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[[left margin circled]] 23 [[/left margin]] in the ΔAEC, As Rad. : Cot. AC :: Cos. ECA : Cot. EC = 55°.. 29'..20 1/2" from which take [[symbol: libra]] C, leaves [[symbol: libra]] E = 43°..35..32 1/2, or [[symbol: leo]] 26°..24..27 1/2 corresponding to [[underline]] August [[/underline]]19[[superscript]]th[[/superscript]]. 2. For its disappearance under the pole. In fig. 35. CQ + QO + OA = CA = 55°..42'..43"; and in the triangle CeA, r. [[superscript]] t [[\superscript]] L. [[superscript]] ed [[\superscript]] at A, As Rad. is Cot. CA :: Cos. eCA : Cot. eC = 75°..1'..20"; eC - [[symbol: aries]]C = 63°..7'..32", or [[symbol: capricornus]] 26°..52'..28"; corresponding to the 16[[superscript]]th[[/superscript]] of [[underline]] January [[/underline]]. 3. [[superscript]] d [[\superscript]] For its first appearance above the pole. In fig. 38 ÆH + HB [[minus sign]] DÆ = DB = 46°..17'..17" and As Rad. Cot. DB : Cos. BDe : Cot. De = 69°..26'..18", to which add D[[symbol: libra]] ([[symbol: aries]]C) gives [[symbol: libra]]e = 81°..20'..6" answering to [[symbol: sagittarius]] 21°..20'..6" and [[underline]] December [[/underline]] the 13[[superscript]]th[[/superscript]]. 4. [[superscript]] th [[\superscript]] For its disappearance above the pole. In fig. 37. in the supplemental triangle BED, HÆ - HB - DÆ = BD = 20°..17'..17", As Rad. : Cot. BD :: Cos. BDE ([[symbol: libra]]CQ) : Cot. DE = 43°..17'..37" to which add [[symbol: aries]]D ([[symbol: libra]]C) gives [[symbol: aries]]E = 55°..11'..25" corresponding to [[symbol: taurus]] 25°..11..25 and [[underline] May [[/underline]] the 16 [[superscript]] th [[\superscript]] Q.E.I. N.B. only the black lines in each figure are used. But all these proportions from the last line on the last page may be more concisely & methodically expressed, when E is put also in the place of e in figs. 35 & 38. thus [[drawn equations; would be best viewed as an image]] [[left]]Rad. : Cot. AC = OQ [[\left]] [[left center]]+ QC - OA, in fg. 36. + QC + OA, in fig. 35. - QC - OA, in fig. 38. + OA - CQ, in fig. 37 [[\left center]] [[right center]] :: Cot ECA : Cot EC = [[\right center]] [[right]]55°..29'..20 1/2" 75..1..20 110..33..42 124..48..35 [[\right] From which take [[top line]][[symbol: libra]]C [[\top]] [[divided by]][[bottom line]] [[symbol: aries]]C [[\bottom]] gives [[left bracket]] [[symbol: libra]]E = 43..35..32 1/2 [[2nd line]][[symbol: aries]]E = 63..7..32 [[next line]] 98..39..54 [[3rd line]][[symbol: libra]]E = 112..54..47 [[right bracket]] [[written sideways]] corresponding to [[\written sideways]] [[left bracket]] [[symbol: leo]] 26°..24'..27 1/2 and [[underline]]August[[/underline] 19. [[2nd line]][[symbol: capricornus]] 26..52..28 and [[underline]]January[[/underline]] 16. [[3rd line]][[symbol: sagittarius]] 21..20..6 and [[underline]]December[[/underline]] 13. [[4th line]][[symbol: taurus]] 25..11..25 and [[underline]]May[[/underline]] ---- 16. [[/equations]] Whence [[underline]] Alioth[[/underline]] appears upon the meridian [[strikethrough]] [[over strike through]]under[[/overstrikethrough] the pole from [[underline]]Aug.[[/underline]] [[superscript]] st [[\superscript]] 19. to [[underline]]Jan.[[/underline]][[superscript]] y [[\superscript]] 16. and above the pole from [[underline]]December[[/underline]] 13 to May 16. Q. E. I. [[in different colored ink in right margin]]Quest. on p. 112 better solved [[symbol: leo]] from line the 23 [[/margin]] * [[red ink]] See the [[strikethrough]] next page [[\strikethrough]] following page [[\red ink]] From a and n let fall the perpendiculars aC, ne, upon EP produced: put m, t, for the Tang. of En, Ea; a, b, x the Sines, and c, d, y the cosines of PE, nEa, PEn, then Rad: y :: m : my = tang Ee. by Prop. VI. B. 1. of [[underline]]Emer.