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 twelfthMathew Carey, and sold by J. Conrad & Company, S. F. Bradford, Birch & Small, and Samuel Etheridge. Printed by T. & G. Palmer, 116, High-Street., 1806 - 518 pages |
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Page 13
... Parallel straight lines are such as are in the same plane , and which being produced ever so far both ways , do not meet . POSTULATES . I. LET it be granted that a straight line may be drawn from any one point to any other point . II ...
... Parallel straight lines are such as are in the same plane , and which being produced ever so far both ways , do not meet . POSTULATES . I. LET it be granted that a straight line may be drawn from any one point to any other point . II ...
Page 34
... parallel . Let the straight line EF , which falls upon the two straight lines AB , CD , make the alternate angles AEF , EFD equal to one another ; AB is parallel to CD . For , if it be not parallel , AB and CD being produced shall ...
... parallel . Let the straight line EF , which falls upon the two straight lines AB , CD , make the alternate angles AEF , EFD equal to one another ; AB is parallel to CD . For , if it be not parallel , AB and CD being produced shall ...
Page 35
... parallel to CD . Wherefore , if a straight line , & c . Q. E. D. PROP . XXIX . THEOR . - notes on this pro- IF a straight line fall upon two parallel straight See the lines , it makes the alternate angles equal to one ano- ther ; and ...
... parallel to CD . Wherefore , if a straight line , & c . Q. E. D. PROP . XXIX . THEOR . - notes on this pro- IF a straight line fall upon two parallel straight See the lines , it makes the alternate angles equal to one ano- ther ; and ...
Page 36
... parallel to the same straight line are parallel to one another . Let AB , CD be each of them parallel to EF ; AB is also pa- rallel to CD . Let the straight line GHK cut AB , EF , CD ; and because GHK cuts the parallel straight lines AB ...
... parallel to the same straight line are parallel to one another . Let AB , CD be each of them parallel to EF ; AB is also pa- rallel to CD . Let the straight line GHK cut AB , EF , CD ; and because GHK cuts the parallel straight lines AB ...
Page 37
... parallel to the given Book I. straight line BC . Which was to be done . PROP . XXXII . THEOR . IF a side of any ... parallel a to the straight line AB ; and be- cause AB is parallel to CE , and AC meets them , the alternate angles ...
... parallel to the given Book I. straight line BC . Which was to be done . PROP . XXXII . THEOR . IF a side of any ... parallel a to the straight line AB ; and be- cause AB is parallel to CE , and AC meets them , the alternate angles ...
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Common terms and phrases
altitude angle ABC angle BAC base BC BC is equal BC is given bisected Book XI centre circle ABCD circumference cone cylinder demonstrated described diameter draw drawn equal angles equiangular equimultiples Euclid excess fore given angle given in magnitude given in position given in species given magnitude given point given ratio given straight line gnomon greater join less Let ABC meet multiple opposite parallel parallelogram parallelogram AC perpendicular point F polygon prism proposition pyramid Q. E. D. PROP radius rectangle CB rectangle contained rectilineal figure remaining angle right angles segment sides BA similar solid angle solid parallelepipeds square of AC straight line AB straight line BC tangent THEOR third triangle ABC triplicate ratio vertex wherefore
Popular passages
Page 28 - Any two sides of a triangle are together greater than the third side.
Page 62 - To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, may be equal to the square of the other part.
Page 28 - IF, from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle. Let...
Page 57 - PROP. VIII. THEOR. IF a straight line be divided into any two parts, tour times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.
Page 26 - If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.
Page 163 - 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 17 - THE angles at the base of an isosceles triangle are equal to one another : and, if the equal sides be produced, the angles upon the other side of the base shall be equal.
Page 189 - In right angled triangles, the rectilineal figure described upon the side opposite to the right angle, is equal to the similar, and similarly described figures upon the sides containing the right angle.
Page 37 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sidef. For any rectilineal figure ABCDE can be divided into as many triangles as the figure has sides, by drawing straight lines from a point F within the figure to each of its angles.
Page 178 - Therefore, universally, similar rectilineal figures are to one another in the duplicate ratio of their homologous sides.