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Scale Lines on the Logarithmic Chart

Nature volume 52, pages 272274 | Download Citation



THE advantages of logarithmic plotting for certain classes of work have for some time been recognised, and now that, thanks to Mr. Human, logarithmically ruled paper can be obtained ready made, the facility of such plotting is greatly increased, so that there is all the more reason on this account why it should become more common than it seems to be at present. It may perhaps be well to point out shortly what the nature and effect of logarithmic plotting is, and to contrast it with the more common method on square-ruled paper. Instead of paper ruled in equal squares, logarithmic paper is ruled first in a series of large equal unit squares representing tenfold changes in the coordinates. Thus two units represent 100, three units 1000, and so on. Similarly the squares are broken up fractionally and unequally into a series of vertical and horizontal lines, whose distance from the left or lower side of the square is equal to the logarithms of the numbers 2, 3, 4, &c., and these are subdivided again logarithmically just in the same way that a slide rule is subdivided. In fact, if logarithmic paper is not available, logarithmic plotting can still be carried out fairly expeditiously by pricking off distances direct from a good slide-rule. The meaning of lines drawn upon logarithmic paper is very different from that upon ordinary square ruled paper. For instance, an inclined straight line ruled in the ordinary way represents the equation y= a+bx, whereas when logarithmic paper is employed the corresponding line gives y = axb. The consequence is that whenever two quantities are related so that one varies as any power, positive, negative, integral, or fractional of another, a straight line drawn in the proper position and inclination represents that relation, the power being equal to the trigonometrical tangent of the angle of slope of the straight line. If the relation that is to be represented is less simple, if the index changes gradually as either of the coordinates changes, so that a curve has to be employed, then the size and shape of the curve represents the law in the abstract, and the position of the curve on the sheet the actual numbers for the particular case and with the particular units; a mere shift of the curve bodily upon the chart, as pointed out by Prof. Osborne Reynolds long ago, being all that is necessary to adopt the same law to new circumstances or new units.

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