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From: -31=© 2009: 1978-x. on 12 Sep 2009 04:17
Proposition 15.

If two straight lines cut one another, they make the vertical angles equal to one another.

For let the straight lines AB, CD cut one another at the point E;
(5)

I say that the angle AEC is equal to the angle DEB,
and the angle CEB to the angle AED.
[Figure]

For, since the straight line AE stands
(10)

on the straight line CD, making the angles CEA, AED,
the angles CEA, AED are equal to two right angles [I. 13]
[p. 278]

Again, since the straight line DE stands on the straight line AB, making the angles AED, DEB,
the angles AED, DEB are equal to two right angles. [I. 13]
(15)

But the angles CEA, AED were also proved equal to two right angles;
therefore the angles CEA, AED are equal to the angles AED DEB. [Post. 4 and C. N. 1] Let the angle AED be subtracted from each;
(20)

therefore the remaining angle CEA is equal to the remaining angle BED. [C. N. 3]

Similarly it can be proved that the angles CEB, DEA are also equal.

Therefore etc. Q. E. D.
(25)

Porism.

[From this it is manifest that, if two straight lines cut one another, they will make the angles at the point of section equal to four right angles.
1. the vertical angles. 26. at the point of section,

1. the vertical angles. The difference between adjacent angles (hai ephexês gôniai) and vertical angles (hai kata koruphên gôniai) is thus explained by Proclus (p. 298, 14-24). The first term describes the angles made by two straight lines when one only is divided by the other, i.e. when one straight line meets another at a point which is not either of its extremities, but is not itself produced beyond the point of meeting. When the first straight line is produced, so that the lines cross at the point, they make two pairs of vertical angles (which are more clearly described as vertically opposite angles), and which are so called because their convergence is from opposite directions to one point (the intersection of the lines) as vertex (koruphê).

26. at the point of section, literally “at the section,” pros têi tomê.

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En tois orthogôniois trigônois to apo tês tên orthên gônian hupoteinousês pleuras tetragônon ison esti tois apo tôn tên orthên gônian periechousôn pleurôn tetragônois.
(5)

Estô trigônon orthogônion to ABG orthên echon tên hupo BAG gônian: legô, hoti to apo tês BG tetragônon
(5)

ison esti tois apo tôn BA, AG tetragônois.

Anagegraphthô gar apo men tês BG tetragônon to BDEG, apo de tôn BA, AG ta ÊB, THG, kai dia tou
(10)

A hopoterai tôn BD, GE parallêlos êchthô hê AL: kai epezeuchthôsan hai AD, ZG. kai epei orthê estin hekatera tôn hupo BAG, BAÊ gôniôn, pros dê tini eutheiai têi BA kai tôi pros autêi sêmeiôi tôi A duo eutheiai hai AG, AÊ mê epi ta auta merê keimenai tas ephexês gônias dusin
(15)

orthais isas poiousin: ep' eutheias ara estin hê GA têi AÊ. dia ta auta dê kai hê BA têi ATh estin ep' eutheias. kai epei isê estin hê hupo DBG gônia têi hupo ZBA: orthê gar hekatera: koinê proskeisthô hê hupo ABG: holê ara hê hupo DBA holêi têi hupo ZBG estin isê. kai epei isê
(20)

estin hê men DB têi BG, hê de ZB têi BA, duo dê hai DB, BA duo tais ZB, BG isai eisin hekatera hekaterai: kai
[phigure]
gônia hê hupo DBA gôniai têi hupo ZBG isê: basis ara hê AD basei têi ZG [estin] isê, kai to ABD trigônon
(25)

tôi ZBG trigônôi estin ison: kai [esti] tou men ABD trigônou diplasion to BL parallêlogrammon: basin te gar tên autên echousi tên BD kai en tais autais eisi
(30)

parallêlois tais BD, AL: tou de ZBG trigônou diplasion to ÊB tetragônon: basin te gar palin tên autên echousi tên ZB kai en tais autais eisi parallêlois tais ZB, ÊG. [ta de tôn isôn diplasia isa allêlois estin:] ison ara
(35)

esti kai to BL parallêlogrammon tôi ÊB tetragônôi. homoiôs dê epizeugnumenôn tôn AE, BK deichthêsetai kai to GL parallêlogrammon ison tôi THG tetragônôi: holon ara to BDEG tetragônon dusi tois ÊB, THG tetragônois ison estin. kai esti to men BDEG tetragônon
(40)

apo tês BG anagraphen, ta de ÊB, THG apo tôn BA, AG. to ara apo tês BG pleuras tetragônon ison esti tois apo tôn BA, AG pleurôn tetragônois.

