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 21
... fides BA , CA do not coin- cide with the fides ED , FD , but have a different situation , as EG , FG ; then , upon the fame base EF , and upon the fame fide of it , there can be two triangles that have their fides which are ...
... fides BA , CA do not coin- cide with the fides ED , FD , but have a different situation , as EG , FG ; then , upon the fame base EF , and upon the fame fide of it , there can be two triangles that have their fides which are ...
Page 28
... fide . Let ABC be a triangle ; any two fides of it together are greater than the third fide , viz . the fides BA , AC greater than the fide BC ; and AB , BC greater than AC ; and BC , CA greater than AB . 83. Ι . Produce BA to the point ...
... fide . Let ABC be a triangle ; any two fides of it together are greater than the third fide , viz . the fides BA , AC greater than the fide BC ; and AB , BC greater than AC ; and BC , CA greater than AB . 83. Ι . Produce BA to the point ...
Page 29
... fides BA , AC are greater than BE , EC : Again , because the two fides CE , ED of the triangle CED are great- er than CD , add DB to each ! of these ; therefore the fides CE , EB are greater than CD , DB ; but it has been shewn that BA ...
... fides BA , AC are greater than BE , EC : Again , because the two fides CE , ED of the triangle CED are great- er than CD , add DB to each ! of these ; therefore the fides CE , EB are greater than CD , DB ; but it has been shewn that BA ...
Page 31
... fides BA , AC are equal to the two ED , DG , each to each , and the b 3. 1 . angle BAC is equal to the angle EDG ; A therefore the bafe BC is equal to the base EG ; and because DG is equal to DF , the angle DFG is equal d to the angle ...
... fides BA , AC are equal to the two ED , DG , each to each , and the b 3. 1 . angle BAC is equal to the angle EDG ; A therefore the bafe BC is equal to the base EG ; and because DG is equal to DF , the angle DFG is equal d to the angle ...
Page 48
... fides which con . tain the right angle . Let ABC be a right angled triangle having the right angle BAC ; the square described upon the fide BC is equal to the squares described upon BA , AC . On BC defcribe the square BDEC , and on BA ...
... fides which con . tain the right angle . Let ABC be a right angled triangle having the right angle BAC ; the square described upon the fide BC is equal to the squares described upon BA , AC . On BC defcribe the square BDEC , and on BA ...
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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 becauſe the ratio biſected Book XI cafe cauſe circle ABCD circumference cone confequently conſtruction cylinder demonſtration deſcribed diameter drawn EFGH equal angles equiangular equimultiples Euclid exceſs fame multiple fame ratio fame reaſon fides fides BA fimilar firſt folid angle fore given angle given in magnitude given in poſition given in ſpecies given magnitude given ratio given ſtraight line gnomon greater join leſs line BC oppoſite parallel parallelepipeds parallelogram paſs perpendicular priſm proportionals propoſition pyramid Q. E. D. PROP rectangle contained rectilineal figure right angles ſame ſecond ſegment ſhall ſhewn ſide ſolid ſpace ſphere ſquare of AC THEOR theſe thoſe 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.