The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh and Twelfth. 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 CorrectedConrad and Company, 1810 - 518 σελίδες |
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Σελίδα 14
... less " than two right angles , these straight lines being continually " produced , shall at length meet upon that side on which are " the angles which are less than two right angles . See the " notes on prop . 29. of Book I ...
... less " than two right angles , these straight lines being continually " produced , shall at length meet upon that side on which are " the angles which are less than two right angles . See the " notes on prop . 29. of Book I ...
Σελίδα 16
... less . Let AB and C be the two given straight lines , whereof AB is the greater . It is required to cut off from AB , the greater , a part e- qual to C , the less . D C B a 2. 1 . A From the point A draw a the straight line AD equal to ...
... less . Let AB and C be the two given straight lines , whereof AB is the greater . It is required to cut off from AB , the greater , a part e- qual to C , the less . D C B a 2. 1 . A From the point A draw a the straight line AD equal to ...
Σελίδα 19
... less , and join DC ; there- A fore , because 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 ...
... less , and join DC ; there- A fore , because 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 ...
Σελίδα 27
... less than two right angles . Therefore any two angles , & c . Q. E. D. PROP . XVIII . THEOR . THE greater side of ... less than it ; it is not equal , because then the angle ABC would be equal a to the angle ACB ; but it is not ...
... less than two right angles . Therefore any two angles , & c . Q. E. D. PROP . XVIII . THEOR . THE greater side of ... less than it ; it is not equal , because then the angle ABC would be equal a to the angle ACB ; but it is not ...
Σελίδα 28
... less than the other two sides of the tri- angle , but shall contain a greater angle . Let the two straight lines BD , CD be drawn from B , C , the ends of the side BC of the triangle ABC , to the point D within it ; BD and DC are less ...
... less than the other two sides of the tri- angle , but shall contain a greater angle . Let the two straight lines BD , CD be drawn from B , C , the ends of the side BC of the triangle ABC , to the point D within it ; BD and DC are less ...
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Συχνά εμφανιζόμενοι όροι και φράσεις
ABCD altitude angle ABC angle BAC base BC BC is equal BC is given bisected Book XI centre circle ABC 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 ratio given straight line gnomon greater join less Let ABC meet multiple opposite parallel parallelogram perpendicular point F polygon prism proportionals proposition pyramid Q. E. D. PROP radius ratio of BC rectangle CB rectangle contained rectilineal figure right angles segment side BC similar sine solid angle solid parallelopipeds square of AC straight line AB straight line BC tangent THEOR third triangle ABC triplicate ratio vertex wherefore
Δημοφιλή αποσπάσματα
Σελίδα 19 - FG; then, upon the same base EF, and upon the same side of it, there can be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise their sides terminated in the other extremity: But this is impossible (i.
Σελίδα 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 sides.
Σελίδα 69 - 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.
Σελίδα 94 - IF a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles made by this line with the line touching the circle, shall be equal to the angles which are in the alternate segments of the circle.
Σελίδα 28 - If from the ends of a 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.
Σελίδα 57 - If 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 of the whole and that part.
Σελίδα 320 - 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.
Σελίδα 24 - If, at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line.
Σελίδα 163 - 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.
Σελίδα 23 - When a straight line standing on another straight line, makes the adjacent angles equal to one another, each of the angles is called a right angle ; and the straight line which stands on the other is called a perpendicular to it.