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addition algebra angle applied Assumption called circle coefficients common complex congruent consider construction contains continuous corresponding curve defined definition denoted derivative determined distance distinct divided division domain draw elementary elements equal equation evident example exists expressed fact factor figure follows formula four function fundamental geometry give given Hence important infinite integral intersect interval joining known least less mathematical means meet method multiple obtained operations opposite pairs parallel pass perpendicular plane positive possible postulates present prime primitive problem projective proof properties propositions proved quantities radicals range rational rays reduced region relation represent respectively root satisfy segment sides solution square statement straight line substitutions symbol Theorem theory triangle true unity unknowns values zero
Seite 93 - That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Seite 41 - If one side of a triangle be produced, the exterior angle is greater than either of the interior, and opposite angles.
Seite 32 - SSS); two sides and the included angle of one triangle are congruent to the corresponding parts...
Seite 259 - A sufficient condition for the maximum number of imaginary roots of an equation of the nth degree,
Seite 251 - In this way we obtain one and only one value for each of the unknowns x\, . . . , xr. The preceding considerations prove the following theorem: A necessary and sufficient condition for a system of linear equations to be consistent is that the matrix of the system has the same rank as the augmented matrix. Since the values assigned to...
Seite 354 - A proposed construction is possible by ruler and compasses if, and only if, the numbers which define analytically the desired geometric elements can be derived from those defining the given elements by a finite number of rational operations and extractions of real square roots.
Seite 111 - B', B, or B" according as KD is less than, equal to, L FIG. 18. B'B B" AB FIG. 19. or greater than LB. In these three cases 2^D is greater than, equal to, or less than, 2±B respectively. Hence: Each summit angle of an isosceles birectangular quadrilateral is less than, equal to, or greater than, a right angle, according as the summit of the quadrilateral is greater than, equal to, or less than, the base. 18. In the Euclidean geometry each summit angle of an isosceles birectangular quadrilateral...
Seite 398 - Ten decimals are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimals would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope.
Seite 98 - If two lines are parallel to a third, they are parallel to each other. We distinguish two cases according as the third line lies between the two lines or not. In the first case, let AK and DQ (Fig. 4) be each parallel to ML. We wish to prove that AK is parallel to DQ. Draw FIG. 4. FIG. 5. AC any line through A in the angle opening DAK. AC will meet ML in some point F since AK is parallel to ML. CF produced will also meet DQ, since ML and DQ are parallel. Hence any line through A in the angle opening...