LOGARITHMIC Lines

LOGARITHMIC Lines. For many mechanical purposes it is convenient to have the logarithms of numbers laid down on scales, as well as the logarithmic lines and tangents; by which means computations may be carried on by mere mensuration with compasses. Lines of this kind are always put on the common Gunter's scale; but as these instruments must be extended to a very great length, in order to contain any considerable quantity of numbers, it becomes an object of importance to shorten them. Such an improvement has been made by Mr William Nicholson, and published in the 77th volume of the Philosophical Transactions. The principles on which the construction of his instruments depends are as follow:

1. If two geometrical series of numbers, having the same common ratio, be placed in order with the terms opposite to each other, the ratio between any term in one series and its opposite in the other will be constant: Thus,

where it is evident, that each of the terms in the upper series is exactly two-thirds of the corresponding one in the lower.

2. The ratio of any two terms in one series will be the same with that between those which have an equal distance in the other.

3. In all such geometrical series as have the same ratio, the property above mentioned takes place, though we compare the terms of any series with those of another: Thus,

where it is plain that 2, 4, 3, 6; also 2, 4, 4, 8, and 2, 4, 5, 10, &c. have the same ratio with that of each series.

4. If the differences of the logarithms of the numbers be laid in order upon equidistant parallel right lines, in such a manner that a right line drawn across the whole shall intersect it at divisions denoting num-

bers in geometrical progression; then, from the condition of the arrangement, and the property of this logarithmic line, it follows, 1st, That every right line so drawn will, by its intersections, indicate a geometrical series of numbers; 2dly, That such series as are indicated by these right lines will have the same common ratio; and, 3dly, That the series thus indicated by two parallel right lines, supposed to move laterally, without changing either their mutual distance or parallelism to themselves, will have each the same ratio and in all series indicated by such two lines, the ratio between an antecedent and consequent; the former taken upon one line, and the latter upon another, will be also the same.

The 1st of these propositions is proved in the following manner. Let the lines AB, CD, EF, represent parts of the logarithmic line arranged according to the proportion already mentioned; and let GH be a right line passing through the points e, c, a, denoting numbers in geometrical progression; then will any other line IK, drawn across the arrangement, likewise pass through three points f, d, b, in geometrical progression. From one of the points of intersection f in the last-mentioned line IK, draw the line fg parallel to GH, and intersecting the arrangement in the points i, k; and the ratios of the numbers e, f, c, i, will be equal, as well as of a, h; because the intervals on the logarithmic line, or differences of the logarithms of those numbers, are equal. Again, The point f, the line id, and the line hb, are in arithmetical progression denoting the differences between the logarithms of the numbers themselves; whence the quotients of the numbers are in geometrical progression.

The 2d proposition is proved in a similar manner. For as it was shown that the line fg, parallel to GH, passes through points of division denoting numbers in the same continued ratio as those indicated by the line GH; it may also be shown, that the line LM parallel to any other line IK, will pass through a series of points denoting numbers which have the same continued ratio with those indicated by the line IK, to which it is parallel.

The 3d proposition arises from the parallelism of the lines to their former situation; by which means they indicate numbers in a geometrical series, having the same common ratio as before: their distance on the logarithmic line also remains unchanged; whence the differences between the logarithms of the opposite numbers, and of consequence their ratios, will always be constant.

5. Supposing now an antecedent and consequent to be given in any geometrical series, it will always be possible to find them, provided the line be of unlimited length. Drawing two parallel lines, then, through each of the numbers, and supposing the lines to move without changing their direction or parallel situation, they will continually describe new antecedents and consequents in the same geometrical series as before.

6. Though the logarithmic line contain no greater range of numbers than from 1 to 10, it will not be found necessary for the purposes of computation to repeat it. The only thing requisite is to have a slider or beam with two fixed points at the distance of the interval betwixt 1 and 10, and a moveable point made to range betwixt them always to indicate the antecedent; then, if the consequent fixed point fall with-

out the rule, the other fixed point will always denote the division on which it would have fallen had the rule been prolonged; and this contrivance may easily be adapted to any arrangement of parallel lines whatever. The arrangement of right lines, however, ought always to be disposed in such a manner as to occupy a right-angled parallelogram, or the cross line already mentioned ought always to be at right angles to the length of the ruler.

