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 10
... wherefore CA , AB , BC are equal to one another ; and the triangle ABC is therefore equilateral , and it is described upon the given straight line AB . Which was required to be done . PROP . II . PROB . From a given point to draw a ...
... wherefore CA , AB , BC are equal to one another ; and the triangle ABC is therefore equilateral , and it is described upon the given straight line AB . Which was required to be done . PROP . II . PROB . From a given point to draw a ...
Page 11
... Wherefore from the given point A a straight line AL has been drawn equal to the given straight line BC . Which was to be done . PROP . III . PROB .. FROM the greater of two given straight lines to cut off a part equal to the less . Let ...
... Wherefore from the given point A a straight line AL has been drawn equal to the given straight line BC . Which was to be done . PROP . III . PROB .. FROM the greater of two given straight lines to cut off a part equal to the less . Let ...
Page 12
... wherefore also the point C shall coincide with the point F , because the straight line AC is equal to DF : but the point B coincides with the point E ; wherefore the base BC shall coincide with the base EF , because the point B ...
... wherefore also the point C shall coincide with the point F , because the straight line AC is equal to DF : but the point B coincides with the point E ; wherefore the base BC shall coincide with the base EF , because the point B ...
Page 15
... wherefore likewise the angle BAC coincides with the angle EDF , and is equal ( 8. Ax . ) to it . Therefore , if two trian- gles , & c . Q. E. D. PROP . IX . PROB . To bisect a given rectilineal angle , that is , to divide it into two ...
... wherefore likewise the angle BAC coincides with the angle EDF , and is equal ( 8. Ax . ) to it . Therefore , if two trian- gles , & c . Q. E. D. PROP . IX . PROB . To bisect a given rectilineal angle , that is , to divide it into two ...
Page 16
... Wherefore , from the given point C , in the given straight line AB , FC has been drawn at right angles to AB . Which was to be done .. Cor . By help of this problem , it may be demonstrated , that two straight lines cannot have a common ...
... Wherefore , from the given point C , in the given straight line AB , FC has been drawn at right angles to AB . Which was to be done .. Cor . By help of this problem , it may be demonstrated , that two straight lines cannot have a common ...
Common terms and phrases
altitude angle ABC angle BAC base BC BC is equal BC is given bisected centre circle ABCD circumference Co-S cone cylinder demonstrated described diameter draw drawn equal angles equiangular equimultiples Euclid ex æquali excess fore given angle given in magnitude given in position given in species given magnitude given ratio given straight line gles gnomon join less Let ABC multiple parallel parallelogram parallelogram AC perpendicular point F polygon prism proportionals proposition pyramid Q. E. D. PROP radius ratio of AE rectangle CB rectangle contained rectilineal figure remaining angle right angles segment sides BA similar sine solid angle solid parallelopiped square of AC square of BC straight line AB straight line BC tangent THEOR third triangle ABC triplicate ratio vertex wherefore
Popular passages
Page 45 - Ir a straight line be divided into any two parts, four 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 41 - If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part.
Page 54 - Ir any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle. Let ABC be a circle, and A, B any two points in the circumference ; the straight line drawn from A to B shall fall within the circle.
Page 18 - ABD, the less to the greater, which is impossible ; therefore BE is not in the same straight line with BC.
Page 10 - From a given point to draw a straight line equal to a given straight line. Let A be the given point, and BC the given straight line: it is required to draw from the point A a straight line equal to BC.
Page 8 - Let it be granted that a straight line may be drawn from any one point to any other point.
Page 256 - Again ; the mathematical postulate, that " things which are equal to the same are equal to one another," is similar to the form of the syllogism in logic, which unites things agreeing in the middle term.
Page 129 - 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 23 - At a given point in a given straight line, to make a rectilineal angle equal to a given rectilineal angle. Let AB be the given straight line, and A...
Page 20 - ANY two angles of a triangle are together less than two right angles.