The first six books of the Elements of Euclid, and propositions i.-xxi. of book xi1885 |
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Σελίδα 17
... ( const . ) ; and things which are equal to the same are equal to one another ( Axiom 1. ) ; therefore AE is equal to C. Wherefore from AB , the greater of the two given lines , a part , AE , has been cut off equal to C , the less ...
... ( const . ) ; and things which are equal to the same are equal to one another ( Axiom 1. ) ; therefore AE is equal to C. Wherefore from AB , the greater of the two given lines , a part , AE , has been cut off equal to C , the less ...
Σελίδα 19
... ( const . ) , and AC is equal to AB_ ( hyp . ) , the two triangles FAC , GAB have the sides FA , AC in one respectively equal to the sides GA , AB in the other ; and the included angle A is common to both triangles . Hence [ Iv . ] the ...
... ( const . ) , and AC is equal to AB_ ( hyp . ) , the two triangles FAC , GAB have the sides FA , AC in one respectively equal to the sides GA , AB in the other ; and the included angle A is common to both triangles . Hence [ Iv . ] the ...
Σελίδα 25
... ( const . ) and AF common ; therefore the two sides DA , AF are respectively equal to EA , AF , and the base DF is equal to B D G the base EF , because they are the sides of an equi- lateral triangle ( Def . xxI . ) . Therefore [ VIII ...
... ( const . ) and AF common ; therefore the two sides DA , AF are respectively equal to EA , AF , and the base DF is equal to B D G the base EF , because they are the sides of an equi- lateral triangle ( Def . xxI . ) . Therefore [ VIII ...
Σελίδα 26
... ( const . ) . There- fore the base AD is equal to the base DB [ Iv . ] . Hence AB is bisected in D. Exercises . 1. Show how to bisect a finite right line by describing two circles . 2. Every point equally distant from the points A , B is ...
... ( const . ) . There- fore the base AD is equal to the base DB [ Iv . ] . Hence AB is bisected in D. Exercises . 1. Show how to bisect a finite right line by describing two circles . 2. Every point equally distant from the points A , B is ...
Σελίδα 27
... shall be at right angles to AB . H Dem . - Join CF , CG . Then the two triangles FHC , GHC have FH equal to GH ( const . ) , and HC common ; and the base CF equal to the base CG , BOOK I. ] 27 THE ELEMENTS OF EUCLID .
... shall be at right angles to AB . H Dem . - Join CF , CG . Then the two triangles FHC , GHC have FH equal to GH ( const . ) , and HC common ; and the base CF equal to the base CG , BOOK I. ] 27 THE ELEMENTS OF EUCLID .
Συχνά εμφανιζόμενοι όροι και φράσεις
ABCD AC is equal AD² adjacent angles altitude angle ABC angle ACB angle BAC angular points Axiom bisector bisects centre chord circles touch circumference circumscribed circle collinear concurrent lines const coplanar cyclic quadrilateral Dem.-Let diagonals diameter divided draw equal angles equal to AC equiangular equilateral triangle escribed circles Euclid Exercises exterior angle Geometry given circle given line given point greater Hence the angle hypotenuse inscribed isosceles less line AC line joining locus manner meet middle points multiple nine-points circle opposite sides parallel parallelogram parallelopiped perpendicular plane points of intersection prism PROP Proposition prove radii radius rectangle contained rectilineal figure regular polygon respectively equal right angles right line segments semicircle sides AC similar square on AC tangent theorem triangle ABC vertex vertical angle
Δημοφιλή αποσπάσματα
Σελίδα 295 - Thus the proposition, that the sum of the three angles of a triangle is equal to two right angles, (Euc.
Σελίδα 182 - When of the equimultiples of four magnitudes (taken as in the fifth definition) the multiple of the first is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth ; then the first is said to have to the second a greater ratio than the third magnitude has to the fourth...
Σελίδα 9 - LET it be granted that a straight line may be drawn from any one point to any other point.
Σελίδα 102 - To divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts, shall be equal to the square on the other part.
Σελίδα 122 - The diameter is the greatest straight line in a circle; and, of all others, that which is nearer to the centre is always greater than one more remote; and the greater is nearer to the centre than the less. Let ABCD be a circle, of which...
Σελίδα 226 - If from any angle of a triangle, a straight line be drawn perpendicular to the base ; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle.
Σελίδα 29 - 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 63 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.
Σελίδα 126 - The diagonals of a quadrilateral intersect at right angles. Prove that the sum of the squares on one pair of opposite sides is equal to the sum of the squares on the other pair.
Σελίδα 194 - If there be any number of proportionals, as one antecedent is to its consequent, so is the sum of all the antecedents to the sum of all the consequents.