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The Sine of 30° or is ,50025189543
The Square of % inverted 16480000000




Whence the Product is ,000000042326 true to eleven Places at least. Wherefore, if according to the Rule, from double the Sine of the middle Arc=1,00050379086 we subtract the said Product,

,00000004232 And from the Remainder

1,00050374854 the Sine of 30° oo the given Extream ,50000000000 be fubftracted


There will remain ,50050374854 for the Sine of 30° 02' the other Extream ; than which, nothing of this Nature can be desired more easy.


Because the Difference of the Differences of the Sines,
or fecond Difference, has always 7 Cyphers before the
significant Figures ; it follows, that the whole Canon
where the Sines consist but of 6 Places, which is as far
as our Tables for common Practice need extend, may
be perform'd chiefly by Addition and Subtraction only,
without forming Multiples of the Square of z by the
nine Digits; tho' perhaps it may be necessary to use the
Method of contracted Multiplication every sth Minute
to confirm the Truth, left in continual doubling and
subtracting an Error should arise in the right-hand Fi-
gure : however, as it may be safely used for 5 Minutes
together, and sometimes more, it will render the
whole very easy.
Note, the Square of z in this Case, vim, the Arch
of 5 Minutes is

[Ve chan


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Multiply the last Antecedent by the second Quotient,

Thus having investigated the Newtonian, and our Author's Series, and exemplified the latter, by making the Sines of 30° or' and 30° oz', and withal shewn how from the Sine of an 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 nearest possible to the Ratio given ; for which I hope to be excus'd, and the rather, because I believe this Method of determining them, was never before publish'd.


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 first Consequent.

For the Terms of the second RAT10.

{} and to the Product add { Voihing}and so will the Rě

fult, be the second Confequent


For all the following Ratlos.

5 Antecedent Multiply the last { Consequent } by the next Quotient

, and to the Product add the laft { concedents} but one; and fo will the Sum be the present Antecedent





EXAMPLE Let it be required to find a Rank of Ratios, whose Terms are integral; and the nearest pofsible to the following Ratio, viz. of 10000 to 31416, which expresles 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 reduc'd to their least Terms.

1250 Whence

and then 3927 divided by 394183927 1250 and 1250 by the Remainder, &c. will be as follows. 5250) 3927 (3 177 ) 1250 (7

11 ) 197 ( 16

1) ulir So the first Antecedent is 1, and the first Consequent 3.

Anteced. I 737tos7 the ad Antecd. Confeq. 33



221 ti=22 the ad Consą. Which 7 and 22 is - Archimedes's Proportion. ${ Conseg. 22 }

112 tisužthe zd Anteci. And

2352 +3=355 the 3d Consen: Which Terms 113 and 355 is Metius's Proportion. .

S 1243 S 1243-77 =1250 XI

239052 3905 +22=3927 Producing the faine 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.


For the Whence'as 1250: 3927

Terms But it must be observd that I to 3 does not express the Ratio fo near as 7 to zz, nor 7 to 22 fo near as 113


Anteced. 7{x16={352

${ Confeq. 3553




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 Dioptricks informs 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 ; found it to be from Air to Glafs, as 300 to 193, or in the least round Numbers, as :1 4 to g. Now if it be as 300 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 (1 107 ) 193 (


107 ( 1
zi ) 88 ( 4
2) 21 ( 10

) 2 (2


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For dividing the greater Number by the less, and the less by the Remainder, &c. the Operation will thew that the Numbers 193 and 300 are Prime to each other;

and that the first Antecedent is 1, as also the firft Consolune quelin

I the second Antecedent


+ 2 the second Consequent. Again {2}x={

ti 2 the third Antecedent

= 3 the third Consequente Again { }}x={i2

8- the 4th Antecedent And

212 +2=14 the 4th Consequent.




And {

2 ti


Hence, the fourth Antecedent and Consequent, make the Ratio to be as 9 to 14, or inversly as 14 to 9, which not only agrees with Mr. Molyneux, but at the same 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 11; but then he speaks of the Red Light. For that great Philosopher in his


Differtations concerning Light and Colours, publish'd in the Philosophical Transactions, has at large demonstrated, as also in his Opticks, that the Rays of Light are not all Homogeneous, or of the same 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 Pro. portion setled 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 so little that it need seldom to be regarded, and either of those mention'd for the most part is sufficient 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 smaller Terms, and the nearest possible, with more Ease and Certainty, than ány Design in the least of touching upon Opticks.

Wherefore, left this small Digression from the Subject in hand, and indeed even from my first Intentions, should tire the Reader's Patience, I Thall not presume more, but immediately proceed to the Construction of Logarithms.

of the Construction of Logarithms. THERE

*HE Nature of which, tho' our Author has suffici

ently explain’d in the Description of the Logarithmical Curve; yet before we attempt their Construction, it will be necessary to premise:

That the Logarithm of any Number, is the Expo-' nent, 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 say i to iso, whereas according to the faid Definition, the Ratio of 1 to 13i, and confequently the Ratio her— mention'd, is of a peculiar Nature, being affirmative when increasing, as of Unity to a greater Number; but negative when decreasing. And


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