Euclid's Elements: Or, Second Lessons in Geometry,in the Order of Simson's and Playfair's Editions ... |
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Page 30
... square upon a given straight line ( AB . ) Constr . From the point A draw AC at right an- C gles to AB ( a ) ; make AD equal to AB ( 6 ) ; draw DE parallel to AB , and BE parallel to AD ( c ) . 1. Of the equal angles . The angle A is a ...
... square upon a given straight line ( AB . ) Constr . From the point A draw AC at right an- C gles to AB ( a ) ; make AD equal to AB ( 6 ) ; draw DE parallel to AB , and BE parallel to AD ( c ) . 1. Of the equal angles . The angle A is a ...
Page 31
... square CH , is equal to the parallelogram CL . Therefore , the two parallelograms BL , CL , viz . the square BE , are equal to the two squares , BG , CH . Wherefore , the square upon BC is equal to the sum of the squares upon AB , AC ...
... square CH , is equal to the parallelogram CL . Therefore , the two parallelograms BL , CL , viz . the square BE , are equal to the two squares , BG , CH . Wherefore , the square upon BC is equal to the sum of the squares upon AB , AC ...
Page 33
... AC , CB . Describe on AB the square AE ( a ) ; draw CF parallel to AD ( 6 ) . Now the square AE equals the two rectangles AF and CE : but AE is the square of AB ( c ) ; and AF is the rectangle of AD and AC , and CE is the rectangle ...
... AC , CB . Describe on AB the square AE ( a ) ; draw CF parallel to AD ( 6 ) . Now the square AE equals the two rectangles AF and CE : but AE is the square of AB ( c ) ; and AF is the rectangle of AD and AC , and CE is the rectangle ...
Page 34
... ( AC , CB ) into which it may be divided , and two rectangles of the parts Constr . Upon AB describe the square AE ( a ) ... square ( h ) on CB . CB C K FE For similar reasons , HF is a square on HG , which is equal to AC . It is also ...
... ( AC , CB ) into which it may be divided , and two rectangles of the parts Constr . Upon AB describe the square AE ( a ) ... square ( h ) on CB . CB C K FE For similar reasons , HF is a square on HG , which is equal to AC . It is also ...
Page 35
... square of the line ( CD ) , composed of the half and the part produced , is equal to the rectangle of the whole line ... ( AC ) . Constr . Upon AB describe the square AE ( a ) join BD ; through C draw CF parallel to BE , and cutting ...
... square of the line ( CD ) , composed of the half and the part produced , is equal to the rectangle of the whole line ... ( AC ) . Constr . Upon AB describe the square AE ( a ) join BD ; through C draw CF parallel to BE , and cutting ...
Other editions - View all
Euclid's Elements, Or Second Lessons in Geometry, in the Order of Simson's ... D. M'Curdy No preview available - 2017 |
Euclid's Elements, Or Second Lessons in Geometry, in the Order of Simson's ... D. M'Curdy No preview available - 2017 |
Common terms and phrases
ABCD alternate angles angle ACD angles ABC angles equal antecedents Argument base BC bisected centre Chart chord circle ABC circumference Constr Denison Olmsted diameter draw drawn equal angles equal arcs equal radii equal sides equals the squares equiangular equilateral equilateral polygon equimultiples exterior angle fore Geometry given circle given rectilineal given straight line given triangle gles gnomon greater inscribed isosceles isosceles triangle join less meet multiple opposite angles parallelogram parallelopipeds pentagon perimeter perpendicular plane polygon produced propositions Q. E. D. Recite radius ratio rectangle rectangle contained rectilineal figure School secant segment semicircle similar sine square of AC tangent third touches the circle triangle ABC unequal Wherefore
Popular passages
Page 90 - If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, the triangles shall be equiangular, and shall have those angles equal which are opposite to the homologous sides.
Page 117 - In the same way it may be proved that a : b : : sin. A : sin. B, and these two proportions may be written a : 6 : c : : sin. A : sin. B : sin. C. THEOREM III. t8. In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference. By Theorem II. we have a : b : : sin. A : sin. B.
Page 92 - IN a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.
Page 79 - THEOREM. lf the first has to the second the same ratio which the third has to the fourth, but the third to the fourth, a greater ratio than the fifth has to the sixth ; the first shall also have to the second a greater ratio than the fifth, has to the sixth.
Page 87 - If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those sides produced, proportionally...
Page 26 - Triangles upon equal bases, and between the same parallels, are equal to one another.
Page 94 - Equal parallelograms which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional ; and parallelograms that have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.
Page 12 - 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 133 - If a straight line stand at right angles to each of two straight lines at the point of their intersection, it shall also be at right angles to the plane which passes through them, that is, to the plane in which they are.
Page 13 - AB be the greater, and from it cut (3. 1.) off DB equal to AC the less, and join DC ; therefore, because A in the triangles DBC, ACB, DB is equal to AC, and BC common to both, the two sides DB, BC are equal to the two AC, CB. each to each ; and the angle DBC is equal to the angle ACB; therefore the base DC is equal to the base AB, and the triangle DBC is< equal to the triangle (4. 1.) ACB, the less to 'the greater; which is absurd.