The next writer whom we can find is Seth Partridge, in a 'Descrip tion &c. of the Double Scale of Proportion,' London, 1685. Ile studiously conceals Oughtred's name : the rulers of the latter were separate, and made to keep together in sliding by the hand ; perhaps Partridge considered the invention his own, in right of one ruler sliding between two others kept together by bits of brass. Coggle. shall's ruler was made in both ways, that is, with the rulers attached and unattached; it appears to have come in at the end of the 17th century. Since that time several works have been written, and various modifications of the ruler proposed. Ward (' Lives of Gresham Professors') is incorrect in saying that Wingate carried the sliding rule into Frauce in 1624: it was Gunter's scale which be introduced there. lu fact the slide was little used and little known till the end of the century. Leybourn, himself a fancier of instruments, and an improver (as ho supposed) of the sector, has 30 folio pages of what he calls instrumental arithmetic in his 'Cursus Mathemnaticus' (1690), but not ono word of any sliding-rule, though ho puts fixed lines of square» and cubes against his lino of numbers in his version of Gunter's scale, Finding so meagre an account on this matter in publications pro fessedly mathematical, we did not at first think of having recourse to any others. When we had finished the preceding however,wo thought — of consulting the Biographia Britaniaica: and there we found, in the middle of a very full life of Oughtred, the whole account of the invention of the sliding-rule, exactly as above, and from the same authorities. On looking at Dr. Hutton's account in the Dictionary, we perceive that he has either used this memoir or some copy of it ; but without giving any information on the subject of the present article.
We shall conclude this article by some account of a new species of sliding-rule, invented by Dr. Beget (` Phil. Trans.' for 1815), which would be very useful in the hands of writers on statistics, and would sometimes save much trouble to the mathematician. The slide con tains a common logarithmic line of two radii, each 10 inches in length. The fixed ruler has not logarithms, but logarithms of logarithms denoted by its spaces. For instance, reckoning from 10 (remembering that log log 10=0), the space from 10 to 100 is log log 100, or rag 2, the Caine space as from 1 to 2 on the slide. And since log log .z is positive or negative, according as x is greater or less than 10, we have the logologarithms laid down on the left for numbers less than 10, and on the right for numbers greater. This instrument is constructed by Mr. Rooker, and in this manner : 10 is on the middle of the upper ruler, which ends on the right at or ten thousand millions ; and on the left at P25. At the ex treme right of the lower ruler we find P25 again, from which we recede to on the extreme left. The upper and lower rulers are so adjusted that from the end of one to the beginning of the other it is exactly two radii, so that a setting on the upper ruler applies also to the lower, but it may be necessary first to slip the slide a whole radius forwards or backwards, in the manner described in the preceding part of the article. And hero again the meaning of the reading en the slide must frequently be determined by common sense applied to the problem.
When 1 on the elide is placed opposite to a on the ruler, we have b on the slide opposite to ou the ruler. Or using the pre ceding rotation— The approximations of this rule are equally easy whether applied to fractional or integer exponents, and Dr. Itoget justly observes that it gives a much better idea of the rapid increase of powers than simple reflexion. It is so little known even to mathematicians, that we put down some of its results as specimens of its powers. Set 1 on the slide opposite to on the rule, and we find for the approximate powers of this number by simple inspection 9'85, 31, 97, 300, 960, 3000, 9500, 29,500, 93,000, &e. The square root is 1172,
the cube root P463, the fourth root P331, the fifth root P257. We must now change the slide, as above directed, so as to put it in con nection with the lower scale, and the proceeding roots are P215, &e. All questions of increase of money, population, &a, are in this manner reduced to simple inspection, and very easy trial gives that approximate solution of exponential equations which the mathema. ticiau must find before ho applies his more extensive methods. Thus, to form the table of logarithms in Seeic the base of which is '4/2 :—Set 12 on the slide opposite to 2 on the ruler, and the table is ready, as far as the instrument will give it. Thus, oppo site to 3, 4, 5, &c., we find 19'0, 24, 31.0, 33'7, &c., almost exactly as in the table cited. It is also worth notice that each division of the upper fixed ruler answers to the hundredth power of the division directly beneath it on the lower fixed ruler. Thus, wishing to know what effect would be produced in 100 years upon a population which increases 3'46 per cent., we set unity to 1.0046 on the lower scale, and find at once 30'925 ou the upper rider, being the number by which the present population must be multiplied.
The late Mr. Woollgar (to whom we were indebted for much information in this article, and who made a particular study of the slid ing-rule) carried to a considerable extent the principle of making the slide or the rule (no matter which) bear, not the logarithms of the numbers marked on its graduation, but those of the values of a func tion of those numbers :(' Mechanics' Magazine,' No. 849, vol, xxxii.) Let a slide be so graduated that the interval from a given point to the graduation r represents log sex. When is then ascertained (by the common scale, if necessary), the formula a4ix is immediately deduced from the common scale and the new slide. Nor need there be a new slide : for any scale being laid down in the groove, the common slide, by having its end made to coincide with one or another division of the scale in the groove, may be rendered capable of answering the pur pose of a new slide.
We long since obtained from Paris a circular logarithmic scale in brass, altogether resembling the one figured and described in the pre ceding part of this article, with the addition only of a clamping screw. This instrument, the scale of which is 41 inches in diameter, is so well divided that it will stand tests which the wooden rules would not bear without showing the error of the divisions. But hero arise disadvantages which we had not contemplated. In the first place, no subdivision can be well made or read by estimation, unless the part of the scale on which it comes is uppermost or undermost, which requires a continual and wearisome turning of the instrument. In the next place, to make the best use of it, and bring out all its power, requires (we should rather say renders worth while) such care in setting and reading, as, unless a microscope and tangent screw were used, makes the employment of the four-figure logarithm card both shorter and less toilsome. For rough purposes, then, a wooden rule is as good ; for pore exact ones, the card is better. We made a fair trial of both on tie tables in SOLAR SYSTEM, and are perfectly satisfied that though the French brass arithmometer did, with great care, bring out the results required, the four-figure card did the work more easily. But, had we wished to abandon two or three units in the last places of figures, there would then have been no doubt that the instrument would have been the easier of the two : but then a straight wooden rule of the same radius would have done quite as well, and been morn convenient still. (' Mechanics' Magazine,' No. 949.)