Geometrically.

B

This Cafe is conftructed after the fame manner with the former, and the Hypothenufe AC is found by taking it's length in your Compaffes, and applying that to the fame line of equal

parts you took AB from.

By Calculation.

ift, By making AC the Radius we fhall have the following proportion for finding A C, viz.

S, C:R::AB: AC

2dly, Making A B the Radius we have this proportion, viz.

R: Sec. A::AB: AC

This may be done without the help of the Secants; for fince (by Art. 76. Sect. 1.) R: Sec. :: Co-S.: R; therefore the former proportion will be

come

Co-S. A: R::AB:AC

i.e. As the Co-Sine of A 34°, 20°

is to the Radius

9.91686

3dly, Making BC the Radius, we have the fol

lowing proportion, viz.

T, C: Sec. C::AB: AC

This likewife may be done without the help of Secants, for fince (by Art. 76. Sect. 1.) T, : Sec. :: S, R; therefore the former Analogy will be reduc'd to this, viz.

S, CR::AB: AC

where no Secants do appear, and it coincides with that in the first fuppofition of this Cafe, fo we shall not repeat the Operation.

ĆA SE - 3.

The Angles and Hypothenufe given, to find either of the Legs.

Example. In the Triangle ABC, fuppofe the Hypothenufe AC 146 equal parts, and the Angle A 36°, 25', confequently the Angle C 53°, 35, requir'd the Leg AB.

Geometrically.

Draw the Line A B at pleasure, and make the Angle BAC equal to 36°, 25' (by Prob. 9. Sect. 1.) then take AC equal to 146 from any Line of equal

parts; laftly from the point C let fall the perpendicular CB on the line A B. So the Triangle is constructed, and AB may be measured from the line of equal parts.

·B

By Calculation.

1ft, Making AC the Radius we shall have the following proportion, viz.

R:S, C:: AC: AB

2dly, Making A B the Radius, we have the fol

lowing Analogy, viz.

Sec. A: R::AC: AB

This may be done without the help of Secants, for fince (by Art. 76. Sect. 1.) Sec. : R:: R: Co-S; therefore the former proportion may be reduc'd to this, viz.

1

R: Co-S, A :: AC: AB

which is the fame with the proportion in the first fuppofition.

3dly, By fuppofing BC the Radius, we have the following proportion, viz.

Sec. C: T. C:: AC: AB

The two Legs being given, to find the Angles. Example. In the Triangle ABC, fuppofe AB 94 and BC 56, requir'd the Angles A and C.

Geometrically.

Draw AB equal to 94, from any line of equal parts, then from the point B raife BC perpendicular to AB (by Prob. 4. Sect. 1.) and take BC, from the former line of equal parts equal to 56; laftly, join the points A and C with the ftreight line A C, fo the Triangle is conftructed, and the Angles may be measur❜d by

rob. 10. Sect. 1.

A

By Calculation.

B

ift, Suppofing AB the Radius we have this Analogy, viz.

2dly, Making BC the Radius we have this pro

portion, viz.

BC: BA::R:T.C

56

94

1.74819!

1.97313

i. e. as BC

is to AB

fo is the Radius

to the Tangent of C 59°, 13'

IQ.22494

CASE 5.

The Hypothenufe, and one of the Legs given, to find the Angles.

Example. In the Triangle DEF, fuppofe the Leg DE 83, and the Hypothenufe DF 126, re quir'd the Angles D and F.

Geometrically.

Draw the line DE 83, from any line of equal parts, and from the point E raise the perpendicular

F

EF, then take the length of

DF 126, from the fame line of equal parts, and fetting one foot of your Compaffes in D with the other cross the perpendicular EF in F; laftly, E join D and F, fo the Triangle is conftructed, and the Angles may be measured by Prob, 10, Sect. I.

By Calculation.

1, Making DF the Radius, we have this proportion, viz.

DF: DE::R: S, F