The Elements of Euclid, Viz: The Errors, by which Theon, Or Others, Have Long Ago Vitiated These Books, are Corrected; and Some of Euclid's Demonstrations are Restored. Also the Book of Euclid's Data, in Like Manner Corrected. the first six books, together with the eleventh and twelfth |
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Page 29
... angle of a triangle is greater than the anterior and oppofite angle , the exterior angle BDC of the triangle CDE is greater than CED ; for the fame reason , the ex- terior angle CEB of the triangle ABE is greater than BAC ; and it has ...
... angle of a triangle is greater than the anterior and oppofite angle , the exterior angle BDC of the triangle CDE is greater than CED ; for the fame reason , the ex- terior angle CEB of the triangle ABE is greater than BAC ; and it has ...
Page 31
... angle BAC greater than the angle EDF ; the bafe BC is alfo greater than the bafe EF . Book I. Of the two fides DE , DF , let DE be the fide which is not greater than the other , and at the point D , in the ftraight line DE , make the ...
... angle BAC greater than the angle EDF ; the bafe BC is alfo greater than the bafe EF . Book I. Of the two fides DE , DF , let DE be the fide which is not greater than the other , and at the point D , in the ftraight line DE , make the ...
Page 32
... angle BAC is not equal to the angle EDF , be- cause then the base BC would be equal a to EF ; but it is not ; therefore the angle BAC is not equal to the angle EDF ; nei- ther is it lefs ; becaufe then the base BC would be lefs b than ...
... angle BAC is not equal to the angle EDF , be- cause then the base BC would be equal a to EF ; but it is not ; therefore the angle BAC is not equal to the angle EDF ; nei- ther is it lefs ; becaufe then the base BC would be lefs b than ...
Page 33
... angle GBC is equal to the angle DEF ; therefore the bafe GC is equal to the bafe a 4. 1 . DF , and the triangle GBC ... BAC to the third angle EDF . Next , let the fides A which are oppofite to equal angles in each triangle be equal to ...
... angle GBC is equal to the angle DEF ; therefore the bafe GC is equal to the bafe a 4. 1 . DF , and the triangle GBC ... BAC to the third angle EDF . Next , let the fides A which are oppofite to equal angles in each triangle be equal to ...
Page 37
... angles BAC , ACE are equal b . A- gain , because AB is parallel to CE , and BD falls upon them , the exterior angle ECD B is equal to the interior and A a 31. I. b 29. I. D oppofite angle ABC ; but the angle ACE was fhown to be equal to the ...
... angles BAC , ACE are equal b . A- gain , because AB is parallel to CE , and BD falls upon them , the exterior angle ECD B is equal to the interior and A a 31. I. b 29. I. D oppofite angle ABC ; but the angle ACE was fhown to be equal to the ...
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Common terms and phrases
alfo alſo angle ABC angle BAC bafe baſe BC is equal BC is given becauſe the angle bifected Book XI cafe circle ABCD circumference cone confequently cylinder defcribed demonftrated diameter drawn equal angles equiangular equimultiples Euclid excefs faid fame manner fame multiple fame ratio fame reafon fecond fegment fhall fhewn fide BC fimilar firft firſt folid angle fome fore fphere fquare of AC ftraight line AB given angle given ftraight line given in fpecies given in magnitude given in pofition given magnitude given ratio gnomon greater join lefs likewife line BC oppofite parallel parallelepipeds parallelogram perpendicular polygon prifm propofition proportionals pyramid Q. E. D. PROP rectangle contained rectilineal figure right angles thefe THEOR theſe triangle ABC vertex wherefore
Popular passages
Page 32 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides adjacent to the equal...
Page 165 - D ; wherefore the remaining angle at C is equal to the remaining angle at F ; Therefore the triangle ABC is equiangular to the triangle DEF.
Page 170 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.
Page 10 - When several angles are at one point B, any ' one of them is expressed by three letters, of which ' the letter that is at the vertex of the angle, that is, at ' the point in which the straight lines that contain the ' angle meet one another, is put between the other two ' letters, and one of these two is...
Page 55 - If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.
Page 32 - ... then shall the other sides be equal, each to each; and also the third angle of the one to the third angle of the other. Let ABC, DEF be two triangles which have the angles ABC, BCA equal to the angles DEF, EFD, viz.
Page 45 - To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.
Page 211 - AB shall be at right angles to the plane CK. Let any plane DE pass through AB, and let CE be the common section of the planes DE, CK ; take any point F in CE, from which draw FG in the plane DE at right D angles to CE ; and because AB is , perpendicular to the plane CK, therefore it is also perpendicular to every straight line in that plane meeting it (3.
Page 38 - F, which is the common vertex of the triangles ; that is, together with four right angles. Therefore all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Page 304 - Thus, if B be the extremity of the line AB, or the common extremity of the two lines AB, KB, this extremity is called a point, and has no length : For if it have any, this length must either be part of the length of the line AB, or of the line KB.