Sidebilder
PDF
ePub

Thus having investigated the Newtonian and our Author's Series, and exemplified the latter, by making the Sines of 30° 01' and 30° 02', and withal fhewn how, from the Sine of the Arc given, to find the Length of that Arc, and confequently the Circumference of the whole Circle; I fhall beg Leave, before I treat of the Conftruction 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, whofe Ratios fhall be the nearest 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 pub

lifhed.

RULE.

Divide the Confequent by the Antecedent, and the Divifor by the Remainder, and the last Divifor by the laft Remainder, and fo on till nothing remains.

Then for the Terms of the first Ratio, Unity will always be the Antecedent, and the firft Quotient the first Confequent.

For the TERMS of the fecond RATIO.

Multiply the laft Antecedent by the 2d Quotient; { Confequent

and to the Product add {Nothing

Unity;

Antecedent.

Refult be the fecond {Confequent.

and fo will the

For all the following RATIOS:

Multiply the laft {Antecedent } by the next Quotient, and to the Product add the laft {Corecedent } but

Antecedent.

one ; and so will the Sum be the present { Consequent.

EXAMPLE.

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 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 muft therefore be reduced to their leaft Terms.

Whence

10000

1250, and then 3927 divided 31416 3927

by 1250, and 1250 by the Remainder, &c. will be as follows:

[blocks in formation]

So the firft Antecedent is 1, and the firft Confequent 3.

[merged small][merged small][merged small][ocr errors][merged small][subsumed][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small]

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.

[blocks in formation]

But it must be observed, that I to 3 does not exprefs the Ratio fo near as 7 to 22; nor 7 to 22 so near as 113 B b 2

to

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 respective 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 ( 1
107) 193 ( 1
86) 107

121) 86 ( 4

2 ) 21 ( 10 I) (

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 alfo the firft Confequent.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

and Confequent make

Hence, the fourth Antecedent 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 nearest of all to the given Ratio, as will appear by repeating the Procefs, according to the Direction of the Rule.

Sir Ifaac Newton himfelf determines the Ratio out of Air into Glafs to be as 17 to 11; but then he speaks of the red Light. For that great Philofopher, in his

Differ

Differtations concerning Light and Colours, published in the Philofophical Transactions, 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 fettled 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 seldom to be regarded, and either of thofe 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 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.

TH

HE Nature of which, tho' our Author has fufficiently explained in the Defcription of the Logarithmetic Curve; yet, before we attempt their Conitruction, it will be neceffary to premife:

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 11; and confequently the Ratio here mentioned is of a peculiar Nature, being affi mative 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 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 fuch Ratiunculæ, 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 Ratiunculæ proportionable to each Ratio, we shall 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 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 refpective 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 Dif ference 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 aflume 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

[ocr errors]

n infinite, then its Logarithm will be as 1+x-1=

[ocr errors][ocr errors]

x3 +

+

n

2

25

3 4 5

,&c. For the infinite

Root of 1+ without its Uncia or prefixed Num

bers,

« ForrigeFortsett »