Sines of the Distances from the Arc of 30 Degrees, be multiplied by 3, the Differences of the Sines will be had.

So likewife may the Sines of the Minutes in the Beginning of the Quadrant be found, by having the Šines and Cofines of one and two Minutes given. For as the Radius is to double the Cofine of 2':; Sine 1: Difference of the Sines of 1' and 3':: Sine 2': Difference of the Sines of o' and 4', that is, to the Sine of 4. And fo the Sines of the four first Minutes being given, we may thereby find the Sines of the others to 8', and from thence to 16, and fo on.

её

PROPOSITION VII.

THEOREM.

In Small Arcs, the Sines and Tangents of the fame
Arcs are nearly to one another, in a Ratio of
Equality.

FOR

OR because the Triangles CED, CBG, are
equiangular, CE: CB :: ED :: BG.

But as the Point E approaches B, EB will vanish in Respect of the Arc BD: Whence CE will become nearly equal to CB. And fo ED will be alfo nearly

equal to BG. If EB be lefs than the

I

10,000,000

Part of the Radius, then the Difference between the Sine

and the Tangent will be alfo lefs than the

Part of the Tangent.

I

10,000,000

Coroll. Since any Are is lefs than the Tangent, and greater than its Sine, and the Sine and Tangent of a very fmall Arc, are nearly equal; it follows that the Arc fhall be nearly equal to its Sine; and fo in very small Arcs it fhall be, as Arc is to Arc, fo is Sine to Sine.

PROPOSITION VIII.

To find the Sine of the Arc of one Minute.

HE Side of a Hexagon infcrib'd in a Circle, that

Tis, the Subtenfe of 60 Degrees, is equal to the

Radius, (by 15th of the 4th,) and fo the half of the Radius fhall be the Sine of the Arc 30 Degrees. Wherefore the Sine of the Arc of 30 Degrees being given, the Sine of the Arc of 15 Degrees may be found, (by Prop. 3.) Alfo the Sine of the Arc of 15 Degrees being given, (by the fame Prop.) we may have the Sine of 7 Degrees 30 Minutes: So likewife can we find the Sine of the half of this, viz. 3 Degrees 45'; and fo on, until twelve Bifections being made, we come to an Arc of 522, 443, 034, 455, whofe Cofine is nearly equal to the Radius, in which Cafe (as is manifeft from Prop. 7.) Arcs are proportional to their Sines And fo as the Arc of 522, 44', 34, 45, is to an Arc of one Minute, fo fhall the Sine before found, be to the Sine of an Arc of one Minute, which therefore will be given. And when the Sine of one Minute is found, then (by Prop. 2. and 4.) the Sine and Cofine of two Minutes will be had.

PROPOSITION IX.

THEORE M.

If the Angle BAC, being in the Periphery of a Circle, be bifected by the Right Line AD, and if AC be produced until DE AD meets it in E: then fhall CE=AB.

=

N the quadrilateral Figure ABDC (by 22. 3.) the Angles B and ACD are equal to two Right Angles DCE+DCA (by 13. 1.) Whence the Angle B DCE. But likewife the Angle E= DAC (by 5. 1.) DAB and DC DB. Wherefore the Triangles BAD and CED are congruous, and CE is equal to AB. W. W.D.

PRO

PROPOSITION X.

THEOREM.

Let the Arcs AB, BC, CD, DE, EF, &c. be equal; and let the Subtenfes of the Arcs AB, AC, AD, AE, &c. be drawn; then will AB AC:: AC:ABAD:: AD: ACLAE ::AE:AD+AF:: AF: AE+AG.

L

ET AD be produced to H, AE to I, AF to K, and AG to L, that the Triangles ACH, ADI, AEK, AFL, be Ifofceles ones; then because the Angle BAD is bifected, we fhall have DHAB, (by the laft Prop.) fo likewife fhall EIAC, FK= AD, alfo GLAE.

But the Ifofceles Triangles ABC, CHA, DAI, EAK, FAL, because of the equal Angles at the Bafes, are equiangular. Wherefore it fhall be as AB: AC: AC: AH=AB+AD::AD: AI=AC+ AE:: AE AKAD+AF :: AF: AL=AE +AG. W. W. D.

