Elements of Plane Geometry According to Euclid |
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Page 110
... multiples whatsoever of the second and fourth , and if , according as the multiple of the first is greater than the multiple of the second , equal to it , or less , the multiple of the third is also greater than the multiple of the ...
... multiples whatsoever of the second and fourth , and if , according as the multiple of the first is greater than the multiple of the second , equal to it , or less , the multiple of the third is also greater than the multiple of the ...
Page 111
... 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 ...
... 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 ...
Page 113
... multiple of A + B by m ; m ( A a multiple of Á . B by m ; and m ( A + B C ) , a mul- tiple of the excess of A + B above C , by m . - Also , mA and mB are equimultiples of A and B by m . The expression m + n is the sum of the numbers m ...
... multiple of A + B by m ; m ( A a multiple of Á . B by m ; and m ( A + B C ) , a mul- tiple of the excess of A + B above C , by m . - Also , mA and mB are equimultiples of A and B by m . The expression m + n is the sum of the numbers m ...
Page 114
... multiple of a greater magnitude is greater than the same multiple of a less . 4. That magnitude of which a multiple is greater than the same multiple of another , is greater than that other magnitude . PROPOSITION I. THEOREM . If any ...
... multiple of a greater magnitude is greater than the same multiple of a less . 4. That magnitude of which a multiple is greater than the same multiple of another , is greater than that other magnitude . PROPOSITION I. THEOREM . If any ...
Page 115
Andrew Bell. will be the same multiple of that magnitude that the sum of the two numbers is of unity . Let AmC , and ... multiples , taken in the order of the terms , are proportional . Let A : B :: C : D , and let m and n be any two ...
Andrew Bell. will be the same multiple of that magnitude that the sum of the two numbers is of unity . Let AmC , and ... multiples , taken in the order of the terms , are proportional . Let A : B :: C : D , and let m and n be any two ...
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Common terms and phrases
ABCD AC is equal angle ABC angle ACB angle BAC angle BCD angle EDF apothem base BC bisected centre chord circle ABC circumference described diameter double draw equal angles equal to AC equiangular equilateral polygon equimultiples exterior angle fore geometry given circle given line given point given rectilineal given straight line gnomon greater hypotenuse inscribed interminate less Let ABC magnitudes multiple opposite angle parallel parallelogram perimeter perpendicular polygon porism produced proportional PROPOSITION radius rectangle AB BC rectangle contained rectilineal figure regular polygon remaining angle right angles right-angled triangle Schol segment semicircle semiperimeter similar sine square of AC tangent THEOREM touches the circle triangle ABC triangle DEF twice the rectangle vulgar fraction wherefore
Popular passages
Page 1 - Things which are equal to the same thing are equal to one another. 2. If equals be added to equals the wholes are equal. 3. If equals be taken from equals the remainders are equal. 4. If equals be added to unequals the wholes are unequal. 5. If equals be taken from unequals the remainders are unequal. 6. Things which are double of the same thing are equal to one another.
Page 73 - 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.
Page 9 - To bisect a given finite straight line, that is, to divide it into two equal parts. Let AB be the given straight line : it is required to divide it intotwo equal parts.
Page 4 - If two triangles have two sides of the one equal to two sides of the...
Page 139 - Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides. Let AC, CF be equiangular parallelograms, having the angle BCD equal to the angle ECG ; the ratio of the parallelogram AC to the parallelogram CF, is the same with the ratio which is compounded of the ratios of their sides. Let BC, CG, be placed in a straight line ; therefore DC and CE are also in a straight line (2.
Page 23 - 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.
Page 129 - 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 80 - A circle is said to be described about a rectilineal figure, when the circumference of the circle passes through all the angular points of the figure about which it is described. 7. A straight line is said to be placed in a circle, when the extremities of it are in the circumference of the circle.
Page 27 - Parallelograms upon equal bases, and between the same parallels, are equal to one another.
Page 44 - If a straight line be divided into any two parts, the squares of the whole line and of one of the parts are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Let the straight line AB be divided into any two parts at the point C : the squares of AB, BC shall be equal to twice the rectangle AB, BC, together with the square of AC.