Euclid's Elements of geometry, transl. To which are added, algebraic demonstrations to the second and fifth books; also deductions in the first six, eleventh and twelfth books, with notes, by G. Phillips. Part 1, containing book i-vi1826 |
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Σελίδα 30
... half the difference between the vertical angle and two right angles . 2. The difference of the angles at the base of any triangle is double the angle contained by the line bisecting the vertical angle and another drawn from the vertex ...
... half the difference between the vertical angle and two right angles . 2. The difference of the angles at the base of any triangle is double the angle contained by the line bisecting the vertical angle and another drawn from the vertex ...
Σελίδα 34
... half of the parallelogram EBCA , because the diameter AB bisects it ; and the triangle DBC is half of the parallelogram DBCF , for the diameter DC bisects it . But the halves of equal things are equal . Therefore the triangle ABC is ...
... half of the parallelogram EBCA , because the diameter AB bisects it ; and the triangle DBC is half of the parallelogram DBCF , for the diameter DC bisects it . But the halves of equal things are equal . Therefore the triangle ABC is ...
Σελίδα 36
... altitude being given ; for as the area of a parallelogram is the product of the base and altitude , it follows that the area of a triangle must be half that product . b C તે angle CEF equal to D ; and 36 [ Book I. EUCLID'S ELEMENTS :
... altitude being given ; for as the area of a parallelogram is the product of the base and altitude , it follows that the area of a triangle must be half that product . b C તે angle CEF equal to D ; and 36 [ Book I. EUCLID'S ELEMENTS :
Σελίδα 37
... half of the given one , and the diameters of this parallelogram shall be equal to half of the perimeter of the other . PROPOSITION XLIII . THEOREM . The complements of any parallelogram which are about the diameter of any parallelogram ...
... half of the given one , and the diameters of this parallelogram shall be equal to half of the perimeter of the other . PROPOSITION XLIII . THEOREM . The complements of any parallelogram which are about the diameter of any parallelogram ...
Σελίδα 49
... half the line . The same by Algebra . Put a equal to the right line AB , and suppose it divided into any two parts f , g ; then shall a2 = ƒ2 + 2fg + g2 . For a * = ƒ + g square each side , and we * Ax . 8. 1 . shall have a2 = ƒ2 + 2ƒg ...
... half the line . The same by Algebra . Put a equal to the right line AB , and suppose it divided into any two parts f , g ; then shall a2 = ƒ2 + 2fg + g2 . For a * = ƒ + g square each side , and we * Ax . 8. 1 . shall have a2 = ƒ2 + 2ƒg ...
Συχνά εμφανιζόμενοι όροι και φράσεις
ABC is equal adjacent angles Algebra angle ABC angle ACB angle BAC angles equal base BC bisected centre circle ABC circumference diameter double draw equal angles equal circles equal right lines equal to F equi equiangular equimultiples Euclid EUCLID'S ELEMENTS exceed exterior angle fore four magnitudes fourth given circle given point given right line gnomon greater ratio hence inscribed isosceles triangle join less Let ABC multiple parallel parallelogram perpendicular polygon proportional Q. E. D. Deductions Q. E. D. PROPOSITION rectangle contained remaining angle right line AB right line AC right line drawn segment side AC similar and similarly square of AC subtending THEOREM three right lines tiple touches the circle triangle ABC triangle DEF whence whole
Δημοφιλή αποσπάσματα
Σελίδα xxx - A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. XVIII. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter. XIX. "A segment of a circle is the figure contained by a straight line, and the circumference it cuts off.
Σελίδα 74 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle...
Σελίδα 33 - The complements of the parallelograms, which are about the diameter of any parallelogram, are equal to one another.
Σελίδα 146 - 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.
Σελίδα 27 - And because the angle ABC is equal to the angle BCD, and the angle CBD to the angle ACB, therefore the whole angle ABD is equal to the whole angle ACD • (ax.
Σελίδα 2 - Things which are double of the same are equal to one another. 7. Things which are halves of the same are equal to one another.
Σελίδα 28 - Parallelograms upon the same base, and between the same parallels, are equal to one another.
Σελίδα 73 - DH ; (i. def. 15.) therefore DH is greater than DG, the less than the greater, which is impossible : therefore no straight line can be drawn from the point A, between AE and the circumference, which does not cut the circle : or, which amounts to the same thing, however great an acute angle a straight line makes with the diameter at the point A, or however small an angle it makes with AE, the circumference must pass between that straight line and the perpendicular AE.
Σελίδα 88 - From a given circle to cut off a segment, which shall contain an angle equal to a given rectilineal angle.
Σελίδα 95 - IN a given circle to inscribe a triangle equiangular to a given triangle. Let ABC be the given circle, and DEF the given triangle; it is required to inscribe in the circle ABC a triangle equiangular to the triangle DEF. Draw (1 7. 3.) the straight line G AH touching the circle in the point A, and. at the point A, in the straight line AH, make (23.