It will be observed that in every construction the logarithmic space: are very unequal, those near the end of the scale being small whet compared with those at the beginning. This is not so great a clisad vantage as might be supposed, for it makes the liability to error increase in nearly the same proportion with the result, so that the percentage of error in the sliding-rule is nearly the same thing in all its parts For example, the scale going from 10 to 100, the interval from 10 tc I I is to that from 99 to 100 as 207 to 22, nearly in the proportion of 10 to 1. The tendency to absolute error will be inversely as these intervals, or nearly in the proportion of 1 to 10 : the tendency to error is therefore about 10 times as great precisely when the result estimated becomes ten times as great. Oughtred appropriated two circles to his logarithms of sines, and it would be easy in his construction of the circles of proportion,' as he called them, to distribute the scale among different circles in such a manner that the graduations should be nearly equal throughout. But the mathematician will easily see that that the most perfect mode of developing this idea would be to lay down the scale on a revolution of a logarithmic spiral, having the pointers joined at its pole. The graduations would then be abso lutely at equal distances from each other on the are of the spiral.
Another modification of the principle of the sliding-rule is as follows : —Let the divisions be all made equal, and the numbers written upon the divisions in geometrical proportion. If this were done to a sufficient extent, any number might be found exactly or nearly enough upon the scale ; the only difficulty being that very small divisions do not give room enough to write the numbers. This modification of the principle has been applied in two very useful modes by Mr. MacFarlane. In the first, two cylinders moving on the same axis, on one side and the other of a third, give the means of instantaneously proposing and solving any one out of several millions of arithmetical questions for the use of schools and teachers. In the second, one circle revolving upon another gives the interest upon any sum, for any number of days, at any rate of Interest under 10 per cent.
The rules for using the sliding-rule, in its most simple form, may be symbolically expressed in the following manner :— Thus, if 1 on either ruler be brought opposite to A on the other, on the first ruler is brought opposite to A ti on the other. But if the slide be taken out and inverted, we have the following rulcai We now proceed to some of the additions which are frequently made to sliding-rules, premising that we do not describe any one in particular, but refer fur detail to the tracts which are afterwards cited. For the extraction of square or cube roots, or the formation of squares or cubes, the following method is adopted :—In the case of squares and square roots, for instance, there is a pair of scales, one on the slide and one on the fixed ruler, of different radii, the radius of one being twice as long as that on the other : for cubes and cube roots the radius of one is three times as long as the other. On the former scale (that of squares and square roots) the rules are now as follows :— The denomination of the answer, or the place of the decimal point, must be determined by independent consideration, as before; but there is one circumstance to be attended to in every case In which two of tho data are to be read on the shorter scale. For example,* suppose it is required to estimate a./(2: 7). By the second formula, 7 on the shorter scale is placed opposite to 1 on the longer, and 2 on the shorter scale is then apposite to 1693 on the longer. The answer from the scale is
then '1693, to all appearance ; but this is not s/(2:7), but s/(2 :70). The place on the longer scale which should give the answer has no slide opposite to It, but only empty groove. But mark where 1 on the shorter scale is opposite to a part of the longer (between 119 and 120), and push the slide in from left to right till the first 1 ou the shorter scale comes where the second now is ; then look under the second 2 of the shorter scale, we have 534 ; and is the true answer so far as the scale will give it. We have taken the most straightforward plan of reading the rule, and have not space for all the details which are in works on the subject, particularly the method of using the slide of numbers with a scale of numbers above and of square roots below. The following is the general principle applicable to the preceding case : It is well known that, whereas in common division the place of the decimal point has nothing to do with the significant figures of the quotient, yet in extracting the square, cube, &c., roots, the figures of the root are altered by a change of the decimal point, unless it be changed by an even number of places in extracting the square root, by three or a multiple of three places in the cube root, and so on. In extracting the square root, a number may either have two figures in its first period, or one ; thus .07616 and 1616 must (in the rule for extraction) be pointed Let us call numbers unidigital or bidigital, according as there are one or two significant figures in the first period. Then the application to the sliding-rule is, that on the shorter scale numbers of the same name must be read either on the same radius or with a whole radius inter vening, while numbers of different names must be read on different radii. In the scale for the extraction of the cube root, numbers must be distinguished into unidigital, bidigital, and tridigital; and signifying these by their initial letters, and taking the succession UBTUDT, there must be the same relation between the scales on which they are read that there is between the places of their letters in the preceding list. Thus, if u be read on one radius, T must either be read on that immediately preceding, or on the next but one. Thus, in the preceding question, which we first solved wrongly, we have 2 and 7 to consider on the the pointing of which is— and both are unidigital numbers. Bringing 7 on the shorter scale to 1 on the longer, we see that the next 2 is on a different radius; it would do then for 70, or .7, or '007, but not for 7. By the process we followed we took not indeed is 2 ou the same radius with our 7, but on the next radius but one, and thus obtained the correct answer. These points, and others (such as the meanings of the lines of sines, tangents, &c., annuities, &c., which are found on several rules) can only be mastered by those who are acquainted theoretically with logarithms, trigono metry, &e. ; for after all the eliding-rule will not teach the method of working any question, but will only afford aid in computation—in common multiplication and division, to any one • in higher to those who understand their principles. Oughtred, the inventor, kept the instrument by him many years, out of a settled contempt for those who would apply it without knowledge, having " onely the superticiall acumme and froth of instrumentall trickes and practises;" and wishing to encourage "the way of rationall scientiallists, not of ground-creeping 3lethodicks: A little distinction between that portion of its use which is generally attainable, and that which requires mathematical knowledge, would have been more reasonable.