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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Page viii
... divided the kingdom among all his subjects , and that he had given each an equal quantity of land , on condition of paying annually a proportionate tribute . If the portion allotted to any one were diminished by the river , he went to ...
... divided the kingdom among all his subjects , and that he had given each an equal quantity of land , on condition of paying annually a proportionate tribute . If the portion allotted to any one were diminished by the river , he went to ...
Page 6
... divided into a variety of cases according to the different positions of the point A , although the construction and demon- stration will , in every respect , be the same . Proclus remarks that some performed it by taking the line AL ...
... divided into a variety of cases according to the different positions of the point A , although the construction and demon- stration will , in every respect , be the same . Proclus remarks that some performed it by taking the line AL ...
Page 21
... divided . PROPOSITION XXIII . PROBLEM . * To a given right line , and to a given point in it , to make a rectilineal angle equal to a given rectilineal angle . Let AB be the given right line , and a the given point in it ; also DCE the ...
... divided . PROPOSITION XXIII . PROBLEM . * To a given right line , and to a given point in it , to make a rectilineal angle equal to a given rectilineal angle . Let AB be the given right line , and a the given point in it ; also DCE the ...
Page 45
... divided into any number of parts , the rectangle comprehended under the whole and divided line , is equal to the rectangles con- tained under the whole line , and the several segments of the divided line . Let A and BC be two right ...
... divided into any number of parts , the rectangle comprehended under the whole and divided line , is equal to the rectangles con- tained under the whole line , and the several segments of the divided line . Let A and BC be two right ...
Page 46
... divided into any two parts , the rectangles contained under the whole and each of the parts are together equal to the square of the whole line . Let the right line AB be any how divided in the point c , the rectangle contained under AB ...
... divided into any two parts , the rectangles contained under the whole and each of the parts are together equal to the square of the whole line . Let the right line AB be any how divided in the point c , the rectangle contained under AB ...
Common terms and phrases
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
Popular passages
Page 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.
Page 74 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle...
Page 33 - The complements of the parallelograms, which are about the diameter of any parallelogram, are equal to one another.
Page 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.
Page 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.
Page 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.
Page 28 - Parallelograms upon the same base, and between the same parallels, are equal to one another.
Page 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.
Page 88 - From a given circle to cut off a segment, which shall contain an angle equal to a given rectilineal angle.
Page 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.