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3. Given the side AC 68, the angle A 124°, and the angle C 37° to construct the triangle.

170. PROB. II. Two sides and an opposite angle being given, to find the remaining side, and the other two angles.

Draw one of the given sides; from one end of it, lay off the given angle; and extend a line indefinitely for the required side. From the other end of the first side, with the remaining given side for radius, describe an arc cutting the indefinite line. The point of intersection will be the end of the required side.

If the side opposite the given angle be less than the other given side, the case will be ambiguous. (Art. 152.)

Ex. 1. Given the angle A 63° 35', (Fig. 29.) the side b 32, and the side a 36.

The side AB will be 36 nearly, the angle B 52° 45', and C 63° 39'.

2. Given the angle A (Fig. 28.) 35° 20', the opposite side a 25, and the side b 35.

Draw the side 6 35, make the angle A 35° 20′, and extend AH indefinitely. From C with radius 25, describe an arc cutting AH in B and B'. Draw CB and CB', and two triangles will be formed, ABC and AB'C, each corresponding with the conditions of the problem.

3. Given the angle A 116°, the opposite side a 38, and the side b 26; to construct the triangle.

171. PROB. III. Two sides and the included angle being given; to find the other side and angles.

Draw one of the given sides. From one end of it lay off the given angle, and draw the other given side. Then connect the extremities of this and the first line.

Ex. 1. Given the angle A (Fig. 30.) 26° 14', the side b 78, and the side c 106; to find B, C, and a.

The side a will be 50, the angle B 43° 44', and C 110° 2′. 2. Given A 86°, b 65, and c 83; to find B, C, and a. 172. PROB. IV. The three sides being given; to find the angles.

Draw one of the sides, and from one end of it, with an extent equal to the second side, describe an arc. From the other end, with an extent equal to the third side, describe a second arc cutting the first; and from the point of intersection draw the two sides. (Euc. 22. 1.)

Ex. 1. Given AB (Fig. 31.) 78, AĆ 70, and BC 54, to find the angles.

The angles will be A 42° 22', B 60° 52', and C 76° 45'. 2. Given the three sides 58, 39, and 46; to find the angles. 173. Any right lined figure whatever, whose sides and angles are given, may be constructed, by laying down the sides from a scale of equal parts, and the angles from a line of chords.

Ex. Given the sides AB (Fig. 35.)=20, BC=22, CD=30, DE=12; and the angles B-102°, C=130°, D=108°, to construct the figure.

Draw the side AB=20, make the angle B-102°, draw BC=22, make C-130°, draw CD=30, make D=108°, draw DE=12, and connect E and A.

The last line, EA, may be measured on the scale of equal parts; and the angles E and A, by a line of chords.



ART. 174. An expeditious method of solving the problems in trigonometry, and making other logarithmic calculations, in a mechanical way, has been contrived by Mr. Edmund Gunter. The logarithms of numbers, of sines, tangents, &c., are represented by lines. By means of these, multiplication, division, the rule of three, involution, evolution, &c., may be performed much more rapidly, than in the usual method by figures.

The logarithmic lines are generally placed on one side only of the scale in common use. They are,

A line of artificial Sines divided into Rhumbs, and marked,

A line of artificial Tangents,


A line of the logarithms of Numbers,

A line of artificial Sines, to every degree,

A line of artificial Tangents,


A line of Versed Sines,

S. R.

T. R.




V. S.

To these are added a line of equal parts, and a line of Meridional Parts, which are not logarithmic. The latter is used in Navigation.

The Line of Numbers.

175. Portions of the line of Numbers, are intended to represent the logarithms of the natural series of numbers 2, 3, 4, 5, &c.

The logarithms of 10, 100, 1000, &c., are 1, 2, 3, &c. (Art. 3.)

If, then, the log. of 10 be represented by a line of 1 foot; the log. of 100 will be repres'd by one of 2 feet; the log. of 1000 by one of 3 feet; the lengths of the several lines being proportional to the corresponding logarithms in the tables. Portions of a foot will represent the logarithms of numbers between 1 and 10; and

portions of a line 2 feet long, the logarithms of numbers between 1 and 100.

