radius ; then AC, which before was a Sine, is now the Tangent of the angle B; and, therefore, to find it, you use the Lines of Tangents, thus :
Take the lateral distance 230 from the Line of Lines, and make it a parallel distance on the Tangent Radius, viz. from 45° to 45°, then the parallel Tangent from 36° 30 to 36° 30, will measure laterally on the Line of Lines 170,19, as before, for the side AC.
5. In the Lines of Secants. In the same triangle, in the base AB, and the angles at B and C, given, as before, to tind the side or hypotheneuse BC. Here BC is the Secant of the angle B.
Take the lateral distance 230 on the Line of Lines, make it a parallel distance at the radius or beginning of the Lines of Secants ; ihen the parallel Secant of 60° 30' will measure laterally on the Line of Lines 287,12 for the length of BC, as required.
6. In the Lines of Sines and Tangents conjointly. In the solution of spherical triangles, you use the Line of Sines and Tangents only, as in the following Example. In the spherical triangle ABC. fig. 11, right angled at A, there is given the side AB=360° 15', and the adjacent angle B=42° 34', to find the side AC. The analogy is radius : sine of AB : : tangent of B : tangent of AC; therefore, make the lateral sine of 36° 15' a parallel at radius, or between 90 and 90; then the parallel tangent of 42° 34' will give , the lateral tangent of 28° 30' for the side AC.
7. In the Lines of Polygons. It has been observed, that the chord of 60 degrees is equal to radius ; and 60° is the sixth part of 300°; therefore, such a cho' i is
the side of a Hexagon inscribed in a circle. So that in the Line of Polygons, if you make the parallel distance between 6 and 6, the radius of a circle, as AC, fig. 12, then, if you take the parallel distance between 5 and 5, and place it from A to B, the line AB will be the side of a Pentagon ABDEF, inscribed in the circle ; in the same manner may any other polygon, from 4 to 12 sides, be inscribed in a circle, or upon any given line AB.
8. Of Gunter's Lines. We have now shown the use of all that are properly called Sectorial Lines, or that are to be used Sector-wise ; but there is another set of lines usually put on the Sector, that will, in a more ready and simple manner, give the answers to the questions in the above Examples, and these are called Artificial Lines of Numbers, Sines, and Tangents; because they are only the logarithms of the natural numbers, sines, and tangents, laid upon lines of scales, which method was first invented by Mr. EDMUND Gunter, and is the reason why they have ever since been called Gunter's Lines, or the Gunter.
Logarithms are only the Ratios of Numbers, and the ratios of alı proportional numbers are equal. Now all questions in Multiplication, Division, the Rule of Three, and the Analogies of Plain and Spherical Trigonometry, are all stated in proportional numbers or terms; therefore, if in the compasses you take the extent, or ratio, between the first and second terms, that will always be equal to the extent, or ratio, between the third and fourth terms; and, consequently, if with the extent between the first and second terms, you place one foot of the compasses on the third term, then turning the compasses about, the other foot will fall on the fourth term sought.
Thus, in Example I, of the three given numbers 4, 7, and 16, if you take the extent from 4 to 7 in the compasses, and place one foot in 16, the other will fall on 28, the answer in the Line of Numbers marked n.
Again, the Artificial Line of Numbers and Sines are used together in Plain Trigonometry, as in Example III, where the two angles B and C, and the side AB are given ; for here, if you take the extent of the two angles 53° 30 and 36° 30' in the Line of Sines, marked s, then, placing one foot upon 230 in the Line of Numbers n, the other will reach to 170,19 the answer.
Also the Line of Numbers and Tangents are used conjointly, as in Example IV. for take in the Line of Tangents t, the extent from 45° radius, to 36° 30', that will reach from 230 to 170,19 the answer as before.
Lastly, the Artificial Lines of Sines and Tangents are used together in the analogies of Spherical Triangles.
Thus, Example VI. is solved by taking in the Line of Sines s, the extent from 60° radius, to 36° 15', then that in the Lines of Tangents t, will reach from 42° 34' to 28° 30, the answer required.
