The

Having fixed upon a convenient length for the entire scale, which must be exactly equal to the length of twenty of the primary divisions of the diagonal or vernier scale, or of the beam compasses (p. 48), by which it is to be divided, bisect it, and figure it 1 at the commencement on the left hand, 1 again in the middle, and 10 at the end. The half line, then, is taken for unity, or the logarithm of 10, and, consequently, the whole line represents 2, or the logarithm of 100. lengths corresponding to the three first figures of the logarithms of 2, 3, &c., up to 9, as found in the common table of logs., may now be taken off from the diagonal scale, or the length corresponding to four or even five figures may be estimated upon a vernier scale, or upon the beam compasses, if the scale be not less than twenty inches in length. These lengths are to be set off from the 1 at the commencement of the line for the logarithms of 2, 3, &c., to 9, and again from the 1 at the middle of the line for the logarithms of 20, 30, &c., to 90. The divisions thus formed are to be subdivided by setting off, in the same manner, the three, four, or five first figures of the logarithms of 1·1, 1.2, 1.3, &c., to 1.9; of 21, 22, 23, &c., to 2.9, and so on, each of the primary divisions being thus subdivided into ten; and these again are to be subdivided each into ten, or five, or two, as the length of the secondary divisions may admit, by setting off the logarithms of 111, 1·12, 1·13, &c.; or of 1·12, 1·14, &c.; or of 1-15, 125, &c.; and the scale is completed.

9. To construct the Line of Logarithmic Sines marked S.The whole length of the scale is taken as the logarithm of the radius, and, since this extent upon the line of numbers represents 2, or the logarithm of 100, it follows that the lines of sines, tangents, &c., are to be scales of the logarithms of the sines, tangents, &c., to radius 100, of which the logarithm is 2: whereas the logarithmic tables of sines, tangents, &c., are set down to a radius, of which the logarithm is 10. By taking 8, then, from each of the tabulated values of the logarithmic sines, tangents, &c., we should obtain the logarithmic sines, tangents, &c., to radius 100, and the three, four, or five first figures of these reduced values are to be set off, from the left hand towards the right, by one of the scales, or by the

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CONSTRUCTION OF LOGARITHMIC, OR GUNTER'S LINES. 27

beam compasses, as explained in the construction of the line of numbers; 1st, for every 10 degrees, then for every degree, and then for every half degree, every 10 minutes, and every 5 minutes, as far as the length of the several primary divisions will admit. The line is then numbered 1, 2, 3, &c., at every degree to 10, and afterwards 20, 30, 40, &c., at every ten degrees to 90, which stands at the extreme right, since sine 90° equals radius.

The tabulated logarithmic sine of 34′ 23′′, being 8·0000669, will coincide, or nearly so, with the zero point upon our scale, and consequently angles smaller than this cannot be taken off from the sines. This remark applies equally to the line of tangents, the tabulated logarithmic tangent of 34′ 23′′ being 8.0000886.

By taking the extents backwards from right to left, and reckoning them as forward distances, the line of sines becomes a line of cosecants *, giving us, in fact, the excesses of the logarithmic cosecants above the logarithmic radius; and, by taking the complements of the required angles, the line of sines becomes a line of cosines when measured forwards from left to right, and a line of secants when measured backwards from right to left.

3. To construct the Line of Logarithmic Tangents marked T. -8 being taken from each of the tabulated values of the logarithmic tangents up to 45°, the extents corresponding to these values are to be set off upon the scale, and numbered from left to right, in a similar manner to that in which the logarithmic sines were set off and numbered upon the line of logarithmic sines. The logarithmic tangent of 45° extends to the extreme right of the scale, coinciding in extent with the sine of 90, since tangent 45° equals radius, and the logarithmic tangents of the angles from 45° to 90 are measured backwards from the extreme right to the complement of the angle required, these extents giving us, in fact, the excesses of the logarithmic tangents sought above the logarithmic radius+

log. tan. of compt.

When, then, the angle is greater than 45, the distance from radius to the angle, though measured backwards upon the scale, must be reckoned a forward distance, and vice versa.

The lines of logarithmic sine rhumbs, marked S.R., and tangent rhumbs, marked T.R., are formed in the same way as the lines of logarithmic sines and tangents, but are set off for the angles corresponding to the points and quarter points of the compass, instead of for degrees and minutes.