En ara tois orthogôniois trigônois to apo tês tên orthên gônian hupoteinousês pleuras tetragônon ison esti tois
(45)

En tois orthogôniois trigônois to apo tês tên orthên gônian hupoteinousês pleuras tetragônon ison esti tois apo tôn tên orthên gônian periechousôn pleurôn tetragônois.
(5)

Estô trigônon orthogônion to ABG orthên echon tên hupo BAG gônian: legô, hoti to apo tês BG tetragônon
(5)

ison esti tois apo tôn BA, AG tetragônois.

Anagegraphthô gar apo men tês BG tetragônon to BDEG, apo de tôn BA, AG ta ÊB, THG, kai dia tou
(10)

A hopoterai tôn BD, GE parallêlos êchthô hê AL: kai epezeuchthôsan hai AD, ZG. kai epei orthê estin hekatera tôn hupo BAG, BAÊ gôniôn, pros dê tini eutheiai têi BA kai tôi pros autêi sêmeiôi tôi A duo eutheiai hai AG, AÊ mê epi ta auta merê keimenai tas ephexês gônias dusin
(15)

orthais isas poiousin: ep' eutheias ara estin hê GA têi AÊ. dia ta auta dê kai hê BA têi ATh estin ep' eutheias. kai epei isê estin hê hupo DBG gônia têi hupo ZBA: orthê gar hekatera: koinê proskeisthô hê hupo ABG: holê ara hê hupo DBA holêi têi hupo ZBG estin isê. kai epei isê
(20)

estin hê men DB têi BG, hê de ZB têi BA, duo dê hai DB, BA duo tais ZB, BG isai eisin hekatera hekaterai: kai
[phigure]
gônia hê hupo DBA gôniai têi hupo ZBG isê: basis ara hê AD basei têi ZG [estin] isê, kai to ABD trigônon
(25)

tôi ZBG trigônôi estin ison: kai [esti] tou men ABD trigônou diplasion to BL parallêlogrammon: basin te gar tên autên echousi tên BD kai en tais autais eisi
(30)

parallêlois tais BD, AL: tou de ZBG trigônou diplasion to ÊB tetragônon: basin te gar palin tên autên echousi tên ZB kai en tais autais eisi parallêlois tais ZB, ÊG. [ta de tôn isôn diplasia isa allêlois estin:] ison ara
(35)

esti kai to BL parallêlogrammon tôi ÊB tetragônôi. homoiôs dê epizeugnumenôn tôn AE, BK deichthêsetai kai to GL parallêlogrammon ison tôi THG tetragônôi: holon ara to BDEG tetragônon dusi tois ÊB, THG tetragônois ison estin. kai esti to men BDEG tetragônon
(40)

apo tês BG anagraphen, ta de ÊB, THG apo tôn BA, AG. to ara apo tês BG pleuras tetragônon ison esti tois apo tôn BA, AG pleurôn tetragônois.
apo tôn tên orthên [gônian] periechousôn pleurôn tetragônois: hoper edei deixai.
The Social Construction of Reality
Language soars into region that are not only de facto but also a priori unavailable to everyday experience.
p. 40. In regarding to the
Automatic link establishment
Automatic Link Establishment, commonly known as ALE, is the worldwide de facto where a priori channel and address coordination is possible.
List of legal Latin terms
a fortiori - a posteriori - a priori - a priori assumption - ab administratis - de die in diem - de facto - de futuro - de integro
Index of law articles (A-E)
a fortiori - a posteriori - a priori - a priori assumption - administratis - de die in diem - de facto - De facto corporation - de
El Eternauta
coincided with the de facto governments of Pedro Eugenio Aramburu a priori, el gran tema que El Eternauta debate, denunciando claramente
Jansenism
contingent, while Calvinists claimed that it was a priori and necessary. distinction between de jure and de facto - that a Catholic was
Gonfaloniere of Justice
Like Florence's Podestà and Priori , it was introduced in 1293 within the republican framework and de facto (though never de iure)
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