Euclid's Elements of geometry, books i. ii. iii. iv1862 |
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Seite 4
... , merely introduced for the purpose of demonstrating some other propo- sition . A scholium is a note or explanatory observation . , 1 PROPOSITION 1. - PROBLEM . To describe an equilateral triangle 4 -H EUCLID'S ELEMENTS .
... , merely introduced for the purpose of demonstrating some other propo- sition . A scholium is a note or explanatory observation . , 1 PROPOSITION 1. - PROBLEM . To describe an equilateral triangle 4 -H EUCLID'S ELEMENTS .
Seite 7
... Demonstration . - 1 . For if the triangle ABC be applied to ( or placed upon ) the triangle DEF , 2. So that the point A may be on the point D , and the straight line AB on the straight line DE , 3. The point B shall coincide with the ...
... Demonstration . - 1 . For if the triangle ABC be applied to ( or placed upon ) the triangle DEF , 2. So that the point A may be on the point D , and the straight line AB on the straight line DE , 3. The point B shall coincide with the ...
Seite 8
... Demonstration . - 1 . Because AF is equal to AG . ( constr . ) And AB is equal to AC . ( hypoth . ) Therefore the two sides FA , AC , are equal to the two sides GA , AB , each to each . 2. And they contain the angle FAG , common to the ...
... Demonstration . - 1 . Because AF is equal to AG . ( constr . ) And AB is equal to AC . ( hypoth . ) Therefore the two sides FA , AC , are equal to the two sides GA , AB , each to each . 2. And they contain the angle FAG , common to the ...
Seite 9
... greater than the other . Let AB be the greater . Construction . - 1 . From AB the greater , cut off a part DB , equal to AC the less . ( I. 3. ) 2. Join DC . Demonstration . - 1 . Because in the triangles DBC BOOK I. PROP . 6 . 9.
... greater than the other . Let AB be the greater . Construction . - 1 . From AB the greater , cut off a part DB , equal to AC the less . ( I. 3. ) 2. Join DC . Demonstration . - 1 . Because in the triangles DBC BOOK I. PROP . 6 . 9.
Seite 10
... Demonstration . - 1 . Because AC is A assumed to be equal to AD . ( hyp . ) D 2. The triangle ADC is an isosceles triangle , and the angle ACD is therefore equal to the angle ADC . ( I. 5. ) 3. But the angle ACD is greater than the ...
... Demonstration . - 1 . Because AC is A assumed to be equal to AD . ( hyp . ) D 2. The triangle ADC is an isosceles triangle , and the angle ACD is therefore equal to the angle ADC . ( I. 5. ) 3. But the angle ACD is greater than the ...
Häufige Begriffe und Wortgruppen
AB is equal AC and CB adjacent angles angle ABC angle AGH angle BAC angle BCD angle EAB angle EDF angle equal angles CBA base BC BC is equal bisected circle ABC circumference Conclusion Conclusion.-Therefore const Construction.-1 Demonstration.-1 describe the circle diameter double equal angles equal to CD equiangular exterior angle given circle given point given rectilineal angle given straight line Given.-Let ABCD gnomon greater Hypothesis inscribed interior and opposite isosceles triangle less opposite angle parallel to CD parallelogram perpendicular point F produced Q. E. D. PROPOSITION rectangle AB BC rectangle AE rectangle contained rectilineal figure References-Prop remaining angle required to describe right angles segment semicircle Sequence side BC square on AC straight line AC straight line drawn touches the circle triangle ABC triangle DEF twice the rectangle
Beliebte Passagen
Seite 25 - If two triangles have two angles of the one equal to two angles of the other, each to each ; and one side equal to one side, viz.
Seite 2 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.
Seite 99 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.
Seite 4 - If a straight line meets two straight lines, so as to " make the two interior angles on the same side of it taken " together less than two right angles...
Seite 66 - ... the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted without the triangle between the perpendicular and the obtuse...
Seite 65 - To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, may be equal to the square of the other part.
Seite 32 - F, which is the common vertex of the triangles ; that is, together with four right angles. Therefore all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Seite 58 - If a straight line be bisected, and produced to any point ; the rectangle contained by the whole line thus produced, and the part of it produced...
Seite 88 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle...
Seite 33 - The straight lines which join the extremities of two equal and parallel straight lines towards the same parts, are also themselves equal and parallel.