## Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with Appendix by Thos. Kirkland. the first six books |

### From inside the book

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**algebraically**by means of their rectangular or polar equations . Prop . XLI . The converse of this proposition is not proved by Euclid ; viz . If a parallelogram is double of a triangle , and they have the same base , or equal bases ... Page 100

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**algebraically**represented by jab . The demonstrations of the first eight propositions , exemplify the obvious axiom , that , " the whole area of every figure in each case , is equal to all the parts of it taken together . " Def . 2. The ... Page 101

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**Algebraically**. ( fig . Prop . 1. ) Let the line BC contain a linear units , and the line A , 6 linear units of the same length . Also suppose the parts BD , DE , EC to contain m , n , p linear units respectively . Then a = m + n + P ... Page 102

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**Algebraically**( fig . Prop . v . ) Let the line AB contain a linear units , and the parts of it AC and BC , and n linear units respectively . Then a = m + n , squaring these equals , .. a2 = ( m + n ) 3 , or a2 = m2 + 2mn + n2 . That is ... Page 103

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**Algebraically**. Let AB contain 2a linear units , its half BC will contain a linear units . And let CD the line between the points of section contain m linear units . Then AD the greater of the two unequal parts , contains a + m linear ...### Other editions - View all

### Common terms and phrases

A₁ ABCD AC is equal Algebraically angle ABC angle ACB angle BAC angle equal Apply Euc base BC chord circle ABC constr describe a circle diagonals diameter divided draw equal angles equiangular equilateral triangle equimultiples Euclid exterior angle Geometrical given circle given line given point given straight line gnomon greater hypotenuse inscribed intersection isosceles triangle less Let ABC line BC lines be drawn multiple opposite angles parallelogram parallelopiped pentagon perpendicular plane polygon produced Prop proportionals proved Q.E.D. PROPOSITION quadrilateral quadrilateral figure radius ratio rectangle contained rectilineal figure remaining angle right angles right-angled triangle segment semicircle shew shewn similar similar triangles solid angle square on AC tangent THEOREM touch the circle triangle ABC twice the rectangle vertex vertical angle wherefore

### Popular passages

Page 93 - If a straight line be bisected and produced to any point, the square on the whole line thus produced, and the square on the part of it produced, are together double of the square on half the line bisected, and of the square on the line made up of the half and the part produced. Let the straight line AB be bisected in C, and produced to D ; The squares on AD and DB shall be together double of the squares on AC and CD. CONSTRUCTION. — From the point C draw CE at right angles to AB, and make it equal...

Page 118 - Guido, with a burnt stick in his hand, demonstrating on the smooth paving-stones of the path, that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.

Page 145 - If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle ; the angles which this line makes with the line touching the circle, shall be equal to the angles which are in the alternate segments of the circle.

Page 88 - If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line.

Page 26 - ... upon the same side together equal to two right angles, the two straight lines shall be parallel to one another.

Page 36 - 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.

Page 144 - 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 92 - If a straight line be divided into two equal, and also into two unequal parts, the squares on the two unequal parts are together double of the square on half the line and of the square on the line between the points of section. Let the straight line AB be divided into two equal parts...

Page xv - In every triangle, the square of the side subtending either of the acute angles is less than the squares of the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall upon it from the opposite angle, and the acute angle.

Page 67 - A proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate or capable of innumerable solutions.