Full text: Volumen secundum. Opera geometrica. Opera astronomica. Varia de optica. (2)

Sit jam # A M = a # Quia ergo æquales Anguli K H E & C H Z, \\ ſive E H G; eſtque E H A angulus re- \\ ctus; erit ut K E ad E G, ita K A ad \\ A G. Quia verò B M ad M K, ut H F \\ ad F K, erit,
# M B = b
# A L = c
# L C = n # ut B M + H F ad H F, ita M F ad F K,
Radius # A D = d # i.e. b + y:y::a - x:{ay - xy/b + y}. add. F A = x
# A F = x # fit K A = {ay + bx/b + y}
# F H = y

Rurſus, quia C L ad L G, ut H F ad F G, erit permutan-
do & dividendo C L - H F ad H F, ut L F ad F G,
n - y: y: :c - x: {cy - xy/n - y}, quâ ablatâ ab A F = x, fit
G A = {nx - cy/n - y}. Eſt autem E A = {dd/x}, quia Proportiona-
les F A, A H, A E: Ergo E A - G A, hoc eſt, E G, = {dd/x} -
{nx + cy/n - y}. Et K A - E A, hoc eſt, K E = {ay + bx/b + y} - {dd/x}

Sed diximus, quod K E ad E G, ut K A ad A G; i. e. {ay + bx/b + y} - {dd/x}: {dd/x} - {nx + cy/n - y}: :{ay + bx/b + y}: {nx - cy/n - y}.

Unde invenitur
2 anxxy + 2bnx 3 - ddbnx - ddnxy = + naddy + bddx
- 2acxyy - 2bcxxy + ddbcy + ddcyy - addyy - bddxy
Eſt autem {2bbc/a}x 3 = {2bbcddx/a} - {2bbcyyx/a}, quia xx = dd - yy. Et quia n = {bc/a}, ergo {- 2bbcxyy/a} - {ddbcyx/a} - 2acxyy
+ ddcyy = - addyy = bddxy:

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