We shall now proceed to explain the uses of Gunter's lines. 1. The Line of Logarithmic Numbers.-The primary divisions upon this line, as explained in its construction, represent the logarithms of all the integers from 1 to 100, while the extents to the first subdivisions will indicate tenths of an unit from the beginning of the scale to 1 in the middle, and units from 1 in the middle to 10 at the end, where the figures 2, 3, &c., stand for 20, 30, &c., as has been explained in the construction. If any of the subdivisions be further subdivided into ten parts, each of these last divisions will indicate hundredths of an unit from 1 at the beginning to 1 in the middle, and tenths of an unit from 1 in the middle to 10 at the end. Upon pocket sectors (p. 34), however, upon which Gunter's lines are now usually placed, affording a greater extent for the purpose than the six-inch plain scale (p. 33), only the part from 1 in the middle to 2 towards the right is a second time divided, and that but into five parts instead of ten, every one of which must be accounted as two-tenths. By this line the multiplication and division of numbers of any denomination either whole or fractional may be readily accomplished, questions in proportion solved, and all operations approximatively performed with great rapidity, which can be performed by the common table of logarithms; but the numbers sought must always be supposed to be divided or multiplied by 10 as many times as will reduce them to the numbers, the logarithms of which are actually set off upon the line of numbers, and these tens must be mentally accounted for in the result.

Multiplication is performed by extending from 1 on the left to the multiplier; and this extent will reach forwards from the multiplicand to the product. Thus, if 125 were given to be multiplied by 250, extend the compasses from 1 at the left hand to midway between the second and third subdivision, in the first primary division from 1 to 2, for the 125. This extent is really the logarithm of 1.25. Set off this extent towards the right from the fifth subdivision after the primary division marked 2, which is taken to represent

the log. of 250, but is really the log. of 2.5, and the compasses will reach to a quarter of the next subdivision beyond the first subdivision after the primary division marked 3. The extent to this point is really the logarithm of 3·125; but in this case it represents the number 31250, because two powers of ten have been cast out from both the multiplier and multiplicand, and therefore the product must be multiplied by the product of four tens, or ten thousand; or, in other words, the first figure of the product must be reckoned as so many tens of thousands.

Division, being the reverse of multiplication, is performed by extending from 1 on the left to the divisor; and this extent will reach backwards from the dividend to the quotient. Thus, if 31250 were to be divided by 250, extend the compasses from 1 on the left to 2.5, and this extent will reach backwards from 3.125 to 1.25. Then, since the divisor contained 2 powers of ten and the dividend 4, the quotient must contain 2, and therefore the result is 125.

Proportion being performed by multiplication and division, extend the compasses from the first term to the second, and this extent will reach from the third to the fourth, taking care to measure in the same direction, so that, if the first be greater than the second, the third may be greater than the fourth, and vice versa. Example.—If the diameter of a circle be 7 inches, and the circumference 22, what is the circumference of another circle, the diameter of which is 10? Ex tend the compasses from 7 to 10, and this extent will reach from 22 to 314, or nearly 31 inches, the circumference required.

The same thing may also be performed by extending from the first term to the third, and this extent will reach from the third term to the fourth (Euc. v. prop. 16). Thus, the extent from 7 to 22 will reach from 10 to 31.4, as before.

To measure a Superficies, extend from 1 to either the breadth or length, both being reduced to the same denomination, and this extent will reach forwards from the length or breadth to the superficial content. Example.-Required the superficial content of a plank 27 feet long by 15 inches broad. Extend from 1 to 125, for 15 inches equals 1.25 feet, and this extent will reach from 27 feet to 33.75 feet, the superficial content required.

Second Method.-Extend from 12 to the number of inches in the breadth, and this extent will reach in the same direction from the number of feet in the length to the number of square feet in the superficial content. Thus the extent for

wards from 12 to 15 will reach forwards from 27 to 33.75, as before; while the extent backwards from 12 to 9 will reach backwards from 27 to 20-25 or 20, showing the superficial content of a plank 27 feet long by 9 inches broad to be 20.25 or 20 feet.

To measure a Solid Content.-The breadth, depth, and length being all reduced to the same denomination, extend from 1 to either the breadth or depth, and this extent will reach from the depth or breadth forwards to a fourth number, which will represent the superficial content of the section at the place measured: then, if the breadth and depth be the same throughout the entire length, the extent from 1 to the superficial content thus found will reach forwards from the length to the solid content. Example.-What is the solid content of a pillar 1 foot 3 inches square, and 21 feet 9 inches long? The extent from 1 to 1.25 reaches forward from 1.25 to 1.56, the superficial content of a section of the pillar; and the extent from 1 to 1.56 reaches from 21.75 to 34, or more accurately to 33.93, the solid content in feet*.

2. The Lines of Logarithmic Sines and Tangents.-These lines are generally used, in connection with the line of numbers, for solving all proportions in which any of the terms are functions of angles, as sines, tangents, &c., and, in fact, all questions in which such quantities appear as factors or divisors. We will exemplify their use by giving the solution, by their aid, of the several cases of right-angled trigonometry.

Case 1. The hypothenuse and angles being given, to find the perpendicular and base.

* Our limits forbid us from entering further upon the uses of the line of logarithmic numbers; but the student will, we hope, from what he sees here, be easily enabled to apply it to every case of mensuration, and, in short, to almost every arithmetical operation. Additions and subtractions, however, cannot be performed by it.

These cases are, in fact, the solutions, by the aid of Gunter's lines, of the following proportions, which will

be obvious to the student upon inspec

tion of the accompanying figure.