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Operation. As the Sine of half the Sum 820 12'C. A. 0.0040361 Is to the Sine of half the Differ. 58° 43' 9.9317679 So is the Tangent of half the Side 25° 22' 9.6758902 To the Tangent of the Diffe

22° 14' 9.6116942 rence of the other two Sides

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Then say again, As the Co-line of half Sum 82° 12' C. A. O 8673703 Is to the Co-fine of half the Differ. 58° 43' 9.7153937 So is the Tangent of half the Side 25° 22' 9.6758902 To the Tangent of half the Sum_61° 08'10.2586542

of the other two Sides

22 14

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Then to the half Sum of the Sides 61 08
Add the half Difference
The Sum is the greater Side Y18322

7 That is in the Ecliptic

2 23 2 2 Substract from There remains

9 06 38 = the Cusp of the first House, viz. in vs 06 38 = the oppo. site Point to which is in Cancer 06 38 = the Cusp of the 7th House, or Descendant.

I 2 00 00

PROBLEM VI.

To find the Cufps of the 2d and 8th Houses.

Practice. In the Oblique Triangle 2 DY, there is given the Angle at Y = 23° 29', the Angle at D = HBC

= 136 521, by Prob. 3, and the Side included Dr 200 44' the Complement of the Oblique Ascension of the ad House, to the whole Circle; to find the Side 2 Y, the Distance of the Cusp of the 2d House from Aries.

1 the Sum of the Angles is

160 21
the Difference thereof

113 23
Then { the half Sum
the half Difference

56 42
the half of the Side

80 10

10 22

Now say; As the Sine of half Sum 80° 10' Co. Ar. 0.0064277 Is to the Sine of half Difference 56° 42'9.9221062 So is the Tangent of half the Side 10° 22' 9.2622921 To Tang. of Dif. of other 2 Sides 89 49'9.1908260

8 49

Then say again ;
As Cofine of half the Sum 80° 10' Co. Ar. 0.7675550
Is to the Co-sine of half the Differ. 56° 42' 9.7395904
So is the Tangent of half the Side 10° 22'9.2622921
To Tang. of Sum of other 2 Sides 302 27' 9.7694375

1
Then to the half Sum of the Sides 30 27
Add the half Difference
The Sum the gr. Side fought 2 Y = 39 16

S
That in the Ecliptic is
which take from

12 00 00 there remains

10 20 44 = the Cusp of the 2d House in Aquarius m 20 44 ; whose opposite Point in the Ecliptic is a 20 44 = the Cusp of the 8th House, as required.

PRO

I 09 16

PROBLEM VII.

To find the Cusps of the 3d and 9th Houses.

PraEtice. In the Oblique Triangle Y E 3, there is given the Angle Y= 23° 29'; the Angle at E=/ AC = 121° 38' (by Prob. 2 ;) and the included Side TE =9° 16' to find the Side r 3 the Distance of the Cusp of the 3d House from Aries.

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Then

the half Sum of the given Angles is 72 33 the half Difference thereof is

49 05 the half of the given Side

4 38

Now fay; As the Sine of half Sum 72° 33' Co. Ar. 0.0204612 Is to the Sine of half Difference 49°05' 9.8783281 So is the Tangent of half the Side 4° 38' 8.9087190 To Tang. of Dif. of other 2 Sides 3° 40' 8.8075183

Then fay again ; As Co-sine of the half Sum 72° 331 C. Ar. 0.5230610 To the Co-sine of half Difference 49°05' 9.8162152 So is the Tangent of half the Side 4° 38' 8.9087190 To Tang. of Sum of other 2 Sides 10° 02'9.2479952

IO 02

Then to the half Sum of the two Sides add their half Difference

3 40 The Sum is the greater Side fought Y3 = 13 42 That is in the Sine r 13° 42' = the Cusp of the 3d House ; whose opposite Point 4 13° 42' = the Cusp of the gth House ; as required.

PRO

ference of the other two Sides 48° 54' 10.0594918

PROBLEM. VIII.
To find the Cusps of the 11th and 5th Houses.

PraEtice. In the Oblique Triangle 11 AY, there is given the Angle at Y = 239 29' ; the Angle A 121° 38'; and the included Side AY = 110° 44', the Supplement of the Oblique Ascension of the 11th House to a whole Circle ; to find the Side 11 Y, in the Ecliptic.

the half Sum of the given Angles 72 33 Then their half Difference

49 05 the Half of their given Side Now fay; As the Sine of half the Sum 72° 33'C. A. 0.0204612 Is to the Sine of half the Difference 49°05'9.8783281 So is the Tangent of half the Side 55° 22' 10.1607025 To the Tangent of half the Dif.

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55 22

;

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Then fay again ; As the Co-line of the Sum 72° 33'C. A..0.5230610 Is to the Co-sine of half the Differ. 49°05' 9.8162152 So the Tangent of half the Side 55° 22' 10.1607025 To the Tangent of half the Sum? of the other two Sides

}72° 27' 10.4999787

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Now to the Half Sum of the Sides

72 27 Add the Half Difference

48 54 The Sum is the greater Side 11. Y = 121 21

S This in the Ecliptic is

4 OT 21 Which take from

12 00 00 There remains

728 39 VOL. II.

Do

Which

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Which is in the Sign Scorpio M 28° 39'= Cusp of the rith House, and its opposite Point 8 280 39', is the Cusp of the 5th House, as required.

PROBLEM IX.

To find the Cufps of the 12th and 6th Houses.

Practice. In the Oblique Triangle 12 BY, there is given the Angle at Y = 23° 29'; the Angle at B = 136° 52'; and the included Side B Y = 80 44'; to find the Side 121, in the Ecliptic.

Then

the half Sum of the Angles given 80 10 the half Difference thereof 56 42 the half of the given Side

40 22

Now fay;

9.9294516

As the Sine of half the Sum 80° 10' C. A. O 0064277
To the Sine of half the Difference 56° 43' 9.9221062
So is the Tangent of half the Side 40° 221
To the Tangent of half the Dif-
ference of the other two Sides

35° 47' 9.8579855
Then fay again ;
As the Co-line of the Sum 80° 10'C. A. 0.7675550
Is to the Co-fine of the Difference 56° 42'9.7395904
So is the Tangent of half the Side 400 22'9.9294516

of half of the other two Sides

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Then to the half Sum of the two Sides 69 49 Add their half Difference

35 47 The Sum is the greater Side

105 36

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