Coroll. 1. Because AB is to AC, as Radius is to double the Cofine of the Arc AB, it fhall also be (by Coroll. Prop. 4.) as Radius is to double the Cofine of the Arc AB, fo is AB: AC:: AC::

AB+AD :: ¦ AD: ¦ AC+AE::AE +AD+AF, &c. Now let each of the Arcs AB, BC, CD, &c. be z'; then will AB be the Sine of one Minute, AC the Sine of 2' Minutes,

AD the Sine of 3' Minutes; AE the Sine of 4', &c. Whence if the Sines of one and two Minutes be given, we may eafily find all the other Sines in the following Manner.

Let the Cofine of the Arc of one Minute, that is, the Sine of the Arc of 89 Deg. 59', be called Q, and make the following Analogies; R: 2 Q:: Sin. 2' :S. 1'+S. 3'. Wherefore the Sine of 3 Minutes will be given. Alfo R: 2 Q:: S. 3′ : S. 2' + S. 4. Wherefore the S. 4' is given; and R. 2 S. 4: S. 3' + S. 5'; and fo the Sine of 5'

will be had.

A3

Likewife R 2 Q:: S. 5': S. 4'+ S. 6'; and
fo we fhall have the Sine of 6'. And in like Man-
ner, the Sines of every Minute of the Quadrant
will be given. And because the Radius, or the
firft Term of the Analogy is Unity, the Operations
will be with great Eafe and Expedition calculated
by Multiplication, and contracted by Addition.
When the Sines are found to 60 Degrees, all the
other Sines may be had by Addition only, (by
Cor 1. Prop. 6.) 1

The Sines being given, the Tangents and Secants
may be found from the following Analogies, (in the
Figure for the Definitions ;) because the Triangles
CED, CBG, CHI, are equiangular, we have

CEED: CB: BG; that is, Cof. : S :: R:T
GB: BC:: CH: HI; that is, T: R:: R: Cot.
CE: CD:: CB: CG; that is, Cof.: R:: R: Secant
DE: CD:: CH: CI; that is, S:R::R: Cofec

SCHOLI U M.

That great Geometrician and incomparable Philofopher, Sir Ifaac Newton, was the firft that laid down a Series converging, in infinitum; from which, having the Arcs given, their Sines may be found. Thus if an Arc be called A, and the Radius be an Unit, the Sine thereof will be found to be

As

A- --|-:
1.2.3 1.2.3.4.5
And the Cofine,

A7

1.2.3.4.5.6.7

+

A? 1.2.3.4.5.6.7.8.9

A6

A8

+

A2 A+ I -+. 1.2.3 1.2.3.4 1.2.3.4.5.6 1.2.3.4.5.6.7.8 Thefe Series in the Beginning of the Quadrant when the Arc A is but fmall, foon converge. For in the Se ries for the Sine, if A does not exceed 10 Minutes, the two firft Terms thereof, viz. AA gives the Sine to 15 Places of Figures. If the Arc A be not greater than one Degree, the three firft Terms will exhibit the Sine to 15 Places of Figures; and fo the faid Series are very useful for finding the first and laft Sines of the Quadrant. But the greater the Arc A is, the more

are

are the Terms of the Series required to have the Sine in Numbers true to a given Place of Figures. And then when the Arc is nearly equal to the Radius, the Series converges very flow. And therefore, to remedy this I have devifed other Series, fimilar to the Newtonian one, wherein, I fuppofe, the Arc, whofe Sine is fought, is the Sum and Difference of two Arcs, viz. Az, or A-z: And let the Sine of the Arc A, be called a, and the Cofine b. Then the Sine of the Arc A+z will be expressed thus:

bz az2 bz

az+

--+- +

I. 1.2 1.2.3 1.2.3.4

And the Cofine is,

bz2 az3

I

1.2

&c.

1.2.3 1.2.3.4

bz5

&c.

1.2.3.4.5

bz4

a z4

bz5

aző 1.2.3.4.5.6

In like Manner the Sine of the Arc A-z is

And the Cofine is,

1.2.3

1.2 1.2.3 1.2.3.4 1.2.3.4.5

The Arc A is an Arithmetical Mean between the Arc A-z and A+z. And the Difference of the Sines are,

1.2 1.2.3.4 1.2.3.4.5.6 Which Series is equal to double the Sine of the Mean Arc, drawn into the verfed Sine of the Arc z, and converges very foon. So that if z be the first Minute of the Quadrant, the firft Term of the Series gives the fecond Difference to 15 Places of Figures, and the fecond Term to 25 Places.

From

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