I shall only further observe, that each pair of Sectorial Lines contains the same angle, viz. 6 degrees in the common six-inch sector ; therefore, to open these lines to any given angle, as 350 for instance, you have only to take 35 laterally from the Line of Chords, and apply it parallel-wise from 60° to 60° in the same lines, and they will be opened to the given angle of 35°.
If to the angle 35o you add the angle 6°, which they contain, the sum is 41°; then take 41° laterally from the Line of Chords, and apply it parallel from 60 to 60, the sides or edges of the sector will contain the same angle of 35 degrees. And, in this case, the sector becomes a general Recipe-Angle, which is an instrument for taking the quantity of any angle contained between two Inclining Planes, as those in fortifications, &c.
9. Of Proportional Compasses. Though this sort of Compasses does not pertain to a common case of instruments, yet a short account of their nature and use may not be unacceptable to those who are not acquainted with them. They consist of two parts or sides of brass, which lie upon each other, so nicely as to appear but one when they are shut. These sides easily open, and move about a centre, which is itself moveable in a hollow canal cut through the greatest part of their length. To this centre on each side is affixed a sliding piece of a small length with a fine line drawn on it serving as an Index, to be set against other lines or divi. sions placed upon the Compasses on both sides. These lines are, 1. A LINE of LINES. 2. A Line op SUPERFICES, ARBAs, or Plans. 3. A LINE of Solids. 4. A LINE of CIRCLES, or rather of Polygons, to be inscribed in circles.
These lines are all unequally divided, the three first from 1 to 10; the last from 6 to 20; their uses are as follow :
By the Line of Lines, you divide a given line into ang number of equal parts ; for, by placing the Index against 1, and screwing it fast, if you open the Compasses, then the distance between the points at each end will be equal.-If you place the Index against 2, and open the Compasses, the distance between the points of the longer legs will be twice the distance between the shorter ones ; and thus a line is bisected, or divided into two equal parts. If the Index be placed against 3, and the Compasses opened, the distances between the points will be as 3 to 1, and so a line is divided into three equal parts ; and thus you proceed for any number of parts under 10.
The number of the Line of Planes answer to the Squares of those in the Line of Lines ; for, because Superfices or Planes are to each other as the squares of their like sides, therefore if the Index be placed against 2 in the Line of Planes, then the distance between the small points will be the side of a plane whose area is 1 ; but the distance of the larger points will be the like side of a plane whose area is 2, or twice as large.--If the Index be placed at 3, and the Compasses opened, the distances between the points at each end will be the like sides of planes, whose areas are as i to 3, and so of others.
The numbers of tho Ling of Solids answer to the Cubes of those in the Line of Lines ; because all solids are to each other as the cubes of their like sides or diameters ; therefore, if the Inder be placed at No. 2, 3, 4, &c. in the Line of Solids, the distances bem tween the lesser and larger points will be the like sides of solids, which are 10 each other as 1 to 2, 1 to 3, 1 to 4, &c. For Example, If the Index be placed at 10, and the Compasses be opened, so that the small points may take the diameter of a bullet weighing 1 ounce, then the distance between the larger points will be the diameter of a bullet or globe of 10 ounces, or which is 10 times as large.
Lastly, the numbers in the Line of Circles are the sides of Polygons, to be inscribed in a given circle, or by which a circle may be divided into those equal parts from 6 to 20. Thus, if the Index be placed at 6, the points of the Compasses at either end, when opened to the radius of a given circle, will contain the side of a Hexagon, or divide the circle into 6 equal parts. If the Index be placed against 7, and the Compasses opened, so that the larger points may take in the radius of the circle ; then the shorter points will divide the circle into 7 equal parts for inscribing a Heptagon.--Again, placing the Index to 8, and opening the Compasses, the larger points will contain the radius, and the lesser points divide the circle into 8 equal parts, for inscribing an Octagon or Square. And thus you proceed for others.