Differtations concerning Light and Colours, publish'd in the Philofophical Tranfactions, has at large demonftrated, as alfo in his Opticks, that the Rays of Light are not all Homogeneous, or of the fame fort, but of different Forms and Figures, so that fome are more refracted than others, tho' they have the fame or equal Inclinations ori the Glass: Whence there can be no constant Proportion fetled between the Sines of the Incidence, and of the refracted Angles. But the Proportion that comes nearest Truth, for the middle and strong Rays of Light, it seems 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 mention'd for the most 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 Opticks. 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 presume more, but immediately proceed to the Conftruction of Logarithms. Of the Conftruction of Logarithms. THE HE Nature of which, tho' our Author has fufficiently explain'd in the Description of the Loga rithmical Curve; yet before we attempt their Conftruction, it will be necessary 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 10, whereas according to the faid Definition, the Ratio of 1 to 1, and confequently the Ratio here mention'd, is of a peculiar Nature, being affirmative when increafing, as of Unity to a greater Number; but negative when decreafing. 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 fuppos'd to be meafur'd by the Number of equal Ratiunculæ contain'd between the two Terms thereof; whence if in a continued Scale of mean Proportionals, infinite in Number, there be affum'd an infinite Number of fuch Ratiunculæ, between any two Terms in the faid Scale; then that infinite Number of Ratiunculæ, is to another infinite Number of the like and equal Ratiunculae 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 compos'd of a Number of equal Ratiunculæ proportionable to each Ratio; we fhall take the Ratio of Unity to any Number to confift always of the fame infinite Number of Ratiuncula, their Magnitudes in this Cafe will be as their Number in the former. Wherefore if between Unity and any two Numbers propos'd 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 Ratiuncula, 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 Logarithms 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 produc'd 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 fuppos'd 100000, &c. we shall have the Logarithms invented by Neper; but if the faid Index be 230258, &c. thofe of Mr. Briggs's will be produc❜d. Wherefore if 1+x be any Number whatsoever and infinite, then its Logarithm will be as Tx is 1+x+xxtxxxxxxx, &c. and the celebrated binomial Theorem invented by Sir Isaac Newton for 1 ; n = 1 x n − 2 x n = 3, &c. or 2 n for being an infinitefimal is rejected; whence the infi nite Root of 1+x=1+*=1+*—** + x3 3n 4n c. and the Excels thereof above Unity, viz. 3 n 4n n 2n 5n + &c. is the Augment of the firft of the mean Proportionals between Unity and 1x, which therefore will be as the Logarithm of the Ratio of 1 to 1x, or a 1 the Logarithm of 1+x. But as 1-1 is a Ra tiuncula it must be multiplied by 10000, &c. infinitely, which will reduce it to Terms fit for Practice, making the Logarithm of the Ratio of 1 to 1+x 1000, &c. But as n may be taken at Pleasure the feveral Scales of 1000, &c. Logarithms to fuch Indices will be as or recipro cally as their Indices. n Again, if the Logarithm of a decreafing Ratio be 1 n fought, the infinite Root of 1-x-x will be found by the like Method to be NXT I *, &c. which substract from Unity, and the 4n Decrement of the firft of the infinite Number of Pro portionals will appear to be xxix2+x+1 *, &c. which expreffes the Logarithm of the Ratio of 1 to 1x or the Logarithm of 1-x according to Neper's Form, if the Index n be put 10000, &c. as before. And to find the Logarithm of the Ratio of any two Terms, a the least and b the greater, it will be as a:b:: b. -a ; or the Difference divided b a a by the leffer Term when 'tis an increasing Ratio, and b-a b when 'tis decreafing. Wherefore putting dDifference between the two Terms a and b, the Logarithms of the fame Ratio may be doubly expreft, and accordingly is either d d2 d: d: &c. d: 46.3 I d d d' 1 n X 262 + 3614 &c. both producing the fame Thing. But if the Ratio of a to b be fuppos'd to be divided into two Parts, viz. into the Ratio of a to the arithmetical Mean between the two Terms, and the Ratio of the said arithmetical Mean to the other. Term b, then will the Sum of the Logarithms of those two Ratios be the Logarithm of the Ratio of a to b. Wherefore fubftitutings for a+b and it will bes. fubftituting for x, and we fhall have for both Ratios + +, &c. is the Logarithm of the 35 5 55 757 Ratio of a to b, whofe Difference is d, and Sum s; which Series without the Index n, is by the bye, the Fluent Fluent of the Fluxion of the Logarithm of std s-d' , af fuming d the flowing Quantity, for the Fluxion of the st-d 2 så d à d'd d2ď ď ď ď ď is Log. of 2X + ++ d +, &c. whofe Fluent 2 x-+ 3 s 555 7 $7 and the fame as above bating the Index n. This Series either way obtained converges twice as fwift as the former, and confequently. is more proper for the Practice of making Logarithms; thus, put a and b any Number at Pleasure, then d b-I which affume > e, and then b= -e, therefore we have for THEOREM I. The Log. of b (==) = — × 2×6+36+ 3 es + e2, &c. n To illuftrate this Theorem. Let it be required to find the Logarithm of 2 true to 7 Places. Note, That the Index must be affum'd of a Figure or two more than the intended Logarithm is to have. |