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These subdivisions are again divided into ten parts, where room will permit; and where that is not the case, the units must be estimated, or guessed at, by the eye, which is easily done by a little practice.
The primary divisions on the second part of the scale, are estimated according to the value set upon the unit on the left hand of the scale : If you call it one, then the first 1, 2, 3, &c. stand for 1, 2, 3, &c. the middle 1 is 10, and the 2. 3. 4. &c. following stand for 20, 30, 40, &c. and the ten at the right hand is 100: If the first I stand for 10, the first 2, 3, 4, &c. must be counted 20, 30, 40, &c. the middle I will be 100, the second 2, 3, 4, 5, &c. will stand for 200, 300, 400, 500, &c. and the ten at the right hand for 1000.
If you consider the first l as its of an unit, the 2, 3, 4, &c. following will be $, td 18, &c. the middle I will stand for an unit, and the 2, 3, 4, &c. following will stand for 2, 3, 4, &c. also the division at the right-hand end of the scale will stand for 10. The intermediate small divisions must be estimated according to the value set upon the primary ones.
Sine. The line of sines is numbered from the left hand of the scale towards the right, 1, 2, 3, 4, 5, &c. to 10; then 20, 30, 40, &c. to 90, where it terminates just opposite 10 on the line of numbers.
Versed sine. This line is placed immediately under the line of sines, and numbered in a contrary direction, viz. from the right hand towards the left 10, 20, 30, 40, 50, to about 169; the small divisions are here to be estimated according to the number of them to a degree. By comparing the line of versed sines with the line of sines, it will appear that the versed sines do not belong to the arches with which they are marked, but are the half versed sines of their supplements. Thus, what is marked the versed sine of 90 is only half the versed sine of 90,
the versed sine of 120° is half the versed sine of 60°, and the versed sine marked 100 is half the versed sine of 80°, &c.
The versed sines are numbered in this manner to render them more commodious in the solution of trigonometrical, and astronomical problems.
Tangents. The line of tangents begins at the left hand, and is numbered 1, 2, 3, &c. to 10, then 20, 30, 45, where there is a little brass pin just under 90 in the line of sines; because the sine of 90° is equal to the tangent of 45°. It is numbered from 45° towards the left hand 50, 60, 70, 80, &c. The tangents of arches above 45° are therefore counted backward on the line, and are found at the same points of the line as the tangents of their complements.
Thus, the division at 40 represents both 40 and 50 the division at 30 serves for 30 and 60, &c.
Meridional Parts. This line stands immediately above a line of equal parts, marked Equal Pt. with which it must always be compared when used. The line of equal parts is marked from the right hand to the left with 0, 10, 20, 30, &c.; each of these large divisions represents 10 degrees of the equator, or 600 miles. The first of these divisions is sometimes divided into 40 equal parts, each representing 15' minutes or miles.
The extent from the brass pin on the scale of meridional parts to any division on that scale, applied to the line of equal parts, will give (in degrees) the meridional parts answering to the latitude of that division. Or the extent from any division to another, on the line of meridional parts, applied to the line of equal parts, will give the meridional difference of latitude between the two places denoted by the divisions. These degrees are reduced to leagues by multiplying by 20, or to iniles by multiplying by 60.
The use of the logarithmical lines on Gunter's Scale.
By these lines and a pair of compasses, all the problems of Trigonometry, &c. may be solved.
These problems are all solved by proportion ; Now in natural numbers, the quotient of the first term by the second is equal to the quotient of the third by the fourth: therefore, logarithmically speaking, the difference between the first and second term is equal to the difference between the third and fourth, consequently on the lines on the scale, the distance between the first and second term will be equal to the distance between the third and fourth. And for a similar reason, because four proportional quantities are alternately proportional, the distance between the first and third terms, will be equal to the distance between the second and fourth. Hence the following
The extent of the compasses from the first term to the second, will reach, in this same direction, from the third to the fourth term. Or, the extent of the compasses from the first term to the third, will reach, in the same direction, from the second to the fourth.
By the same direction in the foregoing rule, is meant that if the second term lie on the right hand of the first, the fourth will lie on the right hand of the third, and the contrary. This is true, except the two first or two last terins of the proportion are on the line of tangents, and neither of them under 450 ; in this case the extent on the tangents is to be made in a contrary direction : For hid the tangents above 459 been laid down in their proper direction, they would have extended beyond the length of the scale towards the right hand; they are therefore as it were folded back up
on the tangents below 45°, and consequently lie in a direction contrary to their proper and natura! order.
If the two last terms of a proportion be on the line of tangents and one of them greater and the other less than 45°; the extent from the first term to the second, will reach from the third beyond the scale. To remedy this inconvenience, apply the extent between the two first terms from 450 backward upon the line of tangents, and keep the left hand point of the compasses where it falls ; bring the right hand point from 45° to the third term of the proportion ; this extent now in the compasses applied from 45° backward will reach to the fourth term, or the tangent required. For, had the line of tangents been
continued forward beyond 45°, the divisions would have fallen above 45° forward ; in the same manner as they fall under 45° backward.
The word Trigonometry signifies the measuring of triangles. But, under this name is generally comprehended the art of determining the positions and dimensions of the several unknown parts of extension, by means of some parts, which are already known. If we conceive the different points, which may be represented in any space, to be joined together by right lines, there are three things offered for our consideration; 1. the length of these lines; 2. the angles, which they form with one another; 3. the angles formed by the planes, in which these lines are drawn, or are supposed to be traced. On the comparison of these three objects,
depends the solution of all questions, that can be proposed concerning the measure of extension, and its parts; and the art of determining all these things from the knowledge of some of them, is reduced to the solution of these two general questions.
1. Knowing three of the six parts, the sides and angles—which constitute a rectilineal triangle ; to find the other three.
2. Knowing three of the six parts, which compose a spherical triangle; that is a triangle formed on the surface of a sphere by three arches of circles, wbich have their centre in the centre of the same sphere—to find the other three.
The first question is the object of what is called Plane Trigonometry, because the six parts, considered here, are in the same plane: it is also denominated Rectilineal Trigonometry. The second question belongs to Spherical Trigonometry, wherein the six parts are considered in different planes. But the only object here is to explain the solutions of the former question : viz.
Plane Trigonometry is that branch of geometry, which teaches how to determine, or calculate three of the six parts of a rectilineal triangle by having the other three parts given or known. It is usually divided into Right angled and Oblique angled Trigonometry, according as it is applied to the mensuration of Right or Oblique angled Triangles.
In every triangle, or case in trigonometry, three of the parts must be given, and one of these parts, at least, must be a side; because, with ihe same angles, the sides may be greater or less in any proportion.