arithmetical complement of any logarithmical tangent under 45°, is what that tangent wants of radius. It has been observed that the division at 40 serves both for 40 and 50, that at 30 for 30 and 60, &c. The reason of this will appear if we consider that the tangent of any arc is to the radius, as the radius to the tangent radius radius co-tangent or logarithmically speaking, the difference between the tangent and radius is equal to the difference between the radius and co-tangent. But the tangent of 45° is equal to the radius, therefore the difference between the tangent of an arc (below 45°) and 45°, is equal to the difference between the co-tangent of that arc and 45°; viz. they are both equidistant from 45°.

co-tangent; or, which is the same thing,

=

(T) The line of tangent rhumbs is constructed in a similar manner as the line of tangents, by taking the arithmetical complements of the logarithmical 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.

(U) The line of versed sines is constructed by the help of a table of logarithmical versed sines extending to 180°. Take the versed sines of the supplements of the arcs, and subtract the logarithm of 2 from them, 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.

(W) 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; sois 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,

(X) 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

the same way will reach from the sine of the difference between the half sum of 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 splaced so close together. This is * Gunter's Rule, and it is more simple in its application than that which Robertson has given in his Navigation, without demonstration (Art. 29, Book IX), for working an Azimuth.

(Y) The line of meridional parts is constructed by the help of a table of meridional parts. Take the meridional parts correspondent to the several degrees of latitude from the table, and divide them by 60; take these quotients from the scale of equal

* 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 whatever, it is demonstrated in Spherical Trigonometry that sine AB X sine BC: square rad. :: sine (AB+BC+AC) × sine (AB+ BC +AC)-AC: square cosine B. But square cosine

BC: square rad. :: sine (AB + BC + AC) × sine (AB + BC + AC) rad. x vers. sine supp1 B

2

, or, sine AB x sine BC: rad. :: sine (AB+ BC + AC) ×

versed sine supp1 B

2

: sine (AB+ BC + AC) :: sine (AB + BC + AC) — AC :

the 7th sine, and multiplying the extremes and means, we have the equation above. Here we see plainly that the versed sines on the scale are only half the versed sines of the supplements of the angles with which they are marked, as has been observed before. He says the mean proportion between the 7th sine and the radius, gives the cosine of half the angle required; this is rad. x versed sine suppt B hence cosine B=

also true, for square cos. B

rad. x

=

2

=

"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."

parts,

!

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 LOGARITHMICAL LINES ON GUNTER'S

SCALE.

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

(Z) 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 pròportional 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

(A) 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.

(B) 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, &c. OF ANGLES.

(C) An angle is the inclination or opening of two straight lines meeting in a point as, A. A

B

-A

A

C

(D) 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 compasses, so as always to remain fixed to each other in B. mity 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.

While the extre

-A

(E) The magnitude of an angle does not consist in the length of the lines which form it, but in their opening or inclination to each other. Thus the angle ABC is less than the angle aBC, though the lines AB and CB which form the former angle, are longer than the lines 4B and CB which form the latter.

B

(F) 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.

(G) 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 aBc and the arc DE is the measure of the angle ABC.

(H) 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;

C

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, &. (I) 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.

A

B

(K) An acute angle is less than a right angle, or 90°, as EDB. (L) An obtuse angle is greater than a right angle, or 90°, as

ADE.

(M) 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°.

With any extent of the compasses

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

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

PROBLEM II.

A

H

D

(P) 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 G 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.

XE

H

(Q) Otherwis