Fig. 7. is a ruler consisting of ten parallel lines.— Fig. 8. a beam-compas for measuring the intervals. B, A, C, are the parts which apply to the surface of the ruler; the middle one, A, being moveable sideways in a groove in the piece DE, so as always to preserve its parallelism to the external pieces DC, which are fixed at a distance equal to the length of the ruler, and have their edges placed in such a manner as to form with the parallel lines which they intersect a ratio, which by composition is $\backslash frac\{1\}\{2\}$; which in the present case requires them to be at right angles to the length. The piece DE is applied to the edge FG of the ruler. The edges or borders H, I, K, L, are more conveniently made of transparent horn, or tortoise-shell, than of any opaque matter.

In using this ruler, apply the edge of either B or C to the consequent, and slide the piece A to the antecedent; observing the difference between the numbers on the pieces denoting the lines they are found on: then, applying the same edge of A to any other antecedent, the other piece B or C will intersect a consequent in the same ratio upon that line, having the same situation with regard to the antecedent that the line of the former consequent had to its antecedent. But if B be the consequent piece, and fall without the ruler, the piece C will show the consequent one line lower; or if C, in like manner, fall without the ruler, then B will show the consequent one line higher.— "It might be convenient (says Mr Nicholson) for the purpose of computation, to make instruments of this kind with one hundred or more lines: but in the present instrument, the numbers on the pieces will answer the same purpose; for if a consequent fall upon a line at any given number of intervals without the ruler, it will be found on that line of the arrangement which occupies the same number of intervals reckoned inwards from the opposite edge of the ruler."

Fig. 9. is an instrument on the plan of a Gunter's scale of 28$\backslash frac\{1\}\{2\}$ inches long, invented by Mr Robertson. There is a moveable piece AB in the slider GH, across which is drawn a fine line; the slider having also lines CD, EF, drawn across it at distances from each other equal to the length of the ruler AB. In using the instrument, the line CD or EF is to be placed at the consequent, and the line in AB at the

antecedent: then, if the piece AB be placed at any other antecedent, the same line CD or EF will indicate its consequent in the same ratio taken the same way: that is, if the antecedent and consequent lie on the same side of the slider, all other antecedents and consequents in that ratio will be in the same manner; and the contrary if they do not. But if the consequent line fall without the rule, the other fixed line on the slider will show the consequent, but on the contrary side of the slider to that where it would else have been seen by means of the first consequent line.

Fig. 10. is a circular instrument equivalent to the former; consisting of three concentric circles engraved and graduated upon a plate of an inch and a half diameter. Two legs A and B proceed from the centre, having right-lined edges in the direction of radii; and are moveable either singly or together. In using the instrument, place one of the edges at the antecedent and the other at the consequent, and fix them at the angle. Move the two legs then together; and having placed the antecedent leg at any other number, the other will give the consequent one in the like position on the lines. If the line CD happen to lie between the legs, and B be the consequent leg, the number sought will be found one line farther from the centre than it would otherwise have been; and on the contrary, it will be found one line nearer in the like case, if A be the consequent leg. "This instrument (says Mr Nicholson), differing from that represented fig. 7. only in its circular form, and the advantages resulting from that form, the lines must be taken to succeed each other in the same manner laterally; so that numbers which fall either within or without the arrangement of circles, will be found on such lines of the arrangement as would have occupied the vacant places if the succession of lines had been indefinitely repeated sideways.

"I approve of this construction as superior to every other which has yet occurred to me, not only in point of convenience, but likewise in the probability of being better executed; because small arcs may be graduated with very great accuracy, by divisions transferred from a larger original. The instrument, fig. 7. may be contained conveniently in a circle of about four inches and a half diameter.

"The circular instrument is a combination of the Gunter's line and the sector, with the improvements here pointed out. The property of the sector may be useful in magnifying the differences of the logarithms in the upper parts of the line of lines, the middle of the tangents, and the beginning of the versed sines. It is even possible, as mathematicians will easily conceive, to draw spirals, on which graduations of parts, everywhere equal to each other, will show the ratios of those lines by moveable radii, similar to those in this instrument."