Multiply the last Consequent by the next Quoti Thus having investigated the Newtonian and our Author's Series, and exemplified the latter, by making the Sines of 30° on and 30° 02', and withal Chewn how, from the Sine of the Arc given, to find the Length of that Arc, and consequently the Circumference of the whole Circle; I shall beg Leave, before I treat of the Construction of Logarithms, to fhew how, from the known Ratio of the Diameter to the Circumference, or any other Ratio whatsoever, that a Set of integral Numbers may be found, whose Ratios shall be the neareft poffible to the Ratio given ; for which I hope to be excused, and the rather, because I believe this Method of determining them was never before published. RUL E. Divide the Consequent by the Antecedent, and the Divisor by the Remainder, and the last Divisor by the last Remainder, and so on till nothing remains. Then for the Terms of the first Ratio, Unity will always be the Antecedent, and the first Quotient the firft Consequent. For the Terms of the second RATIO. Unity; Result be the second Antecedent. Ş Antecedent } Multiply the last {Antecedent { by the 2d Quotient; and to the Product add { Voitung ; } and so will the ent, and to the Product add the laft { confequent} but one ; and so will the Sum be the present { Antecedens . EXAMPLE. H Let it be required to find a Rank of Ratios, whose Terms are integral, and the nearest possible 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: 1250 ) 3927 ( 3 1) II (11 으 1 12 ${ } So the firft Antecedent is 1, and the first Consequent 3. 7+0=7 the second Antec. X75 21+1=22 the second Conf. 112+1=113 the 3d Ant. { Conseq. 22 352 352 +3=355 the3dConfo Which Terms 113 and 355 is Metius's Proportion. Antecedent 113 $ 12431243 + 751250 XII 3905 3905 +22=3927 Producing the same Antecedent and Consequent as at first; which, as it is ever the Property of the Rule so to do, proves, at the same Time, that no Error han been committed thro’the whole Operation. 1:3 ( Whence, as 1250: 3927:: 7:22 Terms 2 of the GO { Ratio, to 355 ; that is, the larger the Terms of the Ratio are, the nearer they approach the Ratio given. Mr. Molyneux, in bis Treatise of Dioptrics, informs us, that when Sir Isaac Newton set about, by Experiments, to derermine the Ratio of the Angle of Incidence, 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 to 9. Now, if it be as 300 is to 193, it will readily appear, by the Rule, whether they are such integral Numbers, whose Ratio is the nearest possible to the given Ratio. 193 ) 300 ( 1 86 ) 107 ( I 1) 2 ( 2 e For, dividing the great Number by the less, and the less by the Remainder, &c. the Operation will Thow that the Numbers 193 and 300 are prime to each other; and that the first Antecedent is i, as also the fiift Consequent. = { ; And{ 1 Whence {:} Ito= the ad Ant. I+1=2 the 2d Con. Again sil 1+1=2 the 3d Ant. Х And 25 1+2=3 the 3d Con. S 8 Again 8+1=9 the4th Ant. And 2 12+2=14the 4th Con. 2 I 2 Hence, the fourth Antecedent and Consequent make the Ratio to be as 9 to 14, or, inverlly, as 14 to 9; which not only agrees with Mr. Molyneux, but at the fame Time discovers, that they are nearer to the given Ratio, 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, according to the Direction of the Rule. Sir Isaac Newton himself determines the Ratio out of Air into Glass to be as 17 to II; but then he speaks of the red Light. For that great Philosopher, in his Dissertations concerning Light and Colours, published in the Philosophical Transactions, has at large demonftrated, as also in his Optics, that the Rays of Light are not all homogeneous, or of the same 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 fo little, that it need seldom to be regarded, and either of those mentioned for the most part is sufficient fo: 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 exsress any given Ratio in smaller Terms, and the neareit polible, with more Ease and Certainty, than any Design in the least of touching upon Optics. Wherefore, left this imall Digression from the Subject in hand, and indeed even from my first Intentions, should tire the Reader's Patience, I shall not presume more, but immediately proceed to the Contruction of Logarithms. of the Construction of Logarithms. THE Nature of which, tho' our Author has fuf fiiently explained in the Description of the Logarithmetic Curve; yet, b fore we attempt their ConItruction, it will be necessary to premise : That the Logari: bm of any Number is the Exponent or Value of the Ratio of Unity to that Number; wherein we consider Ratio, quite different from that laid down in the fifth Definition of the 5th Book of these Elements; for, beginning with the Ratio of Equality, we fay i to i=0; whereas, according to the faid Definition, the Ratio of 1 to 1 5l; and consequently the Ratio here mentioned is of a peculiar Nature, being affi mative 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, fo each Ratio is supposed to be mealured 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 two 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 Raciunculæ proportionable to each Ratio, we shall take the Ratio of Unity to any Number to confift always of the same 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 firft of those Means in each from Unity will be Ratiunculæ ; 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. Consequencly, if the Root of any infinite Power be extracted out of any Number, the Difference of the said Root from Unity fhall be as the Logarithin 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 seek. As, if the Index be supposed 100000, &c. we shall have the Logarithms invented by Neper ; but if the said Index be 230258, &c. those of Mr. Briggs will be produced. Wherefore, if I +xbe any Number whatsoever, and ninfinite, then its Logarithm will be as 1 +* -I= ! 23 x5 + + m, &c. For the infinite 3 4 5 Root of itx without its L'nciæ or prefixed Num bers, n 2 |