I

SECT. 2.

OF THE ARCHES.,

11

A

s. z bc c, the s. Z ccd = s. end, &c, we shall have the fol-lowing proportions; viz,

s. Z ABC s. Z BCD

b/; : cc : : --;— :-----j

s. z abw x s. Z /;bc s. Z bcc x s. Z ccd

■ S. Z ABC S. Z CDE

' S. Z AB/i X S. Z 6bC ' S. Z CDf/ X S. Z £/de’

S. Z ABC

S. Z DBF

%b : ec : :-r-— .-,

S. Z ABO X S. Z OBC S. Z DEC X S. Z CEF

and so on.

Otherwise.

Since c p or b n : B m or nb : : s. z Bbn ,

or s. Z ab6 : s. Z ^bc or s. Z bcc :: • -:-r;

s. z bcc s. Z abo

and cp or qc : c^ or Dr : : s. Z ccq or s. Z cd d : s. Z ccq or1 1

s. z bcc :

S. Z BCC ’ S. Z CD<f’1

it is clear that cp is as

; that is, the forces wib, pc,

s. z BCC

rD, &c. are always reciprocally as the sines of the angles

which they make with the vertical line.

. , . cp x s. Z epe cp x s. Z BCD

And since cc = —— -— —-;

s. Z cap s. Z c cd

. . s. Z BCD

therefore any force or weight cc is as-—-v

J ° s. z ccb x s. Z cca

And this is the same as the property in corol. 4 to the 3d

proposition following.

Corol. If dc be produced to k; then, the sine of the anglehes being equal to the sine of its supplement bcd, thesame weight or force cc will be always proportional to

-; which three angles together make up

s. z bcc x s. z dcc ° ° 1

two right angles.

Properties similar to the foregoing are otherwise deter-mined in the following propositions.