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50, 60, 70, 80, &c. Therefore the tangents of arcs above 45° are 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 represents both 30 and 60, &c.

7. Meridianal parts. This line stands above the line of equal parts, with which it must always be compared, when used. The extent from the brass pin on the scale of meridianal parts to any division on that scale, applied to the line of equal parts, will give, in degrees, the meridianal parts answering to the latitude of that division. Or, the extent from any division to another on the line of meridianal parts, applied to the line of equal parts, will give the meridianal difference of latitude between the two places denoted by the divisions.

8. The line of equal parts is marked from the right hand towards the left; thus, 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 miles. The line of numbers, sometimes called the line of lines, is that upon which most of the others depend. It is usually graduated upon scales, but sometimes upon sectors, &c. It consists of the logarithms of numbers transferred upon a ruler, &c. from the tables, by means of a scale of equal parts, which therefore serves to resolve problems instrumentally, in the same manner as logarithms do arithmetically. For, as logarithms resolve problems, or perform multiplication and division

by addition and subtraction, the same is performed on this line by only turning a pair of compasses in a certain way, or by sliding a slip of wood by the side of another, &c.

A line of this description has been contrived various ways—Mr. Gunter on the two feet ruler or scale, which is generally used at sea; and Wingate on two separate rulers, standing against each other in order to dispense with the use of compasses. Oughtred solved proportions by means of concentric circles. The same was performed by Mr. Milburne, by means of a spiral. Seth Patridge effected the same on the common sliding ruler, for a description of which, see my Mensuration for the National Schools. Mr. William Nicholson has proposed another on concentric circles.

The Construction of the Logarithmical lines on Gunter's Scale.

1. The line of numbers, on which most of the others depend, is constructed thus:-Let half the length of the line of numbers be divided into 1000 equal parts; then, since the logarithm of 1 is 0, (see my Philosophy and Practice of Arithmetic,) the distance from the end to 1 is 0; that is, 1 stands at the beginning of the line. And as the logarithm of 2 is 301, the distance between 1 and 2 is 301 equal parts, taken from a scale of equal parts; the distance 477, the logarithm of 3, is to be set off from 1 to 3; and 602, the logarithm of 4, is to be set off from 1 to 4, &c. These are all the primary divisions on this line, and the intermediate

divisions are formed in a similar manner, by taking the logarithms of the intermediate numbers.

2. The line of sines is constructed by taking the arithmetical complements of the logarithmic sines from the same scale of equal parts from which the line of numbers was constructed, and setting them off from 90 backwards, or towards the left hand; thus the arithmetical complement of the logarithmic sines of 80°, 70°, 60°, 50°, 40°, 30°, 20°, 10°, are 7, 27, 116, 192, 301, 466, 760; and these taken from the same scale of equal parts from which the line of numbers was constructed, and set off from 90 towards the left hand, gives the sines of the above degrees respectively.

As the sine of 90° is equal to radius, and as the several arithmetical complements are what the sines of the arcs want of radius, the reason of setting off the numbers as above is obvious.

3. The line of Rhumbs is constructed in a similar manner to the line of sines, by taking the arithmetical complements of the logarithmic sines of the degrees and minutes which are contained in the several points and quarter-points of the compass, and setting them off from the right-hand end of the scale towards the left.

4. The line of tangents is constructed in the same way as the line of sines, by setting off the arithmetical complements of the tangents under 45° backwards, from 45° towards the left hand. For the tangent of 45° is equal to the radius, and the arithmetical complement of any logarithmic tangent under 45° is what the tangent wants of radius.

It has been before observed, that the division at 40 serves both for 40 and 50—that at 30, for 30 and 60, &c.; the reason of which cannot well be given here.

5. The line of tangent rhumbs is constructed as the line of tangents, by taking the arithmetical complements of the logarithmic tangents of the degrees and minutes contained in the first four points of the compass, and setting them off from the end of the line towards the left hand.

6. The line of versed sines is constructed by means of the logarithmic versed sines extending to 180°.

Take the logarithmic versed sines of the supplements of the arcs, and subtract the logarithm of 2 from them; the arithmetical complements of the remainders, taken from the scale of equal parts, and applied from the right hand towards the left, will give the divisions of the line of versed sines.

The reason of this construction, and the use of the line itself, may be best learned from the teacher. It is by far the most difficult line on the scale, and is very seldom understood.

The preceding lines are the principal ones on the plane scale; and to render them convenient for practice they are transferred from the figure in page 7, to another of convenient length and breadth. These lines are of use in Trigonometry, Navigation, Fortification, Dialing, and in almost all the practical branches of the mathematics.

On these lines all proportions are solved; and when four numbers are proportionals, the ratio subsisting between the two first terms is equal to that

between the two last: that is, the quotient of the first term by the second, is equal to the quotient of the third term by the fourth. Therefore, from the nature of logarithms, the difference between the first and second terms, on the scale, is equal to the difference between the third and fourth. And as four quantities are proportionals when taken alternately, the distance. between the first and third is equal to the distance between the second and fourth. Hence the following General Rule :

The extent of the compasses from the first term to the second, will reach, in the same direction, from the third term to the fourth.

Or, the extent of the compasses from the first term to the third, will reach, in the same direction, from the second term to the fourth.

This rule is general, except the first two or the last two terms of the proportion are on the line of tangents, and neither of them under 45°. In this case the extent on the tangents is to be made in a contrary direction, as has been shewn in my Mensuration.

The reason is obvious, from the construction of the line of tangents. For, had the tangents above 45° been laid down in their proper direction, they would have extended beyond the length of the scale; they are therefore doubled backwards, as it were, upon the tangents below 45°, and consequently lie in a contrary direction to their natural order.

It is further to be observed, that 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

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