On Gunter's scale, the line of the logarithms of numbers begins at a brass pin on the left, and the divisions are numbered 1, 2, 3, &c., to another pin near the middle. From this the numbers are repeated, 2, 3, 4, &c., which may be read 20, 30, 40, &c. The logarithms of numbers between 1 and 10, are represented by portions of the first half of the line; and the logarithms of numbers between 10 and 100, by portions greater than half the line, and less than the whole.

176. The logarithm of 1, which is 0, is denoted, not by any extent of line, but by a point under 1, at the commencement of the scale. The distances from this point to different parts of the line, represent other logarithms, of which the figures placed over the several divisions are the natural numbers. For the intervening logarithms, the intervals between the figures, are divided into tenths, and sometimes into smaller portions. On the right hand half of the scale, as the divisions which are numbered are tens, the subdivisions are units.

Ex. 1. To take from the scale the logarithm of 3.6; set one foot of the compasses under 1 at the beginning of the scale, and extend the other to the 6th division after the first figure 3.

2. For the logarithm of 47; extend from 1 at the beginning, to the 7th subdivision after the second figure 4.*

177. It will be observed, that the divisions and subdivisions decrease, from left to right; as in the tables of logarithms, the differences decrease. The difference between the logarithms of 10 and 100 is no greater, than the difference between the logarithms of 1 and 10.

178. The line of numbers, as it has been here explained, furnishes the logarithms of all numbers between 1 and 100. And if the indices of the logarithms be neglected, the same scale may answer for all numbers whatever. For the decimal part of the logarithm of any number is the same, as that of the number multiplied or divided by 10, 100, &c. (Art. 14.) In logarithmic calculations, the use of the indices is to determine the distance of the several figures of the natural numbers from the place of units. (Art. 11.) But in those cases in which the logarithmic line is commonly used, it will

* If the compasses will not reach the distance required; first open them so as to take off half, or any part of the distance, and then the remaining part.

not generally be difficult to determine the local value of the figures in the result.

179. We may, therefore, consider the point under 1 at the left hand, as representing the logarithm of 1, or 10, or 100; or, or, &c., for the decimal part of the logarithm of each of these is 0. But if the first 1 is reckoned 10, all the succeeding numbers must also be increased in a tenfold ratio; so as to read, on the first half of the line, 20, 30, 40, &c., and on the other half, 200, 300, &c.

The whole extent of the logarithmic line,
to 10,

is from 1 to 100,

or from 10 to 1000,

or from 100 to 10000, &c.

or from 0.1

or from 0.01 to 1,

or from 0.001 to 0.1, &c.

Different values may, on different occasions, be assigned to the several numbers and subdivisions marked on this line. But for any one. calculation, the value must remain the same. Ex. Take from the scale 365.

As this number is between 10 and 1000, let the 1 at the beginning of the scale, be reckoned 10. Then, from this point to the second 3 is 300; to the 6th dividing stroke is 60; and half way from this to the next stroke is 5.

180. Multiplication, division, &c., are performed by the line of numbers, on the same principle, as by common logarithms. Thus,

To multiply by this line, add the logarithms of the two factors; (Art. 37.) that is, take off, with the compasses, that length of line which represents the logarithm of one of the factors, and apply this so as to extend forward from the end of that which represents the logarithm of the other factor, The sum of the two will reach to the end of the line representing the logarithm of the product.

Ex. Multiply 9 into 8. The extent from 1 to 8, added to that from 1 to 9, will be equal to the extent from 1 to 72, the product.

181. To divide by the logarithmic line, subtract the logarithm of the divisor from that of the dividend; (Art. 41.) that is, take off the logarithm of the divisor, and this extent set back from the end of the logarithm of the dividend, will reach to the logarithm of the quotient.

Ex. Divide 42 by 7. The extent from 1 to 7, set back from 42, will reach to 6, the quotient.

182. Involution is performed in logarithms, by multiplying

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