The Elements of Euclid: 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. viz. The first six books, together with the eleventh and twelfth |
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Page 60
... Similar fegments of a circle , are those in which the an- gles are equal , or which contain equal angles . PROP . I. PROB . To find the center of a given circle . Let ABC be the given circle ; it is required to find its center . Draw ...
... Similar fegments of a circle , are those in which the an- gles are equal , or which contain equal angles . PROP . I. PROB . To find the center of a given circle . Let ABC be the given circle ; it is required to find its center . Draw ...
Page 158
... similar to the whole triangle , and to one another . Let ABC be a right angled triangle having the right angle BAC ; and from the point A let AD be drawn perpendicular to the bafe BC . the triangles ABD , ADC are fimilar to the whole ...
... similar to the whole triangle , and to one another . Let ABC be a right angled triangle having the right angle BAC ; and from the point A let AD be drawn perpendicular to the bafe BC . the triangles ABD , ADC are fimilar to the whole ...
Page 168
... SIMILAR polygons may be divided into the fame number of fimilar triangles , having the fame ra- tio to one another that the polygons have ; and the po- lygons have to one another the duplicate ratio of that which their homologous fides ...
... SIMILAR polygons may be divided into the fame number of fimilar triangles , having the fame ra- tio to one another that the polygons have ; and the po- lygons have to one another the duplicate ratio of that which their homologous fides ...
Page 171
... similar rectilineal figures fimilarly defcribed upon four ftraight lines be propor- tionals , those straight lines fhall be proportionals . Let the four straight lines AB , CD , EF , GH be proportionals , yiz . AB to CD , as EF to GH ...
... similar rectilineal figures fimilarly defcribed upon four ftraight lines be propor- tionals , those straight lines fhall be proportionals . Let the four straight lines AB , CD , EF , GH be proportionals , yiz . AB to CD , as EF to GH ...
Page 189
... Similar folid figures are fuch as have all their folid angles equal , See N. each to each , and which are contained by the fame number of fimilar planes . XII . A Pyramid is a folid figure contained by planes that are conftituted ...
... Similar folid figures are fuch as have all their folid angles equal , See N. each to each , and which are contained by the fame number of fimilar planes . XII . A Pyramid is a folid figure contained by planes that are conftituted ...
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Common terms and phrases
AC is equal alfo alſo angle ABC angle BAC bafe baſe BC is given becauſe the angle bifected Book XI cafe circle ABCD circumference cone confequently cylinder defcribed demonftrated diameter drawn EFGH equal angles equiangular equimultiples Euclid excefs faid fame manner fame multiple fame ratio fecond fegment fhall fhewn fide BC fides fimilar fince firft firſt folid angle fome fore 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 leſs likewife line BC muſt oppofite parallel parallelepipeds parallelogram perpendicular prifm Propofition proportionals pyramid rectangle contained rectilineal figure right angles ſhall ſphere ſquare thefe THEOR theſe thro tiple triangle ABC vertex wherefore
Popular passages
Page 156 - 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 3 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.
Page 323 - Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides.
Page 92 - If from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it ; if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square- of the line which meets it, the line which meets shall touch the circle.
Page 80 - EA : and because AD is equal to DC, and DE common to the triangles ADE, CDE, the two sides AD, DE are equal to the two CD, DE, each to each ; and the angle ADE is equal to the angle CDE, for each of them is a right angle ; therefore the base AE is equal (4.
Page 52 - If a straight line be bisected, and produced to any point, the square of the whole line thus produced, and the square of the part of it produced, are together double of the square of half the line bisected, and of the square of the line made up of the half and the part produced.
Page 36 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.
Page 2 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.
Page 54 - AB be the given straight line ; it is required to divide it into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part.
Page 74 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle...