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Again, suppose it required to set off upon any line 2,37 inches ; then place one point of the compasses on the 7th parallel under 2 at h, and extend the other to the 3rd diagonal in the same parallel at i; and the distance hi is that required. Or, if AB be 10, the distance eg is 17,3 ; and hi is 23,7. Also, if AB be 100, then eg is 173, hi is 237 ; and so on.
This diagonal scale has this centissimal division at each end, and the unit in one is just double of that at the other ; thus, if AB be 1 inch at one end, it is į an inch at the other ; or if it be § an inch in the larger, it is inch in the lesser divisions, as is the case upon most of the common Plain Scales.
This unit AB may also be 1 foot, 1 yard, i rod, 1 mile, &c. that every unit, in every kind of measures, is hereby estimated in 100th parts of the whole, which shews the Diagonal Scale to be a most useful invention.
On the other side of the plain scale are the seven decimal lines, which are usually Plotting Scales, because their divisions of an unit into 10 parts, being different in proportion of 4 to 1, the Surveyor has it in his power to vary the scale of his plot or plan of an estate, &c. In that ratio, in seven different drawings; and the superfices, or sizes of the greatest and least plans, will be as 16 to 1. Or, that drawn by scale No. 10, will be sixteen times larger than the plan laid down from scale No. 40.
Also the same variety is to be had in the construction of all other geometrical figures, whether superfices or solids ; and, with respect to the latter, the greatest will be to the least as 64 to 1 ; that is, the Architect can vary the size of his house, temple, &c. in the ratio of 64 to 1 in seven different elevations.
The last Line on the common plain scale is that of Chords ; and much more used than the protractor for laying off or measuring any proposed angle. Thus, let it be required to draw the line BC to make an angle of 35 degrees with the line AC, fig. 3. To do this, set one point of the compasses in the beginning of the line of chords, and extend the other to 60; with that extent (as radius) place one foot in C, and, with the other, describe the arch AD; then take from the chords 35o in the compasses, and set them on the arch from A to B ; and through B draw CB, and it is done.
Again, suppose it required to measure any angle ABC proposed, proceed thus, produce CA indefinitely, take 60° from the chords in the compasses, and, with one foot in C, strike the arch AD, cutting the leg BC in B; then take the arch AB in the compasses, and applying it upon the beginning of the line of chords, it will reach to 35°, the quantity of the angle required. But the Line of Chords is more useful on the Sector, to which we now proceed.
Note.--This line of chords is so constructed on my Navigation Scales, with sexagessimal divisions, that any angle may be measured, or laid off, to a minute of a degree.
VII. OF THE SECTOR. The Sector is the most useful of all Mathematical Instruments; for it not only contains all the most useful lines or scales, but, by its nature, renders them universal, as we shall show in the following description : The lines commonly laid down upon the Sector are these.
1. A Line of Equal Parts, marked L at the end. 2. A Line of Chords to 60 degrees, marked S. 3. A Line of Sines, 90 degrees, marked C. 4. A Line of Tangents, to 45 degrees, marked T. 5. Another Line of Tangents, from 45 to 75 or upwards, marked la. 6. A Line of Secants, marked se. 7. A Line of Polygons, marked POL.
Besides these, when the Sector is quite opened, there is, on one side, 1. Gunter Line of Artificial Numbers, n. 2. Line of Artificial Sines, s. 3. Line of Artificial Tangents, 1. There is, also, a line of 12 inches, and another of the foot divided into 1000 equal parts, placed by it for the purposes already mentioned.
Before a proper idea can be formed of these sectoral lines, and their uses, we must show their construction from the circle. To this end, let AGB, fig. 8, be a quarter of a circle, divided into 90 degrees, described with the radius AC, on the centre C; let AF and CF be perpendicular to AC, in A and C; then, if the radius AC be divided into 10, 100, 1000, &c. Equal Parts, it will be the line so called upon the Sector.
If from A, a line be drawn to any part or division of the quadrant, as G at 65°, then that line AG is the chord of that arch, or of 60°. And if the line AB be drawn, it will be the chord of 90°; and, by setting one foot of the compasses in A, and extending the other to the several divisions 10, 20, 30, 40, &c. they may be transferred from the circle to the line AB, which will then be properly divided into a line of chords in 10, 20, 30, 40, &c. to 90, as on the plain scale.
If from any point G in the quadrant you let fall a perpendicular GI to the radius AC, or GH to the radius CB, then the line GI is called the Sine of the arch AG ; and the line GH is the Sine of the arch GB, the complement of AG to 90°. And if all the divisions of the quadrant were transferred to CB by lines GH parallel to AB; then the line CB will be divided as a Line of Sines in the points 10, 20, 30, 40, &c. to yo, as on the Sector you see it.
By laying a rule from the centre C to the several dirisions of the quadrant, 10, 20, 30, &c. it will cut the line AE in the points 10, 20,30, &c. which will be thereby divided into a Lice of Tangents; and here it must be observed, that the Line of Tangents T on the Sector, extends but to 45°, equal to AD, or BC, radius. And that the Line of Lesser Tangents t, are projected from a lesser radius, and begin from 45° at the distance of its radius from the centre of the Sector.
By drawing the line CL through the division 60°, to the line AE, it makes AL, the Tangent of 60°, and is itself the Secant of 60°, or of the angle ACL. And if, with one foot of the compasses in C, you extend the other to the several divisions in the line AE, and
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 to the radius or Cl. 3. And, therefore, the secant CL of 60°, is equal to twice the radius AC; for CI is to CG as CA to CL. 1. 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. Suppose it required to lay off an angle ACB, fig. 8, 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 Line 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 Tangents. 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,
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 find 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 360°; therefore, such a cho's 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 ali 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
i 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,