EXAMPLE. Let it be required to find a Rank of Ratios, whose Terms are integral, and the neareft poffible to the following Ratio, viz. of 10000 to 31416, which expreffes nearly the Proportion of the Diameter of the Circle to its Circumference. But because the Terms of the Ratio are not prime to each other, they must therefore be reduced to their leaft Terms. 1250, and then 3927 divided 31416 3927 by 1250, and 1250 by the Remainder, &c. will be as follows: 1250) 3927 (3 177) 1250 (7 11) 177 ( 16 II ( II So the firft Antecedent is 1, and the firft Confequent 3. Anteced. I {Confeq. 3 X7= $ 72=S7+0=7 the second Antec. 221+1=22 the second Conf. 21 S Producing the fame Antecedent and Confequent as at firft; which, as it is ever the Property of the Rule fo to do, proves, at the fame Time, that no Error has been committed thro' the whole Operation. Whence, as 1250:3927 ན་ 1:3 Forthe Terms 2 113:355 of the Ratio. But it must be observed, that 1 to 3 does not express the Ratio fo near as 7 to 22; B nor 7 to 22 fo near as 113 b 2 10 to 355; that is, the larger the Terms of the Ratio are, the nearer they approach the Ratio given. Mr. Molyneux, in his Treatife of Dioptrics, informs us, that when Sir Ifaac Newton fet about, by Experiments, to determine the Ratio of the Angle of Incidence, to the refracted Angle, by the means of their refpective Sines; he found it to be, from Air to Glass, as 300 to 193, or, in the leaft round Numbers,as 14 to 9. Now, if it be as 300 is to 193, it will readily appear, by the Rule, whether they are fuch integral Numbers, whofe Ratio is the neareft poffible to the given Ratio 193 ) 300 ( I 107) 193 (r 21) 86 (4 2 ) 21 ( 10 312 For, dividing the great Number by the lefs, and the lefs by the Remainder, &c, the Operation will how that the Numbers 193 and 300 are prime to each other; and that the firft Antecedent is 1, as also the hift Confequent. Hence, the fourth, Antecedent and Confequent make the Ratio to be as 9 to 14, or, inverfly, as 14 to 9;. which not only agrees with Mr. Molyneux, but at the fame Time difcovers, that they are nearer to the given Ratio, than any other integral Numbers lefs than 92 and 143; which are the neareft of all to the given Ratio, as will appear by repeating the Procefs, according to the Direction of the Rule. Sir Ifaac Newton himself determines the Ratio out of Air into Glafs to be as 17 to II; but then he speaks of the red Light. For that great Philofopher, in his Differ Differtations concerning Light and Colours, published in the Philofophical Tranfactions, has at large demonftrated, as alfo in his Optics, that the Rays of Light are not all homogeneous, or of the fame Sort, but of different Forms and Figures, fo that fome are more refracted than others, tho' they have the fame or equal Inclinations on the Glafs: Whence there can be no conftant Proportion settled between the Sines of the Angles of Incidence, and of the refracted Angles. But the Proportion that comes nearest Truth, for the middle or green-making Rays of Light, it feems, is nearly as 300 to 193, or 14 to 9. In Light of other Colours the Sines have other Proportions. But the Difference is fo little, that it need feldom to be regarded, and either of those mentioned for the moft Part is fufficient for Practice. However, I muft obferve, that the Notice here taken either of the one or the other, is more to illuftrate the Rule, and fhew, as Occafion requires, how to express any given Ratio in fmaller Terms, and the neareft poffible, with more Eafe and Certainty, than any Defign in the leaft of touching upon Optics. Wherefore, left this fmall Digreffion from the Subject in hand, and indeed even from my firft Intentions, fhould tire the Reader's Patience, I fhall not prefume more, but immediately proceed to the Conftruction of Logarithms. Of the Conftruction of Logarithms. THE Nature of which, tho' our Author has fuf ficiently explained in the Defcription of the Logarithmetic Curve; yet, before we attempt their Conftruction, it will be neceffary to premise: That the Logarithm of any Number is the Exponent or Value of the Ratio of Unity to that Number; wherein we confider Ratio, quite different from that laid down in the fifth Definition of the 5th Book of thefe Elements ; for, beginning with the Ratio of Equality, we fay 1 to 1=0; whereas, according to the faid Definition, the Ratio of 1 to 11; and confequently the Ratio here mentioned is of a peculiar Nature, teing affirmative when increafing, as of Unity to a greater Number; but negative when decreasing. And as the Value of the Ratio of Unity to any Number is the Logarithm of the Ratio of Unity to that Number, fo each Ratio is fuppofed to be meafured by the Number of equal Ratiunculæ contained between the two Terms thereof: Whence, if in a continued Scale of mean Proportionals, infinite in Number, there be affumed an infinite Number of fuch Ratiuncula, between any two Terms in the fame Scale; then that infinite Number of Ratiuncula is to another infinite Number of the like and equal Ratiunculæ between any other two Terms, as the Logarithm of the one Ratio is to the Logarithm of the other. But if, inftead of fuppofing the Logarithms compofed of a Number of equal Ratiuncula proportionable to each Ratio, we fhall take the Ratio of Unity to any Number to confift always of the fame infinite Number of Ratiunculæ, their Magnitudes in this Cafe will be as their Number in the former. Wherefore, if between Unity, and any two Numbers propofed, there be taken any Infinity of mean Proportionals, the infinitely little Augments or Decrements of the firft of thofe Means in each from Unity will be Ratiunculae; that is, they will be the Fluxions of the Ratio of Unity to the faid Numbers; and because the Number of Ratiunculæ in both are equal, their respective Sums, or whole Ratios, will be to each other as their Moments or Fluxions; that is, the Logarithm of each Ratio will be as the Fluxion thereof. Confequently, if the Root of any infinite Power be extracted out of any Number, the Difference of the faid Root from Unity fhall be as the Logarithm of that Number. So that Logarithms, thus produced, may be of as many Forms as we please to affume infinite Indices of the Power whofe Root we feek. As, if the Index be fuppofed 100000, &c. we fhall have the Logarithms invented by Neper; but if the faid Index be 230258, &c. thofe of Mr. Briggs will be produced, Wherefore, if 1+x be any Number whatsoever, and I n infinite, then its Logarithm will be as 1 + x - I= bers, is 1+x+x+xxx+xxxx, &c. and the celebrated binomial Theorem invented by Sir Ifaac Newton for being an Infinitefimal, is rejected; whence the infinite n + x3 2n 3n 4.22 &c. is the Augment of the first of the mean Proportionals between Unity and 1+x, which therefore will be as the Logarithm of the Ratio of 1 to 1+x, or as the Logarithm of 1+x. But as 1+x-1 is a Ratiuncula, it must be multiplied by 10000, &c. infinitely, which will reduce it to Terms fit for Practice, makeing the Logarithm of the Ratio of 1 to 1 + x = '1000, &c. n x x2 x4 &c. whence if the 3 4 Index n be taken 1000, &c. as in Neper's Form, the 2 3 4 5 &c. But as n may be taken at Pleasure, the feveral Scales of Logarithms to fuch Indices will be as reciprocally as their Indices. 1000, &c. or Again, if the Logarithm of a decreafing Ratio be fought, the infinite Root of 1 found by the like Method to be 1 4n x = 1−x will be &c. which fubtract from Unity, and the Decre ment of the first of the infinite Number of Proportia |