plements 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 logarithmical 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. (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, Leyburn'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, (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 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 more * 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 radius: sine (AB + BC + AC)_x sine (AB+ BC + AC) — AC; square cosine B. But radius x versed sine supp' B square cosine B hence A C B 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 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 Trignometry 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 -AC: sine AB X sine BC; square rad. : : sine (AB + BC + AC) × sine + AC) x sine 2 2 (AB + BC + AC) sine (AB+ BC hence it follows that sine (AB+ BC + AC) × sine (AB + BC ÷ˆ versed sine supp* 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. The mean proportion between the 7th sine, and the radius, gives the cosine of half the angle required; for, square cos. }B= rad. × versed sine supp1 B 2 ✔rad. × 7th sine, the mean proportional. "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 hav"ing found the 7th sine, you might look over against it and there find the angle. Gunter has neither shewn the construction of the line of versed sines nor given the investigation of the above proportions; but he has illustrated them by a prac tical example, wherein the three sides of a spherical triangle are given to find an angle. 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 (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. B 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 (E) 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 -A C (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 aвс 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; 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. (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 C 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°. PROBLEM 1. (N) 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. GA -H D B: With any extent of the compasses greater than AD, and centres A and B, describe arts crossing each other in c; a line CD, 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 H arc in c; lastly, through c and D draw the line CD, and it will be the perpendicular required. PROBLEM II. (P) From a given point c, not in the straight line GH, to draw a staight 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. (Q) 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. IC G -H B A. D H |