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manner to 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 from the end of the line towards the left hand.

(44) The line of versed sines is constructed by the help of a table of 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 same scale of equal parts as the other lines were constructed from, and applied from the right hand towards the left, will give the divisions of the line of versed sines.

(45) The nature and use of this line are, I believe, very imperfectly understood; and in order to explain them clearly, we must have recourse to the inventor, viz. Gunter; he says, p. 231. Leybourn's edition, 1673, that he contrived this line "for the more easy finding of an angle having three sides, or "a side having three angles of a spherical triangle given." He then gives the following proportions:

As radius

Is to sine of one of the sides containing any angle, so is the sine of the other containing side,|| to a fourth sine.

As this fourth sine

Is to sine of half the sum of the three sides, so is the sine of this half sum diminished by the side opposite the given angle, to a seventh sine.

The mean proportional between this seventh sine and the radius, gives the sine of the complement of half the angle required. By the scale,

(46) Extend the compasses from the sine of 90 to the sine of one of the sides containing any angle; that extent applied the same way will reach from the sine of the other side to a fourth sine. From this fourth sine extend the compasses to the sine of half the sum of the three sides, and this extent applied the same way will reach from the sine of the difference between the half sum of the three sides and the side opposite the angle taken, to a seventh sine; immediately under which, stands the angle required in the line of versed sines.

It is for this reason that the line of sines and versed sines are placed so close together. This is * Gunter's Rule, and it is

* As Gunter's works are merely practical, perhaps an investigation of these proportions will be acceptable to some readers:

Let ABC be any spherical triangle, denote the sides opposite the angles A, B, C, by a, b, c, and put s= (a+b+c).

Then Gunter's proportions are,

A

B

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more simple in its application than that which Robertson has given in his Navigation, without demonstration (Art. 29. Book IX.), for working an Azimuth.

(47) The line of meridional parts is constructed by the help of a table of meridional parts. Take the meridional parts corresponding to the several degrees of latitude from the table, and divide them by 60; take these quotient from the scale of equal parts, already described under the article meridional parts, and set them off on the line of meridional parts from the right hand towards the left.

THE USE OF THE LOGARITHMIC LINES ON GUNTER'S SCALE.

By these lines and a pair of compasses, all the problems in Trigonometry, Navigation, &c. may be solved.

(48) 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, in logarithms, 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:

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Hence the mean proportional between the 7th sine and radius gives the cosine of the required angle; and this 7th sine is also equal to half the suversed sine of the required angle.

"But because the finding the mean proportional between the radius or sine of 90° and the 7th sine, is somewhat troublesome," says Gunter, "I have added this "line of versed sines, that having found the 7th sine, you might look over against "it and there find the angle." Gunter has neither shown the construction of the line of versed sines, nor given the investigation of the above proportions; but he has illustrated them by a practical example, wherein the three sides of a spherical triangle are given to find an angle.

(49) GENERAL RULE. The extent of the compasses from the first term to the second will reach, in the 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 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 first two or 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: For had the tangents above 45° 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 upon the tangents below 45°, and consequently lie in a direction contrary to their proper and natural order.

(50) If the last two 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 first two terms from 45° 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.

CHAP. V.

GEOMETRICAL DEFINITIONS AND INTRODUCTORY PROBLEMS,

DEFINITIONS, ETC. OF ANGLES.

(51) An angle is the inclination or opening of two straight

lines meeting in a point as A. A

A

(52) One angle is said to be less than another, when the lines which form it are nearer to each other. Take two lines AB and BC meeting each other in the point B; conceive these two lines to open like the legs of a pair of com

B

C

passes, so as always to remain fixed to each other in B. While the extremity a moves from the extremity c, the greater is the opening or angle ABC; and, on the contrary, the nearer you bring them together, the less the opening or angle will be.

A

(53) The magnitude of an angle does not consist in the length of the lines which form it, but in the extent of their opening or inclination to each other. Thus the angle ABC is less than the angle aвc, though the lines AB and CB which form the former angle, are longer than the lines aв and CB which form the latter.

B

D

E

C

(54) When an angle is expressed by three letters, as ABC, the middle letter always stands at the angular point, and the other two letters at the extremities of the lines which form the angle; thus the angle ABC is formed by the lines AB and CB, and that of abc by the lines aв and CB, &c.

(55) Every angle of a triangle is measured by an arc of a circle described about the angular point as a centre; thus the arc ac is the measure of the angle aвc, and the arc DE is the measure of the angle ABC.

(56) The circumference of every circle is supposed to be divided into 360 equal parts called degrees, each degree into 60 equal parts called minutes, each minute into 60 equal parts called seconds. The angles are measured by the number of degrees cut from the circle by the lines which form the angles; thus, if the arc DE contain 20 degrees, or the 18th part of the circumference of the circle, the measure of the angle ABC is 20 degrees. Degrees, &c. are thus marked, 44° 32′ 21′′ 14"", &c. and read 44 degrees, 32 minutes, 21 seconds, 14 thirds, &c. (57) When a straight line CD standing upon a straight line AB, makes the angles CDB and CDA on each side equal to one another, each of these equal angles is said to be a right angle, and the line CD is perpendicular to AB. The measure of a right angle is therefore 90°, or a quarter of a circle.

C

E

D

B

(58) An acute angle is less than a right angle, or 90°, as EDB. (59) An obtuse angle is greater than a right angle, or 90o,

as ADE.

(60) If ever so many angles are formed at the point D, on the same side of the line AB, they are altogether equal in measure to two right angles, or 180°.

PROBLEM I.

(61) To erect a perpendicular from a given point D in a given line GH, or to make a right angle.

On each side of D take the equal distances AD and BD.

With any extent of the compasses

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greater than AD, and centres A and B, describe arcs crossing each other in c; a line CD, drawn through C and D, will be the perpendicular required.

Otherwise. When the point D is at the

end of the line GH; with the centre D and any opening of the compasses describe an arc; set off the distance AD from A to B; with B as a centre, and the distance AB in your compasses, describe another arc; through a and B draw the line ABC, cutting the second

H

D

arc in c; lastly, through c and D draw the line CD, and it will be the perpendicular required.

PROBLEM II.

(62) From a given point c, not in the straight line GH, to draw a straight line CD perpendicular to GH.

Take any point e on the contrary side of GH to which the point c is, and with the distance ce and centre c describe an arc cutting GH in A and B ; with A and B as centres, describe arcs crossing each other in E, a line CDE drawn through c and E will be the perpendicular required.

Otherwise. When the point c is near the end of the line GH. Take any points A and G in the line GH, with the centres A and G, and the distances AC and GC, describe arcs crossing each other in E, the line CDE drawn through c and E, will be the perpendicular required.

D

H

B

A.

H

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