[[/underline]] Trig. [[numerator]] myc - a [[\numerator]] / [[denominator]] [[square root]] 1 - m [[superscript]] 2 [[\superscript]] y [[superscript]] 2 [[\superscript]] [[\square root]] [[\denominator]] = s. Pe; [[numerator]] my [[\numerator]] / [[denominator]] [[square root]] 1 - m [[superscript]] 2 [[\superscript]] y [[superscript]] 2 [[\superscript]] [[\square root]] [[\denominator]] = s. Ee; whence by a reciprocation of [[underline]] Cor. [[\underline]] 3. to Prop. 28. B. III. [[numerator]] myc - a [[\numerator]] / [[denominator]] [[square root]] 1 - m [[superscript]] 2 [[\superscript]] y [[superscript]] 2 [[\superscript]] [[\square root]] [[\denominator]] : [[numerator]] my [[\numerator]] / [[denominator]] [[square root]] 1 - m [[superscript]] 2 [[\superscript]] y [[superscript]] 2 [[\superscript]] [[\square root]] [[\denominator]] : x/y (tang.PEn) : [[numerator]] mx [[\numerator]] / [[denominator]] myc-a [[\denominator]] = tang. nPE. By: Prop V.B.I. dx + by = S. and; by Cor. 1 to ditto, dy - bx = cos. PEa. [[underline]] then as [[\underline]] before, Rad. : dy - bx :: t : [[bar over]] dy-bx [[\bar over]] x t = tang. CE : [[numerator]] [[bar over]] dy-bx [[\bar over]] x tc - a [[\numerator]] /[[denominator]][[square root]]1 + [[square root] [[bar over]] dy-bx [[\bar over]] x t [[/square root]] [[/square root]][[\denominator]] = S. PC; [[strikethrough]] and [[numerator]] [[bar over]] dy-bx [[\bar over]] x t [[\numerator]] /[[denominator]][[square root]] 1 + [[square root]][[bar over]] dy-bx [[\bar over]] x t [[/square root]] [[/square root]][[\denominator]] = S. EC; and [[numerator]] [[bar over]] dy-bx [[\bar over]] x tc-a [[\numerator]] /[[denominator]][[square root]] 1 + [[square root]][[bar over]] dy-bx [[\bar over]] x t [[\square root]][[/square root]][[\denominator]] : [[numerator]] [[bar over]] dy-bx [[\bar over]] x t [[\numerator]] /[[denominator]][[square root]] 1 + [[square root]][[bar over]] dy-bx [[\bar over]] x t [[\square root]][[/square root]][[\denominator]] :: [[numerator]] dx + by [[\numerator]] / [[denominator]] dy - bx [[\denominator]] (tang. PEa) : [[numerator]] [[bar over]]dx + by [[\bar over]] x t [[\numerator]] / [[denominator]] [[bar over]] dy - bx [[\bar over]] x tc - a[[\denominator]] = tang. aPE = nPE, and ∵[[therefore]] = [[numerator]] mx [[\numerator]] / [[denominator]] myc - a [[\denominator]] : Whence, by reduction, y [[superscript]] 2 [[\superscript]] + x [[superscript]] 2 [[\superscript]] - [[numerator]] a [[\numerator]] / [[denominator]] mc [[\denominator]] y =
Transcription Notes:
There are differences between x used as a variable and x used as a multiplication symbol in this text. I've tried to differentiate - when x is used as a multiplication symbol, it has a space on either side of it. If it is right next to another variable, it can be assumed that it is also being used as a variable.
Additionally, there are many square roots & squares in this text. I've marked square roots using the bracketing format; squares are marked using the carrot notation that is commonly used in modern mathematics. Please note, however, that the text uses the drawn symbols for both of these. I've tried to use notations as much as possible to keep the intent of her formulas as intact as possible.
I found many of the symbols on Wikipedia, "Astronomical symbols."
N.b. means "nota bene", an old term for P.S., or postscript.
In portion beginning "in the ΔAEC" transcription needs to be checked, a "4" replaces a possible Greek cap alpha or Greek cap delta, which may be interpreted as a 4 (which delta often represents). It does not appear to be a 4 in ms.
Check what has been transcribed as commas in mathematical notations, does not really match commas in running word text.
Check out superscript d after/over paragraph beginning with "3."
