E X AMP LE. Let it be required to find a Rank of Ratios, whose Terms are integral, and the nearest poffible to the following Ratio, viz. of 10000 to 31416, which expresses 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 least Terms. 10000 Whence 1250 and then 3927 divided 31416 - 3927' by 1250, and 1250 by the Remainder, &c. will be as follows: ${ So the first Antecedent is 1, and the firft Consequent 3. 7 E 21+1=22 the second Conf, Which 7 and 22 is Archimedes's Proportion. S Anteced. 71 112 112+1=113 the 3dAat. 352+3=355 the3dConf. Which Terms 113 and 355 is Metius's Proportion. Antecedent 113 2 1243 $ 1243 + 731250 X1!= 3905 3905+23=3927 Producing the same Antecedent and Consequent as at first; which, as it is ever the Property of the Rule fo to do, proves, at the same Time, that no Error has been committed thro' the whole Operation. 1:3 Forthe S 1 Whence, as 1250: 3927:: 7:22 Terms 2 113:355 ) of the But it must be observed, that I to 3 does not express the Ratio lo near as 7 to 22 ; nor 7 to 22 fo near as 113 Въ2 3 { confequent 35 } = { Ratio. to to 355 ; that is, the larger the Terms of the Ratio are, the nearer they approach the Ratio given. Mr. Molyneux, in his Treatise of Dioptrics, informe us, that when Sir Isaac Newton set about, by Experiments, to determine the Ratio of the Angle of İncidence, to the refracted Angle, by the means of their respective Sines; he found it to be, from Air to Glass, as 300 to 193, or in the least round Numbers,as 14 tog. Now, if it be as 300 is to 193, it will readily appear, by the Rule, whether they are luch integral Numbers, whose Ratio is the nearest poflible to the given Ratio. 193 ) 300 (1 86 ) 107( ! 1) 2 ( 2 2 And 12+2=14the 4thCon. 12 For, dividing the great Number by the less, and the lefs by the Remainder, &c, the Operation will: fhow that the Numbers 193 and 300 are prime to each other; and that the first Antecedent is i, as also the first Consequent. Whence Ito= the ad Ant. XI { And itis2 the ad Con. Again * { 1+1=2 the 3d Ant. And 1+2=3 the 3d Con. 2 S 8 Again 8+1=9 the4th Ant. XL = 235 =, Hence, the fourth, Antecedent and Consequent make the Ratio to be as 9 to 14, or, inverfly, as 14 to 95which not only agrees with Mr. Molyneux, but at the fame Time discovers, that they are nearer to the given Racio, than any other integral Numbers less than 92 and 143; which are the nearest of all to the given Ratio, as will appear by repeating the Process, ac cording to the Direction of the Rule, Sir Isaac Newton himself determines the Ratio out of Air into Glass to be as 17 to it; but then he speaks of the red Light. For that great Philosopher, in his Dissertations concerning Light and Colours, published in the Philosophical Tranfałtions, has at large demonstrated, as also in his Optics, that the Rays of Light are not all homogeneous, or of the fame Sort, but of different Forms and Figures, so that some are more refracted than others, tho' they have the same or equal Inclinations on the Glass : Whence there can be no constant 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 seems, is nearly as 300 to 193, or 14. to g. In Light of other Colours the Sines have other Proportions. But the Difference is so little, that it need seldom to be regarded, and either of those mentioned for the most part is fufficient for Practice. However, I must observe, that the Notice here taken either of the one or the other, is more to illustrate the Rule, and shew, as Occasion requires, how to express any given Ratio in fmaller Terms, and the nearest poffible, with more Eafe and Certainty, than any Design in the least of touching upon Optics. Wherefore, left this small Digreflion from the Subject in hand, and indeed even from my first Intentions, Thould tire the Reader's Patience, I shall not presume more, but immediately proceed to the Construction of Logarithms. of the Construction of Logarithms. THE Nature of which, tho' our Author has suf ficiently explained in the Description of the Logarithmetic Curve; yet, before we attempt their ConItruction, 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 Jaid down in the fifth Definition of the 5th Book of thefe Elements; for, beginning with the Ratio of Equality, we say I to i=0; whereas, according to the said Definition, the Ratio of 1 to 1=1; and consequently the Ratio bere mentioned is of a peculiar Nature, 1.eing affirmative when increasing, as of Unity to a greater Number; but negative when decreasing. And Bb 3 As as the Value of the Ratio of Unity to any Number is the Logarithm of the Ratio of Unity to that Number, so each Ratio is supposed to be measured 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 such Ratiunculæ, between any iwo Terms in the same Scale; then that infinite Number of Ratiunculæ 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, instead of supposing the Logarithms composed of a Number of equal Ratiunculæ proportionable 1o each Ratio, we shall take the Ratio of Unity to any Number to consist always of the fame infinite Number of Ratiunculæ, their Magnitudes in this case will be as their Number in the former. Wherefore,'if between Unity, and any two Numbers proposed, there be taken any Infinity of mean Proportionals, the infinitely little Augments or Decrements of the first of those Means in each from Unity will be Ratiunculæ ; that is, they will be the Fluxions of the Ratio of Unity to the said 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 thereot. Consequently, if the Root of any infinite Power be extracted out of any Number, the Difference of the said Root from Unity shall be as the Logarithm of that Number. So that Logarithms, thus produced, may be of as many Forms as we please to assume infinite Indices of the Power whose Root we feek. As, if the Index be supposed 100000, &c. we Iall have the Logarithms invented by Neper ; but if the faid Index be 230258, &c. those of Mr. Briggs will be produced. Wherefore, if I +* be any Number whatsoever, and n infinite, then its Logarithm will be as +* I xo x+ *5 + & C For the infinite 2 3 4 5 Root of 1 + x with out its L'iciæ or prefixed Num bers, n bers, is 1 +*+*x+xxx+xXxx, &c. and the celebrated binomial Theorem invented by Sir Isaac Newton for x X Х n n 2n' un 2 x3 n 2n x n 2 in thisCafe rather 1 x x 5 being an Infinitesimal, is rejected; whence the infinite in x2 x3 x4 Root of itx"=+ -t & c. and the 4n Excess thereof above Unity, viz. - + 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 I +* - 1 is a Ratiuncula, it must be multiplied by 10000, &c. infinitely, which will reduce it to Termis fit for Practice, makeing the Logarithm of the Ratio of 1 to 1+x= 1000, & C. Xತಿ ** &c. whence if the 3 4 Index n be taken 1000, &c. as in Neper's Form, the х2 74 Logarithms will be fimply * + 3 4 5 &c. But as n may be taken at Pleasure, the several Scales of Logarithms to such Indices will be as reciprocally as their Indices. Again, if the Logarithm of a decreafing Ratio be fought, the infinite Root of I XI- * will be x2 73 -found by the like Method to be I 32 &c. which subtract from Unity, and the Decre4n ment of the first of the infinite Number of Proportia x3 nals will appear to be Х c. 3 B b 4 which + 2 |